i ( u::]yjiLvi^:i^^j:'i>^ityji'i^|tyj|i^i^y.vi^j!!^i'!tyj: Kansas Citp public 2.ibrarp THE BELL S Y S T E M Jechnical ournal DEVOTED TO THE SC I E N T I FIC^^^ AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION ADVISORY BOARD S. Bracken M. J. Kelly F. R. Kappel E. J. McNe --^Si?? EDITORIAL COMMITTEE ^"^^^ UB^J^'s W. H. Dohertt, Chairman FFR 1 A. J. BUSCH R. K. HONAMAN '"^0 G. D. Edwards F. R. Lack R. G. Elliott H, I. Romnes J. B. FiSK H. iJ. Schmidt E. L Green G.N.Thayer --.-..,. EDITORIAL STAFF J. D. Tebo, Editor M. E. Strieby, Managing Editor R. L. Shepherd, Production Editor INDEX VOLUME XXXIII 1954 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK THIS VOLUME IS BOUND IN TWO PARTS Part I constitutes pages 1-798 Part II constitutes pages 799-1400 LIST OF ISSUES IN VOLUME XXXIII No. 1 January Pages 1-275 " 2 March 277-516 " 3 May 517-798 " 4 July 799-1022 " 5 September 1023-1208 " 6 November 1209-1400 Index to Volume XXXIII AG Relay See Rela3^(s) Abstracts, of Bell System Technical Papers not. published in this Journal 260-73, 505-13 AccouxTixG See Centralized Auto- matic Message Accounting System Almquist, M. L. 1010 Alpeth Cable See Cable (s) Alternate Routing costs 299-301 interoffice trunk networks 279-86 intertoll trunk networks 286-87 principles 278-87 trunk networks, trunk requirements 277-302 Amplifier (s) double-stream 1358-60 magnetic transitors combined with 847 negative resistance and dissected 799-800 "push-pull " DC 849 resistive-wall 1349-50 space-charge wave 1354—55 Analogue Computer 139-40 Analysis of Measured Magnetization and Pull Characteristics (R. L. Peek, Jr.) 79-108 Anderson, Orson L. biographical material 1206 Stress Systems in the Solderless Wrapped Connection and Their Permanence 1093-1110 Anderson, P. W. 1052 Andrus, J. A. 1052 Application of Designed Experiments to the Card Translator (C. R. Brown, M. E. Terry) 369-98 ARC(s) initiation 544-45 interrupted 539-42 reversed 543 welding, percussive 888 Arcing of Electrical Contacts in Telephone Switching Circuits. Part III — Dis- charge Phenomena on Break of Inductive Circuits (M. M. Attala) 535-58 Armature (s) relay direction reversion 17 flux build-up 160 force measurement 11 rebounding 17 saturation 53, 95, 99, 104 Attala, M. M. Arcing of Electrical Contacts in Tele- phone Switching Circuits. Part III — Discharge Phenomena on Break of Inductive Circuits 535-58 biographical material 797, 1398 Open-Contact Performance of Twin Contacts, Theory 1373-86 Attenuation Coefficient (of) circular electric wave in round metallic tubing 1209 Automatic Contact Welding in Wire Spring Relay Manufacture (A. L. Quinlan) 897-923 Automatic Message Accounting See Centralized automatic message accounting Automatic Molding Machine 870-72 Automatic Multiple Head Wire Straightexer 863-64 B Babbitt, Dr. B. J. 923 Ballistic Galvanometer See Gal- vanometer Band Filter See Filter(s) Bangert, John T. biographical material 514 THE BELL SYSTEM TECHNICAL JOURNAL, 1954 Transistor as a Network Element 329-52 Barium Titanate crystal impact forces 12-13 Beedy, H. 923 Bennett, W. R. 1010 Berkeley Timer 15 Blaha, Albert L. biographical material 1019 Electronic Relay Tester 925-38 Bozorth, R. M. 1052 Brannon, Miss M.J. 1194 Brown, Clayton B. Application of Designed Experiments to the Card Translator 369-98 biographical material 514 Brown, D. R. 923 Brunner, A. J. biographical material 1019 Wire Straightening and Molding for Wire Spring Relays 859-84 Busy Hour(s) busy seasons 299 group 294-97 office 294-97 CAMA See Centralized Automatic Message Accounting System Cable (s) Alpeth 559 coa.xial system nationwide dialing 277 sheath non-conductive, continuous incre- mental thickness, measurements of 353-68 polyethylene manufacture 559-77 telephone cables 353 thickness measurement 559-77 Stalpeth 559 Call(s) toll completion 311 Card Translator (See Translator (/ARRIER(s) nationwide dialing 277 Central Office (Telephone) relays 218 Centralized Automatic Message Account- ing System (G. V. King) 1331-42 Chase, F. Harold biographical material 1019 Transistor and Junction Diodes in Telephone Power Plants 827-58 Chatter governor, dial 1292-95 relay, contact, study 17 Chatter Interval(s) relays 17-18 Cioffii, P. P. fiuxmeter 8 motion of individual domain walls 1052 Circuit(s) controller trunk concentrating equipment 309 elements coupled lines 661-719 magnetic multicontact rela.y, new 1016-17 repeater, E2 1075-79 I'epeater, E3 1079-80 welding, ideal, design of 892 Circular Electric Waves See Wave(s) Clark, G. W. 1052 Clogston, A. M. 1052 Coaxial Cable System See Cable (s) Coil, Torcidal magnetic field 42 Communication waveguide as a medium 1209 Computer(s) analogue 139-40 Confluent Band Filter See Filter(s) Connection (s) solderless wrapped diffusion stresses 1099-1108 permanence 1093-1110 properties, affected by plating 1108 stress systems 1093-1110 "Contact Chatter" 17 Continuous Incremental Thickness Meas- urements of A' on-Conductive Cable Sheath (Wojciechowski) 353-68 INDEX Controller Circuit Sec Circuit (s) Converter (s) negative impedance 1055-92 Cosson, H. E. biographical material 1019 Wire Straightonng and Molding for Wire Sprijig liclays 859-84 Coupled Ware Theory and Waveguide Applications (S. K. Miller) 661-719 Cox, Rosemary E. biographical material 1398 electrical contacts, arcing in switching circuits 557 Open-Contact Performance of Twin Contacts, Theory 1373-86 Crossbar System (s) No. 4 A card translator 369 No. 5 connectors 1111 relays 2 Cryst.\l (s) barium titanate impact forces 12-13 nickel-iron ferrite domain walls, motion of 1023-54 Current eddy 176-209 regulators 849 relays 8, 176-209 D Davis, Thomas E. biographical material 1019 Electronic Relay Tester 925-38 Dawson, R. W. 719 Deionization Time 547 Dempsey, F. J. 1052 Dial Telephone, Dialing governor, see Governor nationwide plan 277 relay, electromechanical 1 signaling requirements 1309 toll centralized automatic message ac- counting system 1331-42 nationwide 277-302 operator 277 switchboard 303 Diffraction of Plane Radio Waves by a Parabolic Cylinder (S. O. Rice) 417-504 Dillon, J. F., Jr. 1052 Diode (s) junction telephone power j^lants 827-58 reference voltage 833-34 telephone power ])lants 827-58 regulator 840 semiconductor negative resistance arising from transit time in 799-826 Discharge Phenomena 535 Dissipation, reduction of 330 Distortion, transmission 773, 779 Dixon, J. T. 1010 Domain Wall(s) ferromagnetic defined 1023 motion (in) nickel-iron ferrite 1023-54 E Early, James M. biographical material 797 P-N-I-P and N-P-I-N Junction Tran- sistor Triodes 517-33 Easitron 1350-52 EcHo(es) pulse 752-761 Econojnics of Telephone Relay Applica- tions (H. N. Wagar) 218-59 Eddy Current relays 176-209 Electric Waves See Wave(s) Electrode (s) welding 885 Electromagnet (s) core 53 saturation 97-99 measured magnetization 79-108 pull characteristics 79-108 Electromagnetic Field (s) shadows cast by hills 439 torcidal coil 42 beam field equations 400 magnetically-focusing cylindrical wave propagation 399-416 6 THE BELL SYSTEM TECHNICAL JOURNAL, 1954 Electronic Belay Tester (A. E. Blaha, T. E. Davis) 925-38 Electrostatic Gauge 925 Ellwood Magnetomotive Force Gauge 9 Engineered Alternate Routing 277- 78 Engst, N. K. 884 Enz, R. E. 1052 Eppler, Walter T. biographical material 797 incremental sheath thickness measure- ments 368 Thickness Measurement and Control in the Manufacture of Polyethylene Cable Sheath 599-77 Error, source and measures 397-98 Estimation and Control of the Operate Time of Relays: Part I — Theory (R. L. Peek, Jr.) 109-45 Estimation and Control of the Operate Time of Relays: Part II — Design of Optimum Windings (M. A. Logan) 144-86 Evaporation, in welding 885 Ferrite(s) dielectric constant 1145 measurements on coxial cable technique 1168 nickel-iron, motion of individual do- main walls in 1023-54 Ferromagnetic Domain Wall(s) See Domain Wall(s) Field(s) See Electromagnetic Field(s) Filter (s) confluent band 332 non-inductive 343 Flux relays 7-10, 14-15 Fluxmeter, Recording 8-9 Force relay 4-7, 11-12 Fritschi, W. W. 1330 Galvanometer, Ballistic 8 Gauge (s) electrostatic 925 Ellwood magnetomotive force 9 General Toll Switching Plan See Switching System (s) Glow-Arc Transitions 552, 555-56 Glow Maintenance 552 Governor(s) chatter 1292-95 design, principles 1267-1307 (for) dial, 7-type 1267-1307 input torque dial speed relation 1290-92 motion equations 1273-78 operation 1269-73 (for) regulating speed of rotary dials 1267-1307 Governor for Telephone Dials — Principles of Design (W. Pferd) 1267-1307 Graeco-Latin Square and Factorial Designs 369 Green, E. I. 351 Guetich, T. H. 1131 Guided-Wave Propagation through Gyro- magnetic Media. Part I — The Com- pletely Filled Cylindrical Guide (H. Suhl, L. R. Walker) 579-659 Guided-Wave Propagation through Gyro- magnetic Media. Part II — Transverse Magnetization and Non-Reciprocal Helix (H. Suhl, L. R. Walker) 939-86 Guided-Wave Propagation through Gyro- magnetic Media. Part III — Perturba- tion Theory and Miscellaneous Re- sults. (H. Suhl, L. R. Walker) 1133-94 Gyromagnetic Media guided-wave propagation through 579-659, 939-86, 1133-94 H Gait, J. K. biographical material 1206 Motion of Individual Domain Walls in a Nickel-Iron Ferrite 1023-54 Hamilton, B. H. biographical material 1020 Transistor and Junction Diodes in Telephone Power Plants 827-58 Hamming, R. W. 1194 l.NDK.M ITklix cyliiuirit'al l)6i) non-reciprocal transverse magnetization 939-86 plane 954 Herring, C. 1052 Hills, Shadows Behind calculation 417-504 Hit linger, W. C. 532 Holden, A. N. 1052 Hopper, H. R. 1052 II 111. hard, F. A. 1330 I Impedanxe, negative, theory 1065 Impedance Inversion 349 In-Band Single-Frequency Signaling (N. A. Newell, A. Weaver) 1309-30 Inductance elimination of 342 Instron Tensile Testing Machine 4, 6-7 Interference Effect(s) mode conversion and reconversion 1230-39 Interoffice Trunk Network (s) alternate routing 279-86 Intertoll Trunk Concentrating EquipiuenI (D. F. Johnston) 303-28 Intertoll Trunk Network(s) alternate routing 286-87 Ion(s) See Deionization Time Jensen, J. F. 1052 Johnson, H. E. 1052 Johnston, Donald F. biographical material 514 Intertoll Trunk Concentrating Equip- ment 303-28 Junction Diode See Diode (s) Junction Rectifier(s) See Rcctifier(s) Junction Transitor See Transistor(s) K Kahl, H. 1092 Keller, Arthur C. biographical material 274 Design of Relays 1-2 Kessel, R. L. 923 King, (ierald ^^ biographical material J39S Ccntrali::e(l Autotntitic Mcsaage Ac- counting System 1331 12 Klystron 1348-49 Koehler, D. C. 923 Kompfner, R. 1194 Lambert, Mrs. C. A. guided wave projjagalion through gyromagnetic media 1194 wave propagation along an electron beam 416 Landau-Lifshitz Equation 1023 Lewis, J. A. biographical material 514 Wave Propagation along a Magnet- ically-Focused Cylindrical Electron Beam 399-416 Llewellyn, F. B. pulse transmission, theoretical funda- mentals 1010 repeaters, negative impedance 1092 Logan, Mason A. biographical material 274 Estimation and Control of the Operate Time of Relays: Part II— Design of Optimum Windings 144-86 slow release rela}- design 217 Long Distance Waveguides See Waveguide (s) Lucek, C. W. 1330 Lundry, W. R. 351 M McConnell, J. H. 938 McGlasson, J. 532 Madden, J. J. contact welding in wire spring relays 923 percussive welding 895 Magnet (s) See Electromagnet (s) Magnetic Design of Relays (R. L. Peek, Jr.,II.N. Wagar) 23-78 Magnetic Field (s) See Elect romag- netic Field (s) 8 THE BELL SYSTEM TECHNICAL JOURNAL, 1954 Magnetic Material(s) properties 37-39 Mandeville, D. 719 Mason, W. P. biographical material 1206 Stress Syslems in the Solderless Wrapped Connection and Their Permanence 1093-1110 Matthias, B. T. 1052 Maxwell's Equation 1168 Mazza, L. L. 884 Meola, L. P. 532 Merrill, J. L., Jr. biographical material 1206 Negative Impedance Telephone Re- peaters 1055-92 Meyers, S. T. 1092 Microwave (s) wavelength 417 Microwave Tube(s) operation in terms of waves 1343 using long electron beam low-level or small -signal behavior 1343-72 wave picture 1343-72 Miller, Stewart E. biographical material 798, 1399 Coupled Wave Theory and Waveguide Applications 661-719 Waveguide as a Communication Me- dium 1209-65 Millimeter Wave(s) techniques 1253-56 Millman, S. 1052 Mitchell, G. A. 923 Mitchell, Miss L. 895 Molding automatic wire spring relay block as- semblies 870-83 die 874-77, 879 wire 879-81 Moore, C. R. dial, speed of 1266 Moore, H. R. 1052 Morgan, S. P., Jr. guided wave propagation through gyromagnetic media 1194 percussive welding 895 Morton, J. A. 532 Motion of Individual Domain Walls in a Nickel-Iron Ferrite (J. K. Gait) 1023-54 Multicontact Relay, New, for Telephone Switching Systems (I. S. Rafuse) 1111-32 Mumford, W. W. 719 N Nationwide Dialing See Dial Tele- phone Neel, R. I. 368 Negative Impedance See Impedance Negative Impedance Converter See Converter (s) Negative Impedance Telephone Repeaters (J. L. Merrill, Jr., A. F. Rose, J. 0. Smethurst) 1055-92 Negative Resistance See Resistance Negative Resistance Arising from Transit Time in Semiconductor Diodes (W. Shockley) 799-826 Network (s) interoffice, trunk alternate routing 279-86 intertoU trunk alternate routing 286-87 transistors 329 trunk alternate routing, traffic engineering techniques for determining trunk requirements 277-302 Newell, Norman A. biographical material 1399-1400 In-Band Single-Frequency Signaling 1309-30 Nickel-Iron Ferrite See Ferrite (s) Noise cancellation 1361-62 deamplification 1360-61 N-P-I-N Junction Transistor See Transistor (s) Office busy-hour 294-97 Open-Contact Performance of Twin Con- tacts, Theory (M. M. Atalla, Miss R. E. Cox) 1373-86 INDEX 9 Oi'KRATE Time (of) relays 109-86 OPERATOR Toll Dialing 277 Os^ciLLOscorE, cathode ray 17 O 'Toole, J. L. 369 Parabolic Cylinder ])lane radio waves, diffraction of 417-504 Paulson, C. 884 I'eek, Robert L., Jr. Analysis of Measured Magnetization and Pull Characteristics 79-108 biogra])hical material 275 economics of telephone relay applica- tions 256 electronic rela}^ tester 938 Estimation and Control of the Operate Time of Relays: Part I — Theory 109-43 Magnetic Design of Relays 23-78 Principles of Slow Release Relay Design 187-217 Percussive Welding See Welding Perturbation Methods 1134-68 Peterson, J. W. 532 Pferd, William biographical material 1398-99 Governor for Telephone Dials — Princi- ples of Design 1267-1307 Phenolic Resin See Resin Pierce, John R. biographical material 1399 Wave Picture of Microwave Tubes 1343-72 wave propagation along an electron beam 416 P-N-I-P and N-P-I-N Junction Tran- sistor Triodes (J. M. Early) 517-33 P-N-I-P Junction Transistor See Transistor (s) Polder Relation 1172 Polyethylene cable sheath thickness 559 Positive Resistance See Resistance Power P-N-I-P and N-P-I-N junction tran- sistor triodes 517 Power Plants (Telej)hone) junction diodes 827-58 transistors 827-48 Posin, M. 327 Principles of Sloiv Release Relay Design (R. L. Peek, Jr.) 187-217 Problems in Percussive Welding (E. E. Sumner) 885-95 Pulse Transmission See Transmission Quate, C. F. wave propagation along an electron beam 416 Quinlan, A. L. Automatic Contact Welding in Wire Spring Relay Manufacture 897-923 biographical material 1020 Radio propagation over a succession of ridges on the earth's surface 438 short waves hills, the effect of 417-504 Rafuse, Irad S. biological material 1207 New Multicontact Relay for Telephone Switching Systems 1111-32 Randall, Mrs. K. R. economics of telephone relay applica- tions 256 slow release relay design 217 Read, W. T., Jr. 1052 Rebarber, Mrs. A. 1194 Recorder (s) X-Y 5, 8-9 Recording Fluxmeter See Fluxmeter Rectifier(s) germanium, 65-volt, 200-ampere 855 junction 828-31 magnetic regulated 854 thyraton tube 851-54 Reference Voltage Diode See Di- ode (s) Regulator (s) current 849 series 844 10 THE BELL SYSTEM TECHNICAL JOURNAL, 1954 series current 846 shunt 840 shunt transistor 842 simple series 845 voltage 848 Relay (s) AG 187-217 all busy 321-22 armature direction reversion 17 flux build-up 160 force measurement 11 rebounding 17 saturation 53, 95, 99, 104 central office, telephone 218 chatter intervals 17-18 coil characteristics 25-27 current 8 demagnetization relations 35-37 design of 1-259 dj-namic measurements 10 economics 218-59 eddj- currents 176, 209 electromechanical 1 end 314-15 end of cycle 321-22 failure 1 flux 7-10, 14-15 gauging a 935-37 general purpose 14 group restore 321-22 high speed 11, 53 high speed wire spring multicontact nil identical 169 magnet 4-9, 23 magnetic design 23-78 manufacturing costs 218-59 measuring equipment 3-22 multicontact, new (for) switching sj'stems 1111-32 operate time, estimation and control theory 109-43 windings, optimum, design 144-86 release time 178-84, 187-217 release waiting time 133-38 series 168 series connected 170-76 single 146 slide between parts 12 slow release principles of 187-217 "tens gate" 303 "tens group" 313 tester electronic 925-38 time 15-21, 154-60, 164-66 trunk advance 320 two like parallel relays in series with a third relay 170 unit gates 314-15 \vinding(s) 144-86 cost 223 wire spring block assemblies, automatic mold- ing 870 contact welding 897-923 nickel silver wire 859 phenolic resin blocks, molding 870-84 silicon copper wire 859 springs 897 welding, automatic contact 897- 923 wire straightening and molding 859-84 Relay Measuring Equipinent (H. N. Wagar) 3-22 Relays, Design of (A. C. Keller) 1-2 Release Time relays 178-84 Repeater(s) negative impedance 1055-92 El 1055 E2 1055-92 E3 1055-92 series-type, application 1056-58 Resin (phenolic) molding, for wire spring relay blocks 870-84 Resistance negative 330-32 positive 330 Rice, Stephen O. biographical material 515 Diffraction of Plane Radio Waves by a Parabolic Cylinder (Stephen O. Rice) 417-504 INDKX H Kigrod, William W. l)ingrai)hi('al matcMial olf) Wave Propagation along a Magnet- ically-Focused Cylindrical Electron Beam 399-416 Roach, J. M. 923 K()l)l), 1). T. 368 lioso, Arthur F. biographical material 1207 Xeguiive Impedance Telephone Re- peaters 1055-92 KouTiNG, OF Telephonf. Traffic, Alternate 278-87 trunk networks, trunk nniuinMnents 277-302 Rowen, J. H. 1194 Ryder, R. M. 532 SF See Single-Frequency Schelkunoff, S. A. circular electric wave, loss coefficient 1210 Schenck, A. K. 1330 Schultz, F. A. 884 Seider, J. P. 923 Semiconductor(s) , Semiconducting Materials diodes negative resistance arising from transit time 799-826 junction transistor triodes 517 Shadows, Behind Hills calculation 417-504 Shafer, W. L., Jr. 326 Sheath See Cable: sheath Shockley, William biographical material 1020 Negative Resistance Arising from Transit Time in Semiconductor Diodes 799-826 Shunt Regulator(s) See Regulator(s) Shunt Transistor Regulator(s) See Regulator (s) Signal(s), Signaling Systems (for) intertoU telephone trunks 1309- 12 single-frequenc}- in-band 1309-30 Signal-Loss Effect mode conversion and reconversion 1229 Single-Frequency Signaling See Sig- nals) Smethurst, J. O. biographical material 1207 Negative Impedance Telephone Re- peaters 1055-92 Smith, Donald H. biographical material 1021 Transistor and Junction Diodes in Telephone Power Plants 827-58 Soffel, R. O. 1330 Solderless Wrapped Connections See Connection(s) South worth, G. C. circular electric wave, loss coefficient 1210 Spillar, R. 923 Stalpeth Cable See Cable Storage wire, straightened 861 Stress hoop, relaxation of 1094-99 Stress Systems in the Solderless Wrapped Connection and Their Permanence (O. L. Anderson, W. P. Mason) 1093-1110 Strickland, R. W. biographical material 1021 Wire Straightening and Molding for Wire Spring Relays 859-84 Suhl, Harry biographical material 798, 1207-08 Guided-Wave Propagation through Gy- romagnetic Media. Part I — The Completely filled Cylindrical Guide 579-659 Guided-Wave Propagation through Gy- romagnetic Media. Part II — Trans- verse Magnetization and Non- Reciprocal Helix 939-86 Guided-Wave Propagation through Gy- romagnctic Media. Part III — Pertur- bation Theory and Miscellaneous Results 1133-94 wave propagation along an olocti-on beam 416 12 THE BELL SYSTEM TECHNICAL JOURNAL, 1954 Sumner, Eric Eden biographical material 1021 Problems in Percussive Welding 885- 95 Sunde, Erling D. biographical material 798 Theoretical Fundamentals of Pulse Transmission — / 721-88 Theoretical Fundamentals of Pulse Transmission — // 987-1010 Swickard, A. E. 884 Switchboard (s) toll idle circuits 324 outward 303 Switching System (s) alternate routing methods 287 circuits arcing of electrical contacts 535-58 common-control type special purpose 303 dial, dependence on 1267 four-wire 308 general toll plan 287-90 intertoll trunk concentrating equip- ment 303-28 nationwide dialing 277 No. 4 type 303 relay, see Relay(s) trends 1111 See also Crossbar System(s) Technical Papers, Bell Sj'stem, not published in this Journal 789-96, 1011-18, 1195-1205, 1387-98 - Telephone calls, see Call(s) central office, see Central Office dial, see Dial Telephone power plants, see Power Plants relay, see Relay(s) Telephone Number(s) nationwide plan 277 Telephone Set 500-type dial, 7-type, governor for 1267- 1307 Telephone Traffic alternate routing, engineering tech- niques for determining trunk re- quirements 277-302 group busy hour versus office busy hour traffic 294-97 high usage groups location 290 intertoll trunk concentrating equip- ment 303-28 random versus non-random 298 traffic engineering techniques for determining trunk requirements in alternate routing trunk networks 277-302 See also Busy-Hour (s) Temperature junction diodes in telephone power plants 831-32 Tensile Testing 4-6 Terry, Milton E. Application of Designed Experiments to the Card Translator 369-98 biographical material 515 Test Set(s) (for) relays 925-38 (for) repeaters 1083-85 Theoretical Fundamentals of Pulse Trans- mission—I (E. D. Sunde) 721-88 Theoretical Fundamentals of Pulse Trans- mission—II (E. D. Sunde) 987-1010 Thickness Measurement and Control in the Manufacture of Polyethylene Cable Sheath (W. T. Eppler) 559-77 Thyraton Tube Rectifier See Recti - fier(s) Time See Operate Time; Transit Time Tiner, Mrs. R. M. 1052 Toll Calls See Call(s) Toll Crossbar Switching Office, first 227 Toll Dialing See Dial Telephone Torcidal Coil magnetic field 42 Traffic See Telephone Traffic Traffic Engineering Techniques for Deter- mining Triink Requirements in Alternate Routing Trunk Networks (C. J. Truitt) 277-302 INDKX 13 Transistor(s) ampliH(M-s, magnetic, combined with 847 Transi.stor(s) junction frequencj- range 517 N-P-I-N triodes 517-33 P-N-I-P triodes 517-33 telephone power plants 827-58 typical 839-40 network element 320-52 shunt regulator, multistage 844 Tratisistor and Junction Diodes in Telephone Power Plants (F. H. Chase, B. H. Hamilton, D. H. Smith) 827-58 Transistor as a Network Element (J. T. Bangert) 329-52 Transit Time semiconductor diodes, negative resis- tance arising from 799-826 Translator card designed experiments, ap^ilication of 369-98 Transmission circular electric wave 1211 coupled lines 661-719 distortion 773-779 frequency characteristics 724 measurement, theory of 1085-87 nationwide dialing 277 network transistors 329 pulse theoretical fundamentals 721-88, 987-1010 repeaters, negative imijedance 1055- 92 waveguide 1209 Triode(s) X-P-I-N junction transistor 517-33 P-N-I-P junction transistor 517-33 Truitt, C. J. l)iographical material 515 Traffic Engineering Techniques for Determining Trunk Requirements in Alternate Routing Trunk N^etivorks 277-302 Trunk (s) intertoU trunk concentrating eiiuip- mcnt 303-28 Trunk Network(s) alternate routing trunk requirements 277-302 Van Duyne, C. W. 857 Variance analysis of 395 components of 396 Voltage P-N-I-P and N-P-I-N junction tran sister triodes 517 regulators 848 relays 8, 10 welding 885 W Wagar, H. N. biographical material 275 Economics of Telephone Relay Applica- tions 218-59 Magnetic Design of Relays 23-78 Relay Measuring Equipment 3-22 Walker, Laurence R. biographical material 798, 1208 Guided-Wave Propagation through Gy- romagnetic Media. Part I — ■ The Completely Filled Cylindrical Guide 579-659 Guided-Wave Propagation through Gy- romagnetic Media. Part II — Trans- verse Magnetization and Non-Re- ciprocal Helix 939-8G Guided-Wave Propagation through Gy- romagnetic Media. Part III — Per- turbation Theory and Miscellane- ous Results 1133-94 Walsh, E. G. 1131 Wave(s) circular electric (and) millimeter wave technicjues 1253-56 propagation 1210 (in) round metallic tubing, attenua- tion coefficient 1209 theoretical characteristics 1213-19 14 THE BELL SYSTEM TECHNICAL JOURNAL, 1954 guided gyromagnetic media, propagation through 579-659, 939-86, 1133-94 micro-, see Microwave (s) millimeter, see Millimeter Wave(s) space-charge 1344-48 Wave Picture of Microwave Tubes (J. R. Pierce) 1343-72 Wave Propagation along a Magnetically- Focused Cylindrical Electron Beam (J. A. Lewis, W. W. Rigrod) 399- 416 Waveguide (s) (for) circular electric waves, improved forms 1250-52 circular, filled field patterns 1175-94 communications medium 1209-65 copper bending 1209 coupled wave theory 661-719 dominant-mode 694 long distance practicality 1210 Waveguide as a Communication Medium (S. E. Miller) 1209-65 Wavelength shadows that are cast by hills 417 Weaver, Allan biographical material 1400 In-Band Single-Frequency Signaling 1309-30 Weiss, M. T. 1194 Welding automatic contact wire spring relay manufactured 897-923 automatic multiple resistance 898 contact 897-923 evaporation in 885 guns 918 percussive fundamental problems ^885-95 (in) wire spring relay manufacture 907-09 projection type resistance 899 voltage 885 Wenger, J. A. 532 Whittaker Functions 983 Williams, H. J. 1052 Wire(s) anchoring 878 bends 859 indexing 882 molding 879-81 shearing 882 spooling operation 859 straightened storage 861 straightener, automatic, multiple head 863-64 straightening and molding (for) wire spring relays 859-84 Wire Spring Relay See Relay (s) Wire Straightening and Molding for Wire Spring Relays (A. J. Brunner, H. E. Cosson, R. W. Strickland) 859-84 Wojciechowski, Bogumil M. biographical material 515 Continuous Incremental Thickness Measurements of Non-Conductive Cable Sheath 353-68 polyethylene cable sheath thickness 577 Worley, O. C. 217 Wright, J. P. 1052 X X-Y Recorder 5, 8-9 Printed in U.S.A. HEBELL SYSTEM Jechnical ournal VOTED TO THE SC I E N T I FIC^^->^ AND ENGINEERING PECTS OF ELECTRICAL COMMUNICATION LUME XXXIII JANUARY 1954 NUMBERl DESIGN OF RELAYS Introduction ^ (jj . A. c. Keller 1 Relay Measuring Equipment ^' t;'^ . "• ^- wagar 3 ^lagnetic Design of Relays . r. l. peek, jr. ajTo-'h'. n. wagar 23 Analysis of Measured Magnetization and Pull Characteristics R. L. PEEK, JR. 79 Estimation and Control of the Operate Time of Relays Part I — ^Theory R. l. peek, jr. 109 Part II — Design of Optimum Windings m. a. logan 144 Principles of Slow Release Relay Design r. l. peek, jr. 187 Economics of Telephone Relay Applications h. n. wagar 218 Symbols 257 Abstracts of Bell System Technical Papers Not Published in this Journal 260 Contributors to this Issue 274 COPYRIGHT 1954 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD S. BRACKEN, Chairman of the Board, Western Electric Company F. R. KAPPEL, President, Western Electric Company M . J. KELLY, President, Bell Telephone Laboratories E. J. McNEELY, Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE E. I. GREEN, Chairman A. J. B U S C H F. R. L A C K W. H. DOHERTY W. H. NUNN G.D.EDWARDS H, I. R O M N E S J. B. FISK H.V.SCHMIDT R. K. HON AM AN G. N. T H AY E R EDITORIAL STAFF J. D. T E B O, Editor M. E. STRIEBY, Managing Editor R. L. SHEPHERD, Production Editor THE BELL SYSTEM TECHNICAL JOURNAL is published six times a year by the American Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. Cleo F. Craig, President; S. Whitney Landon, Secretary; John J. Scanlon, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed in U. S. A. THE BELL SYSTEM TECHNICAL JOURNAL V o L u M E X X X 1 1 1 J A N U A H Y 1 9 5 4 n u M b e R 1 Copyright, 1954, American Telephone and Telegraph Company Design of Relays Introduction (Manuscript received September 28, 1953) The electromechanical relay is the basic building block of modern dial switching systems and also of various automatic control systems and computers. All of these systems depend on the action of the relay which is simple in its functions, but has to meet complex requirements placed on it by the systems in which it is used. Perhaps the best illustration of this apparent conflict is to note that a relay has simply to close or open electrical contacts when its coil is energized or deenergized. However, these simple functions must be performed equally well by millions of relays, and each of these must continue to perform reliably for millions and in some cases for more than a billion operations during its lifetime. Furthermore, in many cases the relay must function in a few thousandths of a second, or use little electrical power. The reliability re(}uired in telephone switching systems would be considered unreasonable in many other types of equipment and can be judged from the fact that a single failure in 5,000,000 operations is considered to be poor performance. Stated another way, satisfactory operation for the average relay in modern telephone switching systems is less than one failure in forty years of operation. The measurement of such low troulile rates is, in itself, a difficult and challenging problem. The need for such a high degree of reliability and for the associated requirement of high speed is evident from a few figures relating to 1 2 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 modern cross-bar switching systems. In these, estabhshing a single tele- phone connection between two subscribers makes use, for a very short time interval, of approximately 1,000 relay operations with a total of about 7,000 contacts. As the telephone plant has become more mechanized with its local and toll dial systems, automatic message accounting, automatic trouble recorders, etc., the relays have been required to do more things with less trouble. Accordingly, the steady progress which has been made in the automatism of switching equipment has depended upon improving the performance of relays. To provide this improvement, relay design must be guided by a clear understanding of the physical relations among all aspects of performance and the construction specified by the design. Some of the recent work in the area of relay design, production, service and measurements is covered by this issue which is devoted entirely to the analysis and measurement of relay performance, and to the economic considerations which govern optimum relay design. It is evident from the typical statistics given, that the successful and economical opera- tion of smtching systems and certain other existing automatic control systems demand the best in relay performance. A. C. Keller Relay Measuring Equipment By II. N. WAGAR (M;iiiuscTii)t received Seiitembor 25, 1953) The wide variety of technical problems encountered in telephone relay design calls for quantitative measurements of many kinds, involving static performance and dynamic performance. The present article describes some of the more important rneasuring tools for this purpose, as used in Bell Telephone Laboratories. Instrumentation for evaluation of force, flux, dis- placement, time, or their combination, is described. IXTRODUCTION Msitors to the switching development areas in the Bell Telephone Laboratories are often surprised at the extensive amount of measuring equipment used to study so simple a device as the telephone relay. This is because, though the relay itself may be simple, the problems requiring study are extremely complex. Because relays are used in such large quan- tities, their characteristics affect the economy of the telephone switching system. Not only is their manufacturing cost important; their quality also affects the central office cost. For example, an efficient design lowers the central office power plant cost, and faster-acting relays enable fewer common control units to handle more traffic, which further reduces the cost. These and related objectives pose many complex technical prob- lems invohnng mechanics, magnetics, kinetics, heating, and the like. The scope of relay analysis may be judged from the other articles in this issue. In every case there is need for measurement. Sometimes this (luantitative work is needed to confirm an analytical relation, sometimes to learn more concerning the basic phenomena, and often to characterize the performance of a test model. This article will describe some of the most-used measuring tools for the study of relays. Several kinds of measurements are required. In the first place, there are force measurements. Each relay must press the desired number of contacts together with a suitable force. To do this, a magnet must be pi-()vided which can move the springs, through whatever distance is re- (juired. Thus, measurement is needed of force-displacement characteris- 4 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 tics — both for contact arrangements, and for magnets. Another field of measurement involves magnetics. Both the mechanical output and the electrical characteristics of the electromagnet are determined by its mag- netization relations, requiring the measurement of magnetic flux. Beside such static measurements of mechanical, electrical, and magnetic quan- tities, corresponding measurements must also be obtained as functions of time in studying relay behavior under the dynamic conditions of actual operation. The many instruments which furnish such data may be grouped into those for static measurements, and those for dynamic measurements. Some of these tools are described in the following pages, particularly as they relate to force, current, flux, displacement, time, or their combina- tion. STATIC MEASUREMENTS The Measurement of Force For a complete understanding of the operation of rela}^ designs, knowl- edge of how the forces vary as the magnet air-gap changes, or as the contact members are deflected through their stroke, is of course required. The measurements most needed concern the force versus distance of the contact loads which the magnet must operate; and the force versus air- gap characteristic of the associated magnet. For many years, such meas- urements were made by a process of hanging weights and setting a micrometer screw, point by point, until complete data were obtained. More recently special instrumentation has made available a pendulum- type tensile tester, and a spring balance device, which have greatlj^ increased the convenience of making these measurements, point-by- point. Today, however, many such measurements may be made still more conveniently by a machine which automatically causes the gaps to vary and plots a curve of force ^'ersus distance. Machines of this type have been developed to a high degree of flexibility for tensile testing of materials such as metals, plastics, textiles, or paper. One such machine, which is extremely convenient for relay measure- ments, is the Instron tensile testing machine shown in Fig. 1. This ma- chine is manufactured by the Instron Engineering Corporation of Con- cord, Mass. In conjunction with suitable current supplies to control the behavior of the relay magnet, and with a few special circuits for auto- matically correcting for flexure of the relay parts, and making other similar adjustments, this machine has proven eminently suitable for ob- ser\'ing all "static" force-deflection characteristics of relays. The manner RELAY MEASLRIxXG EQUIPMENT STRAIN GAUGE CURRENT SUPPLY CABINET Fig. 1 — Tensile testing machine for force-displacement measurements. of operation of this machine will be given only in outline as it is described elsewhere. The force system to be measured, such as a relay magnet, is mounted on the crosshead of the machine, whose motion up or down may be controlled from the panel on the right. The other side of the force system being measured is comiected to a strain gauge fastened to the; top framework of the machine, and as the force on the strain gauge progres- sively changes, an electrical indication is given to the pen of the Leeds and Northrup X-Y recorder which then moves horizontally in proportion 6 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 to the force. The crosshead motion is transmitted electrically to a servo system which drives the paper up or down in proportion to the displace- ment of the mechanism involved. Thus, during the motion of the cross- head, the pen moves proportionately to force, the paper moves propor- tionately to distance, and a force-deflection curve is drawn. As a result, the tensile characteristics of mechanisms can be recorded very easily. For example, force variations of relay springs or of relay magnets may be measured, or a combination of the two, as desired. Some typical measurements obtained on this machine are given in Fig. 2 showing on one chart the manner in which electrical contacts change their force against the magnet and the manner in which the magnet pull varies across the gap. Such charts as these, when completely analyzed, enable the designer to choose the proper magnet for a particu- lar spring requirement. Among the useful features of this machine are the accuracy of recording which readily provides one or two per cent accuracy, and the ease of tracing and retracing a particular measure- ment. Also I'eadily obtained are exact information on the mechanical hysteresis losses within the mechanism. For example, in curve 1 of Fig. 2, the load characteristic of the contact springs is seen to have two values. The upper one represents the force on the closure stroke while the lower one represents the force on separation. The area between these two curves is the hysteresis loss; friction at any point is readily estimated as one-half the difference betw^een upper and lower force readings. Special information, such as the exact location of first closure or first separation of contacts can also be included, as the machine is provided with a circuit to insert a "pip" on the record w^hen desired. This instrument has marked advantages because of its accuracy, speed, convenience and wide range of uses. With almost equal ease, mechanisms whose full force variation is 2 grams, and those ranging up to half a ton can be measured. Once a suitable jig has been prepared to properly mount the parts, the Instron machine will furnish a complete set of data in less than an hour, whereas by previous methods it would have re- quired a matter of days. The time required for each indi\'idual cur\'e in the figure is the time allowed for the crosshead to move through the relay stroke. This usually is about 30 to 60 seconds, which adequately simulates the static charac- teristics of the relay force system; i.e., those force properties that would be measured at a particular point if the armature were held there. These static characteristics are commonly used in relay design to establish per- formance criteria for ordinary relays. However, in detailed studies of relay dynamics, these static characteristics must be supplemented by UKLAV MKASrUI.NC IXjll I'M KNT i estimates or measurements of the inertia and other forces which also occur. The (hrect measurement of (l3'namic performance; is described in ;i lat(M' section. The Mcdsuroncnl of Flux The mechanical output of a magnet depends on its maj2;netic ([uality. As is shown in companion articles in this issu(\ it is important for the (lesii;ner to know the \-alues of the closed gap reluctance (iiu , the leakage reluctance (iU , and the effective pole face area A. Beside these important ? 300 V \ . "PIPS" INDICATE Y \"'-^" "'""^ OR SEPARATION n \ \ \ \ \ 1 u \ PULL \300NI W Vo V. \ SEPAR I ATION -\v --VcLOSUF V. CURVES 2 \ MAGNET - '^\ PULL 2 X \k ction\^\ - V ) \1 ^ \ CURVE I ^ \ LOAD OF \ CONTACT \ \. SPRING Vo \ ===*, ^^^ 0.01 0.02 0.03 0.04 ARMATURE TRAVEL IN INCHES Kig. 2 — Tyi)ical recording of rehi}' ix'rfurinaiice made on Instron machine. 8 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 figures of merit, values for saturation fiux, ip" , and leakage fiux ratios at various points in the magnetic circuit, are commonly needed. For such information, flux as a function of ampere turns must be measured. Per- haps the most effective method is to make determinations of the average flux in the magnet structure at a numl^er of different air gaps; then by analysis determine values for the constants just enumerated. A series of magnetization curves for the relay magnet are required similar to those commonly taken on magnetic "ring samples," except that curves are obtained for several different air gaps. The ballistic galvanometer is the most familiar instrument for this purpose, still used for certain problems. In the Bell Laboratories, however, an extremely versatile recording fluxmeter was developed some ten years ago by P. P. Cioffi for use in research on magnetic materials.'' One of these instruments is now in con- stant use for relay measurements because of its accuracy, versatility and the rapidity with which tests can be made. The complete unit is shown in Fig. 3, being made up of a power supply system, galvanometer-integra- tor unit, and X-Y recorder. Its operation will now be described very briefl}'. The magnet to be tested is provided with a search coil. This search coil may be in the form of a winding physically in parallel with the main supply winding, a coil of few turns uniformly spread over the out- side of the coil, or of a specially wound coil mounted at a particular point of interest. When current is progressively ^'aried through the main winding, the voltage induced in the search coil by the magnet flux is transmitted to the galvanometer whose associated optical system divides a light beam between two photocells. The resulting photocell voltage unbalance is amplified and coupled back into the search coil circuit through a mutual inductance, so poled as to tend to restore the gal- vanometer deflection to zero. The feedback current in the mutual circuit is proportional to the flux, and is used to dri^■e the pen on the X-Y recorder. The paper is driven proportionately to the current in the wind- ing, wath the result that flux versus current (or ampere turns) is plotted directly. The instrument gives accuracy of i^a per cent over a wide range of fluxes and currents, with provisions made for readily changing the scale to cover different windings, operating voltages or magnet shapes. Measurements are made in a few minutes compared to a con- siderably greater effort when using the ballistic gah'anometer. Auxiliary features are also provided permitting convenient and complete demag- netization of the test magnets betw'een readings. Readings are extremely stable and repeatable. Typical results are shown in Fig. 4 where a series of magnetization curves for a particular test magnet are shown. By RELAY MEASURING EQUIPMENT X-Y RECORDER POWER SUPPLY GALVANOMETER AND INTEGRATOR UNIT Fig. 3 — Recording fluxmeter. methods of cross-analysis described in a separate article in this issue, these data may be used to find the figures of merit for the magnet, and other information concerning its performance. Often the measurement of magnetic potential at different locations in a magnetic circuit is re(juired. Among the more versatile of such tools is the Elhvood magnetomoti\-e force gauge. A brief description of its operation appears in a companion article.' The above measurements provide information on the static magnetic charaeteristies of a relay, but they do not provide all the information 10 THE BETiL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 a. . 3.5 u. 3.0 2.5 2.0 1.5 1.0 0.5 0 ^^ ,^ ^ 1 ^ / 7 yy y^ ^ ^ / y.' 7 Y> / v V // 7 V r / ''/ 7 7 / / // /a 74 ^ TRAVEL IN MILS AT POLE FACE 1 / / V/ '^/ / 1 1 1 // Ya V/ / i 7 /a m y 1 1 1 // A V 1 1 j 'M W / 1 1 k W / 1 1 1 M V 1 ll 1 //. f 0 20 40 60 80 100 120 140 160 180 200 220 240 260 AMPERE TURNS, NI Fig. 4 — Typical recording of flux-ampere turn data on a relay magnet. needed to describe the action while parts are in motion. For measure- ments in\'olving dynamics, additional methods are described below. DYNAMIC MEASUREMENTS The measurements just described are indispensable in obtaining basic information on the relay's ability to perform its prescribed mechanical functions, which in turn depend upon its magnetic characteristics. The}^ are needed in the every-day calculation of winding designs and the appli- cation of contact spring combinations which will be reliably operated by a given magnet. Cases arise in service, however, where very rigid require- ments must be placed on closure and separation to insure the desired performance. Complete knowledge of the influence of voltage variations, mechanical adjustments, numbers of springs and their natural fre- RELAY MEASURING EQUIPMENT 11 (luencies is needed. Often impacts and slide of parts must be minimized in order to reduce mechanical wear. Furthermore, the design of high speed relays requires an understanding of how the flux rises and decays when current is connected or disconnected. For the experimenter in these fields, many tools are available covering measurements of force, flux, current, and time, or combinations thereof. These measurements may involve shadowgraph, high speed motion picture, or various transducer techniques, and many special methods; some of these are described below. The Measurement of Force There is a definite need to know how forces in a relay structure vary w ith time. It would, for example, be very useful to measure the force experienced by the relay armature as it presses against its spring load during operation. While a satisfactory measurement technique for this problem has not yet been found, there are a number of similar type meas- urements which can be applied to individual mechanical problems in the structure. For example, with the higher-speed relay functions of today, it is found that relays made by older' methods suffer excessive PREAMPLIFIER- CONTROL BOX ■TEST RELAY WITH CRYSTAL INSERTED BETWEEN WEARING SURFACES OSCILLOSCOPE Barium titanate test set. 12 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 mechanical wear. In order to decide whether this is due to impacts, sHde of parts, or vibration, methods ha\'e been de\'eloped to isolate some of the transient forces encountered in relay operation, and measure them individually. Rapidly varying forces, such as impact forces, may be measured with the barium titanate crystal, which acts piezoelectricallj'- to yield a voltage across its surfaces when subjected to compression or shear type forces. Since it can be obtained in very small sizes, it may be mounted within the relay as a substitute for the part to be studied. By observing the voltages that have been developed across it during relay operation, one may readily measure the forces involved.'^ Such an arrangement can be made to provide a faithful frequency response over a vdde range, though it calls for careful amplifier circuit design. The unit shown in Fig. 5 pro- vides for accurate frequency response for impact forces varying as high as 50 kilocycles. Fig. 6 shows a typical force-time relation obtained with this equipment. In some recent measurements on an experimental rela}^, the impact forces between the driving members were measured to be approximately five-fold the static force, but not sufficient to explain pulverizing and other damage to certain of the functioning parts. Imposed Motion Slide between parts has been found to be a very damaging source of wear in telephone relays. It has been studied in some detail by means of ZERO FORCE -STATIC FORCE IMPACT FORCE Fig. 6 — Oscillogram of impact in a relay KELAY AlEASUKING EQUIPMENT 13 Photo of lA recorder setup. transducer elements whose motion is controllable to simulate a wide range of slide conditions typical of possible designs, from which Avear measurements may be taken and analyzed. Such methods were also described in the previous reference.' The measurements may be made either by means of a barium titanate element or by means of a phono- graph recording head such as the Western Electric lA recorder. Because of the feedback coil and amplifier circuit, the latter transducer will generate the precise amplitude of motion desired for any particular test merely by properly setting its input current. It is normally driven at frequencies in the order of 1 ,000 cycles, permitting a large number of slides to be obtained in a short time and affording information to the designer on a very accelerated basis. Through the choice of different amplitudes, different materials to be tested, and different forces between them, a wide range of variables can be covered, to guide the designer in the proper choice of materials and conditions of use. A recorder setup for a typical test is .shown in Fig. 7; data taken on a typical set of parts for an AI'' type relay are shown in Figure 8. 14 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 The Measurement of Flux When attempts were first made to extend the speed performance of general purpose relays much below 8 or 10 milliseconds functioning time, it was found that the simplified eddy-current theory, which treated the core like a single short-circuited turn, was no longer sufficiently accurate. The subject has been clarified through the use of the dynamic fluxmeter which permits a direct determination of how the relay flux changes with time, under actual conditions. This fluxmeter, originally due to E. L. Norton and recently refined by M. A. Logan, was described in a previous issue, and has proven most useful for determining those constants of the relay which the designer must understand in order to work much l)elow 8 milliseconds functioning time. The instrument is shown in Fig. 9. It requires that the relay under test be pulsed, usually at a speed of about 10 to 20 cycles. When the main winding is alternately connected and disconnected, the search coil which surrounds the Annding sends alter- nate positive and negative voltage impulses to a dc ammeter connected across its terminals. Now it can be shown that when the search coil and meter are disconnected during the interval from time zero to the moment of interest, then the resulting reading on the meter is exactly propor- tional to the flux at that moment. This switching function is provided bj^ a timing circuit comprising a 40-kc frequency source driving a 3 decade counting ring, under control of s\\itches permitting selection of the num- ber of cycles of delay time. In this way the flux maybe measured at inter- vals of 25 millionths of a second, with better than 1 per cent accuracy. One interesting measurement has shown that in the short time inter- vals of present-day relays, eddy currents have less effect on the initial 0.4 0.6 0.6 BILLIONS OF CYCLES Fig. 8 — Slide data on AF relay ItKLAV MKAsrUI.NC; K( M 1 I'M KN'T -TEST RELAY Ivnamic fluxmeter. development and decay of the field than was previously thought to be the case. The Measurement of Ttme — Eleclrical Effects The actual functioning time of the relay is one of the more important behavior characteristics which must be tabulated for the particular design. Its measurement can be quite simple or quite complex depending on the problem invoked. One of the simpler time measurements is that permitted by means of the arrangement shown in Fig. 10. This particular eciuipment is composed of a control circuit for studying the various relay timing conditions, and a Berkeley Timer made by the Berkeley Instrument Company. The timer is provided with electronic controls which cause it to start count- ing cycles from a standard f requeue}' when given an electrical impulse and to stop counting upon receiving a second impulse. The circuit may be arranged to give the first impulse when the relay winding is connected or disconnected and the second impulse when the contacts or other 16 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Fig. 10 — Photo of timing setup. members either touch or separate. It is timed by a 100-kc oscillator allowing measurements to be made at intervals of 10 millionths of a second. It is in constant use for the every-day measurement of relay timing. The simple timing measurements just described afford data on the IIKLAV MKASrillXG KQriPNrKN'T 17 basic tiniinji; chafacU'ristics of the slruclui'c, Iml may ^ixc iiisufficiciil information on (lie timing pcrl'ornianco of tlie relay contacts in a paiMicn- lar circuit. For example, contacts may close as desired hut then reoi)en intermittently in a manner to cause either unwanted circuit hehaxior or untlesirahle contact sparking and conseciuiMit erosion. Sometimes the relay armatui'c, in crossing the air-gap, encounters sudden loads as springs are successively picked up in the stroke, causing it to momen- tarily reverse its direction Ix'fore i)ulling home. Also, when the relay armature releases, it may sti'ike the backstop and I'ebound, recrossing a portion of its gaj) and causing undesirable react uating of the contacts. Such momentary opening and closing of the contacts is classed as "con- tact chatter," and special means are needed to detect, measure, and understand its behavior. The string oscillograph has been extremely useful for recording and studying such timing effects when the contact chatter was of compara- tiveh' long duration and low freciuency. For relays whose functioning time exceeded 10 milliseconds and where chatter intervals were in the order of 2 or 3 milliseconds each for possibly 6 or 7 successive times, this instrument was most effective. For many of the faster relays, however, where chatter of this type has been completely eliminated, there remain problems of much higher frequency, shorter duration chatter, which are still of great importance to the circuit designer. In such cases the cathode raj'- oscilloscope is used. A cathode ray oscilloscope arrangement for the study of chatter is shown in Fig. 11. The horizontal axis of the oscilloscope has a calibrated time base and the trace is displaced vertically to mark closing of the contacts or other events of interest. The horizontal sweep may be trig- gered at the initial closure of the contacts, but a variable time delay OSCILLOSCOPE Fig. 11 — Cathodp ray oscilloscope circuit for the stu(l,\- of contact chatter 18 THE BELL SYSTEM TECHNICAL JOURNAL, J.\:XUARY 1954 (a) (b) INITIAL CHATTER SHOCK CHATTER 1 DIV= 10 MICROSECONDS 1 DIV = 100 MICROSECONDS Fig. 12 — Views of chatter obtained by oscilloscope. can be used to trigger the sweep at a later time so that any portion of the chatter can be observed in detail. Typical measurements are shown in Fig. 12. In each of the views, the horizontal lines starting at the upper left represent open contacts, the lower horizontal lines represent closed contacts, and any jumping between them indicates chatter. Fig. 12(a) shows the relatively high frecjuenc}^ chatter which occurs immediately after contacts close. This is called initial chatter and is caused by vibra- tion of contact springs in their higher modes due to the impact of mating contacts. Initial chatter differs in character from the lower freciuency shock chatter shown in Fig. 12(b). Shock chatter is caused by ^^^re vibra- tions induced by the shock of the armature striking the core. The time of each reopen of the contacts and the time between opens is much longer than for initial chatter. Many measurements of the type just described are made in the course of a relay development. They enable the relay designer to relate chatter performance to design characteristics of relays so that contact chatter can be reduced or eliminated. Such measurements also allow one to classify chatter performance according to the circuit application. The Measurement of Time — Mechanical Effects Electrical timing measurements previously described give data on over-all effective performance; to understand these results one often needs to measure displacement-time characteristics both alone, and in relation to current versus time and flux versus time. The string oscillo- RELAY MEASURING EQIII'MKNT Fig. 13 — Schematic of sliado\vgrai)h action graph mentioned alcove, has on many occasions been modified to also record a silhouette of the moving parts of the structure on a moving strip of photographic paper, and thus pro\'lde a simultaneous trace of displacement and current against time. One of the more advanced shadowgraph-oscillograph e([uipments is shown in Fig. 13, and a typical record made on a wire spring relay is given in Fig. 14. In this case the single record shows the current to the winding and the mechanical move- ment at the two opposite ends of the contact-operating card attached to the armature of this relay. From such measurements, ^'elocities of im- pact, location of the parts at a given instant, stagger between parts, relative motion, unbalanced motion, and the like, may all be determined and correlated with electrical changes. Motion of parts may also be studied optical!}' to give displacement or velocity data, using the photocell. Properly placed flags on the moving CARD MOTION ■-^--, TOP OF CARD >■ BOTTOM OF CARD-*" CORE MOTION ^-j WINDING CURRENT—-' 4 H ^ 10 1^ MILLISECONDS Fig. 14 — Typical shadowgram of 2S7 relay armatiiro motion. INSTANT OF IMPACT 5 — I MAXIMUM DISTORTION COMPLETELY RESTORED Fig. 15 — High-speed motion pictures of a, falling relay at the moment of impact. RELAY MEASrinXC IHill PMEXT 21 parts of tlu' slructui'c conti'ol the li.!;ht t'alliii.ii on the cell, whose output can then be oonxerted so as to read (hrectly as (hsplacenient . 'Hie di.s- phieement data may then be (HiTerentialed as a function of time, in electrical differentiating circuits, to give very accurate measurements of velocity. A complete description of this method, originally due to E. L. Xorton, has been published by M. A. Logan.*' As a result of the accuracy and convenience of this method, it has recently been used extensively in relay studies correlating the amount of contact chatter with armature \-elocity. The motion of parts may also be observed with various forms of trans- ducers. One which is now finding application in telephone relay studies is a type of electrostatic gauge, measuring armature displacement by the changes in capacity between it and a fixed electrode. The entire scheme is to be described by T. E. Davis and A. L. Blaha in a forth- coming issue. Some relay motions need to be studied in three dimensions. Then the high-speed motion picture gives best results. By photographing at up to 5,000 frames per second, and viewing at about 20 frames, a time mag- nification of 250 to 1 can be gained. If need be, such pictures may be scaled off to give displacement-time information, a somewhat tedious task. A recent study of the effect of dropping in shipment gives a strik- ing picture of the utiUty of the method Fig. 15 gives views, photographed at 3,000 frames per second, of an AF relay as its mounting plate strikes a concrete block at the end of a six inch fall. Severe distortion, followed by recovery of the parts to normal, may be seen. With such information, relay designers can plan parts to stand the service stresses, and form a clear judgment of the margin of reliability built into the structure, CONCLUSION Although the relay is one of the oldest de^'ices in the telephone busi- ness, many features of its operation are still imperfectly understood, even today. The number and complexity of the continuing technical problems may be judged from the other articles in this issue, on representative relay subjects. As improved relay operation becomes ever more important in the telephone system, the analytical and measuring technology for these devices must progress, in parallel. Some of the typical measuring equipment, as needed for modern relay design, has been described in this article. REFERENCES 1. R. L. Peelv, Jr., Mea-suring the Pull of Relays, Bell Lab. Record, .June, 1953. 2. T. E. Davis, Measuring the Load-Displacement in Relays, Bell Lai). Record, June, 1953. 22 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 3. G. S. Burr, Servo-Controlled Tensile Strength Tester, Electronics, May, 1949, and E. G. Walsh, Continuously Recorded RelayMeasurements, Bell Lab. Rec- ord, Jan., 1954. 4. P. P. Cioffi, Recording Fluxmcter of High Accuracj- and Sensitivity, Rev. of Sci. Instr., July, 1950. 5. R. L. Peek, Jr., Analj'sis of Measured Magnetic and Pull Characteristics, page 79 of this issue. 6. W. B. Elhvood, A New Magnetomotive Force Gauge, Rev. Sci. Instr., 17, p. 109, 1946. 7. W. P. Mason and S. D. White, New Techniques for Measuring Forces and Wear, B. S.T. J., May, 1952. 8. M. A. Logan, Dynamic Measurements on Electromagnetic Devices, B. S. T. J., Nov., 1953. Magnetic Design of Relays By R. L. PEEK, Jr., and H. N. WAGAR (Manuscript received Seplembnr 24, 1953) The mechanical work and the speed of opera! ion of telephone relays are determined in a large measure by the characteristics of the relay magnet. The underlying magnetic principles and the resulting design relationships for magnets are discussed in this article, which is in part a review of the hackground material and in part a description of its use in developing methods for magnet design and the analysis of magnet performance. Basic energy considerations are shown to determine the relations between the work capacity and the magnetization characteristics, and analytical e.rpressions for the latter are given in terms of the dimensions and materials of the magnet. These expressions are developed for the magnetic circuit ap- proximation to static field theory, which is shown to provide an adequate representation of the field relations controlling performance. Methods are given for representing the magnetic circuit relations by means of a simple equivalent circuit. Expressions are derived for the mechanical output of the relay in parameters of this simple equivalent circuit, and these expressions used to determine optimum conditions for meeting specific design objectives. INTRODUCTION The complex switching equipment which handles the telephone traffic in automatic central offices is built up of simple component elements, of which the great majority are telephone relays. The large investment in these relays, of which tens of millions are made each year, has led to intensive effort to construct and use them as cheaply as possible, so that they will perform their function at a minimum over-all cost to the tele- phone system and thus to the subscriber. As the costs of use vary with the efficiency and speed, maximum economy requires the solution of technical problems in magnet design as well as the related problems of mechanical design for economy in manufacture. As a result, the tech- nology of magnet design is under constant study at Bell Telephone Labo- ratories, directed to increased understanding of magnet pcM-formance 23 24. THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 and to its improvement. This article gives the background of this tech- nolog}^ as it apphes to the dc telephone relay. A telephone relay is an electromagnetic switch which actuates metallic contacts in response to signals applied to its coil. Its characteristics as a component of switching circuits are defined primarily by (1) the contact assembly, (2) the coil resistance, (3) the current flow requirements for operation, holding, and release, and (4) the operate and release times. The design of the contact assembly determines the niunber and nature of the contacts, the sequence of their operation, their current carrying capacity, and their reliability and life. Structurally, the design of the relay, including both the electromagnet and the contact assembly, is governed not only by the performance requirements but also by the manufacturing considerations which deter- mine the relay's initial cost, and by equipment considerations relating to the way it is mounted, wired, adjusted, and maintained. The ultimate objective is to perform a circuit function at a minimum over-all cost, including the costs of installation, power consumption, maintenance and replacement, as well as initial cost. The design of the contact assembly determines a force-displacement characteristic representing the mechanical work which must be done by the electromagnet. The relations between this mechanical output and the electrical input to the coil are determined by the magnetic design of the relay, or specifically of its electromagnet. The electromagnetic design is therefore subject to the performance reciuirements, to the design of the contact assembly, and to the manufacturing and equipment con- siderations applying to the whole relay. Magnetic design thus requires the ability to determine the effect on the performance of alternative choices of the configuration, dimensions, and materials of the electromagnet, of alternative load characteristics corresponding to variations in the design of the contact assembly, and of alternatives with respect to the coil dimensions and characteristics. The basic relations required for such prediction of performance are the mag- netization relations, which express the field strength of the electromag- net as a function of the ampere turns, and the armature position. The present article describes the evaluation of magnetization rela- tions, and their use in relating the mechanical output with the electrical input to the coil. The time reciuired for operation, which also depends upon the magnetization relations, is discussed in a separate article.^ While much of the material given here has a broader application, the discussion of its use will l)e confined to the case of the simple neutral (non-polar) relay. MAGNETIC DESIGN OF RELAYS 25 To illustrate the objectives of majiiuMic (l(\si pull exerted by tliis relay's magnet for various values of coil ampere tiu'iis. These pull curves determine the ampere turns required to operate a contact ar- rangement having a specific load characteristic, such as that shown. Hy means of the magnetization relations, these pull characteristics can be related to the design of the electromagnet. The notation used in this article conforms to the list that is given on page 257. 1 THE COIL CONSTANT The relay coil characteristics of interest in its use as a circuit com- ponent are its resistance and the current flow for operation. Subject to some ciualifications as to available voltages and wire sizes, these may be summarized in a statement of the steady state power I^R supplied in operation. As illustrated in Fig. 1, the coil quantity determined jointly l)y the load and the magnetic design is the ampere turn value NI. The power and the ampere turn value are related by the coil constant Gc or X'/R (which ecjuals {Nlf/{l'^R)). This quantity is the equivalent single turn conductance of the coil, and is usually expressed in mhos. It is determined by the coil dimensions, as can be shown as follows: Let A be the area of a cross-section of the coil as cut by a plane through the coil axis. Let m be the mean length of turn, the arithmetic mean of the lengths of the inner and outer turns. Then the coil volume S equals Am. If a is the cross-section of the wire, the number of turns A'^ equals eA/a, where e is the copper efficiency, or fraction of the coil volume occupied by conductor. Substituting S/m ior A, N equals eS/(am). The Avire length is Nm, and hence the resistance R equals pNm/a, where p is the resistivity of the conductor. Hence N/R equals a/(mp), and the coil constant is given by: Gc = ^ = —,. (1) The coil constant is thus independent of the wire size, except to the minor ext(>nt that the copper efficiency c decieases as the wire is made finer. With this qualification, and assuming copper wire to be used, the 26 THE BELL SYSTEM TECHNICAL JOl'RNAL, JAXI'ARY 1954 coil constant is wholly determined by the coil dimensions and the type of insulation used. Thus the power required for relay operation depends upon (1) the load characteristic of the contact arrangement, (2) the magnetic characteristics which relate the pull to the ampere turns, and (3) the coil dimensions as defined by S/m , which determines the coil constant A^ /R, and thus the relation between ampere turns and power input. The external dimensions of relays are largely determined by the coil 2 300 0 t OPERATED POSITION 0.01 0.02 0.03 0.04 ^ 0.05 ARMATURE TRAVEL IN INCHES RELEASED POSITION (AGAINST backstop) Fig. 1 — Typical load and pull characteristics of a wire spring relay with 12 transfer contacts. MAGNETIC OKSKiX OK IJKLAVS 27 Fig. 2 NI, NI2 ABAMPERE TURNS, NI > Energy relations in magnetization. size, or winding space provided. The choice of these dimensions reflects an economic balance between the manufacturing costs of the magnet and its coil, which increase with these chmensions, and the saxings in power cost resulting from increasing the coil constant. 2 MAGXETIZATIOX RELATIONS The magnetization relations of an (electromagnet define the steady state flux linkages of the coil as a function of the two variables: mag- netomotive force JF (equal to 4xA^/), and the gap .v, which specifies the armature position. They may be represented by a family of curves each gi\'ing the average flux linked per turn plotted against NI for a particu- lai- value of x. In Fig. 2, the curves marked .ri and .7:2 represent two such curves, where Xo corresponds to a smaller gap than .ri . To determine how the magnetization relations depend upon the dimen- sions and configuration of the electromagnet reriuires thcii- iii1('rj)retation in terms of static field theory. Such interpretation is needed in deter- 28 THE BELL SYSTEM TECHNICAL JOURXAL, JANUARY 1954 mining the design conditions tor attaining a desired performance. For a specific structure, however, the observed magnetization relations, apart from any other interpretation, provide a record of the part of the elec- trical energy input to the coil which is stored in the electromagnet. The performance of the magnet with respect to the mechanical work which this stored energy can do may be determined directly from the way in which this energy varies with armature position. The experimental determination of the flux ip for particular \^alues of AV and x involves a measurement of the electrical ciuantity AV defined b}^ the ec[uation: N

;ralion of (Miuation (3) gives the followinji; alternatixc expiessions for the liekl energy U: Over the upper curved portions of a magnetization curve, (R is a func- tion of v? as well as of x, and equations (5) do not apply. Decreasing Magnetization In relay terminology, "operation," following closure of the coil circuit, is distinguished from "release," which follows opening of the circuit. The preceding discussion applies directly to the relations for increasing magnetization, as in operation. In release, the field energy and the cur- rent decrease together, giving a decrease in Ntp measured by a voltage time integral similar to the right-hand side of ecjuation (2), but of op- posite sign. The resulting decreasing magnetization curve is obtained by subtracting the decrease in AV from its initial ^'alue. The decreasing mag- netization curve is higher than the magnetization curve, and the field energy recovered electrically is correspondingly less than that stored in magnetization; the difference corresponds to the loss of energy through hj'steresis in the magnetic material. Mechanical Output Referring to Fig. 2, let the current have the steady value /i , with the armature at rest at Xj . If the current increases to lo while the armature moves from .ri to .r2 , the flux (p varies with Ni along some curve such as 1-2 corresponding to the values of x and Ni concurrently attained. The electrical energy drawn by the coil in this process (aside from th(^ heating loss) is given by the integral of i d(N(p), or Ni dip, taken along the curve 1-2. Part of this energy appears in the increase in the field energy from Ui to Uo as given by eciuation (3) for the points 1 and 2 respecti\-ely. The V>alance represents the mechanical work done, the integral of Fdx from Xi to x-2 where F is the pull. Hence: f ' Fdx = f ' Nidif - (Ta - Ti). J Xl '''Pi (6) The first right-hand teim is represented in Fig. 2 by the area 5-1-2-6 while Ui and U- are represented respecti\-ely by the areas 0-1-5 and 30 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 0-3-2-6. Thus the mechanical ^vork, the left-hand term of eciuation (6), is represented by 0-1-2-3-0, the area bounded by the two magnetiza- tion curves and the path follo\ved by (p, x and Ni in the concurrent change from 7i at Xi to h at X2 . If armature motion occurs at constant flux, the first right-hand term in equation (6) is zero, and the mechanical work ec^uals the change in the field energy U.lf cp = (pz in Fig. 2 for example, the work done as the armature moves from xi to X2 is U4 — Uz , represented by the area 0-4-3-0. From equation (6), the pull F is then given bj^: f) TJ F = — — {(p constant). (7) dx If armature motion occurs at constant current, the first right-hand term in eciuation (6) becomes the change in NI(p. If / = /i , for example, motion of the armature from Xi to X2 increases Nltp from NInpi to A"/i^3 . Hence the mechanical work done is the difference between NIi(pi — Uz , represented by the area 0-3-8 and A^/1^1 — L\ , represented by the area 0-1-8. The work done at constant current is therefore the change in the quantity W defined by the equation: W = f ^ -^ J / / / / 6 XIO^ 5432 100 1 2 34 5 PERMEABILITY, >X H IN OFRSTEDS Fig. 5 — M-B and B-H curves for magnetic iron. MAGNETIC DKSKiX OF RELAYS 37 As illustrated in Fig- <>, i^ and // are therefore dotciniincd by the inter- section witli the demagnetization curve of the line luuing a sIojk' gix-en by the right-hand side of (9). The complete magnetic circuit has a residual flux ^p = Ba resulting from the coercive mmf, //c^, which equals the sum of the potential drop II({ — (S{d ma- terial for this type of construction. The properties of nickel are included because of its use as a magnetic separator and hinge member. Hyperbolic Approximation to Magnetization Curves The variation of permeal)ility with density makes it necessary to provide some formulation of the n versus B relation in (le^'eloping an analytical treatment of magnetization relations. The B versus // relation for decreasing magnetization, the loop 2-3-4 of Fig. 4, has a shape similar to that of a rectangular hyperbola, asymptotic to the line repi'esenting B". This curve can therefore be represented approximately by the eciuation : B = ^(Hc- H), (10) B" - B B in which ju", B'\ and He are constants. This purely empirical relation is called the Froelich-Kennelly equation. In general, it does not give a satisfactory fit to the whole loop, but pro- vides a satisfactory approximation for engineering use to the portions of the curve in either the first or second quadrants, using different values for the constants in the two cases. Tn addition, it may be employed to represent the upper portion of the normal, or increasing magnetization curve. The expression for the permeability ^u, or B/{Hc + H) corresponding to (10) is: l^^^Hc + H MM Li Hence, to the extent the B — H curve conforms to equation (10), the reciprocal of the permeability varies linearly with H. Alternately, the 40 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 permeability may be expressed in terms of B, giving the equation: m"(5" - B) M = ^, . (12) If this relation is applied to a part of a magnetic circuit, such as the core of a relay, the reluctance (Re of this part may be written as f/(ij.a), where / is the length and a is the cross-sectional area of the part. Then from (10), (Re is given by: // (Re = cs\c -^ , (13) (f — (f where (Re = (/{ix"a), ip = Ba, the flux through the part, and tp" is the saturation value of = ^o for [F = 0 to eliminalc ;Tr' , the e( [nation may be written in the form <^o m'V <^0 ipo (14) If the length /"and cross-section a of the core are known, together with the magnetic constants of the core material and the rehictance iS\ of the return jiath (external to the core), the constant term.s in (14) may be evaluted. ^p" is equal to aB" , wliei-e />'" is the saluiation density of the core material. .\s previously noted, the \ahies of B :,, in Table 1 may be used as ellecti\-e values of B" . (18) Exact: (R = bin a I 2/-0/ Approximate: (R = — ab Lridis of apj)roximatioii : Less than 1 per cent if a < ,^/o Less than Id per cent if a < /o Fig. 10 — Reluctance between inclined plane surfaces. aud/or x < -I'o In Exact : (R = be Where b = length parallel to axis A])proximate: (ft = , — bi-od Errors of approximation: Less than 1 [jer cent if /'s < 1.4ri' Less than 10 per cent if /••j < 3r,' Fig. 11 — Reluctance of a cylindrical gap. \|^ 46 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Leakage Reluctances An accurate estimate of the reluctance between magnetic members requires a detailed knowledge of the flux paths. Since these are only knoAvn accurately in cases for which solutions of the field equations are available, approximations are obtained by assuming geometrical paths such as straight lines, arcs of circles, ellipses, and so forth. From these assumed paths the reluctance is calculated by means of the expression {/na. The choice of suitable approximations depends largely upon a knowledge of the flux paths in certain simple cases which can be analyzed rigorously, and upon the experimental exploration of more complicated fields. The method is satisfactory provided the separation between the mag- netic members is small. It has been used to derive the relations given in Figs. 10 and 11. A further application of this method gives the reluctance between the side surfaces of coaxial cylinders, as shown in Fig. 13. This is useful in estimating the leakage reluctance shunting an air gap. Where the separation between magnetic members is large, it is difficult to estimate the configuration of the flux paths. It is then necessary to employ the relations applying to the most nearly similar configuration for which a rigorous solution is known. Two such solutions are applicable to a number of problems. The first is the case of two infinitely long, parallel, equipotential, circular cylinders, shown in Fig. 14. The second is the case of two equipotential spheres of equal size, shown in Fig. 15. h- — Fig. 12 — Effective pole face area referred to gap x at f. MAGN'KTIC I)i:si(;\ OF UKLAYS 47 0.8 0.6 ■:(R 0.1 0.08 0.06 0.04 — - eD_. ♦ X > — r<--c--> «--c- - >" — r /c = o.oi ■ ■ — 1 — "•"-f 1 — ■ , — ^_^ — """ 0;! — - H-;;^ ' — ■"" ■^ ^^ ^^p ' - :;:= ..-----^^ic^ V — ' ^Pt ^^^:== ?^ ■^ ^ ^^ ^.o — ' '^ [-'^^--■^Ci^'^l^^^^ ^^^ ^^ -^ P" ^ -- ■"^ p^ ^^^^ ^ 2 ^ ^ ^ ^ 1, rm l + 2-(l + 4/1+^ (R 4 6 6 10 y \rm/ 01 = (n (R = + K \rm/ \/i^'- if cylinders are not circular, let /' = ^ — X perimeter. Fig. 13 — Reluctance between side surfaces of end -on coaxial cj'lindeis. 48 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 bCR 1 \ 1 / ^ I \ _> L^ K-r.*\^ — D — "A k l / /^ / 1 1 1 1 1 15 2 3 4 5 6 8 10 15 20 30 40 50 60 80 100 U where « = (R = /«(« + V?/^ - l) COSH-h 27r6 2x& and 6 = length of each cylinder Fig. 14 — Reluctance between parallel cylinders. For other configurations \vhic'h bear a reasonable resemblance to the above cases, the leakage reluctances can be estimated satisfactorily by judicious modifications of the expressions for the simple cases. For example, consider the case of two parallel rectangular bars. They are roughly eciuivalent to two parallel circular cylinders provided the minimum separation is the same in both cases, and provided the perime- ter of each cylinder is equal to the perimeter of the corresponding bar. Thus, to estimate the leakage reluctance between the rectangular bars, the radius of each equivalent cylinder is taken as \/2iv times the perime- ter of the corresponding bar, and the center-to-center distance between the eciuivalent cylinders is taken as the minimum separation between the two bars plus the two equivalent radii. Leakage between the legs of many magnet forms may be estimated by MAOximr nKSTGx of relays 40 the nppioxiiu;!! inn jusl dcscriltcd. ('oiisidcr tlic idealized loiiii siiowii in Fig. I(), where the magnetic path consists mainly ot two parallel cylinders connected at ()n(> end, th(> armatiu'c and woiking gap being at the op|)()- site end. 'I'he leakagt^ fhix in parallel with the main gap linx is delermined l)y the relnctance of the patli l)etw(M'n these cylind(Ms. l''oi- that poi'lioii of tlie two cylinders appeai'ing ontside the coil and therefore at appio.xi- mately constant potential, the relnctance is fonnd fi'om the relations given in Fig. 14. For the leakage relnctance over the length (Miclo.sed l)y the (oil, the drop in potential along the core results in one-thii-d tlie previous reluctance, as is sliown in S(M'tion 6. The variation in ])er- meance pei' unit length for such cases may l)c found in Fig. 17. Leakage also occiu's between the end sections of many magnet forms, and ma\- he estimated by a procedure similar to those above, assuming the end surfaces to be eciui\'alent to two hemispheres having the same diameter d as the eyhnders. The (luantity C->d in Fig. 17 is one half the permeance between corresponding spheres, as determined from the re- lations of Fig. 15. From this the net leakage permeance of leg and end D AREA S AREA S Exact: (R = ' + '77 + 77' + (^y-©'-(^)' + . . . 1 - AiJpidxiiiiiito: w here (R = (R = D 2-Kr ' 1 1 S = surface area of one sphere. when I) > 6r wlieii /; » Latter apjiroximatioii ma\- Ix' used to estimate leakage r(>luctancc lietwceii hack surfaces of pole ])ieces. Lig. 15 — Keluctance helween spherical surfaces. 50 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ w , USEFUL GAP FIELD • » — CORE LEAKAGE FIELD * — ARMATURE LEAKAGE FIELD Fig. 16 — Distribution of the field of an electromagnet, surfaces may be found as (P =^+ C,f2+ C2d, where values of Ci and C2 are given in curve form, and the equivalent value of d is as shown in Fig. 17. When the structure has two return paths, the reluctance is assumed to be one-half the value given by this last equation. External Reluctance of a Bar Magnet The relations of Figs. 13 through 17 suffice for the evaluation of leak- age in most ordinary electromagnets. In special structures, particularly in polar relays, cases occur where the return path is predominantly an 10 8 6 5 U 4 -1 \ \ v V V ^ ^ ^ \; — — - — ^ 1 >^ TWO LEGS (perimeters P| 8. P2) P1 + P2 ZZZZDICjIZJPs COIL_ D' ^__OP D=D'+c| + C,£2 + C2d THREE LEGS (PERIMETERS P| & P2 ) D/d D~aP2 D' 2.7T yU^i^Y []p, D = D'+d ' ^^"""^^"^ ~^' (P=2(^'+C,£2 + C2d) DPa Fig. 17 - — Effective reluctance between parallel bars joined at one end. MAGXKriC DKSUi.N OF KKLAYS 51 air path. A common case is that of a pormaiioiit magnet magnetized as a separate part prior to assembly in a polar structure. In such ( ases, the reluctance of the return path can be estimated as that of the external field of a bar magnet. The r(>luctance of a bai- magnet is closely represented by the reluctance of the same magnet in the form of a ring, in series with a reluctance representing all the flux return paths. Values for this reluctance may be assigned by measurements of magnetomotive forte to produce a given flux in a ring sample and in a l)ar. The difference in magnetomotive force gives a measure of the rcluctaiu e of the air path. It has been found by l(R 4.0 a . .ENGTH OF MAGNET J.O = CROSS-SECTIONAL AREA FOR RECTANGULAR CROSS-SECTION h = THICKNESS W = WIDTH . ^ li.O ."x"^"x"-""-""x X ; / / \ \ \ N \ Fig. 19 — ]\IaKiif'1i<' fi<'lscribed in the present section cover those procedures which are used most frequently in estimating leakage reluctances. For a more extended treatment of this subject, reference may be made to S. Evershed and H. C. Roters.^ 6 MAGNETIC CIRCUIT EVALUATION In the discussion of the magnetic circuit concept in Section 3, it was noted that the accuracy of the representation varies with the extent to which the network of tubes of induction is sub-di\'ided to correspond to the distributed nature of the actual field. The effect of sub-division upon the accuracy is greater in the high density region, where the reluctance of the iron parts is variable and increasing, than in the low density region, where the iron reluctance is small and approximately constant. Hence the choice of an adequate network is largely contingent upon the location of the iron parts having the highest flux density. Incipient saturation affects the reluctance of these parts, and thus th(> pattern of the field, while parts of lower density remain of low and approximately constant reluctance. In ordinary electromagnets the core is the part of highest density, in which saturation limits the attainable field. It is good design practice to limit the core section to as small a \'alue as is consistent with satisfactory performance, as this results in a minimum inside coil diameter. This is advantageous with respect to the coil constant, as shown bv e(|uation (1). In the special case ot high speed relays, it is advantageous to minimize the mass of the armature and hence its cross-section. In such relays arma- ture saturation ma}^ control, or occur concurrently with core saturation. Saturation elsewhere than in the core or armatun^ is of intei'cst only in the diagnosis of faulty design, since the return members should have a section adequate to carr^' the maximum field at densities well below saturation. Thus in most electromag?iets, saturation occurs in the core, and in- cipient saturation affects its reluctance and the pattern of the associated field. The magnetomotive force \-aries along the length of the coil, and the core is therefore subject to variations along its length in magnetic 54 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 WXHI ■y J (a) TRANSMISSION LINE MAGNETiC CIRCUIT T pAy hAy 2 vAAy 2 hAy H|lh- Fig. 20 (b) DIFFERENTIAL LINE ELEMENT Transmission line analogy to the field of an electromagnet. potential, in flux density, and (at high densities) in reluctance per unit length. To treat it as a single circuit element, as in the discussion of Fig. 3, is thus a highly simplified approximation. An understanding of the extent to which this approximation is valid may be gained by considering a more rigorous analysis, in which, as indicated in Fig. 20, the core and return path are treated as analogous to a transmission line with dis- tributed constants. Transmission Line Analogy In Fig. 20(a), the core is shown as one side of the line, and the return path as the other side. The length of the line is taken as the length of the winding, which is assumed to be distributed uniformly. It is assumed that the line is terminated at its ends by lumped reluctances, (Ri and (Ro . The characteristics of the line and the applied magnetomotive force are expressed in terms of the following quantities: / = mmf difference between the two sides of the line, ^ = reluctance per unit length of line, p = leakage permeance per unit length of line, h = impressed mmf per unit length of line. Instead of trying to represent the entire magnetic circuit by a simple network of a few lumped reluctances, it is assumed only that an infinitesi- mal length (dy) of the magnetic line can be represented by the "7"' MAGNETIC DESIGN OF UELAYh- 55 network shown in Fig. 20(h). The complete magnetic circuit, then, con- sists of an infinite number of these elementary networks < onnectcd end-to-end and terminated in lumped reluctances at the extreme ends. Note that this approximation to the magnetic circuit explicitly recog- nizes the distributed nature of the leakage flux. Consider now the several quantities which enter into the approxima- tion. Each of the terminating reluctances ((Ri and (JI2) includes two com- ponent reluctances in parallel. The first component is simply the leakage reluctance across the end of the line. The second component is the sum of the reluctances of the magnetic members and series air gaps which complete the circuit from one side of the line to the other. For any single value of the working gap, (Ri and (R2 rna.y be taken as constants. The quantit}^ p, which is the leakage permeance per unit length be- tween the two sides of the line, depends upon the geometry of the struc- ture, and may be calculated by the methods of Section 5. Provided the configuration of the magnetic line is uniform throughout its length, p is taken as a constant. The assumption that p is constant is eciuivalent to assuming that all leakage paths between the two sides of the transmission- line lie in planes that are perpendicular to the core. This condition is not satisfied near the ends of the core, Init correction for the end effects may be made in evaluating the terminal reluctances, (Ri and (R2 ■ The quantit}' ^, the series reluctance per unit length of line, involves magnetic material whose permeability varies with flux density. It is a variable whose magnitude depends upon the applied magnetomotive force, upon the terminating reluctances, and upon position along the line, since it is a function of (p. In the low density region, however, its value is substantially independent of lumped approximation correctly represents the limiting conditions. It therefore* provides a rough approximation to the intermediate values. Series-Parallel Magnetic Circuit Subject to the limitations discussed above, the magnetic circuit of most ordinary electromagnets can be represented in the form shown in Fig. 21. The core reluctance (Re is in series with two parallel paths: a leakage path of reluctance (Rl2 , and an armature path of reluctance (R02 + x/A-2 . The subscript 2 is used with the constants of this particular circuit to dis- tinguish them from those of the simpler approximation to be discussed below. The magnetic circuit of Fig. 3 reduces to that of Fig. 21 if the reluctance (R^ of the former is ignored, or considered as part of the reluctance (Ros- The circuit reluctance (ft, or J/V, tan be derived from the circuit by the procedure applying to resistances in an electrical circuit, and is given bv: (Rls (R = (Re + (^«^ + £) (23) (R/.2 + (R,r> + A, The evaluation of the constants of this circuit may be desci'ibed with Fit;. 21 — Series parallel magnetic (•ireuit — tlic u.sual design analogy 58 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 reference to the two structures shown schematically in Fig. 22, where Fig. 22(a) represents the familiar "end-on" armature type of construc- tion, and Fig. 22(b) represents the "flat type" relay of Bell System use. The dashed lines indicate the path along which the length of the parts is measured, while the lined areas are those of the main, heel, and side gaps. The core reluctance, (Re , is determined from eciuation (17), taking the length as between the points 1 and 2 in both figures. The permeability is taken at the nominal maximum value throughout all iron parts. The closed gap reluctance, (R02 , is the sum of the following: A. The iron reluctance, determined from equation (17) taking the lengths of the several parts as measured around the path 2-3-4-5-6-1. In each term the cross-sectional area a is that of the part through which the flux passes. In Fig. 22(b) of course, the two side sections are added to give the total section. B. The contact gap reluctances computed as x/ai and x/a2 , taking X as for an air gap of 0.005 cm. The reluctance of the joint at 1 in Fig. 22(a) is computed in the same way, and included in the sum. C. The "stop pin" air gap reluctance, computed as x/Ai , where A2 is the effective pole face area as determined below, and x is the separation at the measuring point for the stop pin opening. (a) END-ON ARMATURE TYPE (b) FLAT TYPE Fig. 22 — Magnetic circuit components of tj'pical relay structures. MAGNETIC DESIGN OF RELAYS 59 The effective pole face area, Ao , is determined by equation (18), follow- ing the procedure described in the discussion of Fig. 12. The leakage reluctance, (Rlo , is determined in the case of Fig. 22(a) by the procedure discussed in Section 5 and indicated in Fig. 17. In using this, A is the length 1-2, while 4 is the length 2-3 in Fig. 22(b) and ^2 = 0 in Fig. 22(a). The reluctance terms C/2 and C2d correspond to the armature leakage reluctance (Rl.4 of Fig. 3, here taken as in parall(>l with the core leakage reluctance in determining (Rl2 . It should be noted that this procedure provides foi two flux paths across each gap: the flux through the reluctance x/Ai , varying linearly with gap, and the parallel leakage flux through a reluctance calculated as though the armature were absent. This representation allows for the effect of fringing, taking the total field across the gap as the sum of these two fields. It carries the implication that experimentally the two fields cannot be separated by search coil measurements. For the magnetic circuit of Fig. 21, the low density reluctance is given i)y eciuation (23). The reluctance terms are calculated for maximum permeability m', corresponding to density B', and hence for a total core flux magnetization relations is the following expression for the pull F: (20) In terms of the e(iui\'alent circuit constants of Fig. 23, the reluctance (R is given by equation (24). Substituting this expression in (26) gives the equation: F = 6h (iio + (Rl + By comparison with (24), it can be seen that the bracketed term is the ratio ai/^cRo + x/A), which equals the ratio of the gap flux to the total flux if. Hence the preceding expression may be written in the form: F = 8^ (27) By a parallel treatment it can be similarly shown that the application of equation (26) to the magnetic circuits of Fig. 3 and 21 gives expressions identical with (27) except that A is replaced by A3 in the former case and by A2 in the latter. Equation (27) is the famihar expression for ,-b— -»K t -b-^ M{ t b+hx b-hx I — Vv\ ^l| d cr^ Fig. 25 — Magnetic circuit of a polar relay. 04 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 magnetic pull on two parallel planes with a field of luiiform densitj^ ^pa between them. It can be directly derived from the fact that the gap energy is J(/?G/(87r), with 3^ = XipGl Ai . The change in this energy for a differential change dr of the gap equals F dx, so that F is given by equa- tion (27). Derived in this way, it is apparent that (27) depends only on the field in the gap, and is quite independent of the density in the mag- netic material. Pull in Terms of Applied mmf In the low density region, where (R is substantially a function of x only and the magnetization curves are linear, W — U, and hence the expression for F given by equation (8) becomes: dx\6\/ This expression for the pull in the linear region is known as the equation of Perrot and Picou.^ As NI = (R^/(4x), it is identical with (26). For the magnetic circuit of Fig. 23, substitution in (28) of the ex- pression for (R given by (24) gives the following expression for the pull : ^^, ^ 27r(iV/)' a(^(R, -f ^) X y ' (29) In the low density region, the reluctance can always be expressed in terms of the equivalent values of Fig. 25. Using the equivalent values of closed gap reluctance (Ro and of pole face area .4, the pull is given by equation (29). This expression is therefore of general application in the low density region, and is the most convenient equation to use for this purpose. High Density Pidl It was shown above that equation (27) can be derived directly from the expression for the field energy associated with the gap. This expres- sion is quite independent of the reluctance in the rest of the magnetic circuit, and is therefore eciually applicable in the low and high density regions. This essentially physical argument shows that (20) and (27) are applicable through the full range of magnetization. The same result can be obtained from equation (7) by substituting in it the expression for U given by (3), and substituting al^/(47r) for NI. MAGNETIC DESIGN OF RELAYS 65 Tliere is thus obtained: dxJa 0 47r For saturation confined to the core, as in the series-parallel circuit of Fig. 21, 01 = (Re + (Hb . On substituting this expression for (R in the pre- ceding equation, (Re is independent of x and 6{b is independent of i)(Mi(is upon the shape of the load cui'N'c. Il also dcpcMids upon the tfa\('l thi-()ii<>;h whicli llic load must he moved. In principle, the lattei' may he adjusted to any desired \'alue by the ehoiee of a lever arm determining the ratio of the travel at the point of load actuation to the travel at the point where x has been measurcnl in determining th(^ magnetic cireiiit constants, d'he use of a le\'er arm liidt\v('('n load and jiull curves. 68 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 0.6 0.5 O A(Ro=POLE FACE AREA X CLOSED GAP RELUCTANCE Wmax=^^AX work OUTPUT tu u q: o \ PULL CURVE \ , LINEAR LOAD ^X. WORK,V|_ ^^^V CONSTANT rvN^-'" LOAD 1^>-s;n^WORK,V(2 / /1 GAP > < a. ^0.4 tr o §0.3 o _I 0.2 0.1 0 '^y.^x ..^ / /\ ^ 1 Vc Wk^ax / ■ . 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 RATIO X,/Atf?o Fig. 27 — Relation between mechanical output and armature travel. tivity is attained if the lever arm is chosen to satisfy this condition, and in this case Y c = Tl^max/4. For the case of a Hnear load, varying from F2 at .r = 0 to zero at X = X2 , the load is given by: F = 4-l)> where ih = X2/(A(Ro)- For the pull curve to be tangent to this load curve, the values of both F and clF/dx given by this last equation and by (31) must be equal at the point of tangency, Xi . These two conditions give two expressions for the ratio F2/F0 . Equating these, there is obtained the following equation for the point of tangency: Wi = 2W2 - 1 (36) The work done against the load is F2X2/2, and the ratio of this to TFmax is therefore /'Vt2/(2A(RoFo). Substituting the expression for F2/F0 ob- tained as described, and the expression for Ui given by (36), there is ob- tained the following expression for the fraction of TTmax realized against a linear load: Wn (3^1 + 1)^ 4(wi + D' (37) MAGNETIC DEblGX OF RELAYS 69 The ratio Vl/Wu,ixx ^^ shown plotted aj>;aiiist the point of taiigency Ui in Fig. 27. Its maximum vakie is at Ui = 1, when V/, = ir,„ax/2. For this optimum condition, Uo = 2, so that the total travel, x-z , equals '_M(Ro , while the point of tangency, Xi , is at half that travel. For this linear load case, under the optimum condition, the maximum work that can be usefully applied is, then, W = irmnx/2. Actual load curves seldom conform to either of the simple cas(\s of Fig. 2(i, and more commonly have the irregular chai'acter illusti'ated in Fig. 1 . Most of them, however, show a point of closest approach to the pull curve either at the junction of two segments, as in Fig. 28(a), or at a point of tangency to a linear segment, as in Fig. 28(b). In the former case, the coordinates of the junction point may be taken as Fi and Xi , as indicated, and the relations for a constant load for which Vc = FiXi applied. In the other case, the tangent segment may be extended, as indicated, to intersect the axes at Fo and x^ , and the relations for a linear load for which Vl = F^x^ 2 applied. Quite generally, therefore, the work I'^ done against the load cur\'e is proportional to TF^ax , and the proportionality constant is some function of ?/i = Xi/(A(Ro), where Xi is the point of closest approach of the load and pull curves. Writing /(i/i)/(27r) for the ratio V/Wmax , it follows from equation (3-1:) that the ampere turn sensitivity, V/{NI)' is given by: V {Niy (Ro (38) where f(ui) is a maximum for Ui = 1, and is similar to the curves of Fig. 27. For the particular load conditions represented by constant load (b) -H 'PULL 1 LOAD-' v 1 \ ARMATURE GAP, X Fig. 28 — Determination of pull curve to match load. 70 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 (Vc) and linear load (Fl) characteristics, the highest values of useful work were shown to be T{Nlf/(26\()), and Tr{NI)~/6{o , respectively. These relations establish the important design requirement that the point of closest approach of the load and pull curves should be at iii = 1, or where the gap reluctance ih/A ecjuals the closed gap reluctance (Ro . To the extent permitted by space requirements and manufacturing con- siderations, this may be met by a proper choice of lever arm ratio or of pole face area. In the latter case (Ro is not a wholly independent parame- ter but includes a term varying with A, so that allowance must be made for the change in (Ro in changing A. The curves of Fig. 27 show that Vc/Wmax. and ]\/]Fmax dccrcase slowly for values of Ui between 1 and 2. The ampere turn sensitivity is therefore close to its maximum value if «i lies in this range. These relations are particularly useful in preliminary design estimates, as they permit the ampere turn requirement to be estimated for a known load merely from estimates of (Ro and .4. For this purpose it is only necessary to determine F/TFmax by the procedure outlined above. With V known, this determines ir„,ax , which equals ASioFo. Thus f o can be evaluated, and NI determined from equation (32). Core Cross-Section These relations are formally applicable only in the range of linear mag- netization. For the estimates to be valid, however, it is only necessary that this condition be satisfied at the point of closest approach of load and pull curves, as an approach to saturation at smaller gaps will redu( e the pull in a region where, in most cases, it is well in excess of the load curve. For linear magnetization to obtain at the point of closest approach (u = Ui), the core flux should not materially exceed <;p', or aB', where B' is the density for maximum permeability and a is the core cross-section. The flux is equal to 47r.V//(R((/i), and the reciuired \'alue of a is therefore given by: AirNI , a = p, . . • (39) 5(R(wi) Here the values of AU and ui applying are those determined as de- scribed above, and (R(i;ration thorcforo gives an expression for the work W similar to eciiiation (33), with Ui replaced by Ui/Cl and ir,„.,x , or AiRoFo , replaced ])y ITsat. , or Cz,A(RoFo'- There is thus ob- tained : W = -^^^ TF3at. (43) As (43) is of the same form as (33), with u replaced by Ui/Cl , the relations between the load and pull curves are the same in the two cases. Thus the ratios Vc/Wa^t. and FL/TFsat. may be obtained from (35) and (37), or from Fig. 27, by using Ui/Cl to replace ii. Maximum output is thus obtained when Ui/Cl = 1, and the gap reluctance Xi/A at the point of closest approach equals Cl . Thus the optimum leverage for maximum output differs from that for maximum sensitivity by the factor Cl . 9 DISCUSSION A applications The applications of the magnetization relations considered in this article are confined to the static characteristics: the sensitivity and work capacitj' attainable when no specific timing requirements are imposed. As the same relations control both the electrical and mechanical response of an electromagnet under dynamic as well as under static conditions, the material outlined here has further applications in other aspects of magnet performance, as illustrated in the companion articles appearing in this issue of the Journal. ' The sensitivity and work capacity discussed above relate essentially to the operate characteristics. The other static characteristics of relay performance are those for release, and for marginal performance. The release characteristics are primarily dependent upon the operated load in relation to the pull at the closed gap. The rising pull characteristic as the gap is closed tends to give an operated pull well in excess of the operated load, giving a release ampere turn value small compared with the operate value. The coercive force tends to further decrease the re- lease value, and imposes the need for stop pins to assure release. A high ratio of release to operate can be attained by providing a rising load characteristic to match the pull, by using high stop pins (thus increasing (Ro), 01- by using a core or armature section that saturates in the latter part of the tra\'el. A detailed discussion of the provisions for mai'ginal operation is outside the scope of the present article, but ob\'iously in- 7-1 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 volves application of the same relations that apply to the operate per- formance. The procedures outlined for the estimation of sensiti^'ity and work capacity, or of the magnetic parameters required to give a desired per- formance in these respects, have their primary use in preliminary design. They make it possible to determine from the performance requirements definite dimensional criteria which must be satisfied by the design. After models have been made and measured, the load characteristics may be compared directly with the observed pull. Study of the mag- netization relations, and comparison of obser\-ed and estimated magnetic circuit constants, however, is of ^'alue through the whole course of development, and even in the engineering use of an established design. Such study relates the performance directly to the dimensions and ma- terials used. In particular it permits extrapolation of the performance observed on particular models, providing estimates of the range of per- formance variation corresponding to the range of tolerances in dimen- sions and material properties. An essential phase of such studies is the experimental determination of the magnetic circuit constants by the methods discussed in a companion article. Validity The solutions to the static field equations based on the magnetic cir- cuit treatment are inherently approximations. The discussion of this sub- ject in many engineering texts implies that the approximation is rather crude. In principle, as shown above, the magnetic circuit method may be applied to as close an approximation as desired, sul)ject to the ac- curacy with which the pattern of the field can be recognized. The criterion of satisfactory estimation of the magnetic circuit parame- ters is the ability to predict the magnetization relations actually ob- served. Analysis of observed magnetization relations by the method described in the companion article has shown that these relations can be satisfactorily represented by the magnetic circuit relations given here. Satisfactory agreement has been found between obser^'ed and estimated values of the magnetic circuit parameters when the latter are determined by the procedures described above. These conform in large measure to those initiated by Evershed, and are distinguished princi- pally by explicitly recognizing the existence of leakage paths shunting any air gaps, and by recognizing that these cannot be identified by vsearch coil measurements, so that thc^ field between two members l)ounding a gap must })e regarded as the \'ector sum of a constant and a variable field. MAGXKTIC DKSir.X oF RlsLAYS /O I)( s/'i/n Coiisidcralions The I'clal ions coiit i-()lliii<2; the sciisil i\it y and woi'k capacity jjioxidc a direct, indication of the crt'cct of the conlii2,ui'ation and dimensions of an electromato , as shown by the same eciuations, thus decreasing the sensi- ti\-ity. In addition, as chscussed in the companion articles''^" it delays operation and release by increasing the total field linked by the coil. In all these effects, the controlling factor is the ratio (i{/,/(5to , rather than the absolute value of (R/, , so these effcH'ts are mcjst readily minimized by making 6U small. The closed gap reluctance (Ro is the sum of the reluctances of the return meml)ers and of the closed gaps and joints. The reluctance of \hc retiu'ii meml)ers is comparable with that of the c-oi-e but smallcM', and in most 76 THE BELL SYSTEM TECHNICAL JOURNAL, JANUAin l'J.34 cases minor compared with that of the gaps and joints. The latter are equivalent to an air gap of 0.005 cm (2 mil-in) over the area of the joint, giving a marked advantage to one-piece construction of core and return members. The heel and main gaps necessarily introduce reluctances of similar magnitudes, further increased at the main gap by the height of the stop pins required to reduce the residual flux to the release level. It is therefore advantageous to use large areas for both heel and main gaps. When a small value of (Ro is thus obtained, providing high sensitivity, its value is the more sensitive to variations in fit and alignment at the heel and main gaps, and high sensitivity therefore requires close tolerances on the dimensions controlling the fit of the armature to the pole pieces. Heavy section magnets tend to high sensitivity, partly because of the reduced reluctance of the magnetic members, but principally because heavy sections facilitate the provision of large areas at the gaps and joints. As shown above, the effective pole face area is the reciprocal of the coefficient of x in the expression for the variable reluctance term. It therefore depends upon both the heel and main gaps, though the con- tribution from the heel gao is small when the armature is hinged there. For optimum sensitivity, or optimum work capacity, there are optimum values of the gap reluctance Xi/A as shown above, which can be obtained either by a choice of leverage to the load or by a choice of pole face area. It is preferable to attain these optima by varying the leverage, using as large a pole face area as possible, in order to make (Ro small. 10 SUMMARY The material given in this article provides a basis for relaj^ design through the relationships between mechanical output and electrical input. From the indicated relations, one ma^^ find the mechanical work from a given magnet design, or find the magnetic design needed to provide a required mechanical output. Relationships for gaining opti- mum performance are also given. The first step was to show that the mechanical work depends upon the field energy of the magnet, which is a consequence of the magnetomotive force pro\-ided. The magnetomotive force in turn is furnished by the electrical circuit, being determined jointly by the number of turns in the coil and the circuit resistance. The electrical output and the magnetic input were thus equated as fR = -^ IQw'Gc ' MAGNETIC DESIGN OF RELAYS 77 u hero Gc = N'/R, and is shown to be dependent on the coil (hmensions and the conductivity of the wire in the winding. The magnetic field energy is described by "magnetic flux" which, ax'eraged for the entir(> magnet, is related to magnefonioti\'(> force through where ni is an (»\])i-ession accounting for the dimensions and materials of the magnet and its air gaps, called ''magnetic reluctance." The validity of this magnetic circuit concept is discussed at some length, leading to the proof that for many cases a very simplified "equivalent two-mesh circuit" may be used to represent the more complicated actual cases. As a result, performance may be expressed in terms of the equivalent values: olo , "closed gap reluctance"; (Ri, , "leakage reluctance"; and A, "pole face area." Values for these terms may be estimated for cases of initial design (magnet synthesis), or be measured so as to characterize completed modqls (magnet analysis). Evaluation of reluctances of vari- ous magnetic circuit components is described, with numerous relations given for gaps and for magnetic materials, including an approximate re- lation covering the non-linear behavior of ferrous materials. In terms of the magnetic circuit variables, force and work may then be expressed as ,-2 F= ^ ^ w = SttAcRo (1 + w) J" u 2 ' 87r(Ro 1 + W ' where u = x/A6{q , the ratio of air gap to the product A(Ro ■ It is thus found that greatest magnet output for force systems common in i-elays is obtained when the critical load is picked up at a gap Xo = .ICRo , which may be accomplished by choice of lever arm. The maximum work that may thus be considered useful, F^ax , depends somewhat on the particu- lar load characteristic, and for the typical constant-load case, Vc , is related to ampere turns, and to power input as follows: Vc/(Nlf = 7r/2(Ro, Vc/f-R = irGc/'M Greatest useful magnet output is thus seen to depend diicctly on Gc , the "coil constant," and in\'(M'sely on (W,, , the e(iui\alent closed gap i-e- hictance. 78 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Additional cases are given permitting design relations to be extended into the saturated range for the iron. REFERENCES 1. Estimation and Control of Operate Time of Relays, Part I — Theory, R. L. Peek, Jr., page 109 of this issue. Part II — Applications, M. A. Logan, page 144 of this issue. 2. A. C. Keller, A New General Purpose Relay for Telejjhone Switching Sj'stems, B.S.T.J., 31, p. 1023, Nov. 1952. 3. S. Evershed, Permanent Magnets in Theory and Practice, Institute of Electri- cal Engineers Journal (England) 58, 1919-1920. 4. H. C Roters, Electromagnetic Devices, John Wiley and Sons, New York, 1941. 5. S. P. Thompson and K. W. Moss, Proc. Physical Society of London, 21, p. 622, 1909. 6. W. B. f]llwood, Glass Enclosed Reed Relay, Elec. Eng. 66, p. 1104, Nov., 1947. 7. E. B. Rosa and F. W. Grover, Bulletin Bureau of Standards, No. 169, 1912. 8. R. L. Peek, Jr., Analysis of Measured Magnetization and Pull Characteristics, page 79 of this issue. 9. Perrot and Picou, Comptes Rendus, 175, Dec. 9, 1922. 10. R. L. Peek, Jr., Principles of Slow Release Relaj' Design, page 187 of this issue. Analysis of Measured Magnetization and Pull Cliaracteristics |{y K. L. PI-:KK, Jr. (M;inuscri|)t received Sept einher 21,M953) // is shoint in this article thai the observed magnetization relations of most ordinary electroniagnets conform to simple expressions which can be interpreted as the Jinx vs. magnetomotive force equations of a reluctance network, analogous to the current-voltage equations of a resistance network. To the extent of such conform it g the magnetic circuit constants characterizing the network suffice for the evaluation of the field energy and pull charac- teristics of the electromagnet. The agreement between the observed mag- ndization and these simple relations is close in the region of linear mag- netization, and is adequate for engineering purposes at higher flux densities, but the extent of agreement in the latter range varies with the type of struc- ture and the location of the magnetic parts which first approach saturation. Specijlc analytical and graphical procedures are given for the evaluation of the magnetic circuit constants from both pull and magnetization measure- ments. These procedures employ relations which give linear plots indicating the degree of conformity of the observed relations to the expressions used to fit them. The relation of the measured constants to those which can be esti- mated in design is discussed, as is the use and application of the measured constants in development and engineering studies. 1 INTRODUCTION In the design of telephone relays and similar switching apparatus, the characteristics of the electromagnet which serves as motor element may Ije distinguished from those of the mechanical system of contact springs and actuating members which it operates. The performance of the elec- tromagnet is characterized statically by the mechanical work done for a given coil energization, and dynamically by the time recjuired to actuate the mechanical load, including in this the inertia of the moving parts. Both the potential work output of the electromagnet and the energy stored in de\-eloping its field can be evaluated from its magnetization 79 80 THE BELL SYSTEM TECHNICAL JOURXAL, JANUARY 1954 relations. The consequent dependence of the pull and timing characteris- tics upon the magnetization relations makes the measurement and analy- sis of the latter fundamental to the understanding of performance and its relation to design. Procedures are described in this article for the evaluation from meas- ured magnetization relations of a few parameters which suffice for the determination of the pull and of the field energy under given conditions. These parameters, the magnetic circuit constants, characterize the elec- tromagnet to which they apply. They are used as measures of per- formance in comparing different structures, or in studying the effect of dimensional or material variations in a particular design. In addition to their significance as parameters summarizing measured magnetization relations, the magnetic circuit constants may be inter- preted as the observed values of quantities postulated m the magnetic circuit approximation to static field theory. This approximation is dis- cussed in a companion article, which describes methods of estimating the values of the magnetic circuit constants from the configuration, dimensions, and materials constant of the electromagnet. Comparison of observed and estimated values of these constants has served as a guide in developing these methods of estimation. A complete design method- ology is provided by the ability to both estimate and measure the mag- netic circuit constants characterizing the performance of electromagnets. In the experimental evaluation of the magnetic circuit constants de- scribed in this article, the basic method is that employing measured magnetization relations. Because of the dependency of the pull upon the magnetization relations, pull measurements may also be used, subject to certain limitations, for the evaluation of the magnetic circuit constants. The article includes description of procedures for doing this. The notation used in this article conforms to the list that is given on page 257. 2 MAGNETIZATION RELATIONS The magnetization relations of an electromagnet give the average flux ip linked per turn of the winding as a function of the two determining variables: applied ampere turns AT/ and armature position x. The rela- tions are usually shown, as in Fig. 1, as a family of curves giving

considerations of onorgy balance Ihal IIh* mag- net i/at ion relations determine the relations anions; the electrical inj^iut to the coil, the stored energy, and the mechanical onlput attainable. Hence, any generally appli('al)le and simple expression for the mag- netization relations, even if purely empirical, would i)r()vidc a con- \(Miient means for evaluating and describing the performance charac- teristies. The expressions given below for the magnetization relations are not purely empirical, but are those obtained as approximate solutions to the magnetic field ec^uations by the magnetic circuit method. When experi- mentally e\aluated, however, their utility in tlefining the characteristics of the electromagnet to which they apply is independent of this interpre- tation, and depends only on their conformity to the observed magnetiza- 1 ion relations. It is convenient to formulate the expressions for the magnetization cur\-es in terms of the reluctance (R, the ratio 3^/v?, where JF is the mag- netomoti\e force -ixNI. The observations of Fig. 1 are plotted in Fig. 3 in the form of curves giving (R vs. NI for various values of x. A constant \'alue of tp is represented in such a plot by a straight line through the origin. The radial lines used as supplementary co-ordinates are spaced to gi\'e a convenient scale for (p. The reluctance curves of Fig. 3 are similar in character to those apply- ing to most ordinary electromagnets. Each curve has a relatively flat characteristic in the vicinity of a minimum located on a common flux FLUX, yp, IN MAXWELLS 3000 4000 5000 12 16 20 24 ABAMPERE TURNS , NI Fig. 3 — Reluct, ■incc curves. 8-1 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY l'Jo4 line, that marked ip in the figure. At vahies of (^ above this minimum, the reluctance increases at an increasing rate, indicating an upper limit to the value of ^p approached as Nl becomes verj^ large. The observed \-alue of (/?' may be interpreted as that for which the core density cor- responds to maximum permeability, while the indicated upper limit may be interpreted as the saturation flux ip" . The observations plotted in Figs. 1 and 3 were obtained with the sample initially demagnetized, a convenient reference condition for measurement. In actual use, relays and other electromagnets have been previously operated, and the applicable

=a ;a) yc 6 I l^LC l^LP X^ ' 5= / 477-Nll '' ^ II a. < >CRlp -f A, 6 II < < ^(R02 «C LiJ ? * r r- 1 Fig. 6 — Magnetic circuit for armature saturation. 80 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1054 In Fig. 4, (Rn , (Rl , and .1 are constants, so that (R is a function of x only, independent of (p. Equation (3) can then apply only to the low density region, (p < (p'. For most ordinary electromagnets, the magnetiza- tion relations for (p < (p' conform to (3), and hence to the schematic of Fig. 4. For ip > (f', the expression for (R must provide for the variation with (p. It usually suffices to use an expression in which the only term varying with tp is that corresponding to the path in which saturation first occurs. The most common case is that in which saturation first occurs in the core. The reluctance can then be taken as conforming to the schematic of Fig. 5, in which (R02 , (Rl2 and A^ are constants, and (Re is a function of (P only. The total reluctance is given by: (R = (Re -\- (Re, (4) where : (Rl2((R02 + ~ (Re = ^ ^ . (5) (RV2 + (R02 + J If the variation with (p is taken as conforming to the empirical Frohlich- Kennelly equation, (Re is given by: «c = m^^^=(R:.C^^', (0) (f — cp (p — (p where ) applies only for ^ > tp', (Re is the value of (Re not only at (p = (p', l)ut throughout the low density region tp < tp'. In the alternati\'e form given by (G), (R" is defined by this ec[uation, and repre- sents merely the intercept of a plot of (Re extended below the region to which (6) applies. In some electromagnets saturation occurs in the armature rather than in the core. This is the case, for example, in high speed relays in which the armature section is minimized to reduce its mass. In such cases, the reluctance conforms to the schematic of Fig. Ofa), in which (R .4 represents the reluctance of the armature, which varies with (Pa ■ As the ratio (Pa/

on a coniinoii \-ahie of (p in this case. In Fig. ()(iO, ^A may be taken as <2;iven hy an expression of tlie same form as ((>), with ^p' and ip" rephiced by the minimum and saturation \ahies of ip_i , and ol'c rephiced by the minimum \-ahie of 6\a . (Kc may be taken either as given by (6), or as a constant if the variation in (Ha is dominant. The other parameters of Fig. 6(a) are constants. The expres- sion for (U applying to Fig. r)(a) may be obtained in the same manner as those given for the circuits of Figs. 4 and o. The magnetic circuits of Figs, o and 6 are called "design" circuits, liecause their constants may be estimated from the dimensions and material constants of the design, as discussed m the companion article. Estimates of the corresponding \^alues of the equivalent magnetic circuit constants of Fig. 4 are obtained from these design constant estimates by the eciuivalence equations given below. Conditions of Equivalence Whatever magnetic circuit is taken as apphdng, the component re- luctances are independent of ^p in the low density region, tp < (p', and are therefore constants except for the gap reluctance. If this \'aries directly as x, as assumed in the schematics shown, the reluctance for any circuit reduces to an expression of the form of (3), applying to the circuit of Fig. 4. As shown in the companion article, the reluctance of the circuit of Fig. 5 when (Re is constant is given by (3) when the param- eters of this eciuation are given by: A = ^^-, (7) p- (Ro = p'^Mri + }M\c , where : p = 1 + — . (Rl2 These relations are the conditions of e(|ui valence, for which the reluc- tances of Figs. 4 and 5 are identical for all \'alues of x. With them, the I'chictance gixcn l)y (4) can be reduced to the simpler form of (3) when (Sic is a constant, as throughout the low density region. Similar relations apply to the magnetic cii'cuit of Fig. ()(aj. In particu- 88 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 lar, when (Ra is a constant, the reluctance (Rp , or '^e/^a , represented by the series parallel path of (Pa in Fig. 6(a), may be represented by the simple parallel path of ipA in Fig. 6(b). By analogy with equations (7) the constants applying are related by: (7A) (Rlp = (Rla + (Ha , p2 (R02 = = P^(H03 + pdiA , p - 1 + ^^ . (Rla where The circuit of Fig. 6(b) is identical with that of Fig. 5 with 1/(Rl2 equal to the sum of 1/(Rlc and 1/(Rlp ■ Thus the circuit of Fig. 6(a) may be reduced to that of Fig. 5 when (R^ is constant. The resulting circuit may be reduced in turn to the equivalent circuit of Fig. 4 when (Re is constant. Thus the equivalent circuit may be used to represent all cases in the low density region, where the component reluctances are independent of tp, provided the gap reluctance varies linearly with x. 3 ANALYSIS OF MAGNETIZATION MEASUREMENTS In analysing the observed magnetization relations, it is convenient to treat the low density and high density relations in separate and successive steps. For the former, the analysis is based on the equivalent magnetic circuit of Fig. 4. EVALUATION OF EQUIVALENT MAGNETIC CIRCUIT CONSTANTS For a given value of x, the total reluctance in the low density region is taken as constant at the minimum value (R' (x) observed in measurements made from the demagnetized condition. If the observations are plotted as reluctance curves, as in Fig. 3, these values of (R' (x) may be read di- rectly. When the recording fluxmeter is used, (R' (x) can be obtained from the (p versus fF curve as the slope J7/v? of the line through the origin tangent to the curve. The reciprocals P(x) of the values of (R' (x) thus evaluated may be plotted against x. Values of P(x) corresponding to the minimum reluctance values of Fig. 3 are plotted in this way in Fig. 7. The origin of x is taken as for a gap of 0.025 cm., corresponding to the operated position of the actuating card for maximum stop pin height. ANALYSIS OF PULL AND MAGXETIZATIOX MKASIHIOM KNTS SO \'alu('s of ,r incasiu'cd from this oi'i«iiii arc Icnin'il I raxcl, as disl inmiislicd from gap ^■alues, which are measured from I he position of iioii to iron contact. If the relation between (sV (x) and .r confoi-ms to (ii), the \ahie.s of P(.v) must conform to: Pix) = i-+ .^^. ..• (8) (Ri A6\o + X Tlien if .r,- is a c(Mitral reference \ahie of .r, and /'(Xr) the corresponding value of P(x), the expression for J\xr) — P{.i') given by (8) may he written in the form: X — Xc Xc A = -1 UHo + -^^ + (Ro + ^ (x - xc) A (9) P{xc) - P{x) Thus conformity with (3) re(iuires the ratio given by (9) to vary linearly with x. This ratio is plotted against x — Xc in Fig. 7, referred to the upper and right hand scales. The values of this ratio are sensitive to deviations in the values of x and P(x) near Xc , and minor variations from linearit\' are to be expected here. Allowing for this, the results agree with (9), and the relation between (ft' (x) and x therefore conforms to (3). From (9), the slope and intercept of the linear plot may be interpreted as indicated in Fig. 7. Then A may be evaluated by dividing the intercept -Is II Q. - 70 D.08 - 3.06 - 3.04 - X- 3.02 0 0.02 0.04 0.06 1 1 1 X-Xc X 60 50 40 30 20 10 n P(xc)-P(x)/ \ X X \ \ ^^ SLOPE (Ro+V^ \ h> / y /' ^^^ --- ^^P(x) o-^o / y 1 / 1 1 t 0 0.02 0.04 0.06 0.08 O.IO 0.12 0.14 TRAVEL, X, IN CENTIMETERS Fig. 7 — Evaluation of equivalent circuit constants. 00 THE BELL SYSTEM TECHXICAL JOURNAL, JANTAHY \\)')4 vixhie by the square of the slope value, and the latter then serves to evaluate (Ro by subtracting Xc/A. On substituting these values of (Roand A in (8) at x = Xc , (Ri, may be evaluated. For the case plotted in Fig. 7, the values thus obtained are: olo : 0.0300 cm"' A: 1.34 cm' (R,. : 0.0()90 cm""' These values of the equivalent circuit constants, substituted in (3), suffice to accurately characterize the magnetization relations and thus the electromagnet's performance through the low density region. Evaluation of High Density Relations For cases of core saturation, the high density relations may be analysed in terms of the magnetic circuit of Fig. 5. The corresponding reluctance is given by (4), in which (Re = (Re for v? = (^'. If (Re can l)e determined, the three parameters determining (R^ in (5) may be e\'aluated by the equivalence ec^uations (7). To determine (Rr through the high density region from (6) requires evaluation of (Re , (f', and ^ > / y / / 0.1 1 1^ ' / y ^ ^ ^ai48 ^ ^ ^ 1^ ^ .^ ?^^ ^ v^- TANGENT POINTS OF PULL CURVES ^^^ ^ =^ ^-^ ^: ^^ 1 4 5 6 7 8 9 10 n 12 13 14 15 16 17 18 19 20 COIL ABAMPERE TURNS, 'J/^n Fig. 8 — delation of core potential drop to applied magnetomotive force. To the extent that the physical stnictiire may be identified with the corresponding magnetic circuit element, the drop in magnetic potential in the core may be identified with (Rc^ in Fig. 5. Assuming this identifi- cation to apply, magnetic probe measurements may be made at the two ends of the core, as at the points marked .Y in Fig. 2. The potential drop observed in these measurements is the magnetic potential ^e applied to the external magnetic circuit. CF^ is eciual to the applied magnetomotive force 5(or 47riV/) less the potential drop ^r = (Rrv? in the core. Thus ;Tc = ^ — ^E- Values of JFc/(47r) thus determined for the relay of Fig. 2 are shown plotted against JF/(47r) for various values of x. in Fig. 8. These curves are substantially linear in the low density region: their upward conca\at3'' at higher values of JT is evidence that saturation first occurs in the core. Ei'alua(ion of Core Relucatance Consfants The magnetization curves of Fig. 1 and the core magnetomotive force turves of Fig. 8 were obtained with the same model. For given values of ■J and .}; there can be read from these two figures corresjxjudiiig \alues of 92 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 ip and 3^c , giving the corresponding values of core reluctance (Re = 5^c/ " (R' J^ = 47rNl — »- Fig. 9 — Alternative method of evaluating core reluctance. 94 THE BKLL SYSTEM TECHXICAL JOXRXAL, JANUARY 1954 Table I — Evaluation of Magnetic Circuit Constants Equivalent Values (Fig.4) Design Values (Fig. 5) From Fig. 7 (Ro : U.U300 cm-i A: 1.34 cm2 (Ri : 0.0690 cm"! From Fig. 3 (Re : 0.0070 cm-' From equations (7) (Ro2 : 0.0181 cm-i A- : 1.G6 cm2 (Rl2 ■■ 0-0620 cm-i mine(i in Fig. 7, and the \-aliie of (Re determined in Fig. 3. These vahies have been substituted in equations (7) to determine the component terms of (Re . The constants applying to Fig. 5 are of interest primarily for compari- son ^\dth the values computed from the design, as discussed in the com- panion article. While formall}' required to evaluate the high density magnetization relations, they are not explicitly reciuired in estimating the high density pull, as is shown in a later section. The reluctance (R^ of equations (4) and (5) can be evaluated experi- mentally through the full range of the magnetization measurements when the magnetomotive force measurements are available. Thus Figs. 1 and 3 can be used to determine values of

;iiificant increase in (»{/,■ us ^p incr(>ases. Jiy comparison with the plot of (»{,• included in Fijj;. 1, however, this increase in (({a- is minor, and does not affect th(> fact that the limit in_ii reluctance is 5/V", where tp" is the satui'ation flux of the core. It may be noted in passinj>; that the values of 6{k lyinji; on ip' must conform to (o). As this is of the same form as (3), these \-alues of ni^. may be used to evaluate the component terms of (5) by the procedure applied to the values of (R' in Fig;. 7. .Vs the same data are employed, the \alues of (R02 , (R/,2 , and A^ thus determined agree with thos(> com- puted by means of equations (7) within the accuracy of th(> computa- tions. Case of AniialKrc SalKration The analysis of the low density relations is, of course, independent of where saturation occurs, and involves merely the determination of the e([uivalent magnetic circuit constants by the procedure described above. In the high density region, however, the (luantities appearing in Fig. 6(a) must be evaluated when armature saturation occurs. This recjuires measurements not only of JF and ip, but also of 'Je and tpA . ■Je is given by the Ellwood mmf gauge measurements previously described. To measure ; in practice through the low density region. In the high density region, the pull falls off as the result of incipient saturation, and the ratio F' /F increases as 3^ (or ^ttNI) is increased. High Densihj Pull — Core Satiiraiion It is convenient to (^xpress the high density p\i\\ in terms of the I'atio F'/F, where F is the actual pull, and /''' the indicated pull conloi'niing to (13) for the same value of J (or 47r.\7). As F'/F is unit}' at low densi- 98 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 ties, it follows from (12) that F' /F is equal to ipu/fph , where (po is the actual value, and ipo the fictitious value that would obtain if the mag- netic circuit reluctances remained constant at all values of J. For the case of core saturation, to which the magnetic circuit of Fig. 5 applies, the ratio (pa/^p is constant for a given value of x, and hence in this case F'/F eciuals

/■" F is ('([ual to ipa^ ip'a as before. In this case, however, it is if(:,\pA rather than ipa/^p that is constant, so that the expression corresponding to (14) is: ^' = feY, (16) where (Rf is the reluctance JF£/A , and the approximation corresponding to (15) is therefore: In general, there is no simple expression for J^ ;7, and (17) is of Httle interest as a general expression. When saturation is wholly confined to the armature, however, (Re is a constant and usually a minor term, and ^E is then nearly ec^ual to 3^. In this case, the high density pull is given by the approximation: F = T 1 X V [i f^-^Y + f ^' Y" \paJ yy^P^A/ _ (17A) The application of this approximation may be simplified by develop- ing an approximate expression for (R/r in terms of the e([ui^'alent mag- netic circuit constants. The expression for (S{p can be read from Fig. 6(b), where 6^/.? is usually large compared with (sX^i + x/ Ai , so that the latter term is an approximate expression for (s\f ■ As (17A) applies only for (Re small, and as Fig. 6(b) is then of nearly the same form as the ecjuivalent circuit of Fig. 4, the ecjuivalent constants (Rn and .4 are in this case approximately equal to(J^n2 and A2 , respectively. Hence in the case of saturation confined wholly to the armature, to wiii(4i (17A) applies, Up can be taken as given by (Ro + x! A, and (17A) then iiu'olves only (/7.4 , ^pa , and the e([ui\'alent magnetic circuit constants (ii,, and ,1. 5 AXALY.SLS OF PILL MEASUKKMEXTS The following discussion relates primarily to the use of jxill measure- ments in analytical studies as a means of evaluating the magnetic cii'cuit constants. 100 THE BELL SYSTEM TECHXICAL JOURNAL, JANUARY 1954 0.02 0.01 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 TRAVEL, X, IN CENTIMETERS Fig. 12 — Evaluation of magnetic circuit constants from pull data. Evaluation of Equivalent Magnetic Circuit Constants The low density pull is given approximately by equation (13), to which the dashed F' lines of Fig. 11 conform. These lines coincide with the actual pull curves at the points of tengency, which must correspond to the minimum reluctance values. Taking F as F', (13) may be written in the form: Values of NI/F' have been determined for the dashed lines of Fig. lb and are plotted against x in Fig. 12 — the plot marked JF/(47r\/F')- In agreement with (18) these points determine a straight line. The slope and intercept have been used to determine the values of (Ro and .4 listed under 'Tull Results" in Table II. As the reluctance (Re in the circuit of Fig. 5 is identical in form with the reluctance (R of Fig. 4, the pull for Fig. 5 is given by an expression similar to (13), with ff, (Ro , and A replaced by ^e , (R02 , and Ao , respec- tively. As the two circuits arc ec^uivalent at the points of minimimi re- luctance and the corresponding points of tangency for the pull curves, the pull at these points is the same for the two cases. Thus if the ratio of ^E to ^ is determined at the points of tangency, a plot of CF£-/(47r\/F') against x should conform to: ^t^VF' = (R02 / 2 "^ V'27rA2 (19) ANALYSIS OF PULL AND MAGNKTIZATION MEASUREMENTS 101 Table II — Evaluation of Magnetic Circuit Constants From Pull Results From Magnetization Results Olo : 0.03] 9 cm-i 0.0300 cm-i .4: 1.46 cm^ 1.34 fin'^ (Ro2 : O.OISS cm-i 0.01 SI cm-i Ai : 1.85 rm2 1.66 cm^ (Rx. : 0.0630 rm ' 0.0690 cm-i (Rz,2 : 0.0560 cm-' 0.0620 cm-' (Re : 0.0072 cm-i 0.0070 cm-i The points of langcucy arc iiuUcatod on the pull curves of Fig. II. These values of 5 have been marked on the corresponding curves of Fig. 8, from which can be determined the corresponding values of ^c , and thus of -Je ■ It may be noted in passing that these values of Jc are all similar, corresponding to a mean level of about 24 ampere turns, or 30 gilberts. The magnetization results gave vahies of 4,000 maxwells for ^' (Fig. 3) and of 0.007 cm"^ for (Re (Table I), giving a value of 28 gilberts for the product. This agrees with the value of Jc at the tangent points of the pull curves, showing that these points coincide with the points of minimum reluctance ((p = if'). Taking the ratio of Se to ^ as read from the curves of Fig. 8 at the indicated tangent points, the values of ^/(■iiir-y/F') in Fig. 12 have been multiplied by the corresponding values of 'Je/'^, giving the points indi- cated by triangles in this figure. These conform to a straight line relation, in agreement with (19). The slope and intercept of this linear plot have been used to determine the values of (R02 and ^42 listed under "Pull Results" in Table II. The four quantities determined from the two plots of Fig. 12 appear in the three independent equivalence equations (7), and these three equations may therefore be used to determine the other three quantities : (Rl , (Rl2 , and (Re • The resulting values of these latter quantities are listed under "Pull Results" in Table II. For comparison, the table in- includes the corresponding values of these quantities obtained from the magnetization measurements, and given previously in Table I. The agreement between the results derived from these different kinds of measuroments indicates the validity of the analysis described here. High Domtt/ Pull As previously noted, it is convenient to express the pull F in the high density region in terms of the ratio F'/F, where F' is the pull computed from (13) for the \alue of ;T applying. F' 'F is readily evaluated from a logarithmic plot of the observed pull versus ^7, as illustrated by Fig. 11. 102 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 GAP,X, IN CM- / y y / '^ y ,^ ^ / / y ^ 0^ ^ ^ / / ^ ^ __^ /^ / ^ y^ ^ ^ r^ __a;:A8^ " ' p, 277(NI)2 F= OBSERVED I PULL 1 2 3 4 5 6 7 8 \ATf) Fig. 13 — Pull charafteristics for approaching saturation. The dashed tangent Hnes give values of F', so corresponding vahies of F' and F ran be read for various values of 5. Values of F' /F for the results of Fig. 11 have been thus determined, and are plotted in Fig. 13 against (fl/4x)" for the values of x applying to the individual ('ur\"es of Fig. 11. The values of F' /'F thus plotted determine a family of straight lines having a common intercept at the origin of ??". In this respect these re- sults are representative of the pull characteristics of electromagnets when plotted in this way. As an empirical fact, apart from any analysis, this linearity pro\'ides a useful method of plotting pull observations in the high density region, as it facilitates interpolation and the detection of inconsistent observations. The observed linearity in the relation between F' IF and 3^" conforms to the relations postulated in (15) and (17), of which the former should apply for core saturation. If (15) applies, the common intercept of 0.65 at 5" = 0 in Fig. 13 should equal 1 — {ip' /ip")', indicating a value for (/?' V" ('f 0.5!). To determine if the relation fully conforms to (15), the relation between the slopes of the lines and the corresponding \-alues of X must be compared with that indicated by this equation. Let //" })e the observed slope of the plot of F' /F versus 5" for a particu- ANALYSIS OF PTLL AND M ACX imz ATK )\ Ml'-AST HIIM lON'PS 108 I:ii- \;ilu(' of X. If the relation coiifonns to (lo), // = \/(6{'ip"), where (il' is iii\-eii hy (3). Writing (', for the ratio ((il/. + (Ho) VR,, , and '/ for the ratio .(• (.ItRo), 1 + u C L -\- u and hence the \alues of // should confoi-in to the ecuiation: ('/. + u y{i + '0 = («/.¥''■ (20) I'sing the values of A and (Ro determined from the pull results in P'ig. 12 and tabulated in Table II, values of u have been determined for the values of .r applying in Fig. 13, and the corresponding values of // have been determined from the slopes of the corresponding lines. The \'ahies of ij(l + u) thus determined are shown plotted against u in Fig. 14. Tlie xio-'» u - ^ X/A(R -4. 0 '^1 / ^o ^ (Ro + (Rl ^^~ (Ro / / y / / / / v^ SLOPE 1 y"(5^L / ^ / -0.4 0 0.4 .2 1.6 2.0 2.4 2.8 3.2 Fig. 14 — Evaluation of saturation flux io reluctance (il 7, from ])ul characteristics. 104 THE BELL SYSTEM TECHXIC.VL JOl'RX.VL, J.VXUARY 1954 Table III — Evaluation of Magnetic Circuit Constants From Pull Measurements From MagnetizationlMeasurements Figs. 13 & 14: tei'nuiied from coil niau;i let i/.al ion measurements. It is conxcnient louse l-^llwood mmf gauge measurements to determine tlu> core reluctance, hut these may be omitted and the method of Fig. 9 employed, pro\i(led saturation is con- tined to the core. Subject to this same limitation, the design constants ma\' be evaluated from the pull measurements if these are sui)plemented w ith mmf gauge measurements. For armature saturation, the magnetic design circint constants can only be evaluated from magnetization measurements, which must in- clude armature search coil as well as coil flux determinations, and be supplemented by mmf gauge measurements. Limits of Application The \-alidity and usefulness of the procedures described here rest on the agreement of measured ([uantities with the relations used to analyse them. As all the relations used lead to linear plots, the extent of agree- ment in any specific case is apparent from the plot obtained. So far as the presentation of pull results is concerned, this is the only question of \-alidity in\-olved. The relations found for the magnetization results are used to estimate the field energy and the pull. To the extent the magnetization results relate to the flux linkages of the coil, the conclusions drawn from rela- tions fitting those results are valid. This follows from the energy balance considerations discussed in Section 2. The supplementary measurements, such as those with the mmf gauge or an armature search coil, are in principle only convenient means for determining the relations to which the coil magnetization conforms. The relations found b}'^ these means are only \alid to the extent of such agreement. The conformity of magnetization relations to the expressions used here, corresponding to the magnetic circuit schematics, is closest for electromagnets in wliich the reluctance of the iron parts is small com- pared with that of the joints and air gaps. This condition is satisfied for most ordinary relays and similar electromagnets in the low density r;uige, where the expressions given here most closely apply. The expres- sions used for the high density range do not give as close agreement, but provide a satisfactory basis for engineering analysis, particularly when saturation is confined to the core. The treatment is less satisfactory for structures that deviate from these conditions, such as those with arma- ture saturation, or those with long cores of small cross section, where the 108 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 iron reluctance is a major component through the full range of opera- tion, ACKNOWLEDGMENT The procedures described in this article incorporate contributions pro- posed or developed by H. N. Wagar, M. A. Logan, Mrs. K. R. Randall, and others. REFERENCES 1. R. L. Peek, Jr., and H. N. Wagar, Magnetic Design of Relays, page 23 of this issue. 2. A. C. Keller, New General Purpose Relay, B.S.T.J., 31, p. 1023, Nov., 1952. 3. H. N. Wagar, Relay Measuring Equipment, page 3 of this issue. 4. W. B. Ellwood, A New Magnetomotive Force Gauge, Rev. Sci. Instr., 17, p. 109, 1946. 5. Estimation and Control of the Operate Time of Relays : Part 1 — Theorj^ R. L. Peek, Jr., page 109 of this issue. Part II — Applications, M. A. Logan, page 144 of this issue. 6. Kennelly, Trans. A.I.E.E., 8, p. 485, 1891. Estimation and Control of the Operate Time of Relays Part I — Theory By R. L. PEEK, Jr. (Manuscript received August 31, 1953) The dynamic equations applying to the operation and release of elect ro- niagnets are derived in a form in which the armature position and the Jinx linkages of the coil are the variables to he determined as functions of time. The effective magnetomotive force and pull are taken as given by the static magnetization relations. This formulation is valid if the dynamic field pattern is substantially that of the static field, and it is shown that this con- dition is satisfied if the effective conductance of the eddy current paths is small compared with that of the coil or other linking circuits. This is usually the case in operation, but is not the case in normal release. Approximate solutions are obtained for both fast and slow operation, giving the time as related to the 7nechanical work done, the mass and travel of the moving parts, and the steady state power input to the coil. Expres- sions are derived for optimum coil design and pole face area, and design requirements for fast operation are discussed. An approximate expression is derived for the release waiting time, the initial period of field decay prior to armature motion. The effect on this decay of contact protection in the coil circuit is discussed. The sloiu release case is treated in another article in this issue. An analysis is given of the armature motion in release, and the residts of an analog computer solu- tion to this problem reported. These show the effect of design parameters, particularly the armature mass, on the velocity attained in release motion, and- hence on the amplitude of armature rebound. INTRODUCTION' The operate and release times of most telephone relays lie in the range from 1 to 100 millisecs. (0.001 to 0.100 sees.). To use these relays in telephone switching, involving complex patterns of sequential switch 109 110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195-1: and relay operation, it is necessary: (1) to be able to estimate the operate and release times of any relay for which this information is needed in circuit studies, and (2) to develop relays capable of meeting specific- timing requirements, both fast and slow. The essential basis for such design and estimation is a dynamic theory of the operation of electromagnets, of sufficient accuracy for engineering use. The theory presented here is applicable to electromagnets in gen- eral, although the applications discussed are those relating to relays. The theory presented is approximate, partly because of the difficulty of providing a more exact treatment, and partly because simplicity and generality are more important for engineering purposes than accuracy in estimation, which is in any case subject to correction by measured results. The theoretical relations are used alone only in preliminary estimation and in the initial stages of development, as discussed in Part I of this article. In advanced development and in the modification and application of existing structures, as discussed in Part II, the theory is used as a guide in the correlation and extrapolation of observed per- formance. The dynamics of electromagnets involve the concurrent and inter- related phenomena of field development and armature motion, and are accordingly governed by the two differential force equations respectively applying, each containing a coupling term expressing their reaction on each other. These basic equations are formally identical with those of electromechanical transducers, such as loud-speakers, but the treatment has little else in common. The operation of an electromagnet is the tran- sient change from one state of ecjuilibrium to another, as distinguished from the sustained low amplitude oscillations of the transducer case. The coupling terms represent the effect of field energy changes on the coil voltage and on the force causing armature motion respectively. In the steady state, the field energy is a function of magnetomotive force and of armature position alone. In a transient state, eddy currents in the magnetic members aft'ect both field energy and the pattern of the field. In developing an approximate theory, it is assumed that these effects are confined to the total effective mmf, and that the pattern of the field is the same as that of the static field: i.e., that the field energy associated with any portion of the structure, such as an air gap, is fixed by the flux linkages of the coil and by the armature position. The limita- tions on the analysis imposed by this sim])lifying assumption are dis- cussed at the end of the next section. i:s'rTM ATiox AND <'()\'ru()i, OF ()i'i:iv' \ri; timI': ok kklavs 111 1 Tin; 1 )\ \ \Mi(' lOcjiATioxs '['he notation of this article contni-ins to the list t!;i\('ii on paj^c 257. TUl': KLKCTUICAI. IXjlATlON 'riic \()ltai>;(' ('([ualion for a coil of .V tui-ns linkiiiii a fi(>l(l of sti-en<2;th ^p may he written in the form: 0- /)A^ + .vt = 0, (1) (11 \vh(M'(> / is \\\v instantaneous current, R is the cii'cuit resistance, and Hi is the constant voltage ap])iied. Writing ;T, for the instantaneous mmf 47r A'/, and ;T.s, for the steady state mmf \-k NI, the eciuation Ix'comes: (It where G, = X' R, the coil constant, or equivalent single turn conduc- tance. If the field ^ is linked by a number of circuits, a similar voltage eciuation applies to each such circuit. By addition of these expressions, there is obtained: JF - J.S + 4xG ~ = 0, at where t7 = X^' > the total effective mmf, ^s = Yl^si , the total applied mmf, and G ^ ^G; , the total e(|ui\'alent single turn conductance. If the dynamic field has the same pattern as the static field, the instan- taneous mmf 5 must eciual (R<^, where (R is the reluctance 5/V observed in static magnetization measurements. The preceding equation may there- fore be written in the form: (R

/Z)", which varies, by hypothesis, at the same proportional rate as the total field. The resistance of the shell is 2-Krp/{( dr.), where p is the resistivity of the material. The voltage equation for the shell cir- cuit is therefore: ^^"^^ + 1^="' and hence: J = 2r dip pirD- dt The magnetomotive force of the shell is its current multiplied by 47r. This produces a flux increment dtp in the area enclosed inversely propor- tional to the reluctance of the tubes of induction within this area. This reluctance may be taken as inversely proportional to the area, as would be the case in a closed magnetic circuit of uniform cross section. Then ESTIMATION AND CONTROL OF Ol'KllATE TIME OF RELAYS 115 the flux incromciil ])i-o(Iu('(m1 by each shell is ,^ral / (l>p, the total flux ipE prochiccd by the eddy currents is found to he: t dip This is identical in form with the expression for one of the linkiii,<>; circuits assumed in derivino; equation (2), with (JV^ equal to ;Ta- , and AttGe given by (/ (2p). Thus to the extent that the assumption of a uni- form flux distribution is \alid, Ge is given by: This expression applies to a round core. A parallel approximation may be obtained for a reet angular rod by considering it as made up of rec- tangular shells of differential thickness, having their sides in the same ratio as the rod. By treating the perimeter of each shell as corresponding to the circumference of the circular shell in the cylinder case, with the area enclosed corresponding to that enclosed by the circular shell, the following expression is obtained for the conductance of a rectangular rod whose sides are in the ratio Ir. This shows that the effect of using a rectangular section is equivalent to increasing the resistivity of the material. A similar effect is obtained by subdividing the core section, as in laminated construction. If the core, for example, were made of two similar round rods, equation (6) would apply to each, and the effective mmf acting on both would be the vahie of (pKpj? given by this expression, with tp in d^p'dt eciual to half the total flux, so that the resulting expression for Ge would be half that gi\-en by (6). This argument can be generalized to show that, aside from the effect of changes in .section sha])e, the etTect of subdivision is to reduce Ge in proportion to the iiuinbcr of siil)di\-isions. The approximate \-alidity of these expressions for Ge rests on the as.sumption that the eddy current magnetomotive force is minor com- 116 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 pared with that of the coil or other external circuit. From the derivation of (2), the component mmf's are proportional to the corresponding terms in G, so the preceding expressions for Ge are valid only if Ge is small compared with Gc{N /R). In all but exceptional cases, this condition is satisfied in operation, and in release with a short circuited winding or a slow release sleeve. In these cases, the effect of eddy currents is ade- quately represented by a constant Ge term in (2). The exact values of Ge applying are in only approximate agreement with those given by (7) and (8) , as the derivation of these expressions ignores the" variation in flux density along the length of the core, and the eddy current effects in the armature and return path. EDDY CURRENTS IN RELEASE In normal release, the winding circuit is open, and the only effective magnetomotive force is that of the eddy currents. Evidently, the field in the outer layers must collapse almost instantly, while that in the center of the core is sustained by eddy currents in paths whose mean conductance, per line linked, is higher than that applying to a uniform field. The relations applying to a closed path of uniform section can be formulated in differential form, and the solution for a cylindrical section has been given by Wwedensky. For decay from an initially uniform field, the expression for the flux as a function of time is a series of exponential terms with progressively smaller time constants. The first term represents the most persistent part of the flux, and represents a field varying from zero at the surface to a maximum at the center, comprising 69 per cent of the initial field. Its time constant corresponds to an effective value of Ge 35 per cent larger than that given by (7). Somewhat similar relations must apply to an electromagnet, causing a time variation in the pattern of the field, not only radially, as in a closed uniform path, but in the longitudinal variation and in the division of the field between the leak- age and armature paths. An experimental study of flux development and decay in relays has been reported by M. A. Logan. His results agree with this discussion in showing Ge in (2) to be effectively a constant, provided Ge/Gc is less than 0.2, as in the operation of most relays. An empirical expression is given for an effective value of Gc + Ge which provides a correction fac- tor applicable for small values of Ge/Gc ■ The results for normal release show the field decay to have the general character of Wwedensky 's solution, and an empirical expression is given which agrees Avith ob- served results. This is primarily of interest in connection with the voltage ESTIMATION" AND CONTHOL OF OI'KUATK TIMK OF KKT.AYS 11/ induced in the coil and imposed on the contact opening the coil circuit. Because of the changing field pattern, the gap field which determines the pull has a ditVerent, though similar, decay rate. These considerations, as supported by Wwcdensky's relations and Logan's results, indicate that the rate of pull decay in normal release is faster, and only roughly of the same order of magnitude, as that esti- mated on the assumption that (2) applies, with \-alues of Ge similar to those applying in operation. LIMITATIONS OF THE ANALYSIS The validity of equations (2) through (5) rests on the assumption that the pattern of the dynamic field is essentially that of the static field to which the magnetization relations apply. The above discussion of the eddy currents indicates that this assumption is valid when Ge is a minor term in G. This condition is approximately satisfied in normal operation. In open circuit release, howe\'er, the condition is not satisfied, and equa- tion (2) is only a crude first approximation to the controlling relation. The use of (-i) as an expression for the reluctance (R rests on the fur- ther assumption that the magnetization is linear, a condition only satis- fied in the low density region. The initial and controlling stages of opera- tion are usually complete before the field passes out of the lo^v density region, and (4) is therefore applicable to the operate case. A different expression for (R is required in the release case, as discussed in Section 7. 2 Character of the Operate Solution GRAPHICAL representation Some understanding of the relations applying to operation rciiiy be obtained from their graphical representation in Figs. 3 and 4. In these two figures the path followed by the \ariables in dyamic operation is indicated by the dotted lines, with the dots spaced to indicate equal time intervals between them. Fig. 3 shows the ip versus 3^ relation, referred to the steady state magnetization curves for various values of x, with .T = 0 corresponding to the operated position, and x = Xi to the initial unoperated position. Fig. 4 shows the d.ynamic F \^ersus x relation, to- gether with the load curve (bounding the cross hatched area Y) and the steady state pull curve for the applied mmf iFs . The flux and pull increase together with the armature at rest at .Vi until the pull equals the back tension at the point 1. In the earlier mo- tion, 1-2, the velocity is small, and the reluctance (from (4)) changes slowly with x so the motion has little effect on the rate of flux develop- 118 THE BELL SYSTEM TECHNICAL JOIRXAL, JAXIAHY 1054 MAGNETOMOTIVE FORCE, J — i Fig. 3 — Field energy relations in the operation of an electromagnet. ment. In the later motion, 2-3, the reluctance changes more rapidly with X, and the velocity is high, increasing d

for slow o]~)eration. EDDY CUUUKNT CONDUCTANCE DKTEUMINATION Equation (9) is used in one experimental method for the evaluation of the eddy current conductance (/£ . In this, measurements are made of I lie time at which motion starts for various values of the coil constant Gc . The latter may be varied by adding series resistance, and the applied \()ltage adjusted to maintain the current and hence ^s constant in suc- cessive measiu'ements. The time / for motion to start may be determined from shadowgraph measurements. P'or a constant l)ack tension, the pull and hence the value of v or 3^/Js when motion starts is constant in these measurements. From (9), t is then directly proportional to Gc + Ge , so that a plot of t vs. Gc should be linear, with a negative intercept on the Gc axis numerically equal to Ge . Experimentally, an approxi- mately linear relation is obtained in this way, provided the values of Gc covered are in excess of 5Ge , in agreement with the discussion of Section 1. Values of Ge thus determined are consistent with those obtained from other measurements, and in approximate agreement with esti- mates obtained from (7) or (8). COIL CONSTANT In eciuation (9) Ge and (Ri are determined by the magnetic design. The latter, together with the load, determines Jo , the just operate mmf, or 47r(XI)o . As V equals 3^o/3^s , or {NI)o/{NI), the only quantities not fixed by the design and the load requirement are Gc and the steady state ampere turn \'alue NI. As the square of this latter quantity is equal to Gc-I'R, equation (9) may be written in the form: , = i'(e, + (^»), 1 . (10) (Ri \ v-PR / I — V The steady state power I'R is either determined bj' circuit require- ments, or chosen with reference to economy of power consumption. With I'R fixed, the only independent quantity in (10) is v, which is determined by the choice of the coil constant Gc . Neglecting Ge , the time is proportional to 1 , 1 — In v~ I — V wiiich is shown plotted against v in Fig. 5. This figure includes a cur^•e 122 THE BELL SYSTEM TECHNICAL JOUKXAL, JANUARY 1954 giving corresponding values of In 1 '(1 — v), which may be used in specific cases to determine the correction corresponding to the term in Ge . In most cases of slow operation, the operate time is of little im- portance, and Gr is chosen to make NI greater than (A'V)o in all cases: i.e., after allowing for possiljle variations in (.V/)(, resulting from varia- tions in the load and in the magnetic characteristics. These variations result in a variation in v, and the corresponding variation in time is shown by the curve of Fig. 5. If, however, it is desired to minimize the operate time for a given power input, Gc should be chosen to give the value of V (0.715) corresponding to the minimum of the curve. As this minimum is broad, variations in v, corresponding to those in (.V/)o , produce little change in tne operate time. 4 Two Stage Approximatiox Unless the rate of flux de\'elopment is ^•ery slow, as assumed in the single stage approximation, the pull attained in the early travel is in excess of the load, and the kinetic energy T is a considerable part of the total work output V -\- T. As a first approximation to this case, the operate time can be computed as though operation occiu'red in two suc- cessive stages: (1) a stage of flux development with the armature at rest in the unoperated position (x = Xi), and (2) a stage of motion, in which 1.5 0.5 \ \ W>-" .-V J \ ^\ y^ / / f / / •^"T^ ^ ^ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 V Fig. 5 — Relations for estimating time of flux development. ESTIMATION' AXD COXTUOI. OK Ol'KIlATK TIMH OF RKLAYS 123 the flux remains coiistaiit at the \alu(> attained at tlic end of tlic first stage. 'I'his approximation necessarily over-estimates tiie opei'ate lime, as it takes the two stages as occurring in sequence, whereas lliey ac- tually oxerlap. Let /i he the time of the first stage, and /•> the time of the second stage, w here ^> is the time for the motion from Xi to some other armature posi- tion .r_. , measiiringthe completion of ojXM-ation. The first stage ends when the flux has attained some value r integral of (')A). Hence: ^ttA {ClXo + Xi)(ClXo + X2) ' In this equation .r,, .1 may be replaced by the expression for (}^o given by (4A) for .r = Xi , and Gc-I R substituted for (NI)' in the re- sulting equation, which then becomes: where d, is given b}^: (Cl - l)a;o(.Ti - ^2) * The time ^2 of the second stage depends upon the difference between the pull and load curves. As a representative condition, this may be taken as approximately constant. For such uniform acceleration of the effective mass jn through the distance Xi — x^ , the kinetic energy T equals 2wCri — .r->)'/75, giving an expression for ^2 in terms of T. As the time /i of the first stage is given by (9), the total operate time to , or ^1 + /l> is given by: , MG, + Gc), 1 , f2m{x, - X2)y ,. .. '" ^ — w, — ^^r^". + V — r — ;• ^^^^ In this approximate expression for the op(M'ate time, v and T are as yet undetermined quantities, related by ecjuation (12), in which Gc is the value of N' R for the coil used. The two stage operation assumed in deriving this expression corresponds to the existence of a restraint 124 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 which holds the armature at rest until it attains the flux level Vip\ . The approximation will most nearly approach the true relation if the effect of this restraint is minimized, and v taken at the value which minimizes ^0 . This minimum value can be determined by trial, using the curves of Fig. 5 to compute /o for several values of v, and taking the minimum from the resulting plot of U versus v. In development studies, the operate time of interest is usually not that for an assigned winding, but that for the winding giving minimum time for a given power input. Substituting in (14) the expression for Gc given by (12), there is obtained the equation: where Ie is written for 47rC7ij/(Ri , the eddy current time constant. As Gc remains to be determined, this expression contains two independent variables, v and T, which are to be so chosen as to make /o a minimum. Equating the partial derivative Avith respect to T to zero, there is ol> tained the following expression for the value of T for- which the time is a minimum: 7 = 2m{.v-i — X2) I 4C,r In ^ 1 - v/ On substituting this expression for T, the preceding equation becomes: \ V- PR/ I — V \ V- \ — V PR / If the value of v which minimizes t^^ is determined, the resulting value of to is the minimum operate time attainable by optimum coil design. In particular, if Ie is negligible, this minimum corresponds, from Fig. 5, to y = 0.715, for which - In = 2.5. v^ 1 — y The corresponding value of Gc , the coil constant value making to a minimum, is given by (12). In this use of equation (12), r and T are taken at the optimum values obtained as indicated above. OPTIMUM COIL CONSTANT As in the case of the single stage approximation, the optimum coil constant corresponds to a value for v of 0.715 when Ie is negligible. From I KSTIMATIOX AXD COXTHOL OF Ol'lllJ ATI: I'lMl'; OF UlsLA^S 125 the curves of Fig. 5, it can be seen that if Ie is not negligible, the optimum value of V is less than 0.715, and will be smaller the larger the ratio of the first term in (15), that in Ie , to the other two terms. From (12), a reduc- tion in V corresponds to an increase in NI, and hence in the coil constant Go U^R being given) . Thus the optimum coil constant for fast operation is larger for electromagnets with long cores of low resistivity material, for which Ge and ts are large, than for those with short cores of high resistivit}^ material, for which these quantities are small. The physical significance of the optimum coil constant can be deduced from the relations shown graphically in Fig. 4. For a given value of the power input PR, an increase in Gc increases both the steady state mmf IFs and the time constant of flux development 47rGc/(Ri , increasing the upper limit to the attainable pull while reducing the rate at which this limit is approached. If the coil constant were so small that the area under the dynamic pull curve equalled the static work load T^, the operate time would be infinite. On the other hand, an infinitely large coil con- stant would correspond to infinite inductance and an infinite operate time. Between these extremes lies the optimum value of the coil con- stant, giving minimum operate time. This is a broad optimum, near which a change in i^s is compensated by a corresponding change in the rate of flux development, so that the realized dynamic pull curve is not affected by a small change in Gc . In addition to the operate time, the value of Gc affects the final velocity of the armature, and thus its kinetic energy in impact with the core at the end of the stroke. This energy is dissipated in the relay and spring \abration associated with contact chatter. For values of Gc above the optimum, the higher inductance reduces the pull in the early travel, while the higher value of iFg increases it in the later travel. The net effect is to increase both the operate time and the final velocity. Values of Gc ii\)ove the optimum are therefore disadvantageous not only in slower op- eration, })ut in increased impact energy tending to cause contact chatter. FACTORS CONTROLLING SPEED Aside from the term in Ie , equation (15) shows that the minimum operate time, corresponding to the optimum A'alue of v, is determined by the static load V, the inertia load measured by 7n(xi — X2)', the steady state power /-/?, and the constant Cn-. The latter is given by (13), and is the only quantity in (15), aside from ts , which depends upon the mag- netic design. In the range of values applying in practice, Cw is deter- mined ])rimarily by the leakage factor Cl , measuring the ratio of leak- age to useful flux. In most practical cases, X2 is small (zero for complete 120 THE BELL SY.^TEM TECHNICAL JOURNAL, JANUARY 1954 operation), C,, equals or exceeds 4 ((Rl equals or exceeds 3 (Ro), and Xi/xn lies in the range from 1 to 3. For 4 < Cl < 10 and 1 < .Ti/.ro < 3, 1.5 < Cw < 2.G. In practice therefore Cw has a value close to 2.0 for most electromagnets. The term in V, the second term in (15), varies inversely as the power I'R, while the third, or inertia term, varies as the cube root of the power. Hence the second term tends to dominate at low values of PR, and the third term tends to dominate at high values. For a low power input therefore the operate time is controlled by the load V, and varies in- ^'ersely as the power input, while for a high power input, the operate tine is controlled by the inertia term, m{xi — x^)' , and varies inversely as the cube root of the power. In the load controlled case, the time varies directly as the load; in the mass controlled case it varies as the cube root of the mass and as the two-thirds power of the travel Xi — x-i . The eddy current term is very nearly a constant increment to the other two ^ terms, and is therefore relatively more important, the faster the opera- tion. PRELIMINARY TIMING ESTIMATES Within the limits of accuracy to which this two stage approximation applies, the effect of the magnet design upon the operate time appears only in Ie and Cw . If the former is neglected, the optimum value of v is 0.715, and \ In -^ = 2.5 . V 1 — V Taking Cw as having the representative value of 2.0, (15) reduces to: 101' /I35m(-ri - .r2)-Y (,c\ ''^PR^\ — m — ;• ^^^^ This simple approximation provides rough estimates of the operate times attainable with any electromagnet for a given load and power input. As an illustration, consider a typical relay spring load involving a travel of 40 mil-in (0.1 cm) with a level of spring force of 100 gm (lO"" dynes), so that V = 10 ergs. Let the effective mass m of the moving parts be 10 gm. For a steady state power input of 0.5 watts (5 X 10 ergs sec), the two terms of (16) have values of 0.020 sec and 0.014 sec, respectively, so that U equals 0.034 sec. For an input of 5 watts, the two terms become 0.002 sec and 0.007 sec, respectively, and /o equals 0.009 sec. The neg- lected Ie term of (15) might amount to an increment of 0.005 sec for a solid core of magnetic iron, but would be proportionally smaller for higher ESTIMATION AND CONTROL OF OI'KU.VTE TIMi: OF RELAYS 127 resistivity material. In either case, this iucromeiit would he of little consequence in the low power case, hut would niat(M-ially affect the operate time in the hi^h power case. 5 TiiREio Stage Approximation The following analysis ])r()\'ides oreatei' accuracy in estimating operate time than the two stage approximation, together with a more accurate representation of the relations hetween performance and the contr(jlling \';U'iables. It differs from the two stage approximation in treating the initial motion as a separate stage of operation. The three stages of operation thus become: (1) Increase of the flux to the value at which the pull e(iuals the back tension, with the armature at rest, (2) flux development and concurrent armature acceleration, assuming equations (2) and (3) to apply, with the approximation that the variations of (R and J with x are ignored, and (3) the later motion, which is treated in the same manner as the second stage of the two stage approximation. Formally, the second stage should be restricted to a very small part of the total armature motion, in order to minimize the change in (R which is ignored. Relati\'ely minor error, howe\'er, is introduced in ig- noring the variation in (R through as much as half the total travel. The spring force is taken as constant through the second stage. In the relay case, this isapproximatety true for the initial travel prior to the actuation of the contacts, which results in an abrupt increase in the spring load. Thus the second stage can be taken to extend to the tra\^el at which contact actuation first occurs. If all contacts are actuated near the same point in the travel, the end of the second stage coincides with complete operation, and the third stage need not be considered. If the contacts are spread out, the third stage coincides with the stagger time, or time be- tween operation of the first and last contacts. The operate time there- fore consists either entirely or principally of the time for the first two stages. As before let : Xi be the initial (open) gap, for which (R = (Ri , Fi be the back tension (constant load in the first two stages), , ,...,.,. 47r A^' tc be the initial coil time constant, — • — , (Ri Pv .... . 47r //.; he the initial eddv current time constant, ^ Ge, 1 / / // / / /I / 0.004 / 1 / 1 / 1 / 0.003 / 1 < / 1 0.002 1 1 1 /r 0.001 /A /// V 1 1 0.4 0.5 0.6 0,8 1.0 Fig. 6 — Relations for initial motion in operation. FACTORS AFFECTING OPERATE PERFORMANCE The dominant term in the operate time is the time U for the second stage, which also determines the velocity and pull level in the third stage, and hence the final velocity. As U_ = Ziitc + Ie), equation (23) gives the following expression for h : ESTIMATION" AND t'OXTllOL OF OI'KRATE TIME OF RELAYS 131 3 2m.4i(Ri(.ri - X2) tc + ts zl ,^.. h = ^,-r> -, 77 7- V^'*; I-Ii tc hyVi , Z-y) As can he seen in Fig. G, /sCci , z) is, to a rough first appi-oxiinatiou, proportional to z'\ To the extent that this approximation holds, [2 is proportional to the cube root of tnAi(Ri{xi — X2)/{I'R), as for the last term in (15) and (16), except that (.Ci — .^2)' is replaced by Ai(Hi(.Ci — .X2). From (-1A) and (18), Ai(Ri is given by: (C - l).To Here.ro = A(Ro , and (2 is a minimum for that pole face area A for which .li(Ri is a minimum. This condition, as determined by equating the derivative of the preceding expression with respect to Xo to zero, is given by: 2 n 2 xt = Ca 0 > or Xi/A = \/Cl ' ^0 , where A is the optimum pole face area for fast operation. This is a smaller pole face area than that for maximum sensi- tivity, for which Xi/A should equal (Ro , as shown in the companion article cited above. If the pole face area satisfies the preceding condi- tion, the expression for Ai reduces to: Vcl- 1 and the term Ai(Ri in (24) reduces to .Ti(.ri — X2) multiplied by the factor (VCl + l)/('\/Cz, ~ !)• This leakage factor, which increases with the ratio of leakage to useful flux, is then the only term in (24) which varies AATth the design of the electromagnet. If f2ivi , z) were strictly proportional to z^, equation (24) would be nearly independent of the coil constant N^/R, to which tc is proportional. As /2(yi , z) departs from this cube law relation, and also varies with Vi, t2 is not independent of the coil constant. The optimum coil constant is that which minimizes /i + to. In any specific case, a succession of values may be assumed for Gc , and ti and ^2 evaluated by the relations given above. The resulting relation between ^1 + ^2 and Gc is similar in character to that described above in connection with the two stage approximation. 6 Design for Fast Operation The approximations discussed above provide a means for estimating the operate time attainable in specific cases, and for selecting certain 132 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 design variables to obtain maximum speed. In particular, it has been shown that for a given relay, with a specified load and power input, a winding can be selected to minimize the operate time. Aside from this, the relations show that for a given relay, the operate time depends only upon the load and travel and the power input, varying inversely as the latter at low levels of power input, and inversely as its cube root at high levels. Further conclusions can be drawn from these relations mth reference to development studies of relays and other electromagnets. The spring load and travel may be considered as fixed requirements, so far as the magnet design is concerned. The available power may be fixed by circuit considerations, or it may be related to the design by a requirement that the winding dissipate this power in the holding interval, a condition that imposes a minimum size on the coil and the magnet structure. Subject to this and some other limitations, there is a design choice of the dimensions and configuration which determine the magnetic circuit constants, the mass of the armature, and the eddy current conductance. The preceding discussion has shown that, with an optimum choice of pole face area, the magnetic characteristics affect the time only with respect to the leakage factor, the ratio of leakage to useful flux. This factor may be reduced by using a "tight" magnetic circuit, but if this is done the factor tends to vary directly as the length of the magnetic path and inversely as the separation of the core and return members in relation to their cross sections. The leakage may be minimized by using a square outline for the magnetic path. The optimum speed magnet then has a specific configuration in which all dimensions are fixed in re- lation to the cross section of the magnetic members. This dimension then determines the mass of the armature. For this optimum configuration, the power, the spring load, the mass, and the travel determine a level of pull which minimizes the operate time. This pull requires a certain armature cross section if saturation is to be avoided, as the effective pole face area is fixed in relation to the cross section. If the resulting armature mass is minor compared with the mass of the load, the operate time attainable varies with the cube root of the applied power, as in the cases discussed above. With increased power, however, the optimum pull and armature cross section increase, and must eventually reach a level where the armature mass becomes the dominant portion of the mass term. As this condition is approached, any increase in power is offset by an increase in armature mass, such that a lower limit is imposed on the operate time proportional to the travel, corresponding to an upper Umit to the average armature ESTIMATION AKD COXTKOL OF OPEKATE TIAIE OF KELAYb 133 \'elocity attainalile with a neutral electromagnet. This ujoper limit is of the order of 100 cm/sec. Thus, for example, an operate time less than 0.25 millisec cannot be attained with an armature travel of 10 mil-in, no matter how laroe the stead^^ state power applied. 7 Release Waiting Time Like operation, release is made up of an initial stage of flux change with the armature at rest, followed by a stage of armature motion, and the total time is the sum of the waiting time and the motion time. In release, the waiting time is usually larger than the motion time. There are three distinct circuit conditions inider which release occurs. These are: Normal, or unprotected release, in which the coil circuit is open, and the only magnetomotive force maintaining the field is that of the eddy currents. Protected release, in which the coil circuit is closed through a protec- tive shunt, usually comprising a condenser and a resistance in series. The magnetomotive force comprises that of the eddy currents and that of the coil circuit transient. Slow release, in which a sleeve or short circuited winding is used to maintain the field and delay release. The magnetomotive force is pre- dominantly that of the sleeve or winding current. The slow release case is the only one of the three for which the flux decay relation is accurately represented by equation (2). In protected release, the coil current transient is controlled by the condenser, as dis- cussed below. In normal release, the variation in the field linked by different eddy current paths results in the changes in the field pattern discussed in Section 1. As noted there, equation (2) applies to this case only as a very crude first approximation. In all three cases, the relation between tp and ^ applying is that for decreasing magnetization, as illustrated by the curve for x = 0 in Fig. 7. Unlike the linear relation applying in operate, corresponding to the constant reluctance given by (4), the decreasing magnetization curve has a hyperbolic character, and is asymptotic to the saturation flux (p". Residual magnetism results in a residual flux (^o , the intercept of the decreasing magnetization curve on the (p axis. An analytical treatment of the decreasing magnetization curve is given in a companion article,^ where it is shown to provide a satisfactory basis for predicting the release time of slow release relays, the third case above. The other two cases, to w liich the following discussion is confined, involve both non-linear mag- 134 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Fig. 7 MAGNETOMOTIVE FORCE, 7 — *- Field energy relations in release. netization and a variable magnetic field pattern. In an approximation neglecting the latter effect, no advantage is derived from an accurate formulation of the former, and a linear approximation to the demagneti- zation curve may be employed. NORMAL UNPROTECTED RELEASE As indicated above, a rough approximation to the decay of the gap flux, and hence of the pull, may be obtained by assuming that this flux decays in conformity with equation (2), with G = Ge , where the eddy current conductance Ge has the same value as in operate. As a linear approximation to the demagnetization curve, the reluctance (R can be taken as the closed gap reluctance (R(0). Allowance may be made for residual magnetism by postulating a steady state magnetomotive force 57o = (R(0)odinj2; oquation, there is obtained: a+„)(c„+»)g-™;;;^ To reduce this expression to dimensionless form, the following sub- stitutions are made: /' is wi'itten for — - / Fo , where Fo is the operated load, so that / ax I = 1 at the start of the release motion. T is written for 2Cz, tjtE , expressing the time i as a multiple of IeI i2Ci) ■ til is written for 2mA(Sio/Fo. Thus Im is the time for travel of the mass through the distance A(Ro for a constant accelerating force i^o . K is written for 2Cz, ImI^e- The preceding equation then ])ecomes: (l + w)(C. + ^0(/-/v^ H.^((C, + .r(/-K^^)) = 0. (27) This is the form of the release motion equation to which analogue computer solutions were obtained. It is a third order differential equa- tion giving the travel, expressed as a multiple u of ^(Ro , as a function of the time expressed as a multiple r of tsji'lCi). The boundary condi- tions correspond to zero initial values of travel, velocity, and accelera- tion, so that for r = 0, u = du/d.T = d u/dr = 0. ANALOGUE COMPUTER SOLUTION The analogTie computer gives solutions to specific cases, and a specific case of (27) is defined by the values of C l and tM/lE applying, and by the form of the relation between /and u, defining the shape of the spring load. To confine the cases considered to a manageable category, they were limited to those for which ./' = 1 for all values of u: the case of a constant spring load. The constant Cl , the ratio (olo + \ 0.03 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 W V Fig. 11 - Energy absorlx-d by niagiiolic drag in release motion. KS'l'IMAI'ION AND ( ( »\'l'l'.( )1, ()!•' OI'KK A'l'K 'I'lMI') OK l{i;i,ANS \ \',\ As mininuziii<>; rcljound is of major iinpoi'Innce in relay (l('si}i;ii, tlic relations eoiilrolliii^ (he kinetic enei'<2;y to which it is |)i'o])orti()nal are of (lesi.iin interest. Most, of the (|iiantilies a|)|)eariii<; in the al)o\'e relations, such as the spring load and the traxcl, ai'e iixed hy operatiiifi; re(|uire- inents oi- other desitic draji; I'atio can he inci'eased, and rebound I'e- duced, by the decrease in I \, result inii fi'om a decrease in e('fec(i\(' mass. While such a decrea.se increas(\s the lalio / / ,/ , it results in a net decrease in the \'alue of /, the motion time. .\ low effective armature mass is therefore adxantaiicous both lor fast, I'clease and for a hi};h dra)^' I'atio for the reduction of rebound. REFERENCES 1. 1{. I;. I'cck, Jr., ;ui(l II. N. Waj^ar, Manuel ic nosij:;!! of Relays, pa^o 2'.] of this i.ssuc. 2. H. L. IVck, Jr., Analysis ol' Measured .M.i^nct i/,;it ion ;in(l Pull ("liarac(('ri.stics, pajrc 7S of (his issue. :^ H. Wwcdcnsky, .\iiii. d. IMiys., 64. p. (lO'.l, IDL'i. 1. .M. .\. boy;aii, Dviiainic Mcasuicinciils of IMccI roiuat;ii('l ic Devices, P. S. T. J., p. 1113, Nov.,' V.)-)A. "). H. L. Peek, Jr., Priiicij)les of Slow Roloaso Helay Desifiii, i)afi;o 1S7 of this i.ssuc. (i. .\. .\. Currie, The (ieueral Purpose .Xualojr Computer, Bell bah. Record, 29, I)|). 101 lOS, Mar., n)51. 7. Iv Lakatos, Problem Solviiifi witli I lie An.dofi Compuler, I'xdl bah. Record, 29. pp. 1(M)-114, Mar., 1951. 8. K. K. Sumiior, Relay Armature Rehound Aiialvsis, B. S. T. .1., 31, pp. 172 200, J.aii., PJ.W. Estimation and Control of the Operate Time of Relays Part II — Design of Optimum Windings By M. A. LOGAN (Manuscript received September 4, 1953) For each relay structure, there exists a best winding, once the circuit 'power has been specified. The best winding is that one which operates the relay in the least time. Methods for determining the best umuiing are de- veloped. On the basis of the operating time, the relay behavior is classed as mass or load controlled. The design method chosen depends upon this classifica- tion. The design of ivindings for series connected relays is based on a method of determining eouivalent single relays, the behavior of each corresponding to one of the series relays. This method is generalized to allow for differejit magnetic structures for the several relays. Each relay winding is then de- signed in turn, using the design data for its own type of structure. The best winding design is not given directly by an explicit formula. Rather, methods are developed for determining the operate time for any winding. Then by choosing a range of windings, the best one is selected by interpolation. INTRODUCTION The selection of a relay for a circuit application, particularly so in common control systems, involves in part a determination of its operat- ing and releasing time. In Part I of this article expressions are derived for the relations between these times, the design parameters of a relay, and the conditions of operation. These expressions are approximate, and have been developed with primary reference to the selection of favorable characteristics in design. The timing estimates they provide are of sufficient accuracy for the comparison of design alternatives. Once a basic design has been selected, the choice of windings and the prediction of the limiting times occurring in specific applications requires 144 ESTIMATION AND COXTHOL OF ()IM:k All': TIMI'; OF HKLAYS 1 io a more detailed and exact pi'ocedure than can l)e obtained enlii'eiy from the application of these approximate relations. The information further- more has to be (|uite versatile, to i)ermil all new tyjx^ \ariations to be included, such as winding, number of contacts and armature travel. It must, moreo\'er, be in a form that permits determination of the upper and lower limits to the time observed in each specific type of I'elay as actually used, as in applications, interest attaches not only to the nominal conditions, but to all \'ariations that may arise in actual manu- facture and use. Relay operation is complex, and in principle the nonlinear differential equations which describe it can be solved exactly only by computers. Even if this is done, the results are subject to any uncertainty that exists as to the exact form of the relations that apply. The representation of non-linear magnetic material properties, the discontinuous load-travel characteristics, the eddy current effects, etc., do not make such an ap- proach attractive. The relay, however, is a perfect analog of itself. With the magnetic structure set, controlled models can be built to include dimensional and magnetic material variations. With these, exact solutions to a variety of conditions can be determined. With these data available, approximate solutions can be used for interpolation and extrapolation, determining the effect of small variations from the tested conditions. This part of the article will exhibit the form of data presentation in two classes, mass and load controlled operation. It includes the theory used in selection of the forms, and the correction methods used for estimation of variations from the standards. The initial part will be concerned with a single relay operating in a local circuit. The latter part will consider series and series-parallel operation of similar structures but not neces- sarilj' identical windings. This latter problem is solved by determination of an equivalent relay in a local circuit for each of the several relays. The earlier analysis then can be applied to each in turn. The order of analysis can be rcA'ersed. That is, given required operating times, windings can be determined which will provide these times. In this article, a best winding is (1) that winding which, for the speci- fied applied power, results in the minimum operate time or (2) that winding ^vhich, for a specified operating time, requires the minimum powder. A unique solution exists. Existence of Best Winding Fig. 1 shows measured operate time of a relay for two different do power conditions, versus number of turns in the winding. That a best 14G THE BELL SYSTEM TECHNICAL JOURXAL, JANUARY 1954 winding, in the definition of this article exists, as exhibited by the minima on Fig. 1, was shown in Part I to be a consequence of equation (12). This best winding and its determination is the basic subject of this article. OPERATE TIME SIXGLE RELAYS As in Part I, the operating time of a relay is considered as made up of three stages: (1) the waiting time while the armature remains at the back- stop and the pull builds up to equality with the back tension, (2) the motion time beginning at the end of the waiting time and continuing while the armature moves from the backstop to the position of the earliest contact, and (3) the stagger time during which the armature actuates all of the remaining contacts. The mass controlled case, as a practical matter, simplifies to a determi- nation of only the first two without regard for the spring load, with the displacement of the armature taken to the latest contact in the array. The armature pull builds up to values in excess of the load during its early motion and the velocity is so high during the stagger time that this small interval can be included as part of the motion time. This treatment is essentialh^ that of the three stage approximation of Part I. The load controlled case simplifies to a determination of the time required for the pull to build up to the maximum load, including the back tension, with most of the relatively slow motion taking place during this pull buildup. The remainder of the motion time is accounted for as an empirically determined correction factor in the expression for time of pull buildup. This treatment corresponds to the single stage approxima- tion of Part I, with a correction term to account for the additional motion time. Fig. 1 shows graphically the basic characteristics of the two tx'pes of operation. This chart displays the operating time of the same relaj^ under the conditions of 1.0 or 5.7 watts, and three contact load conditions: 12, 18, and 24 contact pairs. For each of the six conditions, a curve is drawn showing the effect of the number of winding turns. It is clear that (1) there is a best number of turns for each case, (2) for the 5.7 watts the best number of turns is the same for all three loads, (3) for the 5.7 watts case the difference in operating time is only 0.1 millisec in 5.6 millisec, between 12 and 24 contact pairs, (4) for the 1.0 watt case the best number of turns increases with the number of contact pairs, and (5) for the 1.0 watt case the operate time is double for 24 compared to 12 contact pairs. These strikingly different behaviors form the basis for di\'ision of relays ESTIMATION AND ('()\TJ;()I. OF ( iI'1;H ATI", TIIMK OF m;r>AVS 117 100 90 80 70 1 \ \ NO. OF CONTACT ^ PAIRS \24 1 WATT \ -^ \ V - 18 -^ N ^ 2 24 18 12 ^ X 5.7 WATTS .^ ^ '''^^ 3 4 5 6 7 NUMBER OF TURNS 20 X 10- Fig. 1 — Chart showing typical mass and load controlled operation. into two classes called mass and load controlled, and furthermore empirically establish which of the three stages of operate time pre- dominates. This distinction corresponds to that made in Part I in the discussion of equation (12). For the lower power cases the term involving the spring load predominates, for the higher power cases the mass term predominates. As this equation shows, there is necessarily a transition range where the operation time merges from one class into the other. Both types of charts are extended to include this range. A rule of thumb for an estimate of whether a time under discussion is in the mass or load controlled case is that if the time is less than U / 27m (xi — rcs) F. (1) it is mass controlled; otherwise it is load controlled. The derivation of this bound will be discussed when mass controlled operation is considered. The single relay design data are general enough to include resistors 148 THE BELL SYSTEM TECHNICAL JOURxXAL, JANUARY 1954 external to the relay winding. Wherever a resistor or power term appears, the sum of all resistances or powers is indicated. Maximum Versus Average Operate Time Presentation In w^hat follows, load controlled operation data are given in terms of maximum times, whereas mass controlled data are average. Either choice could be used for both. The data for converting to average or maximum respectively will be described. However, the choices made here are con- sistent with the normal use of the data. Mass controlled, sometimes called speed, relays operate with rela- tively large power and ordinarily are used in common control circuits where many events occur in succession. The total number of similar common equipments necessary in an office is related to the control circuit holding time. For a system design, the cost of power is balanced against the cost of equipments to arrive at a minimum office cost.^ Now the maximum holding time per call is never the sum of the maximum possible times of each of the several relays operating in succession. It rather is more nearly the sum of the averages, increased by considerably less than the common maximum to average ratio of one of them. For example, if n relays are assumed to have a distribution approaching normal, an analytical expression for the probable maximum is: (2) where 6/av is an allowance for short time deviations of the manufactured product from the long time average. Thus for mass controlled relays the most directly applicable type of data is in the form of average time, and maximum to average time ratio. For load controlled relays just the opposite is true. Here the operating times are relatively long, either because the power drain is to be kept to a minimum, as in a long holding time circuit, or it is an event which takes place while several successive mass controlled e^'ents occur. For either case, it is a single event and only its maximum duration is desired. Here then the most directly applicable type of data is maximum. In addition to these, other data for minimum times are needed for studies of timing when two parallel circuit paths occur. Because how far the relay armature has to move is the outstanding variable in mass controlled relay timing, these charts are prepared for each nominal distance which can be chosen. When there is no concern as ESTIMATION AXD COXTIv'OL OF ( )1'I;h ATH TIMIO OF HKLAYS 149 to relative actuation time of the several contacts on one relay, the small- est armature travel is ordinarily provided. When a definite seciuence is needed for circuit reasons, the actuating means is arranged to guarantee operation of certain contacts before the oth(M-s. This necessitates the pro- vision of a greater armature motion. The principles used in preparing these other types of charts are identi- cal to those used for the two types which are developed specifically in this article. Local Circuit Load Controlled Operation Power Given The best winding for a load controlled relay is not here given explicitly by a formula, but rather is found indirectly by developing a method for determining the operate time for any winding. Then by assuming a range of these, the best one can be selected by interpolation of these derived data. In Part I, the waiting line for a linear system was shown as equa- tion (9) to be: 47r 1 h = — (Go + G, + Gs) In r— , (3) (Ms l — lo/l with lo/I substituted for v. Without motion time, saturation, eddy cur- rents, or non-linear effects, Part I of this article also shows that the best winding is that for which the turns are selected so that NIo/NI — 0.715. Taking the other variables explicitly into account complicates the determination and is at best only approximate. Instead they are included through this interpolation approach. In a previous article a better representation for the core eddy current constant was shown to be: G'e = GeC^^^'''^'^^^^^ = effective core eddy current constant. Substituting this and explicitly indicating a correction term for the small motion time, the form best suited for the present discussion becomes: ^0 = (1 + k/t,)L,{Gc + Gs + G'e) In -J— . (4) 1 - q Except for the correction term for the motion time, t^/ti , this is still a linear equation with all factors known. Omitting the motion time correction, it can be solved either by numerical substitution or by a nomogram, once a value for Li has been chosen. A conservative value is the a\'erage one turn inductance, for the late contact critical load point where x = .T3 . A more accurate value is determined through use of both 150 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 the inductance at the open gap and at the critical load gap. These are weighted in proportion to the ampere turns developed at each location. The ampere turns necessary to overcome the back tension is one factor, and the other is the difference between this first ^-alue, and the total required for operation at the critical load gap. This weighting yields an effective inductance value intermediate between the two extremes for each problem, but a new chart does not have to be prepared. An operate time using the chart is first determined. This time is then adjusted bj'' the ratio of the effective inductance applying and the inductance used for making up the chart. This is exact, as the inductance term appears only as a direct multiplier. A typical nomogram is shown in Fig. 2. It already includes the motion time correction, whose determination will be discussed. The dashed line x10^ 4 5 6 8 10 20 30 50 60 80 100 TIME IN MILLISECONDS 200 300 400 600 Fig. 2. — Nomogram for solution of load controlled operate time. ESTIMATION AND CONTROL OF OPKK A TK TIME OF UKLAYS 151 shows the successive steps taken in detei'iiiiiiiiiji; the iiumerical vahie for a specific case. The dc circuit resistance is detei'mincd by the known circuit voltage and specified power. Entering tiu^ chart on the left ordinate at this re- sistance and proceeding horizonttdly to the right, intersections with increasing number of turns lines, sloping downward to the right, are found, and the appropriate one is chosen. Dropping vertically to the abscissa, the winding time constant tc = LiGr is determined. To this is next numerically added the known effective core eddy cur- rent and a sleeve (if any) time constant by returning vertically to an intersection with the appropriate core indicated as "no sleeve," or core plus slecN'e, curve. Proceeding horizontally to the right hand ordinate scale from this intersection, the time const-int multiplier of the In term in equation (4) is determined. The multiplication of these two factors is accomplished by proceeding to the left along this same horizontal line, to an intersection with the proper q line, sloping downward to the left. Vertically below this last intersection is the operate time. Initially, tentative q lines are drawn, omitting the motion time cor- rection. These lines are straight with a positive 45° slope. Then with an actual relay whose just operate current has been measured, operate time measurements are made, keeping the final current, and hence q, fixed at se\'eral values in tvn-n. This is done by adjusting an external resistance and battery voltage over a wide range, effectively changing X~/R. These measured data are plotted, following the same steps through the nomogram, except the last intersection is with the measured time vertical, rather than the known q. This provides several empirically determined q lines. These are used as templates, to progressively alter the shapes of the tentative straight q lines drawn earlier and shift them to the right. This adjustment then introduces the motion time correction. It has been found empirically that for large operating times, the motion time correction factor has a vsdue of about 0.1 There is no definite di\'ision between mass and load controlled operation, but as the total time decreases, travel time becomes more important. The correction factor increases to a value of about 0.5, in the transition range. For completely mass controlled operation, there is no q effect, so the q lines must all eventually con\'erge. As stipulated above, this chart applies to the current, ftux and pull range w^here the first approximation magnetic constants of the structure are applicable. For this reason, the tests for the motion time corrections are made under conditions meeting these restrictions. These curves should be corrected for magnetic saturation if there is a 152 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 "- 0.80 z o O 0.70 ^ ^ ^ <. ^ N, s^J \ N, % sAG"" N \ \ \ V \, \ 60 260 Fig. 3 80 100 120 140 160 180 200 220 240 AMPERE TURNS REQUIRED (CRITICAL LOAD POINT) Correction of computed operate time for magnetic saturation. large load at the critical load point and the ampere turns needed are in the saturation region for the core or armature. In the saturation region the current builds up faster than the linear time constants indicate, and, therefore, the indicated operate times are too large. Correction factors are determined by graphical integration of the integral form of the flux rise equation expressed as a ratio to the linear relationship: Saturation Correction Factor = i d(p 0 NI - Ni (5) - Li In (1 - Nh/NI) These corrections are plotted as a function of the just operate ampere turns iV7o, with the final NI as a parameter for each curve. Actually, because the correction factor is only of the order of 20 per cent maximum, assuming the final winding ampere turns are well into the saturation region, it is found that a single curve for any one type of relay fits all the computed points to an accuracy of a few per cent, and generally is used. Such composite curves are shown in Fig. 3 for the three types of wire spring relays. A method has now been established for determination of the operate time of a relay with two restrictions (1) that the final ampere turns will operate the relay and (2) that the relay is in the load controlled class. An indication of the latter is whether the operate time determined is in the time region where the q curves are decreasing in curvature. For Fig. 2, 10 milliseconds is taken as the lower bound for load controlled relays. Now the determination of an optimum winding for a particular ESTIMATION AND CONTKOL OF Ol'KHATK TIMK OF KELAYS 153 problem can be finished. Slartiiiii; witli the power gi\'en and choosing an arbitrary number of winding funis, the operate time is determined as described above. If it falls into the load controlled class, then a different numb(M" of turns is next assununl and a second time determined. This procedure is repeated until a time cur\-e \'c>rsus number of turns similar to Fig. 1 can be plotted, including a minimum. The best winding i this minimum. A different curve and optimum number of turns will ai)i)ly to each contact load assumed. Three comi)ut(Hl curves corres])<)nding to the measur(xl 1-watt curves are shown on Fig. 4. These were deter- mined using effective inductance values tlescribed earlier. Finite wire sizes permit only certain number of turns to be physically realizable when the resistance has been specified, without sjilicing two gauges of wire. The nearest gauge on the coarser wire size side is chosen, resulting in slightly too many turns. Note that the curves rise less steeply on the high turn side and the time penalty therefore is less than if too few turns were supplied. 80 75 70 65 60 55 50 Q Z 45 O U 111 <£i 40 _1 _l 5 35 - MEASURED 1 COMPUTED 1 \ 1 WATT \ ( »1 1 k NO. OF \ CONTACT PAIRS ^-i. H y \ I \ \ \ ^ / -J^ J r ^ \ / ^-i>^ ^j' '"i^^ c \ \ . \ ^ L ~7~ _ ^^^ • ^^ - / / f \ V 'v, / P^ *^ / /-. "-^ / 18 20 25 X 10^ 6 7 8 9 10 12 14 16 NUMBER OF TURNS Fig. 4 — Typical computed maximum operate time curves for load controlled relays. 154 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Maximum Time Given For a specified maximum time, the above process is repeated for several assumed circuit powers until the specified time is bracketed. Then by interpolation of the optimum times indicated, the minimum power, maximum resistance and optimimi turns are determined. As an example, Fig. 4 can be used to demonstrate the method. Assume that the three curves were computed for different circuit powers, rather than for different contact spring loads. A line is drawn through the minima. The intersection of this line with the required operate time determines the number of turns. For instance, if 30 millisec were required, the turns would be 12,500. The circuit power at this same intersection can be interpolated for, using the known circuit powers associated with the three curves. It is not economical to have a different winding for each spring com- bination. For this reason, a winding is designed for the maximum spring load and then used for smaller loads. The operate time will always be less with the smaller loads. Measurements of Time Curves Before considering mass controlled operation, the simulation of windings will be discussed. In the above description for establishing the q curves, it was pointed out that by adjusting an external series resistor and the battery voltage to maintain constant final current, the coil constant N'/R was altered without changing q. This can be further extended to permit simulation of any winding for test purposes providing only that the experimental coil fills the winding volume as much or more than the coil to be simulated. For this purpose, a special test winding CIRCUIT TO BE SIMULATED 4 E, N,,R, SIMULATING CIRCUIT -y(^v-i C2,Q2 ^3 = lt^' R. =I^Vr, Fig. 5 — Simulation of winding circuit for timing tests. ^^-) R„= 1^1 R, ^^ = 1-n:-)^' ESTIMATION AND COXTKOL OF OPERATE TIME OF RELAYS 155 always is used which completely fills the winding volume. Hence an.y winding which can ho designed also can he simulated. The conditions wiiich must be I'ulfillcd for perfect simulation dcrixe from Lenz's Law. It is essentially an imjjedance ti'ansformation technicjue keeping the magnetic flux invariant, with the assumption that a winding can be considered as a hniiped ratiier than a distributed network. Tiiis is equivalent to stating that at any instant the cui'i'ent flow is the same in every turn and there is no propagation time in\'olved. This is true for times in\()l\(Hl in electro-magnets. Fig. 5 siiows a cii'cuit to be simulated, in which all the components witii subscripts 1 have been given. The simulating circuit has only the numbei' of turns.Y2 , of the test winding, given. The other four elements must be determined. After switch closure, the exact differential equa- tions appl3'ing are = r.i — nm , t = 0; ii = 12 = 0. (6) at dt Xow for equality of magnetic flux, the two rates of flux change must be identical at all times including the first instant. Inserting the initial boundary conditions and equating the two rates, we have -^ = -^ (7) Ni N2 At infinite time, the same magnetomotive force must apply to both circuits for equality of final flux. Eciuating these, and cancelling the 4Tr factor, Nih = N2I2. (8) Noting that and (9) T E2 we have, after using (7) and rearranging, 150 THE BELL .SYSTEM TECHNICAL JOURNAL, JANUARY 1954 which states that the coil constants must be equal. After steady state has been reached and the switches opened, the two differential equa- tions are: E, = ii{Rn + Ri) -^ n/-^ + ^ f n dt, at Ci Jo E2 = i2{R22 + R2) + ^2 '^' + i / i2 dt, (11) r\ • -^1 • E2 r^ r^ r\ at t = 0, ti = -- , %2 = ~ , Qi ^ Q2 = 0. Ri K2 Multiplying the first equation by A^iCi and the second by N2C2 , and equating term by term for equality at all times and equal magnetomotive forces, two additional equations result: (/?n + Ri) Ci = {R22 + R2) C2 , (12) N,'C\ = N2C2 . The four defining equations, with some further rearrangements using (10), are shown in Fig. 5 as the relations applying for equality of mag- netic flux in the two circuits. As the mechanical beha\-ior of a magnetic structure is completely determined by the magnetic flux, it follows that the mechanical performances will be identical and timing measurements made with the simulating circuit will represent the actual circuit. This is true whether the relay is mass or load controlled. It includes eddy current effects and w^iether or not there is motion of the armature. Local Circuit I\Iass Controlled, Operation Typical mass controlled operate time curves are shown in Fig. 6. These are for two different armature travels, indicated as short and intermediate. The curves are characterized by the circuit power used for each, with the coil constant plotted along the abscissa and the corresponding operate time as the ordinate. As mentioned earlier, these curves are substantially independent of contact spring load, and are plotted for average conditions, including an averaging of the time for the first and the last contact to be actuated. It will l)e noted that the best coil constant is not independent of tlie circuit power, decreasing continuously as the power is increased. Also by increasing the circuit power from 2.3 to 23 watts, the operate time is decreased by a factor of a little less than 3, which is nearer the square ESTIMATION AXD f'OXTROL OF OPKRA'1'1'. 11 Mi; OF IIKI.AVS 157 20 2 2 \ ^ AF RELAY INTfcRMEDIATE TRAVEL k \ "S *<^"l ^^^ ■^ ^~ - \ ^ ^.5^ J--- ^'^^ Li£ ^ ' ^ -_ 11. s:: —~" _,^ 23.04 1 1 1 1 30 40 50 60 80 100 Fig. 6 — Mass controlled operate time tests. root than the cube root of the power ratio, developed in Part I, for the mass controlled case. This is partly attributable to the fact that waiting time is included, and partly to the fact that the 2.3 watts case is in the transition range between mass and load controlled operation. For the operating region where coil constants are larger than the best, the time curves are parallel and increasing. Considering any one vertical line representing a particular winding, increasing the power by increasing the battery voltage will alw^ays decrease the operate time. For curves plotted as in Fig. 1, where the battery \^oltage is kept constant and the time curves are plotted with winding turns as the independent variable, parallel curves again obtain. Thus an increase in circuit power obtained by keeping the voltage constant and reducing the circuit resistance always will reduce the operate time. Conversely, adding any series 158 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 impedance, not including a capacitance, will always increase the operate time. This will be made more evident when the operate times of relays with their windings in series are considered. The curves can be used in either of two ways (1) given the relay circuit for which the applied power and coil constant can be computed, the average operate time can be found from the chart, or (2) given a required average operate time, the necessary circuit power and coil constant can be found from the chart. These curves can be plotted in this form to exhibit the best winding directly because of the independence to contact spring load. For the load controlled case, the contact spring load was an essential parameter. For the mass controlled case, the armature travel becomes the outstand- ing parameter to be considered, but there are only two or three of these. All the other factors except contact load also enter and need to be evalu- ated for two reasons, (1) to provide an estimate of the range in operate time to be expected and (2) to adjust experimental measured time data to average. For any experimental setup, it is seldom possible to provide a structure which is average in every respect. For any one structure, how- ever, all the factors known to affect its performance can be measured. Comparison of these to the manufacturing specifications locates the experimental setup in the universe of all relays as regards each of the factors. It is thus necessary to develop representations relating each of these factors to the operate time, which will suffice for the two uses named above. The development of these relationships will be the subject of the following sections. Waiting Tijne The waiting time, whether an electromagnet is mass or load controlled, is given by the same form of equation used for the total operate time of a load controlled relay: U = U{Gc + G'^) In —^ . (13) 1 - 91 where now qi is determined ]:)y the armature back tension Fi ; Li applies to the open gap; and no sleeve conductance is present. For the present purpose, it is desirable to rewrite this equation in terms of the funda- mental parameters of the relay. For the open gap case the magnetic material is operated in its linear region and the open pole face gap pro- vides additional linearity. For these reasons the expression is quite ac- curate. The sketches of Fig. 7 show the factors to be used. The value of ESTIMATION AND COXTUOL OF Ol'EKATE TIME OF RELAYS 159 Tji ,' the iiuhi('taiic(> foi' one wiiidiiiu; iui'ii is: Also NI = VGcW. (15) From the network of Fig. 7, Tlie armature force developed is -^ = 84- (") When the magnetic pull has reached F\ , the waiting time is over, deter- mining (p« , which in turn determines NIo , and hence q. Substituting (U), (15), (16), and (17) in (13) we have h = 47r J_ 1 (s\l (Ro + ■a/a_ (Gc + G'^) In 1 (18) as the desired expression for the waiting time in terms of the funda- mental constants. Use will be made of this later when an expression for motion time has been developed. Motion Time The motion time immediately follows the waiting time and is the time required for the armature to move from the backstop position Xi , to the location of the last contact xs . From Fig. 1, it is permissible to omit any consideration of contact spring load, and consider the motion to be controlled entirely by the armature mass and magnetic pull developed after the waiting time is over. The determination of motion time is simplified because the initial velocity and net force on the armature are both zero, the latter following from the definition of the end of d . In any problem involving motion, there must be established the initial velocity, position, and a suitable expression for the ensuing force. With IGO THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 these factors, the differential equation of motion can be solved, and the time for a given travel determined. Thus there is now needed an expression for the armature force de- veloped by the pole face gap flux as a function of time. Briefly, the basis for the method to be described is the assumption developed in Part I, that the winding flux continues to rise during the initial motion interval in the same way it would have risen if the armature had not moved. With this assumption, an expression for the motion can easily be derived. With this expression it then can be shown that the armature does spend most of its time in the close vicinity of the backstop, justifying the initial assumption. The present approach differs from the more general one of Part I. It uses the results there de\'eloped regarding the l:)ehavior of mass con- trolled operation, to simplify the initial motion equation and obtain an expression for motion time in a form better suited for the present purpose. Armature Force Rise and the F Concept As an introduction to the method which will be used, consider the general character of flux l)uild-up in a relay. It will have a shape some- what similar to an exponential cur\'e. Now with the armature at the backstop, from the first approximation magnetic network of Fig. 7, a fixed portion of this winding flux will pass through the pole face gap, with the result that the pole face gap flux curve will also have the same general character. Such a measured curve is shown in Fig. 8, as well as the curve when the armature is free to move. Now the armature force developed is proportional to the square of X3 X2 ARMATURE TRAVEL Fig. 7 — Schematics of nomenclature applying to operate time. ESTIAIATIOX AXD COXTKOL OF OPKKATK TIMK OF HKLAYS IGl 5 6000 9-4000 LL 3000 OPERATING, / BLOCKED OPEN ^ ^ ^"^^ X XI y ^ , OPERATir -JG^ / 1 ^^ ■ / Locked open / ^ ^ \ -1 ^ ^ ''^LOPE = F 48 GM PER MS __^ ^ ^J ^ \ \ \ \ \ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 TIME,t, IN MILLISECONDS Fig. 8 — General character of winding flux and armature force rise. the pole face gap flux as given by (17). Hence, the initial force rise will be parabolic, as shown by the lower portion of the curves of Fig. 8. After a long enough time however, the flux will have reached its maxi- mum and the finally developed force will be constant with time. This is shown by the force curves approaching horizontal lines to the right in the same force diagram. As the force curve is continuous, there also must be an inflection point in the curve, and the entire curve has the general shape as shown. The waiting time ^i occurs while the magnetic force builds up to Fx , as indicated, amounting to 2 millisec for the example shown. This time generally includes all of the parabolic part of the curve. Following the waiting time, the initial force l)uild-up necessarily is almost linear be- 162 THE BELL SYSTEM TECHNICAL JOURNAL, JAXl AKY 1954 cause of the inflection of the curve. Two other factors, caused by the ensuing motion, act to provide e\'en more force than the open gap curve indicated by Fig. 8. The first is that a smaller gap takes a larger portion of the winding flux than assumed in the diagram, and the second is that the winding flux rises more rapidl}^ and to a higher value than the open gap curve assumed. The operating force curve shown includes these effects. The relay operates in about 7 millisec and for 6 of the 7 millisec, the two force curves differ very little. Hence, for a relatively long interval after the start of the motion time, a very simple force relationship holds, namely that the force is directly proportional to time, and the proportionality factor is the slope of the straight portion of the curve, designated as F. For the example given, this amounts to about 48 grams per millisecond. By using Xewton's Law, an expression for the motion time can be written. from which, with the initial A'elocity zero 7^3 Ft mill - x) = — . (20) Rearranging, the expression for motion time becomes: The factor F will now be considered. Derivation of Expression for F Experimental — The factor F can be determined graphically. This brings in the second order effects such as quality of iron, cross section, residual magnetism, fit of parts, etc. For this purpose three working curves characteristic of the structure, all at the open gap, are first pre- pared for the particular winding: (a) The dynamic flux rise curve, (b) The static flux curve versus ampere turns, and (c) The static pull curve versus ampere turns. Choosing a time on curve (a), the corresponding instantaneous flux transferred to curve (b), determines the equivalent magnetomotive force. This transferred to curve (c) yields the instantaneous magnetic force. Repeating this for other times in the range of interest, an armature force ESTIMATION AND COXTUt)L OF OPEHATE TIME OF RELAYS 1G3 curve like Fig. 8 is established. The waiting; time is read directly, cor- responding to Fi . The slope of the ensuing linear force range determines Fi . This, with equation (21) completes the determination of mass con- trolled motion time. Of course, the final pull de\-elopcd has to exceed the operated load. This check is made from another i)ull curve taken at tiie armature gap Xz , using the known final ampere turns and the load. Analytical — For oin* present purposes, an analytical relation, expressed in the finidamental constants, similar to that for the waiting time is needed. Its derivation follows: The solution which will be developed is based on linear circuit theory. This necessarily implies exponential flux rise, which is not exactly true. However, the relation is dimensionally correct and accurate to better than first order. Then the use which will be made is to determine the motion time of an electromagnet as the parameters are varied one at a time. These are plotted as ratios to one of them, chosen as a reference. By this means the ratio curves become accurate to better than second order and provide excellent correction factors for actual measured data. For a linear circuit, the pole face gap flux will increase, after the wind- ing circuit is closed to a battery, with the same time constant as the A\inding : e-"^) (22) 47riV7 ., (fa = (1 (Ro + I where T = U {Gc + G'e) and / = E/R. Then the pull: — F— V^Q _ 1 (47riV/)" . _ -tiTs2 SttA' 8tA/ ^ xV^ ^ ^ ' (23) di - ^y./ , xy ^' ^' ' ^^- (24) {(Ro + ly The bracket term is of the form a(l — a), 0 < o < 1, which has a maximum value of 0.25, and from 0.2 jirimc to iiulicato total time, becomes ^^ ^ y^27rn(.T3^ - x,) ^^^^ Tliis CHiuatioii was given a.s equation (1) in the early part of this article. P'or the wire spring relay, this expression has a \'alue of 7.5 millisec. Increasing this by 30 per cent for a maximum value, an estimate of 10 millisec results, agreeing with the earlier lower estimate of maximum time for load controlled operation. Sclcclion of Winding for Mass Controlled Operation One more group of factors needs consideration before a winding is selected. These are (1) the range in dc resistance of the windings, (2) the winding temperature as determined by the duty c,ycle, and (3) the range in the battery voltage. The number of turns of the winding is ordinarily not considered as a variable once it is chosen because of the automatic machine method of winding. An examination of Fig. 6 shows that if the turns are too few, a greater time penalty obtains than if there are too many. Also, decreasing the circuit power, increases the best coil constant. These two considerations indicate that the best coil constant should be chosen under worst circuit conditions. For any other condition the operate time will be reduced. Further, the range between worst circuit and average time will be a minimum. The procedure for choosing a winding is to determine the dc resistance of a maximum resistance winding at the operating temperature set by minimum battery voltage and the maximum duty cycle. This sets the worst circuit power, and by use of Fig. 6, the best number of turns. In no case is a winding specified with fewer turns than will supply sufficient ampere turns to operate the worst relay with the maximum load. In some cases of low power, this sets the number of turns. For some cases of intermediate power, heating requires the maximum winding surface area, also resulting in excess turns. The average resistance with its variation, all at a standardized temperature at 68°F, completes the design. Summary of Single Relay Local Circuit Operation An analytical determination of the operate time of a single relay cannot be obtained in closed form because it requires the solution of two simultaneous, non-linear, non-homogenous, differential equations, without adding the complications of representing the magnetic saturation 1G8 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 and eddy current effects in some convenient form. Approximations for solutions have been developed which predict with good accuracy the order of operate time attainable for a design, but not of sufficient ac- curacy to exactly determine the best winding for a particular case. Thus once a magnetic structure has been established, the operate time data for a single relay necessarily have to be determined empirically by measurements using controlled samples. Actually this procedure is quicker and easier, and besides it gives the correct answer, including all the non-linear effects, such as eddy currents, saturation and motion, no matter how complicated. Complete data for the range of all relays and windings using the given structure can be determined using one such relay with a full winding, through the use of impedance transformation techniques and estimation of small A-ariation effects, using the approximate solutions. By using the approximate solutions only for corrections, the errors become of second or smaller order. Thus single relay operate time data can be determined accurately and presented in a form permitting either analysis or synthesis of per- formance. The design of a best winding for load controlled operation, is accom- plished by a variation of the method of successive approximations. Appropriate battery voltage, winding temperature and resistance of course are considered as part of the solution of the problem. A range of windings is chosen and operate time data established for each winding. If necessary, the range is extended in the appropriate direction until a minimum operate time is included. This minimum determines the best winding. For mass controlled operation, the contact spring load is immaterial, and the data can be presented directly. As part of this type of study, the relative importance of the several parameters affecting the operate time has been determined. These show at a glance whether (1) a change will have a significant effect or (2) what change or changes are neces- sary to effect a necessary reduction in operate time. OPERATE TIME — SERIES RELAYS Series relays are two or more relays whose windings all are connected in series, and energized by the same current, controlled bj^ a single con- tact. The impedance of each one enters into the manner in which the common current will increase, after contact closure. If the procedure used for single relaj^s were followed, there would be a double infinity of combinations to portray, or else experimentally study each combination ESTIMATION AND CONTROL OF t)l'EIlATE TLME OF RELAYS 109 when proposed. This can be avoided, with no loss in generahty, by trans- forming each relay into an equi^^alent single relay in a local cii-cuit. Then the foregoing methods for single relays can be applied to each in lurn. This procedure is the common device of breaking iij) a complicated problem into parts, each of which can be solved by familiar methods. Two assumptions are made. The first is that each winding is a lumped two terminal nc^twork. At any instant the same current is in (>v(ny turn of each relay and there is no propagation time involved. This holds for the times invoh-ed in electromagnets. The second assumption is that, when the relays have different op(M'at(> times, the current reduction caused by the motional impedance of the first one to operate does not significantly extend the operate times of the later ones. In the following transformations, the winding turns, ciu'rents, and hence magnetomotive forces, are kept constant. Identical Relays If the two relays are identical they have some impedance Z(p) which is the same for both. Part of this is the dc resistance and the other part is the ac effect, proportional to the tiu'ns squared. By the extension of Ohm's Law to ac circuits, when a potential source is applied to the two identical devices in series, exactly one-half the source appeal's across either device at any time after application, forming a virtual constant potential point. Thus if a battery of Ea volts is applied, exactly one-half the battery appears across each, including the effect of eddy currents. Now if the voltage across a coil is known, then the response is uniquely determined, knowing just the relay characteristics and the voltage, disregarding the mechanism of how the voltage is applied. For this situation the voltage is in a most convenient form, represented by ex- actly one-half the battery. The operate time can easily be determined for either relay with this information, as the effects of eddy currents and motion are included in the data. The coil constant is already known and the power is one-half the total power. General Case This procedure can be generalized to include any division of dc re- sistance, different turns, and diffei-ent magnetic structures providing, for the latter case, that eddy currents can be ignored. The justification for this will be considered later. The basic problem is: gi\'en the battery voltage E^ , relay No. 1 of .Vi turns, a 1 turn inductance Li , and resistance T^i , in series with relay No. Ni, 170 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Table I — Equation 30 Turns „ ■, „ Relay N Resistance Re Voltage Ee Power W e Coil Constant R1 + R2 E, El A'l' r, , £2 {Nf "* ■+i-;(i;y '^t(m <-+->D + r;(S'] "^"^ '■^'"' 2 of A''2 turns, a 1 turn inductance L-2 and a resistance R2 . What are the two equivalent relays, each in a local circuit and what are their virtual applied voltages? The first step is to reassign the total resistance to obtain two equivalent windings having the same time constant N Li/R. Then determine the two virtual voltages by division of the total in proportion to the impedances. These and their dependent power and coil constant relationships for the case of two relaj^s are tabulated in Table I. The procedure can be extended to any number of dissimilar series structures. For the case of all identical magnetic structures, the 1 turn inductances Li , L2 , etc., are all equal and their ratios become unity, simplifying the expressions. Note that the coil constants are not equal unless the struc- tures are the same magnetically, but that the time constants always are equal. Further, the total power is constant and equal to that of the origi- nal circuit. After determining the effective powers and coil constants, the operate time for each can be read from the applicable single relay charts such as Fig. 6. Tivo Like Parallel Relays in Series with a Third Relay, Identical Structures The equivalent relay method can be extended to include the case of two like parallel relays in series with a third relay, as shown in Fig. 10. The first observation to make is that, because of symmetry, the current flow in the two parallel relays is identical. No winding current change would be made if, for instance, all the dc resistance of the two parallel relays were removed and half of either were connected in series with the single relay. Thus again, the resistances can be assigned as neces- sary to result in equal winding constants and total power. The same net dc resistance gives a second condition : Nl ^ Ni H\E RiE (31) ft + f -ft. + 'l'. ESTIMATION AND C'ONTllOL OF Ol'KK A TK I'lMK OK KKl^AYS J71 Solving RiK = 2Ri + R2 2 + iW (32) /?2K = 2 ( R, + ^' - RlE Because the equivalent coil constants have the same ratio as the e(iui\'a- lent resistances, the voltage division will be: l^iE — RieEo Ri + Ri (33) With these, the equivalent coil constants and powers can be computed and charts such as Fig. 6 used as before. Selection of Optimum Coils, Identical Structures Neither Winding Known, Operate Times Given In the above discussions it was assumed that the windings were known and that the operate time data were sought. The procedure can be re- versed, starting with a desired operate time, or times, and then choosing optimum coils. From time curves such as Fig. 6, the watts corresponding to the desired times can be obtained pro^'ided thej^ are read from the same coil constant vertical because the same current flows through both and hence the effective coil constants necessarily must l)e eciual. E,E ^ E2E — * — Wv — ^M^P — N, R,E ^^WP — VW— R^E N2 — \AA — ^"SM^ — Fig. 10 — Transformation of series-parallel rela}' circuit. 172 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 The particular vertical N'^/R to be used is the one passing through the higher of the two given times, which lies at the minimum point on a watts curve. Satisfying the smaller wattage relay assures sufficient ampere-turns for both winchngs, and also gives a smaller time loss com- pared to optimum for the other higher speed relay, as the higher power curves are flatter for coil constants above the optimum. Thus the speci- fied times determine eciuivalent watts for each coil, and the actual total circuit watts are exactly the sum of the equivalent watts. Hence: Jti\ -f- tti This sets the total resistance {Ri + R'l). Also, knowing the current is the same through the two coils, and that the actual division of the dc resistance has no effect on time, the ratio of the two resistances can initially be chosen the same as the ratio of the two effective powers: 1^^ = ^. (35) W2 R'l Finally, knowing that the effective coil constants for the two relays are equal, the desired turns are determined using the already selected coil constant, Gc • Solving these three equations, the individual relays are: R, = Ei^ , R, = ^ , N, = Vm'c, N, = VR^c (36) W^total ^total The resistance values Ri and R2 can be used as shown above or divided in any other way as long as the total is unchanged. The numbers of turns, however have to be kept fixed. A particularly simple relationship exists when it is desired to have equal times. In this case the turns on the two coils are equal. One Winding Known As is often the case in actual use, a winding must be chosen to be used in series with another known winding and have best operate times. The method described here can be applied to any relay structure, but the numerical values in the analysis are applicable to the wire spring relays only. The first step is to choose the desired resistance for the coil. This is usually set by the heating and power limits, knowing that the higher the total power is, the less the operate time. Then the choice of turns for the coil depends on which of the two coils needs speed the most. ESTIMATION' AXD CONTKOL OF OPERATE TIME OF RELAYS 173 Assume first that we want speed on the coil to be designed, say relay No. 1, rather than the known relay in the circuit, say relay No. 2. Then we want the effective power for this relay as high as possible provided that the coil constant is not too far from o{)timum. From Table I it has already been noted that the two effective coil constants are always equal when the structures are identical. Also the sum of the two effective powers is exactly equal to the total power; that is, as the effecti\'e power to the first relay increases by increasing winding turns, that to th(^ other correspondingly decreases. We see that for relay No. 1, the effective })ower increases as .Vi increases, but also that the effective coil constant increases. Thus an optimum turns value can be found, where on one side the low effective power slows up the operate time, and on the other the high coil constant does. Fig. 11 shows these optimum values. The solid curves for relay No. 1 are plots of time ^'ersus the turns ratio N'2/N'i with total power, E^/{Ri + /?2), and the "series coil constant" of relay No. 2, Ni/iRi + Ri), as parameters. The optimum turns values for relay No. 1 show up clearly on this curve, and are seen to vary with both parameters. This relation of optimum turns to the two parameters is shown on Fig. 12, where optimum N2/N1 values are plotted against total power with the relay No. 2 series coil constant as the parameter. The added series relaj^ always has the most turns when it is designed for least time. Where speed on relay No. 1 is the only concern, the optimum turns can be chosen directly and easily from Fig. 12. Fig. 11, however is of more general use since it actually gives the times and also shows the time values for the second relay (the dotted curves). Thus the turns can be chosen to approach optimum speed on either relay or to choose a com- promise value. Now for relay No. 2 with total power E'/{Ri + Ro), and the second relay series coil constant ^"2/(^1 + ^^2) as parameters, the effective power decreases and the effective G increases when Ni is increased, both increasing the time. In other words, the given relay will always be slowed down by any added series relay winding. The minimum Ai value is limited by sufficient ampere-turns to operate the first relay. As shown by the dotted curves of Fig. 11, which are the time versus N2/N1 curves for relay No. 2, the gain in speed is slight as .Yo/Ai is increased beyond about 2. Thus, although a compromise must always be chosen for speed on relay No. 2, the loss in speed for relay No. 2 is not necessarily great if Ao/Ai can be chosen near 2. For the case of equal operate times, in every case equality of turns of course applies. 174' THE BELL SYSTEM TECHNICAL JOl'RXAL, JANUARY 1954 20 X^ ^1 3 5 7 J 7 10 15 20 30^ / // / - (a) s. / y / 1^ '■/k E^ TOTAL =OWER R, + R2 , ■•* ;y '•^^ / / --^ / / 1 f " -■«.^ *< _10 K - ^s .y y / / "•/ / r^> / ■"■ /. , / / ^^ , ^ ■^>. / / / y •^y ■■"• — _ 15 -~-^_ /' / < ,/ /'•^^ -^^ ^^ - — ^ ^ / ,/- / ^^^^ .20 — — ~ - - ^J ^ -- ^ ~- -^^^_ ■^ 30 N2^XtO-3_ 5 10 3 5 / 7 10 / / 15 / 20 30 f (b) \^ \\ \ /^ / / 1 ? [^ 1 L\ u 7 \^ ^ v "x y / / y y y — . — t-. ._ y > / / / "A-/ -■/ -12. ^;;^». 1^;;;;^^ , ^^ >" / /[^ y -/ / ._ _ .. s^>v. ^^IT j:> < f^ ^ > <^~ — — .!_5_ _20_ ^^ ^^"'"■'^_ ^^— ~*^- ^ < > y^- — ~- "' .^^ -~^^^ ^'""^ "— ^ ~~^__ .30 Nix)0- 5 - - 2 R, + R2 \^ \N^ \ ~ 3 5 7 10 / / 15 > 20 3C / / ) ( c) — . s^^ \ \ \ \ s \ ' \ ^-, •K^ >< / / V ■/) / 7 /.. 3_ -- - -- - \^ s\ ^^ s^ - J^ ^'. ^ y~^j / * ^v, '^^^ ^^__\ i^> K N^J ^ >< /^ / y^ / 1 5 '-^ vT^^O^ V s <>. h- ^ ^ y^-* /> / ■"1 ■^^ «!^^\v^ ^r-^ ^ ^ > =C^/. V r -■ — -— - 7 ^ ■^^ X. > >-c^ ^~ 10 Np^ X 10-3 ~ — ■<' > ^r ""^^^ — — .15. .20_ 30 — — - - — ? = 5 R1 + R2 *** "~~~- — — ..^ RELAY NO.l HAS MOST POWER ■* — LARGE G FOR BOTH 1 1 1 1 1 1 1 RELAY NO. 2 ^->- HAS MOST POWER SMALL G FOR BOTH 1 0.3 0.4 0.5 0.6 0.8 1.0 1.5 2 N2/N, 5 6 7 8 9 10 Fig. 11 — Operate time of ESTIMATION' AM) COXTKOL OF OTEliATK TIME OF KELAVS 17") ^^ $:: ■^^s \ \ \ \ N 'c 3 5 7 10 15 y 20 30 / (e) V <<: §^ x'So ^ ^ 'A t- / 2 — -. — \ \ \ ■ — ^\ n5^ s-> V <^ s ^^ ^^^ ii ^^x^ p- ?^^. / — 3 > •>. \^ ^^ -ii *K > N V, <^ ^ "^^ 5 — ^■*>>. -.... 1 ^ ^ ferre(l to a one turn admit- tance form. 'I'his exhibits the eoi'e eddy current (effect as a, I'c^sistor shunt- ing an ideal inductance, rather than as an inlinite line. Whatever \'alue tile shunting resistor may ha\e, it affects only the transient res])onse of the network, the part with which we are now concerned. In what follows, it will be assumed that the transformation relations have ah'eady been applied, and the (!<■ values applying to Fig. 13 are effective values related to the winding tui'ns in accordance^ with (Hjua- tions (30). For two such networks in series, the voltage division would be in- dependent of time if the ratios of the shunting to series conductances for each structiu-e were eciual. The method de\'eloped for similar struc- tures then would apply with no error. Gpe-^E/G L, = r, = R f NI 1 L, Fig. 13 — Equivalent linear circuit represented by time equation. The ratio is: ^-OeIOc ^ ^g- This function has a maximum value of 1/c; that is, the shunting re- sistor is at least e times larger than the series resistor. An analysis of the range of core conductances and speed windings in use, shows that the actual resistance ratios are somewhat greater than this and hence the ratios are not quite independent of the structure. However, because the maximum is broad, it reduces the actual range to about 10 per cent, including all windings plus a 2 to 1 Ge change. In turn, this signifies that for a suddenly applied voltage, the initial linear network voltage division would differ from the final by less than 10 per cent. This error decreases with time. The above discussion applies to linear networks such as Fig. 13. In an actual magnetic structure the initial voltage division is not affected by eddy currents as they have not had time to })uild up. The voltage 178 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 division starts and ends exactly in the ratio of the effective series re- sistances by virtue of the method used in determining them. For times after circuit closure, a transient voltage error does develop, but it will not approach in magnitude the initial transient error of the linear net- work. The operate time error will be still smaller. The operate time varies as the two-thirds power of the voltage (power varies as the volt- age squared). These considerations lead to the conclusion that ignoring the eddy currents in setting up the equivalent relays, results at the most in errors of operate time estimates for series connected relays of the order of a few per cent. RELEASE TIME The relase time of a relay is not as directly affected by the winding. as is the operate time. For instance, if it is opened by the controlling contact without an RC network, the winding current, ignoring arcing at the contacts, abruptly drops to zero. The winding subsequently plaj^s no part in the flux decay. If the winding has a shunting resistor or RC circuit, then winding current does flow and some effect is present. A resistor always increases the release time. A favorable RC choice can cause a slight decrease in release time. Because of this minor effect, the winding almost always is designed from operate time or sensitivity considerations. Differing slightly from operate time, the release time is divided into two parts, (1) waiting time plus motion time vnitil actuation of the nearest contact and (2) stagger time. For simplicity the combination (1) above is merely called release waiting time. The release time of a relay is a more complicated function than is the operate time. The primary cause of this is the closed gap situation, with little stabilizing effect from an air gap. For the earlier operate time studies the air gap largely contributed to the simple exponential relation- ships. A second effect in release is that the magnetic material is almost always in the non-linear saturated portion of its characteristic. Hence an approximately linear relationship between steady state flux and current cannot be assumed. Finally, for release without an RC network, the only current flows in the core, so it completely controls the flux decay. Even with an RC network, only a minor decrease in release time is possible. For these reasons, except for rough preliminary estimates of release times during preliminary design, all release data are based on measurements. ESTIMATION' AXD COXTKOL OF OPERATE TIME OF RELAYS 179 Conductance Shunt In a companion article, a hyperbolic relationship between the flux and current is used to represent the portion of the hysteresis looj) of concern in release. This, with a conducting sleeve or shunted winding, gives an excellent representation of the release waiting time. It also provides an understanding of the controlling parameters, even with only eddy cui'rents controlling the release. 'I'he form most useful for release consideration is: ^ ^ (.p" - ipo){Gs + Gc + G,) / \nz _ 1\ ^ ^3^^ NIo V^ — 1 2/ where

\ \\ 10 \ \ \, \ V^ 0 \, \ \ \ \ \ V \\ ALUMINUM \ \\ SLEEVE \\V — 0046" f\\\ - \ v \, \ \\\ \ \ \ \Yv \ \ \ \\ \ S, \ \\* \ \, \\\ \ \ \\ 1 s 1 V 1 10 20 30 40 50 60 80 100 200 300 4C RELEASE NI IN AMPERE TURNS Fig. 14 — Release time with a shunted winding. ESTIMATION AND CONTROL OF OPERATE TIME OF RELAYS 181 cross-section. The residual flux is directly proportional to the coercive force times the length of the magnetic material, and inversely propo- lional to the closed gap reluctance. The two largest factors affecting the closed gap reluctance are magnetic permeability, and fit of the joints, including variations in stop pin height. By measuring these factors on the test relay and estimating the corresponding values for the desired reference condition, the measured data can be corrected to the reference ('(indition. Then having the release pull curves after magnetic soak for the same reference condition, the release ampere turns for any load under consideration and hence its release waiting time can be deter- mined. These release pull curves are also shown as part of Fig. 14 for the wire spring relay. The ordinate scale is marked for both contact spring load in grams and releasing time in milliseconds. The particular chart shown is for a constant initial number of ampere turns. For other initial values, a correction chart is provided. If not as desired, the time can be adjusted either upward or downward by changing to a different sleeve, shunting resistor, or both. Of course, in no shunting conductance case can the release time be less than the open circuit time. BC Shunt The above considerations all related to shunting conductances. These serve the purpose of increasing the release time. The shunting resistor also greatly reduces the transient peak voltage developed when the winding circuit contact is opened. For contact protection reasons, RC networks are frequently used across the winding or contact for this same purpose. Such a network also has no power drain when the relay winding is energized. By a suitable choice of capacitance the network can reduce the release time to a value less than the open circuit time. It does this by developing a hea\'ily damped oscillation of winding cur- rent. The frequency must be of the order of the reciprocal of the un- protected release time for a release time reduction. This fixes the choice of capacitance to a small range i.e., there is a best capacitor, one for each winding. Smaller values cause the time to increase toward the unshunted case. Larger values also cause an increase but for this case, if too large, an increased time beyond tlic unshunted case can result. For speed windings, where timing is important, a network is designed for each. For the slower higher resistance windings, because time is not as important, the closest to the best of the available networks is chosen. This results in a time penalty but furthers standardization. To evaluate a network, the transformations developed and shown in 182 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 20 LU 6 1- 5 CRt = 100 /^F -OHMS AF RELAY SOAK = 300 AMPERE-TURNS ^ r 0.006" STOP DISK LILaaaJ 0.006" TRAVEL TO FIRST CONTACT Jj^ p oN K ^ 1 ' ! '■' FOR LAST CONTACT: SHORT TRAVEL -ADD 1.0 MS INT TRAVEL - ADD 1.5 MS V, . ^ Rj = R + Rp ^^ ■■^- RELEASE NI = 20 1 ^„.,^ ^ ^^ O-" .--' > ■ " ^^,^ -"^ u-^ ^^-'"^ ^^^ .^^^ -— ^ „..--- • — \,^ -'^ ^^ ^^'"'^ ^. ' _ - ao^ -- "^ ^ -^^^ ^ ^ ''^ - 60— ' ^ ^ ^ ^-^ ^ ^ ftO_^ ^^ ^^^ ^ — "^ - -■j^ ^ ^^ ^ -^ 2 1.5 1 1.5 2 3 4 5 6 7 8 9 10 15 20 30 40 50 60 80 IOC N^C X I0"6 (tURN2 X MICROFARADS X 10"^) Fig. 15 — ■ Release time as a function of N^C Fig. 5 are used. This permits exact simulation for anj^ winding and net- work from a time standpoint. For presentation of data, an arbitrary separated product form for the more important variables is used. The particular factors chosen are: (1) A^'^C, a measure of frequency, with A^/o (release ni) as a param- eter, (2) /2C, a damping term, with A^/o as a parameter. (3) Soak A^/ with A^ /jR as a parameter. Essentially what is assumed is that the release time can be represented well enough as a function of the form: h = MN'C) X MRC) X MNI). (38) Then by holding two factors fixed and varying the third, each factor in turn is evaluated. Generally the accuracy is better than 10 per cent even with this elementary approach. For a few cases the error approaches 15 per cent. Typical charts for these three factors are shown in Figs. 15, 16 and 17. Release Time for Series Relays and an RC Shunt With two series relaj^ windings it is usual to provide a single RC net- work, as shown in Fig. 18. This results in a double infinity of possible combinations if handled directly. However, it can be broken into two ESTIMATION AXD CONTROL OF OPKUATK TIME OF RELAYS 1.25 183 / r / RELEASE NI = 20/ / / / / / y '40 / ^ y 60 ^° ^ ^ \ \ s. N \ \ \, SK)0 k Sl20 \ 1 1 S 1 \ 40 50 60 80 100 CRt 200 300 400 600 800 1000 IN MICROFARAD-OHMS Fig. 16 — Release time correction for RC damping. 1.15 1.10 1.05 1.00 z ^ o Q Z 0.95 I— ^ ^ S 0.90 O li- 0.65 IN KILOMHOS 200 ^<" ^ '''2 100 \ ■ ^=-^ ^ ^ ;;;;;ml :::l^ 5 _-^£— 1 20__ 10^ ;3;;::: ^^^^^^ ^ ^^;~^ ^ 10 2^ ^ ^^ ^^ "^ ::<^ >«s^ """20^ >" \ 50^ 100^ 200^ 160 180 200 Fig. W 250 300 350 400 500 60 SOAK IN AMPERE TURNS Release time correction for magnetic soak. 184 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 parts, each of which exactly represents, as individual relays, the two series relays. This is done by first assigning the total of the winding resistances for equal A'' IR values. The two equivalent relays then form a voltage divider, independent of frequency. The RC network next is drawn as two series RC networks as shown in Fig. 19. The procedure now is to determine the component RC values to have the identical voltage divider effect as the relays. Then the two equal voltage points can be connected as shown by the dotted line and no circulating current will flow. The resultant is a three node circuit and each branch can be considered independently, using the method of the preceding section. There are four unknowns and hence four equations are needed. They are Rvl + ^22 = R, C1C2 N,' Ni Rn R 22 (39) Ci + C2 , = C, Ni'Ci = #2 C2 The solutions for the four unknowns are shown on Fig. 19. For experimental measurements, each is transformed again by means of Fig. 5. Note that the time constants all are equal, as they must be, because the same current flows through all elements. However, the initial ampere turns A^7 are different by virtue of Nit^N^ . Thus only one circuit like Fig. 5 need be set up. Measurements then are made with two dif- ferent voltages to provide the two different ampere turns and the two release times. By choosing the subscript 1 network to represent the actual circuit, then no subscript confusion results in arriving at the simulating circuit of Fig. 5. Release Time jor Similar Parallel Relays and an RC Shunt Similar parallel relays are split as shown in Fig. 20. Two equal parallel RC networks are first drawn. One is then assigned to each, and then the pairs are divided. The release times are each equal to that of one of the separated circuits. Summary The release time of fast electromagnets is influenced much more than the operate time by the fit of the magnetic parts. For release, the small non-magnetic stop disc introduces a relatively small stabilizing air gap compared to the open gap of the operate case. Secondly, the release always starts after an applied magnetomotive force which differs as ESTIMATION" AND ('()\T!{()T. OF OI'KH A'I'K TIMK OF HKLAYS ISf) between circuits, whereas operation invariably follows an opcMi circuit. Thirdly, with RC protection networks coiniccted, a more complicated wiiuling current is present. Finally, the transmission line beha\-ior of the magnetic core material is less mask(Ml by the winding, and in fact controls completely for tiie o]W]\ cii-cuit case. For the.se reasons, the r I ^ Ri,N, Rp,N 2, '"^2 R -v\v a Fig. 18 — Series relays with one RC network. R,E,N, Rpp.N, ^ ^ ?,' f^" ! ^22 ^2 H( VA » VW \{- R,. = Rp,= R C, = C 1 + Cj = C 1 + Fig. 19 — Transformation to series RC networks. R,,Nj R,,N, R AAA- R,,N, C/2 2R ■A^v c/2 2R AAAr HHww c/2 2R Fig. 20 — Transformation to jjarallel RC networks. 186 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 analytical presentation, as noted in Part I, of fast release time data is not now as advanced as the operate time data. The material presented here describes the present state of the art. For slow releasing relays, the performance can be predicted with ac- curacy. For fast relays, while the general pattern is known, accurate means for estimating variations have not been developed. The present engineering of releasing relay circuits therefore, depends upon specific measured data in chart form, for each condition. ACKNOWLEDGMENTS The analyses leading to the forms of data presentation for the several types of relay timing information are the results of contributions from many people. In particular, the early nomograms for load controlled operation were developed by P. W. Swenson. The node method for separation of series releasing relays shunted by an RC network was developed by R. H. Gumley. The graphical method for design of opti- mum series windings was developed by Mrs. K. R. Randall. To her, I am also indebted for preparation of all the charts of measurements. REFERENCES 1. H. N. Wagar, Economics of Telephone Relay Applications, page 218 of this issue. 2. M. A. Logan, Dynamic Measurements on Electromagnetic Devices, B.S.T.J., 31, pp. 1413-1466, Nov., 1953. 3. R. L. Peek, Principles of Slow Release Relay Design, page 187 of this issue. 4. R. L. Peek and H. N. Wagar, Magnetic Design of Relays, page 23 of this issue. Principles of Slow Release Relay Design By R. L. PEEK, Jr. (Manuscript received Scpteinher 25, 1953) This article presents an analytical treatment of the relations controlling the release time of slow release relays, in which field decay is delayed by the currents induced in a conducting sleeve or slug. An hyperbolic relation between flux and magnetomotive force, fitting the decreasing magnetization curve, is used for the relation between induced voltage and current in the sleeve circuit in determining the rate of field decay. Methods arc given for estimating and measuring the magnetization constants and those appearing in the relation between pull and field flux. These relations are used as a basis for a disciission of the design of slow release relays and of the adjustment procedures employed to meet timing requirements. 1 INTRODUCTION Slow release relays are built and adjusted to provide a time delay be- tween the opening of the coil circuit and the release motion which restores the contacts to their unoperated condition. They are used to assure a desired sequence of circuit operation, as, for example, in maintaining a closed path through the slow release relay's contacts during the pulses sent in dialing a digit, and opening this path during the much longer interval between digits. Slow release relays constitute a minor but sig- nificant part of the relay population in an automatic central office: about ten per cent of the total. For economy in manufacture, installation, and use, slow release relays are made as similar to the ordinary or general purpose relays with which they are used as their special reciuirements permit. Each general struc- tural relay developed for telephone work has had a variant form for slow release use. Thus the Y type^ relay is the slow release form of the U type relay^ widely used in the Bell System, while the AG relay is the slow release variant of the recently developed wire spring relay. The circuit functions of most slow release relays permit considerable \'ariation in release time if the minimum delay specified is assured. This 187 188 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 tolerance in the timing requirements permits the use of more economical practices in the construction and use of slow release relays than would be needed for closer control. As these wide tolerances apply to the great majority of applications, over-all economy is attained by developing slow release relays with reference to them, using other devices for the special applications requiring close timing control. The magnitude of time delay desired is fixed by the operate and release times of the associated general purpose relays. The latter times lie in the range from 5 to 50 milliseconds, with the majority in the lower part of the range. The delays rec^uired to assure sec^uences of events each reciuir- ing time intervals of this order conseciuently cover the range from 50 to 500 milliseconds (one twentieth to one half a second). Delays of less than 100 milliseconds can generally be provided by ordinary general purpose relays having shorted secondary windings or sleeves (single turn conductors) to delay their release. Special slow release relays are used for delays in excess of 100 milliseconds. Factors Controlling Release Delay In terms of circuit operation, release time is the interval from the opening of the coil circuit to the completion of contact actuation during the return motion of the armature. This time is the sum of (a) the time for the magnetic field to decay to the level at which the pull just equals the operated spring load and (b) the motion time for contact actuation. The motion time is never more than a few milliseconds, and is therefore a trivial increment to the long delays of slow release relays. The release time of such relays is therefore, for practical purposes, simply the time of field decay. When the coil circuit is opened, the field decay induces currents in any circuit linking the field. These currents tend to maintain the field and to delay its decay. A short circuited winding or the single high con- ductivity turn provided by a copper sleeve gives a high magneto- motive force for a low induced voltage, resulting in a relatively slow decline in field strength. The time for the flux to reach a given level de- pends upon the conductance of the sleeve, and upon the reluctance of the electromagnet: the ratio of magnetomotive force to flux. The flux level which determines the end of the delay is that for which the pull equals the spring load. The delay is therefore prolonged by a pull char- acteristic such that the load is held until the flux drops to a minor frac- tion of its initial value. Thus the essential features of a slow release relay are a shorted winding SLOW RELEASE RELAY DESIGN 189 or sleeve, a low reluctance magnetic circuit, and a high level of pull for relati^'ely low values of field strength. In the following analysis of slow release performance, expressions are developed for the time of field decay for the decreasing magnetization characteristic, for the reluctance of the electromagnet, and for its pull characteristics. These expressions permit the estimation of release times, indicate the design conditions to be satisfied to attain a desired level of delay, and indicate the effect on the release time of variations in the spring load, in the dimensions of the electromagnet, and in other design parameters. The notation used in this article conforms to the list that is gix'en on page 257. 2 FIELD DECAY RELATIONS If a closed circuit of resistance R and .V turns links a magnetic field of flux (p, the voltage equation is : at where i is the circuit current. IMuItiplying bj^ 4x.¥ and dividing by R this equation may be written as: ^i + ^irGi ^ = 0, at where ^i is the magnetomotive force of this circuit, and d is its value of N~/R, which may be termed the equivalent single turn conductance. If there are several such circuits linking the same magnetic field, a similar expression applies to each, and these may be added to give the equation : g: + 47r(? ^f = 0, (1) at where ;J = ^ 5; , and G = ^Gi . In the case of a slow release relay, one linking circuit is usually a slee\'e, whose conductance may be designated Gs . The applications are identical for a short circuited winding, if the applicable value of N^/R is substituted for Gs . In either case, G also includes a term Ge representing the net effect of the eddy current paths. As the eddy currents at different distances from the center of the core link different fractions of the total field, this representation by a single term is an approximation. As shown in a companion article, however, the approximation is satisfactory when Ge is a minor part of G, as in the 190 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 slow release case. In this same article it is shown that, under these condi- tions, the relation between the changing values of '3- and (p in (1) is sub- stantially identical with that given by static magnetization measure- ments. The static magnetization relation between ff and (p may therefore be substituted in (1), which may then be integrated to give the relation between (p and /. For linear magnetization, or constant reluctance (R = 3/(p, integration of (1) gives the equation. ^ _ ^-«/«o where ^i is the value of

02 b 0 1 1 1 ^ — 10 15 20 NI IN ABAMPERE TURNS 0.1 0.2 0.3 0.4 0.5 i/nI in ABAMPERE-TURNS"' (a) (b) Fig. 2 — Graphical determination of release time. explicit a statement of the general relations applying as does the approxi- mate analytical treatment outlined below. Hyperbolic Approximation to Decreasing Magnetization Curve The decreasing magnetization curve, as illustrated in Fig. 1, has the general character of a rectangular hyperbola, and may therefore be represented approximately by the empirical ec^uation :

K / / / / / / / •X, X / / / / 1 . / / ^-.. 1 1 1 1 1 ^^ ^ y 1 1 1 1.5 2 3 4 5 6 8 10 15 20 30 40 50 Z Fig. 5 — Factors for computing release time. different values of G. For each sleeve size two curves are shown, cor- responding to the limits of variation in magnetic characteristics, or in (Ri and (p" — ^o • The dotted curve included for comparison is the rela- tion of Fig. 6. An important property of the t versus CF relation can be demonstrated by substituting in (6) the expression for (Ri given by (8). There is thus obtained the equation: t = 4:TrG{

^^^ ^**v ^•n^ ^MINIMUM w^^ ^^^ ^fc-'^N. ^^ ^^^ ""S. ^s. • s. ^ ^^ -^ ■v^ ^s. ^ • \ X v*\ W >v* • S. 1 / " ^v Yz COPPER \ K^ sleeve: y^ X MAXIMUM^'^ ^' V \. ^J ^ MINIMUM-' N S. ^V ^ *\. >V^ N. ^\*\. >^ V \.^ V \ \ v\ N, \ x«\ \ \ v\ \ \ \^ - \ <^ - 1 1 \ 8 10 15 20 AMPERE TURNS TO RELEASE 40 50 60 Fig. 7 — Observed release time characteristics. SLOW RELEASE RELAY DESIGN 197 release time is relatively independent of magnetic variations, jirovided it is adjusted to release at a specified ampere turn value. The range in which / is inversely proportional to ;T is thai in which the logarithmic plot of Fig. G has a slope near unity. The point of tangency with a line of unit slope coincides with the value of z for which the l)racketed function of z in (9) is a maximum. By equating the derivative of this to zero, it is found that this maximum occurs for z = 3.09, cor- responding, from (8), to a value of ().4() for \^/{(S{i{(p" — corrcspoiKls to tlH> slopo of the (Icniagnetization cuinc at //,■, as <2;i\(Mi by the initial pcrmoahility m"- Thus the total initial i-cluctancc (U" is ^\vv\\ by the o(iiiation: (ft (ft.; + 1 a (12) \\ hciv X^r a denotes the sum of terms corresponclinj"; to the component parts of the iron path, each term giving the length of the part divided by its cross sectional area. As m" is of the order of 20,000 for soft magnetic materials, the iron reluctance term is small (•omi)ar(Ml with (Hk , which is typically of the order of 0.010 cm~ or less. The value of <^" may be estimated as the saturation fiux for the core, or B"a, where a is the cross sectional area of the core. B" is the satiu'a- tion density, which may be taken as the value of B^f listed in a companion article.^ Estimates of ^" thus obtained are lower than those which best lit the decreasing magnetization measurements. No convenient means for correcting this disparity has been found, other than an arbitrary increase by a factor of 20 per cent, based upon experience. In a particular relay design, however, the oV)served values of (p" vary directly with the core cross section. PULL RELATIONS As sho\A'n by equation (6), the release time varies inversely as the incre- mental reluctance (fti . This is proportional to 6{", in which the dominant - 0.020 5 + 13- 0.010 (R' 0.005 FLUX, 50, IN MAXWELLS 3000 4000 6000 1 / / / / / / / ^ / / / // /^ ^ ^ ^ 1/ / / ^ ■^ ^^^'' '"'■''' U ^ F /^ n -^ -^ ^^ ,^ V // VA g^ ':^ :::^ ^ m ^ >* ^^ 10,000 - 15,000 J + 'Jr IN ABAMPERE TURNS ATT I"ig. 8 — Evjilualiuii uf magnetic constants from measurements of decreasing magnetization. 200 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 term, from (12), is the total air gap reluctance Oi^ . To obtain long delays, therefore, (Re must be made small, and consequently sensitive to small dimensional variations. These may be compensated in an initial adjust- ment, but subsequent changes must be minimized if constancy of per- formance is to be attained. Two expedients have been used to provide a small and stable gap reluctance. The older one is the use of a "residual screw," an adjustable non-magnetic member which serves as an armature stop and assures a small air gap at the pole face. With this scheme, the residual screw is used to adjust the relay. The alternative scheme, used in the flat type relays of the Bell System, is to employ a domed pole face on the arma- ture, providing a spherical surface in contact with a mating plane surface on the core. The only effective air gap at the point of contact is that of the chrome finish on the parts. With this scheme, the relay is adjusted by varying the spring tension, and thus the operated load. General expressions for the pull of electromagnets are discussed in a companion article,^ where it is shown that the pull F provided by a gap flux (p is given by the equation: F=f^-p, (13) Htt ax where x is the dimension in the direction of the pull, and (Rg is the gap reluctance. In the usual case, (Rg varies linearly with x, and dGia/dx has the constant value 1/A, where A is the effective pole face area. This is applicable to any case of plane mating surfaces having an appreciable separation, including the configuration usually employed with a residual screw as separator. Pull for a Domed Pole Face In the case of a domed pole face there is a concentration of the field near the point of contact, which varies with the effective air gap at this point. An expression for the reluctance can be developed for the idealized configuration shown in Fig. 9. In this, R is the radius of a spherical surface mating with a plane over the projected area A bounded by the radius aR. The separation x is that measured at the center of .4. As indicated in the figure, an expression can be obtained for the gap re- luctance (Rg in terms of its reciprocal, or permeance. The latter is given by the integration of the permeances of the differential rings within the projected area. (Rg is conveniently expressed in terms of the ratio (R^/(Rg .SLOW KKLEASE RELAY DESIGN 201 given by the equation: ; = 2'"«"'"0 + 2i)' (14) in which 6\^ = x/A, the vahie of 6\o which would apply it' the .spherical radiu.s were infinite, and both mating area.s plane. On .substituting this expression for (Rg in (13), there is obtained the following e(]uation for the pull at a domed pole face: F = StA mo (R 27r/?(R„ 1 + 2ir72«, (15) The right hand side of (14), and the expression given by (15) for the ratio of F to as shown by (14). Alternatively, 12 may be taken as a function of the ratios (Rf/^(R« and 7?(3lp , and 9. may be represented, as in Fig. 11, by a family of curves gi^^ing fi versus (Rf'/(R„ for various values of /?(R/? . From (17), maximum ampere turn sensitivity is attained by mini- mizing the separation x and the reluctance (R^ external to the gap. Assuming these to be made as small as engineering considerations per- mit, the pull for a given value of A^/ varies as 12. With x and (51^ fixed, 12 now depends only on the dome radius R, to which E(^f is now pro- portional, and on the projected area A, to which (Rf/(R«, is now propor- tional. From the curves of Fig. 11 it is apparent that maximum ampere turn sensitivity is attained by using as large a value of dome radius as possible. For a given dome radius, there is an optimum value of (Rf/(R« , SLOW RELEASE RELAY DESIGN 205 and hence an optimum value of A, corresponding to llic nuixinnun sliown by each curve in Fig. 11. AG Relay Pull 3 The AG relay is a slow release relay \\\t\\ a domed pole face, to which the relations given above apply apiiroximatoly. M. A. Logan and (). C. "Worley developed a more exact expression for the pull in this case, in which an increment to the pull is given by the secondary pole faces on the side legs of the armature, which mate with the side legs of the E .shaped core member. A further increment to the pull is given by the area at the main gap which lies outside the dome proper. Their analysis may be summarized in the notation used here by writing: (Ri - (Rz = (R. + ^^"^'^ (Rg + (Ri where Ri is the iron reluctance, (R^ is the side gap reluctance, and the main gap reluctance is that of the domed gap (Rg in parallel with that of the remaining area, (Ep . Substituting /?/ — fR/ for cRr; in (10), there is ol)tained the following expression for the pull: ^ ^ 2Tr{NI + {NDcY /(Ms ( (Ro \' (^(Rp (Rf \ dx \(Ro + (Rp/ dx , , (Rp \ d(S{G (Ro + ^p) dx (18) An expression for diSia/dx is given in Fig. 9, while dOis/dx and d()]r/dx are given by 1/As and l/^4p , where As and Ap are the effecti^'e pole face areas for these two gaps. For the dimensions applying to the AG relay, these additional terms introduce minor but significant corrections to the values of F computed from (17). Engineering Pull Data In determining the requirements for slow release relays, the estima- tion of the range of variation in pull is a major problem. A procedure for such estimation developed by M. A. Logan makes effective use of the relation between F and (NI + (NI)c) in (Ifi). This relation gives a linear plot of slope two when F is plotted against (AU -f (NI)c) on logarithmic paper. A basic plot of this nature may be experimentally determined for a model having nominal values of coercive force He , to Avhich (.V/)c is proportional, and of finish thickness and side gap scpara- 206 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 tion. The effect of known changes in these two latter factors may be determined by the observed vertical shift in this logarithmic plot, corresponding to changes in 6li and dGia/dx. The effect of a change in He is a change in the value of (A^7)c to be subtracted from (NI + (NI)c) in determining AU. From these results there could be prepared curves giving the relation between F and NI for different combinations of the several variables for which allowance should be made. 5 EVALUATION OF CONDUCTANCE The preceding sections have described procedures for estimating and measuring all the terms entering the expressions for the release time except the conductance G. This ciuantity is the sum of the sleeve con- ductance Gs and the eddy current conductance Ge ■ In some cases a short circuited winding may be used instead of a sleeve : in such cases G is the sum of Ge and the coil constant of the winding, Gc or A^ /R. Sleeve Conductance The conductance of a cylindrical sleeve is the sum of the conductances of the differential shells of length f, radius r and thickness 5r, as indi- .^^'^-1 Sleeve conductance relations. SLO\\" HKLKASK ItELAY DESIGN 207 catod in V\ix.. 12(a). 'I'luis tlic total conductance Gs is given by: {dr JTTpr wlicrc p is the ivsistixity of the nialerial. On integration, there is ob- tained: Gs= -^ In -^ . (19) The vahie of p for copper is 1.73 X 10 ' ohm-cm. If r-y is twice Vi , for example, and f = o cm, the value of Gs given by (19) for a copper sleeve is 320,000 mhos. In the case of a sleeve of rectangular section, as shown in Fig. 12(b), an approximation may be obtained by taking the sleeve as made up of sliells with straight sides parallel to the center hole, connected by quarter circles. Then the perimeter of the shell at a distance r from the center hole is 2(6 + f/ + xr). The total sleeve conductance is therefore given l)v: Gs = r Jo 2(6 + d + 7rr)p in which /-o is the wall thickness. On integration, there is obtained the equation : G, = JL i„ ('i+r'L+in) . (20) 27rp \ 0 -\- d I This expression is identical with that for the cylindrical sleeve, as given by (19), when the ratio ro/ri of the radii is ec^ual to the ratio of (6 + rf + xa) to (6 + d). Coil Conductance For a cylindrical coil, the number of turns N is determined by the area of the coil section cut by a plane through the axis, or {(r-i — ri), where ( is the length of the coil and /'i and r-j, are its inner and outer radii. If a is the cross sectional area of the wire, and e the fraction of the coil space occupied b}' the conductor, Na = e({r.2 - n) . The mean length of turn is x(r2 + n), and the total length of conductor is .V times this. Hence the resistance R is given by: 208 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 The expression for N/R given by these equations gives the following expression for the coil constant N'/R, or Gc : Gc = et Ti — n (21) For unit length of coil, values of Gc are shown in Fig. 13, plotted against /■2/ri for various values of e, together with the corresponding rela- tion for Gs , as given by (19). This shows that the slee^•e provides the maximum value of conductance for the space occupied. It also shows that the space used for a given value of sleeve conductance reduces the value of Gc that can be provided. When the full depth of the winding space is used by both coil and sleeve, called a slug in this case, each oc- cupying part of the length, the value of Gc attainable is reduced in pro- portion to the length used by the slug. When the coil is outside the sleeve, the outer radius of the sleeve is the inner radius of the coil, and the value of Gc is reduced in proportion to the value of Gs . For a given value of e (given wdre size and insulation) the value of Gc is fixed by the winding space and the value of Gs ■ / SOLID SLEEVE^ / ^ / ^. 9^ y A / ^ y ^ / ^ y ^ 0^ - /a y ^ ^ ^ - A p ^^ z>^^ 2 2.5 3 Fig. 13 — Relation between coil constant and coil dimensions. SLOW RELEASE RELAY DESIGN 209 o o OJ 0.10 I H UJ 0.08 tu -I 0.06 LU 0.04 5 y RELAY RELEASE 400 GM LOAD, 0 GAP AFTER SOAK, NI = 350 AMP TURNS / / > / y / A / ^ f y / [« -Ge--* ,/ / y r y / Fig. 14 relaj'. 30 20 10 0 10 20 30 40 50 60 70 80 90 100 COIL CONSTANT, Gc, IN KILOMHOS - E.xpcrimpntal evaluation of ofldj- rurront conductance for release of Edd]) Current Conductance In one of the companion articles it is shown that when the eddy cur- rent conckictance Ge is a minor term in G, as with slow release relays, it is given by the equation: G^E = ^ — ' oirp (22) where p is the resistivity of the material, and ( is the length of the mag- netic path. For iron, p = 11 X 10~ ohm-cm, so for ( = b cm, the value of Ge given by (21) is 17 X 10^ mhos. The equation applies to a path of uniform circular cross section, so that the effective value of t for most relay structures is intermediate between that of the core and that for the complete path. The eddy current conductance of a specific model may be experi- mentally determined by measuring the release time with a winding shorted through an external resistance. A series of measurements are made in which this resistance is varied, while the initial current, which determines the "soak NI" value, is kept constant. The different values of the resistance correspond to different values of the coil constant Gc or N /R. From (6), the time t in this series of measurements varies directly as G, or Gc -\- Ge ■ Then a plot of t versus Gc , as illustrated in Fig. 14, is linear, and has a negative intercept giving the value of Ge . 210 THE BELL SYSTEM TECHNICAL JOURXAL, JANUARY 1954 Values of Ge thus determined agree with those estimated from (22) within the level of uncertaint}^ as to the applicable value of ( in this equation. G DESIGN OF SLOW" RELEASE RELAYS The following discussion is confined to the bearing on design decisions of the performance relations developed in the preceding sections, and does not cover the manufacturing considerations involved. The develop- ment of a specific design depends on the initial choice between certain alternative features which require description. Design Alternatives The features considered here are: (1) the adjustment means, (2) the form of sleeve, (3) the criterion of adjustment. The two methods of adjustment that have been used are (a) residual screw adjustment, (b) spring load adjustment. The former affects the release time by changing the reluctance and the residual flux, the latter by changing the flux or ampere turn value at which release occurs. The differences in the two methods relate more to the stability of the adjust- ment than to its ease or initial accurac}^ Spring adjustment permits the use of the domed pole face, which is inherently stable except as the finish thickness may be affected by wear. The two forms of sleeve are the interior sleeve, which uses the full length of the winding space, but only part of the depth, and the slug, which uses the full depth, but only part of the length. As shown in Sec- tion 5, the values of Gs and Gc , or coil constant N^R, attainable with a given winding space are independent of which arrangement is used. The slug provides some flexibility as to operate time, on which its retarding effect is a maximum when it is near the gap, and a minimum when it is away from it. It is subject to smaller temperature changes, with conse- quent changes in conductance, than the sleeve. The other differences between the two arrangements relate to manufacture, and to the costs of both coil and sleeve. The interior sleeve is used with the flat type relays of the Bell System. The two different criteria of adjustment used are the release current, and the release time. The former is an indirect control, using a measure- ment of release ampere turns to determine the time that will result for the sleeve conductance used: the other is a direct measurement of the quantity to be controlled. In principle, the latter method would appear preferable, but its use is attended with several disadvantages. SLOW hi:lkask rklay design 211 The most important of these is the uncertainty as to the sleeve tem- perntiiro that applies to any measurem(Mit made in central office main- tenance, unless the relay to be tested is cut out of scrNicc tor an hour or more before measurement. The conduct i\il\- of cojjixm- \aries approxi- mately as its absolute temperature, or, for engineei-ing estimates, as o5)0 + Tv , where 7V is the temperature in degrees Fahrenheit. Coil temperatures of 225°F are permitted in normal relay operation. As the time varies as the sleeve conductance, a relay with its sleeve at this temperature would have a release time in the ratio 470/615 or 76/100 to the rated release time for 80°F. Allowance for \'ai'iation in this I'aiige is made in circuit design, but a corresponding uncertainty as to the condi- tion applying in adjustment would effectively double this variation. Cun-ent flow adjustment is free of this difficulty, and the variations in the correlation of release time with release ampere turns are less than tliose resulting from the temperature uncertainty in any convenient pro- cedure for timing measurements. Current flow adjustment has the further advantage of using eciuipment that is employed for other relays in central office maintenance. It is the more commonly used criterion of adjust- ment for Bell System relays. Operate Considerations A slow release relay must not only provide the desired release per- formance: it must also operate its load. The operate pull characteristics are similar to those of other relaj^s of the same general type, as the domed pole face, in particular, gives nearly the same pull at an open gap as a plane pole face of the same total area. Thus the pull characteristics of the AG relay are similar to those of the AJ relay for the same travel. The sleeve retards the flux development, and makes operation slower than that for the same coil input without the sleeve. In most applications of slow release relays, this has little or no effect on circuit operation. Faster operation can be obtained by increasing the steady state power applied, but this is limited by heating considerations. The large part of the winding space used for the sleeve limits the operate sensitivity of slow release relays, and increases the power required for a given load. The load for a given relay design determines a minimum ampere turn value for operation, and this is related to the steady state power by the iden- tity: {Xry- = T-R N-/R. As the coil constant N'-/R, or Gc , is determined by the available winding space aN^ailable foi- the coil, the power require- ments of slow release relays are higher than those of similar i-ehiys having the full winding space available for the coil. 212 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Optimum Design Conditions In considering the design features that are favorable to slow release operation it is convenient to refer again to the expression for the release time given by equation (9): Avhere f{z) is the function shown in Fig. 5 to be nearly a constant in the range of interest. It is also convenient to refer to the expression for the pull given by the equation (16): 2t{NI + {NDcY d(S{a F = (R? dx Together with reference to the operate requirements, these two equa- tions indicate the characteristics that are important for slow release operation. The winding space determines the value of G attainable, as shown by the relations of Section 5. It limits the combined values of Gs and Gc , of which the latter controls the operate power sensitivity, while the former, from (9), determines the release time. Thus both the operate sensitivity and the release time attainable vary directly with the winding space and hence with the over-all size of the relay. From (9), the attainable release time is nearly proportional to ^", and hence to the cross section of the core, assuming (po/(p" to be small. The external core dimension is the internal dimension of the sleeve, so that increases in (p" are offset by decreases in G if this dimension alone is varied. In any case, sufficient section must be provided for (p" to have a margin over the field required to operate the maximum load. In telephone use, the slow release relays are a minority group in a relay population which must, for maximum economy in manufacture and use, have common overall dimensions and as few differences as are con- sistent with the requirements of specific uses. Thus the core section, winding space, and over-all dimensions reflect an optimum choice for the whole relay population, and not for the slow release relays alone. The latter are distinguished by as few special features as are essential to their special function. In the AG relay these are: the armature, the heat treatment of the magnetic parts, the sleeve and coil, and a buffer spring for load adjustment. The material and its heat treatment determine the iron reluctance and the coercive mmf, 4t(NI)c , of which the latter is the more important quantity. It enters the timing relation indirectly in the minor term - 400 — ui 5< '^< 300 uiO i ^ "^ ^ RELEASE time: , MAXIMUM ,-•' MINIMUM y a <^ y^ ^ ^ ^-^^ ^^^ ■=1 — - 15 20 25 30 35 AMPERE TURNS JUST HOLD OR RELEASE Fig. 15 — Engineering data for release time estimation. load attainable in all cases. This corresponds to the case in which the contact force, which is not subject to adjustment, has its maximum value. The NI value read from the hold curve for this load is the lowest that can be specified as a "hold" requirement. Each individual relay has a release F versus Nl characteristic intermediate between the release and hold capability curves. To meet the hold requirement its load must be adjusted, and this adjustment is subject to the tolerance cited above. The minimum load is set by the hold requirement, and the maximum load is set by a release requirement of a lower A^/ value, the difference between the two iV/ requirements corresponding to the tolerance range in load adjustment. Thus adjustment values are determined in the form of limits to the ampere turn value at which release occurs: the lower limit is the release value, the upper the hold. The release time limits can then be read from the timing curves. The maximum time is read from the maximum curve at the release ampere turn value. The minimum time is read from the minimum curve at the hold ampere tiu'n value. This minimum time, when determined for the largest sleeve, is the longest time that can be guaranteed for the load in question, and is subject to reduction in service by the temperature variation previously discussed. When a shorter re- lease time is desired, a smaller sleeve may be used, or a higher hold SLOA\- REI.KASi; i;i;i,\^ DKSIOX 217 \alue specified, subject to the capacity of the adjustment springs to supply the necessary increase in load. In the common case for which only (he mininnnn time is of circuit importance, the release value is chosen without reference to the hold \alue, solely for the purpose of assurinp; that the I'elay will not lock up indefinitely. This procedure takes advantage of the simpler reiiuirements of this case l)y widening \\w adjustment tolerance and rcdncing adjust- ment effort. 7 CONCLUSIONS The relation between the release time of slow release relays and the design parameters can be more accurately expressed in analytical form than the other time characteristics of relays. These analytical relations, as presented in this article, can be used for the estimation of release time, and particularly for the determination of the effect on this time of varia- tions in the design parameters. The need for a low reluctance magnetic circuit makes the performance of slow release relays highly sensitive to dimensional and material variations, and adjustment is required to assure the timing limits required in their use. Such use usually permits a \\ide spread in release time, provided a minimum value is assured. Advantage is taken of this in providing slow release relays which perform their function at a minimum cost in manufacture and use, materially lower than that for the construction and adjustment practices which would be required for closer timing control. ACKNOWLEDGEMENTS The specific references made to individuals and prior studies do not include all the work which has been drawn on in the preparation of this article. In particular, much of the discussion of the applications of the analysis is based on work carried out by M. A. Logan, Mrs. K. R. Randall, O. C. Worley and others in the development of the AG relay. REFERENCES 1. F. A. Zupa, The Y-tvpe Relay, Bell Lab. Record, 16, p. 310, May, 193S. 2. H. X. Wagar, The U-type Relay, Bell Lai). Record, 16, p. 300, May, 1938. 3. A. C. Keller, A New General Purpose Relay for Telephone Switching Systems, B.S.T.J. 31, p. 1023, Nov., 1952. 4. R. L. Peek, .Jr., and M. A. Logan, Estimation and Control of Operate Time of Relays, pages 109 and 144 of this issue. 5. H. N. Wagar, Slow Acting Relays, Bell Lab. Record, 26, p. 161, April, 1948. 6. R. L. Peek, Jr., and H. N. Wagar, Magnetic Design of Relays, page 23 of this issue. Economics of Telephone Relay Applications By H. N. WAGAR (Manuscript received July 6, 1953) Today's telephone central office is largely built around the telephone relay. As each office may use some 60,000 of them, their performance characteristics, as well as their first cost, have a very important influence on the cost of the office. By properly halancing such factors economically, the loicest cost office can be planned. This paper shows how performance features and design factors may all be expressed as ''equivalent first cost," and so be related to manufacturing cost itself. The influence of lot-size on manufacturing cost is considered including a determination of optimum lot -sizes, to aid the relay designer in deciding on standardized models. With a basic performance language formulated — equivalent first cost — optimum relations among the variables may be established. Such results are given for (a) optimum coil-plus-power costs, (b) optimum power-plus- speed costs, (c) optimum number of relay codes, and other related problems. Methods are also given to evaluate how serious the effect may be when opti- mum conditions are not satisfied. Were it not for the application of these methods, central office costs would be much higher. INTRODUCTION The telephone central office of today is in part an enormous computing- machine which, upon instructions from the customer, gi\'en by his dial, calculates how to find the called party; then the remaining mechanism completes the call. This machine is composed largely of interconnected relays, used in enormous quantities. Every dial call in a large city in- voh'es around 1000 relays. A typical large office contains more than 60,000 of them; in fact, they are used so extensively that the Western Electric Company manufactures about ten of them for e\'er3^ new sub- scriber, and their output is figured in tens of millions per year. It is not 218 ECONOMICS OF TELEPHONE RELAY AlM'LirATIONS 219 surprising that relay use has an enormous influence on the cost of the central office — not just because of relay purchase cost but also the rela3''s influence on other office factors. For example, the size of the power plant and the total number of eciuipmcnls for common control, which (k^pend on the functioning time of tiie relays, are decided by the rela}' characteristics. In the application of relays, then, it may well prove that the largest economies can be realized by spending a little more for each relay in the beginning in order to save still more in the cost of the power plant, common control equipment, size of the hnildiiig, and so forth. It has been found that attention to ways of optimizing flu; application costs for relays can lead to an appreciably lower cost central office, and this paper will illustrate a few cases of how such a problem is approached. Because the telephone system is so large, the influence of each relay, taken in total, is also large. Fortunately, at Bell Telephone Laboratories, it has been possible to take the over-all view of the sul)ject through familiarity with all phases of the relay application probUnii; and large economies which will eventually benefit the cust(jmer are resulting. The basic problem to be discussed is how best to realize maximum economy of the central office so far as relays and their uses are concerned. As in most engineering problems, it is necessary to evaluate and com- promise between oppositely varying cost effects of such things as effi- ciency, manufacturing cost, amount of equipment, ease of mounting and wiring, maintenance in all its aspects, and the like. The end result of each of these variables is its economic effect on the office as a whole, and they can onl}^ be compared if their values can be stated on a com- parable basis. It has been found that each such effect may be considered as an incremental cost over and above a reference cost for its particular ideal condition. If properly chosen so as to be independent, then all such incremental costs may be added. They then represent the net "cost penalty," compared to the design with all ideal conditions taken to- gether, and describe the merit of the design. They can also be used to find the optimum design. The methods apply generally to many other similar problems. BREAKDOWX OF THE PROBLEM Consider first how relays are applied in the telephone switching sys- tem. To the greatest extent possible a basic relay structure is chosen, carefully planned for low maintenance effort, all of whose basic parts can be made by mass production methods. On this basic framewoi'k one 220 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 ISOO ECOXOMICS OF TELEPHOXK RF.LAY APPLICATIOXS 221 may apply (a) suitable windings to satisfy the various conditions im- posed by a need for sensitivity, .speed, lowest possible first cost, or some other of the many specialized reciuirements to be encountered; and (b) particular arrangements of contact springs which will provide for the desired functions in the circuit ranging from a single "make" up to complicated sequences such as 12 sets of "transfers." Every different combination of the basic parts is termed a "code," meaning another kind of relay built up from the parts common to the basic type in ciuestion. If an increasingly large number of such codes is encouraged, each tailored to some special circuit function, then the advantages of mass production begin to be lost as the total demand is split into more and more sub- divisions, each with comparatively low demand. Each additional kind of relay is a problem in "coding economics," and can lead to excessive telephone office costs unless it is properly considered. Satisfactory solutions to this coding problem in turn depend on a detailed knowledge of all the factors governing either the first cost of the relays, or the first cost of other features of the system, as controlled by the relays. vVhen all such effects are properly stated, they can be compared and brought into economic balance. Though such studies soon branch out into many fields, the enormous dollar savings are strong incentives to do whatever work is needed. Best economic balance in the system is formed through a series of optimizing steps, generally illustrated by Fig. 1. Graph (a) shows how the total cost of an office may vary with number of codes. The perform- ance part of the cost diminishes with increasing codes because each code will more nearly satisfy the precise circuit needs. But the manufacturing costs of the relays will rise with increasing codes because the lot-sizes grow ever smaller and hence more costly. The lowest point on the sum- mation curve represents a desirable goal. The two base curves of (a) in turn are built from detailed facts about parts costs, coil costs, power plant and eciuipment costs, and many other similar data. The curve of manufacturing cost, for instance, is built from graphs like (b) and (c) which tell how factory costs will vary. The curve of performance cost is l)uilt from graphs like (d) and (e) which tell how office costs will var}' for things like power and speed. In each such case, important individual economies result by following similar optimizing steps. Procedures for the practical application of these ideas have been of particular help in recent redesigns of the Xo. 5 crossbar system to use the newly developed wire spring relay family.^ 1 A. C. Keller, A New General Purpose Relay for Telephone Switching Sys- tems, B. S. T. J., 31, pp. 1023-1067, Nov., 1952. 222 THE BELL SYSTEM TECHNICAL JOURNAL, JAXUARY 1954 In carrying out an actual study, a point of reference is needed first. This involves (a) fixing a standard against which all costs will be com- pared, and (b) expressing the value of all features in a common lan- guage. Once this has been done, every factor may be evaluated as an incremental "cost penalt}^" over and above the reference standard. Finally, in making comparisons, the reference standard will alwaj^s subtract out, so that one needs only to add all incremental cost penalties to get the total cost of the design changes in mind. The evaluation of all variables on a common basis is covered in Part I, which considers various manufacturing costs, power costs, and the cost of functioning time. The remaining parts of this paper then de\'elop relations for maximum economy in the switching system, for some important cases that arise in practice: Coil design for maximum power economy. Economical number of occasionally unused extra parts. Economical adjustment for speed relaJ^ Coil design for maximum combined power and speed economy. So far as possible, results are given in general form, permitting one to work from charts when considering specific application problems. Of further benefit are charts which permit the designer to decide the eco- nomic disad\'antage of a design which may depart from optimum. Design for systems economy through relay design includes many other important topics not considered here, as for example: how to make best use of molding, welding and other manufacturing processes; or how to design for long life, reliability, and other maintenance concerns. This article covers only some typical problems for which optimizing methods are readily applied. Part I — Expressing Systems Performance and Manufacturing Variables in a Common Language In the over-all switching system, one must e^'aluate various effects whose costs seem ciuite different, and a decision must be made as to what design course to follow, no matter how varied the conditions may be. Many of the most important such cases can be handled quite easily by a process of converting actual cost in the telephone plant back to an equivalent cost in terms of the factory production cost of each indi- vidual relay. Once all such costs are so stated, it is possible to examine the incremental effect of each change, and confidently draw conclusions. For example, if it can be stated that a change in a relaj^ coil will save an amount of power that is worth 50 cents per relay equiA'alent first cost, KCOXOMICS OK TKLKl'IIONK KKLAV A I'l'LICATIONS 223 llien there is no doubt about designing such a new coil if its manufactur- ing cost will increase b}^ only 20 cents. So the relay designer needs to understand how various factors, from original manufaclure to hnal application, may be stated on a cost basis that is comparable throughout. Among the most important special cases are the following: 1. Manufacturing cost of winding a coil, 2. Manufacturing cost as a function of ainnial demand, 3. Ecjuivalent manufacturing cost of power consumed by relays, 4. E(iui\'alent manufacturing cost of functioning time of a relay. The method of e\'aluating each on the same basis will now be briefly outlined. In each case, it should be noted that since only comparisons be- t ween variable portions of the system are considered, only incremental ^'alues need be considered. 1.1 THE COST OF A WINDING 111 comparisons between coils, certain common operations, such as soldering the leads, using and cementing spoolheads, etc., will always subtract out, leaving only the difference due to varying amount of copper, which is paid for by the pound, (or per ohm), and the number of turns, which depends on the speed of the winding machine and number of coils wound at one time. Thus, winding costs, which depend only on the electrical design, may be considered as varying only with the cost per ohm and cost per turn, as given in equation (1): Cir = CnR + CsN, (1) where Cw = total cost of the variable part of the winding, Cr = cost per ohm, which may be tabulated by wire size, Cjv = cost per turn of winding, also given in tables by wire size, R = resistance of coil, ohms, N = number of turns in winding. Typical \'alues of Cr and Cn for a particular kind of wire and coil are shown in Fig. 2. 1.2 COST AS A FUNCTION OF ANNUAL DEMAND ^"arious components of relays, as well as their assembly routines, occur in several variants of a general basic pattern. In the course of their manufacture it is necessary to stop the manufacture of one unit, reset the machine, and proceed with manufacture of another. Whenever this happens, pi'oduction stops; and work must be done to reset the machine, all of which may be evaluated as a set-up cost (designated S). As more 224 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 and more variants are required in the over-all general structure, the number of set-ups will increase to the point where the economical manu- facture of the part is seriously penalized. Thus a point will be reached where it is advantageous to make mol:"e parts than are immediately needed, and store the remainder, so as to avoid excessive cost of shut- downs. vVhen the variable effects of (a) set-up costs and (b) inventory costs are fully considered, the ideal quantity to manufacture in one lot may then be found by comparing the results of (a) and (b), as will now be shown. The costs which correspond to these ideal quantities will immediately follow. 1.21 Lot-Size Costs In manufacturing practice there are two outstanding expenses which may affect the cost of any particular item which is classed as a code, or kind, of the basic family of articles. These are the administration costs, and the set-up costs. 1.211 Administration Costs For certain kinds of operations there is a considerable amount of paper- work, drafting, checking, etc. to naintain in normal up-to-date condition. Occasionally, this cost is enough to influence the cost of the °- 0.01 \ s. \ \ \, s \, \ k \ \ \ ^ . - N . 22 2A 26 34 36 38 40 42 WIRE GAUGE Fig. 2 — Typical winding costs. lOCOXOMICS OF 'rKI.i;i'll<).\K IIKLAV Al'l'LK 'A'l'K )XS 225 item. Tlic r(>sultiiit; iii(li\'i(lu;il ('I'toct may tlioii l)c stuted as Ca=^, (2) where Ca = cost penalty per uiiil due to administration, A = anmial administration cost for maintaining one code; in good standing, ti = annual demand for a given code. 1.212 Sr(-up Costs If only ont> kind of part or assembly were needed, it could be con- tinuously built in the same way, with no time lost for changeover to other parts, no particular bookkeeping necessary to control the proper How of differing parts, and M-ith more mechanized action. Usually this condition is far from realized in practice; nevertheless it represents the peak of manufacturing economy, and may be taken as a standard of reference for comparing all other less favorable conditions. The cost ineurred, per item, due to its lot size, as against its cost if there were always but one lot, will be called the "lot-size cost penalty." The manu- facturing lot-size cost penalty per part for any particular process, then, may be stated as follows: C'- = l, (3) = ?■ (4) n where Cl = lot size cost per item, *S' = cost of one "set-up", L = size of lot for which one set-up is made, = n/^, n — annual demand for a given code, C = luimber of lots per year, or lot-frec|uency. Equation (4) gives the cost over and above single-kind manufacture, so far as lot-size is concerned. Hence, when values for each of these variables can be established, the lot-size cost penalty for any particular process can be determined. 1.213 Over-all Lot-Size Costs, Related to Annual Demand Combining the results of e(|uations (2) and (4), the total cost penalty per part due to lot-size is 4. S( C, = --\--. (5) n n 226 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Fig. 3 — Lot-size cost penalty relationships. 1.22 Inventory Costs When more parts are made than are immediately needed, it becomes necessary to store them until they are all used, when the process will be repeated. The resulting cost penalty, compared to no storage at all, may be stated analytically with the help of Fig. 3, which is a plot of production as a function of time. Line A represents the number of parts made in a lot as a function of time, while line B represents the parts used in the next manufacturing stage, as a function of time. The difference between the two lines at any time represents the number in storage, from which their value and hence the interest charges may be found. The cost penalty per part due to storage is: Annual interest factor X value of one part X av. no. of parts in storage (6) Annual production of this part C = The following steps can be taken to supply values for this equation. From the figure, the maximum ciuantity of parts stored equals Ln or 1 - ECONOMICS OF TELKl'lK )\K KKLA^ A I'l'LK ' A TIOXS 227 The average luimher of parts stored is one-half this, or where jV = rate of output of luachiue per year. The peiuUty per part due to lot-size and administration was given in (5), and when added to its base eost Ci gives the total value of one part as C^ + ^+'l. (8) n n The penalty per part, due to storage, equation (6), is then the product of k/n times (7) times (8), where k = interest charges on stored parts expressed as a ratio. Then the storage cost penalty, Cs , is C, = ^(c, + ^f+-i)«(i-»), (9) n \ n n / 2C\ N / 1.23 Total Cost Penalty The entire penalty resulting from lot size charges and inventory charges may now be stated as a function of annual demand by summing equations (5) and (9). Writing q for (1 — n/N), and rearranging terms, the total cost penalty in terms of annual demand is Equation (10) is the basic relation betw^een costs and annual demand, and is of the general shape shown in graph (b) of Fig. 1. It is of greatest interest to determine conditions where this curve has its lowest ^'alue, which occurs when dC/di = 0. The result is which gi\'es the lot-frequency, fo , for which total cost penalty is a minimum. When fo is substituted for ( m equation (10), the resulting optimum cost, as a function of annual demand, Co , is given: ^^ ^ /2kqS{C, + Ajn) _^ kqS ^ A Equations (11) and (12) furnish the means for deciding liow to plan manufacture under differing conditions of annual demand, and how much the product is penalized even after plaiming is carried out as 228 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 efficiently as possible. Two illustrations will be given: for a comparatively low-cost part involving expensive set-up charges and no administrative charges, and for a more costly operation with comparatively inexpensive set-up costs but high administrative costs. For Case I assume a part for which the following constants apply: k = 0.15 N = 2,000,000 S = $10.00 In Cast II assume that A- = 0.15 .V = 2,000,000 S = $2.00 C\ = so. 10 .4=0 Ci = $1.00 A = $100 The results for these two cases are given in Fig. 4. Results such as these are of the utmost value both in manufacturing planning for lowest cost, and in applications planning to show when the lot-size penalties can be tolerated. However, no case is to be expected in practice where optimum conditions can be met more than a part of the time. What with unexpected rush orders, fluctuation of annual de- mands, rescheduling, possible moves, or breakdown of machinery, it is usually impossible to plan production to remain at all times at the optimum value. The variation from optimum cost that results from a departure from optimum lot size is thus of interest; it is readily stated ^,. ^ \ n^. k" Ho \- .^^^ ^ 1 \ \ N -^ K ^ ^"-. \ ^i n"" ■^^"^ V >> \)r \Co 10 1 10 10^ 10^ to'* 10^ 10^ K n, ANNUAL DEMAND FOR A GIVEN CODE Fig. 4 — Optimum lot sizes and costs. ECONOMICS OF TELEPHOXE RELAY Al'l'LIC A TIOXS 229 u O 2 A- ''n =10^ k =0.15 q =0.5 S = 10 A = 0 n = lo'' k =0.15 q = 0.995 S = 2 A = 100 -E \ v A .^ \ \ /, y ^"-., \v r:^ ■^^1^ ^y'^' '-''' 0.3 0.4 0.5 0.6 0.8 1.0 1.5 2 RATIO LOT FREQUENCY TO OPTIMUM LOT FREQUENCY, Fig. 5 — Cost variations with variations from optimum. by rewriting equation (10) in terms of the values of fo and (\ just found in equations (11) and (12). The result is C/C, = 2 + D/U (13) where 2^ S This relationship is seen to depend on annual demands, and also ad- ministrative and set-up costs, which will be peculiar to each case. There- fore, it will be necessary to use the equation with appropriate values of these constants for each case that arises. Typical values for the two previous illustrations are shown in Fig. 5. Inspection of these curves shows that the cost penalty of departing from optimum lot-frequency is somewhat variable depending on the ideal lot-frequency for the particular case. Howe^'er, the higher ^•alues of lot-frequency, which produce the highest penalties, are those most easily under control of the factory, and variations averaging in excess of 2 or 3 to one from ideal are not at all likely. When the ideal lot-frequency is low, manufactiu'ing control is not so easy, but the cost penalty ratio of dejiarting from the ideal is far less. From inspection of these curves, then, the value of C/Cq must be judged by the conditions of each probU^m; a value of cost penalty amounting to about 1.25 Co is often found to be a ciuite satisfactory value to assume. Thus, (a) a method is available to the manufacturing engineer to help plan optimum lot size, and (b) a method is available to the rela\^ 230 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 and circuit designer for deciding how much his designs may be penalized by varying the quantities used. It consists of using the costs determined from equation (12), and modified by an amount indicated by (13), or in ordinary cases by arbitrarily increasing (12) by the factor 1.25. 1 .3 THE EQUIVALENT MANUFACTURING COST OF POWER CONSUMED BY A RELAY The largest part of the power plant used in telephone central offices is the 50- volt equipment needed for the switching apparatus. Because of its special voltage range and the requirements on its stability and reliability, it must be provided as a part of every central office to convert the normal power company voltages into the desired telephone values. This equipment involves generators, banks of storage batteries, and switchboard, bus-bar and control equipment; it is an expensive portion of every central office which, of course, it is desirable to minimize as far as possible. For every relay required in the central office, one must associate a small "chunk" of this power plant; so that each relay needed implies an associated investment in the power plant. In a later section the means for minimizing the power and relay costs are described, and since they will involve comparisons on the same basis, it is first necessary to state the value of the power plant in terms related to the amount of power consumed by any individual unit. The method of evaluating the power plant is given in this section. The problem may be broken into two parts: (a) the equivalent first costs assignable to the power plant equipment, and (b) the equivalent first costs of the charges per kilowatt-hour paid to the power company. Then the figures are combined. 1.31 The Power Plant Power plants furnished for central offices vary over a wide range of size and cost, depending on the size of the office and its estimated ac- tivity. This plant must of course be planned so as to carry the load during the period of peak activity, even though this occurs for only a small part of the time, so that in the long run the cost of the power plant is decided by the share of busy-hour power taken by each relay. To get such costs one must first find the cost of the plant as a function of total power requirements. Present practice in power plant planning results in the purchase of basic power units which may be combined to cover certain ranges of power supplied over roughly a two-to-one range. Within this range, by ECONOMICS OF TELEPHOXK l^ELAY MM'l.K ATIOXS 231 adding generators and batteries in parallel the needed capacity can he attained. Beyond the range, basically heaviei' machines ;u-e lUH'ded. llw resulting installed price of such power plants has the chaiacler shown in Fig. G. Since prices and installation ciiarges change from year to year, the vakies shown here are gi\'en in (qualitative form as illustrations — they must of course be evaluated as concisely as possil)le when considei- ing new central office designs. As seen in the figure, the power jjlant price xai'ies in two wa^'s: in fairly big jumps as basic plant size is changed, and ill smaller luit rather uniform manner as the basic size varies within its lower and upper limits. From Fig. 6 it is seen that for each basic size of plant the rate of in- crease in price with small increases in capacity is about the same for all plants. The average rate for this incremental power capacity, correspond- ing to the expense if relati\ely small power changes are made in any one l)asic type of power plant will be called a. As the power requirements of the office change be.vond a certain point, it is necessary to install basically larger or smaller equipment with correspondingly different base prices. At the points of maximum capacity, there is an abrupt change in the basic price of the plant, amounting often to many thousands of dollars. These abrupt increments, designated .4, represent a lump sum of mone}' that can be saved whenever the power plant size can be re- 80 120 160 CAPACITY IN KILOWATTS Fig. 6 — Installed value of i)0\ver plants. 232 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 duced to the next lower-range unit. Such a saving can be reahzed only part of the time, but it is desirable to attempt to weight its value so that over the telephone system as a whole there is adec^uate encourage- ment to strive for power savings. The method of weighting is complicated by such factors as the fraction of power that may be saved in an office, by the likelihood of certain size offices being representative, and by the e.xtent to which offices are de- signed with safety factors for growth and overload. XcA'ertheless a reasonable estimate can be made by assuming difTering amounts of the office power can be saved, and by assuming that all sizes of office are equall}' likely. This gives the result that the over-all incremental value of power is the average value of power plant per kilowatt saved over the range between Wo and Wo , taking into account the proportionate number of plants that can be converted to the next cheaper size, where C = total price saving, W = total power saving, p = fraction of power saved, not to exceed about .5, (above which point two steps of base price saving might be realized). Wo = lower limit of range, W2 = upper limit of range, a = incremental price of one size power plant, assumed to be about constant for all plants, A = difference in base price to next lower plant, as rasi\ be tabulated for any particular case. By these methods, incremental values of installed power plarit may be found; it is even possible in many cases to assume an average slope for the family of curves in Fig. 6, without greatly changing the results in practice. It is now desired to convert these figures of installed expense to figures comparable to manufacturing cost, in the factory, of the relays which will consume this power. This may be done through standard economic- study practice by which first cost is converted to annual charges through a knowledge of charges for interest, taxes, depreciation, etc. As these factors are not necessarily^ alike for power ec}uipment and relays, the factors used will differ to suit the problem. For the present study, with recognition of the accounting practices then in effect, it was found that ECON'OMICS OF TKLKPIIOXK IHOLAV A I'I'LK'ATIONS 233 OH a basis comparable to relay manufacturing cost, savable power was worth between vSoOO and $2,500 per kilowatt, dependinp; on size, savable power, etc. Now these figui-es must be associated with the power needed by any particular relay. The total switching power during the busy part of the day, which is the period determining the size of the power plant, is the sum of all the individual relay power drains at any particular moment. This in turn may be taken as the sum of the power required by each relay times the probability that it will be energized at any particular time, which is taken as the fraction of tlu^ })usy hour that the relay is expected to be energized. For every relay, this ratio can be determined with some certainty, by consultation with the circuit designers. Summarizing, the switching power capacity required by the office may be stated as the sum, for all relays, of power w for each relay, times m, the fraction of the busy hour that it is energized (i.e., hrs. per busy hr.). The cost of the power plant capacity reciuired for each relay is then C = kmw dollars, (15) where k is the equivalent value of powder. \'ery often it is desirable to state this cost on the basis of the annual power drain. In this way, the plant investment can be easily correlated with the costs paid to the power companies. In telephone offices, the annual power has been found to correspond to 3,000 times the power consumed in a busy hour, and as a matter of convenience power drain in the central office is often stated in terms of the energy per year based on a 3,000-busy-hour year. Then the ratio of use during the busy hour may also be stated as "^ 3000 ' where h = hours per year energized. Writing A'-'/T 000/2 for the power in kilowatts drawn by any relay, the equivalent first cost of the power plant assignable to the relay is then -_^ V J^ l^power plant ^^^^j^ A ^^^^ , (IG) F/h For this case, in other words, tlic (Miuivalcnt lirst cost of power plant has a value Cp^ dollars for each walt-hour-per-year of switching power required. 234 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 1 .32 Cost of Power Purchased from Power Companies The power used by the switching apparatus corresponds to a larger amount of power furnished by the power company, since the power machines are not 100 per cent efficient. Its cost may be stated as Co Cp= -, e where Co = average marginal cost of power furnished, e = conversion efficiency. The annual power bill will then be Co E'h e 1000/2 dollars. Converting from annual charges to first cost, through the same steps as above, gives R U power — Cp, — p^ , 117) or CP2 dollars per watt-hour-per-year. 1.33 Total Power Cost With both power plant costs and purchased power costs now stated on the same basis, they may be combined to give the total power cost, Cp: where Cp represents the dollars per watt-hour-per-j-ear for power con- sumed by a particular relay, and is the eciuivalent first-cost value of power. Thus a method is available to find the value of the power that magnet coils may need — to be balanced against the first cost of the coils (Section 1.1). Procedures for best results are given later. 1 .4 THE EQUIVALENT MANUFACTURING COST OF THE FUNCTIONING TIME OF A RELAY Another large part of the investment in every crossbar-tj^pe central office is the common-control equipment, most important of which is the "marker." This eciuipment has the sole duty of selecting the proper path for each call, and sending it on its Avay. After each such function, ECONOMICS OF TELEPHONE RELAY APPLICATIONS 235 the equipment restores to normal, ready to serve the next call. It takes a. marker about a half-second to do its job; nevertheless many markers may be needed to handle the traffic during each day's heaviest calling period. Since the cost of each marker is measured in the tens of thousands of dollars, it becomes a matter of great importance to shorten the marker work time — in other words, to shorten the time of each relay in the chain of marker events. Since faster relays are usually more expensive it is important to find the dollar value of speed so as to guide an intelli- gent design of each speed relay. As in the case of power, it has been found possible to state the value of speed in terms equivalent to the manufacturing costs of relays. This may be done through a knowledge of the number of markers and asso- ciated equipment needed in relation to their work time, and their cost. Such relationships are shown in Fig. 7, which gives a typical curve for the markers per line needed to handle the traffic. Then, when the value of the marker is known, the value per line per millisecond, Ct , follows. Corresponding to this, a value per millisecond per marker, Cm , can then be found, which varies inversely as the number of markers needed : _ ^' Cm — — , V where p = number of markers per line that are needed. This figure is the value of a millisecond for any complete event or series of events 9 8 7 z " 6 tr UJ D. ^ z \ ^^ LU \ ^^< -J \. ^„,^^ < X^ ^„*-*''''^^ > ^ 3 O ^^^""'^^^-^Cp Ul ' ■ RESISTANCE Fig. 9 — Power and winding costs. / ECONOMICS OF TKLKPIIOXK UKLAV A I'PLH'ATION'S 241 ing cost for a certain wire gauge \'arics while the curve labelled CV shows how power plant cost is changing. ^J'he desirable end result is a coil with resistance value to permit the sum of these to be as low as possible, as in the curve marked Cr ■ I'hc following methods will give such a design. The value of Cr , made up of power cost, Cp , and winding cost, Cw , represents the cosit of all the parts of the central office which are influ- enced by a change in resistance of the coil. Thus we wish to minimize this e(iuati()n: where, from the previous relations, ^ _ ^1 ^^- -R' Cw = K2R, Ki = CphE\ /Vo = Cfl I 1 + — -5- \ Cr K The resulting expression for this total cost is Cr=^ + K2R. (21) In this equation Ki is a constant, but /v2 ^'aries both with resistance and wire size, as the latter affects the values of Cn , Cr , and the quantity N/R, which is determined by the coil design equation R = AN (h + d), in which, using the special notation common to coil design problems: N = turns in winding = h(/K, R = coil circuit resistance, A = IT times the resistivity of wire in ohms per inch length, gi\-en in tables for each size of wire, h = depth of coil space occupied by wire, / = length of winding space, d = inside diameter of coil, K = effective cross-sectional area required by one turn of wire, given in tables for each size. 242 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 For a given relay, the dimensions t and d are fixed, and only the wire size and the coil depth h can be varied. For a given wire size, therefore, the values of h and A^ are fixed by the resistance. As Cn and Cr are known quantities, the values of Ko can be determined and plotted against R for each wire size, as illustrated in Fig. 10. Such a chart applies to a particu- lar relay, as the curves depend on the fixed values of I and d applying. Here it can be seen that for a given resistance the finest gauge wire is the cheapest. Because of this, and since power costs are minimized with the highest resistance, the finest gauge wire would always give the cheapest coil. However, this becomes a theoretical case. As finer and finer wires are used, A = ira becomes larger, and thus N/R or NI/E becomes smaller. In all relay design, an ampere-turns requirement must be met. Thus a practical solution to this problem of optimum costs is to choose the finest gauge wire which will meet the ampere-turns require- ment. If the same relay is assumed as for Fig. 10, the variations of ampere turns with resistance are as shown in Fig. 11. Starting with the NI re- quired, the wire size can be determined from a plot like Figure 1 1 . Then Ki is determined from a plot like Fig. 10. Now, with the gauge of wire chosen, K2 is almost a constant and can be considered independent of R. Thus, upon differentiating (21), Ct will be found to be a minimum when R has a value designated by R, = VkJK^ . (22) The most economical coil, resulting from use of a resistance 7?o , will have a system cost of Co = 2VK^2 • (23) As can be seen by the slopes of the curves on Fig. 11, if the optimum resistance, i^o , is larger than the resistance value at the desired ampere turns, the NI requirement will not be met. Then either the selection of Ro must be made for the next coarser wire, or the resistance value chosen at just the required NI, whichever is cheaper. Conversely, if there is a very large A^7 margin, a finer gauge wire could be tried. In all, two trials should give the optimum values. A procedure has now been outlined for choosing the coil design for optimum power-plus- winding cost. Curves similar to Figs. 10 and 11 should be draw^n for the applicable relay parameters. Then the K2 value is obtained for the finest gauge meeting the ampere-turns requirement, and Ro and Co found. Then, the NI value applying for Ro must be checked and adjustments made as above, if necessary. ECONOMICS OF TELEPHONE RELAY APPLICATIONS 243 rlcr ■" WIF - - !E GAL -34 — I 3^ E NO. , V - -36 ~1 T— ^R ^ = 39 1 — "^0 1 "^ ■ -42 10 10^ 10-^ COIL RESISTANCE, R, IN OHMS Figs. 10 and 11 — K^ and NI values versus resistance for various wire gauges for a typical relay. COST WHEN OPTIMUM RESISTANCE CONDITION IS NOT SATISFIED There will, of course, be cases in practice where the optimum re- sistance is not used, for such reasons as standardizing of certain coil sizes, need for speed, or insufficient winding space. The penalty in cost from departing from the ideal winding size may be found by comparing the cost for any value of resistance, (cfiuation 21), with the cost if re- sistance were optimum, (equation 23): C _ Kx/R + K2R Co " 2 \R "^ 7?„ (24) 244 THE BELL SYSTEM TECHNICAL JOI^RNAL, JANUARY 1954 0.15 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.5 2 3 RATIO OF RESISTANCE TO OPTIMUM RESISTANCE, R/Rq Fig. 12 — Cost penalt.y when resistance is not optimum. Thus, the fractional change in cost caused by any departure from the best resistance is readily estimated, expressed in ratio form as R/Rq . This relation is shown in Fig. 12. SUMMARY In summary of Part II, methods have been given for deciding on the magnet coil design which will provide the cheapest net cost in the switch- ing system. When the conditions of use are known, a series of charts may be prepared such as the one shown, from which the applications engineer may decide the proper coil design, and how much it may be worth to the system. As experience will show, the cost of departing from an optimum design may run well over $1.00 per individual relay, with the result that aggregate savings in the whole central office may run to thousands of dollars compared to the cost when this problem is ignored. New relay development may be guided by the use of these same rela- tions, since through a process of considering various hypothetical de- signs of various sizes, work outputs and magnetic qualities, the potential optimum costs may be found. Design of the recently adopted wire spring relay was very materially guided by considerations of this sort. Part III — Choice of Contact Sets for Maximum Economy One of the many problems involving lot-size costs of relays is the num- ber of standardized sets of contacts that should be provided: i.e., how many kinds of contact arrangements give maximum economy for the product as a whole? By pro\ading only a few choices of kinds of contacts, for example, the annual output is less divided and there are fewer setup charges, with the result that the initial manufacturing economy is large. However, because of the small selection of kinds of contact sets, there will result many cases where extra contacts are provided at greater than ECONOMICS OF TKLKl'llONK Ki;i,AY A I'l'LICA'l'lON'S 245 needed cost. The problem is how l)est to combine these opposing fac- tors. In iUust ration, consider a relay which is to be ecjuipped with molded spring arrangements, capable of providing any number of "transfers" from 1 to 12, each of which is used in approximately equal numbers in the system. At the one extreme, only one molding might be provided, always equipped with 12 transfers. At the other extreme 12 different moldings might be provided, co\'ering each individually needed quantity of "transfers." In the case of the single molding there is no lot-size penalty at all, but on the a\'erage a large penalty in surplus contacts; while for the tweh-e kinds of moldings, lot-size penalties corresponding to one-twelfth of the possible annual output are incurred, but no spare parts at all will be needed. Such a problem may be treated as given below. Let the number of kinds of spring sets chosen be designated by v. Then the annual output of each kind is the total output N divided by V, or N n= -. (25) V Now it was shown in Section 1 .2 that the cost penalty for making things in lots is given by equation (12). For the present problem, this may be expressed in terms of v, by substituting equation (25) into equation (12). The resulting lot-size cost penalty is then V n — _VW(H)i^)^'^ + A (26) The penalt}' due to pro\-iding unneeded springs is the dollar value of the average number of spare springs. The average number of spares is approximately 2v Thus for only one kind furnished, always 12 transfers, the number of spares can vary between 0 and 11, an average of 5.5, as given by the equation. If there were eight kinds, consisting possibly of moldings of 1, 3, 5, 6, 7, 8, 10 and 12 sets of transfers, the average number of extras would be 0.25. The dollar penalty compared to no extras ever needed is 240 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 and the sum of this equation and that for lot size represents the system cost for any one condition. Fig. 13 shows this variation for an assumed case for which iV = 500,000 parts per year. A; = 0.15. S = $100. C\ A 0.20 for spring set cost. 0.005 for spare spring cost. 0. Here it is found that the greatest benefit comes from using in the neigh- borhood of six kinds of spring sets. It is further clearly seen that the sys- tem as a whole will not be particularly penalized if the number of sets chosen is a few different from this quantity — but a penalty of almost two cents per relay, or about two thousand doUlars per year would be incurred by erring too far toward the extreme. Part IV — Choice of Special Adjustment for Maximum Speed Economy Speed can be an exceedingly valuable property of a relay, as shown in Section 1.4. There are cases for example where every saving of 1 2 10-2 5 6 ^ \ N^^" Cq^ ^Ce \, — N.- ^ a z > 2 h- _] < LJ ,0-3 Q- 8 UO 6 O ^ 4 2 10-'' / y^ N / N \ 1 1/ Ce i \ / N / s \ \ \ \ I 23456789 10 NUMBER OF KINDS OF SPRING SETS. V Fig. 13 — Optimum number of spring sets. ECONOMICS OF TKLKIMIOXK HKLAV APPLICATIONS 247 millisecond in workino- time will reduce the average quantity of equip- ment needed by an amount e(iui\'alent to S35, or e\'en more. For such cases a considerable sum of mone}' can be spent on each I'elay in order to make it faster. The following approach helps guide the circuit and re- lay designer in the choice of relay they should make in order to gain the greatest o\'er-all advantage. It is well known to relay designers that one of the most important design parameters controlling operating time of a relay is the spacing between the contacts when they are at rest. This space cannot be less than a certain small distance of about 0.005" to a^'oid contact failures due to ^'ibration, buildups, sparkover, etc. However, because it is costly to hold adjustments accurately to a few thousandths of an inch, the maximum spacing is often quite large, sometimes as much as 0.050". The spread in this distance is primarih' a matter of economics; if it were clear that more money could be spent on the relay to narrow down this spacing while gaining material reductions in operating time as a result, then some closer adjustment would be specified. Such a problem is read- il}^ treated bj' summing (a) the cost penalties for differing values of this spacing and (b) the cost penalties of the corresponding functioning time, and finding the optimum condition that results. One such case is shown below^ The cost penalt}' due to changing spacing may be estimated by con- sidering various actual values, and determining what procedures would be used in the factory in each case. Such a cost study can be made in the factor}'. In illustration, assume a particular relay type deemed to suffer no cost penalty if its armature stroke is allowed to take on any value up to 0.050", but which cannot be less than 0.010" without impairing per- formance. The actual cost per unit of various kinds of manufacturing procedures might be found according to the hj^pothetical conditions given in Table I. Thus a curve can be plotted of the cost penalties for each setting of gap. The design may also be studied for the influence of the gap settings on speed. The operating time of the magnet as a function of spacing is readily determined by experiment, and may also be checked against known operating principles, to be sure the figures are reasonable. For a typical case, operating times corresponding to various spacings might be as given in Table II. At the same time, the value of this functioning time is given by equation (20) to be n _ 9^' Or — • 248 THE BELL SYSTEM TECHXICAL JOURNAL, JAXUARY 1954 Table I — Cost per Unit of Various Kinds of Manufacturing Procedures Assumed Max. Stroke Manufacturing Procedure Cost 0.050" Acceptance of wide dimensional tolerances on all parts, and no adjustment permitted $0 0.045" As above, but crude adjustments added 0.05 0.040" Closer dimensional tolerances, no adjustment 0.07 0.034" As just above, with rough adjustment 0.10 0.025" Very close tolerances on all parts, no adjustment 0.15 0.020" As just above, with touch-up permitted 0.20 0.015" Screw adjustment added, (more expensive parts, but simpler adjustment) 0.35 0.010" Precision setting of all parts 0.65 Assuming for this hj^pothetical case that the values applying are ct = S0.05, w = 100, q ^0.2, and V = 0.0006, then the cost of the resulting time is found in the last column of Table II. A summation of these two sets of cost penalties is given in Fig. 14, which clearly shows that the ideal value of stroke is about 0.020", and that that it va&y be permitted to xsny between the limits 0.015" and 0.025" without a serious economic penalty. In summary, a method has been indicated for deciding how important it maj' be to build in, or omit, expensive design features which have an effect on functioning time. An illustration has been given for an assumed common control cirucit and an assumed change in a design feature of the relay that importantly affects the time. The method, however, is equally Table II — Operating Times Corresponding to Various Spacings Assumed Max. Stroke Functioning Time (Milliseconds) Equivalent First Cost Value of Time 0.050 6.1 SI. 02 0.045 5.6 0.933 0.040 5.2 0.867 0.035 4.75 0.792 0.025 3.75 0.625 0.020 3.2 0.533 0.015 2.55 0.425 0.010 1.9 0.317 ECONOMICS OF TKLKl'llOXE RELAY APPLICATION.S 24i) applicable to other speed problems, once the value of speed is known, and onec the infhuMico of the design chanji-o and its cost arc known. Part \' -C'noicE ov \\'ixdixg fou Maximum Combined IiIconomy OF Speed and Power In the selection of magnet designs for use in circuits where speed is important, much can be gained by modifying the mass, the stroke, the force against the stop, etc., but when all these devices have been ex- hausted there remains the possibility of supplying more power to the magnet coil. If this is done in enough cases, an over-all penalty in central office cost is incurred through the added over-all power plant capacity that may be needed. As already seen in Section 1.3, this cost penalty is proportional to the base value of one watt-hour-per-year, the voltage sciuared, and the hours per year energized, but is inversely proportional to the resistance. In telephone usage, the voltage is usually fixed at 50 \'olts, and etiuation (18) may be rewritten as kph IT' where Cp = (28) l\ CpE' Z 0.70 < 0.30 — '^ \ \ \ Cp + Cadj ^' y ^ \ \ \ \ ^y' ^^^^ / / \ >»^ --'' • > / \ / A ^^^ / V < \Cadj """^ ^^ ^ 0 0.010 0.020 0.030 0.040 0.050 STROKE IN INCHES Fig. 14 — Optimizing a speed relay design. 250 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 The above expression gives the equivalent first cost of the power taken by one relay. For each relay operation, or event, however, there may be several relays energized. Then the total cost penalty per event, due to power, is Cp = 'i^ dollars, (29) n where g = number of relays energized, per marker, for the particular function in question even though only one appears in the time sequence. For the power cost to be compared with the cost due to speed, this cost must be spread over the number of relays, w, involved in the function. Then C, = ^ (30) wH This is the equivalent first cost penalty incurred against each relay, due to its power consumption. As R increases, it is seen to decrease. However, as R increases, the operate time increases. For speed relays, the time t has been found to be a complicated function of the resistance, varying approximately as R^'^. Thus when R increases, time increases, and the equivalent first cost of this action time increases, as already seen in Section 1.4. As power is changed, then, there results two oppositely varying costs — one due to power, and one due to speed. The problem is graphic- ally shown in Fig. 15, where Curve A shows the typical cost variation due to power drain and Curve B shows the variation in cost due to speed. The point where the sum of the two is a minimum is the ideal point to operate, assuming that the coil costs are about equal in each case. Curve C shows how the total cost will vary. Actually, it may be necessary to add the effects of a third variable: the cost of the coil itself as it is redesigned for different power consumption. This can be easily done by the methods shown above, but is ignored in the present treatment for the sake of simplicity. In many actual cases, coil cost has been found to be a much smaller factor than the other variables. CHOICE OF OPTIMUM COIL In practice the above result may be obtained by a knowledge of how a specific magnet design will vary in operate time as its coil resistance is changed. The values may be substituted in the following equation for total cost: ECONOMICS OF TELEPHOXK RELAY APPLICATIONS 2.')! wp wR ' (31) The expression may be evaluated wlien a relation between / and circuit resistance R is established, usually a matter for experiment in any given case. Illustration For a pai'ticular case let us assume that p = O.OOOG, so that equation (31) is C = l f 1670 i c^t + ^4' w \ g R Tj^pical values of operate time, t, for a speed relay in 50- volt telephone applications are given in Table III. From this information, one may readily evaluate equation (32) to cover the practical range of conditions that are met in seri-ice. This has been done, in Fig. 16, to give the total (32) 200 400 600 800 1000 COIL RESISTANCE, R, IN OHMS 15 Optimizing costs of speed plus power by relay coil design. 252 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Table III — Typical Values of Operate Time Coil Resistance (Glims) Optimum Time (Milliseconds) 100 2.45 200 3.1 400 4.15 600 5.0 800 5.8 1000 6.7 cost penalty for the case of very short holding time (h = 100 hr. per year) and for long holding time (h = 2000 hr. per year). The optimum coil resistances for these cases are seen to lie between 200 and 400 ohms for the long holding time condition, and to be as low as possible for the case of short holding time. As a result we see an appreciable economic incentive to plan relay designs which are capable of operating on low resistances. Further study of heat dissipation, operation with holding windings, contacts to withstand heavy currents, and other de^dces to gain fast action, therefore take on added importance to the relay de- signer. Part VI — Choice of the Actual Code With the background of the previous sections, the systems designer can now decide what codes he needs to use in his circuits so as to be < 35 ^ V h = 2000 HR PER YEAR ^ \ ^ ^ 100 y y > v^ / y] / Fig. 16 0 200 400 600 800 1000 COIL RESISTANCE, R, IN OHMS Best coil design for both speed and power economy. ECONOMICS OF TKLKPHOXK HKT.W A PPIJCATIOXS 253 most economical, over-all. The problem will first be discussed in a gen- (n-alizod form, and the moans \'ov <);ottinji- the economy will then be re- \'ic\ved. G.l GENERAL DISCUSSION OF CODING If enough information were available to the circuit designer, he should be able to choose a relay for any particular application by considering (a) The penalties in performance (expressed in terms of first cost) re- sulting from having only a certain limited number of available combina- tions of relays, as against complete flexibility, i.e., the penalties due to standardizing, and (b) The penalties in first cost resulting from many variations of a basic type, as compared with one single standard type, i.e., the penalties due to not standardizing. By weighing both the penalties and the advantages of standardization in each case, they may be maintained in approximate balance, and give the economical number of codes. In the development of an entire new switching system using relays, there will be approximate ideas on the number of relays needed in the system, and on the number of kinds of relays required to fairly well satisfy the circuit functions. Based on past experience, the statistical distribution of kinds of relays for the various uses in relation to annual demand may also be approximated. The number of codes provided will determine the number of cells into which the statistical distribution is divided, and also the relative annual demand for each. In the extreme, only one code might be provided. It would have the advantage of very large lot-size, but also it would have to do the hardest as well as the easiest function. Such a requirement could lead to such absurdities as requiring that all relays have three windings, twelve transfers, best grade of contact metal, and the like; or to a greatly increased number of simpler relays. In addition, with but little flexibility in the design, there would be the added performance disadvantages due to imperfectly matching certain needs for power economy, speed, and the like. Hence, there is a large performance disadvantage, if but one kind of relay were provided. At the other extreme would be cell divisions of kinds of relays to give the perfect match for every circuit use. Then, with no spare contacts, the best possible power economy, and every other condition at its ideal value, no performance penalty at all would be entailed. But every code increase would further subdivide the manufacture, till the lot-size penal- ties grew excessive. These variations are shown by the curves in Fig. 1, and it is their sum. 254 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 as seen in the curve marked c, that should decide where best to work. In planning a whole new central office which may use from 40,000 to 80,000 relays, and a new design of relay whose manufacturing costs are not yet well established, the data to give numerical values for Fig. 1 can only be quite tentative, and the results can only serve as guides. Yet such Avork has been most useful in recent effort on a redesigned No. 5 crossbar system to use the wire spring rela3^ Calculations could be made with enough assurance to determine the general shape of the total penalty curve (Curve c), and showed an optimum value for kinds of re- lays of about 200. Even more important was the indication that the curve w^as extremely flat in its optimum region, giving but a small change in the total penalty if an error, even as great as 2 to 1, were made in the number of codes to be used. 6.2 THE DETAILED PROCEDURES OF CODE SELECTION Background information as gained above helps in guiding the early steps in picking codes of relays for the job, but there is a better method of picking codes when the circuit designer is actually down to cases. By following the steps below, the optimum number of codes will auto- matically result. The first step in the coding program is to pick a list of basic codes. This can be done quite arbitrarily, initially, and should be based on general knowledge of types of circuit functions, together with specific knowledge of certain uses with very large demands. For example, certain coils will be needed to cover the range from very fast operation with no concern for current drain, to slow release and emphasis on econom}^ of power; there will be single and multiple windings as dictated b}^ circuit functions. Also, certain arrangements of contacts may be assumed to span the range of needs from a single make up to the capacity of the design — twelve transfers or twenty-four makes in the case of the new wdre spring relay. Further, certain combinations of springs and coils will be evident at once as ideally suited to particular large-demand uses, and these should be on the original list. Soon the likel}^ minimum demand for each will be quite e\ddent. Such a list of tentative codes and demands is the first step in the coding scheme. Now, as each circuit application arises, this basic list maj^ be consulted. For some cases, an available code will be ideal, but for many others a new arrangement may be desired. In each such case a simple calculation will show the economical choice between a new code or the old code with more features than are really needed. Steps in the calculations are given ECONOMICS OF TKLEPHONK RELAY APPLICATIONS 255 l)('lo\\, where th(^ (^xttMision of the old code is identified as Plan A, and the use of a new code as Plan B. For Plan A, there will be a new demand, (A' + 1'), which is greater than its prex'ious demand hy )', the quantity of the new application. TIu^ demand cost penalty can then be read from charts such as Fig. 4. Performance penalties for pertinent featin"(\s such as speed, power, extra contacts, etc. may be read from charts of the kind described in the earlier sections abo^'e. The sum of these \'ahies, multiplied by Y, is \\w, cost penalty for the now application of an existing code. For Plan B, there will b(> a lot-size cost penalty due to its demand. But there will also be a penalty imposed on the original code, (whose demand was A^), because its demand was not enlarged to value X -f Y. Thus the total lot-size cost penalty is made up of two parts: (a) Penalty due to demand Y, multiplied by Y. (b) [Penalt}' due to demand A" — Penalty due to demand (X + F)] multiplied by X. The unit lot-size cost penalty for Plan B is the sum of these two fac- tors, divided by the quantity Y. The design of relay for Plan B may be chosen by the methods previously outlined to be as nearly optimum as is feasible, and then performance penalties itemized as in Plan A above. The sum of these penalties, added to the lot-size penalties just men- tioned, give a number for comparison with Plan A. After the penalties due to either plan are compared, the least costly should be chosen. In cases where the costs are approximately equal, preference should probably be given to Plan A, as encouraging stand- ardization. In summary, the ideal number of codes in a newly designed system Table IV — Check List for Relay Code Selection Amount of Equivalent Cost Penalty Kind of Penalty Plan A PlanB Lot-size (a) (b) Note (1) Note (2) _ [(2) - (1)]X f = Note (3) Power Speed Extra parts Other Total Note (1): Fill in for demand X + Y. Note (2) : Fill in for demand A'. Note (3) : Fill in for demand Y. 250 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 may be approximated by choosing the more economical alternative re- sulting from filling in the check list given in Table IV. Part VII — • Summary In the telephone central office some 60,000 relays are needed to carry out the automatic switching functions. Since the entire office is built around them, they determine the cost of the office, not only by their first cost, and maintenance cost, but at least equally importantly by their influence on the power plant, and the amount of common control equipment that is needed. There is opportunity for great economy, over- all, when manufacturing cost is put in proper balance with the cost of power, cost of functioning time, and similar performance variables. The preceding pages have shown how each factor in the total system cost may be stated in a common language — the equivalent first cost value. When each variable, expressed as an incremental figure denoting the cost compared to an ideal standard, is so stated, all relay factors in the switching system may be cross-compared. This permits a large num- ber of optimizing procedures to be carried out. Optimizing methods for several important cases have been illustrated, and relations or procedures of value to the relay designer are presented. Particularly outstanding cases are the means for realizing (a) power- plus-coil economy and (b) power-plus-speed economy. On the side of manufacturing cost, relay designers need some guidance as to how their applications problems will be influenced by the lot-size, or annual demand. Optimizing procedures are also given for this prob- lem, to yield approximate values for optimum lot-sizes under various manufacturing conditions, and the corresponding cost penalty as a func- tion of annual demand. The steps to be taken in order to strike the best balance between standardizing for maximum manufacturing economy and diversifying for maximum performance economy are also given. The problem as a whole is a striking example of how design for service can be applied on a very large scale. The economies realized by the ap- proach used here have avoided the expenditure of many additional thousands of dollars yearly to the telephone companies, with corre- sponding economy to the customer. ACKNOWLEDGMENT Appreciation is here expressed to R. L. Peek, Jr., and Mrs. K. R. Randall for helpful technical suggestions, and also to Mrs. Randall for help with the preparation of figures and examples. Symbols (iM:inuscrij)l received October 20, 1953) The following list gives the symbols used through the several articles appearing in tliis issue of the Journal. The list does not include all the variant forms of the different symbols distinguished by subscripts, as these distinctions are indicated in the context where they are used. MECHANICAL : a Cross sectional area C Length m Effective armature mass T Ivinetic energy T' Work done to overcome static load X Armature displacement ELECTRICAL : e Copper efficiency or fraction of the copper volume occupied by conductor E Applied battery voltage G Total eciuivalent single turn conductance Gc Eclui^'alent single turn conductance of coil; N /R Ge Eciuivalent single turn eddy current conductance G'e Effective single turn eddy current conductance; Ges'^^'^^^^^^^ Gs Eciuivalent single turn conductance of sleeve i Instantaneous current / Steady state current m Mean length of turn in winding A^ Number of turns in winding 4x Li Single turn inductance; ,. ^ NI Ampere turns NIq Just operate or just release ampere turn value q Ratio of just operate to final ampere turns; -^-^ 257 258 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 p Resistivity of material R Resistance S Coil volume V Ratio of flux attained at time t to steady state flux W Power; I'R MAGNETIC : A Equivalent pole face area J. 2 Effective pole face area (design value) B Induction (flux density) B' Value of B for maximum permeability B" Saturation density Bm Density at m = 1000 Br Remanence (density at i/ = 0) ^ (Ro 4" (Rl (Ro F Magnetic pull S? Magnetomotive force; -Itt^V/ CFc Coercive magnetomotive force H Field intensity He Coercive force L Inductance H Permeability Ha Permeability of air h' Maximum value of permeability (R Reluctance (Re Reluctance of the core (Re Minimum value of core reluctance (Ri (R(x) for X ^ Xi (Rj Initial incremental reluctance (Ro Equivalent closed gap reluctance (Rz, Equivalent leakage reluctance (Rlo Effective leakage reluctance (design value) (Ro2 Effective closed gap reluctance (design value) (Rl f (Ro + J 6i(x) Equivalent relay reluctance ^^ (Ro + (Rl + I (p flux $ Steady state flux NOTATIONS 259 ifi Initial equilibrium flux V?' Flux for maximum permcalMlity or minimum reluctance ^p" Flux at saturation (fa l?osidual flux tpG Gap flux r Field energy u .r/.4(Ro or x/x^ W Mechanical work done by magnet .To A(Ro TIME : t Time U Operate or release time /i Waiting time t-i INIotion time ^3 Stagger time Ie Eddy current time constant; LxG' e tc Winding time constant; L\Gc is Sleeve time constant; LiG s Abstracts of Bell System Technical Papers Not Published in this Journal Bennett, W.^ Telephone-System Applications of Recorded Machine Announce- ments, A.I.E.E., Trans., Commiin. et Electr(3nics 8, pp. 478-483, Sept., 1953. BiELiNG, D., see D. Edelson. Biggs, B. S.^ and W. L. Hawkins^ Oxidative Aging of Polyethylene, Modern Plastics, 31, pp. 121-122, 124+, Sept., 1953 (Monograph 2155). Thermal oxidation of polyethylene follows the pattern set by lower homo- logues such as paraflinic waxes and oils. It is an autocatalj'tic free radical chain reaction and is subject to inhibition b}' typical antioxidants. The rate of degradation in the dark at room temperatures is found to be extremely low. Photo-oxidation of polj'ethjdene is rapid in contrast with that of sat- urated low molecular weight aliphatic l\ydrocarbons. Furthermore, anti- oxidants are of little benefit in protecting against exposure to light. Opaque pigments are of great value in reducing the effects of light, finely divided carljon black being particularly effective. By proper compounding polj'- ethylene can be made to last many years outdoors. BozoRTH, R. M., see H. J. Williams. Brown, W. L.^ n-Type Surface Conductivity on p-Type Germanium, Phys. Rev., 91, pp. 518-527, Aug. 1, 1953 (Monograph 2173). * Certain of these papers are available as Bell System Monographs and may be obtained on request to the publication;Department, Bell Telephone Laboi'atories, Inc., 463 West Street, New York 14, N. Y. For papers available in this form, the monograph number is given in parentheses following the date of publication, and this number should be given in all requests. 1 Bell Telephone Laboratories. 260 ABSTRACTS OF TKCHNICAL AUTICLES 261 A positive charge on the surfuce of a p-ty\)e geinumiuin crystal induces a net negative space charge witliin tlie crystal adjacent to the surface. This space charge is coni]iosed of ionized accei)tor atoms and also of electrons under certain conditions. When electrons occur they provide a layer of ?i-tyi)e conductivity immediately under the p-type germanium suiface. Such a layer has been found on the p-type region of some n-p-?i-transistors. In the 7i-p-n structure the layer of electrons appears as an extra conducting path — "a channel" — aci'oss the p-type mateiial between the two n-tyi)e ends. The conductance of a channel and the capacity between the channel and the /)-ty])e material have been measured and com))aied with the the- oretical predictions liased on a sin^ple model, Cornell, L. P., see E. P. Smitpl Crane, G. R.^ F. Hauser^ and H. A. Manley^ Westrex Film Editor, J.S.M.P.T.E., 61, Part 1, pp. 316-323, Sept., 1953. This paper describes a film-editing machine which emploj's continuous pio- jection resulting in ciuiet operation. It accommodates standaid-picture and photographic or magnetic sound film as well as composite sound-picture film. Differential synchronizing of sound and picture while running, auto- matic fast stop and simplified threading features in the film gates with finger-ti]) release materially increase operating efficiency. Crissman, h.^ See Both Sides of the Game, Tele-Tech., 12, p. 84, Sept., 1953. DaCEY, G. C.l AND I. M. PvOSSl Unipolar "Field-Effect" Transistor, I.R.E., Proc, 41, pp. 970- 979, Aug., 1953. Unipolar "field-effect" transistors of a type suggested by W. Shocklej- have been constructed and tested. The idealized theory of Shockle\' has l)een ex- tended to cover the actual geometries involved, and design nomographs are presented. It is found that these structures can be designed in such a way as to jdeld a negative resistance at the input terminals. The characteristics of several units are presente<:l and analyzed. It is shown that these character- istics are in substantial agreement with the extended theory. Finally a specu- lative evaluation of the possible future applications of field effect transistors is made. 1 Bell Telejihone Laljorat cries. 3 Western Electric Company. * Westrex Corporation. 262 the bell system technical journal, january 1954 Edelson, D.^, C. a. Bieling^ and G. T. Kohman^ Electrical Decomposition of Sulfur Hexafluoride, Ind. and Eng. Chem., 45, pp. 2094-2096, Sept., 1953. Ehrbar, R. D., see C. H. Elmendorf. Elmendorf, C. H.^, R. D. EhrbarI, R. H. Klie^, and A. J. Grossman^ The L3 Coaxial System, A.I.E.E., Trans. Commun. & Electronics, 8, pp. 395-413, Sept., 1953. The L3 coaxial system is a new broad-band facilit}' for use with existing and new coaxial cables. It makes possible the transmission of 1,860 telephone channels or 600 telephone channels and a television channel in each direction on a pair of coaxial tubes. The principal system design problems and the methods used in their solution are described in terms of its components and their location in the system. Fine, M. E.^ Cp-Cv in Silicon and Germanium, Letter to the Editor, J. Chem. Phys., 21, P. 1427, Aug., 1953. Frayne, J. G.^ AND E. W. Templin^ Stereophonic Recording and Reproducing Equipment, J.S.M.P.T.E., 61, Part 2, pp. 395-407, Sept., 1953. This paper describes new stereophonic recording channel equipment includ- ing a si.x-position mixer and portable three-channel recorder. For re-record- ing, the previously described triple-track recorder-reproducer is available. For review-room and theater reproduction, a theater-type dummy equipped for three-channel stereophonic reproduction is described. Goertz, M., see H. J. Williams. Gramels, J.^ Problems to Consider in Applying Selenium Rectifiers, A.I.E.E., Trans., Commun. & Electronics, 8, pp. 488-492, Sept., 1953. Gray, A. N.^ Room-Temperature Compound Process, Mech. Eng., 75, pp. 625- 628, Aug., 1953. 1 Bell Telephone Laboratories. ' Western Electric Company. ^ Westrex Corporation. ' Sandia Corporation. ABSTRACTS OF TECHNICAL ARTICLES 263 Grossman, A. J., see C. H. Elmendorf. Hagstrum, H. D.^ Electron Ejection From Ta by He+, He++, and He2+., Phys. Rev., 91, pp. 543-551, Aug. 1, 1953 (Monograph 2148). Measurements of total yield (7^) and kinetic eneifry distribution are re- ported for electrons ejected from tantalum by the ions, He+, Hc"^ and Hej"^ in the kinetic energy range 10 to 1000 ev. The evidence ])resented indicates that the electrons are released by a collision of the second kind of the ion with the metal surface (potential ejection). One internal secondary electron is produced per incident ion. The probabilit}- of this electron escaping is reduced by the possibility of internal reflection at the image barrier at the metal surface. 7.- for the slowest ions is observed to be 0.14, 0.52, and 0.10 for He"*", He++, and He2+, respectively. The data presented must be considered representati^•e of gas-covered tantalum, since no gas was observed to desorb from the target on heating to 175°K. yi was found not to vaiy with time after cooling the target indicating rapid re-establishment of the equilibrium gas layer on the surface from within the metal. The work function of the covered Ta surface is found to be ca 4.9 ev, some 0.8 ev higher than that of atomicalh' clean Ta. Considerations based on a theory which includes variation of energy levels near the metal surface show resonance neutrahza- tion of He"^ at the covered Ta surface not to be possible. Thus onh- the so- called direct process of potential ejection occurs, with which conclusion the •measured energy limits of ejected electrons are in agreement. Hauser, F., see G. R. Crane Hawkins, W. L., see B. S. Biggs. Heffner, H.^ Backward -Wave Tube, Electronics, 26, pp. 135-137, Oct., 1953- Electron-stream amplifier utilizing l)ackward-wave mode forms microwave oscillator continuously tvmable o\'er a three-to-one bandwidth b}' a single voltage control. Tubes have been built for frequency centering about 6,000, 10,000 and 50,000 mc. Henderson, O.^ Magnetic Amplifier Controls for Rectifier Protecting Underground Metallic Structures Cathodically, Corrosion, 9, pp. 216-220, July, 1953. This paper covers the initial attempt on the part of the Ohio Bell Telephone 1 Bell Telephone Laboratories. 5 Ohio Bell Telephone. 264 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Company to use a magnetic ami)lifier to give continuous contiol of the out- put of a co])per-oxide rectifier used for cathodic protection of undergi-ound lead covered cables. The controlled rectifier was designed foi- use at a loca- tion where straj' current "end effects" were damaging underground tele- plione caliles and municipal light cables and were threatening high-pressure water mains. When properlv adjusted, the output of the rectifier will in- crease or decrease automatically so that at each instant the amount of forced drainage will be adequate to protect the underground telephone cable sheath but will not be in excess of the value at which neighboring under- ground metallic structures would become anodic and would thus become subject to corrosion. For satisfactory operation the amplifier had to be designed to give a large gain so that a change in the control voltage of onlj' 0.2 volt (+0.1 to —0.1 volt) would be sufficient to vary the output of the rectifier from practically zero current to full rating. In the installation de- scribed in this paper the gain of the magnetic amplifier is in the order of 12,000. The magnetic amplifier type of control is well suited to outdoor in- stallations subject to wide changes in temperature and at remote locations where frequent maintenance inspections are not feasible. The magnetic am- plifier has advantages over other available control devices which accomplish this same purpose in that it has no moving parts, no vacuum tubes, bat- teries, motors, relaj's or contactors. The magnetic amplifier gives a continuous output control which is superior to the stepped increment changes that result from relay or contactor operation. Jerone, M. G., see E. P. Smith. Johnson, J. B.^ and K. G. McKay^ Secondary Electron Emission of Crystalline MgO, Phys. Rev., 91, pp. 582-587, Aug. 1, 1953 (Monograph 2163). Secondary emission is measured from single crystals of ^IgO cleaved along the (100) plane. The maximum ratio of secondary to primary current, Smax , is about 7 at about 1,000 volts and room temperatures. The cross-overs are at 33 volts and far above 5,000 volts. Most probable energj- of emission is 1 ev or less. A definite effect of temperature is established, decreasing with increasing temperature, in accord with expectations for an insulator. Jones, T. A.^ and W. A. Phelps^ Level Compensator for Telephotograph Systems, Elec. Eng., 72, pp. 787-791, Sept., 1953. To eliminate interference in telephotograph transmission through broad- band carrier equipment, it was decided to cancel it from the signal delivered by the carrier facilitj' instead of modif3-ing the carrier ecjuipment. Conse- 1 Bell Telephone Laboratories. ABSTRACTS OK TKCIIN'ICAL ARTICLES 265 ciuently, a I'ocently (lcvel()])0(l teleijhotogi'apli level conipensator, consisting of a pilot channel arrangement tlesigned lor insertion in the tele])li()tograi)h connecting circuits, is utilized. KlSTLER, R. E.'^ Radio Links U.S. and Canada, 'rolophoiiy, 145, pp. 1()-17, 33, Sept. 5, 1953. Klie, Pv. H., see C. H. Elmendorf. KoHMAX, G. T., see D. Edelson. Kruse, p. F., Jr." and W. B. Wallace^ Identification of Polymeric Materials, Anal. Chem., 25, p. 1156, Aug., 1953. KuH, E. S.i Potential Analog-Network Synthesis for Arbitrary Loss Functions, J. App. Phys., 24, pp. 897-902, July 1953. A general method is developed for designing networks with arbitrary loss functions based on the potential analogy. An appropriate potential prob- lem is formed on the basis of the given loss function by introducing con- tinuous charge distribution on the complex freciuency plane. After the po- tential problem is solved, the technique of quantization of charge is used to find the natural modes of the network function. Laxge, R. W.i 40- to 4,000-Microwatt Power Meter, A.I.E.E. Trans., Commun. c^- Electronics, 8, pp. 492 494, Sept., 1953. Lewis W. D.^ Electronic Computers and Telephone Switching, TR.E., Proc, 41, pp. 1242-1244, Oct., 1953. Automatic telephone switching and digital computation have much in common. Both rely upon discrete rather than continuous devices. Develop- ment of recent switching systems with a close functional reseml^lance to large digital computers has increased this overla]). The next big step in 1 Bell Telephone Laboratories. ^Pacific Telephone and Telegraph Company. ' Sandia Corporation. 266 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 telephone switching should be towards electronics. In making this step, switcliing scientists and engineers can be much helped by modern electronic computer technology. To be successful, they must also contribute to this technology. LiNDHOM, P.^ "Feedback" — Heart of Automatic Process Control, Factory, 111, pp. 106-109, Oct., 1953. Logan, R. A.^ Thermally Induced Acceptors in Single Crystal Germanium, Letter to the Editor, Phys. Rev., 91, pp. 757-758, Aug. 1, 1953. Maita, J. P., see M. Tanenbaum. Mallery, P.i Transistors and Their Circuits in the 4A Toll Crossbar Switching System, A.LE.E. Trans., Commiin. & Electronics, 8, pp. 388-392, Sept., 1953. Malthaner, W. A} AND H. E. Vaughan^ Automatic Telephone System Employing Magnetic Drum Memory, I.R.E., Proc, 41, pp. 1341-1347, Oct., 1953 (Monograph 2151). The use of magnetic drum memory in an automatic telephone switching office is described. A capacitive scanner acts as a time-division connector through which information generated by subscribers' telephone sets is con- ve3^ed to storage on magnetic drums. Information thus accumulated is com- bined with "permanent" information on the magnetic drums, processed in accordance with built-in programs and dispatched to control call switching circuits. Technical feasibilitj' of this system has been demonstrated by the construction and successful operation of a large-scale laboratory model. Manley, H. a., see G. R. Crane. Matthl\s, B. T} Superconducting Compounds, Letter to the Editor, Phys. Rev., 91, p. 413, July 15, 1953. McAfee, K. B., see K. G. McKay. 1 Bell Telephone Laboratories. ^Western Electric Company. abstracts of technical articles 267 McCarthy, R. H.» Organization for Production Engineering, Mech. Eng., 75, pp. 785- 788, im. Oct., 1953. McGuiGAX, J. H.^ Combined Reading and Writing on a Magnetic Drum, I.R.E., Proc, 41, pp. 1438-1444, Oct., 1953 (Monograph 2152). This ])aper points out that the characteristics of magnetic recording make it possible to combine reading and writing in the same cell as it passes just once under the head. Amplifier requirements for this method of opei'ation are discussed and a suital)le design i)resented. A single head is used for l)oth reading and writing. The i)rocess can be repeated in every successive cell at a cell rate of 60 kc. The techniques described, which are applicable to either parallel or serial systems, extend the utility of magnetic drums by allowing date processing as well as data storage. McKay, K, G., see J. B. Johnson. McK\Y, K. G.i AND K, B. McAfee! Electron Multiplication in Silicon and Germanium, Phys. Rev., 91, pp. 1079-1084, Sept. 1, 1953 (Monograph 2162). Electron multipUcation in silicon and germanium has been studied in the high fields of wide p-n junctions for voltages in the pre-breakdown region. Multiplication factors as high as eighteen have been observed at room tempei'ature. Carriers injected b}' light, alpha particles, or thermal-genera- tion are multiplied in the same manner. The time required for the midti- plication process is less than 2 X 10"* second. Approximatelj' equal multi- plication factors are obtained for injected electrons and injected holes. The multiplication increases rapidly as "breakdown voltage" is approached. The data are well represented by ionization rates computed by conventional avalanche theory. In very narrow junctions, no observable multiplication occurs before Zener emission sets in, as previoush' reported. It is incidentally determined that the efficienc}- of ionization by alpha particles bomliarding silicon is 3.6 ± 0.3 electron volts per electron-hole pair produced. ISIcSkimin, H. J.i Measurement of Elastic Constants at Low Temperatures by Means of Ultrasonic Waves Data for Silicon and Germanium Single Crystals and for Fused Silica, J. Ai)p. Phys., 24, pp. 988-997, Aug., 1953 (:\Ionograph2171). 1 Bell Telephone Laboratories. 'Western Electric Company. 268 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 Ultrasonic waves (shear on longitudinal) in the 10-30 mc range are trans- mitted down a fused sihca rod, through a polystj-rene or silicone one-quarter wavelength seal, and into the solid specimen. Measurement of reflections within the specimen yields values for velocities of propagation and elastic constants. Data obtained over a temperature range of 78° to 300°K for silicon and germanium single crystals, and 1.6° to 300°K for fused silica are listed. For the latter, a high loss is noted, with an indicated maximum near 30°K. Merz, W. J.i Double Hysteresis Loop of BaTiO:; at the Curie Point, Phys. Rev., 91, pp. 513-517, Aug. 1, 1953 (Monograph 2166). It is known that the Curie point of the ferroelectric BaTiOs shifts to higher temperatures when a dc bias field is applied. If the crystal shows a sharp transition, we expect by applying an ac field at the Curie temperature that the crystal would become alternately ferroelectric and nonferroelectric in the cycle of the ac field. This can be seen in the shape of the hysteresis loop at temperatures slightly above e. In the center of the polarization P versus field E plot, we observe a linear behavior corresponding to the paraelectric state of BaTiOs above 9. At both high voltage ends, however, we observe a hysteresis loop corresponding to the ferroelectric state. A change in tempera- ture causes a change in size and shape of the double hysteresis loops, ranging from a line with curves at the ends (higher temperature) to two ovei lapping loops (lower temperature). The results obtained allow us to calculate the different constants in the free-energy expression of Devonshire and Slater. One of the results shows that the transition is of the first order since the P* term turns out to be negative. The properties of the hysteresis loops are discussed, especially the large spontaneous electrical polarization and the low coercive field strength. Moore, E. F., see C. E. Shannon. Morrison, J.^ Leak Control Tube., Rev. Sei. Instr, 24, pp. 564-547, July, 1953. Phelps, W. A., see T. A. Jones. Prince, M. B.^ Experimental Confirmation of Relation Between Pulse Drift Mobility and Charge Carrier Drift Mobility in Germanium, Phys. Rev., 91, pp. 271-272, July 15, 1953 (Monograph 2168). Experimental data of drift moliilities of minority carriers in germanium are 1 Bell Telephone Laboratories. ABSTRACTS OF TECHNICAL ARTICLES 269 hiouftht into agreement with tlieDretical i)redi('tions l)y distinguishing be- tween grou]) velocity and i)aiti('le velocity of a jnilse of niinoi'ity caiiiers. Corrected high tenii)eiatuie measurements of electron di'ift mobility- are consistent witli the theoietical prediction ^ = AT^^'-. The experimentally determined value of A is 2.0 X 10" cm- deg^/-/v()lt-sec. Reiss, H.i Chemical Effects Due to the Ionization of Impurities in Semicon- ductors, J. C1iem. Phys., 21, p. 1209, July, 1953 (Monograph 2172). This ])aper contains a theoretical account of the possible effects which the ionization of donor and acce])tor im]mrities can induce in the thermod}-- namic phase relations involving their solutions with semiconductors. These effects reflect the energy band structure of the semiconductor. Furthermore, the phenomenon has an interest of its own, for within a certain range of exi)erimental conditions the effects can be attributed to a chemical-like, mass action behavior of the electrons which play the roles of negati\'e ions. Section V is a brief discussion of a fine point concerning the Fermi level. It is shown that although the Fermi level is certainly the electronic electro- chemical potential, it is not the Giljbs free energy per electron unless the density of electron energy levels is linear in the volume of the system. RoBBixs, R. L.i Measurement of Path Loss Between Miami and Key West at 3675 mc, I.R.E., Trans., P.G.A.P. 1, pp. 5-8, July, 1953. Radio transmission measurements ha\'e been made at 3,675 megac3'cles on the 130-mile path between ^Nliami and Key West, Florida, which is largely over watery wastes of the Everglades and shallow sea waters of the Florida Keys. Path loss and fading characteristics for this terrain were not found to differ materially from the characteristics of hilly oi' mountainous paths in the northeastern section of the country. Ross, I. 'SI., see G. C. Dacey. Shanxox, C. E. Turing's Formulation of Computing Machines and Von Neumann's Models of Self-reproducing Machines., I.R.E., Proc, 41, pp. 1235- 1241, Oct., 1953 (Monograph 2150). This paper reviews briefly some of the recent developments in the field of automata and non-numerical computation. A number of typical machines are descril^ed, including logic machines, game-]:)laying machines and learn- ing machines. Some theoretical (luestions and developments are discussed, such as a comparison of com])uters and the brain. 1 Bell Telephone Laboratories. 270 the bell system technical journal, january 1954 Shannon, C. E.^ and E. F. Moore^ Machine Aid for Switching Circuit Design, I.R.E., Proc, 41, pp. 1348- 1351, Oct., 1953 (Alonograph 2153). Design of circuits composed of logical elements may be facilitated by auxili- ary machines. This paper describes one such machine, made of relaj's, se- lector switches, gas diodes, and germanium diodes. This macliine (called the relay circuit analyzer) has as inputs both a relay contact circuit and the specifications the circuit is expected to satisfy. The anah'zer (1) verifies whether the circuit satisfies the specifications, (2) makes sj'stematic at- tempts to simplify the circuit by removing redundant contacts, and also (3) obtains mathematically rigorous lower bounds for the numbers and types of contacts needed to satisfj' the specifications. A special feature of the analyzer is its ability to take advantage of circuit specifications which are incompletely stated. The auxiliary machine method of doing these and similar operations is compared with the method of coding them on a general- purpose digital computer. Smith, E. P.*', L. P. Cornell^ and M. G. Jerome^ Co-ordinating Ml and Nl Telephone Carrier Systems, Elec. Eng., 72, p. 780, Sept., 1953. Tanenbaum, M.^ and J. P. Maita^ Hall Effect and Conductivity of InSb Single Crystals, Letter to the Editor, Phys. Rev., 91, pp. 1009-1010, Aug. 15, 1953. Templin, E. W., see J. G. Frayne. TowsLEY, L. M., see E. A. Wood. Tucker, C. J., Jr.^ Emergency Reporting System Installed by Southern Bell, Telephony, 145, pp. 26, 38, Aug. 29, 1953. New fire and emergency' rei)orting system utilizing telephone facilities for the general public to make verbal reports of fires and other emergencies — the first of its kind in the nation — was put into service in Miami, Fla., on Aug. 1, by Southern Bell Telephone & Telegraph Co. Van Roosbroeck, W.^ 1 Bell Telephone Laboratories. ® Pacific Telephone and Telegraph Company. * Southern Bell Telephone Company. ABSTRACTS OF TECHNICAL ARTICLES 271 Transport of Added Current Carriers in a Homogeneous Semicon- ductor, Phys. Rev., 91, pp. 282-289, July lo, 1953 (iMonograph 2105). Takinji into account the thermal equilibrium minority carrier concentration anil emplovinif the formulation which includes, as one of two fundamental ecjuations, the continuity equation for added carrier concentration Ap, tliis e(iuation is derived in a form which exhibits the ambi])olar natuie of the difYusion, drift, and recombination mechanisms under electrical neutrality. The geneial concentration-dei)en(lent diffusivity is given. The local drift velocity of Ap has the direction of total current density in an n-tj^pe semi- conductor and the reverse in a p-tji^e semiconductor, differing in general in both magnitude and direction from the minoi'ity-carrier drift velocity. Specifying a model for recombination fixes the dependence of a lifetime function for Ap on Sp and the electron and hole mean lifetimes. Negative Ap, or carrier depletion with electrical neutrality, may occur. For known total current density, the continuity equation alone suffices, as for the case of I Ap I small, for which the ec^uation is linear. A condition for this com- parativeh' important case is derived, and theoretical relationships are gi\'en with the aid of a parameter specifying the Fermi level, which determine for germanium the minority carrier-Ap drift velocity ratio as well as the ambi- polar diffusivity and group mobility in terms of resistivity and temperature. Vaughax, H. E., see W. A. Malthaner. VOGELSONG, J. H.^ Transistor Pulse Amplifier Using External Regeneration, I.R.E., Proc, 41, pp. 1444 1450, Oct., 1953. Pulse-regenerative amplifier using a point-contact transistor has been op- erated at a basic frequenc}' of 3 megacycles. To i)roduce regenerated pulses with waveshapes which are practicalh' independent of the wa\-eshapes of the input pulses, a germanium diode circuit has been used in conjunction with an external feed-back path. This arrangement also provides for the s^mchronization of the output pulses with a master clock. Transformer coupling has been incorporated into the circuit to provide dc restoration. Walker, A. C.^ Hydrothermal Synthesis of Quartz Crystals, Am. Ceramic Soc, J., 36, pp. 250-256, Aug., 1953 (Monograph 2146). Research at the Bell Telephone liaboratories on the problem of growing large single crystals of quartz has now i)rogressed to a point where it is possible to grow crystals weighing more than 1 lb. each in a period of 60 daj's or less. Equipment now in use includes autoclaves 4 inches in inside diameter and 4 ft. long, weighing about 1,150 11). each. In developing the 1 Bell Telephone Laboratories. 272 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 liydiothermal process used to grow these quartz crystals it has been neces- sary to soh'e man}- problems in the little-known field of high pressure. The results point to the possibility of growing other t3'i)es of crystals, and the field of usefulness of this process now appears to be much more extensive than was the case at the beginning of the investigation in 1946. Of prime importance is the fact that crystals grown from solution are likely to be better formed and of more perfect quality than those grown from the melt or by other methods. Many of the difficulties inherent in this work have been due to corrosion of steel in alkaline solution. This was a rather unex- pected problem, since in high pressure steam boilers alkali is added in small amounts to prevent corrosion of the boiler tubes. Such corrosion has been shown to be responsible for the appearance of electrical twinning on the growing faces of the quartz crystals. Other causes of such twinning have also been found in the course of this work. Walker, L. B.} Starting Currents in the Backward-Wave Oscillator, J. App. Phys., 24, pp. 854-859, July, 1953. The starting current of a simple model of the backward-wave oscillator described by Kompfner and Williams has been calculated. The effect of space charge is included. The starting current Jo may be written in the form 4Fo 2 .V where Vo is the beam voltage, Zq is the impedance of the circuit, .V is the length of the oscillator in wavelengths measured on the circuit and ao(4QC) is a dimensionless quantity which has been evaluated as a function of the space-charge parameter -iQC. Wallace, W. B., see P. F. Kruse, Jr. Washburn, S. H.^ Application of Boolean Algebra to the Design of Electronic Switching Circuits, A.I.E.E. Trans., Commun. & Electronics, 8, pp. 380-388, Sept., 1953. Werner, D. R.- Effects of Polarization on Telephone Cable Buried Through a Salt Bed, Corrosion, 9, pp. 232-236, July, 1953. Cathodic protection has been applied to a copper jacketed cable in a salt lake bed about one mile wide in which the earth resisti\-it3' was apparently uni- 1 Bell Telephone Laboratories. ^ American Telephone and Telegraph Company. ABSTRACTS OF TECHNICAL ARTICLES 273 form at about 20 ohm-i-outimeters. Cunent of about one ampere flowed on the copi)er jacket into the low resisti\-ity areas on eithei' side and was found to be sustained l)y ])olarization effeets when the cathodic protection current was removed. The current loss on the coi)])er jacket was found to be concentrated in an area about 720 feet wide, 360 feet each side of the point wheie the cathodic protection current had been drained from the cop])er jacket. The copper jacket to soil ])otential tested most negative to a co])per sulfate half cell in the 720-foot area where the current loss was con- centrated and was of the order of —1.0 to —1.1 volts. Permanent remedial measures will consist of installations of magnesium anodes distributed tluoughout the low earth resistivity' area and insulating joints in the copper jacket at locations wheie large changes in the earth resistivity occur. WiLLLX.MS, H. ,1.\ l{. M. BOZORTH^ AND M. (loERTZ' Mechanism of Transition in Magnetite at Low Temperatures, PHn^s. Rev., 91, pp. 1107-1115, Sept. 1, 1953 (Monograph 2149). When magnetite is cooled through — 160°C it is known to undergo a transi- tion (cubic to orthorhombic) that is influenced l)y the pi'esence of a mag- netic field. Our experiments are in agreement with the following mechanism of the transition : The orthorhombic c axis is parallel to one of the original cubic axes and is the axis of easiest magnetization. Generally, different regions of the original crj'stal will transform with their c axes lying along different cubic axes, and when no field is applied there are 6 different orien- tations which different regions assume. When a field is applied during a cooling a c axis tends to lie along the original cubic axis that is nearest to the applied field, the a and b axis having less but different tendencies to lie parallel to the field. Six magnetic cr^-stal anisotrojjy constants are derived from torque curves measured in the (100) and (110) planes. From them magnetization curves ai-e calculated for the 100 and 110 dii-ections, and these are in agreement with ex])eriment. Wood, E. X} and L. M. Towslky' Manganese Film-Shield for FeK X-Rays, Rev. 8ci. Instr., 24, p. 547, July, 1953. 1 Bell Telephone Laboratories. Contributors to this Issue Akthur C. Keller. B.S., Cooper riiiou, 1923; M.S., Yale I'liiversity, 1925; E.E., Cooper Union, 1920; Columbia University, 1920-30; Western Electric Company, 1917-25; Bell Telephone Laboratories, 1925-. Special Apparatus Development Engineer, 1943; Switching Apparatus Develop- ment Engineer, 1946; Assistant Director of Switching Apparatus De- velopment, 1949; Director of Switching Apparatus Development, 1949-. Mr. Keller's experience in the Bell System includes development and de- sign of telephone instruments ; development of systems and apparatus for recording and reproducing sound; and, during World War II, the de- A'elopment, design, and preparation for manufacture of sonar systems and apparatus. His department, in addition to being responsible for a numl^er of military projects, is responsible for the fundamental studies of switching apparatus and the development, design, and preparation for manufacture of electromagnetic and electromechanical switching appara- tus for telephone systems. Member of the American Physical Society, A.I.E.E., Acoustical Society of America, I.R.E., S.M.P.T.E., and the Yale Engineering Association. Representative for Bell Telephone Laboratories in the Society for Experimental Stress Analysis. For his contributions to the Navy during World War II, he received awards from the Bureau of Ships and the Bureau of Ordanance. Mason A. Logan, B.S. in Physics and Engineering, California Insti- tute of Technology, 1927; M.A. in Physics, Columbia University, 1933; Bell Telephone Laboratories, 1927-. His early Laboratories' projects were concerned with wire transmission problems particularly those of circuits, noise and cross induction in local, manual and dial telephone circuits. This was followed b}^ circuit research on alternating current methods of signaling including the use of non-linear elements and electronic terminal equipment. From 1941 to 1948 he worked on military projects, including a mine fire control system, anti-aircraft gun director, magnetic proxim- ity fuses, and guided missiles. For the past Hxe years he has been a member of the SAvitching Apparatus Development Department in which he is supervising a group concerned with static and dynamic behavior of 274 CON ruim TORS to this issue It.) new electromagnets and relays. lie is also engaged in in\('stigalii)ns of the performance of electrical contacts on tele})h()ne rchiA-s. KoHKKT L. Pkkk, ,Ih., A.B. and Met.E., Colnnihia University, Hl'JL and 1923; Bell Telephone Lal)oratories, 1924-. In the C'hemical Research Department and later the Apparatus Development Department, Mr. Peek's work related to the analytical and testing aspects of mate- rials de\-elopm(MU. Since 193() he has been engaged in apparatus de- \'elopment projects, including the wire spring relay and, during World War II, underwater ordnance and magnetostriction sonar. H. X. Wagar, B.S. in Physics, Harvard University, 1926; :\I.A. in Physics, Columbia University, 1931; Bell Telephone Laboratories, 1926-. Mr. Wagar has worked on the design of nearl}^ all types of tele- phone relays as well as magnets, insulating methods and allied apparatus and practices. He was also associated with the preparation of text and presentation of training courses in this field. During World War II his projects for the military included work on an antiaircraft director and a proximity fuse for magnetic mines. He is currently Switching- Apparatus Engineer, in charge of fundamental studies on electromagnetic switching apparatus, including contacts and magnetics. HE BELL SYSTEM Jem / mcai ournai I^W^ AN E VOTED TO THE SCIENTIFIC ^^^ AND ENGINEERING SPECTS OF ELECTRICAL C O M ^JU N I C AT I O N / '■--■'' — ■ — — — ^——M ^^^— — a—pw^—i — 3LUME XXXIII MARCH 1954 NUMBER 2 TraflBc Engineering Techniques for Determining Trunk Require- ments in Alternate Routing Trunk Networks c. j. truitt 277 Intertoll Trunk Concentrating Equipment d. f, Johnston 303 The Transistor as a Network Element j. t. bangert 329 Continuous Incremental Thickness Measurements of Non- Conductive Cable Sheath b, m. wojciechowski 353 The Application of Designed Experiments to the Card Translator c. B. brown and m. e. terry 369 Wave Propagation Along a Magnetically-Focused Cylindrical Electron Beam w. w. rigrod and j. a. lewis 399 DifiFraction of Plane Radio Waves by a Parabolic Cylinder s. o. rice 417 Abstracts of Bell System Technical Papers Not Published in this Journal 505 Contributors to this Issue 514 COPYRIGHT 1954 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD S. BRACKEN, Chairman of the Board, Western Electric Company F. R. KAPPEL, President, Western Electric Company M. J. KELLY, President, Bell Telephone Laboratories E. J. McNEELY, Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE E. I. GREEN, Chairman A. J. B U S C H F. R. L A C K W. H. DOHERTY W. H. NUNN G. D. EDWARDS H. I. ROMNES J. B. F I S K H. V. S C H M I D T R. K. HON AM AN G. N. T H AY E R EDITORIALSTAFF J. D. T E B O, Editor M. E. STRIEBY, Managing Editor R. L. SHEPHERD, Production Editor THE BELL SYSTEM TECHNICAL JOURNAL is published six times a year by the American Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. Cleo F. Craig, President; S. Whitney Landon, Secretary; John J. Scanlon, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed in U. S. A. THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXXIII MARCH 19 5 4 number 2 Copyright, 19S4, American Telephone and Telegraph Company Traffic Engineering Techniques for Determining Trunk Requirements in Alternate Routing Trunk Networks By C. J. TRUITT (Manuscript received November 23, 1953) In 1945 the Bell Sysfejn embarked on an extensive study with the purpose of developing a program for operator toll dialing on a nationwide basis. Operator toll dialing had been done, of course, on a limited scale in various parts of the country for many years, but the concept of this program was one of nationwide proportions carried on with a uniform numbering plan* arrangement and a completely integrated trunking system which would handle traffic at a high speed between any two points in the United States and Canada, even in the busier hours of the day. Implementation of this program required the development of new switching mechanisms and the exploitation of carrier transmission potentialities to a degree never before achieved. Great strides had already been made in these fields, resulting in the practical development of the coaxial cable syste7n arul the first toll crossbar switehing office installed at Philadelphia in 1943. Bid the very core of the nationwide dialing plan ivas the proposed to revo- lutionize the method of traffic distribution so as to combine high speed handling over theinferloll trtink network with a highly efficient iise of facili- ties. The method of aeeomplishing is called "engineered alternate routing^' * W. H. Nunn, Nationwide Numbering Plan, Communication and Electronics, 2, Sept., 1952 and B. S. T. J., 31, Sept., 1952. 277 278 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 in which a first choice direct route for traffic would he provided with trunks in such numbers as to force a predetermiried portion of the load to seek another route where such residue or overflow would he carried at less cost per unit and with little or no delay. The principles of alternate routing and their application to interoffice tr unking systems are the subject of this paper. PRINCIPLES OF ALTERNATE ROUTING Broadly speaking, alternate routing of telephone traffic is a procedure whereby a parcel of traffic is provided a second choice path to its destina- tion which is also the second choice path for one or more other parcels of traffic. The use in common of this second choice path by the several parcels results in an overall requirement of trunk paths to the particular destination which is less than would be the case if each parcel had sole access to its own group of paths — and this is accomplished without impairment of the average speed with which all of the traffic is handled. The familiar graded arrangements of trunk terminals in dial central office switching systems are examples of the trunking efficiency gained from alternate routing. It should be noted at once that the most efficient way to handle traffic from one central office to another is by means of a single group of trunks to which all traffic so destined has access. How- ever, many such loads are so large that the number of trunks required to handle them on a single group often exceeds the practical terminal capacity of ordinary switching systems. Therefore, it becomes necessary to present portions of such a load, each to an individual subgroup. For example, assume that a busy-hour load of 334 calls of 100 seconds dura- tion each (334 CCS) is to be carried at a probability of delay of one per cent (Poisson) from office M to office N and that the equipment permits access to a maximum of only ten trunk terminals for any one trunk group. Since the number of trunks required for the load in question is 17.5 (18) in a single group the establishment of three individual sub- groups is indicated. To maintain P.Ol service with three individual subgroups requires 8.3 trunks in each, assuming equal division of the load. The efficiency of each subgroup is 13.4 CCS per trunk which is some 30 per cent less than that of the larger single group which is 19.1 CCS. Thanks to graded multiple arrangements,* such a heavy loss in effi- ciency need not be accepted. Pursuing our example, consider the load in question to be presented to an appropriate grade. This is illustrated schematically in Fig. 1. * R. I. Wilkinson, The Interconnection of Telephone Systems — Graded Mul- tiples, B. S. T. J., 10, Oct., 1931. TRUNK KEQITIREMENTS IN ALTERNATE ROUTING NETWORKS 279 'J'lu^ capacity of this particular grade is precisely 334 CCS and its ctiicicncy is 1().7 CCS per trunk or only 12 per cent below the single trunk group efficiency. Here the common trunks serve as an alternate route for such portions of the loads a, b, and c, respectively, as can not be handled by the ti\-e trunks which are individual to each. Because it is not likely that a, b, and c will overflow equal amounts of traffic at the same time, the common trunks are kept busy by a more or less com- plementarj^ pattern of greater and lesser overflows at any given moment, emerging from the three subgroups. It is this action of the overflows, amply substantiated by experience, which accounts for the efficiency of graded multiples. Looked at another way, it may be said that all trunks above five in each subgroup of the split-multiple case have been pooled for the common use of all subgroups and that in so doing, it is possible to reduce the number of pooled trunks from 9.9 to five without GRADED TRUNK MULTIPLE INDIVIDUAL- 20 TRUNKS TO OFFICE N OFFICE M Fig. 1 — Typical graded arrangement of 20 trunks on 10 terminals. impairing the speed of service. This very brief discussion of graded mul- tiples serves merely to point out by familiar example, some of the poten- tialities of the alternate routing principle in the economical handling of telephone traffic. ALTERNATE ROUTING IN LOCAL INTEROFFICE TRUNK NETWORKS The effectiveness of alternate routing as illustrated by its action in graded multiples suggests the possibility of improving the efficiency of trunking between central offices by arranging the offices themselves in a sort of grade. Let us carry the analogy as far as practicable and assume that the loads a, b, and c in Fig. 1 are now emanating from central offices A, B, and C and .still destined for office N. Let us assume that A, B, and C are typical offices in a multi-office city which has a tandem office, T, and further, that every office in the city has a group to and a group incoming from the tandem office. Fig. 2 illustrates these condi- tions with respect to offices A, B, C, and N and for simplicity indicates 280 THE BELL .SYSTEM TECHNICAL JOURNAL, MAKCH, 1954 groups carrying traffic in only one direction. The dashed lines represent the direct triuik groups between central offices and will be referred to as "high usage" groups. The solid lines represent trunk groups in the alter- nate route and will be referred to as "final" groups since they constitute the last choice path for traffic to N. It will be seen that the groups AN, BN, and CN correspond roughly to the groups of individual trunks serving a, b, and c in Fig. 1 and that group TN corresponds to the com- mon trimks. But here the analogy breaks down, for the traffic from A, B, and C, respectively, to X will rarely be equal and there is no counter- part in Fig. 1 to the groups from A, B, and C to T. At this point it becomes clear that the technicjues for determining the capacities of simple graded multiples are of no avail in dertemining the proper number of trunks in each group of such a system. The problem is further com- pficated by the fact that offices A, B, and C will offer to their respective tandem groups traffic for which such groups are the only route. Such traffic would consist of a number of relatively small items destined for other offices, not indicated in Fig. 2, and for which direct routing would be uneconomical. To make such a system work there had to be available at the originat- ing office a switching mechanism capable of testing two or more triuik groups in succession for a single call. The Xo. 1 local crossbar office developed in the 1930's pro\ided such facilities and a trial of the alter- nate routing scheme for trunking between local offices was made at two New York City offices in 1941. The basic aim was to so arrange the trunking layout at each office that the traffic to every other office in the city would be carried as eco- nomically as possible over some combination of a\'ailable routes. This objective reciuired determination of the cost of a direct route from the trial office to every other office and also the cost of any potential alter- nate route. The relationship between the costs of the several routes available to a giA-en item, not the absolute values of the facilities was the important factor. It is not the intent here to discuss the derivation of such factors but rather to describe how they were used in determining the interoffice trunk layout. The end result of the examination of line and switching costs for the several routes was a ratio of the cost of each potential alternate route to the cost of the direct route to each distant office. The smallest of such ratios would determine the alternate route to be used unless the ratio were unity or less, a rare circumstance which would indicate that no direct trunks should be pro\-ided. The next step in the problem was to provide trunks in such numbers and arrangements that the cost of TRUNK REQUIREMENTS IN ALTERNATE ROUTING NETWORKS 281 carrying the load from the originating office to each distant office would be at a minimum. This rccjuired some means of determining how much load would be carried by the direct trunks when offered a given load and consequently how much of that load would l)c overflowed to the alternate route. For this purpose, a formula* known as the Erlang B "lost-calls-cleared" assumption was used. This formula states for a given random offered load, the amount of load which will be carried by each of a number of trunks, n, tested in succession provided that the calls failing to be carried on the first attempt (the lost calls) are not reoffcrcd within the hour during which the first offering took 'place. The condition italicized is extremely important to the problem since it requires that calls lost on the direct high usage group, i.e., the calls overflowed to the Fig. 2 — Illustration of sim]jle interlocal trunk network arranged for alter- nate routing. alternate route, must be disposed of without delay on the alternate route or routes. In the New York City trials it was assumed that the then current basis of provision of trunks in each leg of the alternate route (final groups AT and TN, Fig. 2) namely, with a probability of delay of one per cent. (P.Ol) would, as a practical matter, satisfy the condition that calls overflowing from AX should be cleared. It should be mentioned in passing that the results of the trials substantiated the reasonableness of this assumption. A typical Erlang B distribution is shown in Fig. 3, Curve A wherein the load carried by each of n=l-4 trunks is shown for the condition of 240 offered CCS. Thus, assuming the load to be offered in succession to trunks 1, 2, 3, etc., in that order, it will be seen that the first trunk carries the most load, the second trunk somewhat less, the third still less until the fourteenth trunk carries about 0.5 per cent of the total. By * A. K. Erlang, Solution of Some Problems in the Theory of Prohahilities of Significance in Automatic Telephone Exchanges, Post Office Electrical Engineers' Journal, 10, 1917. 282 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 taking the sum of the loads carried bj^ a stated number of trunks and subtracting it from the total offered load, the amount of load "lost" or overflowed can be obtained. It should be noted here that the load carried by a given number of trunks under the pressure of a constant value of offered load, and provided lost calls are cleared, will always be the same whether or not the load is presented to said number of trunks in consecu- tive order. Thus, seven trunks to which 240 CC8 are presented \\ ill carry 185 CCS (Fig. 3, Curve B) whereas six trunks will carry only 165 CCS, all values to the nearest whole CCS. Therefore, it may be said that the "last" trunk in a group of seven carries 20 CCS under the Erlang B concept when the offered load is 240 CCS, since by adding one trunk to a group of six, the capacity of the group to carry load is increased by 20 CCS. The significance of this notation will appear presently. Let us leave the subject of direct high usage trunk group loading for a moment and consider the loading of final trunk groups such as AT and TN, in Fig. 2. We said earlier that an office such as A originates traffic of such small volume to each of many other offices in the city that direct routing would be economically unsound. Such items in the aggregate constitute a load of considerable size and all are routed via a tandem office for distribution to their respective destinations together with simi- lar small parcels originating at B, C, etc. As stated earlier trunks to and from tandem were customarily provided on a P. 01 delay basis and due to the size of the aggregate loads mentioned above such groups varied in size from about 20 trunks upward. Any traflfic overflowing from direct trunks to an alternate route via tandem would, therefore, require the addition of trunks to these rather large groups already established for the handling of traffic for Avhich the tandem route was the first and only route. It now becomes pertinent to inquire into the capacity of trunks added to the tandem groups to carry rerouted load arriving from the direct high usage groups. From the P.Ol trunk capacity table, it can be shown that adding a trunk to a group of 20 trunks increases the capacity of the group by 27 CCS, to the nearest whole CCS. Similarly adding one trunk to a 40 trunk group increases the capacity of the group by 29 CCS. It was decided in the New York City trials to use a constant average value of 28 CCS as the capacity of any trunks added to groups in an alternate route via tandem in order to accommodate rerouted traffic and still keep the groups on a P.Ol basis for all traffic. We are now in a position to compare the efficiency of a triuik added to a final route (two legs constitute one route) with the efficiency of a trunk added to the direct high usage group. The efficiency of the former is, for practical purposes, a constant, whereas the efficiency of the latter TRI'XK TiEQT'IRF.MF.XTS T\ ALTERNATE ROT'TIXO NETWORKS 283 is a function of the number of trunks provided to handle a gi\cn load on a lost-calls-cleared basis. Referring once more to Fig. 2, let us assume that the ratio of tlie cost of an incremental path in ATN to the cost of an incremental path ill AN is 1.4. It may then be stated that it costs 1.4 times as much to handle traffic on the alternate route as on the direct route. The cost ratio is computed on the basis of values of incremental trunks because the ultimate question in the economical separation of load between a direct and an alternate route is whether one more trunk should be added to the direct route or should more trunk capacity be provided in the alternate route to handle a marginal portion of the total load. Let us assume further that the offered load, A to N, is 240 CCS and that the efficiency of incremental trunks in the alternate route is 28 CCS per trunk. With these three factors, the offered load, the cost ratio and the effi- ciency of incremental trunks in the alternate route, the most economical arrangement of trunks for carrying traffic from A to N may now be determined. The first step is to make sure that any trunk in the direct (HU) group will carry load at a cost per CCS equal to or less than the cost per CCS which is characteristic of the incremental trunks in the alternate route. Since the last trunk in a high usage group carries the least traffic, as previously discussed, the significant comparison is the ratio of the load carried by a trunk added to the alternate route to the load carried by the last trunk in the high usage group. The numerator of that ratio is 28 (CCS) and the denominator could be any one of 14 values shown on Curve A of Fig. 3, depending upon the number of trunks provided. If that ratio is made equal to the cost ratio (ATN/AN) there will be determined a value of load to be carried by a last trunk which in turn will determine the most economical number of trunks for the direct high usage group. This value is referred to as the "economic CCS" of the problem and is determined as follows: 1 4 28 Cost ratio -^ = Efficiency ratio -z^ 1.0 ^ X X= 20, the economic CCS On Curve A it will be seen that the sixth trunk will carry 22.5, the seventh, 19.6 and the eighth, 16.4 CCS. Since the loading of the seventh trunk is closest to the economic CCS just computed, the conclusion is that seven trunks should be pro\'ided in the high usage group for the minimum overall cost of handling traffic from A to N. Since the seven trunks as a group will carry 185 CCS (Curve B), there will be 55 CCS 284 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 35 240 CCS OFFERED TO 14 TRUNKS JOG 280 30 ^ . A = LOAD CARRIED ^>^ BY EACH TRUNK 240 220 25 \ ^^ ' \, ^ '^ \ / 200 180 20 \ / ,/ \ Y / \ 160 15 / / / \ V 140 120 10 / 7^ G LOAD ROUP CAR OF n RIED E TRUN iY A KS A \ 100 80 60 40 5 / s \, / s \ / ^>.,^ n ?0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 TRUNK NUMBERS Fig. 3 — Distribution of offered load according to the Erlang B assumption, lost -calls-cleared. (240-185) overflowed to the alternate route where they will be carried at the rate of 28 CCS per trunk. The trunks which need be added to the alternate route are therefore, (55 -h 28) or 1.96. The comparative costs of a number of alternate routing arrangements in which the only variable is the number of trunks provided in the direct high usage group are given in Table I. It is interesting to note that there is no measurable difference in cost between the optimum value for the number of HU trunks (7) and the cost of a 6 HU trunk arrangement. And indeed had five trunks or eight trunks been provided in the HU group under the stated conditions the additional cost above the most economical arrangement would have been nominal. This phenomenon is typical of many alternate routing triangles particularly when the offered load is in excess of say, 200 CCS. The significance of this fact is its effect upon the degree of accuracy with which cost ratios need to be determined. For example, assume that 1.4, the cost ratio used in our discussion, is precisely the right ratio for the facilities involved in the ATN triangle but that an approximate computation had arrived at a TiiU.NK KEQUIUEMEXTS lA ALTEKXATE liUUTlNG XETWOKKS 285 value of 1.12. With the latter ratio the economic CC8 would be (28 -^- 1.12) or 25. On Curve A of Fig. 3 it will be seen that tiunk No. 5 (the last trunk of a 5-trunk group) carries 25 CCS. So, if the lower cost ratio had been used insteatl of the correct one, the effect upon overall trunking costs would ha\e been 1 per cent in excess of the most (u-onomical ar- rangement. Had a cost ratio of 1.25 ])een used, six trunks would have been provided in the IIU group and no cost penalty would have been incurred. On the other side of the optimum point an eight trunk HU gi'oup would meet the requirements of a cost ratio as high as 2.0 with a resulting cost penalty of about 2 per cent. As can be seen from Table I, the cost penalties mount more rapidly when more than the optimum number of high usage trunks are provided than when less are provided. The principles of alternate routing and certain of the technitjues used by traffic engineers in determining (luantities and arrangements of inter- office trunks have just been described with particular reference to the trials that have been cai'ried on in New York City. The latter were very extensi^•e undertakings in which not only single alternate routes were provided but for a majority of items, multiple alternate routes. This was possible because New York City had two tandem systems each with a completing field to all city offices as well as other tandem systems (office selector tandems) each with a completing field to about 20 offices. Thus it was possible in many cases for an originating office to test a direct Table I — Comparative Costs of Alternate Routing System FOR Various Assumptions as to Number of HU Trunks Given: Offered load in CCS 240 Efficiency of trunks added to alternate route 28 Cost ratio, alternate to direct (HU) route 1.4 Nominal Costs No. of HU Trunks HU trunks Alternate trunks All trunks % Deviation from Optimum Per trunk Total Per trunk Total added* 3 1.0 3 1.4 7.50 10.50 7.70 4 i.n 4 1.4 6.10 10.10 3.59 5 5 1.4 4.85 9.85 1.03 6 6 1.4 3.75 9.75 0.00 7 7 1.4 2.75 9.75 0.00 8 8 1.4 1.95 9.95 2.05 9 9 1.4 1.30 10.30 5.64 10 1.0 10 1.4 .80 10.80 10.75 * Overflow CCS from HU Group X 1.4 28 286 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 group, a group to an ofRce selector tandem, a group to tandem No. 1 and finally a group to tandem No. 2 in .succession in its cjuest for an idle path to a given office. It is not the pin-pose here to explore the intricacies of the multi-alternate route system but rather to point out that the basic principles descrilsed for separating a load between a first choice direct route and a single final alternate route on an economic basis were applied successfully to the multi-alternate route arrangements. A thorough anal- ysis of one of the trials disclosed that there was a saving of approximately 20 per cent in the number of interoffice trunks of all types as compared with what would have been required without the use of alternate routes. Another result of the alternate routing system was an improvement in speed of service, since whatever portion of a load was carried by a high usage group was carried without delay and only the portion carried on the final alternate route was subject to about a 2 per cent delay. If, for example, 75 per cent of an item was carried without delay and the re- mainder at P.02 (two P.Ol groups in tandem) the average delay for the whole load would be P. 005 which is one half of the average per cent delay considered "normal" for a direct group without an alternate route. ALTERNATE ROUTING IN INTERTOLL TRUNK NETWORKS The successful application of alternate routing to local interoffice trunking led to investigation of the practicability of applying similar techniques to trunking between toll offices. The problem was different in a number of respects from that of the local interoffice system. First, the physical size of the toll network covering the United States and Canada involved some 2,500 toll centers and lengths of haul between toll centers varying between 20 and 3,000 miles. From this it was evident that many first routes would not be direct but would involve one or more intermediate switching points thus complicating the problem of deter- mining relative costs of first and alternate routes. Second, the development of an alternate routing system for intertoll trunking would have to be predicated on the existence of a slow speed (high delay) trunking system. Whereas the aim of alternate routing within metropolitan areas was to make an already high speed trunking system cheaper the aim in toll would be to make a slow speed system a high speed system at little or no additional cost. Third, new toll office switching facilities of a high degree of mechani- cal intelligence would be rcfjuired. Fourth, improvement in transmission and signaling design of toll circuits would be required to accommodate more links in tandem. And fifth, an entirely new procedure for the correlation and routing TRUNK REQUIREMENTS IX ALTERNATE ROUTING NETWORKS 287 of more than two million items of traffic between toll offices, would need to be devised. Over the years since 1945 practical solutions to the problems raised by the conditions have been attained by Bell System engineers and the fruits of their efforts will be put to the test in 1954 during which year the first practical application of "engineered" alternate routing in the intertoll network will be undertaken. The remainder of this paper will be devoted to the more important aspects of the traffic engineering techniques used in determining the arrangement and numbers of intertoll trunks required in a multi-alter- nate routing system. GENERAL TOLL SWITCHING PLAN A brief description of the General Toll Switching Plan* will be ap- propriate here since any discussion of the alternate routing methods necessarily presumes an understanding of the basic pattern for routing traffic. The plan under which transmission had been designed and traffic routings determined since 1930 comprehended a maximum connection of 5 intertoll trunks in tandem. Early studies of the alternate routing pos- sibilities in toll networks led to the conclusion that a total of 8 intertoll o O D D O o D O RC- REGIONAL CENTER PO- PRIMARY OUTLET O TC- TOLL CENTER Fig. 4 — Illustration of present basic intertoll network showing maximum intertoll trunk linkage. links would provide a more economical arrangement of trunks by short- ening some very long final groups that would otherwise be required. Fig. 4 illustrates schematically the arrangement of switching centers in a maximum connection currently in use. Fig. 5 shows a schematic of the proposed General Toll Switching Plan in which a connection involving 8 links is possible between TCI and TC2. Such a route would constitute the final route between those TC's and each group in the route would be a low delay final group similar to * J. J. Pilliod, Fundamental Plans for Toll Telephone Plant, Communications and Electronics, No. 2, Sept., 1952 and B. S. T. J., 31, Sept., 1952. 288 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 the groups to and from tandem previously described (Fig. 2, AT and TN) for the local trunking situation. It should be noted here that, in the interest of economy, it was proposed for planning to further compromise with the lost-calls-cleared assumption by setting the speed objective on each final group at P.03 rather than P.Ol. Theoretical considerations indicate that the violence done to the Erlang B premise would not be severe enough to invalidate a study of trunk requirements made on that basis nor interfere substantially with the achievement of a high speed trunking system. The switching plan requires that each switching office, i.e., the national center, the regional and sectional centers and the primary outlets be equipped with common control apparatus which would accept the num- ber of digits (10) required by the nationwide numbering plan and direct a call to its designation in prescribed order over a variety of possible routes of which the high usage groups indicated on Fig. 5 are representa- tive. (Alternate routes at TC's would be selected by operators.) The No. 4 A toll switching system* was designed to perform this function and offices so ecjuipped were designated control switching points or CSP's. The CSP's were classified for transmission and traffic routing purposes in order of their relative importance in the network: The national center (NC) has the unique function of switching traffic on the final route between regional centers thus being capable of handling traffic from any point in the country to any other point. In addition the NC is itself a regional center serving a particular area of the country; TC1(>=- ^-^ TC2 D A NC-NATIONAL CENTER RC-REGIONAL CENTER SC-SECTIONAL CENTER PO-PRIMARY OUTLET O TC-ORDINARY TOLL CENTER FINAL GROUP HIGH USAGE GROUP Fig. 5 — Illustration of basic intertoll network for nationwide toll dialing showing maximum final route linkage and typical high usage groups. * F. F. Shipley, Automatic Toll Switching Systems, Communications and Electronics, No. 2, Sept., 1952. TRUNK REQUIREMENTS IN ALTERNATE ROUTING NETWORKS 280 Each regional center (RC) homes on, ie., has a final group to the NC, and has one or more sectional centers homing upon it; Each sectional center (SC) homes on an RC (or the NC) and lias one or more primary outlets homing upon it; Each primary outlet (PO) homes on an RC, NC or SC and has one or more ordinal y toll centers homing upon it; and Each ordinary toll center (TC) is so called because it performs no through switching function but merely serves as the connecting point between the intertoll network and local central offices or tributaries. Thus each toll center (and in the generic sense this phrase includes all CSP's as well as ordinary'' toll centers) was classified with respect to the area served: the TC serving a group of local offices or tributaries, the PO serving a group of TC's, the SC serving a group of PO's, the RC serving a group of SC's and lastly the NC serving all the RC's. Under this arrangement any toll center could home on another of higher rank or classification. Thus a TC or PO, for example, could home upon an RC if so dictated by geographic and economic considerations. Before proceeding with a detailed study of trunk requirements the classifica- tion of toll centers and the homing relationships had first to be estab- lished. This was done in a manner which reflected the known densities and flow of traffic between the larger cities and the relative cost of final routes which in turn reflected the differences in lengths of haul to one CSP as opposed to another, etc. With the classification and homing of each toll center established it was possible to trace the final route, the route of last resort, between any two toll centers in the entire system. Thus was the stage set for determining the location of and number of trunks to be provided in high usage groups whose function would be to move traffic more economically by direct connection between points than could be done by following the final route. It is apparent at once from the illustration in Fig. 5 that the problem of (Uitermining the most economical alternate for a given HU group is different from that encountered in the interlocal situation of Fig. 2 in- asmuch as the latter had only one intermediate switching point in each final route. A further difference not specifically indicated is that intertoll trunk groups handle traffic in both directions whereas interlocal trunk groups handle traffic in only one direction, i.e., there are separate out- ward and inward groups between any pair of local offices. There are special circumstances under which one-way intertoll groups also are estabUshcd but these may l)e ignored for purposes of our discussion. While this paper is not specifically concerned with the transmission aspects of an alternate routing network some mention should be made 290 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 of that subject since it has an important bearing upon how traffic is routed. It is apparent that transmission standards for trunks connecting each class of office with any other class had to be so determined that a conversation between any two telephones in the system could be car- ried on within a specified limit of over-all transmission loss. For example, the possibility of a call being completed over an 8-link final route dictated the transmission tolerances to which each of the 8 links should be de- signed. Similarly, transmission equivalents for the various HU trunks over which calls would by-pass the final route had also to be determined. It became necessary, therefore, to devise a routing pattern which would assure that neither too many links nor the wrong classes of links would be connected in tandem. This interdependence of the transmission and routing aspects led to the adoption of the rule that traffic should be so routed that "when a call fails to find an idle path in the high usage net- work at any switching point (CSP) it can always be offered to the final network at that point." In other words, any CSP used as the intermedi- ate switching point in the alternate route of a high usage group must lie in the final route between the terminal offices of the high usage group. This was the corner-stone of the routing pattern. The Location of High Usage Groups Having established the classification and homing arrangements of all toll centers and the basic routing pattern the next step in engineering the nationwide intertoll network was to determine between what points there should be high usage groups. It should be mentioned here that the following will describe in essential detail the procedures actually used by the Bell System in preparing the first coordinated nationwide estimate of intertoll trunk requirements under a plan deliberately de- signed to take advantage of the speed and economy possibilities of the alternate routing principles already described. The study, which was completed in 1951, involved a look into the future of about 10 years and was predicated on operator dialing of toll calls. Customer toll dialing which is foremost at the present time in Bell System planning may re- quire some further modification of the general toll switching plan but the procedures about to be described \v\\\ be substantially unchanged. A high usage group is established between any two toll centers when the traffic between them in both directions is sufficient in volume to make such a group more economical then any other route available within the routing pattern. In the earlier discussion of the principles involved in determining what portion of a given load should be carried by a direct route it was shown that the ratio of the efficiencies and the ratio of the TKUNK REQUIREMENTS IN ALTERNATE ROUTING NETWORKS 291 costs of the direct and alternate routes are c'oiili()lliii«i. Since load carried l)y a high usage grouj) is a function of the load offered, under the Erlang B assumption, it is apparent that the size of the load between any two toll centers will also limit the possible range of economic CCS values which can be realized. In other words a high usage group to exist at all must have at least one trunk and that one trunk must carry not less than the economic number of CCS required by the cost ratio applying to the case. For example, if a busy-hour load of ()0 CCS is to be carried betw^een offices A and B and the cost ratio indicates an economic CCS of 25 it can be shown that when one trunk is offered 60 CCS it wall carry only 23 of the 00 CCS with the Ijalance 37 CCS being o\'erflowed. Under these conditions no direct (HU) group could be economically estabUshed since the efficiencj' of even the first trvmk of such a group would fail to meet the requirement of the case. This immediately suggests that a prime requirement in determining whether or not there should be a direct group of any size between two toll centers is a level of load at and abo^-e which a group will prove in and below which, of course, it will fail to prove in. As previously stated the cost ratio betw-een the alternate route and the potential direct group is also a controlling factor. Thus, to determine the economic propriety of establishing a direct (HU) group it is necessary to know the following: Cost of path in the alternate route; Cost of path in the direct route; Efficiency of trunks added to the alternate route; and Load offered between toll centers. The last three of these items were available to the engineer but the first item could not be known in all cases because the groups comprising the logical alternate routes in some cases were themselves hypothetical. For example, a high usage group betw-een TCI and TC2 in Fig. 5 might have an alternate route via POI or via P02 which routes in turn would depend upon the existence of groups P01-TC2 and TC1-P02, respec- tively. Therefore, it was necessary to "Cut-and-try" in the process of locating high usage groups. This was accomplished by choosing an average cost ratio (and hence an average economic CCS value) which could be used with the known offered loads b(;tween toll centers to test the feasibility of at least one high usage trunk. With this tentative pat- tern of high usage groups the potentially available alternate routes could then be identified. For each high usage-alternate route triangle thus tentatively selected the test of relative costs was applied to verify the economy of the case. Some proposed high usage groups failed to prove in under such test in which cases the uneconomic high usage groups were 292 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 abandoned and the loads that would have been offered to such groups were assigned switched routes. A switched route for our purposes is one in which the load between toll centers is carried on two or more inter- toll groups in tandem and may serve simultaneously as a first route for some items of traffic and as an alternate route for other items. The cut-and-try process just described resulted finally in the location of all economically sound high usage groups. There remained, then, the problem of determining the number of trunks in each of those groups and in the final groups as well. At this point two rules were estabhshed affecting (1) the order of group load computation and (2) the direction of switched load concentration, respectively. The first of these may best be described b}^ considering the conditions illustrated in Fig. G where a portion of the intertoll network with typical homing arrangements and high usage groups is shown. The follo\\'ing as- sumptions regarding alternate routes may be made: Traffic Item First Route Alternate Route TC1-TC2 Direct Via P02 TC1-P02 Direct Via POl TCl-SC Direct Via POl P01-P02 Direct Via SC TCI and TC2 are not switching centers and the load offered to their interconnecting group will be simply terminal traffic between their re- spective toll areas. Since TC1-P02-TC2 has been determined as the most economical alternate route, calls from TCI to TC2 overflowing the high usage group are offered to the TC1-P02 group. Thus a part of the complete offering to TC1-P02 is traffic rerouted from TC1-TC2. It follows then that the TC1-P02 group can not be engineered until the overflow from the TC1-TC2 group has been determined. Similarlj^, the P0 2 TC1(> Fig. 6. TRUNK UEQUIREMENTS IX ALTKKXATE ROI^TIXG NETWORKS 293 load offered to the P01-P02 group is composed in part of the overflow traffic from TC1-P()2. It is likewise clear that the linal group TCl-POl will l)c offered overflow traffic from both T(M-T(^2, TCf-P()2 and TCl- SC and hence the final group can not he. enginetn'ed until the overflows from all high usage grt)ups terminating at TCI have been detcrminech This transfer of load from group to group with each succeeding high usage group ser\ ing as the base of a lunv triangle in orderly procession from TC to IvC is the essence of the multi-alternate routing system and, therefore, a i)recise order of group load computation was necessary to assun> propel' accounting of the load offered to each group. In practice the order of group load computation retjuires that all TC-TC high usage groups be established first, then TC-PO groups, TC-SC and so on. The process in^'olves selecting a particular TC as a "reference" and examin- ing all traffic invoh'ing that TC to determine what (if any) high usage groups should terminate there. The second rule, affecting the concentration of switched traffic, is related to the order of group load computation in that it postulates a starting point for the process of examining HU group possibilities. The need for such a rule may be best explained by reference to Fig. 7 repre- senting a simple intertoll network in which PO, a CSP serving four TC's, homes on SC, another CSP, serving four other TC's. The question, answered by the second rule, is which of the eight TC's should be used as the first reference TC. The choice will determine the sizes of, i.e., the number of trunk terminations to be provided at, PO and SC, respec- tively. Let us examine the reason for this. Assume the TC's homing on PO are to be used as the first set of reference TC's (it makes no difference which TC is first cho-sen). Assume that the investigation showed an eco- nomical HU group between TC2 and SC Ixit that no HU groups proved into any of the TC's (5, 6, 7, and 8) dependent upon SC. The load offered 294 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 to the TC2-SC group, then, would consist of the sum of the following items: TC2-TC5, TC2-TC6, TC2-TC7, TC2-TC8, and TC2-SC Later examination of HU group possibilities with TC5 as the reference TC would have to recognize that the TC2-TC5 item was already com- mitted to a switched route via SC. Thus the TC5 summation would con- sist of but four items (TC5-TC1, TC5-TC3, TC5-TC4, and TC5-P0) instead of five, thereby reducing the possiblity of establishing a group between TC5 and PO. Even the four-item summation might prove in a group but if it be assumed further that a group proved in between TC3 and SC in addition to the TC2-SC group, the subsequent TC5 summation to the PO area would be reduced to 3 items, namely, TC5-TC1, TC5-TC4 and TC5-P0. It is clear that by using the TC's homing on PO as refer- ence points the tendency is to build up the HU groups to SC and dimin- ish the possibility of establishing HU groups terminating at PO with the result that SC handles more and PO less switched traffic. By thus select- ing TC's homing on the lower class CSP, the PO, as the first set of refer- ence points, an effect of "putting all eggs in one basket" is created since SC is already a larger switching office by virtue of its classiffication in the switching pattern. Therefore, it was deemed advisable to reverse the point of view in order to distribute the switched load more equitabley among the various CSP's. To accomplish this each CSP was given an index number which reflected roughly both its size (total originating toll traffic) and its importance (classification) in the switching pattern. The most important CSP thus determined was assigned No. 1, the next most important No. 2, and so on throughout the hst. Thus by starting with the TC's homing on the CSP with Index 1 as reference points, then mov- ing to the TC's on Index 2 and so on the desired distribution of switching facilities was achieved. GROUP BUSY HOUR VERSUS OFFICE BUSY HOUR TRAFFIC Thus far we have considered the theory of alternate routing and the more important techniques used in applying it to the design of intertoll trunk networks. Another very important aspect of the whole problem involves a difference in load levels between any two toll centers during different hours of the day. It has been the aim in engineering intertoll trunk groups without alter- nate routing to provide a sufficient number of trunks in each group to handle its traflfic at a stated speed of service in the average busy hour of the busy season of the particular group. All significant traffic data for TRUNK REQUIREMENTS IN ALTERNATE ROUTING NETWORKS 295 atlministration and engineering; have been obtained with that end in view. However, in alternate-routing networks the significant level of a trunk group load is not that which is characteristic of its own busy hour but rather that which is characteristic of the hour in which all trunk •iroups in a given network are collectively carrying the greatest aggregate t laffic \'olume. For each group the former level is referred to as the group busy-hour (CIRH) load and th(! latter, for coiu'cnicnce, as the office busy- hour (UBH) load. The reason for using office busy-hour loads between toll centers rather than group busy-hour may be explained with reference to Fig. 8. Here is a represented part of an intertoU trunk network showing 4 HU groups connecting TC with four other toll offices a, b, c and d. TC homes on SC and for simplicity it may be assumed that the alternate route of each such group is the final group to SC. In the hour during which the greatest volume of traffic is leaving and entering TC, i.e., the office busy-hour, the demand for trunk capacity in the TC-SC group will also be greatest since by design the final group to the home CSP of any TC is the route of last resort for all traffic to and from the TC. Thus the group busy -hour of the TC-SC group coincides with the busy-hour of TC as a whole. The group busy hours of the respective HU groups (TC-a, TC-b, etc.) may occur outside of the office busy-hour for TC and during such hours the amount of traffic offered to and hence overflowed by each of the HU groups is greater on the average than that occurring in the office busy- hour. But at any other hour than the office busy-hour there is less total load on the network and hence there is some spare capacity in the final group available for handling the group busy -hour overflow of one or more of the high usage groups. By properly evaluating the average ratio be- tween an}^ given toll center-toll center load in its group busy-hour and its value in the office busy-hour it would be practicable to start with basic data in group busy-hour terms and convert it to equivalent office busy-hour levels before undertaking the procedures for separating loads between direct (HU) and alternate routes. \/l ^ Fio. 8. 296 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 It became necessary, therefore, to develop these relationships from empirical data. Accordingly extensive field studies were conducted in a number of toll offices of different sizes to determine whether there was any consistent relationship between group and office busy-hour load levels of the many groups. Examination of group and office busy-hour loads on a total of almost 300 groups in 6 different offices showed a defi- nite relationship which could be expressed as a function of a given value of group busy-hour load. Fig. 9 shows the group to office ratios finally computed from these data for use in the nationwide trunk study. The ratios for the smaller loads (30-90 CCS) showed considerable lack of consistency as between offices, but an adeciuate set of ratios for these 1000 900 800 700 600 500 400 300 70 ~ ; / / - / / ; / / / / / / - / / / / / y ^ y^ 20 1.6 1.5 1.4 1.3 1.2 1.1 1.0 RATIO OF GROUP BUSY HOUR TO OFFICE BUSY HOUR Fig. 9 — Relationship between group busy hour and office l)us\- hour loads on intertoll trunk groups. TRUNK KEQriHI'^MI>:NTS IN ALTKHNATK KOUTINCi NKTWORKS 2!l7 loads was arriveil at through an averaging? process. The ratio values are <'oiiser\ative, i.e., there was some evi(leiic(> lliat in large toll offices the higher rnlio values apply to larger loads than iiKlicated by the curve. Il(»\ve\('i\ no attempt was made to develop separate sets of ratios for ollices of different size since the use of the common set of values would tend merely to increase slightly the num])er of HU trunks required in a r(^lati\-ely few groups. The elimination of this small distortion did not appear to warrant th(> effort recjuired to a('hie\-e it. The stud}' data also showed that for a gi\en toll office the sum of the respective group busy- hour loads for all groups was approximately 10 1 1 per cent greater than the sum of their respective loads in the office bus\'-hour. The significance of this difference in the GBH and OBH aggregates is at once apparent. The intertoU trvnik network designed on the alternate routing principle is required to handle from 10-14 per cent less busy-hour traffic than is required under the arrangement in which each toll center- toll center load stands alone and must be trunked for its own busy-hour. It is the pooling of trunk group capacities during the hour of maximum traffic flow for a given office plus the increased efficiency due to larger size of final groups that result in a retiuirement of fewer trunks in the network as a whole than would be required with any non-alternate rout- ing trunking system of comparable service characteristics. The alternate routing system will enable the handling of normal busy-hour traffic on \-irtuall3^ a no-delay basis in so far as trunk provision may be controlling. The matter of speed of service potentialities and relative costs of alternate routing ^'el■sus non-alternate routing systems will be treated later. SOME UNANSWERED QUESTIONS There are three ciuestions upon the answers to which depend ultimate judgment of the adequacy and economy of a nationwide toll dialing net- work constructed upon the principles and with the techniques already described. These are: 1. What will be the effect upon trunk requirements of a proper evalua- tion of the non-random characteristic of lost calls, i.e., of the calls over- flowed from high usage groups to other high usage or to final groups? 2. What will be the effect upon trunk requirements of a proper evalua- tion of the effect of non-coincidence of busy hoiu's and busy seasons among the various toll centers?^ 3. What are the relative net costs of a nationwide intertoll dialing system engineered for multi-alternate routing and one designed without engineered alternate routing? 298 THE BELL SYSTEM TECNHICAL JOURNAL, MARCH, 1954 No completely satisfactory answers to these questions have yet been found but some knowledge of the answers can be gleaned both from theory and from much experience in the operation of intertoll trunk networks. Let us consider each question in turn. Random Versus Non-random Traffic Statistical theory states that the portion of random traffic which fails to be carried when offered to a given number of trunks is not itself ran- dom in nature. Such portion is, of course, the overflowed or "lost" calls described in the discussion of the Erlang B formula used in the separa- tion of load between direct and alternate routes. Each such portion should be increased in volume to equal an equivalent random value before being offered to another trunk group since both the Erlang B and Poisson expressions of trunk capacity are predicated on the offering of random traffic. Such adjustment of high usage group overflows was not made for two reasons: one was the absence of any suitable increase factors for the variety of conditions encountered and the other was a definite indi- cation from the New York City trials of alternate routing previously mentioned that such factors were not needed. Analysis of the New York trial indicated that when many parcels of overflow traffic are presented during one hour to the final route the aggregate tends to become random in nature. However, it was deemed advisable to provide some extra trunks in the final route to protect service by absorbing any abnormal peakiness in the offered traffic which might be due to non-random charac- teristics. As a practical matter it was possible to observe and regulate the loading of the final tandem route to the end that a satisfactory over- all grade of service was maintained and from this and other subsequent experience to devise a rule-of-thumb for engineering future requirements in such final routes. A similar approach to the problem was adopted in engineering the nationwide intertoll trunk network. An arbitrary increase was applied to the computed load offered to each final group. Whether this correction was too little or too much is still a moot question but it is believed that it was proper in the sense that it was in the right direction. Statistical theory, however, does not support the view that a combina- tion of non-random parcels tends to produce a random aggregate. The problem is under further study by the Bell Telephone Laboratories with a view to producing some practical means by which field engineers can readily and with reasonable accuracy adjust non-random trunk overflow traffic to equivalent random values for projection purposes. TRUNK REQUIREMENTS IN ALTERNATE ROUTING NETWORKS 299 Non-coincidence of Busy Hours and Busy Seasons With respect to the second question involving the non-coincidence of busy hours among toll centers it should be noted first that trunk recjuire- ments estimated in the nationwide alternate routing trunk study were predicated on a common busy season and a common office busy hour for all toll offices and all intertoll groups. This premise resulted in system- wide requirements which were patently incorrect since it is known that different toll centers have different busy hours and that the busy season for toll traffic in New England for example is in the summer while that for Florida is in the winter. While it was possible to identify the busy season and the average busy hour of each toll center the statistical prob- lem of incorporating such information for some 2,500 toll centers in a completely integrated nationwide study appeared insuperable. The seriousness of the error introduced by the above premise is not as great as might at first appear since the New England Company should know the requirements for the busy season of its territory and the Southern Bell Company is equally interested in the busy season require- ments for Florida. It is only in the trunks required for handling traffic between these two areas that a distortion of requirements could be readily demonstrated as a result of assuming the two busy seasons to be coincidental when in fact they are months apart. The example cited is an extreme case which serves to point up the problem. While no evalu- ation has been made of this distortion, and none seems statistically practicable, it is evident that the direction of distortion is toward over- estimation of trunk requirements.* Thus it may be confidently stated that the general effect of assuming premise regarding the coincidence of busy seasons and busy hours upon the network as a whole was to compute trunk requirements in some groups more liberally than a precise evaluation of all significant factors would indicate as adequate. Proper evaluation of the effect of non- coincidence of busy seasons and busy hours will likely await the findings of field experience. Costs — Alternate Routing Versus No Alternate Routing In planning extensive and radical changes in the methods of handling toll traffic on a nationwide basis it was necessary to explore the economic * In the New York City .studies previously discussed, a similar assumption was made with respect to the coincidence of busy hours and busy seasons of the local offices. Due to the homogeneity of intra-office traffic the degree of distortion in individual trunk group requirements was considered, except for a very few cases, to be insignificant. 300 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, iDol soundness of the whole project. In the early stages of planning the l)ases for making such a judgment were rather meager since the only actual experience with alternate routing trunking had been in local systems, notably in New York City. Furthermore, the cost of the complicated switching mechanism, known to be essential to the plan but not yet developed, was an estimate of uncertain reliability. Added to these were the ever-present hazards of estimating the volume and distril)ution of toll traffic some 10 to 15 years in the future. However, in the light of the information then available it appeared that the overall cost of the proposed arrangments would be approximately equivalent to the cost of continu- ing present methods of engineering geared to a satisfactory trunk speed of service in the busy hour. Busy -hour trunk speed is defined as the aver- age interval of delay* experienced in the busy hour by an attempt at trunk seizure. This interval excludes any effect of operating procedure as such and hence reflects only the adequacy of the trunk plant to handle the offered load. It should be noted parenthetically that the use of busy hour trunk speed as a criterion of adequate intertoU trunk provision is different from the criterion used for interlocal trunk provision. The former refers to the duration of delay encountered by the average at- tempt at trunk seizure whereas the latter concerns the percentage of attempts that will encounter delay of any duration. These different service criteria arose from the traditional need for high speed handling of large volumes of local traffic and the acceptability of a much slower speed for toll traffic. The reference above to a "satisfactory" trunk speed of service pre- sumes a stated objective to be achieved under the current method of engineering. In 1945 the service objective was expressed in different terms which can be interpreted as being roughly equivalent to a busy-hour trunk speed of 30 seconds. It is obvious that a nationwide network de- signed to produce an average busy-hour trunk speed, of, say 20 seconds would require more trunks than one designed to produce an average delay of 30 seconds. With an alternate routing system, however, there is only one speed of service under normal load conditions, namely, one in which the average delay is so small as to be incapable of meaningful expression in terms of seconds per average attempt. It follows then that in any attempt to assess the comparative costs of outside plant and equipment luider the ciu'rent method with those under the alternate routing method of engineering trunks it becomes necessary to first define the service level which the former is required to satisfy. Objective service * S. C. Rappleye, A Studv of the Delays Encountered by Toll Operators in Obtaining an Idle Trunk, B.S. T. J., 25, Oct., 1946. TRUNK REQUIREMENTS IN ALTERNATE ROUTING NETWOKKS 801 levels can and do change with the years tor a Naricty ol reasons thus automatically changing the cost comparison. In spite of the difficulties in ohtainiiig a true and stable conipaiison of the overall costs of the two methods the weight of evidenct; indicated the desirability of proceeding with the plans for achieving an ultimate goal of nationwide toll dialing employing the technicjues of multi-alter- nate routing in the design of the intertoU tiunk ncUwork. With the completion of the study of the \arious intertoU trunk re- (luirements for nationwide operator toll dialing there was established for the first time a bench-mark against which many of the assumptions, theories and procedures which went into its making could be measured for accuracy and practicabilit3^ Among these was the early question re- garding the cost of operator toll dialing with engineered alternate routing compared to its cost without alternate routing. To arrive at a complete answer it would be necessary to restudy the entire network on the current basis of trunking, compare costs of dial switching eciuipment without CSP features with the cost of CSP switching equipment, evaluate changes in the location and tj'^pes of trunk facilities and so on. The under- taking of a stud}^ and analysis of this scope would recjuire a larger ex- penditure of engineering time and effort than would seem justified by the usefulness of the results. It should be noted, however, that analysis of the alternate routing trunk study indicated that the original premise as to trunk economies to be expected were substantially correct. In any event the advent of customer toll dialing with its peremptory require- ment for high speed trunking has rendered the original ciuestion of the relative costs of operator toll dialing, with and without engineered alter- nate routing, somewhat academic. It can be safely assumed that a high speed intertoU trunking system suitable for customer dialing and engi- neered without alternate routing would be prohibitive in cost. CONCLUSION The transition from ringdown (wholly manual) handling of long haul toll traffic to operator dialing of such traffic has been proceeding for many years and at an increasingly greater rate during the last five years until now some 45 per cent of such traffic is dialed by operators. This has been accomplished almost exclusi\'ely on trunk networks operated with- out benefit of engineered alternate routing. Along with this, increasing use of dialing by destination code has been achieved as various cities and areas h&xe converted to the nationwide nimibering plan. The second phase of the transition, now under way, involves the change from non-alternate routing to alternate routing trunking. With 802 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 this, as we have seen, will come a marked increase in the speed of move- ment of toll traffic. The third and final phase of transition toward the ultimate goal now in its infancy, will in^'ol^•e progressive introduction of customer diaUng of all toll traffic. Whereas, for operator toll dialing a high-speed trunking S3'stem is desirable, for customer toll dialing it is a firm reciuirement. Thus engineered alternate routing, which is the basis of the most eco- nomical and effective arrangements of toll plant facilities thus far con- ceived for highspeed trunking, will be indispensable to the further de- velopment of customer toll dialing. There is little doubt that the unsolved problems of the moment will be resolved if not by statistical methods then by the empirical approach which has helped to find the answer to some telephone problems of the past. In the meantime, those who have been intimately associated with the project of nationwide toll dialing through these pioneering years ha^'e faith that the present plans and methods, revised as circumstances and experience may indicate, will result in the ultimate achievement of a toll service which will be fast, dependable and economical. Tiitertoll Trunk Coiiceiitrating Equipment By D. F. JOHNSTON (Mamisoript leceivcd August 25, 1953) This article describes the IntertoU Trunk Concentrating Equipmenl which is a special purpose common-control type switching system. Its function is to combine, at a central point, small groups of trunks serving traffic originat- ing at various outward toll switchboards and to route the combined traffic to a toll or tandem crossbar office in a distant toll center. The operators at the outward toll switchboards are thereby provided with the equivalent of direct access to the intertoll trunk circuits. Both operating and equipment savings are realized by the use of this con- centrating equipment as compared to handling the same traffic via a toll crossbar office at the originating toll center. INTRODUCTION This article deals with a method of handling traffic from outward toll switchboards in a metropolitan toll center to a specific distant toll center with the objectives of (1) providing the efiuivalent of direct access to in- tertoll circuits from the individual switchboards, for traffic for which direct circuits cannot be justified, (2) giving relief to the No. 4 type toll switching system in the metropolitan toll center, and (3) providing a means for the dispersion of toll switching facilities. A metropolitan toll center may contain a number of outward toll switchboards. Some of these are situated in the central toll building and others in decentralized locations. An individual outward switchboard may have a sufficient amount of traffic to a sp(!cific distant toll center to justify a group of intertoll circuits direct from the switchboard to the No. 4 type toll or crossbar tandem office in the distant center. It has been the practice to provide such direct access to intertoll circuits at centralized toll switchboards. Traffic exceeding the capacity of such in- tertoll tnuik groups is handled o\-ei- tandem trunks tVoin llie toll switch- boards via tlie No. 4 typ(^ toll crossbar office in the originating center. The decentralized switchboards in general have reached intertoll cir- 303 304 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH, 1954 o< -I I- T lOl-LU ' >zo > — (£ 0i5 tr DO ujtra a. OQq ZZh- S^ LU ^-^ O-lO ^Iz — (T z o oo cr ^^? zuu tr-iz o o Zis: O 5x: ii|Z Zee g?ut INTERTOLL TRUNK CONCKXTRATIXG EQUIPMENT 305 CLiits only via tandem trunks to the No. 4 type toll crossbar office in the home toll center. Handling such trafhc tiu'ouu;h llu> toll ci'ossbar oflici' involves greater operating effort than handling it hy direct trunks since three additional digits per call must be keyed by the operator to direct the call through the toll crossbar office. Also the cost for an intertoU connection is higher with the tandem trunk method than with the direct circuit method, due to the greater number of switching facilities used in establishing the connection. Fig. 1 shows the components used in both cases up to the point wh(M'(^ they join common facilities. INTERTOLL TRUNK CONCENTRATING EQUIPMENT General The intertoU trvmk concentrating equipment has been developed to provide the eciuivalent of direct access to intertoU trunks for the traffic, from individual outward toll switchboards, which cannot justify the use of direct trunks. It is a small special purpose common control type switching system located at a central point. It gathers small traffic loads, to a specific destination and automatically routes this traffic to a com- mon group of intertoU trunk circuits which terminate in a toll or tandem crossbar office in the distant toll center. The maximum capacity of a trunk concentrating equipment is 100 incoming trunks and 40 outgoing trunks. It may be furnished in smaller sizes. If more than 40 outgoing trunks are required to a particular destination, additional concentrating equipments may be furnished. The field of use for this equipment lies between that of direct trunks and trunks reached through the toll crossbar switching system. The concentrating equipment is arranged only for multifi-eciuency pulsing from the switchboard. This is a system of pulsing in which com- binations of two frequencies within the; voice frequency band are trans- mitted over the talking path to the distant end. Each digit from 0 to 9 employs a different pair of freciueiicies. The intertoU trunk concentrating eciuipment consists of 4 basic circuit components, namely, incoming trunk, trunk selection switches, controller and outgoing trunk circuits. The detached contact form of circuit, prc^sentation is employed in the figures because of its simplicity. In this method the core and winding of a relay may be shown in one location and the associated contacts in other convenient locations. The core and contacts are related by the common designation which appears at the symbols which represent them. 30() THE BELL SYSTEM TECIIXirAL JOURNAL, MARCH, 1954 i^ LL -.1 IT) s ) CD 2 ^ "» >.oi- ;oint>; trunks to which th(\v may l)e switched. Swilchinfj; Ix'twccii inconiiiiii' ;ind outgoinu; trunks is thci'efore jjerformcd on a four- wire basis. Controller CircuH The controUcM' circuit is the control element of the trunk concent I'ating ecjuipmeiit. Its primary function is to select the incoming and outgoing trunks to be interconnected and to operate the proper select and hold magnets of the associated switches to close the required crosspoint. Since the controller circuit is described in some detail latei-, only the general principles of it will be dealt wdth at this time. The controller divides the incoming trunks into ten groups, (0 to 9), of ten trunks each. When idle it admits calls for a very short interval and then closes gates which exclude all other groups until calls recog- nized, within the gate, have been served. The controller serves the groups and trunks within the gate in one of two orders depending upon the direction of selection existing at the time. In one case selection will start with the lowest numbered trunk in the lowest numbered group and progress to the highest numbered trunk in the highest numbered group. In the other case the order will be reversed starting with the highest numbered trunk in the highest numbered group and progressing to the lowest numbered trunk in the lowest numbered group The controller also divides the outgoing trunks into four groups, (0 to 3), of ten. The outgoing trunks are served in order, either from low numbered to high numbered trunks or vice-versa. Once selected, the outgoing trunk remains locked out, after use, until all triuiks have been used, or until a trouble condition causes a reversal of the direction of selection of the outgoing trunks. To avoid comiecting an incoming trunk to two outgoing trunks or connecting two incoming trunks to an outgoing trunk the controller tests both the select and hold magnets for possible trouble conditions, such as crosses, before operating them. Each intertoll trunk concentrating equipment has but one controller. If the controller ceased to function the entire concentrating ecjuipment would be out of service. To insure reliability the philosophy was adopted in the design that no single trouble should disable the controller. This accounts for some of the features, the reasons for which otherwise are not obvious. 310 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 ait] z _|LU . o< ji-T •- .'t^ 0 z ("i-iu 1 Q U tx =>o V 0|-< Q:<(n 0°- — hri ^ u "^ 0 0 z^t Zs:i- 0?=^ O^D o^y u^u Htro^ KCCCC =)'-ij =|i-o 0 0 — — z 0 1 9 < cc H ml^ -il_ "J UJ -i- ^5 g3 (^K ~ 11 zo 0 0 — C>1 - - 1 15 ^'^ ^ ^'^ zx:t z:s:t z^t 5^3 5^=' ^="-' 0(r(r Otrir ^HO ^-u z"^ Q tr < ^Oui ^10 I ^o< t— ^ g (fl 1 — — — — _ — u 0 zx:i- zx:i- Q _nO oZ=' oz^ ai_|Q ^sy e)3u NqCC Hira: _!(-< D^-O Dt-o z|o lii > 1- 0 Q 0 q: q; I y a:u< UJ 3> 1- T :^ o INTERTOLL TRUNK CONCENTRATING EQUIPMENT 311 Completion of Calls The basic circuit components of the intertoU trunk concentrating equipment are shown in Fig. 4. Three incoming trunks in different groups and from different switchboards are shown. Two outgoing trunks in different groups are also shown. Assume that the controller is conditioned to serve both incoming and outgoing trunks in the low to high order, that a call is placed on incoming trunk 14 and a short time later on incoming trunk 21, and that outgoing trunks 04 and 36 only are avail- able for selection. Seizure of trunk 14 at the originating switchboard causes the incoming trunk to place a ground on the start lead of incoming trunk group 1. This ground indicates to the controller that a trunk in incoming group 1 is calling for service. The controller then closes the gate to all other ten groups, tests the select magnets associated with horizontal 04 for crosses (since outgoing trunk 04 is first in order for selection) and finding none operates the select magnets. It then tests the hold magnets associated with vertical 14 and finding no crosses oper- ates the hold magnets, thus closing the crosspoint whose coordinates are (04, 14) connecting incoming trunk 14 to outgoing trunk 04. The outgo- ing trunk transmits a seizure signal over the intertoll facilities to the distant end which then transmits a signal back to incoming trunk 14, which relays the information to the controller. The controller then re- leases from that connection, opens the gates and admits the call waiting on incoming trunk 21. It proceeds to complete this call to outgoing trunk 36 which is next in order of selection in a similar manner. If in the assumed case outgoing trunk 36 was the last trunk then a^'ailable for selection the controller would, at the completion of selection of trunk 36, proceed as follows: (1) If any of the other outgoing trunks which had been locked out were idle the controller would now make these trunks available for selection. (2) If no trunks were idle the controller would wait, and cause a signal to be transmitted to all associated outward switchboards. This signal will prevent the lighting of idle trunk indicating lamps at each switchboard. When one or more outgoing trunks become idle the con- troller will make them available for selection and will permit the idle trunk indicating lamps at the associated switchboards to light as an indication that trunks arc available. Component Circuits of the Controller In the following paragraphs the component circuits of the controller will be described individually. The descriptions of these circuits contain the minimum of detail reciuired to understand how they function. 312 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 I I I I z + I I- I I oi I r 1' O - 3 O D 1 O I ^ — ho:s has a common start lead, (st), (See Fig. 2) to the controller which is grounded when any trunk in the grt)up is calling for ser\-ice. Each ti-inik in the group of ten supplies an indi\-idual lead us to the controlkM' which is also grounded when the particular trunk is calling for ser\ice. 'This lead sei'\'es to identify the units designation of the trunk. In the controller there is a group of tiu-ee relays associated with each group of 10 incoming trunks (Fig. o). These are (1) the tens relay (tn-) which responds to the ground on the st lead from the trunk group when a trunk in that group is calling for service providing that the tens gate is open as discussed below, (2) the units control relay (uc-) which when operated connects the common group of units relays (uo-ug) to the us leads of the grunk group, (3) the hold connect (hc-) which when operated steers the hold magnet operating path to the hold mag- nets associated with the trunks in the particular group. The uc- and hc- relays do not operate until it is the turn of that associated trunk group to be served. Two series chains, carried through transfer contacts on all of the tens relays, control the operation of the units control and hold control r(^lays. The operation of these relays and the manner of advancing selec- tions from one group to another is best explained by the use of the following example. Assume that incoming trunks in groups 1 and 3 have originated calls resulting in the operation of the tni and tn3 relays. Assume also that the controller is conditioned for the low to high direc- rion of selection for incoming trunks. When the tni and tn3 relays operate, they cause the release of the tens gate relays (TCii and tg2) which close the gates to the operation of any other tn relays and operate tiie ucl and hcI relays. When the units gates close as described later, the tnI relay is released and the trunks in group 1 will be served. When the last trunk in this group has been served the ucl and hcI relays release and the uc3 and hc3 relays operate advancing the selections into group 3. The tn3 and uc3 and iics relays then function in the same manner as described for the Txi, I'ci and iici relays. If the direction of selection had been from high to low instead of from low to high, group 3 would have been served first instead of group 1. 314 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Grouping and Individual Relays for Outgoing Trunks In the controller the outgoing trunks are divided into groups of ten circuits. This is done to keep the length of chains on the control relays short (ten rather than forty relays long) and to reduce current drain during light periods since a possible maximum of nine relays will be locked up instead of 39. For each outgoing trunk one out trunk (ot-) relay (shown in Fig. 10) is provided in the controller and for each ten of these a group relay (ciP-) is provided (shown in Fig. 11). When the connection of an incoming trunk to an outgoing trunk is completed the controller operates the ot relay, corresponding to the outgoing trunk, which advances the selection path to the next idle trunk in the group and locks up under control of the gp relay for that group. When the last idle trunk in the group has been selected the group relay is released ad- vancing the selection path to the next group and releasing all ot relays in its own group which are associated with idle trunks. These trunks are then locked out of service until all trunks above or below it (depending upon the direction of selection) have been selected. Two chains are car- ried through contacts of the group relays and through the ot relays of each group for controlling the operation of the switch select magnets, one for each direction of selection. These chains are shown in Fig. 7. Units, Unit Gates and End Relays Ten Unit relays designated uo to U9 are provided to identify the indi- vidual incoming trunks in a group of ten which have calls waiting to be served. These relays, together with the unit gates and lock unit relays are shown in Fig. 6. When the units connect relay (uc-) associated with a group of trunks operates as previously described, it connects these ten u- relays to the US-leads from the incoming trunks. Those of the u relays which find ground on these leads, indicating calls waiting to be served, operate and lock. Any u relay operated operates one of two end relays ei or E2. The END relay in turn operates one of two unit gate relays (ugi or UG2) which opens the operating, but not the locking, path of all u relays thereby preventing a late arrival from stealing preference from an earlier originated call. One end and one units gate relay are provided for each direction of selection. The hold magnets associated with the incoming trunks in the group being served are connected through contacts on the HOLD CONTROL (hc-) relay to two chains of transfer contacts on the UNIT relays (1 chain for each direction of selection). As the connection of the incoming trunk is completed to the outgoing trunk the unit relay INTKHTOLL TRUNK COXCENTRATINO IXilll'.M KXT 315 associated with that iiicoiniiiji; ti'imk is relea.sod a(l\'aiiciiik^ases, signifying the end of selections in that group and enahHng the controUer to advance to the next group, 'ilie rek^ase of the kxd relay releases the units g.vte relay restoring the operating patii foi' all u relays. Select Magnet Operaling Circuit The select magnet operating circuit is shown in V'\g. 7. Two of these circuits are pro\-ided in accordance with the philosophy that no single trouble should block the concentrator. One of these circuits is associated with each direction of selection. When the units gate relay operates as described in the preceding paragraph, battery is connected through the windings of the xs and ss relays, chains on the group relays (gp-), chains on the ot relays, to the select magnet associated with the lowest INCOMING TRUNKS r X LU2 N OR O ■^lll 1 ~ A iR O ODD OR EVEN UNITS RELAYS HT1 OR HT2 r El OR E2 OR LU2 CC1 L OR E OR F y^ — I — w\^ — III *LOCK> UNITS RELAYS 1 X UG1 Et OR OR UG2 E2 El OR E2 UNITS GATES RELAYS E OR F ■AAA lllh 1 *LOCK UNITS RELAY DESIGNATIONS: --1 ARE INCOMING LOW TO HIGH - -2 HIGH TO LOW Fig. 6 — Units, unit gates and lock unit relays 316 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 fe IXTEUTOLL TKUNK CONCENTKATIN'C KQUIl'MEXT 317 or highost iiiioperated or relay in the group dependiug on the direction of selection. If the path is found to Ix^ closed to ground the ss relay will ()p(M"ate. Tht> \s relay is a polar n^ay. 'fiie ixvsistance shown in series with the secondary winding will have a \alue depending upon the num- her of select magnets which are multipled ti)g(ither on each Unci of the l)arti(ular concentrator, which number is a function of the number of incoming trunks. The xs relay will operate on every normal connection since the ciu'rent in the s winding, which is in the direction to operate tli(> rcla}', will be larger than that in the primary winding. When the resistance to ground on the select magnet lead is less than it should be, due to a cross with another select magnet lead or to a direct ground, the current in the primary winding will be greater than that in the secondary winding and the resultant ampere turns will be sufficient in the non- operate direction to prevent the operation of the xs relay. This would cause the selection to be halted, the controller to time out and the trouble registered. A reversal in the direction of selection would occur and selec- tions would then be resumed. If no fault is found with the select magnet operating path the select magnets operate. The controller then introduces a small time interval to permit all parts of the selecting bar operated by the select magnet to come to rest. The hold magnet operating path is then closed as discussed below^ and the crosspoint is closed. The closure of the crosspoint operates a relay os in the outgoing trunk which in turn releases the select magnets and ss relay which releases the xs relay. ffoJd Magnet Operating Circuit The hold magnet operating circuit is shown in Fig. 8. Two circuits are provided, one for each direction of selection to insure against blocking in case a trouble in this portion of the controller. The cross detection part of this circuit is an unbalanced wheatstone bridge, the galvanometer element of which is the polar relay xh. Three of the arms of the bridge are resistances, the values of which are tailored to each particular con- centrator depending upon the number of hold magnets to be encountered on a normal connection. This number is a function of the number of outgoing trunks. The fourth arm consists of the hold magnets. When the TENS GATES relays released as previously described the xh relay operated (at this time the hold magnets are not connected and the bridge is not foimed). Later in the progress of the call, when the units relays operate, the hold magnets are connected and the bridge is formed. If th(> rc^sistance (»f the hold magnet arm of the Ijridge is as expected, the bridge is un- l)alanced so as to keep current flowing through the xh relay in the direc- tion to maintain it operated. This will permit a relay ht, which furnishes 318 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 pHi'r, - -, (M XHl OR XH2 UG1 OR UG2 SCI SC2 — \ h- ECI OR EC2 ^^ n , ^tu ,1: 1"°= T •^ I i| ^^A^^^|lHl' I^AArH|lHl P;II 050 < Oq IS - ;nets, to release and the connection to proceed. If, however, the hold niaji;net lead is crossed to another hold magnet lead the resistance in the hold magnet arm of the bridge will be lower than expected causing the xh relay to release. The release of this relay will prevent the release of the timing relay, thus causing a time out and trouble registration. If no trouble cross is found on the hold magnet lead the ht relay re- leases when the select magnet is operated. This is a slow releasing relay and provides the time for allowing the selecting bar to come to rest be- fore the hold magnets are operated. When this relay has released low resistance ground through the hs relay and 12-ohm resistance is con- nected to the hold magnets as an operating path. The hs relay is polarized and the current flowing from the hold magnets is in the direction to non operate the relay. The operation of the hs relay will be discussed below. Outgoing Trunk Seizure and Continuity Check Up to this point the select and hold magnets have been operated. The crosspoint of the switch is now closed and the outgoing trunk seized. OUTGOING TRUNK TO DISTANT OFFICE VIA INTERTOLL CIRCUIT SINGLE FREQUENCY SIGNALING CIRCUIT RG HilK? I— vw- _^.iH^r';Ta SWITCH CROSS POINTS >h- EORF RTl HS1 OR OR RT2 HS2 HG1 OR HG2 n HGt OR HG2 A OR B RELAY DESIGNATIONS: --1 ARE INCOMING LOW TO HIGH --2 HIGH TO LOW H' CCI OR CC2 E OR HG1 OR HG2 H A OR B J I INCOMING TRUNK CONTROLLER Fig. 9 — Outgoing trunk seizure and continuity check. 320 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 To insure that continuity exists through the switch a check is made by making the advance of the controller dependent upon receiving a signal from the distant end. The os relay, Fig. 9, in the outgoing trunk is operated directly by a ground on one of the contacts of the crosspoint, and opens the operating path of the select magnets and ss relay (shown in Fig. 7). The release of the ss relay causes the operation of the cp relay shown in Fig. 8. The operation of the cp relay changes the potential connected to the winding of the hs relay from ground to about —2.5 volts. The operation of the os relay is the outgoing trunk sends a seizure signal over the trunk to the distant end which then sends a stop signal back to the incoming trunk of the concentrator causing the operation of the SUPERVISORY RELAY (sv) and the cut through relay (ct) of the incoming trunk which locks under control of the operator and connects ground to the hold magnets of the switch. This ground causes the current through the hs relay to be reversed, operating the relay, indicating a successful continuity check. A failure to make a successful check will cause the controller to time out, lock out the selected outgoing trunk and proceed to the next idle outgoing trunk. Outgoing Trunk Lock Out and Advance A TRUNK ADVANCE (ta) relay shown in Fig. 10 is provided for each group of outgoing trunks. The (ta) relays operate in multiple on ev- ery call. They are operated when the select magnets are energized and are released, under normal conditions, when a successful continuity check is made, or in the event of an unsuccessful check, when the con- troller times out. Their function is to prevent the operation of the ot re- lay associated with the outgoing trunk until a connection has been suc- OUTGOING TRUNK 1 I S02 jHili-^ ^ 11-^ S02 ± TO OTHER TA RELAYS ± ^"tJ TO ALL OT RELAYS IN SAME GROUP 1 E Fig. 10 — Outgoing trunk lockout and advance. INTERTOLL TRUNK CONCENTRATING EQUIPMENT 321 cessfuUy completed except as stated above where a continuity check has failed. When the ta relays release, the ot relay associated with the trunk selected, or passed by, is operated and advances the selection to the next idle trunk. Thc^ operated ot relay locks to its own group relay gp until the last ot relay in that gi'oup has operated. When the last trunk in that group is selected the gp rela}'- of that group is re- leased, advancing the selection path into the next group. The ot re- lays in the first group are then controlled only from the individual out- going trunk with \\hich they are associated and will be released when the trunk becomes idle. The trunks in that group cannot be selected again until the group relay has been reoperated which can not occur until all of the outgoing trunks on the concentrator have been selected in tvu"n or until the controller times out. In both of these cases the con- troller will go through end or cycle operation and reoperate the group relays as later described. With the above arrangement the traffic is spread evenly over the whole group of outgoing trunks. End of Cycle and Guard Timing When a connection through the concentrating equipment is released 1)3' an operator, the outgoing trunk sends a disconnect signal to the other end, to release the equipment there. It may take approximately 0.75 seconds for the distant equipment to release after it has received the disconnect signal. If this equipment should be reseized before it is re- leased the new call would be connected to the same subscriber. A guard time could be incorporated in every outgoing trunk to prevent the trans- mission of a seizure signal for this 0.75 seconds, but this would be rela- tively expensive. To avoid this procedure the end of cycle and guard timing feature has been incorporated in the controller. This feature, shown in Fig. 11, together with the outgoing trunk lock out feature insures that no outgoing trunk can be seized for at least 0.8 of a second after it has been released from a previous call. When the last available idle outgoing trunk on the concentrator has l)een selected the last operated group relay gp releases, and two end of cycle relays eci, ec2 which are normally operated also release. Either one of these relays released operates both the group restore relay gr and an all busy relay ab. Both the gr and ab relays cause the idle indicating lamp associated with trunks to the concentrator at all origi- nating switchboards to remain dark as an indication that all trunks are busy. If no trunk outgoing from the concentrator is idle at this time nothing else will happen in the controller. When any outgoing trunk in any group becomes idle the group relays associated with such groups 322 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 will operate. Any group relay operated will operate two guard time relays gti and gt2 which will reoperate the end of cycle relays and enable two timing circuits. The end of cycle relays operated release the group restore (gr) and all busy (ab) relays restoring control of the idle indicating lamps at all originating switchboards to the associated trunks. Timing is obtained by means of cold cathode gas tubes in com- bination with resistance-capacitance circuits. The control gap of the gas tube (terminals 1 and 4) is connected across the capacitor which is normally short circuited thru 510 ohms. To start timing, the gt- relay remove the short circuit around the capacitor and connects ground thru the tf- relay in series with the capacitor and the 0.556 megohm resistor. The capacitor charges from the 130-volt battery. When the voltage across the capacitor reaches the breakdown potential of the gas tube the latter fires and operates the tf relay. The time to fire the tube is determined from the following equation : T = 2.3i?CLog^-^, where E is the voltage of the battery, V is the voltage at which the OT-9 GR 0T-- — I— ♦ TO CONTACTS OF GP- RELAYS OF OTHER GROUPS ^^-AAAH|lh| [-" Fig. 11 — End of cycle and guard timing. INTERTOLL TRUNK CONCENTRATING EQUIPMENT 323 control f>;ap breaks down (which is minimum ()3 volts for this tii])c) R is the value of resistance in ohms, and (' is the ^'alue of the capacitor in farads. The timing circuits measure an inter\al of about 0.8 seconds which is sufficient to ])ermit the distant end of any trunk, whicii has just been released, to restore to normal. When the timing interv^al is com- plete the GUARD TIME relays release, the: direction of selections is re- versed, and selections are resumed. Direction of Selection Control Two directions of selection are provided for both incoming and out- going trunks to (1) by-pass incoming and out-going trunks which are in trouble, (2) to guard against blocking due to a single trouble in the controller itself. (See Fig. f2.) The directions of selection are controlled by two combinations of relays, plus three other relays controlled by these combinations. The aow and aoz relays control the outgoing directions of selection. With these relays in the unoperated condition the outgoing trunks will be selected in the low to high order and with them operated selections will be made in the high to low^ order. The outgoing trunks, in the normal course of events, are served starting with lowest or highest numbered trunks and proceeding to the highest or lowest numbered trunk depending upon the direction of selection at the time. When the last trunk has been selected the direction is reversed by either operating or releasing the aow and aoz relays depending upon the status cjuo ante. In case a short time-out is due to failure to close the crosspoint the direc- tion of selection of outgoing trunks is reversed immediately. It is also reversed whenever a long time-out occurs, whatever the cause. The AW and az relays form the combination which controls the in- coming direction of selection. Controlled by them are the control relays ci and C2 associated with the low to high and high to low direc- tions respectively. With the aw relay unoperated the ci relay is operated and the direction of selection is from low to high. With the aw^ relay operated the ci relay is released, the C2 relay is operated and the direc- tion of selection is from high to low. The incrmiing direction of selection is reversed under the following conditions: 1. When the outgoing direction is reversed upon the last outgoing trunk being selected the incoming direction is also reversed if and when there are no more incoming trunks within the gate waiting to be served. 2. When a short time-out occurs due to the faihu'e of the end relay to operate or failure of continuity check. 3. When a second failure to close crosspoint occurs. 4. When a long time-out occurs. 324 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 USE OF IDLE INDICATING LAMPS AT SWITCHBOARDS A system of indicating an idle circuit in a trunk group is used at toll switchboards to relieve the opei-ator of the necessity of making busy tests on the sleeve of the ti'unk. Associated with each trunk jack in the face of the switchboard is a lamp. The lamp associated with the first idle trunk in the group is normally lighted, while those lamps associated with the other trunks in the group remain dark even though those trunks are idle. When the trunk whose idle indicating lamp is lighted is seized, that idle indicating lamp is extinguished and the lamp associated with AW ^ vw ) AW ^ Wv- (a) INCOMING DIRECTION CONTROL (b) OUTGOING DIRECTION CONTROL Fig. 12 — Direction of selection control. INTEKTOLL TRUNK CONCENTRATING EQUIPMENT 325 the next idle trunk lights. Tlie lump does not necessarily move progres- sively thru the group hut is al\va3's lighted on the lowest numl)ered idle trunk in the group. The diriM't iiileiloU trunks and the trunks to a concentrating equip- DIRECT CIRCUIT TRUNKS TO CONCENTRATING EQUIPMENT IDLE INDICATING LAMPS GROUP CONTAINING TRUNKS CI CI CI CI TO CONCENTRATING C^ C^ C) C EQUIPMENT ONL GROUP CONTAINING DIRECT --SUBGROUP—. TRUNKS AND TRUNKS TO D CI CI CI SINGLE CONCENTRATING OOOO OOOO ^°^^'^^^' SUBGROUP^ D CI CI CI GROUP D = DIRECT CIRCUIT TRUNKS TO CONCENTRATING EQUIPMENT: CI =FIRST C2=SECOND C3=THlRD GROUP CONTAINING DIRECT /-SUBGROUP-. ^SUBGROUP-. TRUNKS AND TRUNKS TO D CI C2 C3 D CI C2 C3 THREE CONCENTRATING OOOO OOOO ^°^'^^^^^ Fig. 13 — Idle trunk indicating control and some examples of groups contain- ing direct trunks and trunks to concentrating equipment. 326 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 meat to the same destination form a single group of trunks in the switch- board multiple. The use of idle triuik indicating lamps is required with such a group for the following reasons: 1. To force the traffic onto the direct trunks, if these are provided, when one or more of these trunks is idle. The direct circuits are first choice and calls should be routed to them when available to prevent overloading the outgoing trunks from the concentrator which might block traffic from other switchboards. 2. To make all trunks to the concentrator busy to operators when all intertoll trunks outgoing from the concentrator are in use. Fig. 13 is a simplified schematic of the idle indicating chain used in an outward switchboard for a group of trunks containing both direct and concen- trator trunks. The arrangement shown is for a group consisting of direct trunks and trunks to a single concentrator. The idle indicating facilities are flexible and any desired grouping arrangement is attainable. A few of these arrangements are indicated by showing the relations of the jacks in the switchboard multiple. USE OF THE INTERTOLL TRUNK CONCENTRATING EQUIPMENT IN AN OP- ERATOR ALTERNATE ROUTING ARRANGEMENT The intertoll trunk concentrating equipment as previously stated is an independent system and its use and location are therefore flexible. ORIGINATING TOLL CENTER OUTWARD TOLL SWITCHBOARDS DIRECT CIRCUITS 7 OR 8 DIGITS CONCENTRATOR CIRCUITS 7 OR 8 DIGITS TANDEM TRUNK CIRCUITS NO. 4 TYPE TOLL OFFICE 10 OR n DIGITS TERMINATING TOLL CENTER NO. 4 TYPE TOLL OFFICE OR CROSSBAR TANDEM OFFICE Fig. 14 — Use of concentrating equipment in an operator alternate routing arrangement. INTKKTOLL TUl \K CONX'ENTRATING EQUIPMENT 327 However, it shouhl he lenieinbcrecl that it contains a single controller which could he ciisal)le(l hy compound Irouhle, a condition which could he .serious if the concentratin<>; (Miuipinent were the sole means of handling traffic from any switchljoard. It is intended that the concentrating equip- ment he used in conjunction with direct intertoll circuits and/or tandem ti'unks to a Xo. 4 type toll crossbar switching system. When used with direct circuits an operator alternate routing system ensues. Referring back to the paragraph which dealt with idle trunk indicating, it is noted that the direct trunks and tlie concentrator tnuiks form a single group or subgroup of trunks to a common destination. The direct trunks appeal' in the multiple at the head end of the group and are followed by con- centrator trunks. The idle indicating lamps direct the operator to a direct trunk, when available, as first choice, and automatically direct the operator to a concentrator trunk when the direct trunks are in use. If the operator also has access to trunks to the toll crossbar office these trunks become third choice for use when the direct and concentrator trunks are busy. Fig. 14 illustrates this situation. The concentrating eciuipment may be located apart from the central toll building without losing the advantages of the alternate routing discussed above. Thus it is available for dispersing the toll plant to minimize the ef!'ect of disaster. A CKNOWLEDGMENT The development of the Intertoll Trunk Concentrating Equipment was the combined effort of many people. Important contributions were made by M. Posin and W. L. Shafer, Jr. I The Transistor as a Network Element* By J. T. BANGERT (Manuscript received Octolier 7, 1053) The development of the transistor has provided an active element having important advantages in space and power. As a result, the question arises whether strategic insertion of such active elements in passive networks might lead to interesting results. This paper gives a theoretical analysis confirmed by experiment^ of certain possible network applications of tran- sistors. Four general areas are considered in which transistors are used as follows: to reduce the detrimental effects of dissipative reactive elements, to eliminate the necessity for inductors in frequency selective circuits, to pro- duce two terminal envelope delay structures having zero loss, and to invert the impedance of reactive structures. The conclusion is drawn that judicious intcrspersion of transistors in a transmission network enables performance to be achieved which would otherwise be unobtainable or uneconomical. INTRODUCTION It has become customary through the years to classify hnear circuits as either active or passive. Tiiis convenient, but arbitrary, division has encouraged a philosophy that regards each as a separate and distinct domain. The recent spectacular advances in active devices suggest that in some cases the traditional boundaries should be erased and that a unified approach should be made. In particular the de\'elopment of the transistor offers the possibility of interspersing small active elements throughout a passive network to achieve certain desirable effects. This paper intends to survey a few of the ways in which a transistor can be used to advantage in trans- mission networks. The discussion is divided into four parts as follows: 1. Reduction of dissipation. 2. Elimination of inductance. 3. Production of delay. 4. Inversion of impedance. * Presented in part at the Radio Fall Meeting, Toronto, Ontario, Oct. 28, 1953. 329 330 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 The fir'st portion offers a new approach to the everpresent problem of imperfect reactive elements. The second portion discusses a method of combining resistance and capacitance with transistors to produce characteristics conv^entionally realized by inductance and capacitance. The third portion proposes a technicjue for obtaining any specified delay characteristic with a two terminal active structure. The fourth portion considers means of using a transistor to transform passive ele- ments of ordinary size into passive elements of greatly reduced size. I. REDUCTION OF DISSIPATION For simplicity in the treatment of network problems it is frec^uently assumed that purely reactive elements will be used. In many cases this approximation is satisfactory; other times it is worthless, and a more reaUstic analysis must be made. In this latter case one possibility is to nullify the unwanted dissipation by means of a bridge balance. This will entail the acceptance of some flat loss. Another possibility is to insert active elements within the network in order to supply just enough energy to offset the inherent dissipation of the elements. This second approach, until now relatively unexplored, is being tried with promising results. To avoid introducing new terminology the discussion will em- ploy the concept of negative resistance which has been studied with interest by many investigators."" Negative Resistance Negative resistance is a misleadingly simple name applied to a com- plex phenomenon. The term implies behavior in some opposite sense to that of an ordinary positive resistance. This is true only for a limited range of frequencies and signal levels. As generally used negati\'e re- sistance refers to a two terminal active network or electronic device in which the voltage-current ratio has a negative real part and negligible imaginary part. Table I — Negative Resistance Parameter Independent variable Shunt Type Series Type Current controlled . Voltage controlled Required external imped- ' Short circuit stable | Open circuit stable ance I Increased magnitude of \ Decreased magnitude of Rn i Rn Parallel capacitance Series inductance Effect of internal gain re- duction Associated reactance THE TRANSISTOR AS A XIOTWORK KLEMENT 331 For convenience the simple forms of negative resistance may be di- \ided into two general classes which are duals in a network sense. Since those classes have been identified in the literature in several different ways, it seems desirable to sunnnarizc the major characteristics in the form given in Table I. 'riiereforc a shunt negative resistance is one whose magnitude is controlled mainly by the voltage across its terminals. It is short circuit stable which means it must operate into a low impedance. When the internal gain used to produce the effect is reduced, the magnitude of a shunt negative resistance increases. In addition it should be associated with a parallel capacitance to predict its behavior outside the working hand of frec|uencies. One method of producing a two terminal shunt negative resistance is to arrange a transistor as shown in Fig. 1(a). To facilitate prediction of the behavior of this combination it is desirable to derive an equivalent circuit. Equivalent Circuit of a Transistor Shunt Negative Resistance An equivalent circuit of the transistor and its associated network is shown in Fig. 1(b). By denoting each condenser reactance as jX, the circuit determinant, A, can be \\Titten as follows. f2X -JX -jX 0 -jx r, + re + Rf + jX -Rf -re -jX -Rf /?/ + i?„ + jX -Ra 0 rm — re — Ra re + re — r,„ + Ra Xext the input impedance is determined as A/An • From this formula the exact general expression for the input im- pedance is found to be very cumbersome and will not be given. A us(^ful appi'oximation can be found by making some simplifying assumptions 7 I >-4Rf - (a) (b) (c). Fig. 1 — Equivalent circuit of transistor sfiunt negative resistance. 332 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 as follows: Let re > > I'b r,n > > n re >>/•« + Ra rm > > /'e + /?a Under these conditions the network can be represented by the eciuiva- lent circuit shown in Fig. 1(c). This circuit consists of a parallel com- bination of resistance and capacitance in which the capacitance is that of the original two capacitances in series and the resistance is negative and equal to four times the feedback resistor, Rj . Hence the magnitude of the generated shunt negative resistance can be controlled b}^ adjust- ment of Rf . One measure of the accuracy of this approximation is how much the "constants" of the equivalent circuit change with freciuency. Calculations of a typical case show that deviations in frequenc\^ of ±5 per cent cause deviations in both capacitance and negative resis- tance of ±0.05 per cent. Hence this approximation is very accurate for narrow band applications. This circuit can now be used to advantage in a band filter. Confluent Band Filter A conventional confluent band filter is shown in Fig. 2(a). In this structure the presence of dissipation in the series branches impairs performance by introducing flat loss, whereas any dissipation in the shunt branch not only produces flat loss, but, Avorse still, causes round- ing of the transmission characteristic at the edges of the band. For narrow filters having a small percentage band width, any appreciable dissiptation in the shunt arm can degrade the transmission characteristic beyond a reasonable tolerance. One good answer to this problem is to use elements having an extremely low resistive component such as quartz crystals. However, quartz is expensive and has other limitations. Another solution is to build a negative resistance into the filter so as to reduce the inherent element dissipation to zero or at least to a tolerable value. In the present case a shunt negative resistance will be used to compensate the shunt branch. This is done by splitting the shunt ca- pacitance of Fig. 2(a) and inserting the circuit of Fig. 1(a). This can be illustrated by an example. When the filter of Fig. 2(a) is designed to give a 5 per cent band at a midfreciuency of 10 kc and impedance level of 600 ohms the shunt branch offers an undesirably low impedance to the compensating transistor circuit and in addition requires cumbersome element values. Both difficulties can be corrected by using capacitative impedance transformations on each side of the shunt branch, thereby THE TRANSISTOR AS A NETWORK ELEMENT 333 ^A\ '■^'0"^ 1 f- ,f L, ' C — nm^ vw- c, L, x^ (a) CONVENTIONAL njw^^V-^^^ (b) ACTIVE Fig. 2 — Confluent band filters, (a) Conventional, (b) Active. rai.siug the impedance of the shunt !)ran('h without changing the im- pedance level at the input and output terminals. At the same time the elements assume much more reasonable values. The modified configura- tion together with the active portion is shown in Fig. 2(b). I'sing the filter described above, a series of transmission curves were calculated and are shown in Fig. 3. When ideal elements are assumed, the transmission is e = kV 'here A" = ip' + kp-{-l) [p4 + kp' + (fc2 + 2) p2 + /cp + 1] Wi and P = JO} Calculation of this expression results in the classical characteristic labeled "Ideal Passive". When, however, typical \'alues of element re- sistance are introduced, the transmission is e = (ao + «ip + a-ip- -\- azp^ + aip^Y — {hp -f 52^"^)^ (J i?2 R, 1\ = R^Cy T, = R2C2 1 22 u Ri ^23 U Rz A, = uc. A, = L2C2 334 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1934: where ao = /?i(l + g) a, = Rr\l\{\ + (/) + 7'o + 1\,] + R2r, a. = i^i[.-li(l + (/) + Ti(ro + Tn) + .4.] + R/r,7\2 a3 = /^,[.li(r2 + 7^23) + -42^,] ai = R1A1A2 61 = /?2T] Sa = RiTiT^i ^1 = Tx(l + g) /33 = TiAs This expression with the compensating negative resistance R^ = ^c prockices the characteristic labeled "Practical Passive". The dotted curves show the effect of adding various amounts of negative resistance to the shunt branch by letting R? assume negative values. The number on each dotted curve is the ratio of the resistive component of the shunt arm at anti-resonance to the magnitude of the compensating negative resistance. For example, for the curve labeled p = 1, the resistance in the shunt arm is entirely compensated so that the loss is only that due to the resistance in the series arms. By increasing the amount of compensation in the shunt arm so that p > 1, called overcompensation it is possible effectively to nullify the losses in the series arm as well. Comparison of the active curve labeled p = 1.21 with the "ideal passive" curve shows that this technique of resistance compensation can produce a practical filter ha\'ing a characteristic equal to that of a filter having ideal elements. Filters of this type have already been successfully used in field test ecjuipment. When the degree of compensation is increased still further, the filter begins to provide gain in the band as shown by the p = 1.36 curve. It is clear that continued increases in the compensation will eventually absorb the terminations causing the structure to become unstable. Although the curves given in Fig. 3 are all calculated, tests on experi- mental models show excellent agreement. To a reader having long ex- perience with passive filters the development of negative insertion loss may seem a little surprising. In order to lend an air of authenticity to this midband gain it is instructive to consider the behavior of a resistive tee section having a negative element. This is a reasonable analogue, THE TRANSISTOR AS A NI<7rW<)RK ELEMENT 335 becauHe at midfreqiiency the reactive component l)ecome.s zeio in each brancli of the filter. Transmission of Synunetrical Tec The insertion loss of a symmetrical tee section with p()si1i\'e resistance is a well known concept. It is doubtful, h()we\-cr, if the l)eha\'ior of a tee with a negatix'e t>lement is e(|ually well known. Consider the section shown in Fig. 4 operating between tei'minations R and ha\'ing series arms, A',i and a shunt arm, Rn . Normalize by letting a = Ra/R and 6 = Rn/R- Ins(>rtion loss is plotted vs. I> with a as the third parameter. For b positive the usual loss pattern results; for 6 negative, a more complex situation de\'elops. When b is very large and negative, the section is still producing a small loss, but as b l)ecomes smaller in magnitude the loss drops to zero and finally becomes a gain. There is a lower limit on the magnitude of 6 beyond which oscillations will occur. This limit is reached when 26 = -(a + 1). 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 FREQUENCY IN KILOCYCLES PER SECOND P'ig. 3 — Transmission of confluent band filters. 336 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Singularities of Confluent Band Filter Recent work on insertion loss design and potential analog methods by S. Darlington and others has fostered the practice of characterizing a network by plotting its natural modes and infinite loss points in the complex frequency plane. In the present case it is instructive to study the effect that reducing dissipation will have on the singularities. A full section confluent band filter has five infinite loss points and eight natural modes. In Fig. 5 the singularities of the passive, confluent band filter discussed earlier are plotted in the complex freciuency plane and identi- fied by the digit one. A single infinite loss point or zero lies on the nega- tive sigma axis, a pair falls at the origin, and a conjugate pair is located near the midband frequency. The natural modes or poles consist of two conjugate double pairs situated at about the upper and lower cut off frequencies of the filter. The distance of the complex singularities from LOSS = -20 LOG,o 2b _(a + l) +2b(a + l) a = 0.5 0.4 0.3 0.2 0.1 ^ ^ ^ -1 0 -e r^"- '^^ ^' -0. b 6 -0 .4 -0 2 -0.1 \ 1 ' 5 LU ZERO LOSS b = 2a ASYMPTOTE b = — Fig. 4. Transmission of symmetrical tee section. THE TRANSISTOR AS A NET-SVOHK ELEMENT 337 6b0 640 630 ALL POLES ARE DOUBLE 1 4 2 ^^•>x^ ^^ , ^^ 620 610 0 1 2 — -X " "-4 -'^3 — y- ' 3 1 4 1,2,3 1 1 5 I ! A'.2,3;4 I 1 1 I ( 1 1 r 1 r 1 -610 -620 1 1 1 4 ""—x 1 --X.. 2 ■'^' 2 3 -630 1 4 „^»'" 2 X 3 -fiSO -30 -28 -26 -16 -14 -12 -10 -8 -6-4-2 0 2 4 ^ RADIAN FREQUENCY IN HUNDREDS Fig. 5 — Effect of reducing dissipation on singularities of confluent band filter. the real frequency axis is a function of the amount of dissipation in the elemets. When a vahie of negative resistance corresponding to the p = 1 curve in Fig. 3 is added to the passive filter, the singularities move from positions marked 1 to those marked 2 in Fig. 5. Adding a larger amount of negative resistance corresponding to the p = 1.21 curve in Fig. 3 produces the singularities marked 3. It should be noted that the infinite loss point on the negative sigma axis as well as the two at the origin have not moved. If the dissipation in the shunt branch is reduced by removing the coil and replacing by one having half as much resistance, the singularities change from position one to position four. In this case the infinite loss point on the sigma axis does move. This illustrates that the change in pattern of singularities resulting from use of negative resistance is similar to, but not the same as, that resulting from use of passive inductors having higher values of Q. M-Derived Band Pass In order to provide a sharp cut-off in a filter use is often made of m- (lerived peak sections. In the configuration shown in Fig. G loss peaks will occur at selected freciuencies al)ove and below the pass band provided the elements are nearly free of dissipation. The closer the attenuation peaks are to the pass band the more nearly free from dissipation the 338 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 elements must be for good performance. As in the previous case, a transistor negative resistance is used to compensate the anti-resonant portion of the shunt arm. The magnitude of this resistance can also be adjusted to serve the additional purpose of compensating for resistance in the series resonant circuit in the shunt arm, as well as the series reso- nant circuits in the series arms. The transmission of a non-dissipative m-derived band filter between unit resistive terminations is .e = kp[{\ — m)'p^ + (2 — 2m -f- /c )p + (1 — 7n)\ {m'p^ + A;p + w) [p4 + km'p^ + (A;^ + 2) p^ -f- km'p + 1] where m = A/ 1 — ij- ) Assuming no dissipation a peak section with an m of 0.86 will give the characteristic shown in Fig. 7 labeled "Ideal Passive". However, when this filter is constructed with typical elements the curve labeled "Prac- tical Passive" results. By introducing a suitable amount of negative resistance the transmission of the practical filter can be made comparable to that of the ideal filter, as illustrated by the curve labeled "Practical Active". For maximum utility active filter sections must be capable of being connected in tandem to form composite filters without instability, re- flections, or interactions. Fig. 8 shows that these filters meet this require- ment by giving the measured transmission of a band filter composed of two dissimilar peak sections. On the basis of attainable electrical charac- c — 1( — ^WT" — ^^/\ j_ \AA/ — ^w^ — 1( — 0 Fig. 6 — Active M-derived band filter. THK THAX8ISTOU AS A NKTWOliK KLKMKX'P 339 9.3 9.4. 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 FREQUENCY IN KILOCYCLES PER SECOND Fig. 7 — Transmission of M-derived band filters. teristics, active filters of this kind appear to offer potential competition to the crystal channel filters used in broad band carrier systems. Working Model To further emphasize this fact the photograph of Fig. 9 shows a model of a composite band filter designed to transmit a 4-kc band at a midfrequency of 98 kc. This model contains seven miniature, adjustable, ferrite inductors, miniature capacitors, and two n-p-n junction transis- tors. The transmission characteristic is shown in Fig. 10. Hence in some cases by employing active cii-cuitry it is possible to use miniature com- ponents thereby gaining at least an order of magnitude in the size and weight of structure. 340 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1951 45.0 42.5 40.0 37.5 35.0 32.5 30.0 10 _l HI 5 27.5 O °25.0 z 1^ ^-^ ^ if) 22.5 O _i g20.0 H a: Lij 17.5 CO z 15.0 12.5 10.0 7.5 5.0 1 { ■\\ **v^^ if 1 1 / / \ \ ^ ^ I M WITH TRANSISTORS \ _ WITHOUT ' TRANSISTORS 1 \ 1 1 \ \ V y, V y I \ .-x' 8.5 9.0 9.5 10.0 10.5 11.0 FREQUENCY IN KILOCYCLES PER SECOND Fig. 8 — Resistance compensation of multi -section filter. Series Negative Resistance For satisfactory performance in many applications a series resonant circuit should approach zero impedance at the resonant frequency. To reduce the residual dissipation in an ordinary tuned circuit a series negative resistance, consisting of two transistors, can be used. This technique is illustrated in Fig. 11 which shows how a purely reactive shunt branch can be achieved in either an /??-derivcd low pass filter or a confluent band elimination filter. Fig. 9 — 98 -kc active channel filter. 40 36 32 If) -J lij (0 28 O lU a z24 ID in 3 20 z o ^ 16 lU i/> Z 12 8 4 0 \ /^ \ 11 V / vJ / / \ / V / 95 96 97 98 99 100 101 FREQUENCY IN KILOCYCLES PER SECOND Fig. 10 — Active channel filter. 341 342 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 II. ELIMINATION OF INDUCTANCE In the practical realization of frequency selective networks it is some- times awkward, difficult, or even impossible to make effective use of coils as inductive elements. This is true, because of severe limitations on space, exacting tolerances on undesired modulation, or necessity for operation at extremely low frecjuencies. It has been well known for some time that inductive elements can be eliminated without restricting the repertoire of the network designer provided he is willing to purchase this freedom by introducing active elements to supply gain. ' It can be easily shown that the transmission through a high gain feed- back amplifier is proportional to the product of the short circuit transfer admittance of the input network and the short circuit transfer impedance of the feedback network : e"" = YiZi In addition it is also known from energy relations that passive net- works containing only one kind of reactance cannot produce complex poles in the short circuit transfer admittance. It is instructive to con- sider the application of these principles to some familiar kinds of trans- mission networks. These networks can be logically divided into two classes: those which are primarily concerned with amplitude such as filters, and those mainly concerned with phase such as delay equalizers. (a) (b) Fig. 11, — Use of series negative resistance, (a) M-derived low pass filter, (b) Band elimination filter. THE TRANSISTOR AS A NETWORK ELEMENT 343 N'on-inductive Filters Low Pass Consider first an iinaj>;t' parameter, constaiit-A', low pa.ss filter which is usually built as a ladder-type structure of series iiiductaiice and shunt capacitance. A full section contains three reacti\-e arms and jii'oduces an asymptotic loss that increases 18 db per octa^•e. The transmission is given by the following expression (1 + coo V + ^0 V) (1 + '^o V) where coo = cut-off frecjuenc}^ in radians per second. The function has three poles, one real and two complex conjugate. The cjuestion now arises how this function can be divided between the input and feedback networks so as to be physically realizable. Since we know that a passive R-C structure cannot have complex poles in the short circuit transfer admittance, there is no choice but to use the impedance function in the feedback circuit for this purpose. It is now found that any R-C structure which will provide the complex poles insists on providing a real zero for good measure. This unwelcome zero can be nullified by supplying its counterpart as a pole in the admittance func- tion. The transmission is now rewritten, as follows: e-' = 1 _(1 + co^V) (1 + ap)_ 1 + ap I -\- coo^p -{- 0)0 y. and the singularities are shown in Fig. 12(a). Since the original transmis- sion function also rec[uires a real pole, the admittance function must now suppl}^ two real poles. A simple ladder structure having three seiies resistances and two shunt capacitances meets this requirement. The complex poles cannot be supplied by a ladder structure, but require some sort of l)ridge such as shown in Fig. 12(a). At low freciuencies the transmission through the filter depends on the ratio of the total series input resistance to the total resistance in the bridge arm of the feedback network. Therefore any amount of flat loss or a moderate flat gain through the filter can be obtained simph' by adjusting the ratio of impedance levels of the input and feedl)ack net- works. Simulation of functions by this technique does not provide a unique solution since there is considerable freedom in choice of configuration and location of the cancelling pole and zero. 344 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 c^AMr -wv — T — vw (a) T I — v\\- X jo; — 0— — X- LOW-PASS (b) X jU -® X Or HIGH-PASS (C) AAV -Q- X jcy BANDPASS X I Cd) I -&■ BAND ELIMINATION JO) Fig. 12 — Non-inductive active filters. THE TRANSISTOR AS A NETWORK ELEMENT 345 High Pass Consider next a high pass filter with cut-off at coo- The transmission is e = cco^p^ (1 + Wo V + '^o V)(l + «o V) o^oY 1 -j- ap _(1 + cooV)(l + apJLl + cooV + WoV. The complex plane plot in Fig. 12(b) shows exactly the same pattern of singularities as the low pass case with the addition of three zeros at the origin. To realize this function the feedback network remains un- changed, whereas the input network becomes a ladder in which the positions of the resistances and capacitances are interchanged. Band Pass A series resonant branch inserted in series between resistive termina- tions is a simple form of band pass filter having the following trans- mission : e = V r -1/^-1 1 + ^mOr^v + t^mV Li«+ «P_ Qv I + ap Ll + co-iQ-ip + OmY. where Wm is the radian frequency of the peak and Q is a measure of the sharpness of the peak. The singularities shown in Fig. 12(c) consist of a zero at the origin and two complex conjugate poles. Once again the complex poles are obtained b}' a bridge circuit in the feedback path. The usual penalty is incurred by the appearance of a real zero which must be cancelled by a real pole. Therefore the admittance function must supply a zero at the origin and one real pole. This is done by a series combination of resistance and capacitance in the input circuit. Band Elimination A parallel resonant branch inserted in series between resistive ter- minations is a simple form of band elimination filter having the following transmission: 2 e = II -2 2 1 + w,„ p 1 + <^m^QV + <^inV~ 1 + co,„ p I -\- ap 1 + ap. 1 -f ap _\ + oiJQp + oiJp-_ The singularities consist of two conjugate zeros on the real freciucncy axis and two complex conjugate poles. A bridge circuit in the feedback 346 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 path supplies the complex conjugate poles and a parasitic real zero, while a parallel tee in the input path provides the conjugate zeros and a real pole. It has been found that the experimental performance of the various non-inductive filters described can be predicted with precision from the theory. Noti-induclive Phase Sections Non-minimum phase networks are used extensively to provide a specified variation in phase with frequency without introducing any change in attenuation. Such all-pass networks consisting only of reactive elements are usually designed as lattices or bridged tee sections. It is theoretically possible and practically desirable to represent any complex all-pass structure by tandem arrangements of two basic all-pass sections called first degree and second degree. The first degree structure pro- t-V\A^ I AAA-- (a) ONE 180° SECTION JCJ ^^-i— VvaH W pvvv-r- (b) I •-AAA^ AW— I -VA--- -X— X- '-AW-' TWO 180° SECTIONS J^ -o-o- JW Fig. 13 — • Non-inductive active phase sections. THE TRANSISTOR AS A NETWORK ELEMENT 347 \i(les ii total change in phase of 1S()° and is characterized b.y a sinf>;le pole-zero pair symmetrically located on the a axis as shown in Fig. 13(a). Only one parametei-, the distance from the origin can l)e chosen. The second degree structure provides a maximum phase shift of 'M\()° and is characterized by two conjugate poles in the left half plane and two symmetrically located zeros in the right half plane shown in Fig. 13(c). The two parameters which can be selected are the rectangular coordinates of one singularity. It has been suggested that these functions can be realized without benefit of inductance. Here again numerous arrangements are possible, but onl}^ a few examples will be given. The basic operation is to perform one division on the original transmission function resulting in a quotient of unit}^ and a fractional remainder of opposite sign. The fractional re- mainder is then synthesized by a RC network in conjunction with an amplifier. Single 180° Section The transmission of a single 180° section is 1 COo p iiCOo - 1 1 + OJo 'p COo + p In this case the fractional remainder consists of only one real pole which is realized by the R-C structure shown in Fig. 13(a). Two 180° Sections The overall transmission of two 180° sections in tandem is the product of each transmission e = 1 coi'p _1 + ^i^pj Ll + ^-i^P 1 W2 P 2(cOi + 0)2 )p (1 -\- Vp)(l + co7ip) - 1 In this case the fractional remainder consists of one real zero and two real poles which are realized by the R-C structure shown in Fig. 13(b). Single 360° Section By far the most common phase corrector is the 360° section whose transmission is € = — _1 + Q-Wp + co-Y: = 9 Q OOmP _1 + ap_ 1 + ^p _1 + Q-^U-^P + U^n-P-- - 1 348 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 In this case the fractional remainder consists of a zero at the origin and two conjugate complex poles which are realized by the R-C structure shown in Fig. 13(c). III. PRODUCTION OF DELAY It has also been proposed that a two terminal active delay eciualizer can be constructed with the help of a negative resistance. As shown in Fig. 14 a two terminal network Z is connected between a resistive source and load, each of magnitude, one quarter R^. The network Z consists of a parallel combination of a reactive network ^X and a negative resistance ( — 7?o)- The transmission through Z is e = Ro "2" 2" i^o - jX jRoX Ro + jX ■So 1^ ^ Ro ^^ 2 +^ ~2'^ Ro~jX This is the desired function, because the amplitude of the transmission 1 1 1 1 jX 1 1 Ro< -Ro 4 < 1 1 0 ____. .J < 120 Q 0,00 OJ 10 § 80 U ^ 60 Z < 40 -I UJ a 20 0 /■ / 1 1 L J / V ^ ^ / \ \, ^S 2 4 6 8 10 12 14 16 18 FREQUENCY IN KILOCYCLES PER SECOND Fig. 14 — Active all-pass section. THE THANSISTOR AS A NMO'inN'OKK ELEMENT 349 is unity regardless of the size of A', and the plmsc and resultant delay are frequency dependent, because X is a function of fretjuency. It is theo- retically possible to produce the most complicated delay equali/.er characteristic by this method proN'ided the negative resistance remains constant over the d(;sired freciuency band. As examples only a single 180° section and a single 3()0° section will be considered. A 300° section results when the reactance is a single antiresonance gi\'en by ,.7" X = 1 - co^LC the transmission is -e ^ ip — k)' + (^l ^ 1 — QT^coZ^p + uZV ^ (p + ky + c,^ 1 + Q-'co-ip + -[!^ i-C ,R2RpC L R2 r\ p (e) (f; Fig. 15 — Impedance inversion. THE TRANSISTOR AS A NIOTWOKK KLEMENT 351 parameters of the transistor and the associatinl oxtcnial cii'cuil will satisfy the following cciuatioii 2ro Rp ^ '1\) eliminate non-essentials it will he assumed that r^ and /'c are neg- hgibly small, and Ha « >',.. Then by a straightforward, hut lengthy, analysis the driving point impedance is found to be Z = R, p==L'C' + p|' + l where L = XL Rf R + 4ro R' = \Rp ^ "X ro = Vc — Tm The circuit representing this impedance is shown in Fig. 15(e). Since negative elements are not convenient a final transformation is made to the circuit shown in Fig. 15(f). COXCLUSION The distinctive properties of the transistor suggest careful considera- tion of a philosophy which regards the transistor as a circuit element to be introduced at strategic points within a network. Initial work indicates that the judicious interspersion of transistors in a transmission network makes possible performance otherwise unobtainable or uneconomical. This paper has presented examples of how transistors may be used to reduce dissipation, to eliminate inductance, to produce delay, and to invert impedance. Undoubtedly this is only the beginning of exploration which should extend the horizons of network design. ACKNOWLEDGEMENT The author is indebted to many associates in Bell Telephone Labora- tories, in particular to E. I. Green for basic philosophy and to W. R. Lundry for much helpful advice, and many suggestions, including the novel concepts of non-inductive phase sections and active delay efiualizers. 352 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 i BIBLIOGRAPHY 1. H. Bode, U. S. Patent 2,002,216, May 21, 1935. 2. A. Hull, Description of the Dynatron, Proc. I.R.E., 6, p. 5, Feb., 1918. 3. A. Bartlett, Boucherot's Constant Current Networks and their Relation to Electric Wave Filters, J. I. E. E., 65, p. 373, March, 1927. 4. H. Mouradian, Some Long Distance Transmission Problems, Journal Franklin Inst., 207, p. 165, Feb., 1929. 5. B. van der Pol, New Transformation in Alternating • — Current Theory with Application to Theory of Audition, Proc. I.R.E., 18, p. 221, Feb., 1930. 6. L. Verman, Negative Circuit Constants, Proc. I.R.E., 19, p. 676, April, 1931. 7. G. Crisson, Negative Impedances and the Twin 21-Type Repeater, B. S.T.J. , 10, p. 485, July, 1931. 8. F. Colebrook, Voltage Amplification with High Selectivity by Means of the Dynatron Circuit, Wireless Eng., 10, p. 69, Feb., 1933. 9. S. Cabot, Resistance Tuning, Proc. I.R.E., 22, p. 709, June, 1934. 10. E. Herold, Negative Resistance and Devices for Obtaining It, Proc. I.R.E., 23, p. 1201, Oct., 1935. 11. L. Curtis, Selectivity Control for Radio, U. S. Patent 2,033,330, March 10, 1936. 12. C. Brunetti, Clarification of Average Negative Resistance with Extension of its Use, Proc. I.R.E., 25, p. 1595, Dec, 1937. 13. E. Schneider, A. New Type of Electrical Resonance, Phil. Mag., 36, p. 371, June, 1945. 14. E. Ginzton, Stabilized Negative Impedances, Electronics, 18, pp. 140, 138, and 140 of July, Aug., and Sept., 1945, respectively. 15. J. Merrill, Theory of the Negative Impedance Converter, B.S.T.J., 30, p. 88, Jan., 1951. 16. H. Harris, Simplified Q. Multiplier, Electronics, 24, p. 130, May, 1951. 17. J. Muehlner, Transfer Properties of Single and Coupled Circuit Stages With and Without Feedback, Proc. I.R.E., 39, p. 939, Aug., 1951. 18. F. B. Llewellyn, Some Fundamental Properties of Transmission Systems, Proc. I.R.E., 40, p. 271, March, 1952. 19. G. Fritzinger, Frequency Discrimination by Inverse Feedback, Proc. I.R.E., 26, p. 207, Feb., 1938.' 20. R. Dietzold, Frequency Discriminative Electric Transducer, U. S. Patent 2,549,065, April 17, 1951. 21. R. Blackman, Effect of Feedback on Impedance, B. S.T.J. , 22, p. 269, Oct., 1943. Continuous Incremental Thickness Meas- urements of Non- Conductive Cable Sheath By B. M. WOJCIECHOWSKI (Manuscript received August 5, 1953) A method has been reccnthj developed for measuring thickness variations of a non-conductive cable sheathing, extruded over a grounded metal jacket. The method translates direct capacitance increments, sensed by probes sliding on the surface of the sheath, into thickness increments. The accuracy of the system based on this method is sufficiently high that the electrical error, which is of the order of a few thousandths of a ^i^iF, can be disregarded. E.xperimental data indicate that accuracy of the new system for absolute thickness measurements of homogeneous samples in stationary conditions is of the order of 0.002". The error caused by translating capacitance to thickness depeiuJs on manufacturing elements and process tolerances, and can be evaluated on a statistical basis. Thus incremental measurements of the cable sheath thickness on the production line yield accuracies of the order of 0.003" . Application of this method to absolute sheath thickness measurements involves assumptions directly related to calibration and man ufacturing process control. These aspects are rather extraneous to a measuring .system per se, and, therefore, are not within the scope of this paper. 1 INTRODUCTION 1.1 The New Cable A tj'pe of telephone cable has been developed in which lead is replaced with a polj'ethylene sheath extruded over a metal jacket. Since descrip- tion of various aspects of this development can be found in the technical literature/'"'^'* only some details of the cable construction and pro- duction that are pertinent to the understanding of the new measuring system, will be briefly outlined here. The cable core, Fig. 1, is covered with a thin layer of a highly con- ductive metal, such as aliuniiuim,^ or two layers of different metals, such as aluminum and steel,^' ^ sealed longitudinally. To achieve the desired 353 354 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 mechanical properties, the metal jacket is corrugated circumferentially. Between the metal layer and the plastic sheathing, a bonding viscous thermoplastic compound is applied (Fig. 2). Normally, this compound fills the depressions of the corrugations on the metal surface adjacent to the surrounding polyethylene jacket. The sheathed cable leaves the extruder with an essentially uniform speed, under pulling force of a capstan. For various sizes of cables and production settings, this speed may range from 30 to 80 feet per minute. After leaving the extruder, the cable is cooled in a trough of water and, before reaching the testing position, dried with compressed air. Fig. 1 — Polyethylene sheath telephone cable. INCREMENTAL SHEATH THICKNESS MEASUREMENTS 355 1.3 Measurement Difficulties Some of \hv prol)l(Miis encountered in tiie nianut'adufe of the new cable were related directly to the lack of reliable methods for measurinp; thickness of the plastic sheathing. Under manufacturing conditions, where sheath thickness cannot be adequately controlled, excess material must be used to assure meeting minimimi thickness reriuirements. Before the new method was de\'eloped, measurements were made by destructive testing of end samples. One or two circumferential strips weie taken from each cable length and microm(>ter measurements were perfonned on each strip, at four to eight points. Unfortunately, the actual sheath thickness ^'aI•ies in a random way along the cable length, even between points only a few inches apart. It was evident that a method, based on a few point measurements, extrapolating long-cable properties which are describable rather in statistical terms only, left much to be desired. 1.3 Preliminary Considerations The following methods of cable sheath measurements were considered: A. Use of an X-ray machine. B. Ultrasonic echo method (radar technicjues). C. Capacitance measurements. For practical reasons as well as for anticipated lack of accuracy, the first of these methods was rejected. The success of the second method was judged doubtful, the main reason being the presence of corrugations and of an irregular layer of the filling compound under the polyethylene sheathing, obscuring delimitation of the reflecting boundary surface. The third method, at first, also had discouraging aspects. In the case under discussion only grounded capacitance measurements are involved, since the metal core cannot possibly be insulated from the corrugating and forming machinery. The recjuired long-time capacitance-to-ground sta- bility and accuracy of the measuring system were estimated to be of the CABLE CORE SUPPLY (IN A CORRUGATED GROUNDED STEEL JACKET) APPLICATOR OF THE BONDING CEMENT POLY- ETHYLENE EXTRUDER WATER COOLING TROUGH TESTING ASSEMBLY WITH MEASURING PROBES COMPRESSED AIR DRYING POSITION FINISHED CABLE TAKE-UP Fig. 2 — Block diagram of the polyethylene extrudinti process. 356 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 order of 0.001 fifj-F and 0.003 mmF, respectively. Meeting requirements of this order, even under controlled laboratory conditions, presents some difficulties — and yet these requirements had to be met on a production line, on moving cable in the climatic and operational conditions prevail- ing in a large cable plant. It was evident, therefore, that conventional grounded-capacitance measurements would not be practical. For instance, a shielded cable connecting the probes with the bridge circuit alone could produce wider random capacitance variations than the capacitance increments under measurement. Thus a new system which would meet all the necessary requirements had to be developed. 2 CIRCUIT DESCRIPTION The measuring system which Avas developed consists of an impedance bridge, a phase sensitive detector, an unbalance indicator (recorder), capacitance probes and associated auxiliary equipment (See Figure 3). 2.1 The Impedance Bridge for Grounded Direct Ca'pacitance Measure- ments. The circuit shown on Fig. 4 employs a bridge having ratio arms magnetically coupled. An application of this type of circuit for capaci- tance measurements has been known for some time.' Such a circuit is capable of performing in one balancing operation direct capacitance measurements while the center point (B) of the transformer ratio-arms winding is grounded. In our case, the ''D" corner of the bridge consists of the metal covering of the cable core, which, as was mentioned above, is necessarih' at the ground potential. Therefore, the "B" corner cannot be grounded. However, by connecting to this ''B"-corner a shielding,^ surrounding the "A-D" and "C-D" measuring arms, including cables and probes, the following results can be achieved: (a) Admittances from the measuring electrodes to the "B"-shielding are not critical. These admittances appear across the transformer-arms and, as a result of a close magnetic coupling reaUzable between these arms, any loading effects across any one of them are sjTnmetricall}^ re- flected at the "A" and "C" corners of the bridge, thus essentially not affecting its balance. (b) Stra}'' admittances from the "B" shielding to ground appear across the opposite corners of the bridge (detector diagonal). Therefore, they also have no essential effects on the circuit balance. (c) As a result of the "B "-shielding, stray admittances-to-ground INCREMENTAL SHEATH THICKNESS MEASUREMENTS 357 Measuring asscMuhly. from the measuring electrodes and from the connecting leads can be reduced to insignificant cjuantities. As a result of the described circuit configuration, the bridge measures capacitance quantities eciuivalent to direct capacitance, in a particular case where one of two measuring electrodes is grounded. Realization of the grounded direct capacitance measurements is made possible by having within the measuring arrangement a three-electrode system in which stray admittances from the third (ungrounded) electrode to either 358 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 of the measuring electrodes do not affeft the fundamental balance con- dition of the bridge network. By the arrangement described not only are the residual effective capacitances between the measuring electrodes and groimd reduced to a desirable minimum (actually below one /x/xF, including calibrating capacitor and balancing networks), but also any adverse capacitance effects of the cables connecting the bridge to the measuring probes are practically eliminated, even though these cables are several feet long. The calibrated grounded direct capacitance range of the bridge ex- tends over 0.32 /x/iF in either direction off balance center position. Any unbalances within the ± 0.25 /x/xF range can be read in increments of 0.005 fiixF per division on a recorder. Since covering such a limited capacitance range directly by an adjustable capacitor could present various practical difficulties, a network, dividing electrically the range of a 100 /X)LtF differential capacitor by the ratio of 150 (approximately), has been applied. Using such a network facilitates calibration and ad- justability and greatly reduces effects of the mechanical instability of the variable capacitor. (Similar networks are applied for capacitance and conductance residual balance controls.) Stationary unbalances of the bridge network can be measured directly in a conventional manner by rebalancing the circuit with the calibrated capacitor. For unbalances rapidly varying in time, however, this null method could not be applied simply. Therefore, a proportional off-bal- ance deflection method had to be used and various means to ascertain overall linearity between incremental capacitance unbalances and indi- cator deflections were provided, so that eventually variations in linearity no larger than 0.4 db over periods of several days and 0.2 db over several hours have been observed in the actual operating conditions. Measurements with the l)ridge depend essentially on the calibrated capacitor. To avoid necessity for freciuent and quite elaborate calibra- tion checking (within a few one-thousandths of a mmF) of this capacitor in a laboratory, a set of supplementary, high stability auxiliary standards has been provided in the test set assembly. The capacitance values (1.05 jixjuF; 1.20 hijlF; 1.35 nnF) of these capacitors are so chosen that differences between any pair of them can be compared directly with the calibrated capacitor in the bridge circuit. Reliability of this system is based on a reasonably high probaliility that change in the calibrated ^'alue of any single capacitor will be revealed in the process of mutually comparing all four capacitors. It was felt that this method of ascertaining calibration accuracy at the operating position was particularly recommended in the case of this circuit as its sensitivity to incremental capacitance unbalances IXCREMEXTAI, SIIKATII TnirK\i:sS MKASTHKMEXTS 350 is actually higher than the sensitivity of the usually available laboiatoiy equipment. 'I'he bridge network is supplied by a 10-kc ac power source. 2.2 Phase Sensitive Detector For eccentricity measurements and control of the sheathing process it is essential to register the direction of incremental deviations from an arbitrary level. For this purpose a phase sensitive detector ' has been provided. Its simplified version is shown on Fig. 4. By proper adjustment of the phase-shifter, the reactive component of the bridge unbalance signal can be oriented to be in-phase with the reference potential (b-a). In this condition, the capacitance unbalance sensitivity of the discriminator is at its maximum, and for a certain range of capacitance unbalances, linearity of- the indicator may be as- sured. Also, when the above phase condition is fulfilled, the circuit is not sensitive to limited conductance unbalances (this fact also renders the cii-ciiit remarkably more stable than a similar circuit using a con- ventional null detector). The dc output from the discriminator is fed through a balanced output stage (V2a and V2b), and an attenuator to a Leeds & Northrup zero- centered recorder. At the operating sensitivity level, each of the 100 divisions of the recorder scale corresponds to 0.005 mmF, or approximatel}^ to 0.001 inch of the incremental sheath thickness. The role of the at- tenuator is two-fold : it provides control of the over-all sensitivity of the measurements (in steps of 0.2 db), and it introduces more than 20 db attenuation into the dc output signal path. This loss is compensated by an added gain within the feedback-controlled ac amplifier (AC-A) pre- ceding the phase discriminator. The net result of this "ac for dc gain- trading" is a considerable improvement of the over-all circuit stability since the range of random drifts, such as usually generated within the phase-discriminator and its direct-coupled output stage, are materially reduced. .'.■! MeaHuring Probes As has been mentioned above, two arms of the bridge circuit consist of a pair of admittances between the grounded metal core of the cable (D corner) and the pi'oljes sliding on the surface of the i)lastic cable sheathing. These probes are connectetl io the "A" and "C" corners of the bridge, respectively, with two shielded flexible conductors (each 360 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 m INCREMENTAL SHEATH THICKNESS MEASUREMENTS 361 about 10 feet long) and are maintained mechanically in the testing posi- tion h}^ the probe assembly (see Figs. 3 and 5(a)). In the design of the probes and their assembly, various difhculties had to !)(' overcome. The probes operate on cables subjected to some luunoidable swings and vibrations while moving with speeds up to 80 feet per minute. The capacitance from either of these probes to the metal cable core, in equivalent conditions, should match each other within approximately one-thousandth of a mmF. This capacitance should not be appreciably affected by limited displacements of the probes with respect to the cable plane of symmetry, such as may occur in actual operating conditions. The first experiments with probes of a conventional design, having flat, or nearly flat, contact surfaces, were quite discouraging. The probe- to-core capacitances fluctuated to an intolerable degree as a result of even minute cable displacements. Eventually, probes were developed which met all the requirements. Each of these probes is in the form of a cut-off segment of a toroid. The major axis of the cut-off elliptical plane is oriented in the direction essentially parallel to the cable axis, while the convex center part of the probe slides on the cable sheathing. This form of probe has the advantage, common with the spherical form, that the capacitance from the probe to the cable core varies but little as a result of displacements and changes of position caused by the cable motion. But the toroidal form has the following advantages over the spherical: first, for the same residual capacitance to the cylindrical cable core, the transverse dimensions of the former are smaller; and, second, the capacitance of the toroidal form with respect to a cj^lindrical cable core can be conveniently adjusted by the simple expedient of twisting the probe element in a plane parallel to the cable axis. (Adjustments with a precision exceeding one-thousandth of a iifiF were actually performed). The probe electrodes, surrounded (except for the contacting face) by the B-shielding, are mounted on mechanically balanced light alu- minum arms [Figs. 5(a) and 5(b)]. There might be one, tw^o, or four probes to an assembly, which can be turned over 3G0° around the cable axis. For eccentricity measurements two probes can be simultaneously used, having a spacing of 180° (foi- measurement of eccentricity across a diameter) or of 90° (for measurement of elipsoidal eccentricity). Also for eccentricity or direct thickness investigations and process settings one probe only may be used, with th(! other bridge measuring arm connected to an auxiliary standard. ^- METAL JACKET Fig. 5 — (a) Mecasuring probe assembl}-. (b) Probe element. 362 INCREMENTAL SHEATH THH'KXKSS MKASUUEMENTS 303 The ax'orajic (•ai)acitaiic(' tiom the ptohc clcnicnt to the ^rouiidcd metal core vai'ics Iroin 1.1 to 1 .ii idtuV t'of cahlcs measured. 3 EXI'EUIMENTAL RESULTS S.l (^ircin'l PcrfornKinrc Under StalioiKinj Conditions Jnrn nu nt(d (■(ipdcihtncc scnsilirilji for grounded direct capacitance measurements, in normal operating conditions with the probes in contact with a cable sam])le: order of ().()() 1 mmI'- Circuil stohiliii/ (uid rrpcdUdn'lili/ for periods over one hour duration: ±0.003 mmF. Orcrall Uncdritij of the unbalance indications, as read on the recorder scale within the range of plus or minus 0.25 ^u/uF off center-balance position: ± (3 per cent + 0.003 mmF). Mechanical Stahiliftj: Moving or twisting of the connecting leads has no effect on balance stabilit3^ Swinging of the cable under measurement, even beyond the limits encountered in actual working conditions, pro- duces barely noticeable effects on the balance indication. Capacitaticc nieasitrement.s on flat poti/elhijlene seimplcs: One of the measuring bridge arms was connected to the auxiliary standard of 1.20 mmF. The other arm was terminated by the probe in contact with a flat l)olyethylene sample [placed on a grounded metal-plate. Thickness of samples at the point of contact was measured with a micrometer to the nearest 0.0005 inch. Capacitance unbalance readings were taken directly on the recorder scale to the nearest 0.005 mmF. In order to avoid notice- able "air-gap" and "surface" effects, which occur when stacking several samples, in no case were more than two flat samples in a stack measured. Under these conditions, repeatability of readings was within one recorder division (0.005 ijluF), eciuivalent approximately to one-thousandth of an inch. In a t\'pical case shown on Fig. (i, of 38 measurements taken in the thickness range from 0.052 inch to O.KiS inch, only three measurements were off from the averaging curve by more than 0.002 inch. (Further investigation disclosed that these three points, marked "A" on Fig. G, were all associated with a particular sample.) Capacitance tneasurcments on stationary cable samples. In order to estal)lish statistical reliability of measurements on actual cables by the described capacitance method, a numbci- of cable samples were tested, varying in core diameter, a^'(M•age poi.xcthyicne sheathing thickness and mechanical construct ion. A typical graph resulting from plotting capacitance increments versus micrometer measun>ments of a cable sample is shown on Fig. 7. Out of 364 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 \ AUXILIARY STANDARD = 1.20 A/^F \ % N": N K ^ >^. Vl \ ^.n N \ \ ^. x v_ ^ D 0.17 0.16 0.15 [2 0.14 I o ? 0.13 z 10 0.12 e \ A V^ -5 0 5 1C 15 CAPACITANCE X 0.005 /i/iF Fig. 7 — Capacitance versus thickness measurements of caMe sheathing sample. 366 THE BKLL SYSTEM TECHNICAL JOUKNAL, MARCH 195-1 KU. ■'■^i---.. "Y' " ■1===-"'"""""' 4- SI 1 ttei^ 1 ^""""*|l "" 2II '''.'■: -Q: acli is in ordei' at this point. The Latin S(iuare is peculiarly well suited to engineering research problems \vher(> many variables exist but the luimber of discrete levels necessary to describe^ the variation of any one is small, where relatively great precision is possil)le in measurements, and where the interaction of the variables is not a factor of the experiment. A key design- for a 5x5 Latin Square is as follows: Column low 1 2 3 4 5 1 A B C D E 2 B C D E A 3 C D E A B 4 D E A B C 5 E A B C D It will be ol)ser\-ed that each Latin letter falls once and only once in every row and column. It is also true that if the logical subgroups .1 , B, C, D, E are considered, each of these sums has each row and column included once. If to the five row- values, column values and letters, the five values of the first, second and third variables are associated respectively, the variables represented by Row, Column, and Letter can be evaluated independently of each other. On the Key Latin Square, another properly chosen sfiuare represented by Greek Letters can be superimposed, and the five values of the fourth variable assigned to these letters at random : 1 2 3 4 5 1 a /3 y b € 2 7 b e a 0 3 e a ^ y b 4 ^ y b € a 5 b € a (3 y itin S(iuare below is then pi •oduced : 1 2 3 4 5 1 Aa B& Cy Db Ee 2 By Cb De Ea A^ 3 Ct Da Ei3 Ay Bb 4 D^ Ey Ab Bt Ca 5 Eb At Ba Cfi Dy 374 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Note that now each of the variables associated with Row, Column, Latin letter, Greek letter has the property that each element of the four categories contains one and only one element of the remaining three categories. For example, consider the five Latin-Greek letter combina- tions, or sample cells, containing e: Row 1 Co\ 5, E; Row 2 Col 3, D; Row 3 Col 1, C; Row 4 Col 4, B; Row 5 Col 2, .4. Then the sum of these five cells will contain the contribution of the 5 Rows, 5 Columns, and 5 Latin letters. It should })e noted that the rows and columns can be permuted with- out affecting the properties of the scjuare. Lideed to protect against systematic effects which may be detrimental, it is usual to assign at random the row and column number, as well as the Latin and Greek letters to the values of variables represented by them. A notable excep- tion to randomization occurs when time is a variable and those measure- ments made under essentially the same conditions within the same unit of time become the experimental unit. In the first experiment, all meas- urements made on one Run become the experimental unit with respect to time. The Factorial Design serves a different purpose. In this discussion only two independent variables A", Z will be predicated but the extension to more variables follows directly. The XZ plane is the plane of the in- dependent variables and we seek the point (Xo , Zq) which gives a value of y (the dependent variable), Y = ^(A', Z), which is optimum in some sense. That is, min Y, max Y may be sought, or the surface f(X, Y^, Z) shown to be a plane. Generally only the region in the neighborhood of y (optimum) is of interest to the experimenter. Hence it is imperative (1) to bracket this point with respect to each independent variable and (2) to have a method of estimating // (optimum). If a factorial experiment has U different values of X and Iz different values of Z, then each replication of the experiment will recjuire U • h units or points (A', Z). The first repetition of an experiment is called the second replication in the same way that the first overtone in music is called the second harmonic by engineers. Since the number of units available for test is usually limited, this places a practical ceiling on the magnitude of 4 and h . As a practical limit in general / should be 7 or less and the values 2, 3, or 4 are far more common. It is generally better to use the smaller values of I and repeat the expei'i- ment, than to conduct an experiment involving only a single replication. In addition to evaluating one variable averaged ovei- the second, we CARD TRANSLATOR EXPERIMENTS 375 ai'o interested in e^■alnaling the interaction of the varial)les on each other, when such intei'action exists and is of interest. In a sense tliis interaction measures the departure of the system y, X, Z from Hnearity. Once the basic designs have been selected and ai)propriately combined to fit most etfici(Mitiy the recjuiremcMits of \\w, pr()])()sed (experiment, and the vahu^s of the variables randomly assigned to th(> schematic layout, a detailed experimental layout must be drawn up. This layout must show concisely and cleai'ly each experimental unit and the makeup of every basic element giving its assigned value of each variable. Explicit dii'ections must be drawn up as to the order of selection of the elements of the unit. It is generally advisable for simplicity to assign the elements at random to the M possible consecutive order integers of the experi- mental unit. Performance Sludy The first seven variables hsted in the Introduction are: 1. The Bin in use, Bins. 2. Position of Test card pair within the bin, Position. 3. The use of 3 digit or 6 digit Cards, Code. 4. Arrangement of Coded and Uncoded Cards, Runs. 5. Load of Bins containing Test Cards, Load. 6. Load of Bins not containing Test Cards, Idlers. 7. Order of repeated measurement. Look. These constitute a system — that is, any or all can be varied at will and hence a design involving all of them simultaneously can be sought. The test set can operate ten test cards; 5 with a 3 digit code and 5 with a 6 digit code. This immediately suggests 5 packages of two coded card pairs, each pair containing a 3 and a 6 digit card. Five pairs can also be handled neatly in 5 bins. The combination of the standard load of 85 cards, with two overloads and two imderloads w^ould give a fair evaluation of load criterion. Budget restrictions force the use of only a limited number of coded cards, with blanks used to fill out the experi- ment. Hence the type of card making up the load must be varied over the loads. It was further found that 5 positions of test cards within the bin covers the range of positions adequately. The pairs of 3 and 6 digit cards now are associated with the Graeco- Latin Scjuare design with Columns identified with Bins; Rows with dis- tribution of coded cards; or Runs; Latin letters with Load; and Greek letters with position within the bin. Now if the position of the 3 digit card is randomly assigned in the pairs, the design abs()rl)s the first five 376 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Table I — Design of Graeco-Latix Square Experiment Each run made with the "x" bins loaded with 0, 50, 85 and 100 cards and consists of four operations (A, B, C, D) of each coded card. Bin No. Photocell Mounting I II III X IV V X 1 X X 1 X 1 X X Lamp Position in Bin a b c d e f^g Bin I Bin II Bin III Bin rV Bin V Card- 1 6 2 7 3 8 4 9 5 10 Run #1 Pos. in Bin d g 2 98 / g 41 44 a b 50 0 a d 15 0 c e No. of Coded Cards . No. of Blank Cards. 2 103 Run ^2 Pos. in Bin c e 85 0 d g 9 41 f g 15 0 a b 2 103 a d No. of Coded Cards . No. of Blank Cards. 2 98 Run ^3 Pos. in Bin / g 2 103 a b 100 0 a d 2 83 c e 2 48 d a No. of Coded Cards. No. of Blank Cards. 7 8 Run #4 Pos. in Bin a d 2 48 c e 2 13 d g 105 0 / g ,1 a b No. of Coded Cards . No. of Blank Cards. 2 83 Run ^5 Pos. in Bin a h 2 13 a d 2 103 c e 2 98 d g 85 0 / g No. of Coded Cards . No. of Blank Cards. 22 28 variables of the list. The layout of these variables in the Graeco-Latin Square design is given in Table I. The remaining two variables are functions of different portions of the machine. The rows of the Sciuare, involving distribution of coded and uncoded cards, represent distinct machine set-ups, and hence if for every set-up the load of the seven not-measured bins is varied, and for every variation in this load 4 observations of the 5 pairs are taken, CARD TKANSLATOli HXPEUIM lONTS 377 consideration of the remaininf>; two variables is achieved. Yet the tedious part of handling the test cai'ds has been reduced to a very reasonable amount. These last two \'ariables, considered by themselves, form a factorial design and where 4 loads and 1 nKvisurements per load are used, 1() measurements being taken for eacli machine set-up. Similarly any of the other five variables considered pairwise with one of the latter two forms another factorial design. This lesulting complex overall pat- tern is shown in Table I. Table II — Translator Tilt Test — Design of Experiment Test Cards Location Bin I II III IV V Cards 6,1 2,7 8,3 4,9 10,5 Position of Test Cards in Bins ab Total number of cards in each bin: 100 Bin # I contains all coded cards. Description of Tilt Tilt ^0 Lamp end lower than photocell end by }{&" in 3 feet. Translator resting on table at all four points. North Lamp end higher than South Lamp end by }{e" in 22". North Photocell end higher than South Photocell end by He" in 22" Tilt $1 Lamp end higher than photocell end by %6" in 3" ft. Lamp end supports blocked up J^" North Lamp end higher than South Lamp end by }/^" in 22" North Photocell end higher than South Photocell end by ^" in 22" Tilt *2 Lamp end higher than photocell end by ^^e" in 3 ft. Lamp end supports blocked up J^" North Lamp end higher than South Lamp end by ^{e" in 22" North Photocell end higher than South Photocell end by ^fc" in 22" Tilt #3 Lamp end higher than photocell end by l}4" in 3 ft. Lamp end supports blocked up %" North Lamp end higher than South Lamp end by ^e" in 22" North Photocell end higher than Souh Photocell end by %6 " in 22" 378 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Tilt Study Routine field practice for the installation of a card translator calls for leveling the table before positioning of the translator. After a trans- lator was placed it had been found that the level was not maintained, and the question of final level reciuirements was raised. Accordingly the test machine was run at 0 in., % in., 1 in., and 1%6 i^i. tilt, Table II. Two test cards were placed in each of five bins. Since the two end card posi- tions on the low side of each bin were suspected as being critical, the two test cards were placed in these positions in the five bins. The experiment is shown schematically below: Bins I II III IV V TiltO cte&i Uih a^hz 0469 aiobi 1 Oebi 0267 as&s ttib^ ttiobh 2 a^hi 0267 a^z O469 ttiobi 3 a^bi 0267 ttsbs ttib^ aio&5 where a is the end position, b is the next-but-end position, and the sub- scripts identify the actual card number. Considering any bin, the two card positions and the four values of tilt may be considered as a factorial design. Similarly, the five bins combine with the four values of tilt as a factorial design. Balanced Loading At the onset of general usage many translators will not be fully loaded. It was desirable to investigate the effects of various patterns of loading at three representative low loads of 200, 400 and (300 cards respectively. Four logical patterns were studied (see Tables III and 1\) all of which are shown below for a load of approximately 400 cards. (Similar patterns follow with loads of 200 and 600 cards.): Bin 1 2 3 4 5 6 7 8 9 10 11 12 W XX X X Y X X XX Z 000000000000 X = 100 cards, 0 = K2 load (approximately) The ten test cards were all placed in Bin One. This experiment involves only two factors; the four loading patterns, CARD TKANSLATOU EXi'EltlMEJMTS 379 Table III — Loading Patterns in Partially Loaded Card Translator — Balanced versus Unbalanced Load Test Load No. of Cards in Bins Treatments Bin I Bin II 100 Bin III 100 Bin IV 100 Bin V 100 Bin VI 100 Bin VII Bin VIII Bin IX Bin X Bin XI Bin XII 600 100 — w 400 100 100 100 100 — — — — — — — — 200 100 100 — — — — — — — — — — 600 100 100 100 — — — 100 100 100 — — — X 400 100 100 — — — — 100 100 — — — — 200 100 — — — — — 100 — — — — — 600 100 100 100 — — — — — — 100 100 100 Y 400 100 100 — — — — — — 100 100 200 100 — — — — — — — — — — 100 600 50 50 50 50 50 50 50 50 50 50 50 50 Z 400 33 33 33 33 33 33 33 33 33 33 33 37 200 16 16 16 16 16 16 16 16 16 16 16 24 Note: All cards in Bin I are coded, see Table IV. Cards in Bins II-XII not coded. and the three loads of cards. A suita})le design is the factorial design with one factor, the loading pattern, at four levels (W, X, Y, Z) and the other factor, the loads of cards at thr(>e lev(>ls (200, 400, 000) with ten test card measurements taken at each of the twelve points. III. data The data on card dropping time were obtained l)y means of siiadow- grams of the light output from two of the light channels of the translator. Each shadowgram comprised the operation of the 10 test cards in se- ([uence. Samples of these shadowgrams are siiown on Figs. I and 2. Vov the purpose of these expei'iments the cai'd dro])i)ing time is defined as the time from the release of the puU-uj) magnets until the full closure of the light channels of the translator exclusive of any cai'd rebound. The data from the various experiments were tabulated and ar(^ gixcn in Tables V to XI, inclusive. 380 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 CARD 2 RUN 3 100 CARDS IN BIN .^_— LIGHT OUTPUT OF LEFT APERTURE iih, lltlllHilHitlllllHIimiMii., •LIGHT OUTPUT OF RIGHT APERTURE CARD 1 RUN 5 15 CARDS IN BIN iiiiiitiiiiiiiiiiiniiiniiiiiiM littllltlMMIlliniHIHHiii. — H 10 MS (< CARD 4 RUN 5 85 CARDS IN BIN iniiiiiiiiiHiiiiitiiiiiiiittii,,. CARD 9 RUN 1 15 CARDS IN BIN |iM(«lliHMllilllilMMH!Hif!!-,.. lillllllHItllllliilllliilHlMnt. CARD 5 RUN 5 50 CARDS IN BIN ;-■--: i^-'-LF^ _ c '-GNET C.jPRE i.O_Fr.O.D L,. ^ = FNT Fig. 1 — Card motion shadowgraphs, card support bars working CARD TRANSLATOR EXPERIMENTS 381 CARD 2 RUN 3 100 CARDS IN BIN liniliiilllMIHIIilHIHIIIIIIiii. iifmiiiniiHiiiHii(finiiiii«M """" CARD 1 RUN 5 15 CARDS IN BIN llllllillllllltlHIIIIIIIHli.. iintiiiitniiintiiliiiiiMh. PUT )F : f h r A^'^-R1 : if^h CARD 4 RUN 5 85 CARDS IN BIN llllllllllllilllllllililllllillh ..... jiniiitiitiHiitiiitiitiiMtiih ... ->- 10 MS ^ .^PD 9 PUN 1 15 CARDS IN BIN UHMUiHttiiiiiiHiiHiiliiiiiK CARD 5 RUN 5 50 CARDS IN BIN liiiiiiiiiHiiiiiiiiiliiilliiii. .,M HiiniiiMiiiMtiiiimHHiii .„ ¥i(,. 2 — Card motion .shadowgraphs, card support l)ar.s out. 382 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Table IV — Card Translator Balanced versus Unbalanced Load Test — Arrangement of Coded Cards in Bin I No. of Cards Arrangement of Coded Cards n Bin I <— Left end of Bin Right end of Bin -► 100 50 33 16 #1 #2 10 #3 #1 #2 5 #3 #1 #2 3 #3 #1 #2 ; #3 15 #4 15 #5 10 #6 15 6 #4 7 #5 4 #6 7 4 #4 3 #5 3 #6 3 / #4 / #5 0 #6 / #7 15 #7 6 #7 4 #7 1 #8 10 #9 #10 #8 5 #9 #10 #8 3 #9 #10 #8 1 #9 #10 Note: Numbers preceded by # represent the test cards. Numl)ers in italics represent quantity of non-test coded cards placed between two consecutive test cards. IV. ANALYSIS OF THE DATA ' After cheeking the raw data for recording errors, they were analyzed and reduced in three distinct steps as follows: 1. Using the techniciues of the analysis of variance* each variable and each measured interaction of several variables was tested as a pos- sible assignable cause of variation. 2. Using the components of variance analysis! on those variables and combinations of variables found to be assignable causes in Step 1 their contributions to the overall variation was estimated, and 3. After tabulating the arithmetic mean for each value of the variables an upper and lower bounds were determined which estimates the allow- ance, or probable range of means, for similar experiments and opera- tions on card translators having the same residual a. The technique used was proposed and formulated by J. W. Tukey.^ The analysis of variance tables were reduced to the summary in Tables XII and XIII for the seven variable studied. Proceeding to the tabulation of the means and the calculation of the allowances appropriate to these means, the data was reduced as in Table XI\\ The magnitude of the several effects can now be noted. The effect of Idler load is c^uite meaningful — as the load on the machine is in- creased to the normal load of 85 we see a decrease in mean dropping time from 3G.5 to 34.4 milliseconds. Also the slight increase for the overload of 100 is not significant statistically or engineeringwise. At first glance the results of Idler loads of 0, 50, 85 and 100 might seem to be inconsistent with those of Operating Loads of 15, 50, 85, 100, and 105. The mean dropping time of the Load of 15 (say) is the a\-erage dropping time of * See Appendix for Discussion and References 2, 4 and 5. t See Appendix for Discussion and Reference 6. CAUD TUAXSLATOU EXTKUIMENTS 383 Tahlk ^' Card Drop Time i\ Milliseconds -- No Cards in X Bins — Card Support Bars Operating Bin * I Bin * II Bin x III Bin « IV Bin * V Card »-* 1 6 2 7 3 8 36.5 4 9 5 10 T^un «1 A 36 38 36 35.5 37 37 37.5 37.5 40 B 36.5 38 36.25 37.5 35 37 36.5 37 37 39.5 C 35.25 37.5 35 36 37 36 36 36 37.5 39 D 35.5 38 37 36.25 36.5 36.5 36.5 36.5 37 39 X 35.8 37.8 36.06 36.32 36.37 36.5 36.5 36.75 37.25 39.37 R 1.25 0.5 2.0 2.0 2.0 1.0 1.0 1.5 0.5 1.0 Run *2 A 34 35 34 33.5 34 34 36 35 37 39 B 33.5 34.5 34 34 34 34 35.5 33.5 37 38 C 34 34.5 33.5 34 35 34 36 35 36.5 38.5 D 35.5 33 33.5 34.5 34.5 34 35.5 34.5 36.5 38 Hull *2 X 34.25 34.25 33.75 34.0 34.37 34 35.75 34.5 36.75 38.37 R 2.0 2.0 0.5 1.0 1.0 0 0.5 1.5 0.5 1.0 l^iii «3 A 36.5 37.5 38 36.5 37 37 37 38 37.5 39.5 B 36 37 36.5 35 36 37 36.5 36.5 38 39 C 36 36.5 37.5 35.5 36 35 37 36.5 38 39.5 D 35.5 36.5 37.5 35.5 36 35.5 35.5 36.5 37.5 38.5 Run «:■! X 36.0 36.37 37.37 35.62 36.25 36.12 36.5 36.37 37.75 39.12 R 1.0 1.0 1.5 1.5 1.0 2.0 1.5 1.5 0.5 1.0 Run #4 A 37 37 35.5 37 36 37 37 37.5 38 38.5 B 36 37.5 35.5 36 36 37 36.5 37.5 38 38.5 C 36 37.5 35 36 35.5 36 37 38 37.5 38 D 35.5 37 36 36 36 36 37 37.5 37.5 37.5 Run «4 X 36.12 37.25 35.50 36.25 35.87 36.50 36.87 37.62 37.75 38.12 R 1.5 0.5 1.0 1.0 0.5 1.0 0.5 0.5 0.5 1.0 Run «5 A 36 36 37 37 35.5 36.5 37 36.5 38.5 38.5 B 36 35.5 37 37 37 37 37 37 38.5 39 C 36 36 36 36 36 36.5 37 37 38.5 39 D 36 35 35.5 36 35 36 36.5 37 38 39 X 36 35.62 36.37 36.50 35.87 36.50 36.87 36.87 38.37 38.87 R 0 0.5 1.5 1.0 2.0 1.0 0.5 0.5 0.5 0.5 Cards in Bins loaded with lo cards in the presence of 4 bins loaded with 50, 85, 100, and 105 cards respectively and of bins loaded as a group with 0, 50, 85, or 100 cards. On the other hand the mean dropping time at- tributable to an Idler load of 50 cards (say) is the average of all dropping cards in the operating bins when the 7 Idler loads are 50 cards. Hence the statistical conclusion that the effect of loads in the operating bins is slight over the range of loads of 15 to 105 cards is reached, and that the 384 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Table VI — Card Drop Time in Milliseconds — 50 Cards in X Bins — Card Support Bars Operating Bin *! Bin *II Bin i((Ill Bin *IV Bin *V Card « - 1 6 2 7 3 8 4 9 5 10 Run ^1 A 35 36 34 35 34.5 35 33 34 33.5 35 B 33.5 35 34.25 34 33 33.5 32.5 33.5 33.5 34.5 C 34.5 34 32.5 34 32.5 32.5 32 33.25 32 34 D 34 32.5 32 34 33.5 33.5 33.25 32.5 33 33 X 34.25 34.37 33.19 34.25 33.37 33.62 32.69 33.31 33 34.12 R 1.5 3,5 2.25 1.0 2.0 2.5 0.75 1.5 1.5 2.0 Run #2 A 35.5 35 34 34 32.5 35 35 35.5 35 35.5 B 35.5 35.5 34 33.5 34.5 34.5 35.5 35 35.5 35 C 35 35 34 35 34.5 34 36 35 34.5 34 D 34 35 34 34 32.5 34 35.5 34 35 35 X 35.0 35.1 34 34.1 33.5 34.4 35.5 34.9 35.0 34.9 R 1.5 0.5 0 1.5 2.0 1.0 1.0 1.0 1.0 1.5 Run #3 A 35 35 37.5 34 34.5 34.5 34 34 34 34.5 B 36 36.5 37 36 35.5 35.5 34 34.5 34.5 36 C 35 35.5 39 35 35.5 35 34.5 34.5 34.5 35.5 D 35.5 37 39 35 35.5 34 34.5 34.5 34.5 35.5 X 35.37 36.0 38.12 35.0 35.25 34.75 34.25 33.37 33.37 35.37 R 1.0 2.0 2.0 2.0 1.0 1.5 0.5 0.5 0.5 1.5 Run #4 A 37 37 35.5 35.5 36.5 35 35 35.5 36 37 B 36.5 37 36 36.5 36.5 36 35.5 35.5 36 37 C 37.5 38 35.5 36 36.5 35 36 35.5 35.5 36.5 D 37 37.5 35.5 37.5 36.5 36.5 36.5 36 37 38 X 37 37.37 35.62 36.37 36.50 35.62 35.75 35.62 36.12 37.12 R 1.0 1.0 0.5 2.0 0 1.5 1.5 0.5 1.5 1.5 Run ^5 A 36 36 35.5 36 35.5 36 34.5 36 34.5 35 B 35 35.5 35 36 35 35 34 34 34.5 35 C 35.5 36 35.5 35.5 34.5 35 34 35 34.5 35 D 36 35.5 35 35.5 34.5 35.5 34 34.5 34 34.5 X 35.62 35.75 35.25 35.75 34.87 35.37 34.12 34.87 34.37 34.89 R 1.0 0.5 0.5 0.5 1.0 1.0 0.5 1.5 0.5 0.5 improvement in dropping time caused by increasing the overall load is generally consistent over the test bin loads. One last estimate must be made — that of the o-' for a single measure- ment where* ''^assignable causes See Appendix for meaning of s\'mbols and discussion. CAKD THAXSLATOR KXPKKIMENTS 385 Table VII Card Drop Time in Milliseconds — 85 Cards in X Bins — Card Support Bars Operating Bin iffl Bin *II Bin «III Bin *IV Bin )((V Card Iff -* 1 6 2 7 3 8 4 9 5 10 T^un «1 A 34.5 34 32.5 32.5 33 34 32.5 34 33.5 33.5 B 35 34 34 32 32 34 33 33 33.5 33 C 33 34 33.5 33.5 32.5 33.5 33 33.5 33.5 33 D 35 34 34 32 33 33.5 33 33.5 32.5 34 X 34.4 34 33.5 32.5 32.9 33.7 32.9 33.5 33.25 33.4 R 2.0 0 1.5 1.0 1.5 0.5 0.5 1.0 1.0 1.0 Run «2 A 33 34.5 32 34 33 34.5 34 33.25 33.5 33.5 B 32 35 34.5 34.5 32 33.5 35 34 33.75 34 C 34 35 33.5 34.5 33 34.5 35.5 35 35 35 D 34 34 34 34.5 33.5 35 35 34.5 35 33 X 33.25 34.62 33.5 33.4 32.9 34.4 34.19 33.55 34.31 33.79 R 2.0 1.0 2.5 0.5 1.5 1.5 1.5 1.75 1.5 1.5 l^iii *3 A 35 35 36.5 32.5 34 34.5 33 34.5 34 34 B 35 35.5 37 34.5 35.5 34 33 34.5 34.5 34 C 35 34.5 36 34.5 35.5 33 33.5 34 34.75 35 D 35 35 37 34.5 35 34.5 34 34.5 34 34.5 X 35 35 36.62 34 35 34 33.37 34.37 34.31 34.37 R 0 1.0 1.0 2.0 1.5 1.5 1.0 0.5 0.75 1.0 Run #4 A 37 37 36.5 36.5 36 36 35 35 36.5 36.5 B 38 37.5 37 36 36.5 37 36 35 36.5 36.5 C 37 37 37 35 36 36.5 35.5 35 36 36.5 D 36 35 36 36 36 36 35 35.5 35.5 36.5 X 37 36.62 36.62 35.87 36.12 38.37 35.37 35.12 36.12 36.25 R 2.0 2.5 1.0 1.5 0.5 1.0 1.0 0.5 1.0 1.0 Rum fHo A 35 35 34 35 33.5 34 34 33.5 33.5 34 B 34.5 34.5 34 34 34 33.5 34 33 33.5 33.5 C 34 34.5 34 34.5 34 33.5 33 32.5 33 34 D 34 35 33.5 34.5 33.5 34 33 33.5 32.5 34 X 34.37 34.75 33.87 34.50 33.75 33.75 33.50 33.12 33.12 33.87 R 1.0 0.5 0.5 1.0 0.5 0.5 1.0 1.0 1.0 0.5 If the assignable causes contributing to this estimate are not removed then this ; is necessary when the trans- intoi' is installed on the table ))ro\ided an uncoded card is next to each separator. W'lien only cards that were at least one removed from the separator were considered it was found that the Translator could he tilted 1 inch in 3 feet without seriously affecting the card drop time. C'onsitlering- the ranges of the several variables considered and the results of the analysis of the data, it appears that there is no major un- known variable having an effect on the card dropping time. It is also believed that the results of the work on this machine can be considered representative of the results that will be obtained on another new pro- duction model translator. Appendices i. analysis of variance The general theory of the analysis of variance has been formulated and discussed at length by several authors.'' *' ^ Basically it reduces to the concept that in any set of data obtained from a statistically designed experiment the total sum of sciuares of deviations from the mean can be partitioned into orthogonal components, and that under certain restrictions the distribution of each component falls into known pat- terns. Hence data taken from designed experiments can be examined for conformance to the known pattern, and a lack of conformity indicates an assignable cause of variation. Further, the distribution of the ratio of mean square deviations under specified conditions has been tabulated as the table of the F ratio. It has also been shown that when a treatment variable is not a parameter or assignable cause of variation in the ex- periment, the partitioned component for that variable must contain only residual variation. Thus, the analj'sis of variance tests the hypothe- sis that the treatment means for a given variable are all eiiual (i.e., the variable is not a parameter) by testing the ratio of the mean s(iuares of mean deviations for the variable to the residual mean s(iuare, i.e., the F ratio. When this F ratio is larger than the critical value at the a" level, th(^ variable is said to be significant at the a ' level. That is, let 396 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 US assume a null hypothesis that the variable in question is not a param- eter or assignal)le cause, and select a critical value, F*, such that the probability of ol)serving an F ratio j^reater than F* (when the null hypothesis is true) is small (say 0.01). Then if the F ratio is computed from our experimental observations and the null hypothesis rejected when this ratio is larger than F*, on the average incorrect decisions will be made not more than 1 per cent of the time. This method of evaluation is not trivial and in complex situations reference should be made to the literature or to experts. II. COMPONENTS OF VARIANCE A basic difference between the estimation of the Components of Variance and the Analysis of Variance above is the concept of the underlying model or law. The Analysis of Variance tests the hypothesis that the treatment variable is not a parameter. In the estimation of Components of Variance we assume that the observed effect of the several values of a given variable is a random sample from a normal population of effects from these values. If w;,, is the true effect of the i'' value of the v variable, then the component of variance due to the y' variable, a„ , is found from 2 Z2 _ k - 1 2 If Uiv = Uov = ■ • ■ = Ha,, , then ct„ = 0, and the mean square for variable V contains only residual variation. If the variable y is a parameter, o-p > 0; and the mean sciuare for variable v contains al -\- Mai (M meas- urements being made at each of the k levels of the variable). It is desir- able to estimate the component of variability of each variable, in order to be able to estimate the variability of a measurement which is affected by these variables. That is, if there are p variables whose components of variance are a} , i = I , ■ • • , p respectively and if the measurement, x, is influenced by all of these variables, then (T; = Z ^I + (tI , and <^x = /l/ ^ each based on N samples of four observations each w(> will detect as significant, differences as small as = ^V^ 4-095 \\'hen N is large we will, therefore detect as significant, differences which may be of no interest engineering-wise. Thus not only is the significance of the effects of interest but also the magnitude of the effect. When the component of variance attributable to each of the sig- nificant effects is estimated only four are so large as to be of interest to the engineer. The four variables are Idlers, Runs, interaction of Idlers on Runs and Bins. The estimates of the components of variance, ^-effect , are obtained by equating the linear combinations of the components of variance shown in the right hand column of Table XII to the mean squares which estimate them and solving. The component estimates are: 0-2 Idlers 3. 2 ms Runs 1.55 2 ms Idlers X Runs 1.60 2 ms Idlers X Runs 1.26 2 ms III. SOURCES AND MEASURES OF ERROR In any experiment a decision must be made as to the number of ex- perimental units to be measured and the number of repeated measure- ments to be made on each unit.^ It is important to note that measure- ments made on the same unit and a measurement made on each of several units give rise to two distinct sources of variation, and that both of these should be estimated. Consider making n measurements on each of /.• units, where the k units are a random sample of units belonging to a normal universe with mean u and variance 0-5 . Further a set of measure- ments on the i' unit is a random sample of measurements from a normal universe of measurements with mean Ui and variance a^ . Clearly, if 398 THE BELL SYSTEM TECHNICAL JOURNAL, MAKCH 1954 the set contains only one measurement {n = 1), we cannot estimate ffy, , and if- there is only one unit (^ = 1) we cainiot estimate al . When both n, k > 1 we can estimate simultaneously both al and crl, . Let Xij be the j measurement on the i' ' iniit, J = 1, • • • , n; I = 1, ... , k. and Xi be the mean of the t' unit, X be the mean of all the units. We can estimate o-^ directly by computing k n E Z (X, - xf k{n — 1) (Xb can only be estimated indirectly by first estimating cri + H(rl from n E {Xi - Xy- k Then the estimate of ab is in t ^^' - - xf 1 m k- - 1 n \ k - I k n \ »-i j-i I (w - l)fc / ■ 1 ^ Since o-^ = ^ (o-b -\ — -), it is clear that if al is large relative to al, , then k n X, for fixed M=tik, will have greater precision if k is large and n is small. The estimate of al, is called the sampling variance or sampling attriluit- able to repeated measurements. The estimate of al is called the component of variance due to experimental variation free of sampling error. For a given experiment the estimate of al, -\- nal is called the experimental error term and measures the precision of measurement of a unit. In the experi- ment ay, = as, and ab = a e • BIBLIOGRAPHY 1. L. N. Hampton and J. B. Newsom, The Card Translator for Nationwide Dial- ing, B.S.T.J., 32, pp. 1037-1098, Sept., 1953. 2. W. G. Cochran and G. M. Cox, "Experimental Designs", J. Wiley and Sons, New Yor,, 1950. 3. J. W. Tukev, "Comparing Individual Means in the Analysis of Variance", Biometrics, Vol. 5, No. 2, 1949. 4. W. J. Dixon and F. J. Massev, "Introduction to Statistical Analysis" McGraw Hill, New York, 1951. 5. R. A. Fisher, "Design of Experiments", Oliver and Boyd, Lontlon, 1950. 6. S. Lee C'rump, "The Estimation of Variance Components in the Analysis of Variance", Biometrics, Vol. 2, No. 1, 1946. 7. C. Eisenhart, "Assumptions Underlying the Analysis of Variance", Biometrics, Vol. 3, No. 1, 1947. Wave Propagation Along a Magnetically- Focnsed Cylindrical Electron Beam By W. W. RIGROD and J. A. LEWIS (Manuscript received August 24, 1953) This paper analyses the nature of wave propagation along a cylindrical electron beam, focused in Brillouin flow by means of a finite axial magnetic field. Two different types of conducting boioidaries external to the beam are treated: (/) the concentric cylindrical tube, forming a drift region; ami {2) tlie sheath helix, forming a model of the helix traveling-wave tube. The field solution of the helix problem is used to evaluate the normal-mode parameters of an equivalent circuit seen by a thin beam, thereby permitting computation of the gain constant of growing waves. The gain constant of the cylindrical beam with Brillouin flow is fourui to exceed that of a similar beam with rectilinear flow, presumably because of the transverse component of electron motion in the former. IXTRODUCTIOX The theory of the heUx traveHng-wave has been treated in previous l^apers/"^ for cases in which the electrons move along straight lines paral- lel to the axis of the helix, as though immersed in an infinitely strong magnetic field. In practice, however, the electron beam is focused by a magnetic field of finite intensity,^" ^ such that the electrons follow spiral paths alxjut the common axis. The purpose of this paper is to extend traveling-wave tube theory to the case of such focused beams, and to compare the gain constants for the two types of electron motion. The motion of the beam in an infinite field is usually described as rectilinear flow; that in a finite focusing fi(>ld, as Brillouin flow. The gain constant of the dominant mode in a traveling-wave tube may be computed from the field solution for the electron beam in the presence of its circuit structure. This procedure, however, recjuires the solution of cumbersome transcendental equations for each particular set of dimensions and operating conditions. A more flexible method of anal- ysis has been provided by Pierce,^ based on an expansion in terms of 399 400 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 normal modes of propagation. For any particular tijpe of beam and cir- cuit, three circuit parameters must be evaluated from the field solution. The performance of the traveling-wave tube is then described quite ac- curately by a cubic equation containing these parameters, over a wide range of dimensions and operating conditions. The usefulness of this normal-mode method has been further enhanced by publication of a nomograph for the calculation of the gain constant. In its initial form, the normal-modes solution for a helix travehng- wave tube was greatly simplified by the assumption that the electron beam is so thin that the electric field acting on it is constant. Employing the field solution for a beam of finite thickness in a helix, Fletcher'' was able to compute the circuit parameters for the solid and hollow cylindri- cal electron beams, respectively, confined to rectifinear flow. This procedure will now be extended to cylindrical beams in Brill ouin flow, in which transverse electron motion occurs. First, it will be neces- sary to solve the field equations for this type of beam in a helix. As a by-product of this computation, the solution of the field equations for the beam in a concentric drift tube will briefly be given. Finally, with some restrictions, the helix parameters will be evaluated, and the gain of helix amplifiers with such beams compared with that obtained with otherwise identical rectilinear beams. FIELD EQUATIONS IN THE ELECTRON BEAM When a small ac field is impressed upon a short length of electron beam, the electrons respond by executing small ac excursions about their steady- state trajectories. These ac motions of charged particles constitute a transverse distribution of ac currents, which in turn excites an ac field distribution. The propagation of an ac signal along a beam depends upon the reciprocal action of these currents and fields. To find the propagation constants for a particular configuration of electron stream and enclosure, we must therefore solve Maxwell's equa- tions in the presence of the ac driving currents in the beam, subject to the external boundary conditions. When the fields and currents possess circular symmetry, these equations may be formally separated into TE and TM groups." In addition, as we are concerned only with "slow" waves, the equations may be simplified by neglecting all terms of rela- tive magnitude k /y , where k is the wave number in free space, and 7 the propagation wave number. TM WAVE Id/ dE\ 2-n T't-iTI^/t-n f^\ -^V-^)-'^^^^ ^'^-+--T- ^rJr) (1) r or \ or I jcoe coe r or WAVE PROPAGATION ALONG AN ELECTRON BEAM 401 ^^^j_dE._j<^j^ (2) 7 dr 7" H, ^'^Er- ^- Jr (3) TE WAVE r dr \ dr / r dr H, = I (^Jh + J.) (5) y \ dr 7 y \ dr I Here (r, 0, 2) are the polar cjdindrical coordinates, co the angular driving frequency, e the dielectric constant and /x the permeability, of free space, in ]MKS units. The ac ampUtudes of the electric and magnetic fields, and the convection-current density, respectively, are represented by the components of E, H, and J. All ac quantities have been assumed to vary as exp j(<^^-7^)- When the assumption is made that the convection current density in the beam is of the same order of magnitude as the displacement current density, equations (2) and (6) reduce to the following: E, = i ?f' (7) 7 dr E, = -^ ?p (8) 7^ dr In order to evaluate the components of / in the beam, it is necessary to determine the velocity and charge distributions, first in the unmodu- lated, and then in the ac modulated beam. The focusing of long cyhndrical electron beams by axial magnetic fields of moderate strength has been fully described by Brillouin^ and Samuel^ This type of electron motion, called "Brillouin flow", can be established when a parallel electron beam abruptly enters a suitable magnetic field. The electrons thereupon acquire an angular ^•clocity component which leads to a balance of radial forces in the beam. The equations of motion of electrons in an axial magnetic field Bo are as follows: r - r^ = v(dVo/dr - rdBo) (9) rd + 2rd = vrBo (10) z = rj-dVo/dz (11) 402 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 In these equations (r, 6, z) is the position of an electron at time t ; dots indicate differentiation with respect to /, following the electrons; rj = e/m., where — e is the electronic charge and m its mass; and Vo is the potential describing the steady, axially symmetric electric field. Relativistic ef- fects and the magnetic field resulting from electron motion have been neglected, as our interest is confined to beam velocities which are small compared to that of light. It is readily verified that a solution of the above eciuation is: ;. = 0, d = do = riBo/2, k = 1(0 (12) rj dVo/dr = reo\ dVo/dz = 0 (13) Thus all the particles in the beam have the same angular velocity, equal to the Larmor angular frequency, and the same axial velocity Uq . From Poisson's equation, we find the charge density: po = - 2ek'/v (14) It is convenient to introduce the angular plasma frequency cop , de- fined by: cOp = —r]po/e = 2^o" (15) In steady-state flow, an electron with initial position (ro, ^o, ^^o) ha^ the position (ro , ^o + di^t, Zo + uJ) at time t. When the beam is mod- ulated by a small ac signal, the electrons suffer small ac displacements from their steady-state trajectories. If we assume that the signal propa- gates along the axis of the beam as exp j(o^t — yz), we can write the perturbed electron coordinates in terms of the Lagrangian coordinates (ro , 9o , Za) as follows: r = ro + f(ro)-expj"[co/ - 7(^0 + ud)] (16) e = Oo -h dot + e(ro)-expjWt - 7(^0 + WoO] (17) z = Zo -{- not + 2(ru) • exp j[a)/ - y{zo + ^'oO] (18) where the tildes indicate ac amplitudes, and the dots indicate, as before, time differentiation at fixed ro , do , ^o- Thus the dots are eciuivalent to multipHcation by j(w — 7?/o), when applied to ac quantities. The equations of motion for the ac modulated beam differ from the steady-state equations (9) — (11), in that the particle coordinates are now given by (16) — (18), and there are ac fields present in addition to the dc fields —dVo/dr and Bo . As is usual in small-signal theory, only first-order ac quantities are retained in any equation. To this approxi- WAVE PIJOPAOATIOX ALOXO AX KLl^rTHOX liKAM 403 niatioii, the ac fields can l)t' cxaluatcd at the iiii|)cilini)('(l particle l)()sit ion. Not all (tf the ac fields need to appear in the toi-ce ('(illations, however. Keference to the field ('(illations shows that the (•t)iitril)uti()n.s of the ac nuijiiietic fields to the force components arc smaller than those due to the electric fields hy a factor of the order of (uo/r)" or smaller (where c is the velocity of light), and hence may he neglected, in addition, the force exerted by Ke i^ <>f the same order as that due to II r , and may be neg- lected tt)(). Omitting the factor exp j\o:t - 7(^(1 + "oO] for brevity from all ac terms, we can write the equations of motion as follows: ? - (/•„ + 7){d, + 'df = -r][-dV,/dr + Kr + (aj + r)ie, + e)B,] (19) (/•u +r)l + 2Hdo -\-'d) = -n^'Bo (20) I = -7,7?., (21) These equations may be simplified with the aid of (12): 7J dVo/dr = (ro + l-)dl (22) and by recalling that the dots may be replaced by multiplication by j(aj — yuo). We obtain, finally: r = -nEr/ioi — yuof (23) 9 = 0 (24) z = 77i?,/(co - 7''o)' (25) Although the foregoing equations deal with the dynamics of individual electrons, the assumption that the beam behaves like a smoothed-out "fluid" of charge, with a single \'el()city at each point, enables us to assign values of velocity and all other ac quantities, to fixed positions in space, (r, 6, z). In these coordinates, the do velocity is given by: vo = (0, rd, , Wo) (26) and the ac velocity by: V = (r, re, i) ^ ^27) = j(w - yuo)[{r, re, z)] Although theac quantities are defined at n , they may be taken to be the same at r, to a Unear approximation. 404 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 The same result, (27), might have been obtained by stating the equations of motion in terms of Eulerian coordinates, in which the per- turbed variables are the components of fluid velocity at any fixed point. In this procedure, the "material" or total time derivative would l)e used in the expressions for acceleration. The ac space-charge density p is found with the aid of the continuity equation: ^ (po + p) = -div [(po + p){v, -f i')] (28) at JPo div V (29) CO — yuo From (23)-(25) and (27), the ac velocity may be written: V = ^"^ {Er , 0, E;) (30) CO — yuo Combining these with Poisson's equation, we find: ""'' dwE= ^ ^'^ -P (31) (co — yuoY (co — yuoy There are two possible solutions to (31): (co — yuo) = cop (32) P = 0 (33) Solution (32) represents two longitudinal space-charge waves of arbi- trary amplitude distribution, with plasma-frequency oscillations about the average beam velocity: 7 = - ± '^ (34) Wo Wo The second solution, (33), however, permits us to evaluate the compo- nents of the ac convection current density J, and thereby solve the field equations (1) — (8): J = Pov -\- pvo (35) Jr = PoV,- = —. T -T- (36) 7(co — 7Wo) or /fl = 0 (37) J, = p,v, = _ -li^ E. (38) CO — 7Wo WAVK I'HOI'AGATION ALOXG AN ELECTRON BEAM 405 The wave (Mniations (1) and (4) for /s, and II, now reduce to the following. These equations have solutions for E^. and Hz , which are finite at r = 0, of the form A-Io(yr), where .4 is an arbitrary constant and /o the modi- fied Bessel function of zero-th order. It is not without interest to remark that the same pair of solutions, given by (32) and (39) — (40), has been found by L. R. Walker for a beam of arbitrary cross-section, with the same longitudinal velocity and space-charge density at every point, in the absence of any impressed do magnetic field. Due to the radial component of electron motion, the beam surface is rippled. For a steady-state radius h, this rippling can be expressed, in a linear approximation, by the perturbed radius: r(b) = 6 + rib) exp j(oot - yz) (41) The rippled beam is equivalent to a uniform cylindrical beam with an ac surface charge density por, or a surface current density whose components are: Gz = poruo (42) Gg = pordob (43) The total ac convection current may be written in a form which applies equalh' well to the cylindrical beam with purely rectilinear flow: Ic = f ./.27rr dr + 2Trbp,n^r{b) •'0 = -jioe-R-2Trb-A-h{yb)/y = -j.eRl E.2.rdr ^^^^ where jR is a beam propagation function which will prove convenient: (co - yuoY {yb - m' ^ and ^e = Oi/Uo , (3p = OOp/Uo (46) 406 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Thus we note that wave propagation along a cylindrical beam with Brilloiiin flow is accompanied by swelling and contracting of its boundary, with constant space-charge density, rather than by space-charge bunch- ing. The second interesting result is that the dynamics and field ecjua- tions for the focused beam are identical with those for a beam with zero dc magnetic field, except for the angular component of surface current density Ge ■ SPACE-CHARGE WAVES We now consider the given beam, of radius h, in a concentric conduct- ing tube of radius a > h. The boundary problem consists of matching the TM wave admittances inside and outside of the beam, at its boun- dary. (The TE fields are of no interest in the drift-tube problem, as they are not excited at the ends of the tube, and are not coupled to the TM fields.) Let I refer to the beam region 0 < r < 6, and II to the space between beam and conductor b < r < a. Then, at r = b, He + Gz _ He tjz til. (47) The beam admittance on the left is evaluated with the aid of (3), (7), (36), and (42): Y 7 /o(7o) (48) In region II, E. = B-Ioiv) + C-Ko{yr) He - ^""'[B-hM - C-K,(yr)\ 7 where 7io and Ki are modified Bessel functions of the second kind. The wave admittance at r = 6 in II is therefore: i c 'h{lb) - {C/B)-K,{yb) (49) 7 L/o(7&) + (C/j5)-Ko(t6)J At r = a, E^' = 0 or: C/B = -7o(7a)/i^o(7«) (50) Equating beam and circuit admittances (48) and (49), we obtain: _ hiya) R = ^ ""^'^1 . ■ --, (51) K,{ya) yb-Ii{yb)- lUyb) - ^''^7^Va^o(76)] WAVE PROPAGATION ALONG AN KLKCTKOX HKAM 407 This cHiuation must ho solved simultaneously with cadi ol' the follow- ing: 7i6 = 0J) + l3,h/\/Ri ^ (52) Thus, for a j^ixen l)eam and froquoncv, the solution consists of two un- attenuated waves, one faster and the other slower than the beam \elo('ity. The wavelength of the'interference pattern is given by: ■4:7r X. = -^^ (53) 7i — 72 For a cylindrical beam, (3 J) = 174VP (54) where P = I/V^'^ amps/(volts)^'^, the perveance. In practice, P and hence /3p6 are usually so small that we can gain a fair estimate of X^ by assuming 7?i = Ro: \s ^ ~— (OO) Pp Fig. 1 shows the variation of R^'^ with yb for several values of h/a. (The "intrinsic" solution (32) is included as a line at R^'~ = 1.) The ordinates of these curves are approximately proportional to the space- charge wavelength, and the abcissae to the frequency, as y c^ ^e = wA/o for small perveance. Space-charge Avaves propagating along a cylindrical beam with rec- tilinear flow have been treated by Hahn and Ramo^. In Fig. 2, their computations have been reformulated in the same way as in Fig. 1, and compared with the results for Brillouin flow, for two values of b/a. The space-charge wavelength is always greater in Brillouin flow, for the principal pair of waves and the same b/a and 76. HKLIX PROBLEM In place of the drift tube at radius a, we now have a helically conduct- ing sheet of zero thickness and pitch angle \l/. In addition to I (0 < r < b) and II (6 < r < a), we shall use III to identify fields in the region (a < r < x). The boundary conditions at r = b are: H/ + a - H/' = 0 E/ - E/' = 0 (56) HJ - Ge - HJ' =0 E/ - EJ' = 0 408 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 A.t r = a, the boundary conditions are: E^ + Ee' cot 1/^ = 0 cot i/' = 0 Er - e!" = 0 Er + Er Hj' + He'' cot yp Hz — Ha cot \p 0 (57) Inasmuch as cot rp ~ t/A-, the contribution of Ee to the field at the helix can conceivably be comparable to that of Ez . The TE fields are coupled to the TM group, in addition, through the angular surface cur- rent Ge , which depends on Ez . AllS equations must therefore be solved simultaneously. The procedure follows that of Chu and Jackson" for the field solution Vr 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 11 M BRILLOUIN FLOW V ' V -1=0.8 V ^-\ -0.6 V \ -0.4 V w \ \ 0,0.2--^ \ \ \, \^ \, \ N, 1 ^ :^ ' ■ . . i^V2 p LASM a,-FR LQUE vICY 0 SCILL ATIO^ JS /b Fig. 1 — Space-charge wavelength X, for cylindrical beam -n-ith Brillouin flow, in a concentric drift tube. Here 6 and a are the beam and tube radii, respectively; R^'^ is a dimensionless parameter; and the waves propagate as exp j{cot — yz). To compute Xs c^ 'ZwR^'-flSp , use I3pb = 174P"2, where P is the beam perveance. The abscissae are approximately given by 7 '^ |3e = oj/uo . (Equations 52-55.) WAVl'; I'KOP AC A ri(».\ ALOXc; \\ KI.KCTKOX Hi: \ M 101) Vr 3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 0.8 i w 1 i \ \ \s -.^"'. -^ 277- y- 7,-72 ~ /3p 1 1 1 1 \ 1 \ 1 \ 1 w» [\ .1 = 0.8 \ \ \\ \ \ V \ \ \ V \ \ \ \ \ \ \ \ \ \ \ X, \ H ^v ^ '<"^ i-^ ^v. :v. ^.^ PLASMA -FREC UENC Y OS CILLA TIONS = ==== — .- — _ yb Fig. 2 — Comparison between space-charge wavelengths for cylindrical beams with Brillouin flow and those with rectilinear (confined) flow, respectively. of the rectilinear beam. The 12 independent variables of (56-57) are re- duced to 6 by expressing He and Ee in terms of E, and H^ , respectively. The latter, however, require 2 arbitrary constants for a complete descrip- tion in region 11, making a total of 8 constants to be determined. The eliminant of the 8 boundary-value equations can be written as a TM wave-admittance equation at the beam surface: where 8 = 6n = 8o + RF hjyh) _jc^eli(yh) - 8-K,iyh) 7 /o(7^) + 8-Ko(yb) Io(yh) K;{ya) LV ya ka cot i/' Ki{ya)h{ya) - Ko(ya)To(ya) (58) (59) (00) 410 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 F = {kh cot ^) i"^) l'(yWr(ya) (gl) \ c / Koiya) In (01), c is the velocity of light. The right side of (58), which is the admittance Hg/E^ looking away from the l)eam surface, contains a term 5 which depends on the hehx geometry and the ampUtude of the TE fields excited by the surface current Ge . Thus, although the TE fields do not affect the electron paths, they are excited by the beam, and coupled to the TM fields at the helix. If Ge were zero, 8 would reduce to 5o , and the circuit admittance in (58) would then be the same as for a cylindrical beam with rectilinear flow. In (59), 5 is expressed in terms of 5o and the product, RF, where R is the beam propagation function, and F a factor dependent on the magnetic field and the geometry. This is the complete field solution of the problem. Equation (58) has four roots: two complex and two real propagation wave numbers, one of the latter representing a backward wave. In addition, there are two un- attenuated space-charge waves, given by (34); or a total of 6 waves in all. EQUIVALENT THIN-BEAM SOLUTION Pierce^ has expressed the admittance equation for an ideally thin beam, interacting with an arbitrary distributed circuit, as follows: 9 ;/3. A^_ .^ E ij^. - r)^2n V^ ^^Lr' - Tl'^ ^e where q = total convection current E = longitudinal electric field r = propagation constant = V'— 7- — A;^ /o = dc beam ciUTent Vo = dc beam potential To , K, Q = normal-mode circuit parameters. For slow waves, F c^ jj. For moderate values of perveance, the ac- celerating voltage may be replaced by the beam potential at the axis: ■Mo ^^ \/2r]Vo WAVE PUOl'AGATIOX ALONG AX KLlXTliON UKAM 411 Then, dividing l)oth sides of (()2) by 2Trb, we may i-e\vrite it as a wave- admittance equation : ■2jQ -ti.y!i.R = -c.vM-A' - rl "^ (3e (64) With the aid of (59), we can suh-e the a(hnittance ecjuation (58) for R, and re-write it as follows: Ye = Y, (65) with r. = -tiy±R (66) 7 2 ^jaseyh hjyb) / 6o \ The solid-c\'lindrical Brilloiiin beam in a helix is thus equivalent to a tliin beam whose circuit admittance is Yb . By equating Yb to the right side of (64), we can evaluate the normal-mode parameters for this ad- mittance, and thereby use all the results of previous thin-beam calcula- tions. ' ' The equivalence of the two circuit expressions, however, requires that we replace the transcendental expression (67) by an algebraic one, with no more than three arbitrary constants. This can be done very effectively, in the region of interest, by means of the approximation:^ Ys^-(y,-y,)(?L') -1^^' (68) \ dy /7=7(, 7 — Tp in which 70 and 7^ are the zero and pole, respecti\'el3^, of Yb : 60(70) = 0 '«^^^^ ^ (- K;S) + 76-A(76VXo(76)l=.. ^^^^ If we were to neglect the term containing F in (70), the error in the magnitude of 60(7^) would be measured by: («, .ot i) ('^) WP^, (71) yh-Ii(yb)-In{yh) \ c / Io{yh) ■ Ko{ya) In most low-power traveling-wa\e tubes, the first factor in parentheses is usually less than 3; the second factor less than O.Ol; and the last factor always less than unity. The error in evaluating yp , moreover, is less than 412 THE BELL SYSTEM TECHNICAL JOURNAL, IVtARCH 1954 this product, for the slope of the curve 5o versus yb increases with 76. Putting F = 0, therefore, leads at most to a very shght error in jj, and dYs\ dy /7=7o Outside of the region (70 , 7p) , 5o grows large rapidly, and the expres- sion for Yb is hardly affected at all by this assumption. Physically, the negligible role played by F in the admittance equation means that 8 O!:^ 80 , i.e., the TM helix admittance is not appreciably per- turbed by the TE fields excited by Go . With F = 0, (67) may be re-written: 7^ /o(7&) V Yn = 2 h{yh) jcoe r/i(76) - 8o'Ki{yb) _ /i(7b) 7 ih{yh) + 8o-Koiyh) h{yh)_ (72) (73) Here Yh is the helix admittance seen by a thin cyUndrical hollow beam, with rectilinear electron flow. As in the case of Yb , it may be replaced in the vicinity of (70 , 7p), by the approximation: Yh (Tp - To) dY, To ^T /7=7o T - T; (74) Fletcher has evaluated the normal-mode parameters for Yh as fol- lows: ■p2 2 To = -To Qh 1 + To. -1/2 1 O 2 2 2 Tp - To T^o = -jTrbyo 1 + To. ,2-|3/2 BYh dy (75) (76) (77) We have used the subscript H to refer the parameters to the hollow beam, and will use the subscript B to refer to the solid-cylindrical Brillouin beam. As Yb and Yh have the same zero and pole, they have the same natural propagation constant To , and the same space-charge parameter Q: Qb = Qh (78) This quantity can be found plotted in Fig. 1 of Reference 4, or in Fig. A6.1 of Reference 1. WAVE PROPAGATION ALONG AN ELECTION BEAM 413 From (68), (72), and (74), we find: (Ys\ ^ (d}\/dy\ yb hiyh) LT/i(76)J,=,„ (79) Kb = Kh (80) The impedance i)arameters for the two beams are therefore related to each other by: "2^ hiyhy _76 /o(7&). Both Pierce^*^ and Fletcher ha\e found the impedance parameter of the hollow beam to be related to that of a thin beam along the axis of a helix, Kr , as follows: Kh = /Vr[/o'(7&)]7=7o The gain parameter C is defined by: C^ = {2K){Io/SVo) (81) (82) Thus, for gi^'en In and Vo , the factor by which the gain parameter of a thin beam should be multiplied to give that of a hollow beam, is: (K^/KrY" = [h"\yb)],^,, (83) Tliis "impedance reduction factor" can similarly be evaluated for the finite cylindrical beam with Brillouin flow: 1/3 {Ks/KrY" = 2 _76 h{yh)-U{yh) (84) Cutler, who calls this quantity F^ , has described how it and the parameter Q can be used to compute the gain of traveling-wave tubes. The procedure depends upon the evaluation of C and QC. The expres- sion for Cb , in Cutler's notation, is: (lu/Kf FiF^ih/SVof" Cb (85) Here K2/K is a factor, of the order of 0.5, which corrects the impedance of the ideal sheath helix for the physical dimensions and support ele- ments of the actual hehx. It is best found by measurement. The factor Fi is plotted in Fig. 3.4 of Reference 1, and obeys the empirical relation: F,{ya) = 7.154 exp (-0.66G4 ya) (86) Finalh% the factor F2 is the impedance reduction factor (84), which is plotted in Fig. 3 of this paper for various ratios of the radii, b/a. It is of interest to compare the relative gain of beams with rectilinear and with Brillouin flow, respectively. Pierce " has computed a first approximation to the impedance reduction factor for the sohd-cyhndrical 414 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 beam with rectilinear flow, by averaging El oxer the Ijeam area (with E, for the empty hehx) : {Ks/Krf" - [Il(yb) - Il(yb)]%^, (87) Fletcher has improved upon this calculation by replacing the solid beam with a thin hollow beam of different radius and dc current. This has the same electronic admittance Ye and deri\'ative dYe/dy when R = 1. The impedance reduction factors for the three types of beams have been plotted in Fig. 4, using a typical value of h/a. For the same h , Vo , and h/a, the gain parameters C are found to be greatest for the hollow beam, and least for the solid rectilinear beam. The high gain of the hollow beam is due to its concentration in the region of greatest field strength. The greater gain of the beam \vith Brillouin flow, relative to that of a similar beam with confined flow, is probably due to transverse electron motion, in two ways: (1) causing electrons to interact with the transverse as well as longi- tudinal fields; and O 4 7 1 I - / > ' - 1 i/ / / A // / o^ // A ^A / // / / f / ^ V {. / y ^ • //a ^z. '/. /^ y y 0> d i -^ ^ - ^ L 7a Fig. 3 — The factor {Kb/KtY^^, or Fi , by which the gain parameter Ct for a thin beam should be multiplied to give Cb , the gain parameter for a cylindrical beam with Brillouin flow, of the same current and voltage. Computation of Cg using ! this factor is described in text following equation (85). WAVE I'UOPAGATION ALONG AN KLKCTHON BIOAM 415 z4 o Q. 2 5 b_ .. / 3-u.o / / / /. y / / / / / / / HOLLOW RECTILINEAR BEAM-^ BRILLOUIN FLOW-^ / y ^ / '/ / / /y / / \^ '/ / y / >') /^ > / ^ ^ /^ J /. :<; >C SOLID RECTILINEAR BEAM: V^"( PIERCE'S FIRST APPROX.) ^^(FLETCHER'S APPROX.) / ^ ^ :^ ^ ^ ^ ^ ^ 0 12 3 4 5 6 7 7a Fig. 4 — Comparative values of impedance reduction factor for several kinds of Ijcams, of the same relative radii hja. (2) ("Ui.sing electrons to move preferentially into regions of retarding longitudinal tields, a process analogous to bundling. CONCLUSIONS Field solutions have been presented for the magnetically-focused cylindrical beam, when modulated by a small ac signal. Two types of beam enclosure have been treated: the concentric drift tube and the ideally thin (sheath) helix. There are two pairs of unattenuated space-charge waves in the drift- tube: one with arbitrary amplitude distribution, and another pair which is coupled to the external field (Fig. 1). The space-charge wavelength of the latter pair is greater than that of space-charge wa\'es in a similar beam with rectilinear flow (Fig. 2). The solution of the helix prol)lem consists of the afoicmcnlioiicd two space-charge waves with arbitrary amplitude, as well as the usual f h = {2ry'' sm h = (2/-) 1/2 cos 2 i = exp (t7r/4), where (r, ru — - TRACE OF_, HALF-PLANE Fig. 2.1 — Diffraction of a plane wave by a half-plane. 422 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 1.2 0.6 / \ / \ r \ 1 \j 1 \ \y i 1 1 SHADOW / ILLUMINATED REGION / / ^^ / ^ ^ y' t, = V2r SIN -^ Fig. 2.2 — The approximate value | e~'^ + *S'i | of | £ | for hp and of \ H \ for vp at a great distance r behind a half-phme. Here, as in all of our work, the wave- length X is 2-K. For an arbitrar}^ wavelength replace r by 2Trr/\, etc. (p~r » 1, exp ( — ^.^•) + S>i has the asymptotic expressions [see Equations (7.7) and (7.8)] + ^1 c(/-)/2sin|, ^ < 0 (shadow) + c(r)/2sin|, ^ > 0 c{:r) = i"\2',rr) '-"'e-'\ (2.4) (2.5) These expressions lead us to picture the field to the right of the half- plane as the sum of the incident wave and a wa\'e, c(r)/Si1p means that the quantity within the brackets is to be computed as though it corresponded to a half-plane with its edge at the crest of the parabolic cylinder, so that p, \l/ are to be used in (2.3) instead of ?•, (p. Also, Jo Ai(u) — iBiiu) (2.11) r" Ai{u) exp {(ut) flu Jo Ai{u) + iBi{u) where Ai{ii) and Bi{u) are Airy integrals defined by equations (13.12) and (13.16), and tabulated in reference." In this paper we find it con- venient to use the Airy integrals instead of the related Bessel functions "The Airy Integral, Brit. Asso. Math. Tables, Part — Vol. B (Cambridge, 1946). DIFFRACTION" OF RADIO WAVKS m A r\i;\H()I,IC ( ^I.IM)1;K rJ.) of order 3-^. As in (2.3), the fractional i)o\voi-s of i arc made precise by takiiii;- / = e.\p {iir/'2). Tln-ee kinds of appro.ximations Ikiac l)een made in the derixation of (2.10), namel}^ those associated with the assumptions (1) that the angle ^ is small, (2) that h is large, and (3) that r/h' is large. The terms in \/^ and 1/r do not cause trouble at i/' = 0 because their infinities cancel each other. The counterpart of (2.10) for vertical polarization is obtained from (2.10) In' replacing E by H and ^(r) by ^r(T), where the subscript v stands for "vertical"; and ^,.(r) is given by (2.11) when Ai{v) and Bi{u) are replaced by their derivatives with respect to ;/. Series for '^(r) + 1/r and ^,.(r) + 1/r which converge for negative (shadow) values of r are given bj^ Equations (7.31) and (7.53), respec- tively, with (j in place of r. Table 2.1 gives values of "^(r) and ^i.(t) which were obtained from the series for r negati\'e, and from numerical integration of (2.11) and its analogue for r ^ 0. When T is large and positive Equations (7.35) and (7.55) show that (/7rr)'''exp (-2tV12), (2.12) ^(r) + 1/r ^,(r) + 1/r ~ (?Vr)''"- exp {iir - trVl2). When r is large and negative the leading terms in (7.31) and (7.53) give ^(r) + 1/r ~ - i'^ 2.03 exp [(2.025 + i 1.169)r], ^,(r) ^ 1/r i^^ 3.42 exp [(0.882 + i 0.509)r]. (2.13) Xow that we have expressions for the field what do they tell us? For one thing, they may be used to show that the field near the shadow Table 2.1 — Values of ^(r) and ^^(r) r l*WI arg. ^(r) l*»(r)| arg. ^^(t) 3 3.13 1 -93.5° 3.16 +104.3° 2 2.21 -1.8° 2.80 192.4° 1.5 1.945 +21.6 2.44 211.3 1.0 1.715 +32.5 1.985 219.5 0.5 1.486 34.2 1.522 218.6 0 1.254 30.0 1.089 210.0 -0.5 1.030 22.9 0.724 193.7 -1.0 0.823 15.2 0.459 167.8 -1.5 0.648 8.48 0.317 130.6 -2.0 0.511 +3.79 0.2S1 92.0 -3.0 0.338 +0.12 ().2S8 15.1 -4.0 0.250 -0.12 0.264 22 M -5.0 0.200 -0.02 0.221 "9^71 426 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 boundary is almost the same as for a half-plane. Away from the shadow boundary, the field in the shadow can be interpreted as a "crest wave" which reduces to the "edge wave" for a half -plane. The crest wave de- creases as an exponential function of \l/ in the shadow instead of as l/

0 were computed from the series, and the ones for 7 ^ 0 w^ere obtained by numerical integration of (2.14). The entries for J^ were taken from the more extensive table given by Fock. In order to express his results m our terms it is necessary to use the fact that the radius of curvature at the vertex of the parabola is 2h. A change in the sign of i is also necessary because the time enters Fock's work through exp { — icct) instead of exp (iwt). The values shown were checked for 7 > 0 b}' the series and for 7 ^ 0 by numerical integration of (2.15) Fock's table shows that by the time 7 has reached —2 the value of J„ exp (ix) has become 1.982 at an angle of +1-45 degrees. This is close to the limiting value of 2 predicted at 7 = — co by (2.17). It will be noted that, for large values of h, J is smaller than Jv by Table 2.2 — Surface Current Densities /ri/'fo/exp (ix + i-r^/3) Jv exp (ix + i7V3) modulus Argument in degrees mod. Arg. -1.0 2.16 -25.9 1.861 -15.43° -0.5 1.38 -16.8 1.682 + 1.52 0.0 0.77 -30.0 1.399 0.00 0.3 0.515 -44.8 1.197 -6.06 0.6 0.327 -62.9 0.991 -14.23 1.0 0.167 -90.1 0.738 -26.63 1.5 0.066 -125.9 0.488 -42.57 2.0 0.025 -161.6 0.315 -57.98 3.0 0.0033 -230.7 0.130 -87.57 4.0 0.00043 -298.0 0.054 -116.75 428 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1<)54 the order of h~^^'^ when 7 is of moderate size. It will be shown later that the current density decreases exponentially as one moves into the shadow, and that its rate of decrease is related to that of the field as shown in Fig. 2.6. We now take up the detailed discussion of the expressions for the field and the current density. It is convenient to consider the current density first. When a plane wave strikes a perfectly conducthig plane, the sur- face current is proportional to the tangential component of H in the incident wave, and flows at right angles to it. In the rational MKS units we are using, the surface curi-ent density is two times the incident tangential H. When we consider the illuminated side of a large parabolic cylinder and calculate the current density by doubling the tangential component of the incident H we obtain the approximations (2.18) which hold when x is large and negative. When h is very large but x/2h small these formulas agree with the leading terms of (2.17) which were obtained from our integrals for the current density. Expressions for the current density deep in the shadow may be ob- tained from the leading terms of the convergent series by letting 7 become large and positive. The exponential decrease is found to be I fo./ I ^ 1.43/i~'" exp (-I.013.i7r'''), (2.19) I J. I ;^ 1.83 exp (-0.44.t/i~'''), where the numbers appearing in these equations are associated with the smallest zeros of Ai{ii) and Ai'iii), respectively. These formulas for a large cylinder are roughly similar to those for propagation over a smooth earth. The radius of curvature at the crest of the cylinder is 2h. Setting this equal to the radius of the earth gives an exponential rate of decrease for / and J„ which is the same as that over a smooth earth for the two polarizations. Of course, the coefficients multiplying the exponential functions are different. This agreement is not surprising since the Airy integrals are closely related to the Hankel functions of order l^3 used in the smooth earth theory. The surface current densities as a function of the distance h — y below the crest for h = 1000 and for h — 0 are shown in Fig. 2.4. The equation of the cylinder shows that h —y = x /4/i so that, from (2.19), fo/ and Jv decrease in proportion to h~^'^ exp [ — 2.02bh~^'^ (h — yY''] and exp [— .88/r^'* (h — yf^j, respectively, far down in the shadow. DIP^HJACTIOX OF KADIO W AVK8 BY A PAKABULIC CYLINDKR 429 1 1 If.j|- ~-W 1 2.0 1.0 ■"■\ -, • *>. 1 1 1 / 1 1 \ \ \ 1 1 1 \ V 0.8 'v', ^, \ v^ 0.6 v, ^ \\ ^, a4 \ \ ^■«*-~ -UvH --^\ N \ \ \ N N, 0.2 0.1 ILLUMINATED SIDE \ \ \ \ \ SHADOW SIDE N ^ ^ — -koJH I \ \ \ ^ \ \ 0.08 \ \ V ao6 \, \ \ \ 0.04 \A ^ \ \ h = o h=iooo not \ \ \ \ \ DISTANCE BELOW CREST h-L) Fig. 2.4 — • The surface current densit}' is plotted as a function of the vertical distance below the crest or edge. The curves for h = 1000 and h = 0 are obtained from Table 2.2 and equations (2.20), respectivel}'. Here, as alwaj^s, the units are such at X = 27r = 6.28 . . . The equations used to compute the curves marked /i = 0 are U-J = (iTrr)' -1/2 + 2^> -it^ (It ./„ = 2(z77r)^'- r c-''\H, (2.20) where the upper signs are for the shadow side and the lower ones for the ilKiminated side. The computations arc made easier bj^ the relations (roJ)- - ao./)+ = 2, (J.)_ + (/,)+ = 2, 430 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 where the subscripts " — " and "-|-" denote opposite points on the il- himinated side and shadow side, respectively, of the half-plane. These relations follow from (2.20).* The radius vector r from the origin is equal to —y on the half -plane. Equations (2.20) follow quite readily from (2.3). The values of J and J^ for an arbitrary angle of incidence and /i = 0 are given by expressions (6.6) and (6.22), respectively. All of the curves in Fig. 2.4, even | ^qJ \ iov h = 1,000, eventually approach the value 2 far down on the illuminated side. It may be shown that | /„ | and | fo-/ | for the half-plane decrease like {irrY^'^ and l/(27r^'V^'^), respectively, deep in the shadow. Hence as we go to the right in Fig. 2.4, the dashed curves will eventually cross over and lie above the solid curves, which decrease exponentially. The larger h, the lower and flatter is the curve for | ^qJ \ . Now we turn to the diffraction field at points far to the right of the cylinder. When 1 1/' ] « 1, so that we are not too far from the shadow boundary, and h is large, the field is given by (2.10), or by its analogue for vertical polarization. In order to get acquainted with (2.10) we first examine the field when | ;/' | « 1 but \p''p ^ 1. When we are so far behind the cylinder that i/'"p ^ 1 even though I i/' I <3C 1, the asymptotic expressions (2.4) show that c{p)/\l/, \p negative (shadow) [e-^^ + S,l Substitution of (2.21) in (2.10) gives + — — , yp positive (2.21) E cip)h"' U(t) + ^) exp (ir'/S), lA < 0 c{p)h"' ( ^(r) + - ) exp (irVS) + e-'\ ^ > 0 (2.22) The presence of c{p) shows that the total wave may be regarded as the sum of the incident wave and a wave spreading out from the crest. The crest wave is the analogue of the edge wave, c{r)/(p, for the half-plane. In fact, when we are in the region where the l/r in (2.22) is the most important term, the crest wave is approximately c{p)/^ (2.23) * They also follow from superposition and consideration of the symmetry of the currents produced on the half -plane ?/ < 0, x = 0 when —/I is impressed. Here A denotes the system of currents which flows in the upper half-plane ?/> 0, x = 0 when the incident wave falls on a complete plane at x = 0. DIFFRACTION OF R.VDIO \V WKS BY .\ P.\R.\BOLir CYLINDKR 481 jl 1 e 1 1 J 1 6 \ / \ 4 / \ / / / ( / \ ^ IV"-^ w ^ / / / ' / 1 \ \ \ ^V M/;^^j^ ^^^ • ^ / / / / f 1 \ \ \ \ / /M^ ^1 / / 0.8 / / / / / / / / / / / 1 ADDITIONAL VALUES / r l^l/(rJ4| ' -2 0.0351 -3 0.00465 -4 0.000617 / f n 5 / V / 0.1 / / SHA DOW / i 1 ILLUMINATED REGIO N VALUES OF T -W }p Fig. 2.5 — ■ The amplitude of the crest wave may be obtained from these curves and expressions (2.24). Here r = /i'/' ^ where \l/ is small. However p must be large enough to make tZ-'p » 1. and this corresponds to a half-plane with its edge at the crest of the cyhnder. t may be small even though we are considering | i/' [ to be large enough to make (2.21) and (2.22) hold, i.e. large enough to make 1 ^ \p'~ >i> 1. Indeed, multiplying by lii^^ gives | t \p'' » ]i^^ which may be achieved for small values of r by making p large enough. It follows from (2.22) and its analogue for vertical polarization that the amplitudes of the crest waves are (hp) (vp) (27rp)-^''^ A^'M ^(r) +1/t1, (27rp)-^'^ /l^'M ^lA) + l/r|, (2.24) where the expression (2.9) for c{()) has l)een used. The last factors in (2.24) may be computed from Tabk; 2.1. They are plotted in Fig. 2.5. When we go deep down into the shadow where r is large and negati\'e 432 THE BELL SYSTEM TECHXICAL JOFRXAL, MARCH 1954 2.03 e we see from (2.13) that |^(r) + 1/r, I ^.(r) + 1/r I ~ 3.42 e^ SO that the absolute value of the field is (2.25) E H S-l/2 ,1/3 1/3, {2Trp)~"' K" 2.03 exp (2.025 K"^p) (2.26) (hp) (vp) I i^ I ~ (27rp)"''' h}'^ 3.42 exp (0.882 h}"^p). where the angle ^ is negative. Thus, as Artmann has pointed out, the field decreases exponentially as we go into the shadow. The larger h is, the more rapid is the decrease. This supports the statement made earlier that it is darker behind a large cylinder than behind a half -plane. Comparison of the expressions for the current density and field strength for the shadow regions shows that there is a simple approxi- mate relation between them. Near the crest of the cylinder, where x is small, the radius of curvature is nearly 2h. Hence the tangent to the parabola drawn from the point P (located at (p, i/') deep in the shadow) touches the parabola at T where x is approximately —2h\f/. This is shown in Fig. 2.6. Replacing x by —2h\l/ in the expressions (2.19) shows that the current density at T is proportional to the field at P as given by (2.26). It follows that (hp) I E/^oJ 1 ~ 1.41 h'" {2irpr"' (2.27) (vp) \H/J,\ - 1.87/i'''(27rp)"''-. This leads us to picture the field at P as being produced principally by the surface currents around T. The effect of the stronger currents (A^) Fig. 2.6 — The field strength at point P (deep in the shadow) specified by the polar coordinates (p, 4^) is nearly proportional to the current density at the tangent point T specified by the rectangular coordinates (x, y). DIFFKACTIOX OF RADIO WAVKS \i\ A l-\l(\H<)Mc (">M\DKR 433 (loser to the crest is perhaps blocked out by the cuiAnliirc of Ihe cylinder. For comparison ^vith the horizontal polaiization case we note that E at (p, 4/) lor an infinitel}^ long current filament at the origin p = 0 is given by l^/fo/ I ~ .5(2tp)-"' (2.28) where / is the current carried by the filament and the frequency is is such that X = 'Iir. There is some difficulty with the pictui-e for vertical polarization because the current element at T points directly towards P and hence should produce very little field there. This is perhaps as- sociated with the fact that the (vp) ratio in (2.27) is smaller than the (hp) ratio by appro.ximately the factor /^^''^ We now leave the shadow region and consider the field at points well inside the illuminated region. Fig. 2.5 shows that for large positive A'alues of T the amplitude of the crest Avave tends to increase with t. The asymptotic expressions (2.12) show that when r is large and positive I ^(r) + 1/r I ^ I ^„(t) + 1/r I ~ (xr)''" = (TnPfV", (2.29) and hence the amplitude of the crest wa\e deep in the illuminated region is, from (2.22) and its analogue, (hp) I E - e-'^ I ^ (27rpr"' (TrrPh)"\ (vp) I H - e-'" I -- (2xp)-''- (x#)'''. (2.30) Since (2.30) is derived from the general expression (2.10) it is subject to the restrictions mentioned just below eciuation (2.11), In particular the angle xj/ should be small (but we must still have \f/~p » 1 as assumed in (2.22)). AVhen \l/ is positive, an application of the laws of geometrical optics to determine the reflection from the curved surface of the para- bolic cj'linder leads to the expressions^' (hp) 1^ (vp) 'h tan (iA/2)" P 'h tan (\l//2)' sec (ip/2), ^ > 0 (2.31) sec (^/2), xl^ > 0 p for the reflected wave. When xp is small these expressions reduce to (2.30) as they should. Expressions (2.31) may also be obtained from our analysis by start- '2 In our two-dimensional case the calculation of the required radius of curva- ture, etc., is not difficult. General theorems dealing with jjroblems of this sort and references to earlier work are given in the paper, ,\ General Divergence Formula, II. J. Riblet and C. B. Barker, J. Appl. Phys. 19, pp. 63-70, 1948. 434 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 ■■ ^ \ .in / \ \ HI < HO , / A • o o o II REFLEC GEOMETR / / / / II *•; / / / / / r / / / / / A f / / EGIO / '/ / TED Rf ^ / ILLUMINA o ^8-3| dmh \ \ ^ ^ \ o Q < \ ^ N. CO \ Nv V N ^v. \ "^ s o o o II X ■s >^ \ "N X ^ > '^ s s s \ o ^ S N \ '^ i-5 ^ la I (JRz\ 9. Q (2.32) (vp) H (2tp)- 1 1 2 sin (i/'/2) 2 cos (yp/2) xP <0 4^ > 0 which follow from (2.1), (2.2), (2.4) and (2.6). From equation (2.21) onward we have been discussing the field for values of i/' and p such that pi/'" » 1. For these values the concept of the crest wave is helpful in visualizing the behavior of the field. Now we consider the field at points close to the boundary of the geometric shadow far behind the cylinder. This is the region in which Artmann was es- pecially interested. His results for the shift of the field may be obtained from (2.10) and its analogue bj" taking | ^ | to be very small. At the shadow boundary xj/ = 0 and [exp ( — ix) + Si]p = }/2- Hence the region of interest at present is in the neighborhood of the point point ti = 0, \ exp ( — ix) + aSi | = ^^ of Fig. 2.2. A magnified view of this region showing the shift of the field is given in Fig. 2.9. The figure shows that, for a given volue of p^, | E | for hp is less than | H \ for vp. As Artmann has pointed out, this is to be expected smce the reflection coefficient for E (hp) is roughly — 1 and the reflected wave therefore 430 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 ^ *5 ^ ^ \ HADOW \ ^ N. en ^ Nv \ v\ \, v \ \. ° V X^ \ \ \ \ o > JZ s s \ ^ V s ^ ■^ k ^ \ ^^^ ''---^^ HI c/HEji J- DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 437 -2024 Fig. 2.9 — Behavior oi \ E \ and \ H \ on tiie shadow boundary far hchiiid I ho half-phme or parabolic cylinder. This is a magnified view of the region around /, = 0, ! exp i-ix) + / - h' = 0.399X (a/A)"' meters. The corresponding shift for \ H \ is — 0.34GX(a/X)' '^ meters. Artmann gives the values 0.39 and —0.20 for the respective coefficients. The discrepancy between —0.346 and —0.20 is cause for worry because it seems to indicate either a mistake in our work, which I have been un- able to locate, or a shortcoming in the approximations made by Artm.ann for the case of vertical polarization. As h approaches zero the paral)olic cylindei- becomes a half-plane and the curves d and e should approach curves b and c, respectively. According to Fig. 2.9 both d and e approach curve a. Tiiis failure is an 438 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 indication of the errors introduced by the approximations used in the derivation of (2.10) and its analogue. 3. RADIO PROPOGATION OVER A SUCCESSION OF RIDGES ON THE EARTH's SURFACE The results mentioned in Section 1 concerning propagation over a succession of ridges may be obtained from the expressions and curves of Section 2 as follows: Consider the situation shown in Fig. 3.1. Let a radio wave start out from a transmitter at T. We assume that by the time it arrives at the first ridge at P it has become equivalent to a plane wave of amplitude K/t traveling in the direction TP, where A is a constant depending on the strength of the transmitter. For the sake of simplicity the waves reflected from the ground are neglected. In a more careful study they would have to be included.* In order to calculate the strength of the wave at the second ridge, we assume it to be a crest wave coming from P. Let G denote the value of I ^ I (we assume the case of horizontal polarization since the reflection coefficient of physical materials approaches — 1 for almost grazing in- cidence) at Q corresponding to a plane wave of unit amplitude incident on P. From Fig. 3.1 we see that the values of ^ and p to be used in com- puting G are ^1/ ^ —i/R, p = 27r^/X, X = wavelength, R = radius of earth. The value of h depends upon the radius of curvature of the ridge: 2h = 2t (radius of curvature)/X. 0 Fig. 3.1 — Diagram showing ridges at P and Q which diffract the radio wave starting from T so that a portion of it is received at *S. A method for doing this (for one hill) is given in Reference 1, page 417. DIFFRACTION' OF RADIO \\AVKS BY A l'ARAHOI,IC CVIJXDKH 13!) The amplitude of the wave striking the ridge at Q is AG/l\/2. The \/2 comes from the horizontal sidewise spi'oading of the wa\^e in going from / to 2(. If we were dealing with the energy instead of the amplitude, the factor would he 2 instead of \/2. When this wave is assumed to be plane and traveling in the PQ direction, similar reasoning shows that the amplitude of the disturbance at the receiver »S is A(f/C-\/^. If, instead of two ridges at P and Q, as shown in Fig. 3.1, there are N ridges between the transmitter at T and the recei\'er at .S, the amplitude of the radio wave at *S is AG^ /C\/N + 1. The distance between T and S is approximately (A'" + 1)^, and the free space amplitude at S is A/ (N -\- l)C. Hence Actual Amplitude at S _ ^n/i^ i iV'- d ^\ Free Space Amplitude at S The actual field at S is therefore 20 N logio (1/G) - 10 logio (N + 1) (3.2) db below the free space field. As an example, let us assume a distance of 280 miles between the transmitter and receiver, and a distance of 40 miles betw^een successiA-e ridges. This gives N — 6. For a wavelength of 10 meters and a radius of curvature of 100 meters for the diffracting ridges, the formulas of Section 2 show that the ridges behave like half-planes and that G ^ 0.227. Equation (3.2) then says that, for a distance of 280 miles and a wavelength of 10 meters the actual field should be about 69 db below the free space field. Although this is in fair agreement with the ex- perimental results, calculations for other distances indicate that the field strengths predicted by (3.2) tend to be smaller than the ones observed. When the work is carried through for X = 1 meter and a distance of 280 miles, (3.2) says that the field is 120 db below free space. The ob- served fields are 70 ± 15 db below^ the free space value. These figures suggest that the roughness of the earth's surface might possibly account for transmission far beyond the horizon for wavelengths of the order of 10 meters. For wavelengths of the order of 1 meter either the approximations leading to (3.2) break dow^n or some other explana- tion is required. 4. SERIES FOR THE ELECTROMAGNETIC FIELD Here we set down series for the electromagnetic field when a plane w'ave strikes a perfectly conducting parabolic cylinder. Since Epstein's 440 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 classical work deals with the general case of finite conductivity, the series we use are special cases of the ones discussed by him. The parabolic coordinates (^, 77) which we shall use are related to the rectangular coordinates (.r, y) and polar coordinates (r, (p) as follows: x + iy = (^ + i-nfm = r e'\ X = ^r] = r cos 770 ^ 0, so 77 is always positive in our work. ^ is positive in the half-plane x > 0 and negative in .r < 0. For much of our work we shall assume the incident w^ave to come in at the angle ^, 0 ^ 0 ^vr, as shown in Fig. 4.1. As mentioned in Section 4 = CONSTANT T] = CONSTANT n^Vo Fig. 4.1 — This diagram shows the angle 6 of the incident wave and the surface of the perfectly conducting cylinder x^ = Ah{h — y) (or r] = 770). DIFFHACTIOX OF RADIO WAVKS HV A I'A l{ A |{< )I,I(' CVMXDKK III 2, the field quantities are assumed to depend upon the time through the factor e\p(icoO where co is the radian frequency. The wave equations for horizontal and vertical polarization are, respectively, S + § + <«' + "'^'^ = 0 (4.3) f + 0 + «^ + "') " = ° (") where, as explained in Section 2, the unit of length has been chosen so that the wa\'elength X = 2ir. On the surface of the perfectly conducting cylinder, i.e. for r] = rio, we must have E = 0 and dH/drj = 0. When E and H are kno^\Ti the remaining components of the field may be computed from Maxwell's equations. Special solutions of (4.3) (and (4.4)) are exp {i(r,' - f)/2] U,X^i'") U„(r,i-'"), (4.5) exp [iin' - r)/2] Uni^i''') W„(nr"'), (4.6) where i^'~ stands for exp (zV/4) and Un(z), Wn{z) satisfy the equation ^^^^ - 2z '!^ + 2nTM = 0. (4.7) Another solution of (4.7) with which we shall be concerned is Vn{z). These three solutions are defined by contour integrals of the form {2Triy^ I exp [/(/)] (If where /(/) = -T + 2^/ - (n + 1) log f. The path of integration for Un(z) comes in from — x where arg t = —t, encircles the origin counterclockwise and runs out to — oo witli arg t = IT. The path for Vn{z) runs from — x where arg < = tt to + x where arg t = 0, and the path for Wn{z) runs from + oo to — x whei'e arg t = —IT. The integrals are Avritten at greater. length in equations (9.1) and the paths of integration are shown in Fig. 9.1. Since the paths may be joined to form a closed path containing no singularities of the inte- grand it follows that lL(z) + VrXz) + Wn(z) = 0. (4.8) When n is a non-negative integer r7„(.) = Hdz)/n! = ^-^ (''" f- e-'" (4.9) n! dz" 442 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 where Hn{z) is Hermite's polynomial. When z becomes very large the leading terms in (9.17) and (9.16) give Un{z) ^ {2zr/n!, (4.10) Wr.{yii-'") ~ i{^"/vy^' e-'''/2'K"\ (4.11) In order to obtain a series for the incident wave exp [ — ix sin 6 + iy cos 6] shown in Fig. 4.1 we consider the special case ^ = 0. In this case the wave is simply exp{iy) or exp [i(rf — ^)/2] and may be obtained by setting n = 0 in (4.5). This suggests that the incident wave may be expressed as the sum of terms like (4.5). The series turns out to be exp [— ix sin 6 + iy cos 6] = exp [ — 1^7] sin d -\- i cos d{ri' — ^')/2] = exp [- izz' sin d - cos e{z~ + z'')/2] (4.12) 00 = e'^sec (6/2) Yi n!{- iw/2yUn{z)Un{z') where w = tan {6/2), z = ^i''\ (4.13) / —1/2 Z = 7]l . This series has been studied by a number of writers. It goes back to Mehler^^ who obtained it by evaluating the integral — 1/2 x2 w e / exp [— (,t — iy)' — {x + tat)'] d( J— 00 first in closed form, and then as a series (by using the generating func- tion exp [— ( — iat)' -\- 2{ — iat)x] for H„(x) and integrating termwise). This leads to (1 - aT''" exp 2xya — {x~ + y~)a 1 - a^ (4.14) which is equivalent to (4.12). Since (4.14) converges when \a\ < 1, (4.12) converges when \ w \ < 1 or | 0 | < 7r/2. ^' Reihenentwicklungen nach Laplaceschen Functionen hoher Ordnung, J. Reine Angew., Math., 66, pp. 161-176, 1866. DIFFRACTION OF RADIO WAVES BY A PAl{AHOLI(' CVLINDKH 443 When the incident wave strikes the cylinder tlic loflected wave has some of the characteristics of a wave spreadinji; radially outwards. Such a wave contains the factor exp ( — ir) = exp [ — r(r + '7")/2] . Consideration of the exponential factors in (4.6) and (4.11) suggests that the reflected wave may be expressed as the sum of terms of the form (4.6). The co- efficients in this series are to be determined so (hat E = 0 or dll/d-q = 0 at the surface r? = rjo, the incident wave being represented by (4.12). For the case of horizontal polarization this procedure gives E = e'" sec (6/2) Z n!{- iw/2yUM [U,Xz') u .^^ 0 (4.15; -Wn{z')USz,)/WrXz,)] E = exp [— ix sin 6 -f iy cos d] - e'" sec (9/2) E n!{- iw/2yUn{z)Wr.{z')Ur.{z',)/Wr.{z,) ^^'^^^ 0 for the complete field. These are special cases of Epstein's results. Here ^■o is the value of z' which corresponds to the surface of the cylinder: z', = i~"\o = (2h/iy" (4.17) The entries for regions II and II' (these are regions in the w-plane {m = n -\- 1) which, as Figs. 11.2 and 12.2 show, contain the large posi- tive values of n) in Tables 12.2 and 12.4 of Section 12 may be used to show that as n — > 00 Vn(zo)/Wr.{zo) - ^-'" exp [2r,o{2iny'% Un{Zo)/M\{zo) = -1 - Vnizo)/Wn{Zo), Un{z)WnKz) 4[r(l + n/2)P {exp[- ^(.2n/iy"] + r" exp mn/iy^']}. (4.18) Since n!/\T(l + w/2)]''-2" (7rn/2)- -1/2 the series in (4.15) and (4.16) converge if | w | = | tan 9/2 \ < 1. Series for H similar to those of (4.15) and (4.16) may ])e obtained for the case of vertical polarization. The boundary condition at t? = 770 is now dll/dr] = 0. It is convenient to introduce the functions 'Un(z), 'Vn{z), 'Wn(z) defined by 'Un(z) = -zUn{z) + dUn{z)/dZ (4.19) 444 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 and the two other equations obtained when U is replaced by V and W. The prime is placed in front of U instead of behind to avoid mistaking 'Un(z) for dU„(z)/dz. The function 'Un{z') makes its appearance when dH/dt] is calculated for the boundary condition. Since iy = —{z'-\- z''^)/2 we have -e'Wniz') = i-'V 'Un(z'). (4.20) or] The analogues of (4.15) and (4.16) for vertical polarization are (as- suming now that H for the incident wave is of unit amplitude). H = c'" sec (6/2) Z n!{-iw/2rUn(z) ^^ 2i) [Uniz') - W„{z7Un{zo)nV,Xzo)l = exp [ — ix sin d + iy cos d] (4.22) - e^^sec {e/2) J2 n!(-iw/2rUn{z)Wdz'yUn{zo)/'Wnizo), 0 and these series converge if \ iv \ < 1. If the parabolic cylinder is merely a good conductor, instead of being perfect, the boundary conditions at r? == ijo are approximately E = — ^Ht, Et = ^H. Here E^ and Ht denote the ^ components of the elec- tric and magnetic intensities and f is the intrinsic impedance r = [io>ui/(g + ic,e)V" (4.23) of the cylinder material, f is assumed to be small compared to the intrinsic impedance fo = (mo/^o)^'^ = co/xo (since X = 2x) of the external medium. In these expressions m, e, y are the permeability, dielectric constant, and conductivity of the cylinder; and mo and eo refer to the external medium. When we set a = l~"\e + Vir ^./^, (4.24) the boundary conditon for hp becomes aE — —dE/dz' at z' = Zq. When a is assumed to be constant we obtain 00 E = e'-" sec(e/2) E n!{-iw/2)"U.Xz) 0 (4.25) [Udz') - W,Xz')[aU,Xzo) + 'U,Xzo)]/[aW.Xzo) + 'W,Xzo)]] " Electromagnetic Waves, S. A. Schelkunoff, D. Von Nostrand Co., N. Y. (1943) p. 89. See also G. A. Hufford, Quart. Appl. Math. 9, pp. 391-403, 1952, where ref- erence is made to the work of Leontovich and Fock. DIFFT{ACTI()\ OF RADIO AVAVKS UV \ 1' \ KAHOI.IC fYLIXDKR llo which reduces to (4.15) when o- — > oo . The constancy of a- may be achieved Ijy either takina; the projierties of the cj^linder material to change in a suitable way or, roiiiihly, by taking Tjn (and hence h) to be so large that only the nearly constant values of a at the crest of the cylinder hav^e an effect on the result (assuming 6 = ir/'2, and restricting our attention to the region neai- the shadow boundary). The corresponding expression for vertical polarization ma}^ be ob- tained from (4.25) by replacing E and a by H and r, respectivelJ^ We shall refer to (4.25) and its analogue later in connection with the field in the shadow (Section 7) and with Fock's investigation of the surface currents on gentl}' curved conductors (Section 6). 5. INTEGRALS FOR THE FIELD When the curvature of the cylinder is small, i.e., when h is many wavelengths, the series of Section 4 converge slowly. The work initiated by G. N. Watson on the smooth earth problem suggests that we con- vert the series into contour integrals with n as the complex variable of integration. When this is done we get an integral with the path of in- tegration Li shoA\Ti in Fig. 5.1. Thus, for example, expression (4.16) for E is transformed into E = exp [ — ix sin 6 + W cos 6] e sec - 2i h\2) si n+ 1) (5.1) sm xn VMWn{z')Vn{z,)dnn\\{z,). At first sight it seems that not much can be done with this integral because the integral obtained by deforming Li into L-i does not converge (this is explained in the discussion of Table 5.1). However, some ex- <5 n- PLANE n = -Lh = -77o72 ,;^ZEROS OF W^, (Zq) Fig. 5.1 — Paths of integration in the complex w-plane. " Proc. Roy. Soc, London (A) 95, p. 83, 1918. 446 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 perimentation shows that if we set Un(z()) = — Vn(zo) —W„(zo) in (4.16), the series splits into two series, one of Avhich may l)e summed and the other may be converted into an integral along the path Lo of Fig. 5.1: E = exp [-ix sin 9 + iy cos 6] + Si + S->(h), 00 ^x = r sec (6/2) E n!{-iw/2rU„(z)W,Xz'), (52) 00 SM = e^'^sec {d/2) J2 nK-iw/2rih{z)Wniz')V„{zo)/WM). 0 The series for *Si may be summed by replacing Un(z) and Wn(z') by their expressions (9.19) in terms of definite integrals, and interchanging the order of summation and integration. The resulting series may be summed and the integrations performed. The result is Sommerfeld's integral for a diffracted plane wave: ,00 CI { • I Nl/2 — j'j- sin fl+ij/ cos 9 / — »'^ Jv *Si = — (t/Tr) e e df, Ti = rj COS - - ^ sm - = {2ry-- sm {~— + ^ ) (5.3) The inversion of the order of summation and integration may be justified when | iij | = | tan (0/2) | < 1 (in w^hich case the series in (5.2) converge) by using (1) the result that | Ru \ < \ a^/N! \ when a is real in A'— 1 / • \ n T- = ^ — Rn-, 0 n! and (2) the ineciuality / r'' exp [-r + 2' '6^]d/ < r e-'^'-^'^'e' dt f exp [-2af + ^%t\ dt Jo J_oo < A2-''h-"'N-'''N! &xp{2hN"-) This inequality holds when N » 6" and at the same time iV ^ 1. The value of A is independent of N and ?; is a number which exceeds | ^ |. In this work the parameter a has been arbitrarily introduced; and has then been chosen so as to make the product of the two integrals a mini- mum w^hen A^ is large. This value of a is bN~^'^. When the series (5.2) for S^ih) is converted into a contour integral taken along the path Li, by the procedure used to obtain (5.1), it is DIFFRACTIOX OF HADIO W \Vi;s in \ I' A H A liol.lc CVIJXDliK 417 seen that Li may be deformed into Lj and we ol)tain c sec 4). Whether a particular integrand, such as the one shown in (5.4), con- \erges at the ends n = ±.i^ of L-i can often be decided from Table 5.1. This table gives a rough idea of the behavior of the various functions in terms of powers of i. For example, if the integrand should turn out to be proportional to z" = exp(z7rn/2) at n = f x , the integral \\ill con- verge Uke exp ( — tt | w | /2). Table 5.1 Order of Magnitude — Rough Approximation Function near n = i» near n = — j« i" 0 00 j-n 00 0 sm -KTl l-ln lin V{n + 1) I" i~" Un{z) ^•-3n/2 {Znll Vn{z) y-3n/2 i-nn lF„(z) i"/2 iZnI'i. The approximations for r(n + 1) follow from its asymptotic expres- sion, and those for the parabolic cylinder functions come from Tables 12.2 and 12.4. The entries for the c^^inder functions may also be sur- mised from expressions (9.4) which hold for z = 0. Table 5.1 may be used to show that the integrand in (5.1) is of the order of z" as n — » —i^c. Hence there is no hope of deforming Lj into L2 in this case. On the other hand, the integrand in (5.4) is of the order of t" as n — > z 3c and of i~" as n ^ — z oc , and therefore (5.4) converges exponentially. In fact, it converges for all real positive values of w = tan 9/2. This enables us to obtain an expression for the field which holds for 0 < 6 < T (i.e. it is not subject to the restriction | li^ | < 1 re- (luired by (4.16)). This expression, which is fundamental for our woik, has the form E = exp [-ix sin 9 + iij cos ^] + Si + S, (h). (5.5) Here Sy and »S'-)(/0 are given b}- (5.3) and (5.4), respectively. In working with (').')) it is sometimes convenient to u.se the expression 448 THE UICLL SYSTEM TECHNICAL JOURNAL, MARCH 1954 exp [ — ix sin 6 + iy cos 9] + Si r^i _^,^, (5.6) = (z'/tt)^'" exp [ — ix sin 6 + iy cos 0] / c ''" f/^. */— CO which follows from (5.3) and exp (-ir) (It = U/iY'-. (5.7) / The development leading to (5.5) shows that is satisfies the boundary condition E' = 0 at 77 = 770 f or 0 < ii' < 1. That (5.5) also satisfies the condition for the extended range 0 < w < ^ follows immediately from (5.8) 2i Jl. \ 2 / sni irn = —{if-n-y'' exp [ — ix sin 6 + iy cos 6] I exp {-it') dt J— 00 = —exp [ — ix sin 9 + iy cos 9] — Si when we note that setting 2' = So reduces Siih) to the left hand side of (5.8) (with z' = zo). Equation (5.8) is due to T. M. Cherr}^ who ol)tained it by expressing the cylinder functions as integrals and interchanging the order of in- gration (he works with the function Dn(z) of our equations (9.2)). Sub- stituting the integrals (9.19) for Un(z) and V„(z') in (5.8) and inter- changing the order of integration leads to a similar derivation. Eciuation (5.8) may also be obtained by deforming L2 into Li when 0 < iv < I and into L3 when 1 < w < go . This leads to the two series e'" sec ^ Z (-ur/2) Vt7„u)F„(/), (5.9) -c'^secl Z (-tW2)1r(n+ DUMWniz') (5.10) which may be summed in much the same way as was (5.2) for Si. An expression for E which is useful in the study of the current density on the surface of the cylinder may be obtained from (5.5) by combining expression (5.8) for exp [ — ix sin 6 + iy cos 6] + Si with expression (5.4) ^^ Expansions in Terms of Paral)olic Cylinder Functions, Proc. Edinburgh Math. Soc, Ser. 2, 8, pp. 50-65, 1948. DIFFRACTION OF RADIO WAVES BY A I'AHAliOhIC CYMXDKR 449 for »S2(/i): sec - i;c\" l\n -\- 1) 2i Jl,\'^/ s sin TT/t Un{z)W„{z') rvr.izo) (o.ll) Vr^iz') .WniZo) Wn(z')] dn. When w exceeds unity (or when w ^ \ and ^ > 7? ^ r/o ^ 0) in (o.ll), it may be verified with the help of Tables 12.2 and 12.4 that Li may be deformed into L3 + L,. When n is a negati\'e integei- the quantity within the brackets in (5.11) vanishes because of (4.8) and because Un{z) = 0. The contribution of L3 is zero since it encloses no poles. The contribution of L4 is equal to the sum of the residues at the poles gi^•en by Wn(zo) = 0. Hence, when w > 1, E ^ —wc sec ,- 2^ 2 s=i lA: r(n+ \)Ur.{z)Wr.{z')Vr.{z,)' -/ sin x/ia]r„(so) 'a/< (0.12) where n = ih is the sth zero of ]Vn{zo). This series also converges when w = 1 and ^ > 77 ^ ryo ^ 0 (which is roughly the .shadow region). The preceding inequality does not necessarily specify the complete region of convergence. Cherry has also pointed out that the expression for a plane wave given by A. Erdeha^ , namely (in our notation) exp [ — ix sin d -\- iy cos 9] sec - 2i LM'^ + 1) U„{z)\\{z') (5.13) sm TT/t + Un{-z)Vn{-z')]dn, may be regarded as the sum of the negative of (5.8) and a similar ex pre.ssion with ^ and 77 replaced by — ^ and —rj. In informal discussions with the writer, Prof. Erdelyi has pointed out that tlie work loading to our expression (5.5) for the field may be considcrablx' sIioiIciumI by starting with some known intcgi'al for the impressed field, such as (5.13) or a related re.sult. One way of doing this is to take exp [ — ix sill 6 + iij cos 6] + Si, '■ Proc. Roy. Soc Edinburgh, 61, pp. 61-70, 1941. 450 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 as given by the left hand side of (5.8), to he the impressed field in the equation E = impressed field + reflected field From the form of (5.8) and the discussion of expression (4.6) (given between equations (4.14) and (4.15)) we are led to assume the reflected field to be an outgoing wave of the form e sec - 2i im r(n + 1) sm Trn V n{z)W r,{^z')a{n) dn where ain) must be chosen so as to make E vanish on the surface of the cylinder. This gives a{n) = F„(2o)/^n(2o) and leads directly to the expression (5.11) for E. When the incident wave is vertically polarized, integrals for H may be obtained from the series of Section 4 in much the same manner as were the integrals for E. The analogues of the earlier results are H = exp( — /.r sin 6 + iy cos 6) iy ^ - 14!^ f /l^y r(^i±l) uMW.iz'YU.U) dn/'W,Xzo), 2i Jli \ 2 / sm irn H = exp {-2X sin 6 + iy cos d) -{- Si + S^ih), e sec Szih) = 2i LM iwV r(/i + 1) sm irn U„{z)Wn{z') 'Vniz'o) dn/'Wn{Zo), (5.14) (5.15) (5.16) H = e sec - 2i iw\ T{n + 1) r iwvn Jl. \2 s sui irn Un{z)Wn(z') H = -xc'^sec (0/2)1; _'W,XZ0) Wniz')_ ■{iw/2)''T{n + l)Un(z)Wniz'yVn{zo sin irnd'Wn{zo)/dn dn, (5.17) (5.18) In these formulas 'Un{z), etc. are defined by (4.19); w, z, z' by (4.13), z'o by (4.17). In (5.14) w is restricted to 0 < w < 1. In (5.15) S^ih) is given by (5.16), and w may be anywhere in 0 < w < co . In (5.17) w DIFFRACTION" OF HADIO WAVKS HV A I'AKAHOLIC CYLIXDKH l")! may also lie anywhere in 0 < iv < oc , but in (5.18) it is resliictiMl lo I < w < =o except when ^ > r? (roughly the shadow region) in which case w may be unity. In (5.18) n = n', is the sth zero of 'ir„(2o). Tlie zeros of 'Wn(zo) interlace those of Wn(zo) shown in Fig. 5.1. When h = 770/2 = 0 the parabolic cylinder degenerates into a half- plane antl our solutions reduce to Sommerfeld's expressions for waves diffracted by a half -plane. It may be shown that if To = V cos - + ^ sm - = (2/-) sni r^— h t 1 , we have S-iiO) = -{i/tY- exp [ix sm 9 + ly cos ^j / c~"' d(, SM = -82(0). (5.19) (5.20) When this expression for ^2(0) is added to (5.6) we obtain Sommerfeld's result for the case of horizontal polarization. One may verify that the series (5.12) leads to Sommerfeld's result as z'o approaches zero. By neglecting 0(2^) terms in (9.3) and setting n^ = — 2.S- + p, for the sth zero of Wn(zo) we may obtain the following rela- tions which are needed in the course of the verification V, = -4:izoTis -f l/2)/7rr(s) + • • • , dWn{zo)/dn at ns = r(s)/4 + • • • , Vniz'o) at n, = -2izoT{s -f l/2)/7r + • • • , (^.21) r Vnjz'o) 1 ^ ^^^ \_s\mrndW„{zo)/dnjn=n, 6. SURFACE CURRENTS ON THE CYLINDER As shown in Fig. 6.1, the surface current J on the perfectly conducting cylinder t? = tjo is parallel to the crest of the cylinder (and to the elec- tric intensity E) when the incident wave is hoi-izontally polarized. We ha\e from Maxwell's equations in parabolic coordinates J = [-//,],.,o = (rTo)-^ (2r)-"'- [dE/drJl,^,,. (6.1) Here Ht is the component of magnetic intensity in the ^-direction. fo is the intrinsic impedance of free space given by fn = (Mn/en)''" = como where the second part of the equation follows from 27r/X = w(jtX(,eo)''^ 452 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH l()o4 and X = 27r. The Eo and the Ho of the incident wave are related by Ho = Eo/^o = 1/fo since £"0 = 1. In rational il//v*S units fo = 1207r ohms. The derivative in (6.1) may be obtained by differentiating expression (5.11) for E and then setting ry = rjo. Use of the Wronskian (9.9) for Vn{zo), Wn{zo) then takes (6.1) into fo./ = M, L dn sni irn {iwrUrXz)/Wn{Zo) (6.2) where iv = tan (6/2), Lo is shown in Fig. 5.1, 2 and Zo are given by (4.13) and (4.17), and Mo = {i/STrY^^e "' sec - , r = {^' + r,l)/'2. (6.3) In this Section r will be restricted to mean a radius vector drawn to the trace of the cylinder on the (.r, y) plane of Fig. 6.1. Closing L2 on the right and on the left gives the two series ro./ = 20/0 Z i-ilVrUn{z)/Wn{Zo), 0 < IV < 1, n=0 (iwyUniz) UJ = -2/7ril/o2] sin irn dWn{zo)/dn (6.4) 1 < 10 < oo, (6.5) where ris is the sth zero of Wn(zo) regarded as a function of n. For the half-plane case 770 = 0, and (9.4) gives WniO) = -r/2r(l + n/2). In this case the series (6.4) may be expressed as an infinite integral Fig. 6.1 — Relationship between surface current density J and electromagnetic field when incident wave is horizontally polarized. E and / are normal to the plane of the paper. DIFFRACTION OF RADIO WAVES BY A rARAHOLIC f'VLINDKH 153 when the integral for r(l + n/2) is inserted and the sum (9.22) [i.e., the sum for the generating function of U„(z)] used. Tntogratiug part of the result gi\'es a fo-/ = {2/iirry - cos - ((3.G) - 2/t sin ^ c"'''""^ / exp (-//') dt . ■i ^5 sin («/2) In (G.6) /• is the distance along the half-plane as measuied from the edge: r = ^~/2 = \ y \. Positive values of ^ correspond to the shadowed side of the half- plane and negative values to the illuminated side. With this interpretation (6.6) agrees with the current density obtained from Sommerfeld's expression for the field. Although (6.6) has been derived from (6.2) on the assumption that 0 < u? < 1 it also holds for 0 < w < oo as may be shown by analytic continuation. Again, (6.6) may be obtained from (6.5). Since the factor r~^'" comes from the multiplier Mo in (6.4), it is pos- sible that (6.6) may give one an idea of how the current density behaves near the crest of a thin cylinder which is almost, but not quite, a half- plane. Of course, r would have to be interpreted as shown in Fig. 6.1. In order to study J when the radius of curvature of the cylinder is large compared to a wavelength we consider the case 6 = 7r/2, i.e. w = 1, in which the incident wave comes in horizontally. In this case most of the variation of the current density occurs near the crest of the cylinder where, as it turns out, ^ is of the order of T?o^'^ vo being large. At the beginning of the investigation rough calculations of the inte- grand in (6.2), based on the asymptotic expressions of Section 12, sug- gested that for small ^ and large r?o: (a) JNIost of the contribution to the integral (6.2) comes from the neighborhood around point C shown on Fig. 6.2 where m = n -{- 1 = zo^/2 = —irio~/2 = —ih. Point C is a critical point associated with the asymptotic behavior of W„(zo). (b) The path of steepest descent for (6.2) roughly corresponds to the line ACD of Fig. 6.2, C being the high point of the path. Along this path Im [fit,) - f(h)] = 0 where /(/) = -T + 2zf - m log /, m = n + 1, is the function entering the definition (10. 1) of the parabolic cylinder functions, and h, h are the saddle points of exp [/(/)]. This path in the n-plane separates the regions in which ir„(^ii) has different asymptotic forms. It is the boundary of region III' in Fig. 12.2 and has been studied in Sections 11 and 12. Once (6) is verified the truth of (a) follows almost immediately since 454 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 the path of integration L^ may be deformed into ACD without passing over any singularities of the integrand of (6.2). In order to verify (b), we note that the entries in Table 12.3 for Wniza) show that along ACD W^„(^:)-.4^^exp[/(^o)]. (0.7) Here the expressions for Wnizo) along ACD are taken to be those cor- responding to the regions / b and II shown in Fig. 12.2. In making the last approximation in (6.7) we have neglected the slowly varying co- efficient of the exponential function in the expression (12.9) for .4n. Since I ^ | « ijo we set ^ = 0 in Un{z). Then upon using the values (9.4) for Vn{^), (6.7) for T^„(2!o), and unity for w, we see that the integrand of (6.2) behaves like rexp[-/(^o)] /Qgx 2 sin (7rn/2)r(l -f n/2) ' On ACD we have, in dealing with TF„(2o), — 37r/2 < arg m ^ — 7r/2. Hence, from ^o + ^i = ^~^'^^o and from J{U) + j\ti) as calculated from (12.9), we have exp [m] = exp (![/(/(.) + fih)] + ^[.f(/o) - m]) (6.9) ~ r'^^(27r)-^'-'r(-n/2) exp (^- ^ + hAm - .m] where we have used the second of expressions (12.10) to evaluate exp [/n(l - log (w/2))/2l. Substitution of (6.9) in (6.8) shows that the integrand behaves roughly like exp (^^ - Um - /(/i)]) . (6.10) The truth of statement (b) then follows from the fact that the lines of steepest descent of (6.10) in the /i-plane are given by Im [f(to) — f(ii)] = 0. To see that C is the high point of ACD we use (12.9) to show that near C we have f(to) - m ^ (2/32o) (zo' - 2mf' where m = n -\- 1. Consequently, /(/o) — f(ti) is real and positive on AC [where, near C, arg (^o" — 2m) = — 7r/6] and on CD [where arg {z'o^ — 2m) = — 37r/2, m being in region IT according to the convention used in (6.7)]. That C is the high point now follows from (6.10). In accordance with statement (a), we must study the form assumed by the integrand of (6.2) when n is near point C. DIF^FHACTIOX OF HADIO W AVKS HY A rAUAHOTJC (AMNDI'.K Wlieii (1) n is near C, (2) 770 is large, and (8) | | | « r?,) we have tor tlie various terms in (6.2) I /Sin irn '-^ 2i r(l + 7i/2)W,Xzo) ~ (770/4)^ '(2x)''-r'' exp [-/77o'/2].4z-(a), 2r(l + n/2)Un{z) ^ "exp :^/n 2 ^S t 6 V2m (fi.ll) (6.12) wliere Ai (a) denotes the Airy integral defined by (13.12), m = w + 1, arg 7)1 is near —ir/2, and a = (2A7?;)^'' (/n + ivl/2), (6.13) da = (2/ir]lY'^ dn. Expression (6.11) comes from (13.21) and expression (6.12) comes from region la of Table 12.2 (strictly speaking, region lb should be used but point C is so close to arg m = — t/2 that the simpler expression for la may be used). In obtaining (6.12) it is necessary to use the terms shown in the expansions (12.5) of to and log ti/to. It may be shown that (6.12) also holds for negative values of ^. We now set w = 1 in (6.2) and change the variable of integration from n to a. Substituting for m in (6.12) its expression in terms of a, expanding in powers of a and neglecting higher order terms, converts the argument of the exponential function into if/2 — i^rjo — iy'^/3 + ayi 1/3 (6.14) B '-oo C IS AT n = -1 -lH D -1.00 Fi^- 6.2 — Patfis of integration used in stuch'ingthe current density and dinrac- tiun jjattern when /) is large. Path BCD is eciuivalont to path Ln of Fig. o.l. AC and CD arc boundary lines which mark a chaiigo in the asymptotic l)cha\ ioi- of H „(2o). Far out towards ^1 the line AC tends to become parallel to BC. 45G THE BELL SYSTEM TECILMCAL JOURNAL, MARCH 1054 where 7 = ^/(2r;o)''' = xl2h'^' (6.15) When our approximations are set in (6.2) we obtain To./ ^ (f/2r7o)"'V-^ exp [-{^770 - V/S] -coexp (-t2,r/3) (6.16) / exp {I'^'iOi) da/Ai{a). J-''--^-^^^^ E ?^^P4L_^ (6.17) 3=1 At {as) The leading tei'm in this series leads to the approximation (2.19) when we use Ai'{a\) = .701 • • • Expression (6.17) gives the form assumed by '6.5) when to = 1 and h ^ -x . For large values of s* a, f37r(4s - l)/8]''' (6.18) Ai'ia,) — (-rT-"'\-asy" When 7 is large and negative an asymptotic expansion for fo«/ may be obtained by .setting the asymptotic expansion (13.19) in (6.16) and using the method of steepest descent. It is found that the saddle point is at a = ao = 7^r' ^ and the slope of the path of the steepest decent through it is given by arg (da) = — 57r/12. This leads to the expression for fo'/ which appears in (2.17). When the incident wave is vertically polarized the magnetic intensity * Reference 11, page 424. DIFFKACTIOX OF 1{.\1)I() WAVKS HY A I'A K \ l<( )1,IC ( "» 1,1 NDKK 457 // is parallel to the crest of the parabolic cylinder. Since (he ( yliiidcr is a perfect conductor, the cui-rent ilensily ./,. on the surface v — rjo is (Hjual in magnitude to H and its dii-ection is that of inci-easing ^. Thus settings' = ^,i in (expression (5.17) for // and usin.u; the U'ronskian (9.9) j>ives -/. = X f -^ {nc)"UMnV,Xz:^, jl., sni Trn / ^\ / ^^^-^^^ N = e-"- (^sec ^j/ 2x> ^ r = {y^l + ^)/2, where 'Wn{z'^) is defined by (4.19). Closing Lo on the right and left leads to the analogues of (0.4) and (6.5): 00 J. = 2iXT. {-iwrUn{z)/'W{z',). 0 < (/• < 1, (G.20) 7i=0 {iwYVniz) J, = -2l7rXYl 1 < w < ^, (6.21) sin Trnd'Wn(zo)/dn_ where n's is the sth zero of 'W„(z'o). The zeros of both 'Wn(zo) and W„(zo) are enclosed b\' the path of integration L4 shown in Fig. 5.1. The current density on a half-plane is obtained by setting ^o = 0 in (6.20), using 'ir„(0) = Wn'(O) = -i"-'/T(n/2 + 1^), from (9.4), and the generating function series (9.22) for Un(z): /„ = 2{i/Ty"e-"-''"'' f c-''' dl (6.22) h sin (9/2) where r has the same significance as in (().6). This agrees with the ex- pression obtained from Sommerfeld's result for the half-plane. When w = 1 and h is large, the path of steepest descent for (6.19) becomes the same as that for the case of horizontal polarization, namely ACD of Fig. 6.2. This follows from the fact that, as may be seen from (12.2), the controlling exponential functions foi- '1T'„(^|') and Wniz'o) are the same. The analogue of (6.16) is obtained \)\ using the approxi- mation (1.3.24) for 'Wn{z',): J, ^ (1/2x0 exp [-/^77o - V/3] 1 (-i2ir/3) exp(i''V) doi/Ai'{a) L =oexp(-i2W3) ^ ^^^ (G.23) '« exp ( i2ir/,'?) where At {a) denotes dAi{a.)/da and 7 is given by (6.15). 458 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 For positive values of 7 (6.23) and d'Ai(a)/d~a = aAi{a) lead to ^ exp (t'^ya's) J, ^ -exp i-i^rjo - n /3] Z /, Y '\ (6.24) where a'l — —1.019, 02 = —3.248, • • •, etc. are the zeros of Az'(a).* When we use Ai{ax) = 0.5357 the leading term in (6.24) gives (2.19). For large values of s* a.: [3x(4s - 3)/8]''', 4 V '\ / \s/ '\-l/4 -1/2 Ai{a,) ~ -(-) (-a,) TT . The expression (2.17) for J„ when 7 is large and positive may be ob- tained by applying the method of steepest descent to (6.23). The asymp- totic expression for Ai'{a) is obtained by differentiating (13.19). When the cylinder is a good conductor, but not perfect, the expression for H analogous to (4.25) leads to an integral for J,,, obtained from (6.23) by substituting Ai'{a) 4- (Ai{a) for Ai'{a), which is ecjuivalent to one given by Fock.j Here ( = — {ihY'^^/^o is assumed to be small compared to unity and f/fo is the ratio of the intrinsic impedance of the cylin- der material to that of free space. Horizontal incidence, 6 = tt/'I, is as- sumed. The analogue of (4.25) for H has the same form as (4.21) except that now 'Un(zo) is replaced by 'L^,(^o) -|- rUnizo), etc. The development leading from (4.21) through (5.15), (5.17), (6.19) to (6.23) may be car- ried out just as before. The work is also related to that given at the end of Section 7 where the effect of finite conductivity on the diffracted wave is briefly discussed. A series corresponding to (6.24) may be derived from the integral. The exponential terms in this series are approximately exp [i^'\(a's — f/a's)], and are similar to those in (7.63). Since fo = (Mo/eo) " is real and f ?^ {iojix/gY'' when g » we (the notation is explained in connection with (4.24); the g denoting conductivity should not be confused with the g defined by (7.20)) the (juantity —I'^f/a's has a positive part. Thus, the attenuation of ./„ in the shadow is decreased slightly when the con- ductivity g of the cylinder is reduced from infinity to a large finite value. 7. FIELD AT A GREAT DISTANCE BEHIND THE PARABOLIC CYLINDER The field at any point, for the case of horizontal polarization, is given by expression (5.5) with S->(h) given by (5.4). Since S-2(h) is the * Reference 11, page 424. t Reference 5, page 418. niFKH ACTION OF RADIO WAVES HY \ l'\l{A|{OLIC ( VI-I NDKH 4.")9 only troublesome term in (5.5) most of this section will he dexoted to its stiuiy. Far behind the cyUnder ^ and rj are lai'j>;e and positiNc, and the corresponding tei'ins in the integrand of (5.4) are r(/. + i)r„(z)]\\(z') = m/r,)"i''c-''\i +/)/2W% (7.1) where the asymptotic expressions (9.10) and (9.17) give 1 + f = 1 + '"^"7^^ + '^" + \^^; + ^^ + O(nVr). (7.2) In writing tlie "order of" term it is assumed that ^ and tj are Ijoth ()(/•''') with /• » n'. When (7.1) is put in (5.4) we obtain .S,(/0 = M, / —. ';. >. dn (7.3) where, from the expressions (4.1) for ^ and 77, .1/, = [(^•/x)^V^sec {d/2)]/4v, . (7.4) l3 = tir/rj = ^[tan (^/2)]/r? = cot ((p/2 + 7r/4) tan (6/2). I Although it is not pro\'ed here, there is good reason to believe that (7.3) can be written as r"/3"F„(.~o) S-2 .m = M. f ■ ^ \:'''\ dn + 00^/r^'') (7.5) when r becomes large and we restrict ourselves to the region | ^ | < 7r/2 in order to get 0(^") = 0(77") = 0(r). The first term contains r only through the factor Mi and is of order r"^'". The "order of" term assumes h to be moderately large compared to unity but h' « r. When h < 1 the h is to be replaced by unity. The general idea leading to (7.5) is that (7.2) may be used over the portion of L2 where the integrand is large and important. On the portion where (7.2) differs appreciably from unity the integrand is negligibly small. The important portion of Lo runs from ?i = 3^ to w = —}4 — ih (approximately). In particular the variation of t "Vn(eo)/sin TnWn(zo) along L2 may be summarized as follows : from —]/^io-\- i ^ it decreases exponentially as i'", from — K to —ih it is ecjual to — 2z plus an oscillat- ing function of order unity, and from —ih to — t^o it decreases slowly at first and then more rapidly until it g(x^s down like 2""" (steepest descent behavior). This may be shown with the help of Fig. 12.2, the entries for regions /'a and //' in Table 12.3, and the following items [see (12.9) and Figs. 10.1 and 10.2 for z = r^'-ri^ with -37r/2 ^ arg {iril 460 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 - 2m) < 7r/2)]: (a) Re [j{t\) — /(/o)] is almost zero between — i.^ and —ih. (b) Im [/(/i) — /(^o)] is almost zero between —ih and — f co. (c) tifta runs from zero to unity as n goes from — 1 to — l — ih. Items (a) and (b) are consistent with/(^o) — fih) ^ (/■izo){z'o^ — ^inf'' which holds when n is near —ih and which was mentioned in connection with (G.IO). This concludes our discussion of the reasons for believing that (7.5) is true for general values of h. Now we shall check it for the special case h = 0. When w^e set /i = 0 (i.e. z'(, = 0) in (7.3), use (9.4) and close L2 by an infinite semicircle, we obtain 1 ^2(0) 1 2iTl + (7.6) This agrees with the first two terms in the asymptotic expansion of the Fresnel integral expression (5.20) for ^§2(0). In expanding (5.20) we need the first of the two asymptotic expansions (both of which hold for T » 1) L -iti i exp (-?T-) e at '^ —- 1 - / -t<2 (It W/iY" + 2T iexp j-iT-) 2T 1 +...1 2iP ^ J ) 1 - 2^ + and also the first of the relations ;r sin 6 + y cos 1 To = 77(1 + 0) cos (e/2). Tl ■r, (7.7) (7.8) (7.9) In much of the following work we shall assume ^ and 77 to be so great that we can neglect the terms denoted by QQi/r^'') in (7.5). We shall use the asymptotic sign '^ to acknowledge this omission. From (7.4), /3 is equal to unity when

1 and in the illuminated region ^ < \. Closing L2 on the right and on the left converts (7.5) into SM ^ 2iMr Z ^"VniZo)/Wn{Zo), (7.10) DIFFHACTIOX OF KADIO WAVES BY A TAl^ABOTJC CYLTXDKK 1()1 S,Oi) 2/.I/1 (/3- D-^-ttZ *"'/3"7„U) sin Trn dWn(zo)/dn (7.11) It can be shown that (7.10) converges if /3 < 1 (see (4.18)) and that (7.11) converges U ^ > 1 (see (12.13)). The term l/(^ - 1) in (7.11) comes from the poles of esc irw inside the path L3 shown in Fig. 5.1 From (7.7) and (7.12) — X sin 6 -\- y cos d — Ti = r, Ti = 7,(1 + 13) cos (e/2) it may be sho^^'n that when ^ > 1, so that Ti is negative, (5.6) has the asymptotic expression exp [-ix sin d + iij cos 0] + aSi -' 2iMx/{l - /S). (7.13) When this is added to (7.11) the 1/(1 — /3) terms cancel leaving a series for E valid in the shadow where j8 > 1 : E 8 = 1 t'"/3"F„(2^) sin Trn dWn{zQ M o)/5nJn=n, (7.14) This series may also be obtained from the more general series (5.12) for E by using (7.1) and neglecting /. We now take up the problem of finding the paths of steepest descent for the integral in (7.5) when /3 is near unity and h is large. When |S = 1 and h is large, the integrand in (7.5) may be expressed in terms of exp [/(^i) — /(/o)] by using Table 12.3. In Section G it has been pointed out that the path of steepest descent for exp [/(/i) — jik)] is the path ACD of Fig. 6.2, with C being the high point. This suggests that the path ACD should be used to deal with the terms in (7.5) leading to exp [/(/i) — j{k)]- These terms are U niz^) /W niz'o) (introduced by the use of (4.8)) for the portion of Lo between B and C, and V n^z'^) /W „{z'^^) for the portion between C and Z). As a further argument supporting the use of the path AC WT note that when n is on AC, i.e., on the edge of region I'h, Table 12.3 gives Un{z,)/Wr.{z,) ~ -i{\ - I'^-W./tof" exp [/(/:) - m] . (7.15) Hence the variation of ^"" esc irn in the integrand of (7.5) is just cancelled by that of (1 — t~*") in (7.15). Consequently f''U„(zo)/[sin irn Wn(z',)] varies as exp [f(^) — f(to)] along AC (the variation of h/to is relatively small). These considerations lead us to write (5.4) as 462 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 S,{h) = Sn + S22 + So, Sn = - [ F (In, J B S,2= -[ i^f',.(^o) dn/Wr.U), (7.16) &3 = / FVniz'o) dn/WnU), Jc F = e'" sec {d/2){iw/2yr{n + l)l\{z)Wn{z')/2i sin ttw. When instead of (5.4) the expression (7.5) for S^ih) is used we obtain S21 '^ —Ml I i'^jS" dn/s'm irn, J B S,, Ml / i"'^"UnU) dn/[sm irnWnU)], (7.17) J A S2Z -- Ml f r"/3'T„(2o) dn/[sm TnWnU)]. Jc In S22 it is permissible to swing AC from its original position BC because the zeros of Un(Z(^) cancel those of sin irn. When /3 = 1, AC and CD are the paths of steepest descent for AS22 and *S23 in (7.17) because Fm [m - f{to)] = 0 on ACD. The asymptotic expression (7.17) for Sn may be evaluated bj^ tem- porarily assuming jS to be a complex number with \ 0 \ < 1 and | arg 0 \ < 'w/2. The integral along BC is the integral along BCE minus the in- tegral along CE (see Fig. 6.2). Deforming BCE into Li of Fig. 5.1 shows that its contribution to S21 is —2iMi/(i — (8). An infinite series for the integral along CE may be obtained by expanding i "/sin irn in powers of exp ( — i2irn) and integrating term wise from n = no = — I — ih to n = 'x> —ih, i.e., from C to E. In this way we obtain S21 2iMi + E exp (no log (8 — 2Tth) (7.18) t=o log ^ — 2x?7 Despite the appearance of the right hand side, it is analytic around /3 = 1 and analytic continuation may be used to show that (7.18) holds f or 0 < /3 < -^ . When h is large only the first term in the series is important and we have S21 - 2iMi[(^ - 1)"' - fS-'-"'Aog ^] (7.19) = 2iMii^ - ir' -f iM2g~' DIFFRACTION OF RADIO "WAVES BY A I'MJABOIJC rVIJXDEH H)3 .1/. = 2M,h''Y'-'' - ^'^'''^''' ^^p ^-'' + ^"^'^''^ where we have introduced two (luantities which will he used later Vtt/ 2^ sin (0/2) ' (7.20) g = -h"'\og0. When )8 = 1 Sn ~ 2iMi(ih + }4). (7.21) When h is large most of the contributions to the integrals (7.17) for ^22 and aSjs come from around n = no = —1 — ih, and we may use the approximations UM)/Wr.(zo) ~ i-*"Ai(ai-'")/Ai(a), (7.22) Vr.(z'o)/Wn(zo) - i"'Ai(ai'yAi{a), (7.23) a = (ihr"\n + 1 + ih), n - no = a{ih)"\ which come from (13.21). Setting these in the integrals (7.17) and using the fact that i'^/sin -wn is nearly 2? around no leads to S.,, ilfo / exp( -l'^'ag)Ai{ai-"') da/Ai{a), (7.24) Joo exn (i2j-/3) /« exp (i2j-/3) ,00 exp (— i2?r/3) S,, f"Mo / exp i-t'''ag)Ai{ai^") da/Ai(a), (7.25) Jo S22 + *S23 ~ ?:i/2^(^), (7.26) where ^(^) = i [ exp (-i'' a^)Az(ar''') da/Aiia) (7-27) + /" exp (-i"'ag)Ai{m'") da/Ai(a). Jo The expression (2.11) for -^(g) is obtained from (7.27) by changing the variables of integration and using the transformations (13.17) for Ai(a). Thus, when h is large and (3 near unity, (7.19) and (7.26) give S.(h) ~ 2i]\h(0 - ly' + Ur-^g-' + ^(g)]. (7.28) In the .shadow, where (5 > 1 and g is negative, (7.13) and (7.28) give E = exp[-w- sin 6 + //y cos 6] + S^ + S,(h) (7.29) ~a/2[^"^ + ^i^((/)]. This and the series (7.14) for E suggest that 'i'(g) + 1/ry may be ex- 4CA THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 pressed as a series in which the paraboUc cyhnder functions in (7.14) are replaced by Airy integrals. One way of obtaining this series from (7.14) is to use the Airy integral approximations (13.21). The zeros Hi, no, • • • of Wniz'o) go into the zeros ai, 02, • • • of Ai(a) by virtue of the relation n — Wo = a{ihy ^ and we have E ~ i-'-"M, E "Ti'/tr"' ' (7.30) .J r \ I 1/ .113 ^ exp [- ttsQi .„„ s where g < 0. Here, as in (6.17), at = -2.338 • • and Ai'(ai) = .701 • • • In obtaining these relations we have used z"'7sin irn ^ 2i, and Ai{asi"') = -i"Bi{a,)/2 = i"'/2TrAi'{a,), (7.32) where the first equation follows from (13.17) and the second from the Wronskian Ai{a)Bi'{ci) - Ai'(a)Bi(a) = I/tt. (7.33) The equal sign in (7.31) holds even though the steps leading to it indicate that ^ should be used. This may be seen from an alternative derivation of (7.31) in which Ai{ai~'^'^) in the first integral of (7.27) is replaced by the right hand side of* Ai(al~"') = -i'"Ai(a) - r'"Ai(ai"'). (7.34) In the first portion the Ai(a)'s cancel and the resulting integral con- tributes — 1/g to (7.27) (g must be negative for convergence). The second portion combines with the second integral in (7.27) to give a contour integral which leads to the series in (7.31) w^hen the path of integration in the a-plane is closed on the left. The closure may be justified by the asymptotic expressions (13.19) and (13.20) for Ai(a) (again g must be negative). Since the integrals in (7.27), and their equivalents in (2.11), converge uniformly for all finite values of g, ^(g) is an integral function of g. When g is negative ^(g) may be computed from the series (7.31). When g is positive I have not been able to find a practicable method of ob- taining ^(g) other than the numerical integration of (2.11). The results are shown in Table 2.1. Since ^(g) is an integral function its Taylor's series about, say, g = — .5 converges for all values of g. The coefficients in this series may be computed from (7.31). However, I was unable to * Reference 11, page 424. niFFI^ACTIOX OF KADK) WAVFS HV A PARABOLIC CYLINDKR K).") obtain useful results by this method because the computation of the coefficients becomes more and more difficult. When g is lai-ge and positi\'e it may be shown that ^(O) f/"' + ii^yf" oxp (-hf/12). (7.35) The procedure used to establish (7.35) is much the same as that used to establish the more general result S-2-1 + *S23 ~ -iMog -1 'h(l - /3)' 1/2 exp [-W- + 2ih{l - /3)/(l +/3)] Lr(l + /3)J sin h(^ + d + 7r/2) ' (7.36) 1 - /3 _ sin U^ - d + x/2) 1 + i3 sin h{

1 (shadow region), ris = sth zero of 'W„{zo), (7.44) DIFFRACTION' OF RADIO WAVKS HY A I'AHAHODIC ('YI,I\1)KH 107 H ~ --liMiTT X ^ > 1 (7.4.3) .sin irn d'\Vn{Z{))/dfl S,, = .S,i defined by (7.16), S,, = .S,, with 'r„(zo)/'Wn(zo) in plaoe of U,.U)/Wn{2o), S,, = S,, with 'V,.U)/"]VnU) in phice of V„{z^)/W„U), ^3., ^ i^'Mo / exp ( - l'^'ag)A i'iar'") da/Ai'{a), (7.47) •'« exp (i2jr/3) (7.46) S3Z '^ ^fi ■'i 00 exp (-i2ir/3) exp ( — ?' ag)Ai'{ai ") da/Ai'{a), Szi + Szz ~ ^■i¥o^,(^), ^((^r) = ? ^^ / exp { — l^'^ag)Ai'{m '*^^) rfQ:/.4/'(a) •/« exp {iiir/Z) ^00 exp (— i2ir/3) / exp { — i'^ag)Ai'{ai^^^) da/ Ai'{a), Jo H ~ lAhlg ' + ^.(^)J, ^ < 0 (7.48) (7.49) (7.50) (7.51) H -2/3 M,Z exp(-a,^i ) '^.{g) + g-' = -^'''I: rf {-as)[Ai{a's)f' exp(-o,grz ) g < 0 (7.52) ^ < 0 (7.53) ^(-aOUKaO]^' a,' = sth zeroof Ai'(a), a[ = -1.019, -47(ai') = 0.5357, Ai'ia^i"^') = -z^''5/'(a:)/2 = -^''V[27^.W(flI)], (7.54) A/"(«) = aAi(a). When g is large and positive, ^.(g) fi'"' - a-^gy" exp (-^VVl2), (7.55) 'S32 + *^'33 '^^ —iMig + "Kl - ^)" Lr(l + 0). "- exp [-ir + 2^fe(l - i8)/(l + &)] (7-56) sin K^ + ^ + V2) The change in sign of the second term on the right in going from (7.36) to (7.56) comes from (12.2) and the analogous expre.s.sion for 'Wn{z[^) 468 THE BELL SYSTEM TECHNICAL JOTTRNAL, MARCH 1954 (only ii contributes to Un(zo) and only to to Wniz'^) at the saddle point nil of the second integral in (7.38)). So far in this section the parabolic cylinder has been assumed to possess infinite condu(;tivity. When the cylinder has a finite (but very large) conductivity, it may be shown that the field far out in the shadow is approximately hi sm irn TV„(4n) + a 'Wn\ZQ) Equation (7.57) is suggested by (7.14) and (4.25). The analogue of {1 .bl) for vertical polarization may be obtained by replacing E, a in (7.57) by H, t so that 'Vniz'^) + rVnizo) appears in place of Vniz'a) + 0""^ 'V{z'(t), and so on. When the parabolic cylinder functions are replaced by Airy integrals according to (13.21) and (13.24), equation (7.57) may be written as E ~ f'M, f fe'^P <-"f .71'''m "^ '''^'"'' *» (7-58) JL'i At{a + k) where g and M2 are given by (7.20), a by (7.23) and k = -{ihr"'UU- (7.59) I fc I is small compared to unity. L4 is a path of integration in the a plane which encloses the zeros of Ai{a + /c) in a clockwise direction. Changing the variable of integration in (7.58) to u = a -\- k enables us to con- clude that E for finite conductivity ^'^p ' -^/j E for infinite conductivity _ (7.60) Since we have assumed d = ■7r/2, the relation (7.60) holds in the region where the angle ^p defined by Fig. 2.3 is negative. The analogue of (7.58) for vertical polarization is obtained by re- placing E by H, omitting the i"'^, and replacing the ratio of the Airy integrals by Ai'[{a + (/a)l"]/Ai'{a + (/a) (7.61) where 1= -iihf'UU. (7.62) Even though h is large, f/fo is assumed to be so small that ( is small compared to unity. The path of integration L4 must now enclose the zeros of Ai'{a + l/a) which are close to those of Ai'{a) at a = a's^s = 1, DIFFHACTIOX OF R.VniO WAVFS BY A l'AHAT?OMC CYUNDFU If)!) 2, • • • .It must not pass close to a = 0 since the work leading to (7.61) assumes (/a to be a small number. Changing the variable of integration to /' = a + (/a, approximating i/a, (/a by (/i\ i/v, and evaluating the integral by considering the residues of the poles at y = al gives // ^ ,-'M/. E (l - -4)"" ""' '"''-^rl'^f "•" • (7.63) This shows, to a first approximation, how the expression (7.52) is modified when the cyhnder is a very good, but not perfect, conductor. Of course g must be negative in (7.03). Since ( in (7.62) varies as li'"^ while k in (7.59) varies as IrT^^^ it appears that the field for \'ertical polarization is much more sensitive to changes in the conductivity than it is for horizontal polarization. It may be verified that the change in the exponential terms in the series (7.30) and (7.52) produced by finite conductivity, namely as changes to a^ — k (7-64) tts changes to a^ — S/us, agrees, to a first approximation, with the change produced in the cor- responding series (given, for example, by the series (27) and (28) on page 45 of Reference 7) for the propagation of radio waves over the earth's surface. 8. FIELD AT A GREAT DISTANCE BEHIND THE PARABOLIC CYLINDER WHEN 6 = x/2 AND h IS LARGE In the work of Section 7 the angle of incidence 6 may lie anywhere between 0 and tt. Here vve take d = t/2, w^hich corresponds to the case shown in Fig. 2.3 and described in Section 2. Some simplification is ob- tained thereby. For example, the incident wave is now simj^ly exp ( — ix). We shall write the expressions for the horizontal and vertical polariza- tion cases as E = (e-''^ + S{)r + Sn + (^22 + ^23), (8.1) H = (e-'' + S,)r + Sn + (Sz2 + ^33), (8.2) respectively. Here >S2i, • • • are defined by (7.16) and (7.46) in which e = 7r/2, w = 1, (8 3) /3 = t/r, = cot (

— 1, Un(z) = Fr f e^''--"'e dt, J— 00 Vn{z) = -FT" [ e"^'+'"Vf/r, (9.19) Jo Wn(z) = -Fi' [ e-'"-''" r" dr, Jo F = 2VVr(n + l)7r'''. When n is not an integer the path of integration in the integral (9.19) for Uniz) is indented downward at the origin. Equations (9.19) mav also be obtained from (9.1) by using (9.13) and (9.18). When n is an integer Un{-Z) = (-)" Un(z), Vn{-Z) = (")" W^iz), (9.20) and when ?i is a positive integer Uniz) = s,{n,z) = {-y'l:fe-'\ n\ dz" UUz) = 0, (9.21) V.n(z) = -W^niz) = -is2{-n,z) = - -—- ^—^ e\ From Maclaurin's expansion and (9.21), E ^"f'n(2) = exp [-r + 2zt]. (9.22) 478 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 10. FORMULAS FOR THE SADDLE-POINT METHOD Much of our work involves the behavior of the parabolic cylinder functions as functions of n when n is a large complex number. Although this subject has been studied by several writers," •" • " their results are not in the form we require. As the work of Sections 6 and 7 shows, the paths of steepest descent for the integrals in our electromagnetic problem are intimately connected with the function /(/o) — f(h)- In turn, this function is closely related to the saddle point method of e\'alu- ating Un(z), etc., for large values of n. For the sake of completeness, we shall outline this method. We shall pay special attention to the rela- tive importance of the two saddle points as n moves about in its complex plane. When we write the integrand of the integrals (9.1) as exp [f(t)] we obtain expressions of the form Uniz) = 1-f exp [fit)] dl, Zirl Ju (10.1) /(/) = -f- ^ 2zt - m log /, m = n + I. The saddle points of the integrand are at the points U and t\ in the com- plex ^plane where /'(/) is zero: 2/o - 2zto + m = 0, tl - zio = -m/2, z + {z - 2my" 2 z -{z' - 2my" to + h = z, h = ^^^ — , 2/o/i = m. (10.2) Let the path of integration U of (10.1), for example, be deformed so as to pass through a saddle point, say ^o, along a path of steepest descent. Let 00 fit) = fito) - E hit - t,Y/k!. (10.3) 2 Then, if 62 is not too small, the contribution of the region around to 20 Nathan Schwid, The Asymptotic Forms of the Hermite and Weber Functions, Amer. Math. Soc. Trans. 37, pp. 339-362, 1935. References to earlier work will be found in this paper. Schwid's work is based on R. Langer's studj' of the asymptotic solutions of second order differential equations. -1 O. E. H. Rvdbeck, The Propagation of Radio Waves, Trans, of Chalmers Univ. of Tech. 34, 1944. 2^ G. N. Watson, Harmonic Functions Associated with Parabolic Cylinder Functions, Proc. London Math. Soc. (2) 17, pp. 116-148, 1918. DIFFRACTION OF RADIO WAVKS BY A I'AKMfOlJC ( ' V !.! NDKR 70 to the \-aliie of tlio integral is exp lj(/(i)] times 2ir J exp - Z I'l^U - /o)'/'A-! (// ~ (27r6..)~''-[l + ! -/>,/?, + lO^'B,} (10.4) + 1-/^/^. + I35';i + oGb,h,]B, - 2\00hl(Mlh + 55(280)6356} + •■•] w here Bk = {'Iboy'' / k! . The sign of (27r62)~^'" is chosen so that the argu- ment of the right hand side of (10.4) is equal to arg (dt) at t = to on the path of steepest descent. The derivatives oi f(t) at to give ^2 = 2('o - /i)//c, b, = 4/1//0' , 64 - -12/1//,^ , -biB-2 + lO^afi;} = 24/0(^0 - ^1)=^ The values of these quantities at the saddle point ^i may be obtained by interchanging to and /i. If more terms of (10.4) are desired they may be obtained from the formal result ~ exp - E«a7VA-! 2ir J-x A-=2 dt (27ra2) ^ (10.6) 1 + E y-^i^iO, 0, -aa, -«4, • • • , -a2.)/A-!(2a2)' where F„ (oi, a^, . . . , ««) is the Bell exponential polynomial." It is neces- sary to rearrange the terms given by (10.6) in order to get them in groups ha\'ing the same order of magnitude. A more -careful treatment of the terms in the asymptotic expansions for Dn{z) has been given by Watson." His method is similar to that used by Debye for Bessel func- tions. In our woi'k we shall deal with two different complex planes, and the reader is cautioned against confusing them. One is the complex /-plane, shown in Fig. 10.1, which contains the paths of integration for integrals such as (10.1). The other is the complex w-plane, shown in Fig. 10.2, which is introduced because we are often more interested in Uu(<^), etc., as functions of m = n -\- I than as functions of z. In the eai'lier sections " E. T. Boll, Kxj)()iipiiti:il Polynomials, .Vnii. of M.ith. 35, pp. 25S-27y, 1> ^ .^^ BOUNDARY BETWEEN to HALF-PLANE AND t, HALF-PLANE Fig. 10.1 — Diagram showing the half-plane regions to which the saddle points to and ti are confined in the /-plane. One might wonder why cuts in the w-plane are required since it has already been pointed out that Un(z), etc., are one-valued functions of m = n + 1. The trouble is that the asymptotic expressions for Un{z) are many-valued functions of m even though Un{z) itself is not. Now that we have considered the saddle points ta and /i, we turn to a consideration of the paths of steepest descent in the /-plane which pass through them.* The path of steepest descent which passes through /o, for example, is that branch of the curve Im im - f(U)] = 0 (10.8) for which to is the highest point (i.e., Re [f(t) - f(to)] ^ 0 on it). The * Watson^s has studied paths corresponding to Re(n) > 0 when z is any com- plex number, and has given curves which are related to some of those shown in Section 11. DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 481 paths of steepest descent may be shown to have the following prop- erties : 1. Let t = tr ■{- iti = r exp {id). Then the paths of steepest descent either run out to /r = + °^ with ti -^ Im z or spiral in to i = 0 as r = (constant) exp { — irird/mi). 2. The steepest descent path through U may be computed by a graph- ical method based on* arg {dt) = diVgt - arg {t - h) - arg (t - h). (10.9) ^ KVz2-2m Z2 - _^ /A~~\ARG(z2-2m) ^^)^--V*-^ POINT m -2-' 2 ARG Z ^ X 1? . m 2ARGZ-77 Fig. 10.2 — Diagram showing the cuts in the complex w-phme, w = n + 1. It' we draw the triangle to 0 ti and bisect the interior angle at ^o by the line 6o^o then arg (dt) at to = angle t if oho. (10.10) If one goes clockwise in traveling from the side tJi to tJh) then arg dt is negati\'e. Likewise, arg (dt) at ti (on the path through ti) is the angle between the side tJo and the bisector ^i6i of the interior angle at ti. 3. When 7n has the critical value z'/2 the saddle points coincide: ^0 = ^1 = z/2, and the paths of steepest descent start out from t = z/2 in the three directions arg (t — z/2) = (arg z)/3 + 5 where 8 is 0, 27r/3, or -2x/3. 4. Some of the paths of steepest descent change their character as m goes from one region of the m-plane to another. This is illustrated in Section 11 for the case z — p exp (iV/4) where it is shown that the * A similar method was used in 1938 by A. Erd^lyi in an unpuljlishcd study of the asymptotic behavior of confluent hypergeometric functions. 482 thf: bell system technical jourxal, ^l\rcii 1954 boundaries are given by Im U(to) - m] = 0, (10.11) or a similar equation involving another pair of saddle points, e.g., ti and ti exp (i2Tr). In this equation z is regarded as fixed and ^o, h are functions of m defined by (10.2). It should be noted that although (10.8) defines a path of steepest descent in the ^-plane, (10.11) defines curves (bound- aries of regions) in the ?/? -plane. 5. If m is such that the path of integration for a particular function, say Un{z), passes through both to and ^i, each one will contribute to the value of Un{z). Furthermore, if m is such that Reim -m] =0, (10.12) U and t\ have the same height and the two contributions have a chance of cancelling each other and giving a value of zero for Vniz). Thus (10.12) or some similar equation defines the lines in the »i-plane along which the zeros of Uniz), etc., (regarded as functions of m) are asympto- tically distributed. 6. The lines in the w -plane defined by (10.11) and (10.12) may be obtained by substituting the values (10.2) for U and h in m) - Kh) = h' - t{ - 2Wi log (/o//i), (10.13) and setting the imaginary and real parts, respectively, to zero. How- ever, instead of dealing with m directly it is easier to use w = u -\- iv \ defined by w = log {h/h) = log I k/ty I -f tXarg h - arg ^i), (10.14) | m = ^'/(cosh w + 1), (10.15) | where (10.15) follows from (10.14) and (10.2). Then (10.13) becomes /(/o) - .f{h) = /«(sinh w - w) ^ z'ismh w - w) (10-16) cosh w -\- 1 The inequalities (10.7) show that M ^ 0, I V I ^ TT. 7. For the special case z = p exp (?7r/4), (10. IG) gives (cosh u + cos V — V sin v) smh w = (cosh u cos v + 1) u, (10.17) (cos V + cosh u -{- u sinh u) sin v — (cosh u cos v + 1) v, DIFFRACTION OF RADIO WAVES UY \ 1' A I{ A H< )I.I(' CYLINDER 483 respectively, for Im\f(to) - /(/i)] = 0 ami Re [/(/o) - fih)] = 0. These equations are plotted in Fig. 10.3. It will l)e noted that a curve is shown for V > IT even though this puts w outside the allowed rectangle. This is done because one of the paths of integration, W, passes through both ^0 and /i exp ( — t27r) when m is in a certain region, and the correspond- ing zeros of Wn(z) lie on the curve defined by Re If (to) - f(h exp I -i2r})] = 0. It may be shown that a curve corresponding to KU) - fihexp {-^•27^}) with — TT < y < TT may be obtained from the curve corresponding to /(^) ~ /(^i) ^vith TT < y < Stt by simply subtracting 27r from v. This is done on Fig. 10.3. ARG m =-90° BOUNDARIES ■lm[fao)-f(ti)] = o LINES OF ZEROS (EXCEPT CURVE (a)) ■Re[f{to)-f(t,)] = o ARG m = 90° / Re[f(to)-f(t,e-L27r)] = o (a) ARG m = 270 m — »»0 270 Fi|2;. 10.3 — Boundaries of the regions shown in Kiff. 11.2 and lines of zeros shown in Fig. 12.1 as they appear on the u' = n + ir plane when z = ?'/V- 484 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 1 1 . DESCRIPTION OF PATHS OF STEEPEST DESCENT In this section we shall study the paths of steepest descent when z = i^'^p in some detail because it is one of the two cases (the other is iT^'^p) encountered in our diffraction problem. Curves related to some of those shown here have been studied by Watson (for Re{n) > 0, as mentioned in Section 10) and by Rydbeck* (for Re(n) = —}4)- The affixes to and ti. of the saddle points are given by (10.2) and are made definite by the two cuts in the 7n-plane w^hich now run from m = ip'/2 torn = i^ and from m = 0 to w = — z'co. From Figs. 10.1 and 10.2 (drawn for z = I'^p) we obtain, -7r/4 < arg {ip^ - 2mf'^ ^ 3ir/4, — ir/2 < arg m ^ 3x72, (11.1) -7r/4 < arg U ^ 37r/4, -37r/4 < arg h ^ 57r/4. A convenient equation for the path of steepest descent through to is obtained by setting t = r exp (id), n -\- 1 = m - nir -\- imi in (10.1) and (10.8), and dividing through by p^: - {r/pf sin 2e + 2(r/p) sin {6 + 7r/4) - {nu/p') log {r/p) - {mr/p')d = Im [j{to) + m log p]/p'' Replacing ^o by ti gives the equation for the path through ^i. Equation (11.2) and its analogue for ti were used to compute the paths of steepest descent shown in Figs. 11.1 and 11.6. When m/p' is small, so is ti/p. It may be shown from (11.2) (for ti) that (r/ 1 ii I ) sin {d + 7r/4) - {rm/ \m\) log (r/ | ^i | ) (1 1 .o ) — B rrir/ I w I ?:b {mi — rur arg ti)/ \ m \ gives the behavior near ^ = 0 of the path through ^i. Paths computed from (11.3) are shown in Figs. 11.3 and 11.5. Here ^i ;^ 'mi'^'^/2p. In computing the paths shown by the figures of this section, the work was simiplified by taking m to be purely real or purely imaginary, or by assuming m/p~ to be small. Even so, this often required the solution of a rather simple transcendental equation [as (11.2) and (11.3) show]. The graphical method based on (10.9) was not used, although it might Pages 26-36 of Reference 21 cited on page 478. DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 485 have been if we had elected to study a case in which neither nir nor m,- were zero. Most of the vahies of m/p' studied in this section are listed in Table 11.1. The point numbers are those listed below and shown in Fif>;. 11.2. Paths of steepest descent are shown for all but the last two entries of the table. We now take up these values of m one by one. 1. m = pV2. Fig. 11.1 (a). Since Re f(h) > Rej{h), h is higher than ^0. In all of the figures dealing with the /-plane in this section, solid lines mean | arg t | < tt while dashes indicate | arg t\ > ir. 2. ni = p\ Fig. 11.1 (c). As m goes from p^/2 to p^ the path through ti changes its type, h is higher than ^o- ■INDICATES ARG t >77 UL.Wf Vf, W(. WP Fig. 11.1 — Paths of steepest descent in / = i, + ?7, phme when z = I'/^p and m^n + 1 is real and positive. Ui and U/ denote initial and final branches of the path of integration for f/„(i"2p), and so on. to and ti are saddle points. 486 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 O < > > I— I Eh < ;? CO « Oh H O CO W h^ Eh I— I o Plh H Q Q o iC t^ 0 0 ro -H re CC =D "O lo 0 0 I^ !>• a. 0 0 0 0 0 0 0 0 .-H C^ .-^ .T^ .'-; .-^ .•;H S ++++++ 1 1 +++ + + 1 1 + 10 -+< »C 0 'M ^ ^ rt rt 0 0 0 0 0 -H r-H 0 --H 0 CO C^ 1 1 1 1 1 + iC- T-H 05 < r^ ■* 00 10 0 ■* 10 ic (M CD 0 10 10 lo 10 CD 10 ^ ^ ^ 'f rf CO (M Oi 0 -^ 'J^ ^ ■6 0 1^00500i000050-H ^ l-l^ 0(M 00C50000(M t^ t^ mcOOO ^,-1 ,_|,-l 0 '-H ^ -^ ^ rt 0 0 O^i-H c^ 1 II II < ^^ 1 --g 1 g ^1 3 3 ^ M s ^ ■6 0 OiO io>o 10 >c 0 »0 0 00 0 0 lO iC >0 0 0 0 0 0 lO iO O--! 0000000.-H i-H ■-< o^t-H a ■3 i-hCQCOtTOOOOO-hC^ICO [ -t<| 1 T3 a + ,(_3 t^ C1 0 U2 +^ « :^ ^ a ci

- Points 4, 8, 9 correspond to Fig. 11.5 and points 11, 12, 13, 14 to Figs. 11.6 (a), (b), (cj, (d). 488 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 I, II, III shown in Fig. 11.2. It may be verified that for large negative values of nii the boundary lines in Fig. 11.2 are given approximately by mr = dz 2"'7r''p\mi\''\ (11.4) There was a period, while these curves were being worked out, during which it appeared that the regions I, II, III told the entire story. How- ever, when small \'alues of 7n were studied it was found that region I splits up into the two sub-regions, la and lb, such that the boundary between them is given by Imf/(^i) -Khe-'n] ^ 0. (11.5) Vl 'iu. to//5 W, Wl m -o.oobp^ U/p Fig. 11.3 — • Paths of steepest descent for \m\ = 0.005 (P-, z = i^'^p. Away from t = ii all paths look much like Fig. 11.3(a). DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 489 This is of the same form as (10.11). That /i exp ( — 27ri) is a saddle point follows from (lifT(M-(Mitiation of the ofiuatiou lite^"") = /(/) + 2irimr - 2irmi. (ll.G) Combining (11.5) and (11. G) shows that the boundary between la and lb is given by lUr = 0. This is indicated on Fig. 11.2. We now examine the paths of steepest descent when m is small. Fig. 11.3 (a) gives a large view of all the paths, irrespective of arg m, when in/p is small. 4. m = O.OOop". Fig. 11.5 shows the vicinity around ^i. 5. \m\ = 0.005p', arg m = t/2 -0.05. Fig. 11.3 (b). 6. I w I = 0.005p", arg m = 7r/2. Fig. 11.3 (b). After passing through ti the path encircles the origin clockwise and runs down into the saddle point at t = h exp ( — 27ri). Since rrii is positive, (11.6) shows that ti exp ( — 27ri) is lower than h. The path for arg m = 7r/2 — 0.05 sug- gests that from h exp ( — 27rt) the path runs out to co exp { — iri) along the path of steepest descent which lies directly under (on the Riemann sheet for —Zir < arg ^ < — tt) the path which runs from ti to t = =o exp (ix). It follows from (11.6) that, as t traces out a path of steepest descent through ^i, t exp ( — 2iri) traces out a path of steepest descent through ti exp ( — 2xf) directly under the path through ^i. 7. I m 1 = 0.055p', arg m = 7r/2 + 0.05. Fig. 11.3 (b) shows that after passing through ^i the path of steepest descent spirals in to f = 0. According to (11.3), the spiral is given by r ^ (constant) exp ( — mrO/^ni) (11-7) when r is small and d large. Two things are to be noted. First, the type of path is different from that for arg m = 7r/2 — 0.05. Hence arg m = 7r/2 marks a change of type similar to that shown in Fig. 11.1 (b), except that here h exp ( — 27rf) takes the place of ^o. Condition (11.5) takes the place of condition (10.11), and is satisfied by virtue of nir = 0 when arg m_ = ir/2. The second thing to be noted is that up until now all of the paths of steepest descent have ended at ± «> and U, V, W could be deformed into them without difficulty. How can Ave deform U, for example, into a path of steepest descent when the path through ti spirals in to < = 0? The way to deal with this problem is shown in Fig. 11.4 where U is continuously deformed into two portions, one coinciding with the path through ti, as shown in Fig. 11.3 (b), and the other with the path of steepest descent through ti exp ( — 2iri). The second portion lies directly "underneath" the first portion. 490 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 In Fig. 11.4 the dashes mean, as before, that the path of steepest descent is on a sheet of the Riemami surface other than | arg t | < ir. The alter- nate dots and dashes are used to indicate that | arg t | > tt and that in addition the path lies directly under the curve it parallels. Although in Fig. 11.4 the two kinds of dashed curves are joined at about arg / = — Sir — 7r/4, they actually should spiral in to / = 0 before they connect. 8. I m I = 0.005p", arg m = x. Fig. 11.5. For arg w = tt, (11.6) shows that ^1 and ti exp ( — 2xi) are of the same height. 9. m = 0.005p", arg m = 37r/2. For tt < arg m < Sir/2, li exp ( — 27rz) is higher than ^i and the paths spiral into t = 0 comiterclockwise. At arg m = 37r/2 the rate of spiralling is zero and we ha\'e the path shown in Fig. 11.5 (which is the path for arg m = x/2 rotated by 180 degrees). Here arg h = 57r/4. 10. m = 0.005p", arg m = — 7r/2. The paths for arg ?n equal to — 7r/2 Uf >t, ^\ t-PLANE Fig. 11.4 — Deformation of path of integration U into path of steepest descent through ti when m = 0.005 p^ e.xp (iV/2 + iO.Ob). DIFFRACTION OF RADIO WAVES KY A I'AHAHOLIC CVI.INDKH l!M and 37r/2 have the same shape and bolli are highest at the saddle ])()iiii whose argument is — 37r/4. In botli oases the contributions are the same and hence liie vahie of U^iz), for example, is the same for ai-g m = — 7r/2 as for 37r/2 (as it must be since our paraboHc cylinder functions are one- valued functions of ???)• Before lea\-ing the region around m = 0 we point out that when I tn/p' I « 1 the path of steepest descent through ^o is almost inde- pendent of arg m. Also, the curves of steepest descent for l/r(m) =~ f e't 2Tn Ju dl (11.8) PATHS SUPERPOSED BUT ARGUMENTS OF t DIFFER BY 277 ^x ^K 1 tr/|t,| 1 / > h> \ 1 / N <\ -2 -1 •*^^^m = 0.005/3 ^ e"* "a"- / J / = + 57r/4 t,/lt, m =0-005/5' Fig. 11.5 — Paths of steepest descent for \m\ = 0.005 p^, z = ?"2p- These curves are much the same as those in Fig. 11.3(b) except for the values of arg m. 492 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 behave in much the same way as those just described. The Hne nir = 0 divides the m-plane into two regions corresponding to different types of paths, and the negative real axis is a line of zeros corresponding to Re U(ii) - J{k exp {-2-Ki))] = 0 where ii = m is the saddle point. 11. I w I = 0.55p^ arg m = it. Fig. 11.6 (a). This value of m marks a change in the type of path. 12. \m\ = p^ arg m = ir. Fig. 11.6 (b). 13. 1 m I = p\ arg m = 7r/2, arg {ip~ - 2m,) = 37r/2. Fig. 11.6 (c). The complication of the paths in Fig. 11.6 (c) is due to the superposi- tion of two boundaries in the m-plane. Im f(to) = Im f(h) accounts for U//5 y^MF ___Jo^____Jil3 u,^^^"'"^*''^^-^ 1 -^ ^ (a) 1 fo 1 ARG (Z^ -2m) = 375/2 Fig. 11.6 — Paths of steepest descent for miscellaneous values of m with z - i^'^p. DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 493 the path running from ^o to ^i, and Im f(ti) = Im /[^o exp ( — 2iri)] for the one running from h to to exp ( — 27rt). The saddle points in order of their height are k, h, U exp ( — 27rt), to being the highest. 14. m = ip'/2. Fig. 11.6 (d). Here ^o = ^i and the dashed Hnes go into the saddle point at ^i exp ( — 2Tt). The paths of steepest descent change their directions upon passing through the saddle points. 12. ASYMPTOTIC EXPRESSIONS FOR Un{z), F„(2), Wn{z) The asymptotic expressions given here are for z = I'^p and z = i~^'^p [with i'' = exp (tV/4)] when n is not too close to z^/2. As mentioned earlier, there is a close relation between our results and those given by Schwid.* The main difference is that we regard n as variable and z as fixed while Schwid regards z as variable with n fixed. Another point of difference is that in place of the ?n = n + 1 which appears in our expressions for to and ti the quantity n + 3^ appears in Schwid's work. The quantity 2n + 1 appears to enter naturally when the asymptotic values are obtained from the differential equations. This is seen when the WKB method is applied to equation (9.7). By examining the paths of steepest descent shown in the figures of Section 11 we can determine the saddle points corresponding to Uniz), etc., (for z = i^ ^p) for various values of n. The contributions to the integral (10.1) from the saddle points ^o to /i were discussed in Section 10. The contribution from the saddle point ^i exp ( — 27ri) (which enters when z = t''p) is, from (11.6), exp {i2irm) times the contribution from ti. Although we shall be concerned mainly with asymptotic expressions for the parabolic cylinder functions themselves, expressions for their derivatives may be readily obtained. Thus U'n{z) = dUn{z)/dz has the asymptotic expression U'n{z) ~ 2/o [contribution of ^o to Un{z)] + 2tx [contribution of ti to Un{z)] (12.1) + 2/i [contribution of ^i exp ( — 27rz) to Un{z)] and similar expressions hold for V'n{z), W'n{z). These follow when Ave note that differentiation of the integrals (9.1), which define the functions, introduces a factor 2t into the integrand. Of course, if the path of in- tegration does not pass throught a particular saddle point, its contri- bution to (12.1) is zero. Upon replacing ^o and ti by their expressions * Reference 20, page 478. 494 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 (10.2) and subtracting the corresponding expression for zUn{z) we obtain 'Un{z) = U'n{z) -Z Un(z) ~ {z^ — 2riif''^ [{U contribution) — {t\ contribution) (12.2) — ihe''"' contribution)] where 'Un(z) is the function defined by (4.19). The same is true for 'Vn(z) and 'Wn(z). Consideration of the various paths of integration shown in Section 11 leads to the results shown in Table 12.1. The leading terms of the asymptotic expansions are listed for the various regions of the m-plane Table 12.1 — Leading Terms in the Asymptotic Expansions for Un{z), Vn(z), W„{z) WHEN Z = I'^p, p > 0 Region in j»-plane m = n+ 1 l^«(»"V) l\{i"'p) »'»(»■" V) la II lb III A, Ai - Ao (1 - i^")Ai (1 - i*'^)Ai Ao Ac Ao Ao - Ai -Ao - Ai -Ai -Ao - Ai + i4"Ai -Ao + i^"Ai shown in Fig. 11.2. If the next order terms are required, they may be obtained from (10.4) and (10.5). The notation used in Table 12.1 is as follows: z = i^'p, m = n -\- 1, i — exp (iV/2), - 7r/2 < arg yyi ^ 3x/2, - 7r/4 < arg ^o ^ 37r/4, - x/2 < arg Up- - 2m) ^ 37r/2, - 37r/4 < arg ^i S 57r/4, t, = [(''■' p + dp' - 2?ny'']/2, h = [t"p - Up' - 2mf']/2, Ao = [dip' - 2m)-"y2i7r"'] exp/(/o), ■ (i2.3) Ar = [tr-(ip' - 2mr'''/2r"'] expfiU), Hio) = zto-i- - - m log /o = 2(1 - log -^ - log j) + ^ P'c, f{h) = zti + - - m log /i = _M - log - - log - ) + I ph. DIFFRACTIOX OF RADIO W W KS HV A I'AUAHOLIC CYLIX DKK l!)') Sometimes it is helpful to use (27r) " exp loy; m 1/r , (m + 1 ~2 l/r(l + n/2) for -7r/l^ < arg m < 7r/2, (12.4) r (^^')/27r = r"-^r(-n/2)/2T, 7r/2 < arg m < 37r/2, where the last line is obtained by setting m exp ( — Tri) for /// in the second line. The asj^mptotic expansions for regions 76 and III may be obtained from those for la and II by using equations (9.11) and (9.13). However, the work is more difficult than one might suspect at first glance. Incidentally, the leading terms in the asymptotic expansions (9.15) and (9.17), which hold when p ^ x and ii remains fixed, may be ob- tained by considering the entries for la and Ih in Table 12.1. It is sometimes convenient to use the limiting forms of the asymptotic expressions when \ m\y> p .\n this case, for z = i ' p, 2'o -^ 2 + i(2m) 2'i -> 2 + 'i{2my'^ 1 2m + 0(>n-"'-) \-£\+00n-") (12.5) log (i/to — > + tT + iz{2m)' -1/2 2 + 6m + 0(m-^'0, where the upper signs hold when —ir/2 < arg m < 7r/2 and the lower ones when 7r/2 < arg m < 3x/2. Substituting (12.5) in (12.3), neglect- ing the higher order terms, and setting D 0-3/2 -1/2 B = 2 w exp I 1 -104')+. 72 ao = exp [-p{2m/iy^^], «! = exp [p{2m/iy'^] = 1/ao , (12.6) converts Table 12.1 into Table 12.2. In this table B, ao, ai, are defined by (12.6); m = n -{- 1; — 7r/2 496 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Table 12.2 — Leading Terms in the Asymptotic Expansions for Uniz), Vn{z), Wn{z) WHEN Z = I'^p AND [ 2n ] » p' Region in w-plane (m = n+\) yn(i"V) F,.(t>'V) W,.{i"'p) la II lb III v^Boii Bii-^ao + i"ai) (1 - i"')i-"Bao (1 - i^^)i'^Bao -i-^Boio — i~^Bao i^Bai B{i"ai — i~"ao) B{i~^ao — ?"ai) — i'^Bai Bii^^ao — i~"ao - i"ai) B(i^"ao — i^ai) < arg m ^ 37r/2; and in regions la and Ih arg m is approximately — 7r/2 and 3x/2, respectively. Gamma functions may be introduced into the expression for B with the help of (12.4). It may be verified that the functions do not change, except for neghgible terms, in crossing over the boundary from la to lb (ao and ai are interchanged and B is changed by the factor exp { — m-wi)). Since the zeros of our functions, regarded as functions of ?i, occur (asymptotically) when the contributions from two saddle points cancel each other, we may look at Table 12.1 and pick out regions which may possibly contain zeros. Thus, Ao may equal Ai along the line | Ao | = I Ai I, i.e. very nearly Re\j{h) - j{t^] = 0, in the m-plane. These Hues were discussed in Item 7 of Section 10 and are plotted on the auxiliary w- plane in Fig. 10.3 When plotted on the ??i -plane the lines appear as shown in Fig. 12.1 The condition Re\j{h) - fih exp (-2x2))] = 0 gives the line I Aq I ?i^ I i*"Ai I for some of the zeros of Wn{i'^p)- yn (Z) Fig. 12.1 — When Un(z), V„{z), and Wniz) are regarded as functions of n their zeros lie on the lines indicated when z = i^'^p. The three branches coming out from 7n = ?pV2 are lines along which \ Ao\ ~ \ Ai\ and the branch for IF,, (2) coming down from m = 0 is a line along which | Ao 1 = 1 i*"Ai | where Ao and Ai appear in Table 12.1. DIFFRACTION OF RADIO WAVES BY A PARACOLIC CYI.IXDKR 407 The location of the zeros far out on the lines of Fig. 12.1 may be ob- tained by writing the appropriate expressions of Table 12.2 as propor- tional to B times a cosine or sine. Examination of the trigonometrical terms shows that Unii^'^p) has zeros at n ;^ 2A; + 1+ i^'^k^'^/r, 7„(i'"p) has zeros at 7i ;^ -2/c + i"%k"^/r, (12.7) TF„(i"V) has zeros at n ^ -2A; + {'"^k'^^Tr, where fc is a large positive integer. Of course, Un(z) also is zero when n is a negative integer. So far we have been dealing with z = I'^p. Now we consider the case z = i~^''p. Asymptotic expressions which hold when z = i'^'^p may be obtained from Table 12.1 by using the relations (9.11) between functions of z and of its complex conjugate z*. Thus, for example, V a+ib{i~^'" p) is equal to the complex conjugate of W a~ib{i''^ p) ■ These relations, and relations such as [^0 for z = i p, 71 = a — ib] * = U for z = i p, n == a -\- ib [f(to) for z = i''p, 71 = a — lb] * = f(to) for z = -T^^'p, 7i = a -\- ib (12.8) have been used in constructing Tables 12.3 and 12.4 from Tables 12.1 and 12.2 The interchange of V„{z) and Wn{z) should be noted. The rriL m -PLANE ~~'i'.\ . I'b la 'm = n+i m' ^^"» \"~~--.^ n' Un_(_Z). \_ "n 0 / I'b I'd EI' Wn (Z) _,^- n' m' m = -L/?2/2 ^'^^ZEROS OF Un (Z) U' Fig. 12.2 — Regions in the comple.x m-plane corresponding to different asymp- totic expressions when 2 = z~"V- The lines on which the zeros of the various functions lie are shown bj' the dashed lines. The corresponding information for z = i^'-p is shown on Figs. 11.2 and 12.1. 498 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Table 12.3 — Leading Terms in the Asymptotic Expansions for Un(z), Vn(z), W^(Z) WHEN Z = l^"'p, p > 0 Region in »»-pIane m = n + 1 f^»(t-"^p) K„(x-"V) fr„(t-i'2p) la' II' lb' III' ^A[ A.I — Ao (1 - i-'-)A[ (1 - i-*^)A[ Ao A\ —a: -A', - a - ?■-"") A.' -A„' + i-'^A[ Ao — Ai regions in the m = n + 1 plane corresponding to the different asymptotic expressions are shown in Fig. 12.2. The boundaries are simply those of Fig. 11.2 reflected in the real /M-axis. The lines of zeros are also shown in Fig. 12.2, and are reflections of those of Fig. 12.1 except for the inter- change of Vn{z) and Wn(z). Table 12.3 may also be constructed by returning to the paths of in- tegration shown in Section 11. It may be shown that corresponding to every path of steepest descent for z = i^'^p, n = rii there is another path, obtained from the first by reflection in the real /-axis, which gives the path of steepest descent iov z = i p, n = Wi The notation used in Table 12.3 is as follows: z = i~^''^p, m = n + 1, i = exp (iV/2), -37r/2 ^ arg m < x/2, -3x/4 ^ arg to < t/4, -37r/2 ^ arg (-zp' - 2m) < 7r/2, -57r/4 ^ arg ti < 37r/4, t, = [r^'> + i-ip' - 2m)"-]/2, k = [i'~p - i-ip' - 2m) ^'1/2, Ao = [k"\-ip - 2mT"'/{-2iir"')\ expfito), (12.9) A[ = [h"\-ip' - 2m)-^'V2x '1 expM), f{to) = zto -\- ^ - m log /„ = -( 1 - log - - log - m zh + m m 2 "2 Sometimes it is helpful to use m log /i = -I 1 - log - - log - 1 + i ph . -1/2 {2^r exp 2 log 7n 1/r (j^^-^j = l/r(l + n/2) for -x/2 < arg m < t/2, (12.10) 27r = r+'r(-n/2)/2x, -37r/2 < arg m < - 7r/2. 1 — m DIFFRAT'TIOX OF RADIO WAVKS ^\\ A l'\i;\HOI,I(' CYLINDKH 499 Table 12.4 — Leading Terms in the Asymi'totic Expansions for Un(z), Vn(z), Wn{z) WHEN Z = iT^'-p AND I 2/1 I » p' Region in m-plane « = « + 1 I^n(.-"V) Vn(i-^'*p) Wniir^'^p) la' IV lb' iir i-'^B'a', B'{i"ao + i-^a'i) (1 - i-*'')i"B'ao (1 - i-*")i''B'ao B'(i"a!, — i-na'i) -i-^B'a'i B'ii-^"a'o - i"ao B'(i-^"a'o - i-^a',) — i"B'a'o -i^B'a, i-"B'al B'{i~"a[ — i"a'o) The notation used in Table 12.4 is as follows: m = n -\- I, i = exp (i7r/2), —3ir/2 ^ arg in < 7r/2, nr n-zn —1/2 B = 2 IT exp m 1 - 1 og 2 V °2 a'a = exp [ — p(2im)^''"^], a[ = exp [p{2imf''^ = l/a'o, (12.11) B' may be expressed in terms of gamma functions with the help of (12.10.). Approximate expressions for the zeros are given by the complex conjugates of (12.7). For example, if fc be a large positive integer such that 2/v » p", the zeros of Wn{i~^'~p) are at n = n(k) where i^"ai — exp (iTk) and }i(k) ■2k -\- i ■'4p/i-'''/7r — 47pV7r'. (12.12) Here the approximation has been carried out one step further than in (12.7) We also have for the quantities in (7.11) [aiF„(r^>)/a«] „=,.(.) ^ (-Y^'B'i[7r - p(i/ky% (12.13) [i""F„(r^^-p)/sin xn] „=„(,) ^ (-y-''2B'i. 13. asymptotic expressions for r„(2), ETC., WHEN fl IS NEAR Z-/2 The asymptotic expressions given in Section 12 fail when n is near z /2. Expressions for the parabolic cylinder functions which hold for this region have been given by Schwid.* More recent studies of this sort, based on differential equations, have been made by T. M. Cherry" and F. Tricomi.^" Their results suggest the possibility that our expressions * Reference 20, page 478. ^* Uniform Asymptotic Expansions, J. Lond. .Math. Soc, 24, pp. 121-130, 1949. Uniform Asymptotic Formulae for Fund ion.s with Transition Points, Am. Math. Soc. Trans.; 68, i)p. 224-257, 1950. " Equazioni Differenziali, Einaudi, Torino, pp. 301 308, 1948. 500 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 for the electromagnetic field which contain Airy integrals may be re- placed by more accurate, but also more complicated, expressions. In dealing with our functions we shall work with the integrals and our procedure is somewhat similar to that used by Rydbeck.* First however, we point out that when we WTite (as suggested by the work of Cherry and Tricomi) y = e-''"Tn{z), ax = z-{2n-\- lf\ 2(2n -f if "a = 1, (13.1) the differential equation (9.7) for the parabolic cylinder functions goes into Pi-xy = 2-''\2n + lV"x\j. (13.2) ax- The Airy integrals Ai{x) and Bi{:x) (and also Ai[x exp (± f27r/3)]) discussed later in this section are solutions of % - ^y = 0. (13.3) and therefore we expect that approximate solutions of (13.2) are given by, for example, y = CiAiixOil + 0(rr~")] (13.4) where the 0(rr^'^) term corresponds to the particular integral of (13.2) when the y on the right hand side is replaced by its approximate value Aiixi"). Here Ci is independent of x (or z) but may depend on n, and v may be 0 or ±4/3. Since the labor of computing Ci is considerable, we shall work out the approximations directly from the integrals. We shall consider the case z = i^''p, p > 0, first. When n + 1 = m = mo ^ ip^/2 the saddle points to and ti coincide at ti = i^'^p/2. Con- sequently only those portions of the paths of steepest descent which lie near ^2 are of importance. This is true even if m is not exactly equal to mo. We therefore regard f{t) = -t' -{- 2zt - m log t (13.5) in (10.1) as a function of the two variables t and m (linear in m) with z fixed at i^'^p. Expanding (13.5) about t ^ t-i , ni = m^ gives ^ '"wio Wo , mo (m — mo) , viq — — — log — - — log 2 2^2 2 ^ 2 J (13.6) - 4(/ - ttf/^z - 2(m - mo){t - /o)/^ + • • • /(o = I + * Page 87 of Reference 21 cited on page 478. DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 501 where we have used t, = z/2 = i"'p/2 = {m,/2f" (13.7) and have arranged the terms within the brackets so that they represent the first two terms in the expansion of m m . m , (27r)"^ / p/ W + 1 (13.8) about niQ . The paths of steepest descent in the ^plane when m = rrio are shown in Fig. 11.0(d). The three branches start out from I = t-i in the directions arg (t — /s) = 15°, 135°, and — 105°. In this section we take the paths of integration to be those of Fig. 11.6(d) even when m is not exactly- equal to 7??o . Since we are dealing with asymptotic expressions we may confine our attention to the region around t = (2 where the paths of integration are essentially straight lines [the contributions from to exp (— 2x1) are neghgible]. When (13.6) is set in the integral (l/27r0 / exp [/(/)] dt (13.9) we see that the initial directions of the branches are such as to make {( — Uf/z positive (arg z = 45°). Some study of (13.6) and of the Airy integrals we intend to use suggests that we change the variable of inte- gration from t to s and introduce the parameter h where ^-to = s(z/4:y'\ b = (m - mo)(2/zy" = (m - m,)/m,"\ (13.10) This and (13.6) converts the integral (9.1) for Vn{i'^p) into y.(,"V) = ^^/^V^,. f i(2.)-r(^i )•'-«><■="" (13.11) exp [ — bs — s' /3 + • • ■] ds. When we use the Airy integral defined by .4/(0;) = T M cos {xt + t^/Z) dt Jo = {i"/2Tr) f exp [-r^"xs - sV3] ds, Jooexp (i27r/3) (13.12) we obtain / /A\113 22/2 r, T/ r -1/2 N U/4j e Zt -f.Ws V„{l p) '^ ; --TT- Trr, Al{bl ) i(2ry-r['^Y « ' '' (13.13) 502 THE BELL SYSTEM TECHXICAL JOURNAL, MARCH 1954 In order to obtain expressions corresponding to (13.13) for Un(z), T^n(^) we examine Fig. 11.6(d). We have already seen that the limits of in- tegration for s, in the integral (13.11) for Vn(i^'^p), are [co exp (t27r/3), oo] . In the same way it follows that the limits for Unii^'^p) and Wn(i^'^p) are [=» exp (— t27r/3), cc exp (z27r/3)] and [ CO , CO exp (— t27r/3)] , respectively. When we take s' = s exp (+ i2Tr/3) as new variables of integration (with the upper sign for Un(z) and the lower one for Wn (z)), the integrals corresponding to (13.11) go into Airy integrals. We can write our results for z = i^'^p, when n is close to ip/2, as follows : Vn{i"p) ~ Ci~"'Ai{U^"), (13.14) Wn{l"p) ~ Ci'"Ai{hl-"'), where c = (p/4)^^^(27r)''V'''^^Vr(^^^) , h = i2/pr\m - ipV2)z-''\ i = exp {iTr/2), m = n -\- 1. (13.15) The asymptotic expansions whose leading terms are given b}^ (13.14) may be obtained by the method used by F. W. J. Olver"^ to study Bessel functions. Ai{;x) and its derivative have been tabulated for positive and negative values of x* Here we shall use the definitions and results as set forth in Reference 11. These tables and (13.14) enable us to obtain values of Un{i^''p) along the rays in the //(-plane defined by arg {m — ip'/2) = tt/G and — 57r/6. Along the tt/G ray hi"'^ is negative. Since the tables show that the zeros of Ai{x) occur when x is negative, it follows that the zeros of Un{i'~p) occur on the tt/G ray. In the same way it is seen that the zeros of Vn{i''p) and Wn{i''p) occur on the 57r/6 and the — 7r/2 rays, respectively. This agrees with Fig. 12.1. The Airy integral defined by * Reference 11, page 424. 28 Some New Asymptotic Expansions for Bessel Functions of Large Orders, Proc. Cambridge Phil. Soc, 48, pp. 414-427, 1952. Bi DIFFHACTIOX OF RADIO WAVP:S BY A I'AHAHOLIC CYLINDKR 503 ,— 2/3 r ^oocxp (t2?r/3) !t/3) 3 — 2/3 p |.oocxp(t2?r ^TT [_ J« fxp (— i2« •'« exp (— ''- TT Jo )r/3)J exp [-I'xs - s73] ds (13.16) exp ,3 -~ + xt\ + sin f ^^ + a;/ dl is also tabulated in Reference 11 where it is shown that Ai{xi"') = f%ii{x) - Wi{x)]/2, Ai{xi^*") = r'"[Ai(x) + iBi(x)]/2. (13.17) With the help of these relations we may evaluate the expressions (13.14) for Un{i 'p), etc., on any one of the six rays arg (m - ip-/2) = ±57r/6, ±/7r/2, ±t/6. When 6 is a general complex number the expressions (13.14) may be evaluated with the help of the modified Hankel functions hi(a), h2(a) tabulated in Reference 27 for complex values of a. The relation needed is Ai(a) =l-h,(,-a) -^±Ji,{-a), k = (12)'"{-"\ When I arg a | < tt we have the asymptotic expansion Ai(«) ~2-V"''V' (exp [- (2/3)a'^'1)(l - 5/48a:''' + • • and when ] arg (— a) \ < 27r/3 we have (13.18) ) (13.19) -1/2 \-l/4 Ai(a) - TT-^'i- a)-"* shi [(2/3)(- aY" + 7r/4] . (13.20) Both of these expansions follow from the discussion of the asymptotic behavior of hi{a) and h2{a) given by W. H. Furry and H. A. Arnold.^^ Asymptotic expressions for Unii'^'^p), ■ • • valid when n is near —ip/2 may be obtained by applying the relations Un{z*) = [Un*(z)]* ■ ■ ■ given by (9.11) to the expressions (13.14) for Un(t''p), • • • : (13.21) Un(f -%) /^■^ a -'"Ai{h' r'% F„(r -'%) ^ C'l -"'Ai{h' n, TF„(^" -'%) ^ a "'Aiih'i -"\ "Tables of the Modified II:iids;ci Funcfioiis (jf Order One Third and of Their Derivatives, Harvard Univ. Press, 1945. 504 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 where h' = (2/pY\m + ip'/2)i"\ ^^^-^^^ i = exp (^7^/2), m = w + 1. In (13.14) hi"^ = -h and in (13.21) bT'" = -h' since Ai(a) is a single-valued function of a. It is interesting to note that the factor i^^^ in the expression for U„{i^'^p) gives the direction of that one of the three paths of steepest descent (in the ^plane) which is not traversed in getting Un{t'^p). The same sort of thing is true for the remaining expressions in (13.14) and (13.21). The functions 'Un(z) = exp (z'/2)d[Un(z) exp (- z'/2)]/dz, defined by (4.19), may be computed from (13.21) when z = C^'^p. We need the relations d/dz = I'^d/dp and m = -tpV2 + h'{p''/2if\ (13.23) dV/dp = i2/Z)(2ipy'\i - m/p) = i(2ip)"' - 25V3p, which follow from the definition of 6'. When the differentiations are carried out we obtain 'UniiT'^p) - (2py''C'r"'Ai'(b'i'-"'), 'VniiT'^p) ^ i2py"C'i"'Ai'(h'i''"), (13.24) 'Wr.{t~"'p) ^ {2pf"C'i'"'Ai'{h'i-^"), In these expressions the prime on the Airy integral denotes its deriva- tive: Ai'{a) = dAi{a)/da. (13.25) Abstracts of Bell System Technical Pa])ers Not Published in this Journal AiKENS, A. J./ and C S. Thaeler.- Control of Noise and Crosstalk on Nl Carrier Systems, A.I.E.E. Trans., Commun. A: Electronics, 9, pp. 605-Gll, Nov., 1953. Benedict, T. S.^ Microwave Observation of the Collision Frequency of Holes in Germanium, Letter to the Editor, Phys. Rev., 91, pp. 1565-1566, Sept. 15, 1953. Bexxett, "W.i Telephone System Applications of Recorded Machine Announce- ments, Elec. Eng., 72, pp. 975-980, Nov., 1953. Applications of voice-recording equipment discussed in some detail can be divided into four general groups: Announcements made directly to and providing a ser\'ice to subscribers, such as weather forecasts and the time of day; announcements to assist subscribers in connection with telephone service, that is, intercept announcements when an individual calls a vacant or disconnected terminal, or emergency announcements if an unusual condition prevents normal service; announcements to expedite service and assist operators in completing calls, including completion of calls from a dial to a non-dial phone, and advising operators of the time delay for completing long distance calls; and si)ecialized announcement or record- ing services, such as price quotation and ticket reservation. * Certain of these papers arc available as Bell System Monofjr.'iphs and may be obtained on re(iuest to the puhHcation Department, IJcll Tch>])lu)n(' Laboratories, Inc., 463 West Street, New York 14, N. Y. For ])apers avaihible in this form, the monograph number is given in parentheses following the date of pubHcation, and this number should be given in all recjuests. ' Bell Telephone Laboratories. 2 American Telephone and Telegraph Company. 505 500 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 Briggs, H. B.,1 and R. C. Fletcher.^ Absorption of Infrared Light by Free Carriers in Germanium, Phys. Rev., 91, pp. 1342-1346, Sept. 15, 1953. The absorption of infrared light associated with the presence of free carriers in germanium has been measured by injecting these carriers across a p-n junction at i-oom temperature. The absorption is found to be proportional to the concentration of carriers. The absorption as a function of wave- length shows the same rather sharp maxima previousl}' ol^served in normal p-type germanium. These l)ands are found to change with temi)eratuie. An explanation of this absorption is offei-ed in terms of a degenerate energy band scheme. Briggs, H. B., see M. Tanenbaum. Carlitz, L.,1 and J. Riordan.^ Congruences for Eulerian Numbers, Duke Math. J., 20, pp. 339-343, Sept., 1953. Clark, M. A., see H. C. Montgomery. Crabtree, J.,1 and B. S. Biggs. ^ Cracking of Stressed Rubber by Free Radicals, Letter to the Editor, J. Polymer Sci., 11, pp. 280-281, Sept., 1953. Dickinson, F. R., see L. H. Morris. Felch, E. P.,^ and J. L. Potter.^ Preliminary Development of a Magnettor Current Standard, A.LE.E. Trans., Commun. c^' Elec, 9, pp. 524-531, Nov., 1953. In the wartime development of the air-liorne magnetometer, a method of detecting extremely small changes in magnitudes of magnetic fields was developed. The principle involved was the use of a second-harmonic type of magnetic modulator now known as a magnettor. Tliis instrument can detect changes in magnetic fields in the order of 10"^ oersted. A study was made at Rutgers University under the sponsorship of Bell Telei)lK)ne Labo- ratories to detei'mine the feasibility of obtaining a standard of curi'ent using the magnettor ])rinciple. Fletcher, R. C, see H. B. Briggs. 1 Bell Telephone Laboratories, Inc. ABSTRACTS OF TIXIINRAL ARTICLES 507 GoERTZ, M., see H. J. Williams. Gray, M. C.^ Legendre Functions of Fractional Order, Quart. Appl. Math., 11, pp. 311-318, Oct., 1953. Grisdale, R. O.^ Formation of Black Carbon, J. Appl. Phys., 24, pp. 1082-1091, Sept., 1953. Electron microscopic evidence is presented in support of the hj^pothesis that black carbon resulting from pyrolysis of gaseous hydrocarbons is produced through the intermediate formation of droplets of complex hy- drocarbons. Electron diffraction studies further confirm the hj-pothesis if, as has been found for joarticles of carbon blacks, the droplets consist in l)art of gra])hitic nuclei arranged with their basal ])lanes tangential to the droplet surface. The carbonization of small sohd splierules of highly cross- linked oi-ganic ])olymers is described, and it is sliown that the morpholog}' of the carl)onization products is wholly analogous to those for pyrolytic carbon and carbon blacks. It is suggested, therefore, that the formation of carbon by the carbonization of solids and by deposition from the gas phase occurs through similar mechanisms and that the two processes are simply two extremes in an infinite series of ])rocesses which are all fundamentally ahke. Grisdale, R. O.^ Properties of Carbon Contacts, J. Appl. Phys., 24, pp. 1288-1296, Oct., 1953. Microphone carbon has been ])roduced l)y deijosition of ])yi()l\-tic cai'bon films over the surfaces of small sphei'ules of silica. The ]iro])erties of con- tacts between these spheiules are shown to be dependent on the structure and geometr}' of the carbon surface as determined by election diffi'actif)n and microscopic .studies. The gra])hite-like civstallites in ])yi(jlytic carbon surfaces are more or less ])referentially oriented with their basal jilanes parallel to the surface, and the contact piopcrties depend systematicalh' on the degree of orientation. This is explained in terms of the anisotropy in properties of these ciystallites which are closely approximated by those of single crystal giaphite which were determined. The contact resistance and its tem])erature coefficient and the "bui'uing voltage" for carbon con- tacts are exijlicable on this basis. Howevei', the microphonic sensitivity of carbon contacts is indei)en(I('iit of the suifacc sti-uctiiie and dc])ends only on the surface geometr^^ ' Bell Telephone Laboratories, Inc. 508 the bell system technical journal, march 1954 Harris, C. M.^ Speech Synthesizer, Acoust. Soc. Am., J., 25, pp. 970-975, Sept., 1953. "Standardized speech" constructed from building blocks called speech modules has been described; it was sjmthesized by piecmg together bits of magnetic tape containing recorded speech sounds. An electromagnetic device, a "speech module synthesizer," is described here which performs the synthesis automatically. When buttons on a kej'board are pressed, a sequence of corresponding speech modules are automatically recorded on tape exactly in tandem. The modules are selected from a group "stored" on a rotating magnetic drum. The pressing of a button causes an electrical signal corresponding to a module to be reproduced — the electrical switch- ing is so arranged that onlj' one complete module is reproduced for a single button-pressing. This electrical signal is amplified, biased, and then fed into a constantly rotating head which makes contact with stationary mag- netic tape and records the signal on it. A 10-kc signal superposed on each stored speech module controls an electromagnetic clutch which (a) measures the length of the recording accurately', and (b) advances the tape at the completion of the recording by the correct amount so that the next record- ing forms a connected sequence with it. The same module may be used any number of times and in combination with different stored modules, thereb}- introducing wider experimental control in standardized speech studies. The principle of this type of device could be applied to other classes of problems involving communication of information, as the conversion into speech of typing or of electronicallj'-red printed matter. Harris, C. M.^ Study of the Building Blocks in Speech, Acoust. Soc. Am., J., 25, pp. 962-969, Sept., 1953. Identification of the information-bearing elements of speech is important in applj'ing recent tl linking on information theory to speech communica- tion. One way to study this pi'oblem is to select groups of building blocks and use them to form standardized speech which then ma}- be evaluated; a method having the advantage of simplicit}' is described. Individual re- cordings of the building lilocks were made on magnetic tape and then var- ious pieces of tape were joined together to form words. Experiments indica- ted that speech based upon one building block for each vowel and consonant not only sounds unnatural but is mostly unintelligible because the influences on vowel and consonants aie niissing which ordinarih* occur between ad- jacent speech sounds. To synthesize speech with reasonable naturalness, the influence factor should be included. Here these influences can be ap- ])roxi mated l)y em])loying moi-e than one l)uilding block to represent each linguistic element and by selecting these blocks properly, taking into account the spectral characteristics of adjacent sounds so as to approximate the ^ Bell Telephone Laboratories, Inc. ABSTRACTS OF TECHNICAL ARTICLES 509 time i)attein of the formaiit structure oocuiriug in ordinary speech. Tliere is no a prion method of deteiinining how many l)uil(hn<>; l)locl W ^ Xm -> P 0 — INTRINSIC < 3 X 10'3 D ,0'6_|o17 EMITTER B AS E DEPLETION LAYER COLLECTOR (a) STEP-BASE p-n-L-p lO'S-10'9 10'8-io'5 EMITTER COLLECTOR (b) STEP (alloy) p-n-p ,10'''-10'8 COLLECTOR (c) GROWN (graded) p-n-p Fig. 2 — Impurity den.sit}' profiles. 522 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 The high donor concentration in the base region leads to low ohmic base resistance (r^') and fixes the position of the base face of the deple- tion layer. In the depletion layer, the concentration of impurities is so low that the field region (space-charge layer) extends from the n-type base to the p-type collector at low voltages. Depletion Layer The properties of the depletion layer which are important at high frequencies are the capacitance across it (Ce) and the carrier transit time through it {tc). These are determined primarily by the impurity density, the thickness of the region, and the base-to-collector voltage. Potential and field distributions in the depletion layer for both small and FIELD DISTRIBUTION POTENTIAL DISTRIBUTION DEPLETION REGION (a) Nd=Na (b) NdNa Fig. 3 — Field and potential distributions in depletion region of p-n-i-p transistor. P-N-I-P AND N-P-I-N JUNXTION TRANSISTOR TRIODES 523 typical applied voltages are shown in Fig. 3 for p-n-i-p structures in which the depletion layer contains no net impurities (a), a small acceptor dominance (b), and a small donor dominance (c). When collector voltage is increased from zero, the space charge layer thickens until it extends from base to collector. Further increase of voltage simply increases the field strength in the region, without significant further increase in its thickness. The capacitance initially changes inversely as the square root of collector potential, but becomes constant when depletion region thick- ness becomes constant. The time required for holes to drift from base to collector decreases with increase of depletion region field until scattering- limited carrier velocities are reached (about 5 X 10 cm/sec for holes, at 10,000 volts/cm). 1" It should be noted that normal operation does not occur until the depletion layer extends from base to collector (particu- larly if the depletion region is slightly n-type so that effective base thickness is large at low collector voltages, see Fig. 3(c)). The breakdown voltage of the collector is very high,* since the field strength in the deple- tion region is relatively uniform by comparison with that in older types of units, the region is wide, and strong fields are required to produce carrier multipHcation. Base Region Base region design seeks the conflicting objectives of short diffusion transit time, requiring a thin region, and low ohmic base resistance, requiring a thick region. In practice, the region is made as thin as feasible, but of low resistivity material, and base contact geometry is chosen to minimize the ohmic resistance. In the p-n-i-p, very low base resistivity is practical, because the collector breakdown potential is fixed by the thickness of the intrinsic depletion layer rather than by the baise re- sistivity as in fused junction p-n-p's. The large donor density in the base region together with the very high frequencies of operation make the emitter depletion layer ca- pacitance (Cxe) both larger and more important than in previous tran- sistors. In order to reduce this capacitance, the emitter junction area is made small, thus leading to emitter current densities of 1 to 100 amperes/cm . In general, as the dc current density is incrcuised, the minimum dc collector voltage must also be increased in order to preserve * An avalanche mechanism similar to a Townsend discharge in gases is now believed responsible for reverse voltage breakdown in junction structures. See Reference 3. 524 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 emission-limited current flow. Insufficient voltage may result in space- charge limited operation. ' ° Three structures which may be used to obtain low ohmic base re- sistance are sho^Mi in Fig. 4. Obviously, the base contact ring may be placed arbitrarily close to the emitter, as in Fig. 4(a), so that the base resistance is that of the region beneath the emitter. Since this is some- what difficult, the ring may be placed at a distance from the emitter, and the emitter imbedded in the base n-region as in Fig. 4(b), reducing the resistance between the emitter periphery and the base ring at only a small cost in alpha cutoff frequency. In addition, as shown in Fig. 4(c), the n-region used may be of graded resistivity such as results from im- purity diffusion from the surface. The large impurity concentration at the surface minimizes both edge emission and radial base resistance. (a) CLOSE-SPACED RING (b) IMBEDDED EMITTER (C) IMBEDDED EMITTER - DIFFUSED SURFACE LAYER Fig. 4 — Low-base resistance structures. r-X-I-1' AND X-P-I-\ JUN'CTION TRANSISTOR TKIODES 525 These advantages are, however, balanced in part ])y an increase in the (Miiitter depletion rojiion capacitance associated with the low resistivity base material. DESICX TIIKORY General The piiiicipal objectives in the initial i)-n-i-p desif>;ii liave been high alpha cntoff l're(inency, low collector capacitance, and low ohmic base resistance. The eqnivalent circnit employed is shown in Fig. 5. The output and feedback admittances which are important in earlier jnnc- Ypo — qle kT inh 2 Fig. 5 — Equivalent circuit of the p-u-i-p transistor. tion triodes are omitted, since the space charge layer widening factor (Hi" or Hec , ■ — — F7-) is very small. ' ^Fhe transfer admittance is shown qw dVc as a current generator (aie) with cutoff fre(|uency (| a' | '^3 db down) of /a because this gives explicit recognition to base region diffusion transit time Tb and allows it to be combined with space charge layer transit time Tc . Km) tier Region Design Emitter region acceptor concentration should be very lai'ge (10 — 10 atoms/cc) in order to keep the injection ratio y close to unity at both low and high frequencies.^ At low frequencies, y is determined by emitter resistivity and carrier life path or diffusion length, base resistivity and width, as 1 7 = To = 1 + (TbW O'eLne 526 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 At high frequencies, 7 is determined by the ratio of acceptor density in the emitter to donor density in the base as 1 7hf = " 1 4- ^ /I Obviously, since the effective donor density in the base must be large to give low ohmic base resistance, the effective acceptor density in the emitter must be even larger if high frequency 7 is to be close to unity. Base Region Design Base region thickness, w, and the diffusion constant. Dp , determine the diffusion transit time for holes from injection by the emitter to collection by the field of the depletion layer. For circular electrodes, which are useful, easily made, and easily ana- lyzed, the ohmic base resistance for the active region of the base between emitter and collector depends on base resistivity, pb, and base thickness as follows : ' = ^^ = 1 (2) If w is made small, n' can be reduced only by making No large. Although large reductions in Vb can be made, increasing Nd is ultimately a self- defeating procedure for several reasons: as No is increased both Dp and the electron mobility, iin, decrease, thus increasing hole transit time and also partially off-setting the reduction in ri by Nd • In addition, the capacitance of the emitter depletion region varies approximately as Nd''^, thus diverting more ac emitter current from hole injection. This capacitance is where Ve is the average electrostatic potential across the emitter de- pletion layer. Equations (1) to (3) show the conflicts which necessarily arise in base region design for very high frequencies. The limiting design combines very small w, large Nd , small emitter area A « , and relatively P-N-I-P AND X-P-I-N JUNCTION TKANSISTOU TIUODES 527 large dc emitter current /« so that the minority carrier emitter admit- tance ijee is at least of the order of magnitude of jwCxe • Total emitter admittance is |^(l+jW)'«coth Vee + JCoCtc = (1 + J(^t)w' 2-|l/2 1/2 coth + M\e (4) Depletion Layer Design As mentioned previously, the most important characteristics of the depletion layer in the p-n-i-p are the transit time for holes, Tc , and the capacitance, Crc • The minimum voltage for normal operation, Fmin , and maximum or breakdown voltage, Fmax , are also significant. The minimum voltage for "normal" operation is reached when the electric field between the n-type base and p-type collector is strong enough so that the holes drift at their limiting velocity of 5 X 10^ cm/sec* The collector to base voltage required for normal operation is the product of the minimum field strength for the limiting velocity and the thickness of the depletion layer and is given by F:ni„ = 10,000 X,n (5) in which Xm is depletion layer thickness in cm. The maxunum field ob- tainable before reverse voltage breakdown is not known exactly, but is in practice near 100,000 volts/ cm, so that F:„ax ^ 100,000 x„ . (6) Depletion layer capacitance is nearly independent of collector voltage in normal operation and is inversely proportional to layer thickness. Cro = "^ (7) Xm Transit time for holes increases directly with layer thickness, however, being Since increase of Tc decreases the alpha cutoff frequency /„ , the choice * At lower field strengths, the transit time for holes is longer, giving a lower alpha cutoff frequency. The "normal" is the best, rather than the only possible, operating condition. 528 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 of Xm is a design balance between Cc and fa , with any desire for low voltage operation weighting the scales toward smaller Xm . As collector voltage and therefore field strength is reduced below that required for normal operation, transit time is increased because of the reduced drift velocity. In addition, the holes in transit interact with the ac field of the layer, thus increasing the output conductance Qcc ■ Further, a larger density of holes in the layer is required to carry the same cur- rent, disturbing the field distribution. If the voltage is reduced greatly, space-charge limited emission may occur, ' producing much longer effective transit times. If output voltage is reduced sufficiently, the collector field will not extend all the way from base to collector. If the layer is someAvhat n-type, the field region collapses toward the p-region of collector. If it is some- what p-type, the field collapses toward the n-region of the base. The latter arrangement has the advantage that /„ is less drasticall}^ reduced. Further, in normal operation, the negatively charged acceptor atoms of a slightly p-type layer will neutralize the charge of the holes in transit, thus making the field more nearly constant from collector to base. The effects of low voltage on the collector field distribution are indicated ap- proximately by the dashed lines of Fig. 4.* Collector Region Design Acceptor concentration in the collector should be large for several reasons. This gives a low collector body resistance, which virtually eliminates internal series loading of the collector, and it aids operation by fixing the position of the collector edge of the depletion layer. The advantages obtained may be seen by considering a unit in which the collector body is made somewhat p-type and a collector contact is at- tached at some distance from the depletion layer. If 10 ohm-cm p-ma- terial is used for the collector body and a collector contact fastened 2.5 mils from the collector resistance of 250-500 ohms will result. In addition, because of the weak drift field at the collector edge of the depletion layer, the hole transit time is about twice that for a true p-n-i-p. Alpha Cutoff Frequency A current transmission cutoff frequency /„ for the p-n-i-p is given approximately byf * The field distributions occurring in an intrinsic depletion layer at low field strengths have been discussed in Reference 11. t It is assumed that alpha is given by a = a o(l + jf/fa)- Equation (9) repre- sents the phase of this expression quite well, but the amplitude rather poorly. P-N-I-P AND N-P-I-N JUNCTIOX THAXSISTOU TRIODKS 529 U = ., ( 1 u-,. (9) Equation (9) implies (correctly) that the delay time for total current passing through the depletion layer is about one-half the transit time for the carriers. This results from the induction of charge on the base and collector electrodes by the carriers in transit. If ^ = core is carrier transit angle and Jc = e'" is the conduction current of holes entering the depletion layer from the base, the total current entering the depletion laver from the base can be shown to be which reduces for small ^ to It may be noted that the total current / of eciuation (3.6-2), when written in the form Jmax Z 0 in which 6 is the phase shift of the total current ^\dth respect to the conduction current entering from the base, is approxi- mately 0.973 Z -22.5° for ^ = 45°, 0.901 Z -45° for ip = 90°, and 0.636 Z -90°for 0.96, fa ^^^ 25 mcps, Tb c^ 60 ohms, and Cc ^^^ 1-8 mmf. These values agree quite well with those expected from the resistivities and layer thicknesses employed. The unit oscillated at 95 mcps with Vc = —30, Ig = 1.0 ma. Connected in a common emitter \ddeo amplifier working from a 75-ohm generator im- pedance into a load resistance of 2,150 ohms shunted by 5 mmf of ca- pacitance, this unit produced a power gain of 23 db at 500 kc, falling to 20 db at 3 mcps and 15 db at 10 mcps.* In an uncompensated common emitter tuned circuit, this unit gave 20.5 db at 10 mcps with 3 mcps band- width between the three db points.* It has been operated with a collec- tor voltage of —90 volts. SUMMARY The designed elimination of donors and acceptors from a thick col- lector depletion layer introduces a new design variable in junction tran- sistor triodes. The new structure (p-n-i-p or n-p-i-n) is believed capable of development into the microwave frequency range. Several factors which were of second order importance in p-n-p and n-p-n units such as emitter depletion layer capacitance and collector transit times become significant in limiting ultimate performance. The thick depletion layer permits operation at higher voltages than were previously possible in any but low frec^uency units. Moderately good results have been obtained already. Units having 10 mil emitter diameter, 15 mil collector diameter have produced stable gains without compensation of 20.5 db at 10 mcps and have oscillated at 95 mcps. The junction transistor now promises to be a serious competitor to high vacuum triodes over a much larger range of frequencies and power levels than before. ACKNOWLEDGMENTS J. A. Morton and R. M. Rj^der have strongly supported and en- couraged this work. J. W. Peterson and W. C. Hittinger have collaborated in and contributed to the experimental studies. The models constructed and tested are the products of the persistent efforts and many useful suggestions of J. A. Wenger, J. McGlasson, and L. P. Meola. Many others, particularly those engaged in semiconductor materials research * These measurements were made by L. G. Schimpf. P-N-I-P AND N-l'-I-X JUXCTIOX TUAXSlSTOli THIODES 533 and development, have also assisted us. Discussions with colleagues have been most hel[)ful in preparation of this report. REFERENCES 1. J. S. Saby, Fused Impurity ])-ii-]) .Juiiclion Tiaiisistors, I.H.lv I'roc, 40, ])|). 1358-1360, Nov., H)52. 2. R. L. Wallace, Jr., L. G. Schimpf and E. Dicktcn, A Junction Transistor Tetrode for Iligh-Freciuency Use, I.R.K. Proc, 40, pp. 1395-1400, Nov., 1952. 3. K. G. McKay, and K. B. McAfee, Electron Multiplication in Silicon and Ger- manium, Phys. Rev., 91, pp. 1079-1084, Sept. 1, 1953. 4. W. Shockley, and R. C. Prim, Space-Charge Limited Emission in Semicon- ductors, Phys. Rev., 90, pp. 753-758, June 1, 1953. 5. G. C. Dacey, Space-Charge Limited Hole Current in Germanium, Phys. Rev. 90, pp. 759-763, June 1, 1953. 6. J. ^L Early, Effects of Space-Charge Laj'er Widening in Junction Transistors, I.R.E. Proc. 40, pp. 1401-1406, Nov., 1953. 7. J. M. Early, Design Theory of Junction Transistors, B. S. T. J., 32, i)p. 1271- 1312, Nov., 1953. 8. W. Shockley, M. Sparks and G. K. Teal, The ])-n Junction Transistors, Phys. Rev., 83, p. 151, July, 1951. See also Reference 7. Nov. 1953, op cit. 9. M. B. Prince, Drift Mobilities in Semiconductors. I. Germanium. Phys. Rev., 92, pp. 681-687, Nov. 1, 1953. 10. E. J. Ryder, Mobilities of Holes and Electrons in High Electric Fields, Phys. Rev., 90, p. 766, June, 1953. 11. R. C. Prim, D. C. Field in a Swept Intrinsic Semiconductor, B. S. T. J., 32, pp. 665-694, May, 1953. 12. W. E. Bradley, et al. The Surface Barrier Transistor, I.R.E. Proc, 41, pp. 1702-1720, Dec, 1953. 13. C. W. Mueller and J. I. Pankove, A p-n-p Alloy Triode Transistor for Radio Frequencj' Amplification, RCA Review, 14, pp. 586-598, December, 1953. Arcing of Electrical Contacts in Telephone Switching Circuits Part III — Discharge Phenomena on Break of Inductive Circuits By M. M. ATALLA (Manuscript received November 16, 1953) This is a presentation of a study of the discharge phenomena occurring befiveen contacts on break of an inductive load. The main objectives are: (1) to forward some detailed explanations of the main components of a break transient in terms of basic conduction and emission processes, and (2) to establish the conditions that determine the nature of the transients. The study covered the following: (1) occurrence of interrupted and steady arcs, {2) initiation of reversed arcs in one breakdown, (3) arc initiation under dynamic conditions, (4) initiation and maintenance of glow dis- charge, and (5) glow-arc transitions. INTRODUCTION An important phase in the study of discharge phenomena between contacts is that involving the break of an inductive circuit. A typical switching circuit in its simplest form consists of a battery in series with a coil (electro-magnet), a cable or lead and a pair of contacts. Coils now in use may have inductances of the order of tens of henries and may store as much energy as 10 ergs. On break of the circuit an appreciable portion of this energy may be dissipated between the contacts through a steady arc, a series of interrupted arcs, a glow discharge or any of their combina- tions. In most cases, the energies involved are too high to provide satisfactory contact hfe from the standpoint of electrical erosion. The discharge transients obtained are usually complex in nature.' A close examination of these transients reveals a great deal of rather curious effects that have not been previously considered in detail. This is a presentation of a recent study of the break transient with the primary objective of furnishing some explanation of the more pertinent phenomena involved in terms of the basic concepts of surface emission and gas conduction. 535 536 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 NOTATION a Arc radius or equivalent characteristic length of cross section c Local capacitance at the contacts e Electron charge la Current densit.y in the arc ith Thermionic emission current density ing Normal glow current density tag Abnormal glow current density (^^sLimit) Limiting glow current density preceding glow-arc Limit transi- tion k Boltzman constant I Local inductance at the contacts m Alass of contact metal atom n Number of consecutive arcs in otic breakdown r Resistance of the local contact circuitry s Separation between the contacts / Time ich Charging time between breakdowns tdei Deionization time following an arc tg Glow duration Us Velocity of contact separation Uch Charging velociy defined as s/tch Uat Velocity of the metal atoms V Arc voltage Vn Residual voltage at the contacts following a breakdown of n- consecutive arcs z Impedance {l/cf^ A Constant in the thermionic equation Aa x\rea of arc spot C Circuit capacitance E Battery voltage F Field strength / Current Ig Current in a glow discharge Im Minimum arcing current Jo Liitial closed circuit current L Circuit inductance R Circuit resistance T Absolute temperature T h Absolute boiling temperature ARCING OF ELECTRICAL CONTACTS IN TELEPHONE SWITCIITNC CIRCUITS 537 T'o Absolute initial temperature T^ \''oltage Vai Arc initiation ^•oltagc Vgi Glow initiation voltage Vg Voltage drop across the contacts wtih normal glow a Thermal diffusivity (f Work function oj Angular frequency (fc)~^ ' GENERAL A typical circuit consisting of a battery, a coil of an electro-magnet, a cable or lead and a pair of contacts is shown in Fig. 1(a). Due to the usual magnetic core of the coil, this circuit presents some unnecessary complications in making interpretations of the observed contact phe- nomena. Since our main objective is an understanding of the basic phenomena occurring between the contacts, it appeared justifiable to restrict our work to circuits and circuit elements that lend themselves to simple treatment. Figure 1(b) shows the circuit used in most of this work. All coils used have air cores. When the contacts are closed, a steady state current lo = E/R is established in the circuit. At the first physical separation between the contacts, the circuit current will charge the capacitance C causing a ^'oltage rise at the contacts at an initial rate of lo/C. In the meantime, the separation between the contacts will increase. The first breakdown will occur when the voltage across the contacts first reaches or exceeds the arc initiation voltage corresponding to the separation attained, the atmosphere in^'ol^■ed and the contact surface condition. Fig. 2 represents diagrammatically the occurrence of the first discharge, abc is the arc initiation voltage versus separation line for a "normal" contact." The COIL CABLE E"^ JL _L CONTACT I (a, ^ ^ I L R I I ^WT^ "vVv' 1 ^WO"^ 1 E"^ ^C CONTACT ^ . O'' . . - . . - Fig. 1 — (a) Tj'pical relay circuit in i)racticc. (b) Linear circuit used in this stud3\ 538 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 CONTACT SEPARATION Fig. 2 — Initiation of the first arc between contacts on break of an inductive circuit. portion be corresponds to the sparking potentials in the atmosphere, ab corresponds to the range of small separations, of the order of or less than the mean free path of an electron in the atmosphere, where the arc is initiated by field emission through the influence of surface contamin- tions or films. As was shown in Reference 2, when the cathode surface was carefully cleaned, the constant field line was not obtained and the arc was initiated at the minimum sparking potential of the atmosphere. It occurred on the sides of the contacts along a path much longer than the minimum separation between the contacts.* Lines 0-1 and 0-2 represent the voltage rise at the contact with small and large shunt capacities. Points 1 and 2 are the respective first dis- charge points. In the first case, the arc is initiated at a smaller separation and higher field strength without direct influence of the atmosphere. In the second case the arc is initiated at a lower field strength at the spark potential of the atmosphere. f The first arc established may or may not be maintained depending on conditions that are discussed in the next Section. When an arc is inter- * With Pd contacts a gross field of 20 X 10^ volts/cm was reached between clean contacts without initiating an arc along the shortest gap. According to the Fowler- Nordheim equation a field of about 50 X 10^ volts/cm is required to give the neces- sary initiatory electrons. It is possible, however, that before such a high field is attained a metal bridge is pulled electrostatically' to short the gap. The electro- static stress is roughly given by 0.5 X 10"^^ F"^ Kg/cm^ where F is the field strength in volts/cm. At F = 50 X 10® volts/cm, the stress is 1250 Kg/cm^ which may exceed the yield stress for the contact metal. t The first arc may be initiated at an appreciably lower voltage than predicted by the above static consideration. The first break at the contacts usually follows the explosion of molten bridge drawn between the contacts. Thermionic emission can then furnish the initiatory electrons of the arc. This is only possible, however, if the voltage across the contacts exceeds the ionization potential of the metal atoms before e.xcessive cooling of the cathode has occurred. ARCING OF ELECTRICAL CONTACTS IN TELEPHONE SWITCHING CIRCUITS 539 rupted, it is followed by a recharging process to a new arc initiation voltage when a second arc is initiated. Under certain conditions, the second arc may be initiated at a lower voltages than the first arc due to residual effects of the first arc which may alter the conditions in the gap. This effect is discussed later. A transient on break with a series of interrupted arcs is shown in Fig. 3. The first arc was initiated at 230 volts and a gross field of 2.5 X 10^ volts/cm. All the following arcs were initiated at the spark breakdown potentials in air corresponding to the separations involved. Fig. 4 shows a transient where the arc w-as sustained with occasional interrup- tions. In addition to arcing, one may obtain glow discharge. Fig. 5 shows a transient where glow discharge predominates. Glow initiation and glow-arc transitions are discussed in a later Section. Fig. 6 shows the methods used for current and voltage measurements. As indicated, direct voltage measurements at the contacts were avoided to eliminate the unnecessary complications of the measuring circuit. INTERRUPTED ARCS Conditions for Obtaining Interrupted Arcs A breakdown from a voltage Vai into an arc corresponds to a rapid voltage drop at the contacts from Vai to the arc voltage v. For most prac- 50 -6 TIME,t, IN 10~^ SECONDS Fig. 3 — -Typical contact voltage transient on break of an inductive circuit. Pd contacts in atmospheric air, E = 50 volts, L = 0.2 henry, R = 950 ohms and C = 510 X 10^'- farad. Velocity of contact separation = 40 cms/sec. 540 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 o o > o < h- z o o 300 TIME,t, IN 10"^ SECONDS Fig. 4 — Contact voltage transient with sustained arc on break of an inductive circuit. Pd contacts in atmospheric air, E = 50 volts, L = 0.025 henry, R = 115 ohms, C = 20 X 10~'^ farad. Velocity of contact separation 40 cms/sec. tical purposes one may neglect the voltage drop time which is the initia- tive period of the arc. For the circuit in Fig. lb, the current through the arc is the summation of the main circuit current and the transient current from the l-c circuit. The transient current is (Vai — y)(T)^'%in ^'{Icf^. Fig. 7, (a) and (b), represent diagrammatically the voltage and current transients for lumped and distributed circuits. In both cases the arc is 300 lU < O > o z o o 500 TIME,t, IN 10"^ SECONDS Fig. 5 — Contact voltage transient with glow discharge on break of an inductive circuit. Pd contacts in atmospheric air, E = bO volts, 700 ohms relay coil and C = 200 X 100~i2 farad. Velocit^y of contact separation = 40 cms/sec. ARCING OF ELECTRICAL CONTACTS IN TELEPHONE SWITrillNG CIRCUITS 541 tcrmiuated when the curiciit drops to the minimum arcing current /, It is e^•i(^eIlt that the condition for ohtainin<>; an interi'uptcvl arc is: /o — {Vai — v) < In (1) It may t)c pointed out tiiat surface contamination, such as organic acti\ation, tends to decrease both /^ and V aC ■ According to equation 1, one may conchide that contact surface contaminations usually tend to cause a transition from an interrupted arc transient to a steady arc transient. The latter is usually associated with appreciably higher (Miergy dissipation between the contacts and much lower contact life due to erosion. L R ^WT' WV V-OSCILLOSCOPE yV\ \ :iox C Fig. 6 — Voltage and current measuring circuit. Residual* Voltage Folloiving an Interrupted Arc At the interruption of the first arc the voltage at the contact is v, the arc voltage, and the voltage at the capacitor C, Figure lb, is vi which is usually negative. If the local contact circuit is non-dissipative, the residual voltage is vi = 2v — Vai . For a dissipative circuit with a resistance r corresponding to the frequencies involved: h = v - (Vai - v)e-''"''-'"'' (2)t for an oscillating circuit, as is usually the case, where z = {l/Cf^. The capacitor C at v\ will then recharge the local contact capacity c, c « C, through the inductance I. If the voltage attained at the contacts is sufficient and the conditions in the gap and at the contact surface are favorable, a reversed arc may be re-initiated, as previously discussed. This process may repeat several times and the residual voltage y„ will change sign and decrease progressively. At the end of n arcs, it can be shown that the residual voltage Vn is given by: * The term "recovery has also been used in the literature, t Equation 2 and 3 are valid only for small values of r/z. These are ai)pro.\ima- tions of the more general expression given by Germer." 542 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 n=0 This equation indicates that Vn is negative for odd numbers of arcs and positive for even numbers of arcs.* If r/z is neglected, Equation 3 is re- duced to i-iyvn = Vai - 2vn (3a) For Vai = 300 volts and y = 14 volts, the residual voltages following the first four arcs are respectively —272, +244, —216 and +188. These Vai Io+iVa-L-Vj(^]'''^ Io+(VaL-vJ Zc t :o I Im ^1 ARC TIME (b) Fig. 7 — -Mechanism of interruption of an arc. (a) Lumped circuit elements (b) Distributed elements. values are numerically higher than measurements due to neglecting the term r/z. For the circuits used in our experiments r/z ranged be- tween 0.1 and 0.5 and as many as 4 or 5 consecutive arcs have been ob- tained in one breakdown. Figure 8 shows a transient with both positive and negative residual voltages corresponding to even and odd number of arcs respectively.! * Except when Vn is not too much higher than the arc voltage v. t The following alternative explanation for the occurrence of high positive residual voltage was considered : the first arc may be extinguished by the formation of a metal bridge due to the arc-. This may occur before the capacitor C has at- tained a negative voltage. This possibility, however, was eliminated. From the measured residual voltages the energies in the arcs were calculated. The heights of the bridges produced were computed (reference 2) and were found to be too small compared with the contact separations. ARCING OF ELECTRICAL CONTACTS IN TELEPHONE SWITCHING CIRCUITS 543 LLI (i) < O > I- o < f- z o u 300 0 50 TlMEjt, IN 10"^ SECONDS Fig. 8 — Contact voltage transient with interrupted arcs on break of an in- ductive circuit. Pd contacts in atmospheric air, E = 50 volts, L = 0.010 henry, R = 40 ohms and C = 900 X lO""^^ farad. Velocity of contact separation = 40 cms/sec. Initiation of Reversed Arcs in One Discharge In one breakdown from a voltage Vai it is commonly observed that a succession of reversed arcs may be obtained. It was shown in equation 3 that the residual condenser voltage Vn progressively decreases, numeri- cally, with the number of arcs n. Following the interruption of the first arc, the condenser voltage is — | Vi | and the contact voltage is +v, the arc voltage. The capacity C \vill then recharge the local capacitance at the contact through a small lead inductance /. If the circuit resistance is neg- lected, the maximum voltage the contact will acquire is — (2 | Di | + v). If this equals or exceeds the original arc initiation voltage Vai , a, second arc is obtained. For illustration, consider a breakdown initiated at Vat = 300 volts and y = 14 volts. From Equation 3, Vn was calculated for the first four arcs at r/z = 0.0 and 0.2 The corresponding maximum contact voltages acquired after each arc were also calculated and the results are given in Table I. For r/z = 0, column 3, one may obtain, ac- cording to this simple circuit consideration, more than 4 arcs, actually 5. For r/z — 0.2, which is a reasonable practical value, only 2 arcs may be obtained, column 5, since following the second arc the maximum voltage attained at the contacts is only 256 volts which is less than the initial arc initiation voltage. It is possible in some cases, however, to obtain a few more arcs than 544 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 Table I — Initiation of Reversed Arcs by Overcharging of Contact Capacitance (Calculated) r 1 = 0 r z 0.2 (1) (2) (3) (4) (5) Arc No. Vn Max. Cont. Voltage Vn Max. Cont. Voltage 1 -272 -558 -195 -404 2 +244 +502 +121 +256 3 -216 -446 -60 -134 4 + 188 +390 +22 +58 Vai = 300 volts, i; = 14 volts. predicted alcove. These additional arcs have appeared to be initiated at lower voltages than the first arc. This is undoubtedly due to the residual surface and gap effects of the previous arc* These are discussed in the following section. Arc Initiation Under Dynamic Conditions — Introduction In Reference 2 measurements have been presented of the arc initiation voltage between contacts at different separations and surface conditions. These tests are "static" in the sense of allowing enough time to elapse between two arcs to obtain a complete reconditioning of the contact surfaces and gap. With successive arcing, as obtained on break of an inductive circuit or during one breakdown, it was observed that the arc may be initiated at appreciably lower voltages compared with static test results. One arc may enhance the initiation of a shortly following arc possibly through the effects of: residual ions in the gap or on a cathode surface film, residual metal atoms in the gap and residual thermionic emission. Exactly how each of these effects can enhance the initiation of the arc can be determined only after an understanding of the mechanisms of initiation of the first arc, its maintenance and its termination. It is in order at this point to present a sketchy outline of some plausible mecha- nisms which are largely of speculative nature. This discussion is also limited to short arcs initiated and maintained with no direct influence of the surrounding atmosphere. * The additional arcs observed may be partially accounted for by a considera- tion of the actual value of the arc terminating current which was taken as zero in the above calculations. AKCIXG OF ELECTKICAL CONTACTS IN TELEPIIONK SWITCHING CIRCUITS 545 a. Arc Iniiiaiion (1) The first initiatory electrons are produced by field emission. The necessary field strength is largely dependent on cathode surface con- ditions. It is highest for perfectly clean cathode surfaces and appreciably lower in the presence of cathode surface films. ' ' This is probably due to lower work functions or due to the presence of positive ions on a cathode film causing local field intensification.^ (2) The field emission electrons will travel to the anode where, to qualify for setting the second step in arc initiation, should be able to produce, through evaporation, some anode metal atoms* or possibly atoms of an adsorbed gas or a surface film. (3) The potential drop across the contacts should exceed the ionizing potential of the evaporated atoms to allow ionization by electron collision. (4) Ions produced, on approaching the cathode, will cause local fields high enough to produce electron avalanches. (5) the above processes will rapidly multiply leading to the establishment of an arc. h. The Established Arc One main characteristic of the short arc is its very high cathode current density.! This high emission rate indicates that the short arc is not only initiated hut also maintained by field emission.X^ Since the total voltage drop across the arc is only of the order of 10 volts, the cathode drop thickness should be very small compared to the total arc length. The cathode drop is followed by the arc column or plasma which is a high conduction medium with equal electron and ion densities, a small potential drop and a relatively high neutral atom density. To maintain the arc: (1) enough metal atoms should be produced to maintain the necessary ionization medium, (2) ions lost by collection at the cathode, by recombination and by lateral diffusion should be replaced by an * The arc may also he initiated without the assistance of tlie anode atoms or ions*. The field emission current density at the cathode in this case, was found to reach a critical value before the arc is initiated. It is thought' that at this current density the emission spot can attain its melting point through resistive heating. The cathode in this case will furnish the necessary metal atoms for the subsequent steps of arc initiation. t Recent measurements by the author obtained from arc tracks on Pd contacts produced by short duration conslanl current arcs indicated current densities as high as 50 X 10^ amp/cm^. X Paper by P. Kislink to be published in the Journal of Applied Physics. § Recent analytic considerations, to be j)ublished by the author, indicate that in such arcs the current density should be dependent on the work function of the cathode material as well as on the product "pressure X sejjaration" in the arc. For instance, for work functions of 2 and 5 volts, our calculations show that the minimum current densities are, respectively, 5 X 10^ and 1.4 X 10' amp/cm^. 546 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 equal number of ions obtained by electron-atom collision in the arc column. c. Arc Termination In general, the arc may be terminated by disturbing one or more of the steady state conditions discussed above. For instance, if the potential across the contacts is decreased to or below the ionization potential of the metal atoms, the necessary ionization process will stop and a de- ficiency of ions in the arc will result. The negative space charge will immediately upset the arc potential distribution interrupting the high electron emission, etc. The arc is also interrupted when the current drops to the minimum arcing current value. This is a well established experi- mental characteristic of the arc which has yet to be explained in terms of the more basic concepts. It is thought, however, that a decreasing arc current decreases the pressure and the atom density in the arc col- umn. It is possible that when a limiting current is reached the ionization rate becomes too small to maintain the condition of equal space charges in the arc column. One should expect, accordingly, that providing the contact surfaces* with a film of low evaporation energy should furnish a more adequate supply of atoms to the arc which may then be main- tained at lower currents. This is in accordance with observations ob- tained for active contacts. Arc Initiation Under Dynamic Conditions; Observations on Break It appeared of interest to examine the relations between arc initiation voltage and contact separation during the break transient and compare them with measurements made under static coditions. In Fig. 3, the increase in arc initiation voltage with separation is in accordance with the static relation shown as a broken line. During the period 2-3, the breakdowns occurred along longer paths than the minimum contact separation and at the minimum value of the sparking potential. By measurement ^3 = 20 X 10~ sec, S3 = 8 X 10~ cm and ps = 0.61 mm Hg X cm. This is roughly the ps value at the minimum sparking poten- tial in air.^". By gradually decreasing the charging times of the transient, by adjust- ing circuit parameters, it was observed that a point was generally reached when a portion of the breakdowns was initiated at voltages well below the coresponding static initiation voltages. Fig. 9 illustrates this * The necessarj^ atoms may be obtained from either electrodes or both. Arc transfer observations generally indicate signs of evaporation from both electrodes. AKCING OF ELECTRICAL CONTACTS IN TELEPHONE SWITCHING CIRCUITS 547 300 UJ o > o < I- z o o 0 80 TIME,t, IN 10~6 SECONDS Fig. 9 — Lowering of arc initiation voltage under dynamic conditions. Tran- sient on break of Pd contacts in atmospheric air. E = 50 volts, L = 0.010 henry, R = 40 ohms and C = 270 X 10~'^ farad. Velocity of contact separation = 40 cms/sec. effect. In contrast to the static line, shown as a broken Une, the break- down potential shows little change with separation for a major part of the transient. Towards the end, it shows a gradual increase which in this particular case fails to reach the static line. Figure 10(b) is a plot of the ratio (Vai)dyn/(Vai)stat versus time along the transient. This phenomenon is attributed to residual effects in the contact gap or on the contact surfaces. In this section, are discussed the possibilities of the presence of residual ions, residuad atoms and residual thermionic emi.ssion. a. Deionization Time This is determined by calculating the transit lime of an ion across the contact gap under the applied field corresponding to the charging of the contact capacitance. For simplification, the initial motion of the ions and the initial field are neglected, the voltage rise is approximated by V/Vai = t/tch and the field is taken as V/s. 113 (4) 548 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 20.0 O 17.5 UJ 01 a. 15.0 UJ Q. 5 12.5 o ? 10.0 J 7.5 in II 5.0 I o ^ 2.5 0 • > • • • »f — • • • « (a) ^0.8 '0.7 0.6 0.5 0.4 1 1 1 1 I 1 ' • > (b) 10 20 30 40 50 60 70 80 Xl0"° TIME,!, IN SECONDS 10 20 30 40 50 60 70 80 XlO"6 Fig. 10 — Lowering of arc initiation voltage under dynamic conditions. Defining a charging velocity s/tch = Uch and a deionization velocity s/tdeio = Udeio and substituting in equation 4 gives Udeio — bm (4a) Following an arc, the contact voltage increases until a new breakdown occiu's at Vai . At this instant residual ions from the previous arc could be present in the gap only if Uch > Uddo , or if Uch > 6m j i'^y This is a convenient expression to apply to our measurements, Fig. 9. For any breakdown point on the transient Vai is measured and Uch is calculated from the corresponding circuit current, capacity C and con- tact separation. For illustration, for Pd contacts and Vai = 300 volts, equation 5 shows that for the presence of residual ions, the charging velocity Uch must be greater than 10 cms/ sec. For / = 0.3 amp. and C = 10^ farad, tch = VaiC / 1 = 10~ sec and for the presence of residual ions the separation between the contacts must be greater than 1.0 cm. This sep- aration is much greater than most separations involved in our field of study. In Fig. 10(b) are plotted the values of Uch during the transient. Uch reaches a maximum of about 1.8 X 10* cms/sec. This maximum occurs because Uch is proportional to si which is a product of two monotonic functions one increasing and the other decreasing. It is of interest to note that the decrease in Uch caused an increase in the ratio {Vai)dyn/{Vai)stat ■ * Deionization by recombination and lateral diffusion were neglected. ARCING OF ELIOCTIUCAL CONTACTS IN TELIOI'IIONIO SWITCHING CIRCUITS 549 From a oroup of transients similar to Fij^. 9, obtained at different con- ditions, the plot in Fig. 11 was made. It indicates that in general, the ratio (ra,)d2/n/(F„,).s/„( starts decreasing at about //,/, = 2 X 10'' cms/sec and at 2 X 10 the arc initiation voltage is only 50 per cent of the corres- ponding static value. As shown in the iiguiv a deionizing velocity of lO" cms/sec is just about two orders of mtigiiitude too high to account for this phenomcMion. It should be added, howi^ver, that while all the ions have cleai'ed the gap, it has been proposed" that the life time of an ion on a surface (ihn can be long enough to enhance the initiation of the next arc. If this mechanism is accepted, our data would indicate that the life time of the ions was only of the order of 10"'^ second. b. Residual Atonis After an arc, the contact gap contains some metal atoms e\'aporated from the electrodes by the arc. These atoms will clear the gap by travel- ling to and condensing on the electrodes and by lateral diffusion. A crude approximation is given here of the time of recollection of the atoms on the electrodes based on their initial momentum. One may visualize the arc spot on an electrode to have a temperature distribution extending from submelting temperatures to a range of boiling temperatures, corresponding to the arc pressures. The lowest temperature is probably the normal l)oiling temperature of the contact metal. At the termination of the arc, the metal atoms produced at the lowest l)oiling temperature are the slowest and last to recondense on the 1.0 0.9 ^ -£>■ Cr > 0. \ \ • 800x10 o 1000 -12 f \ \ \ \ □ 1400 A 2800 fl k a °i o \ > o a ;h • o *8 T= 2500° K^ N Udeio VaL=300 -/ -^^ UcH = S/tcH IN CM PER SEC Fig. 11 — Apparent relation l)et\veon arc initiation voltage and velocity of charging. E = 50 volts, L = 0.010 henry, A' = 40 ohms and C as indicated for Pd contacts in atmospheric air. 550 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 opposite electrode. An estimate of their velocity may be obtained by assuming thermal equilibrium to have preceded the arc extinction and by using the Maxwellian velocity distribution. The most probable velocity of the metal atoms at the boiling temperature 2' & is : Uat = I (6) \ m / Due to subsequent collisions of the atoms, the velocities thus obtained are probably too high. For Pd at Tt = 2500°K, Uat = 6.4 X 10* cms/sec. In Fig. 11 is plotted a portion of the velocity distribution at the above conditions. It appears that residual atoms can still be present in the gap at the initiation of the next arc. If it is assumed that the presence of Pd atoms in the gap is alone responsible for the lowering of the arc initiation voltage, one may conclude that the sparking potential in Pd vapor is lower than in air. No evidence, however, is available to support this. On the other hand, at least for contacts with gaps short enough to ex- clude the surrounding atmosphere, or for vacuum contacts in general, it is quite probable that the presence of metal atoms in the gap could enhance arc initiation. This, as pointed out previously, is because the arc cannot be initiated until atoms from the electrode surfaces are evaporated, by electron bombardment or otherwise, to be subsequently ionized. c. Cooling Time of The Arc Spot, Maintenance of Thermionic Emission At the interruption of the first arc, the arc spot initially at the boiling temperature of the metal, will start coohng mainly by conduction to the bulk of the surrounding metal. For a certain period, however, it will remain at temperatures high enough to furnish enough thermionically emitted electrons that may enhance the initiation of the following arc. Assuming the arc spot to be a hemisphere of radius "a" initially at a temperature Tb while the rest of the metal is at To, the temperature T at the center of the hemisphere is given by : {T - To)/{n - To) = ^^ I z'e-' dz (7) Numerically, ior T i = 2500°K and To = 300°K, T drops to 2400°K and to 1600°K at a/2 {at)"' - 2.0 and 1.2, respectively. It is evident that the cooling time is proportional to the area of the arc spot. If the current at which the arc is terminated is /„ and the arc current density is ia, the area of the arc spot is ^o = Im/ia and a = (Jm/T^iaf"- For ARCING OF ELECTRICAL CONTACTS IN TELEPHONE SWITCHING CIRCUITS 551 ia = 10^ amp/cm"^ and /„ = 0.5 amp one gets: T = 2400°K at t = 4 X 10~^ sec and T = 1600°K at / = 1.1 X 10"^ sec for Pd. The corresponding thermionic emission is obtained from ith = ATe 2- ~ kT _■> with .1 = 60 amp cm~" deg.~ and

.19 >23 >4.8 350 — — — 1.0 1.0 >.10 >13 >2.6 320 — — — 1.0 0.4 >.04 >5 >1.0 600 18,000 8 660 0 3.8 <.015 <2 <0.4 500 — — — 0.22 2.5 .19 24 4.8 450 — — — 0.44 1.9 .23 29 5.8 400 — — — 1.0 1.2 >.15 >19 >3.8 350 — — — 1.0 0.6 >.076 >10 >2.0 600 18,000 15 906 0.25 2.0 .23 29 5.8 550 — — — 0..30 1.7 .23 29 5.8 500 — — — 0.35 1.3 .20 26 5.2 450 — — ■ — 1.0 1.0 >.17 >22 >4.4 400 — — — 1.0 0.7 >.ll >14 >2.8 600 18,000 20 1050 0.40 1.5 .28 36 7.2 550 — ■ — — 0.40 1.2 .23 29 5.8 500 — — — 1.0 1.0 >.19 >24 >4.8 450 — • — — 1.0 0.8 >.14 >18 >3.8 * No glow was detected with a time resolution of 1 per cent of a half period 7r(LC)i/2. t Uninterrupted glow occupied the entire half period. t Obtained by dividing {Ig)max bj' the total cathode area. transient time, to a full transient time. By calculation, the corresjDonding limiting currents and limiting current densities were obtained, columns 7 and 8 respectively. The ratios of the limiting current densities to the normal glow current density are also given in column 9. They show that at the interruption of the glow discharge the current density was 5 to 7 times the normal glow current density. This indicates a transition from normal glow to abnormal glow before the final transition into an arc. One may, therefore, conclude that if glow discharge is obtained it starts as normal glow which may occupy only a small fraction of the cathode area. By increasing the current the cathode glow area expands at con- stant current density until it covers the entire cathode area. Further current increase leads to a transition into abnormal glow with higher current densities. Transition of the abnormal glow into an arc occur.s when the current density reaches a limiting value. This limiting ciu'rent density is extremely sensitive to surface contamination and generally AKCIXG OF ELECTIUCAL CONTACTS IN TKLErilONlO SWITCHING CIKCUITS 557 increases with surface cleaning.* For clean Pd contacts in atmospheric air an average limiting current density of 30 amps/cm , or about G times [ho normal glow current density, was obtauied. This sudden transition fiom the low current density glow to the very high current density arc represents a high rate of change in the emission process. With con- taminated contacts, this is probably due to the presence of low work function high emission spots on the cathode. These spots may be elimi- nated by proper cleaning thus allowing glow discharge to be maintained at higher current densities. The observed glow-arc transitions for clean contacts, consistently occurring at about 30 amps/cm for Pd, may still be attributed to the formation of a surface film on the cathode through a cathode-atmosphere reaction. f JMeasurcments have also indicated that under certain conditions, glow discharge cannot be obtained even at currents much below the limiting currents discussed above. It appears that there is a limiting rate of rise of ciu'rcnt with time above which glow discharge cannot be main- tained. In Table IV, column 6, the initial rates of current rise are given. In all cases where the rate of current rise was greater than about 3 X 10 amps/sec, lines 1, 2, 3 and 7, no glow was obtained. The experiment was repeated with two other cathode diameters of 0.2 and 0.05 cm. The limiting rates of rise obtained were approximately the same as given above, indicating that the limiting rate of current rise is independent of the cathode area. This seems reasonable since at the beginning of the transient the currents are very small and the emission area is only a ver}' small fraction of the cathode area. No detailed explanation, how- e\-er, can be furnished at this time as to why such a limit of the rate of current rise does exist. It is obvious, nevertheless, that while the rate of current rise can be increased without limit by manipulating the ciivuit parameters, the conduction mechanism in the contact gap, will, in gen- eral, have its own limitations as determined by the emission processes involved. ACKNOWLEDGMENT I am indebted to ]\Iiss R. E. Cox for assistance with many of the ex- periments and calculations reported here. * With a contaminatod cathode surfaco a transition into an arc may (jccur (hir- ing the noiDial gh)\v ]j(Mi()(l woU Ix'forc the curi'cnt is high enough to allow normal glow to cover the entire cathode surface. This is particularly true with larger cathod(; areas which are usually hai'd Xa clean satisfactorily 1)\' the above pro- cedure. t A recent unpublished study !)>■ V . K. Haworth has shown that in the absence of the usual surface contaminants, glow discharge is capable of activating pal- ladium and silver contacts through the formation of surface films. These surface reactions appear to be strongly clej)endent on the atmosphere. 558 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 BIBLIOGRAPHY 1. A. M. Curtis, Contact Phenomena in Telephone Switching Circuits, B. S. T. J., 19, p. 40, 1940. 2. M. M. Atalla, Arcing of Electrical Contacts in Telephone Switching Circuits — Part II, B. S. T. J., 32, pp. 1493-1506, Nov., 1953. 3. G. H. Pearson, Phys., 32, pp. 1493-1506, Rev., 56, p. 471, 1939. 4. L. H. Germer, Arcing of Electrical Contacts on Closure — Part I, J. Appl. Phj's., 22, p. 955, 1951. 5. M. M. Atalla, Arcing of Electrical Contacts in Telephone Switching Circuits — ■ Part I, B. S. T. J., 32, p. 1231, 1953. 6. F. E. Haworth, Experiments on the Initiation of Electric Arcs, Phj-s. Rev., 80, p. 223, 1950. 7. F. L. Jones, Initiation of Discharges at Electrical Contacts, Proc. Inst. Elec- trical Engineering I 124, 169, 1953. 8. W. P. Dyke, J. K. Trolan, E. E. Martin, and J. P. Barbour, The Field Emis- sion Initiated Vacuum Arc — I, Phys. Rev., 91, p. 1043, 1953. 9. W. W. Dolan, W. P. Dyke, and J. K. Trolan, The Field Emission Initiated Vacuum Arc — II, Pliys. Rev., 91, p. 1054, 1953. 10. J. J. Thomson and G. P. Thomson, Conduction of Electricity Through Gases, Vol. 2, p. 487. 11. H. S. Carslow, Introduction to the Mathematical Theory of the Conduction of Heat in Solids, 2nd Edition, p. 150, 1921. 12. S. Dushman, Rev. Mod. Phys. 12, p. 381, 1930. 13. F. E. Haworth, Electrode Reactions in Glow Discharge, JI. Appl. Phj-s., 22, p. 606, 1951. 14. L. H. Germer, Erosion of Electrical Contacts on Make, J. Appl. Phys., 20, pp. 1085-1109, 1949. Thickness Measurement and Control in the Manufacture of Polyethylene Cable Sheath By W. T. EPPLER (Manuscript received October 22, 1953) 'The manufacture of multiple sheath for Alpeth and Stalpeth cables re- quires the application of a sheath of polyethylene over a sheath of corrugated metal which is flooded with a rubber asphaltic compound. For high quality and minimum cost, this outer sheath must be of uniform thickness through- out its length. One of the problems in cable sheath manufacture is to maintain the concentricity and average thickness of the extruded polyethylene sheath to close limits during manufacture. This article reports on: (1 ) The applica- tion of a capacitance sensitive bridge to the measurement of the eccentricity and average thickness of the sheath on cables moving at speeds of 20 to 100 feet per mimde; {2) The method of thickness calibration; and (S) The use of the thickness measurements in maintaining the sheath concentricity and average thickness within close limits during the sheathing operation. mSTOEY In the manufacture of multiple sheath for Alpeth and Stalpeth cables, an outer sheath of polyethylene is applied. It is desirable for high quality and low cost to make this outer sheath of a uniform thickness throughout. The construction of these cables is shown in Fig. 1. In both designs, the outer sheath is polyethjdene extnided onto a corrugated metal under- sheath which has been flooded -with a i*ubber asphaltic compound. The extrusion art had been unable to obtain a high degree of control, primarily because measurements of the thickness could not be obtained until after the sheath was applied to the cable core. Eccentric sheath must have a greater average thickness than concentric sheath, if the thickness of the thin side is not to fall below a required minimum thick- ness. The s^Tnmetrical design of a tj'pical core tube and die for sheathing is shown in Fig. 2. Concentric set-up of these extrusion tools around the 559 560 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 cable core "will not produce concentric extruded sheath. This is caused by an unbalance in the plastic flow in the extruder. The flow makes a ninety degree turn from the extruder cylinder into the die head, and to reach the far side of the die, must flow around the core tube. The flow resistance also varies with changes in the temperature of the plastic and of the extruder screw speed. The core tube is fixed in position in the extruder head. The die is located around the core tube and can be moved in any direction eccentric to it. Fig. 3 shows a core tube and die mounted in the extruder head and indicates the location of the four die adjustmg screws by which move- ment of the die in relation to the fixed core tube is accomplished. The die must be located at some one eccentric position in relation to the core tube to compensate for the differences in flow resistances in the head. To set the die for concentric sheath and to adjust for specified thick- ness the ijrevailing practice of the cable art of measuring the wall thick- ness of a sample taken from the lead or finish ends of the sheathed cable Avas of necessity resorted to because it was the best technique available. The cutting of a ring of sheath and the micrometer gage are shoAMi in Fig. 4. These end samples only approximate sheath conditions because CONDUCTORS CORE WRAP — ALUMINUM SHEATH SOLDERED STEEL SHEATH- SEAM CEMENT FLOODING COMPOUND POLYETHYLENE JACKET ALPETH DESIGN STALPETH DESIGN Fig. 1 — (Left) Telephone exchange cable of Alpeth design; (right) Stalpeth design. POLVETIIYLEXE CABLE 8IIEATI1 THICKNESS 561 Pig. 2 — Typical core tulie and die. they are onl}- short pieces to represent cables up to a few thousands of feet in length. Sheath eccentricity is expressed as a percentage and is the difference between the thicknesses, of the thickest and the thinnest sides of a cross section, in relation to the specified Avail thickness expressed in mils. Control from end sampling resulted in most cables ha\'ing eccentricities of 30 per cent to 60 per cent. Also, it was difficult to keep the average thickness to within ±0.010 inch of the specified average thickness. The need for a better gaging method than end sampling, led to an investigation of detennining the wall thickness in terms of the capaci- tance that would be fomiod l)}^ the metal luidersheath and a probe sliding on the sheath svu-face. A test set as shoA\Ti in Fig. 5 was developed which I'osponds to changes 562 THE BELL SYSTEM TECHNICAL JOUKNAL, MAY 1954 tube and die assembled in extruder head and die adjusting in capacitance. The capacitance response is in turn calibrated in thou- sandths of an inch of sheath thickness. The electronic system of the set has been described in the Bell System Technical Journal previously.* It is practical from test set measurements to control the concentricity of Alpeth cable to within 35 per cent and Stalpeth to within 20 per cent. Average thicknesses within ±0.005 inch are maintained. Formerly, the safe practice was to use an excess of approximately 10 per cent over specified average in order to keep the thin side of eccentric sheath ^\'ithin the minimum spot limit. Control from test set measure- ments eliminated the necessity of using an excess of polyethylene because sheath of improved concentricity maintained close to the specified av- erage thickness does not vary below the specified minimum spot thick- ness. The quality of the sheath is improved because it is of consistently high dimensional unifonnity not previously obtainable. Also, concentric sheath has better flexing characteristics since eccentric sheath concen- trates the stresses of flexing in the thin side. * Continuous Incremental Thickness Measurements of Non-Conductive Cable Sheath, B. M. Wojciechowski, B.S.T.J., 33, pp. 353-368, Mar., 1954. POLYETHYLENE CABLIO SHEATH THICKNESS 563 Fig. 4 — (Left) Removing test strip from end of cable; (right) performing micrometer measurements on test strip. CALIBRATION OF THE TEST SET FOR SHEATH THICKNESS MEASUREMENTS Calibration of capacitance into thickness was difficult because the capacitance is not a simple function of polyethylene thickness. It de- pends also on the curvature of the sheath surface, the size and shape of the probe, the amount of flooding and the height and shape of the cor- rugated metal. For a given probe, it depends chiefly on the thickness, the flooding and the sheath curvature. The flooding sometimes varies from a thin film to an e.xcess that overfills the corrugations. The surface curvature is not unifonn because the soldering of the metal overlap of Stalpeth cable generally produces a flattened sector and the capstan at the soldering operation results in an elliptical shape. Changes in the sur- face curvature and in the amount of flooding can be compensating or cumulative in varying the capacitance. To deteiTiiine whether a correlation between jacket thickness and capacitance existed, extensive spot checks for three sizes of cable were made. Marked points on cable were measured for capacitance and then with a micrometer. A slight error can exist because the micrometer measurement is only one spot in the center of an area which is effective to capacitance. This condition is sho\\Ti by Fig. 6. Also, it is difficult to determme accurately the surface curvature associated with the capaci- tance measurement. The relation of thickness to capacitance conditions in the samples is 504 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 19.54 Fig. 5 — Capacitance test set and unit for tracking prol)c.s on cable surface. shown by Fig. 7. The sheath thickness is specified as the distance be- tween the outside surface of the sheath to the bottom of the corrugations formed into the polyethylene by the crests of the corregated metal sheath, as indicated by dimension 7\ The top sketch shows the normal amount of flood. The capacitance will be different in each of the three conditions of equal thickness shown. With excess flood, center sketch, the distance between plates is increased and the capacitance is decreased. POLYKTIIYLEXE CABLE SHEATH THICKNESS 565 PROBE TOP VIEW ^-ESTIMATED EFFECTIVE AREA PROJECTED SIZE-^^. ^ ^^ AVERAGED BY TEST SET '^(^^ SPOT MEASUREMENT ----3 BY MICROMETER ^ FRINGE LINKAGE ■■. \_ MAJOR LINKAGE -^ -- Fig. 6— Thickness measured hy direct calibration; spot hy micrometer- urea by capacitance, ictti, -FLOODING PLATE (ESTIMATED LOCATION OF EFFECTIVE RESULT OF CORRUGATION PLATE EXCESS FLOOD NO FLOOD POLYETHYLENE J'iR. 7— Iv|ii:il Iliickiicsses. (iillcrcnt capacitances. 56G THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 Insufficient flood, bottom sketch, alters the dielectric from polyethylene plus some flood, to all polyethylene. The capacitance is decreased. A typical plot of points and a calibration curve are shown in Fig. 8. Each of the three cable sizes measured revealed a wide band of plot points. In each curve the points were more dense toward the left side of 0.15 0.05 CAPACITANCE IN ///^ F 1.30 1.25 1.20 1.15 / f. C 7o . 7 o 5 0 CD 0 \J =o / 3 CD o / ° ° °M 3 O 0 0 c^ oo c o/ ^ °o o > / C 5/ 3 o o X< 0 °\ o: o o ]/o "^ 'o o 3 o t ° A '£ ° A yboo TOcn> (DO ° / ^ ( A / < 5 A ( y -40 -30 -20 20 30 40 -10 0 10 RECORDER SCALE Fig. 8 — - Measured points of sheath thickness versus recorder readings and developed calibration curve. POLYKTIIYLEMO CABLE SHEATH THICKNESS 567 the band, beoominp; progressively less to the right across the band. The majority of points to the extreme right were found to be cases of excess flood. Many of the points, near the extreme right had insufficient flood- ing. Points close to the curve had the flood just filling the corrugation \alleys. Other points consist of \'arions other amounts of flood and/or arc the result of deviation from correct surface cur\-ature. 0.24 0.23 0.22 0.21 0.20 0.19 a. 0.13 0.11 0.10 0.08 0.07 1 / FLAT SAMPLE I ) / t A / A / / A > / A A J A o A \ A f / r J. r r 0 k ■100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 U RECORDER SCALE *\ Fig. 9 — Calibration of flat sample thickness versus recorder scale. 568 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 Fig. 8 also shows that the greatest percentage of points are ^\'ithin a thickness range of approximately 0.010 inch. In moving downward from maximum thickness the concentration of measured thicknesses increases rapidly over approximately 0.003 inch and then becomes progressively less covering an additional 0.010 inch. The calibration curve was placed at about the location of maximum point concentration. By averaging the thickness indications along a short length of the cable, a measurement adjusted for the occasional extremes in flooding and surface variations is obtained. The accuracy for practical use is therefore within limits of d= 0.005 inch from the mean. Investigation was also made of flat samples of Pol^^ethylene placed upon a flat metal plate. Flat samples eliminate the ^'ariables introduced by the cable surface curvature, the corrugated metal undersheath and the flooding material. A plot of capacitance against thickness for fiat samples is shown in Fig. 9. Each point represents an individual molded flat sample. The majorit}- of points are within ±0.003 inch of the curve. The measurement of sufficient points to obtain cur\'es for the many cable diameters would invoh'e an impractical amount of work. The calibration curves for the three cable sizes and the curve for flat samples drawn to the same capacitance versus thickness scale ha^-e simi- lar form, but are displaced one from the other. The displacement of the calibration curves for cables of core diameters of 1.39 to 2.38 inches is shoAAii by Fig. 10. The displacement is approximately 1 meter division for a diameter change of 0.1 inch. Calibration curves for other cable diameters than the three measured were obtained by an approximation formula based on measurmg a few points from each sheath diameter to determine the displacements and slopes and multipljang the flat sample curve values by the displacement and slope correction factors. The curve for flat samples and the curve for 2.38 inch diameter cable plotted to the same scales is sho^^^l in Fig. 11. The two curves are suffi- ciently alike so that by multiplying the flat sample curve thickness values by a constant (Ki) obtamed from the ratio of the cable sheath thickness to the flat sample thickness at zero recorder scale, the amount of curva- ture of the resultant curve and the measured sheath curve are essentially the same, and they have the same thickness and capacitance values at zero recorder reading. A multiplier (Ko) can then be added to adjust the slope of the percentage curve to make it practicall}^ coincide with the sheath thickness curve. Actually, there is a slight difference between the cur^'ature of the flat sample curve and those of cable sheath. The amount of curvature increases as the cable diameter decreases. POLYETHYLENE CABLE SHE.VTH THICKNESS 569 \liiim mil 1 \m ' M/////J/ ' / i ' i //ft J W/M i/m COF ?E Dl 2. 2. AMETER A W///// / 30-.. ?0~ ~'> w///// 2.10-^ 2.00.. A m// /^ m / -1.90 ~^~-1.80 -^-1.70 ^-1.59* -^ ^M t~ mw ■--1.50 ---1.39* — mm MWa / j^M^ / A ^^^j / * MEASURED CURVES (OTHERS CALCULATED) //. m. ^^ .^^ m ^^ y ^ -50 -40 -30 -20 10 20 30 40 -10 0 RECORDER SCALE Fig. 10 — Calibration curves hy core diameters, thickness versus recorder scale. )70 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 CAPACITANCE N ///iF 1 45 1.40 1.35 1.30 1.25 1.20 1.15 1.10 1.05 1.0 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08 / f I 1 1 / 1 1 1 1 1 / / I / / f 1 1 1 t / / 1 1 1 / 1 ft ft / 1 FLAT SAMPL ACTUAL (Tp) '/ 2 38 // ACTUALi'/ ii LU I o 7 / // it If // // f/ Z LU z o I 1- / f // // t / / ^ (- < liJ X (0 / / // // // /y / • / / © CURVE OF FLAT SAMPLE THICKNESS TIMES K, K|= RATIO OF 2.38 THICKNESS TO FLAT SAMPLE THICKNESS AT ZERO RECORDER READING TpK, 0.07 0.06 /® (D Ka CHANGES SLOPE OF CURVE © FOR + SCALE K2 = T(AT +35 METER) -To TpK,(AT+35METER)-To FOR -SCALE K2 USE T AND TpK, AT -35 METER .X f^ y2.38 ACTUAL / AND /^T=(©-To)K2+To / 0.05 -20 -10 0 10 RECORDER SCALE Fig. 11 — Adjustment of flat sample ilirect calibration to obtain calibration of 2.38-inch diameter cable sheath. The result is the following approximation formula, from which the thickness calibration can be calculated within 0.001 inch with the error negligible over most of the working range. T = TrKiK^ where T = Thickness in thousandth's of an inch of polyethylene cable sheath. POLYETHYLENE CABLK SIIEATII THICKNESS 571 Tf = Thickness fo flat polyethylene sample at same recorder meter reading as for T. Ki = Ratio of actual cable sheath thickness to flat sample thick- ness at zero meter reading. 7v2 = Constant to change slope of TpKi curve. ■^ (at +35 meter) i 0 For + meter readings K2 = For — meter readings K2 J F-tVi(at + o5metor) — ■/ 0 J (at — 35 meter) -i 0 i FjK.l(at- 35 meter) — -/ 0 To = Thickness in thousandth's of an inch of cable sheath at zero meter reading. The A'l factor accounts for the dimensional differences between the capacitor formed by a flat thickness of polyethylene on a flat plate com- pared to the actual capacitor constmction of cable at zero meter. Both have the same capacitance of 1.20 uuF at zero meter reading. Ko ac- counts for changes resulting from the curved surfaces of cable. 7li and K2 are different for each cable diameter. Since zero meter is used as a reference point, the formula becomes: T = (TfK, - To) K. -f- To ACCURACY CHECK UNDER OPERATING CONDITIONS A check* was made of the accuracy of calibration and of the response under operating conditions of applying the sheath to the cable. The upper graph in Fig. 12 was obtained with the test set probe tracking at a cable sheathing speed of 50 feet per minute. The probe was shifted to different octant locations on the circumference for lengths of the cable as indicated on the graphs. The track of the probe was marked on the sheath surface and the sheath then removed, cleaned of flooding com- pound and the micrometer measurements of the thickness taken at sbc-inch intervals along the length. The lov»-er graph is a plot of the thick- ness obtained by micrometer. The ability of the test equipment to track and respond to the thickness variations is apparent from comparison of the two graphs. APPLICATION OF TEST EQUIPMENT FOR EXTRUSION CONTROL The test set is placed at some distance after the extruder to prevent the probe from marking the plastic polyethylene. The machinery of the * Test and measurements by courtesy of J. L. O'Toole, Bell Telephone Lab- oratories. 572 THE BELL SYSTEM TECHXICAL JOURXAL, MAY 1954 HONI lOO'O X SS3NM0IHi HDNI 1000 X SS3N^DIHi POLYETHVLKXK CABLE SHKATIU THICKNESS 573 sheathing hue is diagrammed in Fig. 13. At the top left is the supply reel of metal jacketed cable. The cable is pulled through the flood tank where the hot rubber asphaltic compound is flowed over the corrugated metal sheath. It then progresses through the die head of the extruder where the polyethylene sheath is extruded over the flooded metal sheath. The cable with plastic polyethylene then enters the cooling trough where it is cooled and solidified. At the exit of the cooling trough is an air blower for drying the water from the sheath surface. The test set is located after the dr3'cr. The next unit is the capstan which pulls the cable. At the final unit to the right, the sheathed cable is taken up on the shipping reel. A typical recorder graph taken along 360 feet of cable length with the sensing probe held at one location on the sheath circumference is shown in Fig. 14. With apparently stable conditions of extrusion the spot thick- ness indications will vary as much as plus or minus 0.010 inch while the lengthwise average remains stable as sho^\^l in Fig. 14. These fluctuations are sheath thickness variations \\'hich result from the complex interaction of the many sheathing line variables, but they may be increased or de- creased b}^ response to luieven flooding distribution and/or variations in surface curvature. However, it is practical to visually average this graph to within ±0.001 inch. For die adjustment, thickness measurements are obtained visually by estimating the average of the fluctuations of the recorder's visual indica- tor. Measurements are taken at quadrant locations corresponding to the locations of the four die adjusting screws. Opposite thicknesses give the amount of eccentricity. Die adjustments can be made accurately because the amount of eccentricity is known and the amount of die movement is governed b}- the adjusting screw pitch. Adjustment to specified average sheath thickness is made by averaging measurements at eight positions equally spaced around the sheath. In- creasing the speed of the cable in relation to the speed of extrusion in- creases the stretch of the polyethylene and decreases the average thickness. Decreasing the cable speed' increases the average thickness. APPLICATIOX OF TEST EQUIPMENT FOR SHEATH INSPECTION The thickness test provides an accurate gage for the inspection organi- zation to mea.sure compliance of the sheath to specified requirements. Inspection possibilities with the thickness test set are many and the problem becomes one of an economic procedure that will assure the re- quired quality. Continuous recording of the entire cable length is prac- tical but is unnecessary from a manufacturing viewpoint. Recorder chart speed is one half inch per minute and cable speeds are from 20 to 100 574 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 POLYETIIYLKNE CABLE SHEATH THICKNESS 575 4c, •^ 130 MO 150 160 170 180 190 200 Fig. 14 — Recorder graph of single octant variation and thickness scale in thousandths of an inch. feet per minute. It was found that the fluctuations or variation peaks of one Une along the cable length could be averaged from chart lengths of }/^ inch. Also that by taking measurements consecutively by octants around the circumference a practical measure of the entire circumference is obtained and is sufficient coverage to locate the minimum wall thick- ness. The graphs of Fig. 15 show typical inspection recordings of two cable lengths. Four thicknesses are specified for inspecting sheath, all of wliich are obtained from a graph of the consecutively recorded octants. These checks are: 1. The minimum spot thickness. 2. The average thickness lengthwise along the thinnest side. (Average of minimum octant.) 3. The average cross sectional thickness. (Average of octant averages). 4. The maximum difference between the lengthwise average of the thickest side (average of maximum octant) and the lengthwise average of the thinnest side (average of minimum octant). 576 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 Fig. 15 — Inspection graphs of two reel Uiiiu,tli.s — octant graphs witli (cli- mated octant averages — calculation of average thickness and eccentricity; location of specified thickness and actual thickness, in thousandths of an inch. roLVKTHYLKNK CABLK SIIKATII IIIICKNESS 577 The location of the four major thickness limits have been indicated below the test graphs. CONCLUSIONS This test equipment lias proved to be a practical means for the control of the concentricity and the average thickness of the polyethylene sheath on Alpeth and Stalpeth cables. It is accurate, reliable and of rigid con- struction suitable for continuous shop use. It measures the sheath wall thickness directly in thousandths of an inch both visually and as a re- corded graph and does so non-destructively as the sheath is applied. Concentricity is maintained within 35 per cent on Alpeth and within 20 per cent on Stalpeth cable. Average thickness is controlled to within ±0.005 inch of specified average thickness by the practice of visually averaghig graphs of about twenty-five feet of cable length. Polyethj'lene is conserved in two ways which reduce manufacturing costs. First, improved control pemaits operating at specified average thickness \\'ithout varying below minimum spot limit. Previously, an excess over specified average thickness was necessary to prevent the wider range of variation from going below the specified minimum spot thickness. Second, the sheath is of consistently unifoiTn dimensional ciuality not previously obtainable which made it practical to reduce the average wall thickness 11 per cent below previously specified thickness. ACKNOWLEDGMENT The writer wishes to express his appreciation of the co-operation of B. M. Wojciechowski of the Western Electric Company, who designed the capacitance test set and of Bell Telephone Laboratories cable en- gineers, in establishing the sheath requirements and for their encourage- ment in this project. Topics in Guided- Wave Pro2:>agation Through Gyromaguetic Media Part I — The Completely Filled Cylindrical Guide By H. SUHL and L. R. WALKER (IManuscrijit received January 26, 1954) The characteristic equation for the propagation constants of waves in a filled circular guide of arbitrary radius is written in terms of magnetizing fi£ld and a carrier density, which are shown essentially to determine the dielectric and permeability tensors for a gas discharge plasma and for a ferrite. The complex structure of the spectrum of propagation constants and its dependence upon radius and the two parameters are analyzed by a semi- graphical method, supplemented by exact for midae in special regions. Thus the course of individual modes may be charted with fair accuracy. 1. INTRODUCTION Any material medium which propagates electromagnetic disturbances possesses a local electric or magnetic structure and it is just the motion of the electric or magnetic carriers under the fields of the disturbance that determines how the propagation takes place. If a dc magnetic field be applied to the medium one may expect the local response to be altered and, consequently, to find changes in the character of the propagation. Gyromagnetic media are those for which such changes are sufficiently large to be experimentally significant. For plane waves and for optical frequencies the experimental effects and their explanation have been familiar for a great many years. The non-reciprocal rotation of the plane of polarization of light travelling parallel or antiparallel to an applied dc magnetic field, Avhich is known as the Faraday effect, is such a phenom- enon. So also is the fact that the medium becomes doubly refracting for arbitrary directions of propagation. Interest in gyromagnetic media at longer wavelengths first arose in connection with radio propagation in the ionosphere. The ionosphere is essentially an ionic cloud and the earth supplies a magnetic field, which, for the charge densities involved, is sufficient to produce a large effect 579 580 THE BELL SYSTEM TECHXICAL JOURNAL, MAY 1954 upon propagation. Here, as in the earlier optical cases, the disturbances considered are essentially plane wa\'es. In recent j^ears, with the ex- tensive development of microwave techniciues, two gyromagnetic media have been investigated using guided waves. One of these is the gas dis- charge plasma, an ionic medium like the ionosphere, in which, however, the charge density may be varied over wide ranges in a controllable manner. The magnitude of the effects observed in such ionic media are governed by the relation of the applied frequenc}^ to the cyclotron fre- quency of the ions in the dc magnetic field. Goldstein and his associates^ have studied the propagation of waves in a cylindrical wa\'eguide within which a discharge is supported and to which a longitudinal magnetic field is applied. Among many effects which they have obser-\-ed is a large Faraday rotation. The other medium being actively investigated is the low-loss ferro- magnetic medium, as exemplified by the ferrites. In this case the pe- culiarities of the medium have their origin in the precession of the magnetization of the ferrite about the applied field. This precession takes place with a frequency dependent upon the applied field strength and large changes in the nature of the propagation occur when the fre- quency of the r.f. applied field approaches this. Polder'- worked out the effective properties of such a medium for plane waves and Hogan^ has made various experimental studies of the propagation in C3dindrical guides containing ferrite. Here, again, Faraday rotation and other non-reciprocal effects have been observed. In this paper a variety of topics associated with the theory of guided waves in gyromagnetic media is considered, with the main emphasis laid on the ferrites. The exposition does not attempt to be systematic. A'ery few problems in this field admit of a thorough analytic treatment and, frequently, the more closely allied they are to the practical uses of ferrites in microwave devices the more fragmentary is the anal3^sis. On the other hand since the problems can always be formulated it is always possible in specific cases to resort to a purely numerical solution. The problems considered here all arise in the effort to analyze the operation of various devices and different idealizations are utilized in particular cases. In Part I the general properties of gyromagnetic media are discussed and the connection between tRe phenomenological constants of the medium and the underlying molecular model is derived for the ferrite and for the plasma. The assumptions necessary to render the ferrite problem tractable are discussed at some length. Maxwell's equations are written down for a general gyromagnetic medium and some of the salient features of their solution are noted. The propagation of circularly po- Gl^IDKD-WAVK l'U( )1'A(; ATIO.N TllUOlHiU CY UOMAGNETIC MEDIA 581 larizcd waves in circularly cylindrical ^iiide filled with ferrite or plasma is then considered. The characteristic equation connecting frequency and propagation constant is first derived. For the purpose of obtaining results which can be compared with experiment, a specific molecular model is chosen for the ferrite. In this way the ferrite itself is specified by a single parameter, its saturation magnetization, and its state by an- other, namely the applied field. The object of the calculation, then, is to find, for a given ferrite and a gi\''en guide radius, the mode spectrum of the Avave guide and the A'ariation of propagation constant with mag- netic field. This is done by a semi-graphical method supplemented by exact analj'tic formulae in the neighborhood of certain critical points, series expansions in certain regions and some numerical computations in others. A sketch of a similar procedure applicable to the plasma is given. It should be pointed out that the filled cylindrical waveguide is not a topic of the highest importance from the technical standpoint. It is for this reason that no effort is made here to obtain a comprehensive body of exact numerical information about the modes. One wishes, on the other hand, to exploit the simplifying features of the problem (as con- trasted with the more useful case of a cylinder of ferrite not filling the guide) so that the discussion may be exhaustive, in the sense that the complete mode spectrum is exhibited. In Part II we deal with cases of transverse magnetization. By that term we mean the following: the microwave fields propagate in a direc- tion normal to the dc magnetization and they do not vary along the magnetization direction. They may then be separated into two inde- pendent sets of field components, of which only one explicitly depends on the dc magnetizing field. For these two fields wave impedances are defined which can be used for matching purposes. A few simple examples are then given. One special case, that of the ''non-reciprocal helix" utiliz- ing ferrite, is of importance in traveling-wave tube work and is discussed at length.'^ The slow-wave propagation along both a cylindrical and a ''plane" heUx are treated; magnetic loss is analyzed in some detail for the plane case, and general rules are given for its approximate deter- mination in the cjdindrical case. In Part III pertur})ation theory and some miscellaneous topics are taken up. Suital)le perturbation methods are developed for cases in which the wave guide fields are drastically modified over small volumes (as occiu's if thin pencils or thin discs are inserted) and also for situations in which the local properties of the medium are but slightly disturbed over finite volumes. Among the miscellaneous topics dicusssed is the 582 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 propagation between infinite parallel planes filled with ferrite in a longi- tudinal magnetic field. The effect upon Faraday rotation of multiple reflections is considered. 2. THE PHYSICAL PROPERTIES The propagation of electromagnetic waves in a medium is governed by Maxwell's equations which connect the space variations of E and H, the electric and magnetic intensities with the time \'ariations of D and B, the electric displacement and magnetic induction. To characterize the particular medium relations may be given of the form D = \\ e\\E and B = 1 1 M 1 1 ^ where 1 1 e 1 1 and 1 1 ju 1 1 are the dielectric and permeability ten- sors. For disturbances whose amplitude is in some appropriate sense small, the elements of these tensors will be independent of rf ampli- tude, but will depend upon the dc state of the medium, upon the fre- quency of the signal and in unfavorable cases upon the wavelength of the latter. With the assumptions made in this paper the dependence upon wavelength will not arise. The form of || e || and || m || may be known experimentally or it ma}^ be deduced from some molecular model of the medium. If the e(iuations of motion of the parts of the medium are known under applied electric and magnetic fields, the displacement and magnetic induction resulting from this motion may be found explicitly. In isotropic media and in the ab- sence of applied dc fields, each component of the displacement or of induction depends in the same way upon the associated component of E or H. The tensors then become diagonal with equal elements. The ap- plication of a dc magnetic field, say in the 2-direction, causes ions to circle about this field or magnetic dipoles to precess about it. It follows that a rf electric field in the ionic case or magnetic field in the ferrite, normal to the dc magnetic field, will produce a component of motion at right angles to itself and in time quadrature with it. From symmetry and from the equations of motion in a magnetic field the tensors may be expected to be now of the form (1) where a is an even function of magnetic field and h an odd function, c, in general, will be independent of the magnetic field. That a and 6 at a given frequency and for a given sample of the mediinii are not independent but are related through the magnetizing dc field, Hq , is a fact of which we need not take cognizance when solving Maxwell's a -Jb 0 jb a 0 0 0 c GUIDED-WAVE PKOPAGATION THROUGH GYROMAGNETIC MEDIA 583 equations subject to the appropriate boundary conditions. Their sohition will determine the propagation constant (3 of a wave as a function of a and b, no matter what their interrelation. On the other hand, in a given experiment jS is generally determined as a function of one parameter only: the magnetizing field II o . Comparison of the family of calculated results 13 = /3(o, 6), with the results /3 = ^{Ho), found experimentally will, of course, determine a and h as functions of Ho . If, however, we have a prior knowledge of a and b in terms of Ho , either through postulating the correct dynamical model for the medium, or through independent experiments, we can utilize the functional form of a and b in our analysis of /3, and thus arrive directly at /3 as a function of Ho . The distinction between the two methods is by no means aca- demic; early introduction of such a functional form of a and b into the waveguide problem actually simplifies the analysis. Aside from this prag- matic consideration the latter method seems to us more appropriate for another reason: it is hardly the task of analysis of technical devices to check on the physical theories that give a and b as functions of Hq ; such checks are made by experiments specifically designed to avoid the ana- lytic complexities attending the solutions for most of the technically important structures. Accordingly we adopt the more direct approach of expressing a and 6 in terais of Ho (and, of course, in terms of the magnetic or electric carrier density of a given sample) throughout these papers, even in those few cases in which /3 can be expressed analytically as a function of a and b. 2.1 Ferrites Most ferrites used in microwave applications are fully saturated in do magnetic fields that are small compared with the dc field with which they are biased in operation. We shall therefore always postulate a fully saturated sample. Accordingly the magnetization vector Af at a point in the sample will always be of constant magnitude, although its orienta- tion will change in the ac field. One equation of motion for M that takes this into account is ^ = y[M X IJr] - r^ [M X [M X IJr]] (2) at \ M \ where Ht is a total effective magnetic field seen by the spins that make up M, t is the time and 7 is the gyromagnetic ratio appropriate to elec- tron spins, whose ^-factor is close to 2. The expression on the right hand side of (2) is in the nature of a torque; the force on M is always at right angles to M, thus leaving its magnitude unchanged. The first term on 584 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 the right of (2) is quite Avell su))stantiated by quantum mechanical con- siderations. It is a vector normal to M and to the force Ht and is re- sponsible for the precession. The second term is also a vector normal to M, but is in the plane of M and ^ in a sense such as to reduce the angle of the precession. It thus represents a damping. Not much is known about the precise mechanism of the damping, so that its phenomenological representation by the second term of (2) is still in doubt. Ht , the total field acting on the electron spins, is made up of terms not all of which are of electromagnetic origin. It consists of the dc field ^0 within the sample, the ac field H, the anisotropy field, and the field ascribed to the quantum mechanical exchange forces between spins. Hq in the sample must be calculated from the applied dc field ^ext by a purely magnetostatic calculation, which, in the case of sufficiently simple shapes, can be carried out with the help of the appropriate de- magnetizing factors. Throughout this paper it is assumed that this problem has been solved, so that H^ is given. Furthermore it is assumed that ^ext and ^o are uniform. Boundary effects due to non-uniformities of ^0 are neglected. The microwave field H in the sample is one of the unknowns of the problem of propagation, and will appear in the solution of Maxwell's equations subject to the appropriate boundary conditions. The anisotropy field, a property of a single crystal of ferrite, arises from the fact that through the medium of spin-orbit interaction, the electron spins can "see" the orbital wave-functions. Since these have the symmetry properties of the crystal, it is to be expected that the aniso- tropy field will be a vector function of M, with the symmetry properties of the crystal. The samples of ferrite used in practice contain a great many small crystals randomly oriented, so that the net effect of the anisotropy field on microwave propagation must be obtained by means of an averaging procedure. The integrations involved are laborious and have not been carried out so far. We shall therefore neglect anisotropy altogether. Since anisotropy fields are usually of the order of a few hundred gauss, this will put our results in error below frequencies of about 3,000 mc/sec. (Corresponding to a precession frequency of yH^ = 3,000 mc/sec. Ho is about 1,100 gauss.) The field between two spins ascribable to exchange forces w^ill be zero when the two are parallel, and thus arises out of differences of spin ori- entation (that is, differences of M) from place to place. In fact, analysis shows that this magnetic field is proportional to S7'M for cubic crystals. Thus equation (2) really involves position coordinates as well as time. Hence the ac part w of M at a point will depend not only on the ac field GUIDED-WAVK I'KOPACiATIOX TIIHOIXJH (; YK(XMA(i.\KTK' MIODIA 585 H at that point, but on values of // Ihroiiohout the vokime of the sample. Therefore B, which is luoU + wt, ^vill likewise be a functional of H over the whole sample. Fortunately it turns out that the spatial \'ariation of H in a microwave structure is so much slower than that characteristic of the "spin waves" to which V"A/ gives rise that this effect is finite negli- gible at microwave frequencies. Only in th(> most immediate vicinity of gyromagnetic resonance could such effects become significant. Thus, we shall regard Ht simply as the sum of the dc and ac magnetic fields, ^0 + U, ai^d correspondingly M as the sum of the dc magnetiza- tion (directed along Ho in a saturated sample when anisotropy is neg- lected) plus an ac part ?«. Ecjuation (2) must now be solved for m in terms of H. It is a non-linear ecjuation, whose solution m will depend on H non-linearly, as will B. Even if m could be determined in this way, ]\Iaxwell's eciuations would become non-linear, and hope of their solu- tion remote. It is therefore necessary, and in the great majority of ap- plications also quite sufficient, to assume that the ac quantities in (2) are so small that their products can be neglected and only linear terms taken into account. The terms m and H may now be assumed to vary as exp jo)t. Under these circumstances, (2) becomes "^-i = 7([m X ^o] + [Mo X H]) at ay 'Wo ([l/o X [m X ^o]] + [Mo X [Mo X H]]), and is easily solved for m in terms of H, and of the dc quantities Hq , Mo which we shall assume to point in the ^-direction. Each of the components mx , rny is a linear function of both H^ and Hy and when they are sub- stituted in the components of the equation B = ^iqH -{- m, lead to ex- pressions of the form (1) for B in terms of H: (3) Bx = nH, - JiifJy , By = jxHx + m/4 , and B. = IJLoHz is con\ ^enient to introduce two i auxiliary cjuantities .= 1^1 Ho. > V = 1 7 1 Mo Mow 586 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 and in terms of these one obtains the relations first derived by Polder: (4) Ho o-^l -\- a^) — I -Jr Ijaa Sgn p K _ — p Ho (t\1 +«") — !+ 2ja 0 = -1 p < 0 a is the ratio of the natural precession frequency — | 7 1 i?^o to the Zir signal frequency, p is the ratio of a frequency ^^ | 7 1 Mo/ no , associated ZTT with the saturation magnetization Mo , to the signal frequency. Note that a and p always have similar signs : if Ho is reversed, so is the satura- tion magnetization. Equations (4) are true only for a fully saturated sample. Therefore they hold good only for values of a greater than the very small value corresponding to the amount of Ho reciuirecl to saturate the sample. In practice that value of Ho is generally so small that this restriction is trivial. In the text a number of formulae will appear which apply "near a = 0". These are to be understood as applying near the very small value of 1 — 0- and these equations describe the loss-free case. If in equations (5), a is replaced by (a + ja sgn p), the resulting expressions check (4) to order a. For small a, it follows that any propagation problem need be considered for the loss-free case (5) only.* The first order change due to loss in any formula so obtained can be deduced by differentiation of the formula * A form of the damping term in Equation (2), no less justified experimental!}' than the one used above, is — , : I M X —^ ). When this expression is used the 11/1 V- dt ) permeabilities are exactly functions of the variable, 4 nhl 4 \ '^J / -/ fVJ 3 / / / 0 / / / / 2 H 1 / / f / / i <>^ ^ y y f / 0 9'^^ :== L-__i,^^ ^ L=^ - "'- -0.5 ^ ^^ ^ 0.8 0.6 0.4 -1.0 -1.5 //i -2.0 -2.5 // I / 1 /i ' -3.5 / 0 fo/'s/ -q^I M -4.0 ff \°\ O ^/ 1 -4.5 -5.0 \ \ 1 0 0.1 0.2 0.3 0.4 0.5 0 Fig. 1(c) — The ratio ph = k/m versus a 6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 (T 590 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 generalized in some of those cases to take into account the existence of groups of electrons with different "effective masses" 7n. 3. THE SOLUTION OF MAXWELL's EQUATIONS Maxwell's ecjuations will now be solved in a cylindrical waveguide filled with a hypothetical medium which contains the ferrite and the plasma as a special case. It will be supposed therefore that both its per- meability and dielectric constant are tensors of the form previously considered. 3.1 Field components The following notation will be found convenient. The projection of a vector A upon the plane normal to the ^-axis will be written At . If the components of A, are a, ^ then an associated vector having components (/3, —a) is devoted by A*- A similar notation is used for differential operators. Thus, if V denotes {d/dx, d/dy), V* denotes (d/dy, — d/dx)'\. Denoting scalar products by a dot, the following identities are evident At*-At* = ArAt ; (At*)* = -At ; ArAt* = 0; AfBt* = -At*-Bt) and At-Bt* = ^-component of [A X B]. Also if k is a unit vector along the positive ^-axis, k X A = —At*. Similar relations hold for differential operators. If one denotes the star- ring operation by the symbol P then clearly where o is a number. Maxwell's equations may noAV be written, for that case in which the dependence of any component upon t and z is of the form e^'''^'^~^^\ in the form: V*//. + mt* = jo^eEt + c^vEt*, \/-Ht* = jo:e.E., \/*E, + j^Et* = -jo^iiHt - o^KHt*, and y-Et*= -j^n^H,, where use is made of equations (3) and (6). t The operator V* is called "flux" by SchelkunofT. Strictly, one should write V« and V«*, rather than V and V*, but this is needlessly cumbersome. (9) Gl•lDKD-^^•.\VI•: I-KOPAC ATION IIIKOI (ill (IVKOMAiiXK'riC MKDIA 5!)1 2.0 1.6 Iv 1 \ W "^ ^^ ^ ^ \l \ 1 \ 1 -^ ~ 0 WHEN 1 fo ' 1 ^=Vi-q2 i 1 1 1 1 1 (a) \ w \ \ ^ ==^ "^^ xV\ u>\ > } (b) 0.4 0.8 1.2 cr .6 2.0 0 0.4 0,8 1.2 1.6 2.0 Figs. 2(a) and 2(b) —The relative dielectric constants e/eu and v/tu versus a. It is desirable to remove scale factors as far as possible. A unit of length given by /3o ^oa/m^c^ will l)e tised to measin-e lengths. This unit is Xo/27r, where Xo is the wave- length in an unbounded, unmagnetized medium. It will be assumed that (3 is in future measured in units of i3q . Finally all magnetic fields will be multiplied by y/n^/e, to give them the dimensions of electric fields. Using the definitions of the p's and p's given in Section 2, Maxwell's equa- tions may be put into the form: (10a) (10b) (10c) and V-^,* = -Uh. (]0d) 592 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 Et and Ht may now be eliminated yielding two simultaneous second order ecjuations for Ez and H, . These, in turn, may be combined to produce two independent second order equations each of which is satisfied by an appropriate linear combination of Ez and H^ . These equations may be solved and Ez and Hz expressed as linear combinations of the solutions. The transverse fields are then written in terms of E^ and Hz and, finally, the boundary conditions are applied leaving a transcendental equation in /3 . Operating on (10a) and (10c) withV- and taking account of (10b), (lOd), one finds that j^V-Ht* = veUV -Et + jpeHz) = -iS^. , and jl3^-Et* = -VH{j'7-Ht+ JphEz) = m^ Operating on (10a) and (10c) withV*-, usingV*- V* = V^ and so on, one obtains, using (lOb) and (lOd), V'^. + j^y-Ht = ve{-H, + PEV-Et), and (12) V'Ez+Jl3V-Et = -ph(Ez + pnV-Ht). Now, elimination oi V -Et and V -Ht between (11) and (12) yields V'Hz -\- pe (l - Pe^ - ^)Hz = jKpe + Ph)Ez , and ; "r; (13) V'-E/. + VH il - Ph" - ^]E, = -Mpe + Ph)Hz, equations which demonstrate that pure TE or TM fields no longer exist, as the result of the presence of p's. Hz or Ez might now be eliminated between these equations giving a single equation in V" and ( V')", but it is more convenient to find those linear combinations of Ez and Hz which satisfy a first order equation in V . Writing such a linear combination as xP = Ez-h jKHz , (14) and adding jA times the first of equations (13) to the second, it is found that this is an ecjuation in xp alone of the form VV + xV = 0, (15) provided that A is a root of the quadratic A - 1 = 0. (16) 2 Ve {'- 2 - PE - VeVh/ \ 2 - PH - VeVh) Kpe + Ph) GUIDHD-WAVE PUOrAGATlOX TilKOlCill (I VliO.MAfiNETlC MKDIA 593 The value of x" is then given by X1.2' = ve (1 — pe' — ) — /3(p£ + pn)M,i , (17a) \ VeVh/ or Xi/ = Vh(i - p/ - — ) + Kpe + Ph)Ai,2 , (17b) \ VeVh/ where \i and A2 are the roots of (IG) and xf, X2" are the corresponding x^- The labehing of the roots is not important, but consistency must be maintained. From (14) E, and Hz must satisfy E, + jAiH, = yp, , and E, + jK^H, = ,^2 so that & = ^f ~ t"^' . (18a) A2 — Ai and H. = i -^^ . (18b) A2 — Al Solutions of (15) may now be sought in cyhndrical coordinates. To satisfy the boundary conditions in circular guide it will be necessary to assume the solutions to vary as e^" , where 6 is the polar angle and n is any integer, positive, negative or zero. Equation (15) then becomes if r is the radius. Solutions which are regular within the guide ^^'ill have the form of constant multiples of J„(xi.2^), where J„ is the n^^ order Bessel function. The solutions of (15) are, then, V'1.2 = A^,2Jn(xl.2r)e'"\ (19) where the ^4's are constants. E^ and Hz can be found now from (18), but further equations must be found to express Et and Ht . Using P to denote the starring operation, (10a) and (10c) may be re-written as (JpfP - PEVE)Et + mt = - V7/. , and 594 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 -mt + (JPhP - PHVH)Ht= V^. , which yield {[veVh{^ + PePh) — iS ] — JvhVe{ph + Pe)P]Ei = -i/3V^. - v„{jP - PhWH,, and {[V£j'fl(l + p^p//) — iS"] — JvhVe{ph + PE)P]Ht The term in parentheses may be removed by using the rule for inverting such expressions in P which was given earlier. This process gives i2 ^Et = 8 1 + PePh — ) + j{pH + Pe)P VeVh, (20a) and mt = where 12 = veVh = VeVh 1 + PePh — VeVh [- i|3VE. - VhUP - Ph)VH.], + jipH + Pe)P (20b) MjP - pm)VE, - i^VJYJ, 1 + PePh - I\ VeVh/ (pe + PhY _VeVh - (1 + Pe) (1 + Ph) &- VeVh — (1 — p£)(l — Ph) It may be noted that for plane waves in the unbounded medium along the z axis, which have Ez = H^ = 0,0, must vanish and that the propaga- tion constants for such plane waves are evidently given by I3~ = VEVnil ± Pe) (1 ± Ph)- (21) The values of E^ and Hz given by (18) may now be substituted in (20) and the operator P removed. This gives, finally, (Ai — A2)UEt = j ( 1 + PePh — ) il^M — PhVh) + Vh{ph + L\ VeVh/ Pe [ — (i8A2 — PhVh)(pe -\- Ph) + Vn [l + PePh VeVh (22a) v*h minus the same expression with suffixes 1 and 2 interchanged. GUIDED-WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA 595 (Ai - A.,)i2//, = ( 1 + PePh — ) {i\.2VEPE — 0) — VeA2(pb + Ph) A VeVh/ (22b) VeM ( 1 + PePh — ) — (p£ + P/f)(AoI/£Pfc. — 13) V*>/'1 \ VeVh/ J minus the same expression with suffixes 1 and 2 interchanged. Equations (22a) and (221)) may be written in a ^■ariet3' of equivalent forms by making use of the relations V)etween Ai and Ao . The manipula- tions which have been used in deriving (22a) and (22b) assume the use of rectangular coordinates, but the results are \'alid in polar coordinates — , — - ) . That this is the case may dr r dd be seen from the consideration that the rotation, —d, which carries the vector {Ex , Ey) into the vector {Er , Ee) also transforms ( — , — ) into \dx dy/ d 1 d dr r dd^ 3.2 The characteristic equation The boundar}^ conditions of the problem are that E, = 0 and Eg = 0 at r = ro , the radius of the guide. E^ is given by [see (18)]. (A.> - Ai)^., = [A2.4:J.(xir) - AiA2Jn(x2r)W"\ (23) and vanishes at r = ro if . /n(X2''o) . ./n(Xl/'()) Ai = ; A2 = . A2 Ai Hence the relations hold: '/'i,2 = -— /n(x2.iro)J'„(xi.2r)e^"^ A2,l From (22a) it follows that Jnix2ro)e inO d2 - S ( 1 4- PhPn — ) (/3A2 — PhVb) r \\ VeVh/ (Ai - \2)^Ee + VnipH + Pe) \JniX\r) + U/3A2 — PhVh){pe + pa) -\- Vh\\ + PePh — — ) ( XvL'iXir) \ VeVh ' J * In Ai)i)en(lix III the field components in polar coordinates are written out fully for the ferrite and })la.sniu cases with some changes in notation which are introduced in Sections 4.11 and 4.2. 596 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 minus the same expression with the suffixes interchanged. Hence (Ai - Ao)QEe{ro) To + hipE + ph) - (Ai - A,)m>H r-^ + pAl - Ph') \VeVh , / 1 2 i3^ \\ XiroJn'ixiro) AiVh \l - Ph - I > ^—. r— (24) - 0, one in X < 0 and these are called /„ , /„' respectively. All /„ curves pass through X = 1, cr = 1, all/,/ curves pass through X = —1, a = — l.The lines X = 0, X = — 1 are also infinity curves to be denoted by I a , Is respectively (As X -^ +1, (? tends to a finite value). Zero curves of G are given by or in a more readily computable form by _ 1 _ /V(l - X^) X \{F-\\)f (35) The branches of F ^(X) may be labelled according to the scheme: "0" for - X < \F~\\)\- < J?; "1" forii' < [/^'(X)]' < J2 and so on. The ?ith branch of F~\x) gives rise to an 0„ curve for X > 0 and to an 0,/ curve for negative X. All 0,/ curves pass through X= — l,o-= —1; all save one of the 0,^ curves pass through X = 1, o- = 1. The exceptional one, seen to be Oo , is associated with the "0" branch of F~\\) on which F~ (1) = 0. For fixed 1 — 0 and for X > 1 as crX — > 1 +0, the argument of /' tends to infinity and remains real. Therefore G passes through all \alues an indefinite number of times and (t\ = 1 is a limit hue of all contours, G = constant. For X < 1 as o-X -^ 1 + 0 and for X > 1 as o-X —> 1 — 0, the argument of F is 602 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 195-i GUIDKD-WAVE PHOPAdATiOX TIlKorcill (I YK().MA( iXETIC MKDIA ()03 Fig. 6 — Qualitative behavior of G{\, a) at large distances from the origin as a function of arc tan 0. If X remains fi.xed, then for X> — l,(7-^T^asa--^±cc; and for X < — 1, G' — > ± ^ as o- -^ ± oc . As X ^- 0, the curves of constant G are asymptotic to Xo- = f 1 — -\j — B\, where B goes from — -^ to + X with G. Interleaved with these families of curves are the curves 604 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 (7 = ± 00, which are Xo- = (1 — ^ ) + 0(X^)- More detailed information on these matters will be found in Appendix II. From the G^-diagram it would be possible to determine pairs of X-values with opposite signs, which, for a definite o--value satisfy the characteristic equation, but, for a given y such pairs would not necessarily satisfy the Polder relation (31b). It is necessary to have a procedure which takes account of the latter systematically. Such a method may be based upon the fact that if, for a and y positive, the Polder relation is solved for Xi in terms of X2 it can be thought of as a rather simple mapping of the whole X2-quadrant upon a part of the Xi-quadrant (Xi > 0). Similarly for 0" and p negative there is an analogous mapping of the Xi-quadrant onto the X2- quadrant. Considering first the case cr, p > 0, the Polder relation may be written in the forms X, = '-±J^ = 1 + l + v^ = r(x,). (36) 1 — (7X2 cr 1 — (xX-i From (3G) it may be seen that the curves X2 = const, transform into a bundle of hyperbolae passing through the intersection of a = 1/Xi and 0- = Xi — 7); that is, through Xio , ao , where = -p/2 + yl'+l , Ai. = p/2 + |/ ?+' These hyperbolae have the vertical asymptotes Xi = — I/X2 , and intersect 0- = 0 at Xi = p — X2 . For a fixed positive a less than o-q , Xi decreases from l/cr to c + p as X2 increases from — =0 to 0, but when 0- is greater than ao , Xi increases from l/cr to o- + p under the same circumstances. Thus the whole X2-quadrant is transformed upon that part of the Xi- quadrant which lies between the hyperbola Xi = l/a and the straight line Xi = o- + p- It follows that points in the Xi-quadrant which are, for a given p, excluded from this region, cannot be the site of acceptable so- lutions of the G-equation. Since as has already been stated, the Polder relation is unchanged by the substitution Xi -^ — X2 , X2 ^ — Xi , o- — > —a, and p to — p, it follows that for cf and p negative a similar mapping of the Xi-quadrant upon part of the X2-quadrant takes place. The transforms of the hues Xi = const, and so forth may easily be found by using these substitutions in the formulae already given. Reference to Fig. 1(a) and (b) will show that ±0-0 are the values of o- at which ;u reverses sign. Therefore we may expect o-q to play a special role GUIDED-WAVK PKOPAGATIOX THKOl (UI (iYROMACiXETIC MFODIA (iOo in the propagation theory, as also does a = I. The following scheme exists: for 0 < 1, ^ is negati\'e and ji posilixc. if a is changed to — (7, /x goes into ju, and k into —k. The procednre which will now l)e used to discuss the solution of the characteristic ecfuation, ohserxing the Polder relations, begins by writing the equation, for a, p positi\'e, in the form G(Xi , 0, X > 0, and of the function ^(Xo , a, To) in the quadrant X < 0, o- > 0. The latter surface has now to be transformed into one in the Xi-(|uadrant by the relation X, = T(Xi) = (a -h p - Xi)/(1 - aXi) (or equally well, Xi = T{\2). This may be effected by considering the transformation of curves G(\2 , o-, Vq) = constant, onto the Xi-quadrant. For the /' curves whose analytical expression in terms of a and X2 is very simple, the corresponding explicit expression of the transformed curve in Xi and a is simple. Contours other than /' are most easily transformed l)y replotting G(\2 , c, ro) = const, in the hyperbola-mesh formed by the lines r(X2). However, information about particular points and about asjTiiptotic behavior of these transformed curves is available in analytic form and is stated in Appendix II. The two surfaces so obtained vAW in- tersect in various curves, along Avhose projections on the X — o- plane both Polder relation and (r-equation are satisfied. For each such projection Xi is a function of 0, X > 0 (luadrant now being transformed on to the cr < 0, X < 0 quadrant. 606 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 It is possible to translate such solution curves into ^ — a curves in a direct graphical manner if a mesh of constant ^ lines is drawn in the first quadrant. From (30) these are given by \x - {p + a{\ - /3')]Xi - /?' = 0, or Xi^ 1 - ^ + V = 1 ^\ Xi 1 - )82_ The contour, /3 = 6, is just the contour x = \ — l)' displaced along the = 0.5. \ — a\ GUIDKD-WAVH I'KOPACJA TION TIIKOUCiH Ci YKOMAGNETJC MEDIA G07 course in \\\c tliii'd (|ua(lraiit is iiniucdiatcly found by rofloction in the origin. When p = 0 the magnetization of the ferrite vanishes and it is clear that we should then obtain just the modes of a guide filled with isotropic material (n = n^ ; k = 0). Superficially it might appear that, since the equations (31b) and (33) depend upon a, even for p = 0, this result might not be attained. We now show that I3~ is indeed independent of a for p = 0. It may first be noted that in this case if o- 5^ 1, the Polder re- lation (31b) transforms 1 - o-X, The G-equation reads into °o, iS" tends to the value appropriate to the TEu-mode in an isotropic medium (^t — > jUz = /xo , k ^ 0 as a- ^ 30). Thereby the whole solution curve is classified as specifying part of a TEn-limit mode. The remaining section of the TEn-limit mode in the upper half -plane is again found in the region between (O/)?- and (7b') r foi" «■ < ao . Any Ime 0- = constant < o-q cuts these two curves at two values of Xi . As \\ varies between these values, G(T{\i), a) varies from 0 to — 00 ; it is, thus, (;riDKD-wAVE PROi'AGATiox TJiKorcu <;yi{()ma(;n"kth' .mi;i)I.\ (117 /^ / ^ ro = 5.75 /.' /y y X ^ ^ .^. # ^ ^ ^ ^ ^ ^^'-■^S' i^y ^ ^^ ^ ^ ^ /X y^ ^ Vy y} Tor^J^O __ — ^ -0.4 -0.2 0.2 0.4 P"ig. 10(a) ■ — /S^ versus p for small values of a — the TEu-limit mode. clearly equal to the finite (negative) G(Xi , a) somewhere between. This situation persists up to a- = o-q — 0 and a solution curve therefore exists between o- = 0 and a — o-q . It meets a- = 0 for Xi satisfying 1 Xi- 1 X.^ ^ F(ro \/r^^-) - 1 LA2 and Xi + X2 = p. im These equations have been solved numerically; the corresponding (3"^ = — X1X2 is shown in Fig. 10(a). For ro between Ui and^i a value derived for (3^ from the first three terms of an expansion of jS" in powers of p, equation (01), turns out to be in very good agreement with the numerical calculation up to p = 1, for a = 0 and presumably is good for small a. At (To (the point at which /x becomes negative), the solution curve is "cutoff". However, the corresponding jS^ is not zero. As o-q is approached from below G(Xi , a) -^0 and so G'(X2 , a) tends to zero. Thus, X2 tends to 618 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 0.5 P>' 0.2 / — To =5.75 ^ / / / / V^-OO / / f \ / / 1 / / ~^ ^4.2 5 \ \ M a r4.825 / Y \ \ / -t.O -0.8 >: -0.6 -0.4 -0.2 'To=5.33 0.4 0.6 Fig. 10(b) — /32 versus p for small values of er — the TMu-liniit mode, the negative root X20 (unique for the present radius) of Fin — (TfiK'i = \1. The associated ^^ is -X20X10 = -~ and is shown in Fig. 11(a). The way Co in which jS^ approaches this value as cr ^- o-q can be found and is one of the more subtle examples of behavior of a mode near a special point. Writing o- = (to — 5cr, Xi = Xio — 5Xi , we observe that, since cto + p 0, the Polder relation in the form o-Q 1 0- -f p - l/o- Xl = - -t- ' r a 1 — aK2 d\i fully determines -— ^ ; any variation due to 6X2 vanishes at o- = ao . SXo can a l),G(Xi,o-) is given by ro Xi 1 - o-Xi Xi^ - 1 ' which near ao , Xio may be written GUIDED-WAVK riU)l'A(iA TION TllKorCII GYKOMAGNETIC MEDIA (ill) -VSa dG TIui perturbed (^(^2 ,o") which (since 6'(A2o , co) = 0) is 5X> + 0(6(t) 5X20 equals the preceeding expression and gives dXo = -\/5 Aio TT r 0*0 oA] According]}', 6/3" dGV' axTo, + 0(5a-), a result which shows that /?" tends to its terminal value along the vertical. It is clear analytically and graphically that this mode persists as p ^ 0, and must be identified with the only isotropic mode for this radius, namely TEu . No other branches exist below cr = o-q , since G{Xi , o-) and G(T(\i), a) have opposite signs except in the region just considered. The two solution curves considered so far are not the only ones; in fact the infinity of sheets of the surface G(Ai , a) in the region bounded by 7i , Oc and {Ia')t , Fig. 9(a), intersect the transformed sheets G(T{\i)cr) in infinitel}^ many more curves. In the blank areas of that region the G'-functions have equal sign, and all these areas must be carriers of solu- tion curves, since in every one of them every single contour G(\i , a) = g To = 5.75 / /" \ y / \ \ / \ / \ 1 1 \ / \ / \ 0.02 -1.0 -0.6 -0.6 -0-4 -02 0 0.2 0.4 0.6 0.8 1.0 P Fig. 10(c) — (3^ versus p for small values of " //// Iw IF V i (a) 1 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 To (\JU "^0.2 y /O., 518 = ^0 P ='-0 / / '0.677 / Ch y' ^ / 0.781 o.e 61 / Vy / 0.5^ .905 0.951 /; / > y ^ 5V :^ 1.000 // / ^^, ^ >^< D /^ ^/ /y ^ ^ ^ ^ ^ ^ /A w ^ (b) w- 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 Fig. 11 — /32 at the tj-pe 2' cutoff as a function of /o for various jo. 2o' cutoff, TEu-limit mode; (b), 2/ cutoff, TMn-limit mode. The presence of a curve for p = 0 is clarified in the text. 620 GUIDED-WAVE PROPAGATION TllUOl (ill O YUO.MAGXKTIC MKDIA 021 crosses all contours of G{T(\i), a), in particular G(T{\i), a) = g. All the additional solution curves arising in this way start at c = 1, Xi = 1; the ?i"' of them threads its way from one blank region to another, first through the intersection of /„+i with (Ih')t , then through the intersec- tion of 0„ with {Oi)t , and finally comes to an end at the intersection of /„ with (Ia')t ■ At the end point (a- = 1, Xi = 1), X2 and, therefore, /S" are infinite, (just as for the TEn solution curve). At the end point (/„ , (7.4') 7), Xo , and, therefore, (3' are zero. The a and Xi values corresponding to the latter are obtained from the equations l-J^rn' ^ J? . ^^^ = a^ + p. (40) 1 — ffnXln ro" It is possible to derive the slope 3(3' /8a of the (f — a curves at these cut- off points. Near cut-off, the infinity /„ of G{Xi , a) is matched by the infinity /./ of G(Xo , a). The G'-equation therefore degenerates to Fin) = Xo , /l — Xi^ . Xl„ jn Writing a = an — .rXo , Xi = Xi„ — //X2 , expansion of the right hand side of this equation to order I/X2 furnishes one relation between x and ij\ the Polder equation furnishes another. The two can be solved for .r, and so, since to first order 2 x^ -.0" — an \ ^^ 6/3 — —AlvA2 ~ Xin = Xi — jO jo d(3^/8a may be found. It is found that for convenience in computation, the results of this calculation are best presented parametrically. Ecjua- tions (46-8) represent equations (40) and 8l3'/8a in this way. Fig. 12 (a) and (b) show the result of some computations. Near a = 1, /3" = CO , these added solution curves behave rather like the TEn curve. The 2 1 . leading term in the expansion of jS in powers of _ ^ is now K-^-^1 >^ •>rS >^ y 1 i/^ ^r^ in ^ — o -I- + OJ 01 OO rvj / ^ ^i^ UJ r- !o := /^ ^ /^ — ' d 3 H- \ do y /^ + + ^^ ■o oo y /^ \ to OO f*?d + + ^ o to o o o 1 1 y'?/^^ ^ + V ^^ ^ .. \ do 1 1 (O*" "?«> y ^^ + + ^ ,^ -<: ^^ IT) O C\J o €\JO >/0 o O A 1 ^ o — • 1 1 1 1 1 1 2 t- _L II 1 1 .- ^ bC H ^ r^ — ' rt H ,^ 0) o c ^ o :S — ^ 1^1 o .2; - £ > Si c3 £. V c 2 c rt ^ GUIDED-WAVE PROPAGATION' THROUGH GYROMAGNETIC MEDIA 023 from which higher modes are drawn as the guide radius is increased. That the propagation of modes which for larger guide radii correspond to higher TM and TE modes is possible for limited ranges of c might be ascribed to the larger /i-values in those ranges, which cause the wave to see an effectively larger guide. This explanation is convincing only when (T > 1 . When cto < o" < 1 , m is negative, and the propagation must then be the result of an interplay between ju and k. In passing we remark that we are here dealing with the propagation analog of so-called "shape resonances," which physicists sometimes encounter in resonance experi- ments on small spheres of ferrite in cavities. We now turn to a discussion of the solution curves for o- < 0 which lie in the third quadrant. Fig. 9(b) shows the partition of the region allowed by the Polder relation (again for p = %) into positive and negative regions by the various /', 0', (Z)r and (0)r curves. Regions in which G(X2 , 0. The singular area is that part of the region bounded by Ib' and Oc in which the G'-functions are both positive. Here both G(X2 , (t) and G(T{\2), : as Xo becomes more negative, it follows that GiTiXo), 0, is given by jS" = 0, a = — (1 + p). (When p < —1, the branch does not exist at all.) We note that a left- circular plane wave is cut off at exactly the same value of a as the TE- mode is in this particular case (see, however, the following sections). The slope at cut-off is determined b}^ expanding the G functions near their infinities at /„ and Ib' and utihzing the Polder relation. The slope is found to be da p(l - F{ro)) ' ^ ^ A further solution curve lies in the region between Oc and (Oo)r for 0- > — o-Q . It has no analogue in a guide with isotropic material and will be discussed later. In the discussion of the mode spectrum for radii between tii and ji three distinct types of cut-off point have alread}^ been encountered. When larger radii are treated it is found that no other types arise.* In Section 4.17 formulas rele^'ant to the three types are given. An examination of the field components in the neighborhood of the cut-off points is of some interest. Cut-off points of type one (intersections of /„ and (Ia')t or In and {Ia)t), at which jS" = 0, have Ez = 0 and the field is of a pure TE-type. The medium behaves transversely as though it had a permeability, /x — k /n. Although the field is purely TE at cut-off the mode terminating at such a point may in the limit of vanishing magne- tization be either a TE- or a TM-mode. This impartiality extends to cut- off points of the other types. Cut-off points of type II [(O,/)?- — 0^ or (On)r — OcT occur at 0" = ±0-0 , where n = 0 and here /S" does not van- ish. In such cases one of the x's is finite and the corresponding contri- butions to the field pattern quite normal. The other, however, tends * There is an exception to this statement. This is the type designated in Sec- tion 4.17 as 2o=o which cuts off an isolated mode having no TE or TM analogue. GUIDKD-W AVE PROPAGATION THROUGH GYROMAGNIOTIC MIODIA ()25 to an infinite imaginary value and the associated fields are confined very closely to the guide walls. The wall currents are very large and essentially longitudinal. Type III cut-off points [1 ,i — (/,i)r] at Avhich n = — ^:have|CJ' = 0, but the fields are not of a purely TE- or TM- type. They consist essentially of a rotating, transverse, //, which is uniform ovvv the guide. The components 11^ , Ee and Er are smaller by one order of a — x^ , In tends to the line X = 1, /„' to X = —1. No /„ curves ever enter the region X > 1, a < 0; no /„' curves enter X < —1, cr > 0. It is also im- portant to relate the /„ , /„' curves to the boundaries of the Polder regions. /„ curves cut the Polder boundary o- = X — p, of the first quad- rant in at most one point. As ro increases from 0 to j„ , this point moves from (T = (To to (T = 00 . Thereafter, no intersection occurs at fixed p until ro equals j„/\^l — p^; it here reappears at cr = 0 and moves steadily to 0- = 1 — p as ro increases indefinitely. The only intersection with the other Polder boundary cr = 1/X, is at X = 1, cr = 1, regardless of ro . The 0„ , 0,/ curves are gi\'en by ;d 2 ro 1 -X^ 626 thp: bell system technical journal, may 1954 if the n*** branch of F~ (X) is used. Thus, as ro increases, the successive cur^'es either all pass through a fixed point (which can only be X = ± 1 , or = ±1, 71 > 0) or move steadily up or down without further intersec- tion. An 0„ curve starts from trX = 1 at ro = 0 and falls for X < 1, rises for X > 1, as ro increases. For large X, since a ^^ jn/fo (X — 2), n > 0, the 0„ and /„ curves move together with a constant separation. Oo is singular, since it does not pass through X, o- = 1 and falls steadily for all X; it tends to o- = 0 for large X. TheOn' curves rise from crX = 1 at ro = 0 for — 1 < X, fall for X < —1. They run parallel to /„+/ for —X very large. For small X there is an expansion ,= U-rLy J'i + o(x) holding for 0„ and for 0„'. This indicates that for ro < u„ , 0„ goes to -f ^ and 0„' to — 2o for small X, but at ro = ?i„+i , 0„ and 0„' merge momentarily at 0, 2ro' •U„+l2(w„+i2 — 1) < 0. For larger ro , 0„ goes to — » and 0„' to + =o . Since the union of 0„ and 0„' takes place at a negative o-, it is clear that 0„ curves, unlike /„ curves, may cross the line o-o , since 0„ and /„ have a fixed separation for large | X |, this pair escape intersection with the boundary at the same value of ro , namely jn ■ Similarly 0„' and /n+i escape together at ro = jn+i for a < — ao . When ro is less than iii , a case in which the iso- tropic medium would not propagate, no part of Oo' Ues in the upper half plane and there is then no (Oo')r curve. The solution curve w'hich in the previous discussion of Section 4.12 was assigned to TEn , after passing the intersection [/i — (Ib) t] can no longer escape to infinity and terminates on [/i — (Ia)t]- Thus, the TEn mode at this radius has become an incipient mode with cut-off and other properties given by the formulae already quoted for such modes. As ro approaches ?/i from below, the j3' — a curve is double valued between o-cut-off and some larger value. This is borne out by the fact that d/3 /da becomes positive at cut-off, and by the observation that the solution curve bulges towards large a- between /i — (Ib')t and its terminus. The part of the /3 — o- CriOKD-W AVK PU()l'A(iATl().\ 'nillorcJll (iYK().MA(ii\KTIC MKDIA (127 d3- curve along which -j— < 0, will tend smoothly towards the ff — a ciirxe acr for ro just greater than Ui . The course of the TEn solution ('ur\-e remains qualitatively unchanged for all ro > iii . When ro passes through Ui , and the TEu solution curves escapes dis- continuously to infinity, the solution curves below it disengage from their former end points In+i — {Ia)t and instead end at the point In — {I .a.)t ■ When ro exceeds ji , the curves /i and Oi escape intersection with (X = \ — p simultaneously, for a > cto , and the curve (Ii)t makes its first appearance. From the asymptotic formulae (App. II) the latter runs to infinity between 7i and Oi , and now the solution curve which ended for Ui < ro < ji at /i — (Ia)t is carried to infinity between 7] and (/i)r • The asymptotic expression for /3 versus a, given in formula (56) indicates that /3- tends to the isotropic value for the TMn mode. No further qualitative changes w411 take place in behavior of this mode as ro increases. As /"o increases through H2 (the value at which the isotropic medium supports the TE12 mode), the (Oi)t curve makes its appearance, an event accompanied by the escape of the uppermost incipient solution curve (the one ending at (Ia)t — 1 2) to infinity. The escape takes place in the same way as that of the TEu solution curve as ro passed through Ui . The newl}^ escaped cur^^e, of course, represents the TEi2-limit mode. The end points of the remaining incipient solution curves also jump discontinuously to their next higher neighbors as they did at ro = ih . The course of events as ro is increased further should now be abundantly clear, and is summarized in Table I on page 642. We now turn to the region 0 < o- < o-q and consider first the situation 0 < To < th . It is clear that in the area bounded by 0^, Oo , (/aOt and {I' b)t both G functions are negative. There is no simple geometrical argu- ment which determines the existence of a solution curve in this region. It is therefore necessary to use a type of analytic argument, which is use- ful in a number of other cases, although fully discussed only in the pre- sent instance. We show that the least ^•alue attained by G(\i , a) in the admissible region for p = 0 (which contains all regions admissible for other p-values) is greater than the maximum value of G(X2 , cr) in the range — 1 < X2 < 0, (T > 0. Consider the variation of G(ki , c) as the point Xi = 1, (T = 1 is approached along a line of constant x iii the admissible region f or p = 0 (see Fig. 4). We have the relation G{\ 0, x is between 0 and l,and the X curves run from o- = 0 to o- = cc . G' will clearly decrease as a increases from zero on any one of these curves. Thus G attains its maxi- mum on 0" = 0, where its value is 'F{ro Vl - X^) ■ I X I ^ \ Since F(ro\/l — X-) is positive for n < Ui and | X | < I, G is clearly less than — 1 . In passing we note that for ro = ?ii, both G functions may attain the value — 1 . As ro passes through Ui , the (Oo')t curve appears in the region under discussion and together with (Ib')t delimits the region carrying the TEu- solution curve already discussed at length. No qualitative changes occur in that curve as ro is increased indefinitely. When ro exceeds ji , the (Ii)t curve appears between (Oo')?- and {Ia')t- Between {Ia)t and (/i')r the G functions have a region of common sign, yet no solution curve arises there for a given p until ro reaches Ji/a/I — V~* From then on, the 7i curve cuts {Ia)t , see Fig. 9(c), and a solution curve exists between (//)r and 7i . It is cut off at the intersection (Ia')t — h; there, pr = 0 and a, -^— are given by the same parametric formulae (46-8) da applying to the cut-off of incipient modes, the parameter 6 being nega- tive. The curve begins at cr = 0, where it satisfies the usual ecjuation, which for this radius has two solutions. The solution with the smaller X, belonging to the present curve, tends to the isotropic TMn-limit as p -^ 0. At a fixed ro , sufficiently below U2 , this mode does not exist at * There are some exceptions to this statement. When 4.82 < ro < id = 5.33 and p exceeds ■\/T^^~j^^/r^, a double-valued /S^ — o- curve exists between two positive a values. For values of n still closer to U2 further regions of common sign maj^ arise as a result of the interplay of the {Oi')T and (/a')?" curves. We have not examined these regions closely. Such dubious regions are confined to the immedi- ate neighborhoods below the Un . GUIDED-WAVE rKOPAOATKIX TIIIIOTCH CVUOMAOXETir ArKDIA 020 all when i, v> \/i-% If To is greater than u^ , the (Oi')t curve has appeared. A new region of like signs of the G's arises between it and (/i')r , see Fig. 9(e), and con- tains a sokition curve. This ends at ctq , Xio and begins at cr = 0 at a value of Xi pertaining to the TlMn-mode. Thus, it is clear that as ro passed 11-2 , the end-point of the TMn curve jumped discontinuously from {I a)t — h to (To , Xm . This jump is anticipated as ro approaches V2; the (3' — a curve first bulges beyond (J a')t — h towards its later course and returns to that point with positive slope. As ro increases further no change occurs in the qualitative behavior of the mode. It may be noted that above No the mode exists for all p. Be^'ond ro = ih , at least part of the area betAveen 7i and {Ia)t is an achnissible region and does in fact contain the TE12 solution curve. It begins at cr = 0 and Xi given by that solution of eqn. (39) which is, in the limit p = 0, the TE12 solution. It is cut off A^ith /3^ = 0 at {Ia)t — I\ , the end point relinquished by the TMn-solution curve. As ro passes j-i , the TE12 solution retains its cut-off point, but, beyond ro = Us , it will transfer this point discontinuously to ao ■ This is also true of their progress with changing radius and of the escape process. The singular character of the (Oo)t curve and the presence of Ib lead to some local changes in the progress of the modes but have no effect on their more salient features in this particular range of a. The scheme of progression of the end points is shown in Table I. In contrast with the state of affairs in the region just discussed, the mode structure in the area between a = 0 and a = — ao is verj^ mark- edly affected by the presence of (Oo')r and Ib'- When ro < Vi , a solution curve exists betw^een a = —I — p, and a = — (To . It starts with /3' = 0 at the intersection of (/..Or and {Ib') with a slope given by (41). For sufficiently small ro , 13 tends to infinity as (T — > cTo , since the solution curve approaches the line 0/ or (0^)r • Its shape is then given by (52), see Section 4.17. As ro increases, Oo falls steadily. Eventually, for sufficiently large p, its minimum falls below See footnote on page 628. G30 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 — crn . (Oo)r HOW has two branches for cr > —ao, which pass through the point a = — Co , X2 = — — making there a finite angle with each other. (Oo)r is completed by a loop in a < — o-q , Fig. 9(g), which does not affect the incipient modes appreciably. The mode in question now has two branches. The first starts as before and ends at or == — cro , where the associated /3" is given by (3a- = K/o'o and \a is the smaller root of It resumes at 1. Fig. 13(c), ro = 2.75, TEu-limit mode, a < -1. Fig. 13(d), (e) and (f), ro = 5.75, TEu- limit and TEi2-limit modes; Fig. 13(d), -1 < o- < 1; Fig. 13(e), 1; Fig. 13(f), a < -1. Fig. 13(g), 13(h) and 13(i), ro = 5.75, TM„-limit modes; Fig. 13(g); -1 < o- < 1; Fig. 13(h), (7 > 1; Fig. 13(i), crn continuously to a- = — cro , X2 = 1 (To and, simultaneously, the cut-off point of the TMu curve occupies the position relinquished by the for- mer. The TMu mode now exists for all p. A new solution curve (TE12) appears in the region bounded by 0/, (Oi)r and //, if p is not too large, terminatuig at (Ia)t — I\, the point left by the TMi terminus. (If p exceeds -y/l — ji^/uz' this curve will not exist at all.) Figs. 13(a) to (i) and 14 show the approximate course of the /3" — o- curves for the TEn mode at ro = 2.75 and for the TEu , TMn and TEi2 modes at ro = 5.75. The incipient TMu mode at ro = 2.75 is shown for positive 0-, p only. They were computed b}^ the methods outlined above. 4.14. Guides of large radius. It is of some interest because of the high dielectric constant of ferrites to examine the behavior of the modes as the radius, ro , is allowed to become very large. The two sides of the G-equation will remain determinate for inilimited ro provided ro /[^IX-VI^J JL 1+73' i 1 1 0.9 y / 1 0.8 1 ill 0.7 i Ifj- / ^ - 0.6 l\\ \ d \\ \, 0.5 M Sn s N p =1.0 TEn - MODES 1 ^ ^ [^ 5i; ■^ ^0.2' 0 ^ 0 4 1 ^ ^ ^ ■^ ■ ^rd = — ^n _— \ ^^ ' -— _-_ — - — ^~~~ 1 — — 0.3 V V y - l\\ \ 0.2 A \^ V \\ \ \ INCIPIENT TM,| - MODES 0.1 \ \ \ 0 \o .4X0. ^ ).8^ P = 1.0 n 0.2 1 - \ > ^ \ /3^ Fig. 13(b) — See Fig. 13. 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 i M m Ij In II fj / 1 1 /> 1 1 - ^ <^ ^ )\ 1 p = -1.0 ,^ ^ ^ > / / / _- "^ '^ y^ y / y f ■■■"^ _— z^ ^ % A^ y / — " ^""^ ^ -0 2 , - ^ 0 ~ -2.6 -2.4 -2.2 -2.0 -1.8 -1.6 -I 0" Fig. 13(c) —See Fig. 13. 632 .0 -0.8 criDKD-WAVK IMIOI'ACATIOX Tlli;()r(ill f;YKOMAr.NETlC MEDIA 633 1.6 /i' N^Eii MODES \ '^ S;L ">» ^ 0.2 - — . \ 0 0 -->» TE„ MODES 0.4^ ^ / / "^-1.0 A r -y TE,2 ^ 10DES f 0 0 ^ U^ .^^o!4 06^ o""?"^^ 02 ^ -1.0 -0.8 -0.6 -0.4 -0.2 Fig. 13(d) —See Fig. 13 remains finite, while '■o \f •; ^- I or ro 0 0.2 0.4 0.6 0.8 1.0 1 - Xi^ 1 — a\i \ " '\ \ — (s\\f becomes infinite imaginary. Examining tlie solutions obtained under these conditions it is possible to find expansions for ^ in inverse powers of )\) . These are as follows: for p > 0, cr > 1 or /; < 0, — (To < cr < 0 2V2 \ .r„" a — \ \ (T — 1/ To^ (43a) where the .r„ are the successive roots greater than zero of F(.r„) = 1 and the modes are associated with the Xn by the scheme: TEn — ^ Xi , TMii -^ X2 , TE12 — > X3 , etc : for p > 0, 0 < < 0, 0- < — 1 p/2 \yn^ I, V n = ±1, ^--^ 1 + (J 0 < (T < (70 , n = ±l,i3' -; for (To < 0- < 1, no w = ±1 modes; for p 1 + , \. These, in turn, may be classified in the following way. For cr > 1, /i = — 1, and for 0 < cr < ao , " = +1 which correspond to /x and k both positive, the propagation constant tends to the value for a plane wave whose direction of circular polariza- tion coincides with that of the wave guide pattern. For o-o < o- < 1, \ i 1 '- — /^ TE,, - MODES 1, ?i = +1, the propagation constant tends to that for a plane wave whose polarization is in the opposite sense to that of the field pattern and here m is positi^'e, but k is negative. An examination of the field pattern in this last case shows that most of the field energy is indeed associated with a circular polarization opposite to that of the pattern as a whole. The discussion of the preceding sections shows that the complete structure of the mode spectrum for a guide filled with lossless ferrite is very complex. It is also clear that for some combinations of guide size and magnetic parameters the course of an individual mode in the 13" — a plane may be quite involved. In particular, two values of /3" associated with the same mode often occur at a given a. The extent to which the complexity of the spectrum will be observed in practice will depend principally upon the loss of the real ferrite and upon the guide radius. The effect of loss near a = 1, where the incipient modes are crowded will be to cause simultaneous excitation of many of these and conse- quently^ a confused z dependence of the guide excitation. For values of ro just below jn , the point of escape of the TE modes, the latter exist over considerable ranges of a, see Fig. 12(a), and would probably be observable. The TE modes near Un also persist over a wide range, but are double-valued. Concerning such double-valued waves it may be ob- ser\'ed that from the results of the subsequent treatment of losses, it is ., dl3- . . clear that if y-j — ; > 0, it is necessary to put the source of power at the opposite end of the guide. 4.15. Losses, Faraday rotation and merit figure. So far the analysis has been concerned with the loss-free medium. It is of some interest to determine the attenuation constant (the imaginary part of jS) that arises when losses are taken into account. As long as these are small, this can be done rather easily; in fact, sufficiently far from resonance (o- = 1), for each formula giving /3 , we can establish one giving the attenuation constant. If the losses are of magnetic origin we utilize the fact (already demon- strated in section 2) that to first order in a, the permeabilities n, k are functions of o- + ja sgn p, and of no other combination of o", a. Since 0", (X enter ^Maxwell's equations only through /u and k, /3 , which is derived from them, must likewise depend on a through o- + ja sgn p. Any formula for |8" derived for the loss-free medium can, therefore, be generalized to the lossy case by replacing a with o- -F ja sgn p, to first order in a. To this order, then, we find G38 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 /3^ = /3'^ + ja sgn p da where /3' is the propagation constant for the loss-free case. Thus j^ = j^ - :w 2^'d\cT\ (44) and the last term on the right, (multiplied by our scaling variable /So = co\/jLi.eo) is the attenuation in nepers per meter. The present con- vention is that the waves propagate in the positive z direction, as exp {—j^z). It follows that they will decrease in that direction only if 5(/3')V^I o- I < 0. Occasionally this is not the case, and presumably indicates that the direction of the power flow opposes that of the phase velocity. For small dielectric loss, too, it is possible to derive formulae for the attenuation constant from those already obtained; obviously the latter depend on e only through e = en — jei , and can therefore be expanded. But it must now be remembered that /3 was defined as /Sactuai/wv/Mce. and ro as ractuai wVTiIe so that the scaling parameter oi\^ix,e will make contributions to the imaginarj^ part. It is then readily verified that /3a = /3 - i 1 26o/3'(9(ro') {r.Y (45) A few words may be said about the relation of Faraday rotation to the ^~ — a curves. A linearly polarized plane wave travehng in the un- bounded medium along the magnetizing field can be regarded as the Fig. 14 ■ — The course of the fully developed modes (solid lines) and of some of the lower incipient modes (dotted lines) as a function of 0, written as 1„ and of In — (Ia)t , a < 0, written as 1/ /3^ = 0 There is a parametric representation: (46) Xl,2| = e, IpI = 2 sinh d ( 1 — -— and 3n-/ 1 '^ 1 1 Jn-/ 640 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 The slope at cut-off: ("^ - l) coth 6 •sgno-, d^'\ t^l^^-^j^< Ahd d =0 and near —ao we have P' = V 1 (52) ro^dcTol - |a|) (2 - I p Hero I)- Type 3. Intersections of Is ~ (Ia)t; for — cro < o- < 0 only; no sub- script is needed. /3' = 0, (T = -1 -p, and OP') (a/3') 1 F(ro) (see Fig. 15) (54) {da)p (dp). p 1 - F(ro) The cut-off points of the modes follow various schemes in different ranges of a as indicated below. For 0- > 0-0 we have Table I. When a < — o-y , 1,/ replaces 1„ in the Table I. "None" indicates that the mode exists, but has no cut-off. For 0 < 0- < o-u we ha\'e Table II. "N.P." in Table II indicates that the mode is not propagated. For — au < cr < 0 wo have Table III. In this range of a one has also the mode without classical analogue. For ro < ;/i this is cut-off at 3 and 2^ and may have a second branch from 2o2 to 2ox . For fo > wi the second branch only exists. 642 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 Table I Mode Radius TEu TMu TE12 TM12 TEu etc. ro < vi ll I2 \ I3 I4 Is Ui < ro < ji None ll I2 i I3 I4 ji < ro < U2 None None lo \ I3 I4 1l2 < /-O < J2 None None None I2 I3 J2 < ro < W3 None None None None I3 etc. II. Asymptotes, genesis of the modes and spot poirits For I cr I ^ CO there are asjnnptotic formulae: For TEin-modes 2N 1 - ro' 1+^ For TMi„-modes /3^ (55) (56) For 1 0- 1 > (To , all modes have their origui in the points cr = 1 , Xi = 1 oro-= — 1, X2=— 1, where /S" -^ co. The variation of jS with a in the neighborhood of these points is described by the two expansions: for 0- ~' 1 0n' = V an + 1 J; , jl - ftn , 2a^ Z'? _i_ 1 ^ . ^ Xn"- \P + 0((T - 1) (57) where and with the scheme an a„ + 1 2ro" F{Xn) = 1 Xn > 0, n 1 2 3 4 etc. Mode TEu TM„ TE,2 TMi2 etc. GlIDED-WAVE PHOPAGATION TMUOUGII GYKOMAGNETIC MEDIA (US Table II Mode Radius TEii TMu TEi2 TM12 TE18 etc. 0 < m < III X.P. X.P. N.P. N.P. N.P. III < )■(, < ji 2n' N.P. N.P. N.P. N.P. ji < ;-o < II-: 2o' li \ N.P. N.P. N.P. 11 ■! < !■() < J2 2o' 2/ ll i N.P. N.P. ji < '•() < ih 2o' 2/ ll \ I2 N.P. V3 < )-o < ji 2o' 2/ 22' ll \o etc. for ff '^ — 1 0,: = an -\- li(T + 1 Table III Mode Radius TEii TMu TEu TMi2 TE13 etc. 0 < /-o < III N.P. N.P. N.P. N.P. N.P. "1 < ''0 < Jl 3 i N.P. N.P. N.P. N.P. ^1 < ro < Hi 3 \ ll' N.P. N.P. N.P. n-2 < ro < j-2 2i 3 i 1/ N.P. N.P. ji < '-0 < Ui 2i 3 \ 1/ I2' N.P. «3 < '-0 < ji 2i 2, 3 ll' W etc. 2\ an V) 2 (58) + 0((r + 1) where and Vn an + 1 2ro2 ' /^(Z/n) = -1, with the same identification as abov^e. For 0- > o-fl a spot-point is given by /„ — {Jb)t with 644 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 ^' = Xl = 1 + 1 + 0-' and (59) 1 \ 2 2 1 — 0-Xi ro^ ' The identification scheme is again that shown above. For — o-Q < 0- < 0 an isolated identifiable point arises from Ib- {In)T which is expressible in inverse form by the relations 2 1 - + -r-; iS" and J2 Jn' p = (^'- 1) 1 - (60) for To" \^i- The identification of the modes with n proceeds as in the earlier parts of this section. III. Small p. To order p" there exist the following expansions for jQ": for the TEi„- mode 3^ = /3o^+ ^' 1 - a"" 2 Un^ — 1 ^0 V + + 2/3o (1 - o^r 5 + 5Un — 2Un (Un' -ly ' HUn' - 1) 2 .. , X (^0^ - 1) (1 - Uj) ^ • • • ' ^""'^ where /3o^ = 1 - ^ To' For the TMi„-modes ^2 o 2 0" , 1 3^0 — 2 2 /3 = /3o - :^ -, p + 7:, :;^, ^^, ^^ p 1 - a-" (1 - (7^)2 2(1 -^0^) (62) GriDKD-WAVK PROPAGATION THROUGH GYUOiMAGNETlC MEDIA 645 where ^o' = 1 - ' n' The radius of convergence of these series is not known. It is clear that it will depend on a and Avill become smaller as o- -^1. 4.2. The Plasma (pr, = 0, vn == I). The characteristic equation (26) may now be written — , [XiF„(xiro) - 7i] = i- [X2F„(x2ro) - n], (63) xr X2^ where (in contrast with the ferrite case) Xi,2 = i8Ai,2 . The X satisfy ^^j. _ {vE - 1)(1 - ^'M - v.p/ ^^^ _ ^2 = 0, (64) Pe and the x's are given by Xi.2^ = (1 — ^'/vn) — Ph\i.2 (65) From the equations for ps and ve in terms of a, q given in Section 2, equation (64) may be written Xi/ - f j-^ - -/5^) X,2 - /3^ = 0, (66) X1X2 = -^' (67a) (67b) or Xi + Xo = -^ - 0, X2 < 0 will be adopted. Equation (67b) will hereafter be called the plasma relation. The transformation Xi — > — X2 J X2 — ^ — Xi , (T — > — (T leaves the plasma relation unchanged and changes n to —n in the H-equation. As more fully explained in connection with the ferrite sec- tion, it is therefore necessary to consider positive n only if a is allowed to take on negative as well as positive values. As before, only the first azimuthal mode number (n = ±1) is considered in this paper. The method of analysis is the same as that used for the ferrite. Here we shall only sketch the most important steps; the reader will have no difficulty in completing the analysis by referring to Section 4.11. For fixed ro , a contour map of H is drawn in the X, a plane (see Fig. 16 drawn for ro '^ 2.2). The gross features of this map are determined by the lines H = 0, H = ± =0 . For greater detail recourse is had to the 1 — X^ lines = constant, along which values of H are readily generated . 1 — ffX Further help is obtained from a knowledge of the location of the saddle point of H. The infinity curves are given by the same formulae as for the ferrite, except that the line X = 0 is no longer an infinity line: along X = 0, H = —1. Zero curves are given by 0" = — — 1 _ ro^(l - X^) X X[F-KX-0? The branches of aX = 1 are also zero curves in the same restricted sense as for the G function. In the same notation as for the ferrite, all In curves pass through a = 1, X = 1; all /„' curves through —1, —1. The same is true for all 0„ , 0„' curves (n > 0) . The only exception is denoted by Oq , it arises from that branch of F'^ along which F~^(l) = 0. GUIDED \VAVK PKOPAGATION THROUGH GYROMAGNETIO MEDIA G47 o'o — Vl — VT e These hyperbolae have vertical asymptotes Xi = — — , and cut the line X2 0- = Oin — Xa.For a fixed positive o-q , Xi increases from l/o- to o-/(l — q). Thus the second quadrant transforms into the region between a = X(l — g") and a = 1/X in the first quadrant. Simi- larly, the inverse transformation X2 = T'(Xi) transforms the fourth quadrant into the region between o- = X(l — g) and o- = 1/X in the third quadrant. Points outside these regions cannot be site of acceptable solutions of the H equation. In order to locate acceptable solutions, the H = equation is now written in the form H(ki , a, n) = H(T{\i), a, To) when cr > 0, and in the form H{\2 , c, n) = H{T{\i), cr, n) when a- < 0. These equations represent the curves of intersection of the //-surfaces. Along each such curve, both //-equation and plasma relation are satisfied. Their projections onto the first (or third) quadrant give Xi (or X2) as a function of a, and hence X2 (or Xi) from the plasma re- lation. Thus jS" = — X1X2 is known along each solution curve. The rough locating of the solution curves, and the estabhshment of precise analyti- cal formulae near special points on them proceeds in complete analogy with the ferrite case. Here we shall consider only the radius ro '^ 2.2, as typical of radii large enough to permit propagation of the TEu mode GUIDED-WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA 649 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 Fig. 17 — Geometrical exploration of solution curves for the plasma. The dotted regions are excluded by the jjlasma relation; the cross-hatched regions are those in which H{\. a) and H{T{\), a) have unlike sign. Solution curves may lie only in unshaded parts of the 1st and 3rd quadrants, q^ = 0.25, ro ~' 2.0. through the unmagnetized plasma, but too small to admit higher modes. The solution curves for ro '^ 2.2 are indicated roughly in Fig. 17. In the first quadrant, for a > o-q , a solution curve starts at Xi = 1, 0- = 1, passes through the intersection of 7i , (Ib')t and proceeds to infinity as indicated. The formulae in Section 4.21 describe the cor- responding curve near o- = 1, and at o- — > =», showing that the solution 650 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 curve describes the TEn-limit mode. Incipient modes also exist, just as in the ferrite; their end-points on a = (1 — g")Xi (or, briefly, on (X2 = 0)7-) are now the points for which H{\i , a) = —1 and, simulta- neously, a = (1 — g")Xi . Below o-Q , there is only one solution curve for Vo '^ 2.2. It begins at 0- = 0, Xi = jSiso (= — X2 , by the plasma relation), where jSiso is the propagation constant of the TEn mode in the unmagnetized plasma. (In contrast with the ferrite, the plasma becomes isotropic as o- -^ 0). It is cut off at the intersection of the contour H(ki , a) = — 1 in that region with (X2 = 0)r . At that point j8 = 0 and a is best stated, thus: (r = (1 - g-) V(l + y')/[l + (1 - q')y% where y is the (unique) real root of F(jyro) = V(l +2/^)[l+ (1 - q')y']- Alternatively these two equations may (by varying y) be used to generate j'o's and the corresponding cut-off values of a. Of course, the two equa- tions are merely a re-statement of the equations H(\i , a) = — 1, o" = Xi(l — q^), heed being paid to the fact that the argument of F is imaginary in the region considered for the radius under discussion. In the third quadrant f or cr < — o-q , we also find the TEn-limit mode. Its solution curve begms atX2=— 1, 0-=— 1, and proceeds to o- = -co without passing through any easily computed intersections of I curves. Formulae pertaining to the TEn mode in this range are stated in Section 4.22. Again the incipient modes are found in their usual region. For 0 > a > —ao , the solution curve corresponding to the TEn mode begins at o- = 0, X2 = — Aso(= — Xi) and is cut off at the intersection of i/(X2, 0") = — 1 withX2(l — q) = (^ (or (Xi = O)?-). At that point fi =0, and a is given by cr = -(1 - q') V(l - 2/^)/[l - (1 - q')y% where y is the least real root of F{uy) = -V(l -y')[l - (1 - q')y']- Alternatively, this equation can be used to generate tq , and the asso- ciated 0-, if ?/ is regarded as a parameter, which for Ui < n < ji is between zero and unity. At a fixed Vq the higher roots of the last equation with sign reversed and the corresponding a are associated with the cut-offs of the incipient GUIDED-WAVK PUOPAGATIOX THHOIGH GYIJOMAGNETIC MKDIA Gol 0.96 0.94 0.92 0.90 0.88 0.86 t 0.84 o (3 0.82 0.80 0.78 0.76 0.74 0.72 0.70 (a) q'-. = 0.1 ■ r f °;2, ^ / ^ ( 1 ^ 1 / ^ / 1 ' 1 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 t -0.7 o -0.5 -0.4 -0.3 -0.2 -O.t 0 (b) q2= o.\. ■^ "~ 0.2, ^ — 0.3 ^ -- /" 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Fig. 18 — Cutoff value of a for the TEn-limit mode in the plasma as a function of Tss for various 7^. modes in the lower half plane. Similarly the real roots of Fi^mS) = -V(l - y'){\ - (1 -g^)^^j, and the corresponding 0- = +(1 - 9 ) V(i - y^)/[i - (1 - (i')y\ are associated with the cut-offs of the incipient modes in the upper half- plane. These equations have been solved for the TEu mode and their solutions shown in Figs. 18(a) and 18(b). 4.21. Some formulae relating to the plasma (chiefly for TE -modes). The formulas given here emplo}^ dimensionless variables (Section 3) except where otherwise stated. Approximations for extreme values of a or q (a) small a, q not near unity TEim mode: = ,8„' + .1,„(T -f B,J -^ (70) G52 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 where , _ 2 d ( 1 - V ^^- uj-ll- 2 B - ^ 2 J iXllKX [l 1 ^^ ^ -Dot ,, ,., (1 - §2)2 L ' uj - 1 1 2i(,„V/J TM modes show no first order variation with a. (b) small q , a" not near unity. Here ^a and Va are the actual propaga- tion constant and radius, without scaling factors : TEiTO mode: 2 Pa = (1 — g JW MofO — — ;• 2 aq' I 2 \ , ^^4^ 1 — (r2 Vwm^ - 1 (71) TMim modes: /3„^ = (1 - g^)coVoeo - %' - co^o T^ (l " "T^, ) • (72) (c) Approximation for large o- ; q not near unity. TEi„ mode: 1 uj' 2q 1 - g2 ^„2 ^(1 _ ^2) (^^2 _ 1) (73) All formulae in (a), (b), (c) apply to both positive and negative a (right and left circular waves). Formulae 70 and 73 show that the first order changes in /3 , whether due to very large or very small a, have coefficients that differ only in sign. The TEn mode near resonance {q < 1) Near cr = +1, q' \i^' " V (74) /3 = - 1 - g2 0- _ 1 GUIDED-WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA G53 Near a = —1, Cut-off of the TEn mode {q < \) («i < n < ji) (X positive: ^' = 0; cr = (1 - g') V(l + y'/[l + (1 - q')tf-], (76) where y is the only real root or the smallest imaginary root of J^Um) = V(i + y')[i + (1 - q')y''']- a negative: ^^ = 0; cr = -(1 - q-) V(l - y') /[I - (1 - q')y% (77) where ij is the smallest real root of Fim) = -V(l - y')[l - (1 - q')tf]. (See section 4.2 for further explanation). APPENDIX I. THE F-FUNCTION The function F(x) has been defined by the equation F{x) = X Ji{x) ' Using the infinite product for Ji{x) and differentiating logarithmically one finds x' Fix) = 1 - 2 Z ^-^±—^ , (78) n=l Jn X where Ji{jn) = 0. Near one of its poles, jn , F(x) behaves as jn/ix — jn). It is also useful to know the form of F{x) near one of its zeros, ■?/„ , which are also zeros of J\{x). Such an expansion may conveniently be found by using the Ricatti equation satisfied by F(.r), which is dt t 2 7-f2 /wr.N X— = l-x-F. (79) ax The expansion near w„ is then F{un + y) ^ y\ - - Un - ^ — ^ + 1 + higher terms. (80) G54 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1934 c3 <0 CO oo VA .s ° 3 ,< ^ b /< O O 01 o £?E ° ^H ?r ^ oo.t: ^ o 1 ^ :3 AV« J^£ b b X =5 te fe b & GUIDED-WAVE PKOPAGATION TIIKOUGII GYKOMAGNETIC MEDIA G55 The equation may also be used to furnish an expansion near x = 0. This is .2 ,4 F(x) = 1 - ^ - ^ + higher terms. (81) Finally, putting x = jy, one finds from Eq. (79) for large y 1 3 1 FUy) = 2/-:^ + g-+ higher terms. (82) APPENDIX II. INFORMATION PERTAINING TO THE CONSTRUCTION OF G^-DIAGRAMS The accurate construction of the contours G = const, is conveniently based on the contours (l — X )/(l ~ o-X) = const, along any one of which G is a function of X alone. These contours are shown in Fig. 4. Their asymptotic properties are almost self-evident. The curves G = g = const, have various asymptotes. These, together with their range of ^'alidity, and their Polder transforms where needed are stated in Table IV. The formulas given in Table IV show that the curves G = const, generally have two kinds of as3rmptotes; linear and hyperbolic. Formula (83) shows the behavior of G along a line of constant finite slope unequal to 7*0 /jn , the asymptotic slope of the /„ curves. Parallel to a line of slope ro /jn all G contours must be found, not just the restricted range given by the first formula. Writing o- = ( ro^/jn) X + x in the equation G = g, and expanding F near its pole j„ , we find x in terms of g and ob- tain (84) which holds for all g, from — cc to + =o . When g = 0 it also gives the linear asymptotes of 0„ , 0„' curves except Oo , as is readily verified from the equation _ 1 _ ro\l - X') "" X x[F-Hx)? ' for the zero curves. Formula (85) shows how the G-contours tend towards o-X = 1 from the side o-X > 1 as X -^ 0. Formula (86) relates the asymptotic behavior of the curves G = g to the zero curves, g = 0, for small X. All G curves approach zero curves arbitrarily closely as X — > 0. The only exceptions are the infinity curves whose form near X = 0 is 656 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 When n = u„+i , the 0„ curve merges with the On' curve at -2 ro (w„+i2 — 1) Un+l^ ' Similarly all of the contours G = g oi the sheet to which 0„ belongs merge with the corresponding G = g oi the sheet to which 0„' belongs, at _-^(1+4J^^. (88) These remarks apply to all 0„ , 0„' curves, Oo included. The Oo-curve, for large X, behaves as 1. (In fact, for ro < 1, Oo lies wholly in the first quadrant; when Ui > ro > 1, Oo cuts cr = 0 once.) The saddle points of G are most easily found by considering G in the coordinate net formed by the curves x = const, and X = const. At a saddle point dG d: and simultaneously ^x ^x Lx^ \X ^-2 = 0 = 0 The only saddle points that might be missed in this way are points at which the two derivatives are not independent, that is points where the X contours have vertical tangents, and it is easily verified that no saddle points exist there. Proceeding with the differentiations, we find that — - = 0 gives dX F(rox) = 0 or rox = Un (89) and so — =0 gives dx - + ? F\u:) = 0 X X GUIDED-WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA 657 or 2 X„, = !^!L__i (90) The corresponding o-„s are given by 1 — Xns u 2 1 — a-„sX„s r(? (91) and are all positive. Thus all saddle points lie in the first quadrant. At a saddle point G = —ro/u„ and therefore it is the intersection of two contours G = —To /u„ . Forn > 1, one of these obeys the asymptotic formula (83), the other is asymptotic to 7„ and /„_i (see Fig. 5), and obe3's (84), with n and n — I, near those curves. For n = 1, one of them still follows formula (83), but the two "arms" of the other are asymp- totic to 0- = 1/X and 7i , and so follow (85) and (84) with ri = 1 re- spectivel3^ Three further facts useful in the construction of G^-diagrams are: 1 — X^ Un^ r\? Along a curve r = — 2", G equals — — 2; thus the zero curves of F 1 — (tX ro Un are contours of constant G. Along X = +1, (92) r 1-0- ^" 2 - 2 4 (T, X - -> 1 along (o- - 1) = «(X - 1); ^-^ 2 F As 0-, X — > — 1 along (o- + 1) = «(X + 1); « + 1 (93) G^ - ('•" Z^) As for G(T(\), cr, To) we have, in addition to the asymptotic formulas in the table : Ib' transforms into = 1 + — 7—- . 0- + 1 The intersection of (/„)r or (/„')r with cr = 0 is given by % X., = p±y/i-4! 658 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 APPENDIX II. THE FIELD COMPONENTS The field components are given here for the ferrite and for the plasma. They are normalized in such a way that E; takes a simple form. It should be noted that the X's appearing in these equations are those defined in Sees. 4.11 and 4.2 for the ferrite and plasma respectively and have a different significance in the two cases. We write A,{r) = Jnixir) Jn (xiro) A,(r) = Jnixir) Jn (X2ro) Then, for the ferrite, E. = lUr) - .4,(r)]e'"' , E. = -j, ' -^'('■) 1 Ee = -/3 H. = i/3 Fnixir) - ^ r xr I Ai r X2^ 1 ^2 I'nB 'A,{r) _!_ iFnixir) _ 1 _ Mr) ± (KM . r xi^ I X] / r x2-\ X2 in$ I A^{r) - 1 A,(r) Ai A2 .jn0 "Ai(r) 1 xr He = -j 'Ai{r) 1 ; p . N Xi^ 1 2 1 — Xi Xi Mr) _1_ r xi^ n Fnixir) n — 1 ^ 1 - X2 i^n(x2r) 7n9 , and Xi A2(r) 1 (1 - xi^) , /n(x2r) - f (1 - X2O j'nfl and, for the plasma, E. = [Ai(r) - A,{iW' , Er = -''- AM 1 /3 L r xi"' {(1 - x')Fn{xir) +nXi} ^ A {(l-X2V„(x2r)+nX2} ?• X2^ ^jne GUIDED-WAVE PKOPAGATlOiS' THliOUGII GYKOMACNETIC MEDIA G59 Es = ¥r -^ (Xi Fn(xir) + n(i -xr)} . {r) xr ^^^'^A{X2/^«(x2r)+n(l-x/)}' r X-l- ine H.= -^-[XuUir) - X,.Ur)] e'"\ Hr = - and He = -j ^'^"^ % {XiFn(xir) + n} - ^ -1 {X.i^„(x2r) + n\ r xr r X2- ,jn9 L ?" Xi" ?• X2^ )>iO REFERENCES 1. Goldstein, L., Lampert M. A., and Henev, J. F., Plijs. Rev., 82, jj. 956, 1951. 2. Polder, D., Phil. Mag., 40, p. 99, 1949. 3. Hogan, C. L., B.S.T.J., 31, p. 1, 1952. 4. Suhl H., and Walker, L. R., Phys. Rev., 86, p. 122, 1952. 5. Kales, M. L., J. Appl. Phys., 24, p. 604, 1953. 6. Gamo, H. J., J. Phys. Soc. Japan, 8, pp. 176-182, 1953. 7. Cook, J. S., R. Compfner and H. Suhl, Letter to the Editor, I.R.E. Proc, to be published. Coupled Wave Theory and Waveguide Applications By S. E. MILLER (Manuscript received February 2, 1954) Some theory describing the behavior of two coupled waves is presented, and it is shown that this theory applies to coupled transmission lines. A loose-coupling theory, applicable when very lilile power is transferred be- tween the coupled waves, shows how to taper the coupling distribution to minimize the length of the coupling region. A tight-coupling theory, appli- cable when the coupling is uniform along the direction of wave propagation, shows that a periodic exchange of energy between coupled waves takes place provided that the attenuation and phase constants (a and /3 respectively) are both equal, or provided that the phase constants are equal and the dif- ference between the attenuation constants (on — 0:2) is small compared to the coefficient of coupling c. Either {a-i — a^j/c or (^i — /32)/c being large compared to unity is sufficient to prevent appreciable energy exchange be- tween the coupled waves. Experimental work has confirmed the theory. Appli- cations include highly efficient pure-mode transducers in multi-mode sys- tems, and frequency-selective filters. INTRODUCTION This paper describes some theoretical relations in coupled transmission lines, and the use of coupled lines as circuit elements. In order to illus- trate the points of interest in the theoretical material, several applica- tions will be stated first. Detailed discussion of experimental models will be given after the theoretical sections. The theory of coupled transmission lines may be used to determine many properties of a multi-mode transmission system in which there is distributed coupling between modes. In round pipe, for example, the individual modes of propagation can be considered as separate trans- mission lines which in the perfect waveguide are completely independent. Geometric imperfections in the waveguide, if distributed ()\'er many wavelengths, cause a transfer of power between modes which in general 661 662 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 form is predicted by coupled transmission line theory. As a consequence, analysis of the mode-conversion effects associated with circular-electric- wave transmission in commercial round pipe has been aided materially by applying the coupled-transmission-line concept.' In another problem, the transmission of the circular-electric waves through bends,- the coupled-wave theory of subsequent sections has also provided valuable insight. Coupled transmission lines can be employed as circuit elements to exchange power between one mode of a multi-mode line and a designated mode of another transmission line. Consider Fig. 1, which shows a rec- tangular Avaveguide having entries 1 and 2 coupled through a series of apertures to a parallel round waveguide having entries 3 and 4. The rectangular guide may be made single mode for convenience, and for the configuration shown may be made to couple to any TE mode of the round guide. Input power at entry 1 may be transferred in whole or in part to the selected mode at entry 4, the remaining portion of the power appear- ing at entry 2. Very little power in any mode will appear at entry 3 for excitation at 1, and very little power in undesired modes will appear at entry 4. Thus the structure has the h3^brid property in addition to being mode selective. A matched impedance is presented at all entries to all modes over a very broad frequency band. Recently, coupled transmission lines have found use as input and out- put circuits for travelling-wave tubes. In this instance a helical input (or output) line was electromagnetically coupled to the travelling-wave- tube helix, with conditions adjusted for complete energy transfer be- tween the helices. The result is an input-output circuit requiring no metallic connection to the tube helix and requiring no connection through COUPLING APERTURE 3' Fig. 1 — Coupled transmission line transducer. COUI'LKI) WAVK THKOKY AM) WA V Kdll I)l': A Tl'LlCATIOXS G03 the vacuum seal. R. Kompfner conceived this form of connection to travelling-wax'e tubes while working Avith the Admiralty in England, and demonstrated the usefulness of the idea here at the Laboratories. Similar work was done by the group at the Ele(;tronics Research Lab- oratory at Stanford University, and was described by S. T. Kaisel at the August, 1953, West Coast LR.E. Convention. Both groups re- (|uested pre-publication copies of this paper for use in their research. LOOSE COUPLING THEORY On the assumption that negligible power is abstracted from the driven line of two coupled transmission lines, the magnitude and mode content of the forward and backward waves in the side line may be written. With reference to Fig. 2, there is assumed coupling between two uniform Hues in the interval — L/2 to +L/2 along the axis of propagation, and no coupling elsewhere. On the basis of loose coupling a normalized voltage wave on line 2 may be written £-2 = i,o,-'^'^i'^^^^+^'^-\ (1) in which the phase reference is taken as x = —L/2. The forward current // in the side line at the point x = L/2 is L If = KFM J I 4>(x)e''''^"^'-"^'^' dx, (2) 2~ where — iirtd/Xl+l/Xz) " = '— z ■ (f){x) = a couplmg function. More precisely, 1/0 (.x) is the ratio of the voltage on line 2, Ei{x), to the equivalent voltage generator in series with line 1 at x. K = fraction of the transferred current which travels in the forward direction. M = the transfer constant for the various modes which can propagate, relative to the mode for which If! >Z,o 1 LINE 2 1 Z20S 1 SZ20 ^ *-X Fig. 2 — Schematic of coupled transmission lines. COUPLED ■\VAVK TJIHOUV AM) W A VKi; f 1 1)1'; Al'l'LICATIOXS ()()5 A simplified example will illustrate the application of these relations. Suppose the coupling function 4>(x) is constant in the inter\-al —L/2 to L/2 and zero for other values of x. Then the discrimination function is, from (4) Uniform Coupling Discrimination = . (5) sni 6 Let us further assume, in the hypothetical example, that line 2 (Fig. 2) is a single-mode line having a guide wavelength X2 equal to 1 .2X0 , and that line 1 is the three-mode line ha\'ing guide wavelengths Xi , Xo , and X3 equal to l.lXo , 1.2Xo , and 1.3Xo respective^. Assume the coupUng length L equals 20Xo . For equal coupling to all modes in a differential unit of length, the relative current waves travelling in the forward direc- tion in the three modes of line 1 are obtained from (4). For the ratio of the Xo forward current to the Xi forward current, for which (5) gives a discrimination of about 13.5 db. For the ratio of the X2 forward current to the X3 forward current corresponding to a discrimination of about 14 db. For the I'atio of the Xo forward current to the X2 backward current, corresponding to a discrimination of about 43 dl). The backward currents in modes Xi and X3 can similarly be verified to be very small compared to the forward-travelling X2 current. Thus, directivity and mode purity in a simplified case have been shown to be of the desired form. It may be noted that the denominator of (4) is the Fourier ti'ansform of the coupling function (f)(x). Since the numerator of (4) is independent of 6, the discrimination is maximized by minimizing the denominator. An analogous prol)lem exists in the time versus frequency domain rela- tions, and experience with the latter can be used to predict the discrim- inations to be expected using various coupling distributions. In the simple example cited al)ove, a length of coupling interval of 20X0 yielded a discrimination l^etween the desired versus imdesired for- 666 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 Fig. 3 — Discrimination versus O/iz for linear taper coupling. ward wave components of about 13 db. How can this discrimination be improved? If the difference between the wavelengths of the desired and undesired wave types is increased, the value of Q is increased and greater discrimination results. In practical cases, however, there frequently is very little that can be done about the wavelength difference because it is inherent in a structure Avhich is fixed by other considerations. By increas- ing the length of the coupling interval L the value of Q is also increased; in the case of uniform coupling, (5) shows that a value of B/r equal to about 8 is required to get 30 db discrimination. In the above example this corresponds to L approximately equal to 125Xo . The latter coupling length is probably impractical, and is certainly inconvenient. The final alternative is to alter the distribution of coupling between the lines, and considerable can be done in this manner. Suppose a Hnear taper of the strength of coupling is used, as sketched in Fig. 3. Then the discrimination becomes Linear Taper Discrimination = g/2 sin ^/2 (6) which is plotted in Fig. 3. The first peak in discrimination occurs at QJ-k equals two, compared to a value of 0/t equals one for the first peak using uniform coupling; however, for all values of ^/tt greater than about 3, the linear taper provides superior discrimination. This illustrates a gen- eral trend; tapering the coupling distribution improves the discrimina- COUPLED WAVE THEORY AND WAVEGUIDE APPLICATIONS GG7 tion for large Qj-w values at the expense of an increased S/tt value for the first discrimination peak. The first two lines of Fig. 4 gi^•e the discrimination functions for two forms of cosine taper; Fig. 5 shows a plot of the first function and Fig. G shows a plot of the second function for a particular case. These figures illustrate the importance of the slope at the ends of the coupling distri- bution. Comparing Fig. 5 with Fig. 3, Fig. 5 has a larger end-slope, shows a lower value of Q/-k for the first peak in discrimination, but provides FIGURE FOR DISCRIMINATION DEFINITION 95(X)=COs(j^) S(X) SHAPE A DISCRIMINATION COS e 95(X) = i+||i+cos(^)[ 2 ■f) SIN e Ck SIN (kg) ^-o)Vm f95(x) = l + C L"_kL Z 2 2 2 (i+ck) SIN e ^rk siN(ke) 9 * (ke) rs^(x)=i !^| I*— -*| 0.273 L Fig. 7 — Discrimination versus Q/-K for two uniform couplings superposed. the discrimination is greater than 38 db for ^/tt between 1.95 and 3.0, and is greater than 50 db for ^/tt larger than 3. Below QJ-k = 1.95 the discrimination is similar to that shown in Fig. 6. Linear superposition of two uniform coupling distributions yields a structure which is easy to fabricate and, in cases where the requirements are not too complex, may provide satisfactory discrimination. The third line of Fig. 4 gives the general relation, and Fig. 7 shows the discrimina- tion plot for a case of interest. Discriminations on the order of 30 db are available in a broad region between d/ir equal to 1.3 to 2, an attractive abscissa value compared to the d/ir = 8 required for simple uniform coupling. Linear superposition of a linear taper and uniform coupling also yields a structure which is easy to fabricate, and the theoretical discrimination plot for an interesting set of conditions is shown in Fig. 8. High discrim- inations are provided over greater ranges of 6 than for the case of two uniform coupling functions superposed. The general relations involved in the superposition of coupling func- tions may be summarized as follows: Let c/)i(.r), 4>2{x) • • -ipnix) be kno^^'n coupling functions and let (f)r = <^i + 2 + (7) Let the maximum length of the coupling interval be L. Then, designating the transforms of 01 , 02- • •„ by Fi and Fi- • -Fn respectively, Avhere .L/2 ''-L LI2 (t>n(x). .i(2fl/L)i-' dx, (8) 670 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 60 Fig. 8 — Discrimination versus B/v for a linear taper and uniform coupling superposed. and letting /^T- = i^l + /^2+ •••i^n, (9) the discrimination function for the composite couphng distribution ^t{x) is given by Discrimination Ft{Q = 0) Ft (10) Another useful theoretical approach to the employment of multiple distributed coupling functions is illustrated in Fig. 9. The top sketch represents any coupling function <^i(a;). The lower sketch shows a new coupling function 2(^) formed by locating a 4>iix) at ±d/2 on the "x" axis. Using F\ to denote the transform for <\>\{_x), and Fi to denote the transform corresponding to 02(a;), wherein Fa = 2iPi cos Q', ''-a -a (11) COITPLED AVAVE THKOHY AND WAVEGUIDE APPLICATIONS G71 for the forward wave discrimination and for the (hrectix'ity as defined earlier in connection with (4). The discrimination fnnction for the composite couphng function P,. . . ,. F,{d = 0,d' = 0) F,{d = 0) 1 ,^_. Discrmimation = ^ = -. (12) Fo Fi cos 6' The factor 1/cos d' is the discrimination function associated with two point couplings, and the overall discrimination is the 'product of that discrimination and the discrimination associated Avith a single distributed coupling function 2,di = L/3, and di = L. Then (14) yields Discrimination = 7- = — - — — . (15) 6 cos e/d -f 2 cos 0 cos^ d/S CENTER OF ARRAY J 80 3 1 |„ L H Fig. 10 — Schematic of point coupling distributions. COUPLED AVAVE THEORY AND WAVEOUIDE APPLTOATIONS 673 38 in LU CD UJ4J36 zo -0-34 .in ^tr ujuj 32 -J CO LU< gi 30 m5 28 / y / / / / / / / / / / / / / 0 1 RATIC 1 1.2 1.3 1.4 1.3 1.6 1.7 1.0 1.9 2.0 %■ AT FIRST NULL WITH EQUAL SIDE LOBES %■ AT FIRST NULL WITH UNIFORM ARRAY Fig. 11 — For all spurious mode responses down ordinate db, abscissa is the ratio of the required coupling length to the length required for constant amplitude coupling to produce a first null at the same value of (1/Xi — IA2). (After C. L. Dolph, Reference 4). which is the relation given by Mumford. The approach is perfectly gen- eral, and henceforth the coupling distribution only will be given with the understanding that the corresponding discrimination function can be obtained from (13) and (14). For the case of tapered amplitudes and an even number of equally spaced couplings, (13) can be simpUfied to 2ai cos 2n - 1 + 2a2 cos 3d 2n - 1 + • • • 2a„ cos 6. (1(3) This case is of interest because a solution has been worked out for the analogous antenna problem to bring the spurious responses (the peaks of the side lobes in the antenna case, or the peaks of the undesired mode responses in the wave selector case) to the same lex'el relative to the de- sired response. This makes the total length of the coupling array a minimum for a given required degree of discrimination. The solution includes specifi- cation of the Tchebysheff distribution of coupling strengths Oi , 02 , a^- • • ttn that are rec^uired to achieve various levels of spurious response, and the resulting increase in total array length required to place the first null 674 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 in undesired mode response at the same value of L _ LViY- +- Ai A2/ \Ai X2 as for uniform strength couplmgs equally spaced. Fig. 11 shows the latter relation, a very useful yardstick with which to evaluate the extra coupling length required by less ideal but more easily constructed coupling distri- butions. An important practical question is "What is the smallest number of point couplings which will satisfy requirements in a given stiuation?", for it is time-consuming and expensive to fabricate the coupUng holes or probes in some circumstances. The large range of possible mode condi- tions and discrimination requirements makes it difficult to give an answ^er in closed form, but the general restrictions involved may be stated. In the case of n equally spaced couplings (of any amplitude taper) the dis- crimination vanishes at d/w = (n — 1). This is illustrated by the dis- crimination plot of Fig. 12. Moreover, it is found that equally spaced couplings produce discrimina- tions which are periodic in d/w on the interval (n — 1), and which are symmetrical about O/t = (w — l)/2. The implication of the discrimination zero at d/w = (n — 1) is that a large number of point couplings are required to get good directivity and good forward wave discrimination. In the simple case cited above in which L = 20Xo , the O/t value for directivity was shown to be 33.3. t 20 .L 1 1 > I i I 1 J, 1 1. i\ \ [ A k A - J J ^ y V f V \j \ v/ V J 0123456789 e 77 Fig. 12 — Discrimination for 8 equal-strengtli point couplings equallj^ spaced. COUPLED WAVE THEORY AXD WAVKGUIDE APPLICATIONS 675 Thus, something; on the order of 50 or 00 (Mnially spaced couiihngs might be needed. Simulation of continuous coupling functions with eciual strength couplings may he carried out as follows: the coupling amplitude \-ersus distance plot may be di\-ided along the distance axis into a number of inter\-als of equal area, and a point coupling placed at the center of each interval. The more efficient continuous coupling functions recjuire more point couplings to get a good simulation in this manner. For example, the function of line 2, Fig. 4, "with c = 22.4 and /.• = 1 has been simulated with 12 and 40 equal strength couplings (as described above) and the exact discrimination plotted using (13) and (14). The results are given in Figs. 13 and 14. The original continuous coupling function yields dis- criminations greater than 38 db for all values of ^/tt greater than 2 ; the 40-point simulation approximates this well in the region of Q/-K = 1.7 to 4.5, but thereafter begins to fail. The 12-point simulation (Fig. 13) never matches the original but does best in the region of small Q/t. It is more efficient to seek high discriminations by tapering the strength of equally spaced couphngs than by tapering the spacing be- tAveen equal strength couplings. However, when Ioav discriminations are acceptable, the relative efficiency of tapering the spacing between con- [<— 0.341 L — >| I 0.0682 L I U a503L A I U 0.700L >l I ME Fig. 13 — Discrimination for 12 oqiial-strcngth point couplings arranged to simulate the continuous distribution of Fig. 4, line 2. 676 THE BELL SYSTEM TECHNICAL JOURXAL, MAY 1954 Fig. 14 — Discrimination for 40 equal-strength point couplings arranged to simulate the continuous distribution of Fig. 4, line 2. < 20 r ^2 V ^2 ' Xi = 0.208 L X2 = 0.103 L X3 = 0.170 L X4 = 0.038 L 1 1 M i 1 [ \j\ l\j 1 f / , ' / I / \ / J V y V -^ 4 0/7T Fig. 15 — Discrimination function for 8 equal strength couplings arranged to maximize the bandwidth (e/ir range) for moderate discrimination. COUPLED WAVE THEORY AND WAVEGUIDE APPLICATIONS C)77 stant strength couplings is much greater than when high discriminations are required. Fig. 1.3 shows a (hstribution wliich i)ro(hi('es about 20 db discrimination from QJ-k = 1 to 3.25. Eight coujolings arranged with the Tchebj'sheff amplitude taper for 20 db discrimiuation would i)roduce that discrimination from 6/w = 1.05 to 5.95. It is possible to obtain directivity or mode discrimination at smaller d/ir values than made available with uniform coupling. This situation is analogous to the superdirectivity problem in antenna design, with similar results — the lobes of spurious response are increased. In particular, if the coupling near the ends of the third array of Fig. 4 is made larger than the coupling in the center region, making "c" a negative quantity, the first peak in discrimmation occurs at d/ir less than one, and the first minmaum in discrimination becomes less than 13 db. B}^ implication, emphasis has been placed on obtaining both mode dis- crimination and directivity simultaneously. However, by employing a relatively short coupling length it is apparent that the discrimination associated with may be kept small when the directivity associated with is in suitable range for good discrimination. Consequently, one can de- sign a directional coupler with little mode discrimination. Conversely, when using a relatively small number of point couplings, the mode dis- crimination in the forward wave may be good when the directivity is poor. TIGHT COUPLING THEORY* We now consider the case in which a significant amount of power is taken from the driven transmission line by the line coupled to it. To simplify the problem the coupling is assumed uniform along the length * An analysis of coupled transmission lines was given by W. J. Albersheim," and the effects of coupling between waves on certain particular forms of trans- mission media were analyzed by Meyerhoff' and Krasnushkin and Khokhlov.*" The treatment given here is intended to be more general and is believed to de- scribe the effects of wave coupling under a greater variety of conditions. 678 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 axis. The space variation of the wave ampHtude may be written '^-^ = -(ri + kn)E, + k-nE, , (17) and '^ = k,,E, - (r, + /:22)^2 , (18) ax in which kn , A-22 represent the reaction of the coupUng mechanism on lines 1 and 2 respectively ^21 , ki2 represent the transfer effects of the coupHng mechanism ri,2 are the uncoupled propagation constants of line 1 and 2 respectively ; -E'1,2 are the complex wave amplitudes on lines 1 and 2, and are so chosen that \ Eif and | E2 \ represent the powder carried by lines 1 and 2 respectively at the input or output of the coupling region. The usual transmission-line equations are of this general form, except for second derivatives iii place of the first. The first derivatives appear here because we deal only with the forward travelling waves, which the preceding section has shown are the only significant waves when small coupling per wave length is employed. Limiting our interest to the cases for Avhich reciprocity holds and noting that there is alwaj^s a transverse plane of symmetry midway between the ends of any pair of uniforml}^ coupled lines, we ma}^ transform the wave amplitudes to make ki2 = kii = k. We may further simplify the equations without loss of essential generality by submerging the differences (kn — k) and (k22 — k) into a modified propagation constant for lines 1 and 2 respectively, yielding and in which dEi dx dE2 dx = -(71 + k)E, + kE2, (19) = kE, - (y, + k)E2 , (20) 71 = Ti + An - k, and . 72 - To + A-22 - k. For some cases kn = kn = k and for all cases of interest here 7„ differs very little from r„ since we are concerned only with loose coupling per wavelength. COUPLKD WAVK THKOin AND W A \ i;( ; T 1 1)|; A I'l'I.K Al'K ).\S (170 The solution, for Ei = ].0 and 7^2 = 0 at x = 0, is 1 (Ti - T2) Ei = and 2\/(7i -72)2 + 4fc2_ + 1 , (ji - 72) 2 2v^(7i - 72)2 + 4/o2_ (21) k k ."here n = -H(2A- + 71 + 72) + }W(7i - 72)^ + 4ib2, (23) r2 = -1/^(2/.- + 71 + 72) - i.i\/(7i - 72)- + 4A-2. (24) The nature of the couphng eoefhcient A- is the first thing to investigate. Assume no dissipation in either the transmission Hne or in the coupling mechanism. Then it follows that for any value of x, 1 ^1 I" + I -^2 I" = constant (25) on the basis of energy conservation. It may be determined that (25) leads to the requirement that the coupling constant k be purely imagin- ary. This is a very important result. In all of the following discussion A- is taken to be purely imaginary. Even where dissipation in the trans- mission lines themselves is important, it is still assumed that the coupling mechanism is non-dissipative. The simplest case is 71 = 72 = 7, coupling between identical trans- mission lines. Then (21) and (22) reduce to El = cos ex e-^''^'^\ (26) and E. = i sin ex r^''+'^\ (27) where k = ic. The exponential of (26) and (27) shows that the coupling modifies the average phase constant, and that the attenuation in the driven line (Ei) is the same as in the uncoupled case for ex (coupling length times coupling strength) eciual to nw radians. The amplitude and phase variations due to the coupling are plotted in Fig. 16. Complete power transfer between lines takes place cyclically, A\ith a jjeriod of ex — IT, and with suitable choice of the product ex, an arljitrary di\'ision of power between the lines may be selected. 680 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1054 90 \ ^^^ \ V ^ V \ N x \. 'n \ X \^ X X ^ X X N X X ^ x N !i \ l.U :.^»- -!<-■ ^^v^ /' N y^ ^\ ^ ^V \ X X ^ \ /|Ed \ \ /|E,| \ / / LU o D t 0.5 / / / \ \ \ / / \ / / \ \ / \ < / \ / \ 5 / 1 1 1 \ \ / / / 1 \ 0 1 ' f 1 "424 42 |cxi IN RADIANS Fig. 16 — Wave amplitude and phase factors versus the integrated coupling strength ex for tightly coui)led transmission lines having identical propagation constants. Let us now assume that the phase constants of the two lines are un- equal, l)ut the attenuation constants are the same. Then cxi — a? — a, and (ti - 72) = i(^i - /32), (28) and equations (21) and (22) reduce to 77- _ -ta+>{c+(/3i+/32)/2)]x r' * /2Q^ where E* = cos ^(^1 - /S.) 1 y^ - 02y ^c" -f 1 ex 2c /^ ^2)^ + - sin r/(A 1 ^ ^ - 02)' 4c2 + 1 ex 4c2 TJ _ -[a+t(c+(/9i+^2)/2)]xjp * (30) (31) COUPLED WAVE THEORY AND WAVEGUIDE APPLICATIONS G81 1.0 0.8 ^ N ;,---■ ^-•■ X;^ (a) \ ^^'' / \ 0.5 f / ^<-- -■■/ A-./3.-/32 ^0 \ / A f^^^' """"x ^ ^-^'-^2=0 r N N X 0.5 N X (b) /'" f — "'^ ^-v. \\v / / ^> \ V /^ \ \ / \ 2^ \\ / '^ \ \ \ / \ \\ / \ \ 4 / \ \ \ / \ / >\ / \ / N \ / N / / \S 0 0.t77 0.2 77 0.3 77 0.4 77 0.577 0.677 0.777 0.877 0.977 77 CX IN RADIANS Fig. 17 — Wave amplitude and phase factors versus ex when the coupled lines have equal attenuation constants but unequal phase constants. where E.y /^ - ^^y 4c2 + - - [^(A + /32)- 4c2 + 1 CX (32) The major effects of couphng in this case are represented by £"1* and E2*, which are plotted in Fig. 17 for several values of (/3i — (32). As (0i — ^2) becomes different from zero, the maximum power transferred from the driven line to the undriven line decreases, and the period of the cyclical \'ariation in amplitude is reduced. The latter period is the value of ex given by / (gi - ^2)^- 4c2 + 1 CX = TT (33) The driven and undriven-line wave amplitudes E* and E-* at the max- imum power transfer point, namel}^, at \/'- (81 - Bo)- TT (34) 682 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 100 60 40 20 10 6 . 4 1.0 0.6 0.4 El* E£ 0.001 0.004 0.01 0.04 0.1 0.2 0.4 1.0 2 4 6 10 20 40 20 LOG IeI Fig. 18 — Wave amplitude factors at the maximum power transfer value of ex versus (/3i — ^2)/c when the coupled lines have equal attenuation constants. are plotted in Fig. 18 as a function of the ratio (/3i — ^2)/c. It is evident that this maximum energy transfer may be made very small for suitably large values of (/3i — /32)/c. The behavior of £"1* and E2* as a function of coupling length x is shown with greater accuracy in the wide amplitude range of interest in Figures 19 and 20 respectively. Consider now the case in which the coupled lines have identical phase constants, |Si = 182 = (3, and unequal attenuation constants so that (ti "" 72) = («i " «2). Then (21) and (22) reduce to iii = e -[ai+i{c+d)]x 1 2 (q!i — 0:2) + 2V(ai - «2)2 - 4c2_ (ai — 0:2) 1 , .2 2\/lai - a ,[(ai-a2)/2+}.^V(«l-«2)2-4c2]j; ,[(ai-a2)/2-M\/(«l-«2)2-4c2]x and P _ -[ai + i(.c+fi)]x 2)2 - 4c\ ic (35) (35') |e[(«l-«2)/2+J^\/(«l-<*2)2-4c2]x V(ai - a^Y - 4c2 ^ (36) — e[(«l-a2)/2->^V(«l-«2)2-4c2]xl or XV2 — e ii2 (36') COUPLED WAVK TIIKOIIV AM) W A\ IOC I 1 l)K A I'l'LlCATIONS G83 The amplitude factors Ei** and E2** have been defined in such a way as to reflect the principal effects of attenuation difference in the two lines; for the case in which the driven line attcnual ion constant ai is negligible, note that £"1** and E^** contain all the anii)litude variations of Ei and E2 respectively. In general, Ei** and E-** are the ratios of the wave am- plitudes actually present in lines 1 and 2 respectively lo the wave am- plitude which would exist in lino 1 at the same value of x in the absence of coupling. -80 - _ -60 -40 -20 - - \ /, W^ = o -10 -8 -6 -4 //^ ^\. \ - // \\ - / 0.5V y - / / \\ - /^ ^s5 \ _ -2 I — // \ ^ L ' -1 -0.8 7 ^ -V- w / \ \ -0.6 -0.4 -0.2 -0.1 -0.08 -0.06 -004 / ^_^ \ \ ■ // \ \ \ - / \ \ \ \ \ \ / \ _/ I \ - f" U \t — I -j — -0.02 V V ' — 4— -0.01 \ V 1 1.5 2.0 CX IN RADIANS Fig. 19 — Driven line wave amplitude versus cx_ with unequal phase constants and equal attenuation constants. The curves are periodic for larger values of ex. 684 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 -40 -2 iH-0.8 o O -0.6 O O -0.4 -0.2 -0.1 -0.08 -0.06 -0.04 -0.02 \/. ^ / / \ /\ \ / ' \ / / \ / \ \^ / / \ S. / / - v^ / / ^ - \ «~-^-^^ / \ > / - / / - /o.5 - / - \/ 'f-o - - 1 - 1 1.5 2.0 CX IN RADIANS Fig. 20 — • Undriven line wave amplitude versus ex with unequal phase constants and equal attenuation constants. The curves are periodic for larger values of ex. We consider first the case of {a\ — ao) negative, i.e., a lower attenua- tion constant inthe driven line than in the undriven line. The effects of unequal attenuation constants may be illustrated at the integrated coupling strength ex = 7r/2 which, as Fig. 16 shows, results in complete transfer of power to the undriven line when ai = ai and iSi = ^2 • Fig. 21 shows that the driven line wave amplitude £"1** is very small when {ax — a-^lc is small, but is only Y^ db below unity when (ai — a2)/c is about 55. Fig. 22 illustrates the way the undriven line wave amplitude £"2** decreases as {ax — a^)lc increases. For integrated coupling strengths less than 7r/2, the effects of unequal attenuation constants are not pronounced at small (ai — a-i)lc, but again for large (ai — 0:2)/^, Ex'* approaches unity and 7?2** becomes small. COII'LKD WAVIO TIIKOUV AND WAVIOCiUlDlO APPLICATIONS 08; -60 -40 -10 -8 -6 q -2 -1.0 -0.8 -0.6 -0.4 - 1 — 1 - ""^ c K 2 j V ^ y \ - 37r 4 y - y ^v - ^ '^ S. — - -«. ^ \ \ s ... 77- 4 ^'"*"^«. N. \ s \ - S, \, \, - — — N N , N - 77- 8 ~~ >« \, \ \, - N s \ \, ^ \ s. 1 , 1 1 s 1 \ S 1 1 -0.02 -0.04 -0.1 -0.2 -0.4 -1.0 -2 «, -a-2 -20 -40 -100 Fiy;. 21 — The effect of unequiil attenuation constants on the driven line wave ain])litu(le for equal phase constants, and ex constant. Negative (ai — a-{) indicates tliat the undriven line has the larger attenuation constant. -100 -80 -60 ^-10 * -6 -0.8 -0.6 — n ,^ ^ ^ K ^ ^ ^ '^'^^^ ~ 377 y^ 4, ^^ ::y '— 4 y / cx=f k - J / / / / . 1 1 -0.02 -0.04 -0.1 -0.2 -0.4 -4 -6 -10 -20 -40-60 -100 0(.x~0(.2 Fig. 22 — The effect of un('(|u;d attenuation constants on the undi-ivcn line wave amplitude for ecjual jjhase constants and ex constant. 086 THE BELL SYSTEM TECHNICAL JOURNAL, ^L\V 1954 -100 -60 ^-^ _Si^ -40 \ y \ :=^ ^-^ -10 oj F -^ -^^:z. =- -= ■ -6.0 ^-4.0 Uj -2.0 ZA p ^ '^X— =To^ — ^ ==^ 8 -'0 -1 -0.6 —7 r 7^ . . /// / «1 - «2 , — ■ -0.2 -0.1 t- ^^ c -0.06 -0.04 -0.02 -0.01 V 7^ 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4 0 4 5 5 0 5.5 6.0 6 5 7.0 CX IN RADIANS Fig. 23 — Driven line wave amplitude versus ci with equal phase constants and (ai — a2)/c as a parameter. The curve for {a.\ — a-^^/c = 0 is periodic. For integrated coupling strengths greater than 7r/2 the effect of small values of (ai — a^^lc is to increase the loss to £"1**, as shown by the curve for ex = 37r/4 in Fig. 21. However, for sufficiently large values of (ai — a-2)/c the loss to El** is made small. The variation in £'1** and £"2** as a function of coupling strength (ex) is given in Figs. 23 and 24. The periodicity of Ei** is removed for -100 -60 -40 -20 -10 -6 -4 -0.6 -0.4 -0.06 -0.04 -0.02 -0.01 - \ -100 \ ^ .^ -10 / ^ ' ^ \c — _ — J ■--jz. f- " ' V^ P iS- r^ \ -1 \ ar«2 V c - \ \ \ \ - \ I 1 1 11 5.0 5.5 6.0 &5 7.0 0 0.5 ID 1.5 2.0 2.5 3.0 3.5 4.0 4.5 CX IN RADIANS Fig. 24 — Undriven line wave amplitude versus ex. with equal phase constants and (ai — a.'i)/c as a parameter. The curve for {ax — a.i)/c = 0 is periodic. COUPLED WAVK THKOlfV AND WAVKCUIDK APPLICATIONS G87 (ai — a-i)lc as small as —1, but a value as large as —10 or more is re- quired in order to reduce the loss to is'i** to a moderate value for large integrated coupling (ex) \-alues. When {ax — ai) is positive, the attenuation constant for the undriven line is less than that for the driven line, and under these circumstances £"1** can exceed unity. Physically this means that the power loss line is caiiying the energy for a distance and returning it to the driven line at a more distant point. The curves of Fig. 25 and Fig. 26 show the varia- tion of E** and £"2** versus positive {ai — a2)/c values, at fixed values of integrated coupling strength ex. For ex equal to 7r/4, the driven line wave magnitude £'1** decreases as the ratio (ai — q:2)/c assumes small positive values and goes through a balanced type of null near (ai — a-i)/c = 3.5 (see Fig. 25). Again this is the resultant of the lower loss undriven wave carrying power for a distance and returning it to the driven wave in the proper phase to cause cancellation of the straight- through component of the driven wave. For ex between 7r/4 and t/2 the null would move from (ai — a2)/c near 3.5 toward (ai — a2)/c = 0. Figures 27 and 28 show the variation of Ei** and E2** versus the integrated coupling strength ex at fixed values of (ai — q;2)/c. In these -60 —\ ! -40 ' ^ - -~-<^ 2 / -20 \ / \ / -6 -4 \ k y / 77 \ 1^ ^\ 4 \ \ -^ ^ -1.0 V. N 377- \ \ \ \ \ \ 06 04 0.2 -ni \ \ I \ \ 0.2 0.4 0.6 Fig. 25 — Driven line wave amplitude versus (ai — an)/c with e([ual phase fonstants and ex constant. Positive (ai — a2) indicates the undriven line has the smaller attenuation constant. -4.0 -3.0 n -0. ° -0. cx--^ -ii^V4 - ~ 3;7N 4 \ N \ > V \ \ 4 \ \ \ \ \ \ \ \ I \ \ 0.04 0.06 0.1 0.2 0.4 0.6 1.0 Fig. 26 — Undriven line wave amplitude versus (ai — a2)/c with equal phase constants and ex constant. -100 -60 -40 -20 -10 -6 -4 -1 -0.6 -0.4 _i o 0.2 (\J 0.4 0.6 1 1 , \ 1 u \ j t 11 1 II 1 "/ f' // «-1-0t2_g 1 //"T"~- — j 1 \ 1 \ \ \ \ \ A \ \ /, \ \ \ \ / N \, / ' \ \, s / t \ s -\ »». , ^^ ■ — ^ v 1 ^ ^ :^ ^^ 20 LOG 6-CX X ■- 20 LOGfiSCX --«. ^^ ■ — ^^^f^-=-= ■^-*- — _ __ ~-^ 1 i 1 1 ~" •" 40 60 100 0 G5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4 5 50 5.5 6.0 6.5 7.0 CX IN RADIANS Fig. 27 — Driven line wave amplitude versus ex, for equal phase constants; (ai — a->)/e as a parameter. 688 COUPLED WAVK THEORY AXD WAVEGUIDE APPLICATIONS (iS9 figures a double logarithmic scale is used on the ordinate to represent amplitude ^•ariations from 50 db below unity to amplitudes 50 db abo\'e unit}'. An arbitrary break in the scale has been made at ±0.1 db which for practical purposes will be assumed to correspond to amplitudes of unity. With reference to Figure 27, small positive values of (ai — a-^jc move the first null in isi** from ex = ir/2 toward lower \alues of ex. For abscissa values greater than 7r/2, Ei** exceeds unity. For (ai — aoj/c = 1, 7^1** again has a minimum in the vicinity of ex = 37r/2 but this second null has disapp(nu'ed for (ai — ao)/c = +2 and presum- ably also for larger positive values. With reference to Figure 28, E^** grows at a more rapid rate as a function of ex when (ai — a2)/c takes on positive values. The null in the vicinity of ex = tt is still present for (ai — a2)/e = 1 but has disappeared at (ai — a2)/c = 2. For (ai — a2)/c equal to +2 (and presumably for larger positive values) the undriven wave amplitude £"2** is greater than Ei** for ex larger than about 0.5. The (luestion comes to mind in connection with this case in which the -30 -20 -10 -6 -4 0.4 0.6 4 6 10 20 40 60 too 1 \ ^ « \\ \\ ai-a.2_^, r '^'^ =+2 / , Y j 1 I \ 1 1 i\ J s\ V ) \ * A ^ / \ ^ V V^ ^ 20 LOG €^^ V 1 ^" 1 / -^ •fc ^ s:!-^- ^ ■ — cJ -^ 20 LOG f 2CX — — . ' " — . "~"1 «»^ — - - _ 1 1 ""^^•J _ 0 0.5 1.0 1.5 2.0 2.5 3.0 35 4.0 4.5 5.0 5.5 6.0 6.5 70 CX IN RADIANS Fig. 28 — Undriven line wave amplitude versus cz, for equal phase constants; (ai — a2)/c as a parameter. 690 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 undriven wave has a smaller attenuation coefficient than the driven wave, "How much less is the undriven line wave amplitude than would have existed at the same value of x if the same incident wave had been launched in the lower loss line and in the absence of coupling to the higher loss line?" This amphtude difference for the condition (ai — a-2)/c = 1 is represented in Fig. 28 by the difference between the cur\-e for E-/* and the curve labeled 20 log e". Similarly, for the condition (ai — a2)/c = 2, this amplitude difference is represented by the difference between the curve for £2** and the curve labeled 20 log e ". The general case of 71 ^ 72 is important both in interpreting undesired mode coupling effects in multi-mode systems as well as in e\'a]uating -40 -30 -0.6 -O.S -0.4 -0.1 -0.08 -0.06 -0.05 -0.04 i 1 i \ i\ \ A A \ / \ -\ /\ / \ \ / \ / \ / ^ / ^ \ / \ \ ' \ \ \ w \i V y A A A \ / \ /\ \ \l \i / Vy \ J 1 \ \J / - \ \J 1 ^E,*- \ 1 V hr* - 1 10 0 I 2345678 CX IN RADIANS Fig. 29 — Driven and undriven line wave amplitudes versus ex with (ai — a2)/c 0.03 and (/Si - 02)/c = 0.5. COUPLED WAVE THEORY AND WAVIXIUIDE APPLICATIONS ()91 O -0-6 ^ -0.4 -0.04 -0.02 - \ E*^ __ _ — * — ^ ^r* / " / r / / / / / 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 CX IN RADIANS Fig. 30 — ■ Driven and undriven line wave amplitudes versus cz with (ai — a-ii/c = -2 and (/3, - ,32)/c = 2. ~~ errors iii construction of devices intended to produce 71 = 72 . To facili- tate discussion of this case we define and EJ _ 77- *** -|ai + i(c + ((3i-|-^2)/2)]x (37) (38) where Ei and E2 are defined by (21) through (24). The relation between £"1*** and El (or £"2*** and Eo) is the same as described in connection with (35) and (36). Small deviations from 71 = 72 are represented in Fig. 29, which shows El*** and E.*** versus ex for (ai - a2)/c = -0.03 and (^1 - ^2)/c = 0.5. At ox = x/2 radians, the first complete power transfer point in the 71 = 72 case, the above values correspond to a phase difference ((3i — 182) x = 7r/4 or 45°, and an attenuation difference (ai — 0:2)0; = 0.03 7r/2 or 0.047 nepers (0.41 db) for the path length of the coupling distance. In the ab- sence of the dissipation difference, but for the same difference in phase constants, Fig. 20 shows that E2* reaches a maximum at —0.26 db near ex = t/2, whereas the value including the dissipation difference (Fig. 32) is —0.46 db. The latter two values differ by 0.2 db or one-half of («! — 0:2).'^; when («! — 0:2) /c is small compared to unity, this is a general result. More sizeable deviations from 71 = 72 are represented in Fig. 30, which shows El*** and E2*** versus ex for (ai — a2)/c = — 2 and (/3i — /32)/c = 692 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 -80 -60 -40 - ^ ^ ' -10 -8 O -4 O E| »4 ^ - .^ ^ ^ ^_ E.*** O _p OJ 1^ l^"^.- ^S, N N, -0.8 -0.6 -0.4 -0.2 -0.1 - V \ S \, \ \ , i. L_ 1 \ 1 -0.02 -0.04 -0.1 -0.2 -0.4-0.6 -1 -2 -4 -6 -10 -20 -40-60 -100 c Fig. 31 — Driven and undriven line wave amplitudes versus (ai — a.2)lc for ex = 7r/2V2 and (/3i - ;32)/c = .2. 2. At cx = 7r/2, the phase difference is therefore x radians and the at- tenuation difference x nepers. The result is appreciable attenuation for El*** and only a moderate ratio of £'i***/£'2***. Fig. 31 shows the way dissipation differences counteract the coupling forces when there is a phase constant difference (/8i — ^^/c = 2. This may be compared with Fig. 21 which represents the case of (/3i — ^i) — 0. Very little change in E^*** occurs until {a\ — cf^/c exceeds (/3i — ^2)lc\ this is again a general result. Finally, we may inquire as to how much power is dissipated in the system when attenuation constant differences are utilized to mitigate the effects of coupling. A measure of the power preserved is £i^ + 1 E-{ and this ciuantity is plotted in Fig. 32 for cases previously discussed in connection with Figs. 21 and 31. Either in the absence or presence of a phase constant difference, the attenuation constant difference shows a maximum effect in reducing the available power at {a\ — 0:2) /c = 2. This is probably a general result brought on by the factor V(7i - 72)- - 4c2 found in the exponent of terms describing E\ and Ei . rOT^PLKD AV.WE TITF.OTIV .WD AV.\ VKni^IDE \PPT>Tr.\TTOXS OO.S t.o 0.9 0.5 '~' ^ ^ ^ k" u^ \ V CX=f; A=/32 N. FIG. (22) ^ 'A s \ \, / / ^ 2V2 c FIG. (31) 1 1 1 -0.02 -0.04 -0.1 -0.2 -0.4 -t.o -2.0 -4.0 -10 -20 -40 -100 c Fig. .'52 — Available power versus (on — 0.-1) /c for several cases of interest. TIGliT COUPLING EFFECTS OF MULTIPLE DISCRETE COUPLINGS In practice it is convenient under some conditions to produce the de- sired coupling between transmission lines using multiple discrete cou- plings. It is then of interest to know the relation between the total power transferred and the number and strength of the individual couplings. It is the purpose of this section to state these relations. We assume two transmission lines having identical propagation con- stants, with coupling units located at intervals along the lines as shown schematically in Fig. 33. A coupling unit may be a single point coupling, or an array of point couplings, but is always assumed to have the property of low reflection in the driven line and low back-wave transmission in the undri\'en line. If there are n\ couplings of magnitude ax , n-2 couplings of magnitude a^ , and Hk couplings of magnitude ak located along the lines in any order whatsoever, the wave amplitudes in Vo = 0 Eo=i.o V2 ETC. E, Ez Fig. 33 — Schematic of transmission lines with multiple jjoint couplings. 694 THE BELL SY8TEM TECHNICAL JOUKNAL, MAY 1954 Z 7 f. / / / / / / / / / / / / i / f / 3 3 / / / y* NUMBER OF COUPLING UNITS / / / / / y / y / / 1 A / f / / Y \ J / / / / v\ ^^ ^ SJ / / 0 2 4 6 8 10 12 14 16 18 20 22 24 CC. LOSS PER COUPLING UNIT IN DECIBELS Fig. 34 — Overall loss to the undriven line versus loss per coupling unit, with the number of coupling units as a parameter. the driven and luidriven lines respectively are E — cos [ui sin^ a\ + rii sin~ 0:2 + • • • Uk sin aj , (39) and V = sin [jii sin~ ai + n-i sin~ a-i -\- • • • Uk sin~ a/] . (40) These are amplitude factors due to coupling, and the normal attenuation effects in the uncoupled lines must be added separately. For complete power transfer we set the bracketed quantity of (39) and (40) equal to 7r/2, which gives the desired information about number and strength of point couplings. Other transfer losses may similarly be prescribed or determined. For multiple coupling units of the same coupling strength, Fig. 34 shows the overall transfer loss to the undriven line versus loss per cou- pling units as a parameter. The shape of these curves from the complete transfer point toward higher losses is very nearly the same. Fig. 35 shows the loss per coupling unit versus number of coupling units, with overall transfer loss to the undriven line as a parameter. SOME RESULTS OF EXPERIMENTS IN DOMINANT-MODE WAVEGUIDE In a pre^'ious paper on dominant-mode waveguide directional cou- plers, complete power transfer between dominant-mode rectangular COUPLED AVAVK TIIIOOKY AND "WAVKCUIDE APPLirATIONS (iOf) 35 30 yy^ <^yy y / z>' / / OVERALL LOSS ,_ TO SIDE ARM "^ / < / ^. f / 1 1.0 DB / / / ^ ^ \/ / \ / Y 1 osc\ — ^ 1 ,Zo 1 . 1- -*■ 1 RECEIVER L 1 /— \ / ■^, / '^^ V ' \ V^ y^ \ V oscj (Zo 7o, , ^^ ^ Zo 1 ^—•-RECEIVER ) 120 160 200 240 280 320 360 AZIMUTHAL ANGLE IN DEGREES Fig. 40 — Distribution of radial electric field at the guide wall for the forward and backward waves of the transducer of Fig. 39. power in any other mode of the multi-mode guide, the radial probe technique narrows down the possible mode types to a very few. Measure- ments of this type, recorded in Fig. 40, indicate that the forward wave has the radial electric field distribution to be expected for the TMn wa^'e. PTowever, the forward wave might have the same radial field distribution at the wall and actually be the TEn wave instead of TMn . The TEu wave is very simply generated from a dominant mode rectangular guide, by means of a long taper transition along the axis of propagation from the rectangular cross section to the circular cross section. Such a trans- ducer was used to measure the output wave of the TMu transducer and it was found that the TEn component was down on the order of 30 db below the value which would be present if the radial field intensity ob- served at the top of Fig. 40 had been due to TEn • By a process of elim- ination, therefore, and by virtue of the fact that we have a pure pattern suggesting the presence of a single mode, we have established that the mode generated is actually T~Sln • Other checks can of course be made, such as measurement of the phase constant of the output wave. The backward wave shown at the l)ottom of Fig. 40 has a maximum field more than 20 db below the maximum field of the forward wave and ro2 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 has a six-peaked variation with angle Avhich indicates the presence of TE31 . The transfer loss of the TMn transducer was derived by (1) calibrat- ing the receiving probe on a known amount of power in the TEn wave, (2) inserting this same amount of power in the rectangular waveguide of the coupled wa^•e transducer and, with the probe at the transducer output, observing the change in the receiver response, and (3) correcting the observed loss using the theoretical difference in the rachal electric field at the wall for the TEu versus TMu waves in the known wa^'eguide diameter. (This technique is described in more detail by Aronoff.') The result gave a transfer loss of about 25 db to the TMn wave. The insertion loss for the rectangular guide of the transducer was less than 0.2 db. Coupled-wave devices of the type shown in Fig. 39 were built for several of the modes in 2" round w^aveguide. The one built for the TE31 mode in 2" waveguide (mechanically similar to the TMu model of Fig. 39) has several characteristics worthy of mention. Fig. 41 shows the So a: J -60 UJ(D ?Q -64 ^ ^ , Zo RECEIVER r\ r\ r\ /^ r \ 1 r\ \ \ \ / \ \ \ 1 1 \ 1 1 /^ s, / ^ /"■>> /^ ^~\ V \ / \ /^ r \i ^ / ' \ ' osc\ fZo n n 7. j \l \} 1-^ \l \ 1!=!' — *- RECEIV 1 1 ER 80 120 160 200 240 280 AZIMUTHAL ANGLE IN DEGREES Fig. 41 — Distribution of radial electric field at the guide wall for the forward and V)ackward waves of TEiqCI to TE31O coupled wave transducer. r()TM'M:i) WAVK TIIKOUY A\D AVAVKCriDK APrMr'ATTOXS r()8 Table II Observed Discriminations Ratio of Forward Traveling TMn Power to Xo = 3.1 cm Xo = 3.3 cm Xo = 3.5 cm db db db T.Mu H.nckwaid >20 >20 >20 TEii Foi'ward 28.5 28 26 TE„ Backward 37 35 39 TMoi Forward 46 46 41.5 TMoi Backward 49 51 45.5 TEm Forward 24.5 21 23 TEm Backwaril 29.5 35 31 TEsi Forward 14 26 21.5 TEo, Forward 45 46 45 TEoi Backward 64 69 67 measured forward and backward wave patterns in the roinid guide, for excitation in one of the rectangular guides of the transducer. Only TEsi of the six modes possible in the 2" pipe at 3.3 cm has a six-lobed pattern of azimiithal distribution of radial electric field at the wall, and hence the clean pattern with equalU^ spaced deep nulls indicates the pre.sence of a rather pure TE31 mode. The six maxima of the forward wave were eciual within ±0.15 db. The backward Avave had a peak electric field at least 23 db down on the peak electric field of the forward wave. Using coupled transmission line technirjues and the familiar geometric taper techniques, transducers were built for all of the six modes possible in 2" diameter pipe at 3.3 cm for use in the circular electric \\ave research program.^ These transducers were used to measure the forward wave and backward wave output of the TlNIu transducers, as given is Table II. In reality, imperfections in either one of the two transducers involved in a measiu-ement could result in the recorded A^alues of discrimination. For example, if the TMn transducer were perfect and the TEm output transducer contained some T]\Iii . then the insertion loss measurement involving the two transducers face to face would produce an indication of mode impurity. Since we do not have independent information on the mode ]:)urity of any one of the transducers at the level of the obser\'ed wave im{)urities, we can only state that both transducers involved in a discrimination measurement are probably at least as good as the number tabulated. It should l)e noted that very high discriminations between TEoi and TMu were achieved, despite the fact that this one discrimination de- pends solely on the mode-.selective nature of the coupling orifice. Similar discriminations can be employed effectively to augment the wave-inter- 704 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 ference discrimination even in cases where there is difference between the desired and undesired modes' phase constants, to achieve very large discriminations. In the TMn discriminations listed above, the values for TE31 are not great but are consistent with computed values for the coupling length and the coupling function employed; longer coupling lengths would produce better TMn versus TE31 discriminations. A TIGHTLY COUPLED TEio° TO TEoi° WAVE TRANSDUCER* A highly efficient means of transferring power from dominant-mode rectangular waveguide to one of the higher modes of a multi-mode wave- guide would be essential in a waveguide transmission system. When several modes can propagate in one or both of the guides, the problem of achieving complete power transfer is more difficult and requires some new techniques. This section describes these techniques and gives ex- perimental data for a circular-electric-wave (TEio° — TEoi°) transducer. The desired transducer was reciuired to make the wave transformation between a single-mode rectangular waveguide and the circular electric mode (TEoi°) of an 0.875" round waveguide at a nominal frec^uency of 24,000 mc. The 0.875" round waveguide at this frequency will support 10 modes of which the circular electric mode and its degenerate partner TMii° are the fourth and fifth in order of appearance. The minimum length of the coupling interval reciuired to achieve mode discrimination may be estimated using loose coupling theory (equation 4). The mode nearest to TEoi° in phase constant is the TEsi^ and for this mode a coupling length of about 0.18 meters is recjuired in order to produce a A'alue of ^/tt equal to unity. As shown by equation (5) for uniform coupling, it is necessary to have ^/tt ec^ual to unity or greater in order to develop discrimination against the undesired mode. The maximum coupling coefficient permissible for a given amount of mode impurity at the complete power transfer point may be estimated using the tight coupling theory of the preceding sections. For example, equations (31) and (32) show that for the ratio (/3i — /32)/c equal to 10, the transfer loss to the undesired wave will always be greater than 14 db (regardless of the length of the coupling interval), corresponding to an energy loss for the desired wave of less than 0.2 db. For the TEoi° and TEsi^ modes the calculated values of /3i and ^2 lead to the conclusion that the coupling coefficient c between TEsi^ and TEio° must be less * When discussing the modes of hoHow metallic waveguides of different cross- sectional shapes, it has been found convenient to use a superscript to designate the shape of the cross section. (See G. C. Southworth, Principles and Avplicatinns of Waveguide Transmission, D. Van Nostrand Co., 1950). Thus, TEio^^ refers to the TFio mode in rectangular waveguide. COUPLED WAVE THEORY AND WAVEGUIDE APPLICATIONS 70.') than 3.45 radians per meter. If the coupling coefficient for TEio° to TEoi° is equal to that for TEio° to TEsi^ it follows that the total coupling length must be greater than 0.455 meters, because complete power transfer requires that the product of coupling-length times coupling- coefficient be exactly 7r/2 (see Fig. 17). Actually, the TEio° - TEsi^ coupling may be greater than the TEio° — TEsi^ coupling Avhich leads to the requirement for longer coupling intervals. It is evident that the shorter coupling intervals may be employed at the sacrifice of greater mode impurities. The preceding calculations were made for the TEio° — TEsi^ and TEio° — TEoi° transfer ratios as though only one mode of the multi-mode waveguide were present at a time, i.e., using a theory based on coupling between two waves instead of a theory for the simultaneous coupling between a plurality of waves. It is felt that this is probably Fig. 42 — An experimental circular electric wave (TEioi^ to TEoiO) transducer for 24,000 mc. justified provided that the coupling per unit length is weak and only one mode in each guide carries an appreciable amount of power. Fig. 42 shows a photograph of one of the models used to obtain experi- mental data. The coupling holes w^ere located in the narrow wall of the rectangular waveguide, thus avoiding coupling to all of the TM modes of the round waveguide. The total coupling length was 0.55 meters. The coupling orifices were spaced about 0.3 wavelengths in the dominant- mode rectangular waveguide, which assured reasonable directivity in the transfer of power between waveguides, provided that two or more coupling elements were employed. The transfer loss between the rectangular wa\'eguide and the circular electric mode of the round waveguide was measured as a function of the number of coupling elements, using the structure of Fig. 42 with the addition of a mo^•able thin-walled metallic cylinder. The latter could be moved inside the transducer in such a way as to cover up a varial)le number of couphng holes, and contained a long wooden termination so that all the power entering the mo\'able cylinder was absorbed. The inner diameter of the movable eyhnder was large enough to propagate the 70G THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 44 4 2' 40 38 36 34 en UJ 32 m u 30 Q Z 28 in 26 O ^ 24 ct UJ U. 22 CO z < 20 cr 18 16 14 12 10 8 L \ \ \ s ? i ^, /\ f / \ \ / V 'TE°-TE° / / N 1 \ / / \ w L y 5 y V \ 1 L \*'TEg,-TES \ Zo^ RECEIVER ^ \ \ \ \ \ ^.. 1 > IS t-l ^ 3 4 5 6 8 10 20 30 40 NUMBER OF COUPLING HOLES Fig. 43 — Transfer losses versus freciuencj^ for the transducer of Fig. 42. circular electric wave but did cut off some of the waves which could propagate in the round guide of the transducer itself. The measured transfer loss under these conditions is recorded in Fig. 43. It is seen that the TE n _ TEoi° coupling was so weak as to be in the region where power from successive coupling elements should add inphase all the way up to 40 coupling elements. The observations show the inphase addition for less than 30 coupling elements but show a marked deviation in the vicinity of 40 to 66 coupling elements. This is evidence of inequality of the phase constants for the TEoi° and TEio° waves. More will be said about this matter presently. The transfer loss between the rectangular waveguide and the TEn mode of round waveguide, is also recorded in Fig. 43. As expected, the power from successive coupling elements did not add inphase and no appreciable build-up of power in the TEu mode took place. One way of evaluating the total power in all modes other than the circular electric mode, is to measure the value of the transverse magnetic intensity at the wall of the round waveguide. The circular electric wave has no such field component and all other waves do possess such a field COITPLKD WAVK THKOHY A\n WAVK(iriDK Al'PI.ICATIONS r()7 comjioneiit. Thus tlic total value of the t I'ansx-crso magnetic intensity at the round \va\ei»;ui(le wall is a measiu'e of the impurity associated with the circular electric wave. (This is very similar to the radial probe t(H'h- niciuc described by 'SI. Aronoff.') Using this method of e\alualioii, the mode impurities ])resent at the output of the transducer were measured as a function of the number of coupling elements, and the results are recorded in Fig. 44. The absolute calibration of the ordinate relates the observed magnetic intensity to that which the same power input used at the rectangular guide would have produced if placed in the round wave- guide in the TEn mode. These measurements show that for all of the modes other than the circular electric mode, the energy components from successive coupling elements suffer destructive interference. Al- though curves are shown only for one and for 66 coupling elements, the patterns for intervening numbers of coupling elements were similar in shape and never exceeded an intensity value greater than about 6 db above that given for the 66 coupling element case; thus the mode dis- criminating property of the coupled wave transducer was verified ex- perimentally. Returning to the question of TEio° — TEoi^ transfer loss, it is clear from Fig. 43 that the rectangular Avaveguide has a phase constant which is not equal to that of the circular electric mode in the roimd waveguide. One reason for this inequality lies in the fact that the coupling elements disturb the phase constant in the two waveguides unequally, a conse- quence of the fact that some of the power transferred to the round wave- 5 25 PtU / / 1 1 \ / \ \ 66 COl JPLING HOLES k (A IP \ / 1 ^ \l \, A 7 \ / /] U \ \ V ' \ / 1 CO JPLING HOLE \ ^ 60 120 160 200 240 AZIMUTHAL ANGLE IN DEGREES Fig. 44 — Distribution of transverse magnetic intensity at the wall for the transducer of Fig. 42. 708 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 guide on a single coupling element basis, appears in modes other than TEoi . Thus, the total coupling to TEio° is greater than to TEoi^. The total coupling modifies the phase constant of each line, per (20'), and since the total coupling coefficient is unequal for the TEio'-' and the TEoi° modes, the perturbed phase constants should be expected to be unequal when the unperturbed phase constants are made equal. A method of determining the magnitude of this phase-constant disturbance has been suggested by S. A. Schelkunoff. In this method the reflected wave from a single coupling orifice is measured in the dominant waveguide and in the single mode of interest in the multi-mode waveguide. Having de- fined the ratio of the incident to the reflected power in the same mode by the sympol p, Schelkunoff determines that the disturbed phase constant i8', is related to the undisturbed phase constant jS by the relation /3 = ^ + (41) in which "d" is the distance between the coupling orifices in the coupling arrangement which one wishes to evaluate. This relation may be used to evaluate the change in the phase constant for the circular electric mode and for the wave in the dominant waveguide, and the change of wave- tr 6 UJ \ f= 24,000 MC VS \ ^ \ \ \ 1 \ \ \measured \ COMPUTED^ \ s \ \ \ \ ^N V ; Fi W\ f^ ^ N -]y~i-23,A00 MC □ "- -24,000 6- 25,000 1/ 30 40 50 100 110 120 Fig. 60 70 80 90 NUMBER OF COUPLING HOLES 46 — Rectangular guide insertion loss for the transducer of Fig. 42. guide dimensions required to correct this phase constant difference may be computed as though the coupling elements were not present. For the small phase constant disturbances which are associated with the Aveak couplings employed, this procedure was found very accurate. The reflection measurements and associated calculations for the model of Fig. 42 indicated that the rectangular guide width should be 0.340" for equality of phase constants instead of 0.359" as computed neglecting coupling effects. The measured value of the transfer loss when the in- dividual coupling holes had been enlarged and the rectangular guide width had been altered to the 0.340" value is shown in Fig. 45. It is evident that the theoretical value of 0 db transfer loss was approached, and that the shape of the transfer loss versus number of coupling ele- ments, was reproduced very well. The 0.75 db minimum transfer loss consisted of no more than 0.3 db heat loss, the remaining loss being due to power present in other modes. The measured insertion loss in the rectangular waveguide is shown as a function of the number of coupling holes at the three frequencies in Fig. 46. Complete power transfer would, of course, correspond to an in- finite insertion loss in the rectangular waveguide. It is interesting to note that at 24,000 mc the peak in the rectangular guide insertion loss occurred at 85 coupling elements whereas the maximum in the TEio'-' — TEoi° transfer loss characteristic occurred at about 96 couphng elements (Fig. 45). This difference is likely to be the result of power transferred back to the rectangular waveguide from round waveguide modes other than circular electric. Additional evidence of deviations due to the coupling between a plurality of waves was obtained; the rectangular-guide in- sertion loss as a function of number of coupling elements did not increase 710 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 OoQ D9 (a) \ ^ \ I ^ o ^ o V in 6 (b) ¥' ,11=01 D -o.__ -:: / f "=10 y ' / r -'■^ 20 30 22 23 24 25 26 27 28 FREQUENCY IN KMC Fig. 47 — Transfer and insertion loss versus frequency in the utilized modes of the transducer of J'ig. 42 using all (112) coupling holes. m 50 (\ A J y "^ \te--teo / \ N \ A, f \ \ / TE°-BACK WAVE TE^, / J < N \ v' s N \ K \ y ^ o..,_. / \ i/i D 19 20 21 22 23 24 25 26 27 28 29 30 FREQUENCY IN KMC Fig. 48 — Circular electric wave directivity and one unwantetl mode (TEuO output versus frequency for the transducer of Fig. 42. COUPLED WAVE THKOPvY AND WAVKOT'IDE APPLTCATIOXS 11 15 ^-•s v^ ,^ \ j—N / \ \ 1 ■'Eg,! f r / \ 1 V / \ \ d \ \ A \ 7 \ X \ h ^-wj Cr-*> r\ / \ < y K .»■»*' > 1^; J ^ /te,? \ J / \ v^ Y 19 20 21 24 25 26 22 23 FREQUENCY IN KMC Fig. 49 — Impedance characteristic of the transducer of Fig. 42. smoothly according to a cosine amplitude function as would be expected for two coupled waves of identical phase constant, but instead exhibited ripples. The remarkable thing about the data of Figs. 45 and 4G is that it agrees with the theory for two coupled waves as well as it does. The coupling per individual orifice decreases with increasing frequency and this is verified hy the observation (Fig. 46) that a greater number of couphng elements are required to reach the maximum insertion loss in the rectangular guide at the higher frequency. Some indication of the overall bandwidth of this first experimental model is given in Figs. 47, 48 and 49 which show respectively the TEio° — TEoi° transfer loss, the insertion losses in the TEio° and TEoi° modes, the TEio° - TEuO and TEio° - backward wave TEoi° transfer losses, and the TEio° and TEm^ return losses in the frequency range 20,000 to 30,000 mc. Xo one of these characteristics represents the degree of ex- cellence which is achie\'able but they do demonstrate that good im- pedance match, low transfer losses to the desired mode, and appreciable discrimination against unwanted modes, can be achieved over frequency ratios on the order of 1.5. FREQUENCY SELECTIVITY In the case wherein the coupling is so weak as to not affect the total phase constant appreciably, all modes of hollow conductor waveguides of any cross section have the same phase constant at all frequencies pro- vided that these modes have the same cut-off frequency. This results 712 THE BELL SYSTEM TECHNICAL JOUKNAL, MAY 1954 in very broad band mode-selective characteristics, as has been demon- strated. The transfer loss characteristics are in general a function of frequency, since the individual coupling holes are somewhat frequency selective. There may be applications wherein less variation in transfer loss as a function of frequency is required. One approach to this problem is to make the coupling holes individually have less coupling variation with frequency; since the total coupling loss between two identical transmis- sion lines is a function only of number of coupling holes and the loss per hole (equations (39) and (40)) constant coupling per hole will produce constant coupling overall. Riblet and Saad have reported on this ap- proach. There is another approach to obtaining flat coupling versus frequency despite variations in the coupling per hole, and that is to intentionally create a difference between the phase constants of the two coupled lines. Fig. 17 illustrates the transfer characteristic when the coupled lines have unequal phase constant, and either identical or negligible attenuation constants. Near the maximum for the transferred wave | E2* \ there is a region wherein the transfer loss is independent of coupling strength, and the transfer loss in this flat-loss region is under control of the ratio (iSi — ^2)/c. Hence for a given transfer loss there is an optimum ratio of phase constant difference to coupling strength in order to minimize the overall transfer loss variation. For the distributed coupling case, equa- tions (31) and (32) represent the transferred wave amplitude and show that the transferred wave goes through a maximum as a function of integrated coupling strength ex, when /• <»^'+„..; + „ w The transferred amplitude at this maximum point is 1 1/^ -^.r^, (43) 4c2 The integrated coupling strength at the maximum point is Co-'Co = /v^ ^ x^f^ • K • (44) IT /^^T^+^ ' For the important case of an optimum 3 db transfer loss coupler, E->* is 0.707. Then (^i — l32)/c equals 2 and CoXo equals 7r/2-\/2 from (43) COUPLED WAVE THEORY AND WAVEGUIDE APPLICATIONS 713 and (44). Assuming a coupling length .To of two wavelengths in the line with the smaller phase constant, it follows that /8i//32 is about 1.18 show- ing that a phase-constant difference of 18 % is required. This phase-con- stant difference is quite readily attainable in the wa\'eguide structure of Fig. 50(a). The two modes coupled together are given slightly different cut-off wavelengths in the coupling region, and may be tapered to the standard wa\'eguide size outside the coupling region. The desired phase- constant difference can also be obtained in two identical metallic guides by inserting a piece of dielectric into one of the guides in the coupling region as sketched in Fig. 50(b). Although rectangular waveguides are used in Fig. 50 to illustrate the method of obtaining frequency inde- pendent transfer characteristics, the approach is general and may be applied to an}' form of single or multi-mode transmission line. SECTION A-A (a) Cb) Fig. 50 — Examples of structures in which flat transfer loss may be obtained despite coupling loss variations. In either dominant-mode directional couplers or in multi-mode cou- pled-wave devices such as the one illustrated in Fig. 1, one may obtain much more frequency selecti^•ity than occurs incidentally due to the f requeue}^ sensitivity of the coupling elements used. This may be done by coupling two transmission lines which have the same phase constant at one frequenc}^ but unequal phase constants at other frequencies. Then, as shown by equation (31), the midband transfer loss may be set at any desired \'alue by adjusting the integrated coupling strength ex at midband (where /3i — (So = 0), and at other fre(iuencies where (^i — 02) ^ 0, the transfer loss will increase. For the particular case of ex = 7r/2 (fixed) for which complete power transfer occurs when /3i = 02 (and as- suming ai = ao or both as are negligible). Fig. 51 shows the shape of the filter characteristic, Eo* versus (0i — 0-i)f2e. This plot is valid for any form of transmission line. A very simple configuration for realizing such a frequency-selective filter involves coupling between two hollow conductor waveguides, one 714 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 24- \ t \ \ t 20 \ 1 \ *- \ \ \ \ / -16 \ \^ y \ \ v_ 1 \ / \ / / -4 -3 -2 "'-1 k ^ 0 / / / 1 i / '' I ,^— --' y ^ r 0 D 0.2 0.4 0.6 0.8 1 0 12345678 A-/^2 2C Fig. 51 — Transfer loss E-2* versus (0i — 02)/2c for coupling strength ex = 7r/2, the value required for complete power transfer. of which is air-filled and the other of which is filled with a material of dielectric-constant e. The phase constants for these waveguides have the form sketched in Fig. 52, in which /3o is the phase constant in free space. At the frequency /,„ the two waveguides have identical phase constants and, in a typical case, negligible loss constants so that complete power transfer can be obtained. For the case e = 2.55, Fig. 53 shows the com- puted frequency characteristic on the assumption that the integrated coupling is set for complete transfer (ex = 7r/2) and is independent of frequency. (Actually the usual coupling mechanisms are somewhat fre- quency sensitive and would increase the selectivity somewhat.) This filter 'Cl 'C2 FREQUENCY — »• Fig. 52 — The general form of the phase constants for two hollow conductor waveguides, one of which is filled with a dielectric. COUPLKD WAVE THEOKY AXD WAVKOnOK APl'LICATIOXS 715 "A 10 < 4 ''1 t / i / \ / / \ / \ / \ / \ i / V / f \ V / 1.00 Fig. 53 — The transfer loss E2* versus normalized frequency for two coupled hollow conductor waveguides, one of which is air filled and has a guide wavelength ■y/2 times the free space wavelength at/,,, , and the other of which is filled with a material of dielectric constant 2.55 with dimensions chosen for equality of phase constant with the air-filled guide at /„, . Coui)ling ex assumed constant at ir/2. characteristic applies regardless of the shapes of the hollow conductor waveguidess (which may be dissimilar) and regardless of the modes selected. It is apparent that frequency selectivity in the transfer characteristic E2* can also be obtained without recjuiring that the phase constants be unequal by using coupling elements which are freriuency sensitive. DIELECTRIC WAVECiUIDE CONFIGURATIONS The coupled-wave approach to circuit design is applicable using any form of transmission line, the only important variant associated with different forms of line being the physical structure associated with intro- ducing the desired coupling between lines. In a recent publication*' A. G. Fox showed that dielectric waveguides are very attractive for use in the millimeter wavelength range, and this section points out how dielectric waveguides can be used in various forms of coupled wa\'e devices. Fox showed that dielectric waveguides arranged in the configuration sketched in Fig. 54 are coupled by the electric field components only, and that 7iG THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 periodic energy exchange of the type described by equations (26) and (27) is observed. ]\Ioreover, he also showed that if one line were made very lossy the energy exchange phenomena disappeared and, despite sufficient coupling to cause complete power transfer when both lines were loss-free, power passed through the coupling region in the low-loss line with less than 0.25 db attenuation. This verified the predictions of equations (35) and (36). Other implications of the coupled wave theory can also be utilized in dielectric waveguides. If the two lines (Fig. 54) are made of materials having different dielectric constants and their cross-sectional dimensions set so as to secure identical phase constants at a frequency fm , then a frequency-selective coupled-wave filter results and the selectivity charac- teristic of Fig. 53 applies. As an alternative to using materials having different dielectric constants, the same dielectric may be used for both lines by making one line solid and the other hollow. If both lines are made of the same material and the cross-sectional dimensions are set so as to obtain a known difference between their phase constants, the result is a directional coupler having a region of flat trans- fer loss (of any desired magnitude) and equations (42), (43) and (44) apply. Both of the preceding appUcations can be carried out in dielectric waveguides having arbitrary cross-sectional shapes. Fig. 54 — Coupled dielectric waveguides. COUPLED WAVE THEORY A.ND WAVEGUIDE APPLICATIONS 717 If one of the transmission lines (Fig. 54) is round and the other is rectangular and if their cross-sectional dimensions are set for equal phase constants, then the power in one of the two polarizations of the lound line may be transferred to anj^ desired extent to the rectangular guide, and power in the other polarization of the round guide will pass the coupling region undisturbed. Two such rectangular-rod to round-rod coupling configurations arranged in cascade along the round-rod, with the two rectangular rods coupled in planes at 90° to each other, consti- tutes a means for independently connecting to the two polarizations of the round-rod. This type of de\'ice depends upon the fact that the phase constants of the two polarizations of round-rod are identical, whereas the two phase constants for the rectangular rod are different. Thus a wave interference occurs in the transfer characteristic for one of the polarizations, and for suitable values of (/5i — ^i)/c (see Fig. 18) the power transferred in this polarization can be made small. SU.MMAKY Two approaches to a theoretical description of the behavior of two coupled waves have been presented. One, based on the assumption of negligibly small coupling, is applicable in cases where very little power is transferred between the coupled waves. The other, a solution based on uniform coupling between waves in the coordinate of propagation, is valid for any magnitude of total coupling. The loose coupling theory shows how to taper the coupling distribution in order to minimize the length of the coupling interval required for a given degree of directivity and/or for a given magnitude of mode im- purity. In particular, it is possible to shape the coupling distribution so as to discriminate sharply against one or more undesired modes in a coupled-wave arrangement involving just a few modes. (See Figs. 7 and 15 for examples). The theory indicates that significant exchange of power takes place provided that the attenuation and phase constants of the coupled waves are ecjual, or provided that the difference between the attenuation con- stants and the difference between the phase constants are small compared to the coefficient of coupling. A suitable difference between either the attenuation constants or the phase constants of two coupled waves is sufficient to prevent appreciable energy exchange (equations 29-32 and 35-36). It follows that substantially single-mode propagation is possible in a multi-mode structure even though geometrical effects tending to cause coupling between modes are present. A gradual transition in the boundary 718 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 of a multi-mode waveguide will not cause an appreciable exchange of power between modes provided that the quantity ((81 — /32)/c is suffici- ently large for the modes which are coupled by the boundary change. Similarly, for disturbances in the coupled-wave system which takes place OA'er a large number of wavelengths in the direction of propagation, the coupled-wave theory indicates that all conversion will take place in the forward direction and very little reflection in any mode will result. The tight coupling theory shows that for the case of identical complex propagation constants, a periodic exchange of energy between waves takes place along the coordinate of propagation. The only effect of the existence of an attenuation constant for both waves (compared to the dissipationless case) is to add the same exponential attenuation factor (to the periodic energy exchange phenomenon) which would have existed for a wave traveling on one of the lines in the uncoupled state. When the phase constants of the two coupled waves are not equal (and the attenuation constants are either equal or negligibly small compared to the coupling coefficient), the exchange of energy between waves is no longer complete but remains periodic (Fig. 17). The quantity (0i — ^2)/c determines the fraction of the total energy Avhich is exchanged, and also modifies the period of the energy exchange phenomenon along the axis of propagation. When the phase constants of the two lines are equal but the attenua- tion constants are unequal, the energy transfer phenomenon differs only slightly from that associated with equal propagation constants pro\'ided that the quantity (ai — a2)/c is less than about —0.1. For (ai — a-2)/c more negative than about —1, the periodicit}' of the energy transfer phenomenon has largely disappeared (Fig. 23) and as (ai — a2)/c be- comes on the order of — 10 or more, the principal effect of the coupling for the low loss line is a minor alteration of the phase and attenuation constants. The wave amplitude for unit input on the low-loss line be- comes [from (33) for | (ai — 0:2) \/c » 1] 7^ _ —lai—c-l(.ai—a2)+iic+0)]x (A^\ Through proper choice of the phase constants relative to the coupling coefficient in two coupled transmission lines, it is possible to make di- rectional couplers having an arbitrary transfer loss that is independent of frequency despite ^'ariations in coupling strength with frequenc}^ (equations 43-44). It was also shown that the coupled- wave approach may be utilized to create highly freciuencj^-selective filters which may operate between single-mode media or between selected individual modes of a multi-mode system. COT'PLKD AVAVK TTIKOHY AXD AVAVKfU-IDE APPUrATK )XS 710 The experimental data given for two dominant-mode rectangular waveguides showed that the periodic energy exchange theoretically pre- dicted for a t'oupled-wave system can he achieved in coui)led transmis- sion lines. Performance characteristics were given for some loosely coupled trans- ducers between a dominant-mode rectangular waveguide and one mode of a six-mode waveguide. A tapered coupling distribution wuh used to achie^•e the mode selectivity in a limited length interval. The problems associated with a coupled-wave transducer for ti'ans- ferring all of the power from a dominant-mode rectangular waveguide to the circular electric mode in a ten mode waveguide, were discussed and the obserx'ed characteristics of an experimental model were gi^•en. The application of coupled-wave techniques to other types of trans- mission systems was illustrated by pointing out analogous structures using coupled dielectric waveguides. ACKXOWLEDGMEXT The writer is indebted to W. W. Mumford for helpful discussions during the early stages of this work; to R. W. Dawson who made most of the measurements on the models of Figs. 36 through 44, and to (I. D. Mandeville who made measurements on the model of Fig. 39. BIBLIOGRAPHY 1. S. E. Miller and A. C. Beck, Low-Loss Waveguide Transmission, Proc. I R E 41, pp. 348-358, Mar., 1953. 2. S. E. Miller, Notes on Methods of Transmitting the Circular Electric Wave Around Bends, Proc. I.H.E., 40, pp. 1104-1113, Sept., 1952. 3. W. W. Mumford, Directional Coujjlers, Proc. LR.E., 35, pp. 160-165, Feb , 1947. 4. C. L. Dolph, A Current Distril)Ution for Broadside Arrays Which Optimizes the Relation Between Beam Width and Side Lobe Level, Proc. LR E 34, pp. 335 348, June 1946. 5. S. E. Miller and W. W. ]\Iumford, Multi-Element Directional Couplers, Proc I.R.E., 40, PI). 1071-1078, Sept., 1952. 6. H. J. Riblet and T. S. Saad, A Xew Type of Waveguide Directional Coupler, Proc. LR.E., 36, i)p. 61-64, Jan., 1948. 7. ^L Arnoff, Radial Probe Measurements of Mode Conversion in Large Round Waveguide with TEoi Excitation, (submitted to Proc. I.R.E.). 8. A. G. Fox, Xew Guided Wave Techniques for the Millimeter Wavelength Range, given orally at the March, 1952, LR.E. National Convention. To be sulmiitted to the Proceedings. 9. Alan A. MeverhofT, Literaction Between Surface-Wave Transmission Lines, Proc. LR.E., 40, pp. 1061 1064, Sept., 19.52. 10. P. E. Krasnushkin and U. \' . Khokhlov, S])atial lieating in Coupled Wave- guides Zh. Tekh. Fiz, 19, pp. 931-942, Aug., 1949, (in Russian). 11. W. J. .\lbersheim, Propagation of TEm Waves in Curved Wave Guides, B.S.T.J., 27, pp. 1-32, Jan., 1949. Theoretical FiiiKlanientals of Pulse Transmission — I By E. D. SUNDE (Manuscript received 8ci)teinl)er 23, 1953) A compendium is presented of theoretical fundamentals relating to pulse transmission, for engineering applications. Emphasis is given to the con- sideration of various imperfections in transmission systems and residtant transmission impairmcjifs or limitations on transmission capacity. In Part I of this paper, Sections 1 to 11, fundamental properties of trans- mission-frequency characteristics are discussed, together with general rela- tions between frequency and pulse (ransmission characteristics and special transmission characteristics of importance in pulse systems. This is fol- lowed by a presentation of engineering methods of evaluating pidse distortion from various types of gain and phase deviations. In Part II, Sections 12-16, transmission limitations imposed by charac- teristic distortion will be discussed. Part I 1. Properties of Transmission-Frequency Characteristics 724 2. Frequency and Pulse Transmission Characteristics 730 3. Idealized Characteristics with Sharp Cutoff 736 4. Idealized Characteristics with Gradual Cutofi" 741 5. Idealized Characteristics with Natural Linear Phase Shift 743 6. Pulse Echoes from Phase Distortion 752 7. Pulse Echoes from Amplitude Distortion 761 8. Fine Structure Imperfections in Transmission Characteristics 764 9. Transmission Distortion bj^ Low-frequency Cutoff 773 10. Transmission Distortion by Band-edge Phase Deviations 779 11. Band-pass Characteristics with Linear Delay Distortion 784 Part II 12. Impulse Characteristics and Pulse Train Envelopes 13. Transmission Limitations in Sj-mmetrical Systems 14. Transmission Limitations in Asymmetrical Sideband Systems lo. Double vs. Vestigial Sideband S3'stcms 16. Limitation on Channel Capacity by Characteristic Distortion Acknowledgements References 722 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 INTRODUCTION Pulse transmission is a basic concept in communication theory and certain methods of modulating pulses to carry infoiTaation approach in their characteristics the ideal performance allowed by nature. In certain applications, such as telegraphy, pulse signalling and data transmission, it has the advantage of great accuracy, since the information is trans- mitted in digital form by "on-off" pulses. This at the same time facili- tates regeneration of pulses to avoid accumulation of distortion from noise and other system imperfections, together with the storing, auto- matic checking and ciphering of messages, as well as their translation into different digital systems or transmission at different speeds, as may be required in extensive communication systems. Another characteristic of pulse systems is that improved signal-to-noise ratio can be secured in exchange for increased bandwidth, as in pulse code, pulse position and certain other methods of pulse modulation. Finally, pulse modulation systems permit multiplexing of communication channels on a time divi- sion basis, which under appropriate conditions may have appreciable advantages over freciuency division in the design of multiplex terminals. In pulse modulation systems, pulses are applied at the transmitting end in various combinations, or in varying amplitude, duration or posi- tion, depending on the type of system. Pulses thus modulated to carry information may be transmitted in various ways, or undergo a second modulation process suitable to the transmission medium. The received pulses will differ in shape from the transmitted pulses because of band- width limitations, noise and other system imperfections. The performance of the system in the absence of noise can be predicted if the "pulse trans- mission characteristic" is known, that is, the shape of a received pulse for a given applied pulse. Although the pulse-transmission characteristic suffices for detennina- tion of system performance it is customary for various reasons to relate it to the "transmission-frequency characteristic," that is, the steady- state transmission response expressed as a function of fiequency. For one thing the transmission-frequency characteristics of various existing facilities and their components are known, and for new facilities can be determined more readily by calculation or measurements than the pulse- transmission characteristic. But the more fundamental reason is that the transmission-frequency characteristics of various system components connected in tandem or parallel can readil.y be combined to obtain the over-all transmission characteristic, while this is not the case for pulse transmisssion characteristics. It is thus possible to analyze complicated systems with the transmission-frequency characteristic as a basic THEORETICAL FUXDAMEXTALS OP PULSE TRAXSMLSSIOX 723 parameter, and to specify reciuirements that must be imposed on the transmission -frequency characteristic of the system and its compoiuMits for a given transmission performance. A fundamental problem in pulse modulation systems is transmission distortion of pulses by system imperfections in the form of phase and gain deviations over the transmission band or a low-frequency cut-off, usually referred to as "characteristic distoi'tion," which may give rise to excessi\-e interference between pulses and resultant crosstalk noise or errors in reception, depending on the type of system. Because of such interference, characteristic distortion limits the number of pulse ampli- tudes peiTuissible in the transmission of information or messages over a given channel, and may reduce the rate at which pulses can be trans- mitted in sj'stems emploj'ing only two pulse amplitudes, the minimiun number. It thus places a limitation on channel capacity which, unlike signal distortion by noise, cannot be overcome by increasing the signal power. Characteristic distortion is an important consideration particularly in wire systems where there is a low-frequency cut-off caused by trans- formers, and where the transmission band may extend over several octaves with substantial variation in attenuation and phase shift, or may be sharply confined by filters. In wire systems there are also fine structure deviations from a smooth attenuation and phase characteristic of a more or less random nature, resulting from small random impedance variations and mismatches along the lines. Gain and phase deviations remaining even after fairly elaborate equalization may be appreciable and difficult to overcome, especially in S3'stems comprising a large num- ber of repeater sections. The purpose of this paper is to present a compendium of theoretical fundamentals on pulse transmission in a form suitable for engineering applications, both from the standpoint of design of new pulse transmis- sion systems and pulse transmission over existing facilities. Emphasis is placed on considerations of various system imperfections, because of their importance from the standpoint of transmission performance, and since literature on this question is rather limited. Certain fundamental properties of transmission-frequency characteristics are discussed, to- gether with general relations between frequency and pulse transmission characteristics and special transmission characteristics of importance in pulse systems. This is followed by a presentation of methods of evaluat- ing pulse distortion from various types of gain and phase deviations, to- gether with resultant transmission impairments or limitations on pulse transmission rates in low-pass, symmetrical and asymmetrical sideband 724 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 systems. Converselj^ these methods raay be used in the design of pulse modulation systems to evaluate requirements imposed on the transmis- sion characteristics for a given transmission perfoiTnance. Transmission impaiiTnents may result from S3'stem imperfections other than characteristic distortion, which require a different theoretical ap- proach and are not considered here. Among them are erratic timmg of pulses, thermal and other noise within the transmission system and in- terference from outside sources, such as other communication systems or atmospheric disturbances. 1. PROPERTIES OF TRANSMISSION-FREQUENCY CHARACTERISTICS A basic parameter of transmission systems is the transmission-fre- quency characteristic T(tco) = A(o:)e-''^^"\ (1.01) in which w = 27r/ is the radian frequency, ^4(co) is the amplitude and \l/(u) the phase characteristic. The transmission-frequency characteristic may designate the ratio of received voltage to transmitted current, of received current to transmitted voltage, of received to transmitted cur- rent or of received to transmitted voltage. The two latter ratios are not the same except for sjTnmetrical networks \\ith impedance matching at both ends. For symmetrical structures having appreciable attenuation, such as transmission lines between repeaters, the ratios are virtually the same with impedance matching at the receiving end. In the following, T{io}) will designate any of the above ratios, as the case ma}^ be. When a number of networks are connected in series, as is usually the case in transmission systems, the resultant transmission characteristic is where Ti , To ■ • ■ Tn are the transmission characteristics of the individual networks with the same impedance terminations as encountered in the series arrangement, i.e. as measured in place or with equivalent termina- tions. The phase characteristic \l/ can in general be regarded as the sum of three components. The first is the minimum phase shift component, 1^", which has a definite relation to the amplitude characteristic of the system, and is of particular interest in connection with phase distortion with different types of amplitude characteristics. The second is a THEORETICAL FITNDAMKXTAT.S OK rrLSK TRANSMISSION t'2.) linear component cur,/ , which represents a constant transmission delay Td for all frequencies, as in the case of an ideal dela.y netwoi'k. Ladder tjq^e structures and transmission lines have phase characteristics which can be represented by the a])ove two components. The third component can be represented by a lattice structure ^vith constant amplitude chai'- acteristic but varying phase. Such a network component may he i)resen1 in a transmission system or may be inserted intentionally for i)hase equalization, i.e. to supplement the first component above so as to secure a linear phase characteristic without altering the amplitude characteristic of the sj'^stem. The following discussion is concerned with the relationship of the first component to the amplitude characteristic of the .system, or conversely. The natural logarithm of the transmission-frequency characteristic given by (1.01) is lnT(ico) = fnA(o)) - ti/'(co). (1.03) The component t7iA(co) is referred to as the attenuation characteristic, and when expressed in decibels equals 8.69 hiA(oj). The followdng relations exist between the attenuation and phase char- acteristics of minimum phase shift systems or system components: '" fnA{.) = -l f ±^ du = ^ I 2}^^ a,^ (1.04) and TT J-00 u — u IT Jo u^ — co- in the evaluation of these integrals, the principal \'alues are to be used, i.e., re.sults of the form fn( — ii) are to be taken as In \ —u\ rather than ^/i I w I + tV. As an example consider an attenuation characteristic as shown in Fig. 1, with A(o}) = Ao between co = 0 and coc and .li between co = coc and X. Equation (1.05) then becomes ,0f N 2co TT Jo U" = - fn{Ao/A{)(n T Jo n- — CO- Ju COc -|- CO (LOG) In Fig. 1 is shoAvn the phase characteristic for Ao/A^ = 100, correspond- ing to a 40 db cutoff at co = co^ . 726 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 z ■= 10 12 / \ ATTENUATION / / \ \, / ^ \PHASE PHAS ^ ^~- X 0.2 0.4 0.6 1.6 U 30 S Z 20 I < 10 z 0 < 1.8 2.0 0.8 1.0 1.2 1.4 Fig. 1 — Low-pass transmission frequency characteristic with sharp cut-off. In Fig. 2 the attenuation and phase characteristics are shown as a function of co/wc for w < coc and as a function of the inverse ratio coc/w for CO > coc . It will be noticed that for the above case the phase charac- teristic is infinite for co/coc ^ 1 and has even sjonmetry about this point, while the attenuation characteristic has odd symmetry with respect to the midpoint of the amplitude discontinuity. The phase characteristic may be modified by a gradual cutoff in the attenuation characteristic, as illustrated in the figure. It is possible to shape the attenuation char- acteristic to obtain a linear phase characteristic in the transmission band, i.e. between co/wc = 0 and 1. Since transmission systems with a 5 \ /, , \ / / \ \ 1 4 // \ \ ATTENUATION | // \ V ' PHASi:// \/\ 1 // '^O 3 / / \ i* /> 7^ vV // / // / ^\ 2 1 —7^ <" / — ^ ^^ X / 1 >^HASE ^ "S ^ 1 \^ ^ / \ ^ / "■^^ 0 ^ ^^ ^^.^ 0.8 0.6 0.4 0 0.2 0.4 0.6 Fig. 2 — Solid curves same as in Fig. 1, l>ut with inverse scale for aj/oj^ > 1. Dashed curves illustrate modification in phase characteristic with gradual cut-off in attenuation (not computed). THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0 00/ 00^ OJc/UJ Fig. 3 — Low-pass transmission frequency characteristies with natural linear pliase shift foi- w/wc < 1. linear phase characteristic in this range are of particular importance in pulse transmission, this case will be considered further. Tt will be assumed that the phase characteristic has even symmetry when expressed in the scales of Fig. 2, in which case the phase charac- teristic as shown by the solid lines in Fig. 3 is given by V'"(") = ^T j/cOe < 1, (1.07) = coc r/co oi/(x)c > 1. With these expressions in (1.04) the attenuation characteristic becomes: (nA{o:) = 1 + 1 /a; Wc (ri 1 + Cj/cJc~| 1 — Oo/WcJ For CO = 0, the latter expression approaches the limit (nA{()) = 4ajcr/7r, so that /'n.l(a;)A4(0) 1 + 1 Z \Wc in 1 i/Wc_ (1.08) wliich is the attenuation characteristic shown in Fig. 3. Other attenuation characteristics with a linear phase characteristic between co/wc = 0 and 1 are possible with other types of variations in the attenuation or phase characteristic for co/co, > 1 than assumed above. For example, the attenuation characteristics may be assumed 728 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 constant for w/coc > 1, in which case the attenuation characteristic will be somewhat different for w/wc < 1 and the phase characteristic different for co/coc > 1, as illustrated in Fig. 3. (The solution for the latter case is given in Reference 2.) It will be noticed that there is a comparatively minor difference betw^een the attenuation characteristics for w/ojc < 1 in the above cases, so that the attenuation characteristic for co/coc > 1 has a relatively minor effect, provided there is no discontinuitj^ near w/coc = 1. The transmission loss characteristics shown in Fig. 3 represent a close approximation to the type of characteristic employed in pulse transmission systems, as will be sho\\Ti later. In the above examples low-pass characteristics were assumed. For high-pass characteristics the algebraic sign of the phase is reversed with respect to the amplitude characteristic as indicated in Fig. 4, which also illustrates relationships for band-pass characteristics. The band-pass characteristics are obtained by connecting low-pass and high-pass net- works in tandem. The resultant attenuation and phase characteristics are obtained by adding the low and high-pass attenuation and phase characteristics, as illustrated in the figure. In the second case showTi in the figure, the band-pass characteristic is assumed to have a linear phase characteristic in the transmission band, in which case the attenuation characteristic will not be symmetrical about the midband frequency, unless the latter is high in relation to the bandwidth. The third case illustrates the type of band-pass characteristic encountered in wire systems with a low-frequency cutoff. There will then be phase distortion at the low end of the band, since it is not feasible with a fairly sharp low-frequency cutoff to obtain a linear phase characteristic in the trans- mission band. If the amplitude or attenuation characteristic of a transmission system is modified, it will be accompanied by a modification in the phase characteristic. Of basic importance are cosine modifications in the attenuation and amplitude characteristics. Let the modified amplitude characteristic be of the form A(co) = ^o(co)e'"=°'"^ (1.09) where ^o(w) is the original amplitude characteristic. The modified at- tenuation characteristic is then ^nA(co) = ^nAoiu) + a cos cor. (1-10) In accordance with (1.05) the modified phase characteristic becomes, ,0/ N 1 r* /n.4.o(co) , a f°° cos cot lA (co) = - / du + - / du, . . TV J-oo Oi — U TT J-M OJ — U U-ii/ — ^o(w) + a sin wr, THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 729 where \po(u) is the phase characteristic of- the original ampUtude charac- teristic Ao(co). Thus, for any consine modification in the attenuation characteristic there is a corresponding sine modification in the phase characteristic, and for any sine modification in the phase characteristic a corresponding cosine modification in the attenuation characteristic. In general any modification in the attenuation characteristic may be represented by a Fourier cosine series, in which case the modification in the phase charac- teristic will be the correspondhig Fourier .sine series. With a cosine modification in the amplitude rather than in the at- LOW-PASS (a) HIGH-PASS (b) BANDPASS (a-b) ATTENUATION PHASE/' ATTENU- ATION N u^o ,.- ^ 1 / \ ' / ^ 1 /PHASE M / \ ' ATTENUATION / \ 1 \ \ 1 \ / \ / ^^ / \ t^o / '^^l \ 1 / \ / \ / PHASE \ / FREQUENCY, CJ — *- Fig. 4 — Attenuation and phase shift for various types of transmission fre- quency characteristics. 730 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 teniiation characteristic * A(o:) = Ao(co) [1 + a COS cor], (1.12) and the corresponding phase characteristic becomes (nAoiu)[l + a cos ut] TT J-o dii. = iPM + 2 tan"' u r sin cor 1 + ^^ cos COT ' lAoCw) + 2[r sin cor + — sin 2 cor (1.13) 3 7' + - sin 3 cor + . , o where = - [1 T Vl - a^], (1.14) and the minus sign is to be used. Thus, a cosine modification in the ampHtude characteristic is accom- panied by an infinite series of sine deviations in the phase characteristic. For sufficiently small values of a, r = a/2 and (1.13) reduces to (1.11). 2. FREQUENCY AND IMPULSE TRANSMISSION CHARACTERISTICS In dealing with pulse transmission, it is customary to consider three basic types of time variations of currents and electromotive forces, a cisoidal variation, a unit impulse and a unit step. The cisoidal variation, e'"', is basic in the solution of network and transmission problems in terms of complex impedances and admittances. The unit impulse is a current or electromotive force of very high intensity and short duration, such that the area under the impulse is unity. The unit step is a current or electromotive force which is zero for if < 0 and unity thereafter. The time responses of networks or transmission systems to these three basic time functions are interrelated so that each may be obtained when one of the others is known. Furthermore, the time responses for electro- motive forces or currents of arbitrary wave shape may be obtained from the response characteristic for any one of these basic time functions. The pulses applied in pulse systems can usually be approximated by impulses. Furthermore, with impulses certain simple relationships can be established which are either obscured or more complicated when a THEORETICAL FUXDAMEXTALS OF PULSE TKAXS.MlSSlOX 731 unit step is assumed. For these reasons, only the transmission charac- teristic for impulses will be considered here, or for pulses of sufficiently short duration to be regarded as impulses. Corresponding to any transmission-frequency characteristic is an impulse transmission characteristic, P{1), which designates the received pulse as a function of time for a transmitted unit impulse. The impulse and transmission frequency characteristics are interrelated by the follow- ing Fourier integral relations P{t) = ~ I TMe'"' do:, (2.01) T(ii,) = [ PiOe-'"' dt. (2.02) •'—00 The transmission characteristic for an applied pulse or signal of arbitrary shape G{t) is given by H{t) = ~ [ ^(^■a;)>S(^•a;)e'"' c/co, (2.03) 27r J- 00 where Siico) is the frequency spectrum of the applied pulse and is given by S{io:) = f GiDe""'' dt (2.04) J— 00 In the case of a symmetrical pulse S(io)) is a real function. In view of (1.01), expression (2.03) may also be written H{t) = - [ A{oo)S{oo) cos M - '/'(c^)] f/co, (2.05) T Jo where the relations A (-co) = A(ui), »S( — w) = AS(aj), \l/i — o:) = — '/'(w) have been used, and it is assumed that S(io}) = ».S(co) is a real function, as for a symmetrical pulse. In most pulse transmission systems, the applied pulses can be ap- proximated by short rectangular pulses. Rectangular pulses of unit amplitude and duration 8 have a frequency spectrum SM^S'-^^. (2.06) coo/ Z The same pulse transmission characteristic as when an impulse is applied is obtained with a rectangular pulse if A(o)) is modified by the factor (a;5/2)/sin {co8/2). In the following it will be assumed that the applied pulses are of sufficiently short duration to be regarded as im- 732 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 A(ajr+u) = t¥(u) PHASE AMPLITUDE A(a;r--u) = 1 1 / 1 / 1 / 1 / / \ / /J^r r(-U)^'' / / 1 • 1 / 1 ^ 1 • 1 / Cfr 1 y .f \ip{CCr -U) = >ir(- U) + ^r Fig. 5 — Transfer of reference frequencj'^ from co = 0 to co = cor . pulses or that otherwise the above modification is applied, in which case P(0 = - [ A(w) cos M - '/'(w)] d<^. (2.07) In the latter equation A(co) can also be regarded as the frequency spectrum of a pulse applied to a transmission system having a constant amplitude characteristic and a phase characteristic i/'(co) over the band of the pulse spectum. Equation (2.07) applies to any type of transmission-frequency char- acteristic and is convenient in this form for low-pass characteristics. For band-pass characteristics as shoA\ai in Fig. 5 however, it is convenient from the standpoint of general analysis as well as for numerical evalua- tion to use a reference frequency cor within the tranmsission band, that is, to employ the transformation 00 = 0;^+ » db^ = du. With the notation a(w) = yl(co) = A{U + COr), ^(lO = i/'(w) - i/'(w,) = 1^(0;) - l/'r, (2.08) THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 733 equation (2.07) can be written: Pit) = cos [o^rt - rPr)[R-{t) + /?+(/)] + sin U( - 'Ar)[Q-(0 - Q+m /?_ = -/ a(- u) cos [ut + ^'(— u)] du, TT Jo R+ = - a{u) COS [(// - ^(?/)] ^^<, X Jo Q_ = - I (i(— It) sin [ui + ^(— w)] d«, and IT Jo Q^ = - / a(?0 sin [w/ — ^(w)] (^it. TT Jo (2.09) (2.10) (2.11) The envelope P(t) of the impulse transmission characteristic is given by Pit) = [(i?_ + 7^)"' + iQ- - Q^)V". (2.12) Comparison of (2.09) with (2.07) shows that R- and R+ can be identi- fied with the impulse characteristics of low-pass systems having the same frequency characteristics as the bandpass system below and above av . The impulse characteristics Q_ and Q+ which arise from asymmetry in the transmission characteristic with respect to av are not present in low-pass systems, since by definition the amplitude characteristic has even symmetry and the phase characteristic odd sjnnmetry with respect to zero frequency. The first and second components of (2.09) are referred to as the in- phase and quadrature components of the impulse characteristic of band-pass systems.^ The transmission-frequency characteristic may cor- respondingly be regarded as made up of a component with even sym- metry and another component with odd symmetry about Wr , as indicated in Fig. G. These two components, together with the in-phase and quadra- ture components, will depend on the choice of w, . How'ever, P(t) as given by (2.09) and the envelope as given by (2.12), will remain the same, since a single impulse characteristic is associated with a given transmission- frequency characteristic. With the customary pulse transmission methods, the reference fre- quency cor may be identified \\ith a modulating or carrier frequency, which has a special significance when the envelope of a sequence of received pulses is considered. Although for a single pulse the envelope ■34 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 is always the same, for a sequence of pulses the resultant envelope of the recei\'ed pulse train will depend on the in-phase and quadrature com- ponents.^ The reason for this is that one has even and the other odd symmetry about the peak amplitude of the envelope for a single pulse, when the phase characteristic is linear. In order to compare the transmission performance as the reference or carrier frequency is changed, it is necessary to determine the in-phase and quadrature components for each carrier frec^uency under considera- A J-, ^. FREQUENCY, a; Fig. 6 — Decomposition of amplitude characteristic CEi asymmetrical with respect to wr into a component di of even symmetrj' and a component (Js of odd symmetry about wr . When the phase shift is linear, (Ji = d-i + Cls . tion. One method is to evaluate integrals (2.10) and (2.11) for each Carrie frequency, which may be facilitated by resolving the transmis- sion-frequency characteristic into symmetrical and anti-symmetrical components as indicated in Fig. 6. This, however, is a rather elaborate procedure which can be avoided with the aid of a simple translation from one reference or carrier frec^uency to another, as shown below, provided the in-phase and quadrature components or the envelope has been determined for one reference frequency. Equation (2.09) may also be written, with tp = (p(t): P{t) = cos(co./ - ^r - 00 . The impulse characteristic is zero when coiio = ±nT, or to = ±Ti , ± 2ri , • • • ± UTi where 1 (3.02) Impulses can thus be transmitted at the latter intervals Avithout '' " 2f; (a) AMPLITUDE AND PHASE CHARACTERISTICS (b) IMPULSE CHARACTERISTIC -to"* — 1 — •■to K-PHASE AMPLITUDE SINT^toA FREQUENCY, CV , , , , - , ; , - , , i i Fig. 7 — Idealized low-pass characteristic with sharp complete cut-off. THEORETICAL Fl'XDAlVIEXTAES OF PULSE TRANSMISSION 737 mutual interference between the peaks of the received pulses. This is a basic theorem underh'ing the determination of the transmission capacity of idealized systems. For an idealized bandpass characteristic between wo and wi , it follows from (2.09) with ^(») = iiTd and '^( — u) = —iiTd that the impulse characteristic with respect to the midband frequency w, = co„, is P(t) = 2 cos[co„,^o - M P(t), (3.03) where P(t) is given by (3.01) and i/'o = ^m — (^mTd is the phase intercept at zero frequency. For the transmission characteristic to be ideal in the sense that the peak pulse amplitude occurs when ^o = ^ — r^ = 0, it is necessary that \{/o = ±nT, where n is an integer. This is not necessary if the bandwidth is small in relation to the midband frequency. There will then be a large number of cycles of the modulating frequency Um within the envelope P(t), and the latter can be recovered by envelope detection regardless of the phase of the modulating frequency. With \po = zknir, P{t) = cos coJo , (3.04) X CO Jo oJiS sin coi^o ojo8 sin wo^o /„ ^.j-v IT Ciilti) IT Uoti where Um = (coo + wi)/2 and Wg = (coi — coo)/2. The shape of the impulse characteristic as given by (3.04) is illustrated in the upper haK of Fig. 8. Alternately the impulse characteristic may be regarded as made up of two components in accordance with (3.05). The first component corresponds to a low-pass characteristic of band- width coi , the second component to a negative low-pass characteristic of bandwidth ojq , as indicated in the lower part of the Fig. 8. The factor sin = t — Td and Qi = - Gtiiii) sin uto du. (4.02) The function Pi(t) will be zero at the same points as the original pulse transmission characteristic with a sharp cut-off at coi and under certain conditions also at other points. It will modify the original impulse char- acteristic by reducing the oscillatory tail, as illustrated in Fig. 10, but the zero points remain unchanged. With the above modification, the resultant impulse characteristic ob- tained by superposition of (3.01) and (4.02) becomes P(t) = - sin o)ifo ( — — 2 / (li(w) sin uto du. I , TT V^ii Jo / = - sin o}itoFii), (4.03) where Fit) = 1 r"^ — — 2 / (li{u) sill 7/^0 du to Jo (4.04) In the following the expression for F(t) is given for the case when the 742 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 band-edge is modified by a supplementary characteristic of the form Giiiu) = 2(1 ~ sill Tni/2(j)x) u < to^ , = 0 U > coj (4.05) This form of Qi represents a close approximation to actual modifications of band-edges by a gradual cutoff and also results in rather simple ex- pressions for the modified impulse characteristic With (4.05) in (4.04), 1 f"-" . F{t) = - — (1 — sin Tni/2wx) sin uto du, to Jo 1 — cos COx^O I cos COxto cos COx^O = Wi cos ooxto IT + 'IWxk TT — 20ixU. 1 1 (4.06) 1 _OixU T — 2oOxto TT + 2cOxio J ' cos CCxU ^0 1 — {2u)xto/iry- The impulse characteristic obtained from (4.03) is boix sin coi^o cos oixk Pit) T uito 1 — (2cOxio/7r)2 (4.07) For the particular case shown in Fig. 11 the value of Ux is taken to be wi/2. For a symmetrical bandpass characteristic, as shown in Fig. 12, P(0 = 2 cos {o:Jo - ypo) Pit). (4.08) Pit) is obtained by replacing coi by cos in (4.07), and \po is the phase intercept at zero frequency as in connection wdth (3.03). This gives Pit) = 5(j)s sin oisto cos COxto coJn 1 (2cOxV7r)2- (4.09) For the particular case shown in Fig. 12, the value of Wx is taken to be w,/2. The in-phase and quadrature components with respect to anj^ fre- quency are obtained from (2.19) with xpy = (jiTd and are shown in Fig. 12 for the particular case in which the reference frequency is displaced from the midband frequency by co^ = cos . THEORETICAL FITNDAMEXTALS OF I'lLSK T1{AXSMIRST().\ i;-! 5. IDEALIZED C'lIAUACTERISTICS WITH NATURAL LINEAR I'JLV.SE Sill FT With the type of amplitude characteristics discussed above it is necessary to employ phase equaUzation to obtain a linear phase charac- teristic. Furthermore, oscillations of appreciable amplitude remain in the impulse characteristic. A virtually linear phase characteristic to- gether with a reduction of these oscillations can be attained by a further extension of the gradual cut-off in Fig. 10, such that co^ = coi . An ampli- tude characteristic of this type, together with the corresponding impulse 1.0 0.8 0.6 0.4 0.2 0 1 N 1 1 1 \ 1 1 \ 1 1 1 1 V (a) FREQUENCY CHARACTERISTIC 0.2 0.4 0.6 0.8 1.0 1.2 1.6 Fig. 11 — Low-i);iss characteri.stio with gradual cut-ofl' and as.sociatcd impulise characteristic. Linear phase characteri.stic as.sinnod. 744 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 1.00 0.75 UJ \ V \ \ / \ t 0.50 ^u;r \ 5 / \ ^ 0.25 /k-^s-^ K UJ% - -^\ 0 / V FREQUENCY (a) FREQUENCY CHARACTERISTIC r, = 77/Ws = — ^ TIME >■ (b) IMPULSE CHARACTERISTIC Fig. 12 — Sjmimetrical band-pass characteristic with gradual cut-off and associated impulse characteristic. In-phase and quadrature components shown with respect to cor = wm — w» • characteristic is shown in Fig. 13. The supplementary amplitude charac- teristic and the impulse characteristic are obtained by making co^ = wi in (4.05) and 4.07). The resultant amplitude characteristic between co = 0 and co = 2coi in this case becomes A(a;) = 1 1 + COS ZO)] = COS 2 TTCO 4coi' and the impulse characteristic: fwi sin 2coi/o P(0 = (5.01) (5.02) TT 2coi^o[l — (2coiio/7r)-] ' where coi is the bandwidth to the half -amplitude point on the trans- THEORETICAL FVXDAMEXTALS OF PULSE TRANSMISSION J4-) mission freqiieiK-y characteristic and 'Jcoi the bandwidth to the point of zero ampU tilde. In Fig. 13 is also shown the amphtude characteristic given by (1.08), which Avil) have a hnear phase characteristic in the transmission ])and. i.e. from w = 0 to 2coi . Because of the close approximation of (5.01) to the proper type of amplitude characteristic as regards phase linearity, the phase characteristic associated with (o.Ol) may for practical purposes be regarded as linear. For a sjmimetrical band-pass characteristic as shown in Fig. M, the impulse characteristic is given by (4.08) and the envelope by (4.09) with cox = Ws , or Pit) = i^ sin 2o)Jo TV 2w.ai - (2co,./o/7r)2]" The in-phase and quadrature components shown in Fig. 14 with (a) IMPULSE CHARACTERISTIC (b) FREQUENCY CHARACTERISTIC r\\ 0.8 UJ §0.6 11 a. 5 0.4 < 0.2 0 \ 1 1 \ \ \ V 1 \ \ \ \\ V \ \ V 1 \ ■ 3 (^1 FREQUENCY ct'w / \ / \ / 1 \ \ / \ ^^ ^^ -0.75 -O.bO -0.25 0.25 0.50 0.75 tnfK. :2tof, Fig. 13 — Low-pass transmission frequency characteristic, 1, and associated iniinilse characteristic. Fre([uency characteristic, 2. is same as shown by solid lines in Fig. .3 and ha.s a linear phase characteristic between w = 0 and wmox . 746 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 l.UU 0.75 - y / 1 \ 0.50 ^ / \ 0.25 / - -a;s- -*+«-- -itj%--*\ N. 0 y^ ^^ ujr o FREQUENCY, 00 *- (a) AMPLITUDE VS FREQUENCY CHARACTERISTIC \ ^-ENVELOPE P(t) (b) IMPULSE CHARACTERISTIC Fig. 1.5 — Particular case of band-pass characteristic with tiradual cut-off in which inij^ulse characteristic is zero at intervals ro = 2/0 748 THE BELL SYSTEM TECHXICAL JOURNAL, MAY 1954 by the dashed hnes in Fig. 15. With a gradual cut-off, however, the phase characteristic will be nearly linear and have a finite slope, so that the above pulse transmission rate can be realized provided ^o = zLmr. The same pulse transmission rate can also be attained with vestigial side-band transmission, discussed in section 14. Another particular case of interest is that shown in Fig. 16, in which Um = 2us ■ In this case (4.08) becomes with \po = zLnw and with P(t) as UJm = 2COS FREQUENCY (a) FREQUENCY CHARACTERISTIC ENVELOPE P(t)/ / coso^mtPlt) — -/- -1^"- T, = l/2fs WITH PULSES TRANSMITTED AT POINTS I, 2, 3 THERE IS NO MUTUAL INTERFERENCE BETWEEN PULSE PEAKS (b) IMPULSE CHARACTERISTIC Fig. 16 — Particular case of symmetrical band-pass characteristic for which , = 2w, . THEORETICAL FUNDAMENTALS OF IMLSE TRANSMISSION 740 0.9 N^v -*1 (f ^- 1 1 0.8 \\ i i 0.7 _ 1 lii 0.6 a 1- a -^ 0.4 ^ N^/ri^^o.s - ^ 0.3 - \\ 0.2 - \\ O.t - ^\^ 0 ^^^5s»_ 0 a;, 2a;, FREQUENCY Fig. 17 — Modification of frequency characteristic to obtain same response as for impulses, when pulse duration is in-olonged to half the pulse interval. given by (5.03) Fit) = — ^ cos COmto sin cjmto 5cOr sin 2o3mto (co„,^o/7r) -] (5.05) TT 2Umto[l — (o^mto/Tr)'^]' Pulses can in this case be transmitted witliout mutual interference be- tween the pulse peaks at the points shown in the above figure. The effective pulse transmission rate is the same as for a low-pass characteris- tic between w = 0 and co = 2a) ,„ with haK amplitude at co,„ .* As mentioned in Section 2, when pulses of finite duration are employed, the same response as for impulses is obtained if the amplitude charac- teristic is modified by the factor (co5/2)/sin (co8/2). In Fig. 17 is shown the resultant minor modification in the amplitude characteristic (5.01) when the duration of the pulses is equal to half the pulse interval. The low-pass and band-pass amplitude characteristics considered above can also be regarded as the specti'a of pulses applied to a trans- mission system ha^'ing a constant amplittide characteristic over the * W. R. Bennett and C. B. Feldniaii originally i)roposed this type of charac- teristic in an unpuljlished memorandum, as a means of matching the bandwidth economy of baseband transmission without inclusion of frequencies near zero. 750 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 band of the spectra. If the phase characteristic of the system is Hnear over this band, the received pulses will have the same shape as the impulse characteristics. It should be recognized, however, that there may be appreciable phase distortion within the transmission band or pulse spectrum, if there are amplitude discontinuities beyond the band resulting from a sharp cut-off by filters. Nevertheless, the type of am- plitude characteristic or frequency spectrum considered above has de- cisive advantages from the standpoint of transmission distortion of the pulses, as shown later, since appreciable phase distortion Avill ordinarily be confined to the edges of the band where the frequency components of the pulse spectrum have low amplitudes. Another type of amplitude characteristic resembling that shown in Fig. 13 and frequently considered in connection with pulse transmission is a Gaussian characteristic: yl(co) = e-'"'\ (5.06) The corresponding impulse characteristic is P(0 = ^-A_ e-'-'''\ (5.07) 2(7rcr) 1/2 If it is assumed that the amplitude is reduced to 1 per cent of the peak value after an interval ta = tt/wi , corresponding to the first zero point of an ideal impulse characteristic, it is necessary that U /4o- = 4.6, or (X = .54/col^ The corresponding amplitude and impulse characteristics are A(co) = e-o-^^("/"i)', (5.08) and P{t) = -^ e-oA^ito<^0\ (5.09) In Fig. 18 a comparison is made of the two frec^uency characteristics (5.01) and (5.08) considered above, and of the corresponding impulse characteristics (5.02) and (5.09 V The comparison shows that for the same pulse transmission rate and with negligible intersymbol inter- ference, a somewhat wider band must be provided with a Gaussian amplitude characteristic. This is a disadvantage, particularly when the band is restricted within prescribed limits by considerations of inter- ference in adjacent transmission bands, as radio pulse systems. THEORETICAL FUXDAMEXTALS OF PULSE TRANSMISSION 751 5 < 0.5 // X 1.0 (a) FREQUENCY /t 7 ^^ 0.8 liJ Q0.6 D _l 5 0.4 < 0.2 0 NX CHARACTERISTICS // // f \ A \\ k 1 \ // / / / / \ \ \ \ \ / / / \ \ \ > \ y ^:- 1 1 1 1 A ^ \ «v^ ' ^ \ \ 0 Cc"! ZU)\ FREQUENCY \ \ \ \ \ \ V (b) IMPULSE CHARACTERISTICS \ \ \ \ \\ s. \ V ^ -0.4 -0.2 0 0.2 0.4 0.6 0.8 I.O 1.2 1.4 1.6 1.8 2.0 2.2 'tofMAX - 2tof) Fig. 18 — Comparison of two representative frequency and impulse transmis- sion characteristics. Frequency characteristic 1 : T{ui) = ^[l -(- cos 7rc<;/2wi]. Fre- quenc}' characteristic 2: T'(co) = e.xp — 0.54(co/a)i)2. Amplitude characteristic 1 of Fig. 18 has certain properties, aside from the linearity of the associated phase characteristic, which makes it preferable to a Gaussian as well as other types of amplitude characteris- tics for most pulse systems. The corresponding impulse characteristic has zero points at intervals n = l/2/i with the minimum possible oscilla- tion consistent with this property for a given bandwidth. This permits the use of this impulse characteristic for pulse systems with discrete pulse positions with minimum intersymbol interference and considerable tolerance on synchronization. Since the oscillation in the impulse char- acteristic is inappreciable, it can also be used for pulse systems without discrete pulse positions and with other methods of detection than syn- chronized instantaneous sampling. In view of these attributes, an ampli- tude characteristic of the above type, rather than a constant amplitude characteristic with sharp cut-off, may be regarded as ideal when various physical requirements for practicable pulse systems are taken into con- sideration. 752 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 6, PULSE ECHOES FROM PHASE DISTORTION For any transmission — frequency characteristic the corresponding impulse characteristics can be determined from the Fourier integral relation (2.01). This, however, may involve the evaluation of compli- cated integrals, which in general would require numerical integration and would be a rather elaborate procedure. A preferable method of sufficient accuracy in most engineering applications is to employ the theoretical solutions given previously for various ideal transmission characteristics with a hnear phase shift as a point of departure or first approximation. A satisfactory second approximation can in many instances be secured by evaluating the transmission distortion resulting from a sinusoidal deviation in the phase characteristic. Furthermore, any type of deviation in the phase characteristic can in principle be represented by a Fourier series in terms of harmonic sinusoidal devia- tions. Aside from the circumstance that in many cases a sine deviation in the phase characteristic affords a fairly satisfactory approximation to actual phase distortion it has the advantage in theoretical formulation that it permits determination of the resultant pulse distortion by the method of "paired echoes." In the usual application of this method only small phase deviations are considered resulting in a single pair of pulse or signal echoes of small amplitude, and the method is then particularly simple.^' ^ When delay distortion is appreciable, however, as is fre- quently the case in wire circuits, it becomes necessary to consider a large number of pulse or signal echoes of considerable amplitude. Since the amplitudes of the pulse echoes may be obtained from available tables of Bessel Functions, the determination of the echoes is, nevertheless, simple in procedure and the determination of the shape of the distorted pulses or other signals not too elaborate. A given amplitude characteristic within the transmission band may be associated wdth various phase characteristics, depending on the shape of the amplitude characteristic outside the transmission band and also on whether or not a minimum phase shift system is involved. It is therefore permissible to consider the effect of various departures from a given phase characteristic independent of the amplitude characteristic within the transmission band. With a sinusoidal departure from a given phase characteristic \{/q(o:) as shown in Fig. 19, the modified phase function becomes ;/'(to) = ^o(w) — h sin wr. (6.01) With THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 753 \ho modified ti'ansinissioii-frefiiieiicy cluiracteristic becomes which, inserted in (2.01) gives -<'>=^j_: ro(/co)e ib sin icT iwt 7 e f/c (6.02) (6.03) are (6.04) The following relation (Jaeobi's expansion) in which Ji , J2 • Bessel Functions in their usual notation can now be employed e = Jo{b) + Ji(h)[e - e J J2{b)[e -\- e T + j.me'''" + e-'n + • • • Let Po(0 designate the shape of the received pulse or other signal for a transmission frequency characteristic To(i(jo) obtained from (6.03) with 6 = 0. In view of (6.04) the solution of (G.03) may then be written P{t) = Jo(b)Po(t) + Ji(b)[Po(t + r) - P,{t - r)] + J2{b)[Poit + 2t) + P,(t - 2t)] + J,mPo{t + St) - P,{t - 3r)] + Ji{b)[Po(t + 4r) + Po{t - 4r)] + (6.05) The shape of the received pulse or signal P{t) is thus obtained by superposing an infinite sequence of pulses or signals of shape Po(t). The peak amplitudes of the pulse or signal echoes and the times at which they occur with respect to ^ = 0 are given in the following table. The reference point t = 0 is arbritrarily selected to coincide with the peak of the pulse Po(t) : Time -3t -It — T 0 T 2t 3t Amplitude .... Jz(b) J2ib) J lib) Jo(b) -Ji(b) Mb) -Mb) A sufficient number of echoes must be considered until their peak amplitudes become negligible. The superposition of echoes to obtain the resultant pulse is illustrated in Fig. 20. Instead of plotting the various echoes and combining them 754 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 into a resultant pulse or signal as in Fig. 20((') the equivalent and less laborious method shown in (d) can be employed. With the latter method the pulse Po is plotted with reversed time scale and its peak made to coincide with the point for which the amplitude of the resultant pulse P is to be determined. The amplitude of P is determined as indicated in the figure. In the particular case where the original phase characteristic ^0 is linear, the pulses Po(t) will l)e symmetrical with respect to their peak amplitude, and this assumption will be made in the following applications. For amplitudes 6 « 1, the Bessel Functions become negligible except for Jo and Ji , which are given by JqQ)) ^ 1 and /i(6) ^ b/2, so that (6.05) becomes Pit) = Poit) + I Poit + r) I Poit - r). (6.06) p (oj) = coTd - h SIN cor FREQUENCY, U) —»' (a) LOW-PASS CHARACTERISTIC S'Cul^urd-bsiN ur (b) BANDPASS CHARACTERISTIC Fig. 19 — Low-pass and band-pass characteristics with sinusoidal phase dis- tortion. THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION I .)0 For amplitudes h > 1, it is necessary to consider a greater number of echoes, as "will be evident from Table I for b = 1, 2, 5, 10 and 15 radians. The preceding equations apply to low-pass characteristics and also to sj-mmetrical bandpass characteristics, as shown in Fig. 19. In the latter case a(ii) = a(-u) and ^(-ii) = -^(u) in (2.10) and (2.11) so that R+ = R- and Q+ = Q- and (2.09) becomes P{t) = cos (c^rt - ypr) R{t), (6.07) where R(t) = R+ + R- and co, = w,„ = midband frequency. The en- velope R(t) is accordingly obtained by replacing Po(t) in (6.05) by Ro(t), the envelope in the absence of phase distortion. In Fig. 21 is shown a particular case of a sine deviation in the phase characteristic and the corresponding delay distortion, which approxi- mates that encountered in many instances. For a low-pass system the phase and delay distortion would be as shown for u > 0. In this particu- Jdi. AMPLITUDES OF PULSE ECHOES ±k. (a) -Ji Po(t+2r) Po(t+r) Po(t) Po(t-r) Po(t-27-) J2Po(t+2r ;— RESULTANT PULSE P(t) .JoPo(t) J2(Po-27-) -^ P= Joa 0+ Jiai-j,a-i y/ ip 1 \ + Jgag+Jaa-s ai| J, 1 Jo laoS^ a-, = o |J2 a-2 = o ~Ji (c) (d) Fig. 20 — Determination of resultant pulse by superposition of pulse echoes. 756 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 Table I — Amplitudes of Echoes, /„(6) b = 1 2 5 10 15 n = 0 0.7652 0.2239 -0.1776 -0.2459 -0.0142 1 0.4401 0.5767 -0.3276 0.0434 0.2051 2 0.1149 0.3528 0.0466 0.2546 0.0416 3 0.0196 0.1289 0.3648 0.0584 -0.1940 4 0.0340 0.3912 -0.2196 -0.1192 5 0.2611 -0.2341 0.1305 6 0.1310 -0.0145 0.2061 7 0.0534 0,2167 0.0345 8 0.0184 0.3179 -0.1740 9 0.0055 0.2919 -0.2200 10 0.2075 -0 0901 11 0.1231 0.1000 12 0.0634 0.2367 13 0.0290 0.2787 14 0.0120 0.2464 15 0.0045 0.1813 16 0.1162 17 0.0665 18 0.0346 19 0.0166 20 0.0073 lar case the maximum amplitude h is at the maximum frequency wmax = 2wi , so that sin wr = 1, or wr = 7r/2, for w = 2wi . Hence the interval between pulse echoes is r = 7r/4coi = l/8/i . The interval r is accordingly }4, the interval n •= l/2/i required for the pulse Po(t) to reach zero ampli- tude in the absence of phase distortion. PHASE CHARACTERISTIC AMPLITUDE CHARACTERISTIC ENVELOPE DELAY '^-brcos ur ■-- — t "' Fig. 21 — Particular case of sinusoidal phase deviation. THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 757 For the particular case illustrated in the above figure, the delay dis- tortion is given by When CO - 0 \\'hon COT = TOJr Hence dyp/diAi = — 6r cos cor. dxp/do) = —br = — C?max • = 7r/2 #/dco = 0. dyp/doi = —dmvx COS (c07r/2cOmax)- (6.08) (G.09) With r = 7r/2comas and br = dmax , the following relation is obtained 0 — — COmax C^max — ^/max O^n (6.10) In Fig. 22 are shown the positions of the pulse echoes for the above case on a numerical scale ^/ma^ , together with their amplitudes for 6 = 5 radians. On this scale the interval between pulse echoes 7 = 1/4/max is }i. In the same figure is shown an assumed pulse shape in the absence of phase distortion, which is the same as shown in Fig. 13, except that the small tail has been neglected. The peak of the pulse is taken at (fmax = —0.75, and the amplitude of the resultant pulse at the 1.00 0.85iJ 'bO.85 *o.70 \transmitted pulse AMPLITUDES OF 0.39 .-'PULSE ECHOES 0 "t 'wax Fig. 22 — Illustrative example of cak'uhition of impulse characteristic shown in Fig. 23, by method illustrated in Fig. 2()(d). 758 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 1.0 0.9 5 0.4 < CO 3 0.3 O r\ r \ DELAY DISTORTION / -^ ^ L / \ ~*^ '"max [*" / A \ / /b=i5 J 10 \i 5 / ° / \ Y 1 QmaxTmax = 3.75 I 2.5 \ ,i V 0 / 1 \ \ fl / / J 1 J / / / \ \ / / J / / \r ^Z" >\ 10 15 / J \ A 10 "^ ] / J \ 4 \j v/ -5 -4 -3-2-1 0 1 2 Fig. 23 — Impulse transmission chamcteristics with cosine variation in delay. corresponding point obtained by the method illustrated in d of Fig. 20 is P = 1-0.365 4- 0.85 (0.39 + 0.05) -H 0.5 (0.26 - 0.328) + 0.15 (0.131 = 0.70. 0.178) In Fig. 23 are shown the resultant pulses obtained by the above method for various values of h and the corresponding values of c?ma,x/mas • Since the interval between pulse echoes is small in relation to the duration of the pulse Po(t), as seen from Fig. 22, the individual pulse echoes cannot be discerned in the resultant pulses shown in Fig. 23. It will be noticed that as h increases, the pulses are received with decreasing transmission delay, which is due to the choice of reference delay in the delay distor- tion curve. That is, as dmax or h is increased, the delay becomes increas- THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 759 ingly negative with respect to (/,„ax = 0 used for reference. The curves apply to u band-pass system as indicated in the figure, and also to a low-pass system having the delay distortion shown above the midband freciiiency of the band-pass sj'stem. An improved approximation to phase distortion is sometimes obtained by considering two sine deviations in the phase characteristic. If the phase characteristic is given by \l/((jj) = i/'o(co) — h' sin o)T — b" sin ut, (6.11) b =-0.0435 b (a) PHASE DISTORTION // (b) DELAY DISTORTION d MAX = 0.85 d MAX UJ Q -0.2 LL o Q -0.4 5 -0.6 < p -0.( < \\ (c) 1 1 \ \ \ / 1 f / \ S // \ 1 / / v^ ^^^ y^ K^ '-.. i .,'- ''" -1.0 -O.S -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 f/fMAX = f/2fl P'ig. 24 — Shape of delay distortion with combined fundamental and third luirmonic cosine variation in delay. 760 1.0 0.9 0.8 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 5 0.2 z r DELAY DISTORTION C^MAX^^MAX =3.75 _/ \ k "'fwAx --H ■V ? // ,/ \\ H.I.... / / 1 1 1 1 \ \ 1 1 / / / / i \ClMAX■f^^J1AX = 3■18 1 1 Umax V^ 1 >v i »^ * ^ / / / / / / 1 1 1 I ^_Ay 1 1 1 / M 1 1 / M 1 I 1 ' '0 / 7 1 / / \ / / \ / A ^_ / V / > ^ ^ ^^ ^' " 1 » / / / \/ v_/ -5 -4 -3-2-1 0 12 tf MAX = 2tf, Fig. 25 — Comparison of impulse characteristics with fundamental and com- bined fundamental and third harmonic cosine variation in delaj^ as in Fig. 24. the combined effect of the two sine deviations is obtained by first deter- mining the effect of —V sin cor' from (6.05). The vakie of P(0 = -Pi(0 thus obtained from (6.05) Avith 6 = 5' and r = r' is next substituted for Po(0 in (6.05), with h = h" and r = r" to evaluate the effect of -h" sin cor". That is, the system is considered to consist of a tandem arrange- ment of two components, the first with a phase distortion —6' sin cor and the second wdth phase distortion —5" sin cor". In Fig. 24 is sho^vii a particular case in which the second component is a triple harmonic of the first with amplitude h" = —0.04356'. This results in an improved approximation to the delay distortion encountered in certain wire facilities, where the band is sharpl}^ confined by filters. In Fig. 25 is shown the pulse shape for this case with b' = 15 radians, together \xith. that for a single sine deviation of 6 = 15 radians. It AAdll be recognized from the above that as the number of sine com- TIIEOKETICAL FUNDAMENTALS OF PULSE TRANSMISSION 761 ponents required to represent a given phase distortion increases, the determination of the resultant pulse becomes rather laborious, unless the sine deviations are all small in amplitude. In the latter case each sine deviation corresponds in a first approximation to a single pair of echoes, so that the effect of a number of sine deviations can be obtained by direct superposition. 7. PULSE ECHOES FROM AMPLITUDE DISTORTION Departures from a given amplitude characteristic may in certain cases be approximated by a single cosine variation, as illustrated in Fig. 26. Since the amplitude characteristic is an even function of co, any departure from a gi\'en amplitude characteristic may be represented by a cosine Fourier series. The effect of a cosine variation in the amplitude charac- teristic is therefore of basic interest. A cosine variation will in general be accompanied by a change in the phase characteristic, as discussed in Section 1, but it will first be assumed that phase correction is employed to maintain a fixed phase characteris- tic. Let Ao{o)) be the original amplitude characteristic and let the modified amplitude characteristic be of the form A(co) = i4o(w)[l + a cos ut] (7.01) Equation (2.01) for the impulse transmission characteristic then be- fomes, ^nth To(zco) = Ao(w)e~'^"^"\ P(0 = ^/ ZtT Jco To(ico) I + -{e 4- e ) t iO t J e rfoj. (7.02) = Poit) + I Pod + r) -f I Poit - r). (a) RATIO OF AMPLITUDE CHARACTERISTICS (b) IMPULSE CHARACTERISTIC O I- < 1 A {cj)/Ao(.co) = 1 +a cos cj T FREQUENCY, OJ \* T *\* T >] Fig. 2G — Pulse echoes from cosine variation in amplitude characteristic with- out change in phase characteristic. 762 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 There will thus be pulse or signal echoes of amplitude a/2 at the time T before and after the main pulse as illustrated in Fig. 26. With a cosine variation in the attenuation rather than in the amplitude characteristic, the modified amplitude characteristic becomes A (a;) = ^o(co) e"*^"' "^ (7.03) and the modified impulse characteristic p(t) = ^ r ToMe'"'°"'V''' dw. (7.04) 27r J- CO The expansion corresponding to (6.04) is in this case: a cos cor t / \ \ t / \ /' ^^t t — ia)T\ e = Io{a) + Ii{a){e + e ) + /2(a) (e"'"^ + e~''"0 + /3(a)(e''"^ + e-^'"0,+ ••• (7.05) where h , h • ■ ■ are Bessel functions for imaginary arguments in their usual notation. The resultant modified impulse characteristic in this case becomes Pit) = /o(a)Po(0 + h(a)[Poit + r) + Po(t - r)] + h(a)[Po(t + 2t) + Po(t - 2r)] (7.06) + h(a)[Po(t + 3r) + Poit - St)] + • • • which can be interpreted in a similar way as discussed for (6.05). For small values oi a, h (a) = 1, h (a) = a/2 and the remaining terms in (7.06) negligible, so that (7.02) is obtained. As discussed in Section 1, when the amplitude characteristic is modi- fied in accordance with (7.01), the resultant modification in the phase characteristic is in accordance with (1.13) _i r sin wr , . 4/1 = 2 tan -— — ; . (^.07) 1 + r'' cos cor The modified transmission-frequency characteristic is in this case r(iw) = ro(2co)(l + a COS coT)e~''^S (7.08) which can be transformed into r(zco) = TS^) — 1-, (1 + re-n\ \ -\- r- = Toiic) — ^, (1 + 2,-e-'"^ + r'e-'n. 1 -f r (7.09) THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 7G3 Thus, with II eosino variation in the amjilitudc characteristic in ac- cordance with (7.01), accompanied by a minimum phase shift change in the phase characteristic in accordance with (7.07), the modified impulse characteristic becomes Pit) = :r4-, [n(0 + 2rPo(t - r) + r'Po{t - 2r)], (7.10) 1 -|- r- where , = i [1 - vr^^^]. (7.11) The received pulse or signal P(t) will thus consist of three components each having the same shape as the pulse or signal Po{t), but differing in amplitude and displaced in time, as indicated in Fig. 27. For small values of the amplitude a of the cosine deviation, r = a/2 and I -{- r~ = 1, so that Pit) = Poit) + aPoit - r) + ^ Poit - 2r). (7.11) The solution for a somew^hat similar case given elsewhere,^ has an infinite number of echoes, with the second echo given by a Poit — 2t) rather than (a /4:)Po(t — 2r) as above. In the case referred to, the ampli- tude deviation is in a first approximation a cos cor, but there are addi- tional terms in cos 2oot, cos Scot etc, which are responsible for the different amplitude of the second echo and for the infinite sequence of echoes. With a cosine modification in the attenuation characteristic as given by (7.03), there will be a corresponding sine modification in the phase characteristic in accordance with (1.11). The modified transmission- (a) RATIO OF AMPLITUDE CHARACTERISTICS (b) IMPULSE CHARACTERISTIC FREQUENCY, Cv Po(t-27-) 7 *^ r- a«i, r = -a/2 Fig. 27 — Pulse echoes from cosine variation in amplitude characteristic with associated minimum phase shift variation in phase characteristic. 764 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 frequency characteristic is in this case rr(^ \ rp ( ^ \ a (coswT— tsin ojt) = ro(tco)e , 2 3 l+a6 +-e +3^6 + (7.12) = ro(ico) The modified impulse characteristics is in this case P(0 = Po(0 + an(< - ^) + ^ ^o(^ - 2r) + I Po(^ - 3r) + (7.13) For small values of a both (7.11) and (7.13) give for the modification in the impulse characteristic resulting from a small cosine deviation in the amplitude or attenuation characteristics accompanied by changes in the phase characteristic: P(0 = Po(0 + a Po(^ - r). (7.14) In certain appUcations it is convenient to regard Po(0 as a pulse or signal apphed to a transmission line and P(0 as the received pulse or signal w\\h a cosine deviation in the amplitude characteristic of the transmission line. In the lower part of Fig. 28 is sho■^^'n the modification in the received pulses resulting from a slow pronounced cosine deviation in the ampli- tude characteristic shown at the top. In Fig. 29 is showTi the effect of positive and negative cosine variations when the ampHtude at zero fre- quency is held constant, a condition which may be approximated in wire systems as a result of variation in attenuation over the transmission band with temperature. Curve 1 would correspond to a 3.5 db smaller loss at the maximum frequency 2wi than at zero frequency, and curve 2 to a 6 db greater loss at the maximum frequency. It will be noticed that pulse distortion as well as the variation in the peak amplitude of the pulses is greater under the first condition, i.e. curve 1. Pulse overlaps can in both cases be avoided by a moderate increase in pulse spacmg, and in the first case can be substantially reduced also bj' a decrease in pulse spacing. 8. FINE STRUCTURE IMPERFECTIONS IN TRANSMISSION CHARACTERISTICS As a result of imperfections in the transmission medium and in equali- zation there may be fine structure departures from a nominal transmis- sion characteristic, as illustrated in Fig. 30. They are often caused by THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 7G5 2 0.5 < ^;— — — ______^^^^- A/Ao- (h-0.5 COSo;/-) ^^\ ^ -^^ r= 77/4 0;, ^">$k i"^ r^'^^i FREQUENCY 0. PULSE CORRESPONDING TO Aq 1. FIRST PULSE ECHO 1.4 y \ 2. SECOND PULSE ECHO 3/ \ 3. RESULTANT PULSE FOR / \ FREQUENCY CHARACTERISTIC A uj 1.2 ^ / \ a / \ D / \ ►— / \ Zl / \ a 1.0 — / \ 5 / 0 \ < / .^^'^""'^^ \ en 1 / ^\ \ O 0-8 - / / \ \ z / / \ \ < 1 / 1 \ \ Z 06 - / / ' \ \ < 1 / 1 \ \ 1- 01 / / ' --•^^""■■^-««.. \ \ Z / / '*^^ 1 "^ \ \ - 0.4 / /\ \A 0.2 - / ^^' \ nV ^ y ^ .^""^ L_--|--^--r---\3^V^ »-^/:/4»T^Aj/4*n --- ^ Fig. 28 — Modification of impulses characteristic by slow cosine variation in amplitude characteristic. echoes in very long lines resulting from impedance mismatches. Fine structure deviations from a specified amplitude characteristic may in principle be represented by a cosine Fourier series, since the amplitude function is an even function of o). Thus, if the specified amplitude char- acteristic is .4o(co), the actual amplitude characteristic ^(co) may be represented by an infinite cosine Fourier series as: A(co) = .4o(ci;)[l + Oi cos COT + «? cos 2ajr + • • • + a^n cos mwT + • • •] . (8.01) 7()() THE BELL SYSTEM TECHNICAL JOURNAL, AL\Y 1954 The coefficients ai , a2 •••««•• • are determined in the usual manner by Fourier series analysis to represent the function (8.02) over the frequency band. If Aoiu) closely approaches A(co) the fine -0>^ 2 \ \ (a) RATIO /\{CO)//\q{CJ) NUMERALS ON CURVES CORRESPOND TO THOSE ON IMPULSE CHARACTERISTICS BELOW CO, FREQUENCY 2U}, Fig. 29 — Effect of slow cosine variation in amplitude characteristic when amplitude at zero frequency is held constant. THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 767 structure departures a(w) in the transmission characteristic and luMice the coefficients Oi , 02 ■ ■ • a,,, • • • will l)c small. In the above representation .lo(w) can also he refi;arded as th(> ampli- tude characteristic of a terminal network or as the frequency spectrum of a pulse applied to a transmission system witfi an amplitude charact- teristic /(oj) = 1 + «(co). In a Fourier series analysis of the deviation in the amplitude charac- teristic, the fimdamental period of the amplitude variation would be selected so that there is one complete cycle betAveen — coi and a;i , the cutoff frequency, in which case ojit = tt or r T = — COi This is the interval between pulse echoes when the amphtude charac- teristic is represented by (8.01). It is identical with the interval ri given by (3.02) at which pulses can be transmitted without mutual interfer- ence with a constant amphtude transmission frequency characteristic. A/Aq = i + oc(a») Aq = NOMINAL OR IDEAL CHARACTERISTIC : ACTUAL CHARACTERISTIC FREQUENCY ^^MAX (a) TRANSMISSION FREQUENCY CHARACTERISTIC ^-Pq = NOMINAL OR IDEAL ^^" IMPULSE CHARACTERISTIC * --P = ACTUAL IMPULSE CHARACTERISTIC TIME AP= P-Po'' (b) IMPULSE TRANSMISSION CHARACTERISTIC Fig. 30 — Fine structure imperfections in transmission frequency cliaracteristic and resultant i)rolongation of impulse characteristic. 768 THE BELL SYSTEM TECHNICAL JOURNAL, AL\Y 1954 Assume that pulses of unit peak amplitude but varying polarity are transmitted at intervals t = n and consider the interference with a given pulse from all pulses. As illustrated in Fig. 31, the first preceding and following pulses will in accordance with (7.02) give rise to a pulse echo dzai/2 and the second preceding and following pulses to a pulse echo ±a2/2 etc., where the signs of the echoes depend on the polarity of the pulses and on the signs of the coefficients Oi , 02 . The resultant intersymbol interference Ua{t) will depend on the polarity of the various PULSE ECHOES FROM INDIVIDUAL PULSES a,/2 4 82/2 ♦ PULSE ECHOES 1 a,/2fIfai/2 ♦ a 1/2 32 T^T !t-a2/2 n Ja3/2 1-92/2 ta3/2 1-33/2 -33 xAJfa^A RESULTANT PULSE TRAIN AND INTERSYMBOL INTERFERENCE U;,= Fig. 31 — Combination of pulse echoes into intersymbol interference for a particular case. THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 701) pulses and will thus varj' with time. It can have any value assumed by the expression UM) = ±|-^±|^±|?±|-.. ±|:±^+ ••• (8.03) The maximum possible intersymbol interference will thus be the sum of the absolute values of the coefficients a„, . f'a = I «! ! + 1 ao I + I as I + • • • + I o,„ I + • • • (8.04) In certain pulse systems, such as PAM time division systems, rms intersymbol interference is of main importance, while in others, such as PCM or telegraph systems, peak intersymbol interference is of principal interest. If the fine structure imperfections are regarded as of random nature, in the sense that they are not predictable and vary between systems having the same nominal transmission characteristics, peak intersjTnbol interference can be estimated from rms interference by applying a peak factor of about 4. With random variation in the ampli- tude of intersymbol interference, the probability of exceeding 4 times the rms value is in accordance with the normal law about 5 X 10~^ Peaks in excess of 4 times the rms value will thus be so rare that they can for practical purposes be neglected. The rms intersymbol interference is equal to the root mean square of all the different values which can be assumed by expression (8.03). This turns out to be equal to the root sum square of the amplitudes am/2 and —am/2 of the pulse echoes, or Ua = 1/2 When Urn are the various coefficients in the Fourier representation of a{u) over the frequency band from — cci to coi , the following relation holds. ^ 2_/ ^m = o~ / a (w) dco = ~- / a^((jo) do I 1 ZCOl J-ui COi Jo (8.06) where a(w) in the present case is given by (8.02) and represents the departure in the ratio /l(aj)/^lo(co) from unity. With (8.06) in (8.05) the following expression is obtained for rms intersymbol interference due to amplitude deviations a:(co) not accom- panied by phase deviations C/a = g (8.07) 770 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 where g is the rms deviation in a(ui) over the transmission band as given by ni/2 a = 1 f y \ 1 — / a \w) aoi OJi Jo 1 r ^ o Ji -^0 1/2 (8.08) The rms amplitude deviation expressed in db is a'' = 20 logio(l + g) ^ 8.69 g when g < 0.1 (8.09) A corresponding analysis can be made for fine structure imperfections in the phase characteristic. The deviation /3(aj) = ypio:) — t/'o(w) from a prescribed phase characteristic i/'o(co) may in this case be represented by a sine Fourier series since the phase characteristic is an odd function of co: /3(co) = 61 sin COT + 62 sin 2cor + • • • + 6», sin ??icor + The resultant peak intersymbol interference becomes ^6 = 1 &1 1 + 1 62 1 + • • • + I &« 1 + • • • and the rms intersjrmbol interference U, = \tK 1/2 = h, (8.10) (8.11) (8.12) where h is the rms phase deviation in radians as given by [1 /•"! "]l/2 - / |8'(co) do: , COl Jo J (8.13) In the above derivation, the amplitude and phase deviations were assumed independent of each other. The resultant rms intersymbol interference from both is in this case u = {u: + u^Y' (g^ + i')' (8.14) This relationship, applying to an ideal transmission characteristic, has been established by a different method in a paper by W. R. Bennett. THEORETICAL FUNDAMENTALS OF PULSE TRANSMLSSION 771 From (7.05) it will be seen that with minimum phase shift relation- ships a small cosine deviation of amplitude a„, in the amplitude charac- teristic will be accompanied by a phase deviation bm = am. Hence in this case (8.14) gives U = 2'" a (8.15) This also follows when it is considered that in this case all the pulse echoes occur after the main pulse, and have ampUtudes Oi , a^ • • • a^ . The root sum square of the amplitudes is in this case [X* '^''«T ^ which is greater than Ua as given by (8.05) b}^ the factor 2"". The above analysis was based on an infinite sequence of pulse echoes, which combine to give the proper pulse distortion but may be regarded as fictitious in nature. The assumption of an infinite secjuence of pulse echoes can be avoided by a different method of analysis outlined below, which does not involve the assumption that the coefficients are known from a Fourier series analysis, and furthermore, does not assume an ideal amplitude characteristic with a sharp cut-off as above. Let Ae and Aoe~**° designate tw^o transmission — ^ frequency charac- teristics, where A, Ao , \j/ and i/'o are functions of w, which for con- ^'enience is omitted in the following. The squared absolute value of the difference in the transmission frequency characteristics is then A,e-'^' 1' = I A e"' ^0 [2(1 - cos ^) (! + «) + a], (8.16) where a = a^u) = {A — .4o)A4o represents the deviation in the ratio of the amplitude characteristics from unity and /3 = /3(co) = xp — \po the deviation in the phase characteristic. Let P and Po designate the impulse characteristics corresponding to the above transmission frequency characteristics, and let AP = P — Po • Assume that unit impulses of varying polarity are transmitted at uni- form intervals rj . The rms value of AP over the interval n in relation to the maximum amplitude P(0) of the received pulses, or the rms inter- symbol interference U, is then given by U — [ A'[2(l - cos i3)(l + a) + a] do: rri Jo 1 P(0) L^rri Jo (8.17) For small values of a and /3, this expression becomes u = (8.18) 772 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 If a and /3 are random variables representing fine structure deviations uniformly distributed over the transmission band, it is permissible to simplify (8.18) to: U = v(-^T {a+hy'\ (8.19) where 1 / /•Wmax \l/2 a = I / adc^), and (8.21) / 1 /""niax \l/2 b = 1 / /3' dco , (8.22) \Wmax •'O / where Wmax is defined as in Fig. 30 and coi is the bandwidth at the half amplitude point. For a transmission characteristic with linear phase shift, aside from small random imperfections as considered here: P(0) = - / ^0 do. (8.23) T Jo For the particular case of a transmission characteristic with constant amplitude between co = 0 and coi = Wmax , 17 = 1. Pulses would in this case be transmitted at intervals n = tt/oji so that tt/ojiti = 1 and (8.19) is identical with (8.14). For a transmission characteristic of the type sho\vn in Fig. 13, pulses would also be transmitted at intervals n = tt/wi so that tt/coiti = 1. In this case comax = 2wi , and evaluation of (8.20) gives 77 = 3^'^/2 = 0.866. Rms intersymbol interference is thus reduced by the factor 0.866, for the same values of g and b. However, these are now the rms devia- tions taken over a band which is twice as great as with a sharp cut-off at coi . Expressions (8.14) and (8.19) can also be applied to localized imper- fections in the amplitude and phase characteristics confined to a narrow portion of the transmission band. This follows when it is considered that such deviations can be represented by Fourier series containing a large number of coefficients, so that the resultant intersymbol inter- ference can attain a great number of different values depending on the sequence of transmitted pulses. A particular case of a localized imperfec- tion in the amplitude characteristic in the form of a low-frequency cut-off is considered in the following section. THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 773 9. TRANSMISSION DISTORTION BY LOW FREQUENCY CUT-OFF A low-frequency cut-off in the transmission frequency characteristic of wire systems is unavoidable with transformers as employed for in- creased transmission efficiency or other reasons. In single sideband frequency division systems, there is a low-frequency cut-off in individual channels caused by elimination of the carrier and part of the desired sideband. The effect of a low-frequency cut-off can be avoided by em- plojdng a symmetrical l)an(l-pass characteristic as illustrated in Fig. IG, or more generally by double sideband transmission with a two-fold in- crease in bandwidth as compared to a low-pass system. It can also be overcome by vestigial sideband transmission with inappreciable band- width penalty, but with complications in terminal instrumentation. The effect of a low-frequency cut-off can, furthermore, be reduced without frequency translation as involved in double or vestigial sideband trans- mission, by certain methods of shaping or transmission of pulses, as discussed in the following, and by certain methods of compensation at the receiving end or at points of pulse regeneration not considered here. The nature of the pulse distortion resulting from a low-frequency cut- off is illustrated in Fig. 32. If the phase characteristic is assumed linear, the amplitude characteristic may be regarded as made up of two com- ponents, in accordance with the following identity: A(co) = Ao(a:) + [A(co) - A^], (9.01) where ^4o(a;) is the amphtude characteristic without a low-frequency cut-off and [A((j}) — Ao(co)] a supplementary characteristic of negative amplitude, as indicated in Fig. 32. The impulse characteristic may correspondingly be written P(t) = Po(0 + [^(0 - Po(t)]. (9.02) If the cut-off is confined to rather low frequencies, the impulse charac- teristic AP{t) = P(t) — Po{t) will extend over time intervals substan- tially longer than the duration of Po(t) or the interval at which pulses are transmitted. The total area under the resultant pulse is always zero. When a sufficiently long sequence of pulses of one polarity is trans- mitted, the cumulative effect of the pulse overlaps resulting from the modification P(t) — Po(t) in the impulse characteristic will be a dis- placement of the received pulse train, as illustrated in Fig. 33 for various intervals between the pulses. This apparent displacement of the zero line, often referred to as "zero wander," will reduce the margin for dis- 774 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 tinction between the presence and absence of pulses in a random pulse train. In the particular case when pulses are transmitted at the minimum interval ri = l/2/i possible without intersymbol interference in the ab- sence of a low-frequency cut-off, the pulse train will ultimately vanish when an infinite seciuence of pulses of one polarity is transmitted, as illustrated for the last case in Fig. 33. The number of pulses of one polarity, or nearly all of the same po- larity, which can be transmitted before the limiting condition illustrated in Fig. 33 is approached depends on the extent of the low-frequency cut-off. If the low-frequency cut-off is inappreciable, this number may be sufficiently great so that the probability of encountering such a sequence in a random pulse train and resultant errors in reception may be so small that it can be disregarded. The reciuirement of the low- frequency cut-off which is necessary to this end is evaluated below for pulses transmitted at intervals n = l/2/i . TRANSMISSION FREQUENCY CHARACTERISTIC WITHOUT LOW-FREQUENCY CUTOFF IMPULSE CHARACTERISTIC WITHOUT LOW-FREQUENCY CUTOFF P-(^ LOW-FREQUENCY CUTOFF COMPONENT Fig. 32 — Separation of low-frequency cut-off componente A-Ao and P-Po in transmission frequency and impulse characteristics. TIIEOKETICAL FUNDAMENTALS OF I'ULSE TUANSMISSIOX //5 PULSE TRAIN ENVELOPE ZERO LINE WITH LOW-FREQUENCY CUTOFF "ZERO LINE WITHOUT LOW-FREQUENCY CUTOFF ^/__ ^X^ Y^ X A-. I J_r 1 , PULSE TRAIN ENVELOPE AND ZERO LINE n 2 I / WITH LOW-FREQUENCY CUTOFF ?~v 7~N f"'^ /~N 7~s /•"% / \ / \ -' \ / \ / \ / /^ > / / /^ / / \ / \ / \ / \ / ^ ' ZERO LINE WITHOUT / / \ / \ / \ / \ / \ y LOW-FREQUENCY CUTOFF ^1 ^1 S^<. s^w^ \ / r Fig. 33 — Effect of low-frequency cut-off on recurrent pulses as pulse interval is decreased. If it is assumed that positive and negative impulses are applied at random to the transmission systems at intervals n , the rms intersymbol interference resulting from a low-frequency cut-off can be evaluated by essentially the same method as employed in Section 8 for fine structure imperfections in the transmission characteristic, provided wo is much smaller than oji . On this basis, rms intersymbol interference in relation to the peak amplitude Po(0) of the pulses in the absence of a low-fre- quency cut-off becomes: For a transmission characteristic with linear phase shift Po(0) = - f AM Jco. (9.05) TT Jo 776 THE BELL SYSTEM TECHNICAL JOURNAL, ^L\Y 1954 For the particular case of sharp cut-offs at coo and coi Ao(a}) = 1 0 < w < coi , A(aj) — Ao(cji) =—1 0 < a PHASE ^^\ / -> 1- CHARACTERISTIC/^ | / I '^ ./ 1 / n. >*^^ / 5 jy^ DELAY ^ y DISTORTION — ^ \ — i_ 1 — 1 0 a;' OJ, FREQUENCY, OJ Fig. 37 — Constant amplitude characteristic with hand-edge phase distortion. If there is no phase distortion, i.e., /3 = 0, between co = 0 and co', equa- tion (10.03) becomes X = a'l COS /3 dbi 112 I r ^Wmax- (10.04) As an example, consider a parabohc deviation from a Hnear phase characteristic between w' and comax , in which case delay distortion would vary linearily in this band, as indicated in Fig. 37 for a constant ampli- tude characteristic for which Wmax = wi . In this case /3 = /3i COi — cu (10.05) where /3i is the maximum phase deviation, obtained for w = coi . Equation (10.04) in the above case becomes: ' \2n 9 (•" 1 2(coi - 1 — cos /3i 2(aji — a; ) CO 1 — CO Jo d!co, cos u du (10.06) COi 1 _ 1 ( — ) (K + X) 2\2i3i/ where R -\- iX = erf {^-^'"^e"'^) in which erf is the error function. For a constant amplitude transmission characteristic as assumed above, (r/coiTi) = 1, so that (10.02) becomes C7 = X, which may also 782 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 be written: U = I ^^ ~ " Y "-TOO, and (10.07) COi F(/3i) = 2 1/2 1/2 ^-H^J ''^ + -^' 1/2 (10.08) For ^'ariolls values of the maximum phase deviation /3i in radians the function F becomes: /3i 0 0.25 1 4 00 F 0 0.14 0.43 1.24 1.42 If, for example, phase distortion were confined to 10 per cent of the transmission band, then (wi — co')/wi = 0.1. For a maximum phase deviation of 1 radian at the edge of the transmission band, F = 0.43 and U = 0.135. For a maximum phase distortion of 4 radians, F = 1.24 and U = 0.39. Since peak intersymbol interference may exceed the above rms values by a factor of about 3, and the maximum tolerable peak intersymbol interference in a system employing two pulse amplitudes would be less than 1, it is evident that band-edge phase deviations must be held at rather small values, less than about 3 radians, in the upper 10 per cent of the transmission band. The above severe tolerances on band-edge phase distortion can be overcome by employing a transmission frequency characteristic of the type shown in Fig. 38 and previously discussed in Section 5. If the phase characteristic is linear between co = 0 and wi , and phase distortion between coi and 2coi varies as ^ = 01 2coi — wi = /3i 1-^ (10.09) equation (10.04) can be written 1 + cos 2coi 1 - cos ^1 1 - 0)1 do), (10.10) 1 — sin -z u) (1 — cos 3iU~) du. Pulses may also in this case be transmitted at intervals n = tt/wi without intersymbol interference in the absence of phase distortion, so THEORETICAL FI'XDAMEXTALS OF PULSE TRANSMISSION 783 that (10.02) becomes U = \ or ,•1 r - sill :^ u } (I — cos iSiH.') (hi 1/2 (10.11) The maximum delay distortion at the edge of the transmission liand, i.e., CO = 2coi , is t/max = 2/3i/coi . The product of this delay distortion with the maximum frequency /max = 2/i is dmax/max = 2j(3i/7r. For various \-alues of maximum phase distortion /3i and the corresponding product (/max/max , tlic followiug \-alues of rms intersj^mbol interference are ob- tained b}' numerical integration of (10.11). (This integral can be ex- pressed in terms of a number of Fresnel integrals, but numerical integra- tion is simpler and sufficiently- accurate for the present purpose.) /3i IT 27r 47r «. f^max /max 2 4 8 00 u 0.070 0.120 0.185 0.330 The particular case c/max/max = 8 is similar to that shown in Fig. 36, except that this figure applies to a Gaussian characteristic, for which the amplitude at oj = coi has been taken as 0.32 rather than 0.5 in the case considered here. For this reason rms intersymbol interference from phase distortion would be greater in the present case. FREQUENCY, CJ Fig. 38 — Typical transmission frecjuency characteristic with phase equaliza- tion over 50 per cent of transmission band. 784 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 FREQUENCY *- fm Fig. 39 — Sub-channel with nearly linear delay distortion. Peak intersymbol interference may exceed the above rms values by a factor of about 3. In a system employing two pulse amplitude (1 and — 1), the maximum tolerable intersymbol interference is 1. This value would thus be attained in the above case for dmax/max = '^ • Hence, in a system employing two pulse amplitudes, and in the absence of noise and intersymbol interference from other sources, there would be no limitation on phase chstortion for co > wi , provided the phase charac- teristic is linear between co = 0 and wi . 11. BAND-PASS CHARACTERISTICS WITH LINEAR DELAY DISTORTION In Fig. 39 is shown a transmission frequency characteristic together with an assumed delay distortion d\}//doo. When a portion of the trans- mission band is employed for pulse transmission, as for example in pulse signalling, data or telegraph transmission over portion of a voice channel, there may be an appreciable component of substantially linear delay distortion, as indicated in the above figure. The departure from a linear variation can usually be approximated by a cosine variation in delay, and the system can then be regarded as made up of two components in tandem, one with linear the other with cosine variation in delay. The effect of the latter can be evaluated by the methods outlined in Section 6, and the effect of a linear variation by methods established in this section. In Fig. 40 is shown a symmetrical amplitude characteristic with linear delay distortion over the transmission band. Phase distortion with respect to the midband frequency is in this case ^(m) = ^u and ^(-w) = ^u, (11.01) THEORETICAL FUNDAMENTALS OF PT'LSE TR ANSAIISRION and delay distortion d^(u)/du = 2^u,d used later in this section do not have the same meaning as in earlier sections.) With (11.01) in (2.10) and (2.11), the in-phase and quadrature components in (2.09) become 9/^ r°° /?_ + -R+ = — / Q:(w.) cos ut cos /3?(^, and T Jo 9^ r" Q_ + Q4. = — / (i(u) cos ut sin i3n^. TT Jo (11.03) The in-phase and quadrature components can accordingly be identi- fied with the real and the negati^'e imaginary component of the integral 9^ r"^ J=— a{u) cos ute''^"\hi. (11.04) X Jo The solution of this integral is rather simple for the particular case of a Gaussian transmission characteristic a (m) = e' in which case J = - I e IT Jo -{a + i^)ii^ COS ut du, (11.05) (11.06) -t^H{a+i0) [T{a+mY'-' AMPLITUDE CHARACTERISTIC /PHASE CHARACTERISTIC, /}U2 DELAY DISTORTION, 2/3U FREQUENCY -2/3u Fig. 40 — Symmetrical band-pass amplitude characteristic with linear delay distortion. 786 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 The real and nesati\e imaginary components of this expression are /?_ + R+ = 25 (^\" e-"'" cos (6 - U'), and Q_ - Q+ = 25 p j e-"'" sin (6 - hf), where tan 29 = 0/a = 6/a The impulse characteristic obtained with (11.07) in (2.09) becomes p(t) = 25 (-\" e-"'" [cos (a^rt - 4^r) cos (6 - br) ^^/ (11.08) + sin (cort — \pr) sin (9 — bf)]. From (11.08) it is seen that the envelope is P(t) = 25 (-\ e^"'". (11.09) The peak of the envelope obtained with ^ = 0 is smaller than Avithout delay distortion (^ = 0) by the factor ^ (11.10) 1 + i^/ayv' The constant a is smaller than without delay distortion by the factor 77 . If ^0 designates the time required for the instantaneous amplitude of a pulse to decay from its peak to a given value ^nthout delay distortion, the time ^1 to reach the same amplitude with delay distortion is ^1 = t,/rf = toil + (^/a)T\ (11.11) If Wniax indicates the frec^uency at the 40 db down point on the trans- mission frequency characteristic, acomax" = 4.6. The corresponding delay distortion is f/max = 2a),„axiS. Thus l3/a = .68 c?,„ax/max so that (11.11) becomes : k = ^o[l + 0.40 (f/„.ax/max)T''. (11-12) The effect of a linear delay distortion across the transmission band is thus to disperse or broaden the envelope of the received pulses, as illus- THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION ■87 tratcil in Fig. 41. For a specified pulse overlap or iiitersyinl)()l inlerfereiu'e the pulse spacing must accordingly l)e increased by the factor ti/d, , so that for a given transmission performance the transmission capacity is retlucetl by the factor (o/d . About the same effect would be expected for other pulse shapes or amplitude characteristics resembling the (laussian shape assumed in the above deriA-ation. Comparison of (11.08) with (2.13) shows that the function (^(/) with respect to the midband frequency is ^(0 = e - hr. (11.13) If the reference or carrier fre(iuency is displaced from the midband by AMPLITUDE CHARACTERISTIC FREQUENCY — ^ DELAY DISTORTION (a) TRANSMISSION FREQUENCY CHARACTERISTIC WITHOUT DELAY DISTORTION-,/ ■\WITH DELAY DISTORTION (b) IMPULSE CHARACTERISTICS WITHOUT AND WITH DELAY DISTORTION t, =to[l + 0.46(d^^^f^^,f]2 ^MAX = FREQUENCY FROM MIDBAND AT WHICH AMPLITUDE OF TRANSMISSION FREQUENCY CHARACTERISTIC IS REDUCED 40 DECIBELS ^MAX = DELAY DISTORTION AT f^^X IN SECONDS Fig. 41 — Lengthening of impulse envelope l)y linear delay distortion for Gaussian transmission characteristic. 788 THE bp:ll system technical journal, may 1954 o}y , the in-phase and quadrative components are in accordance with (2.18) RJ + R+' = cos {d - bt- + coyt - \Py) Pit), and (11-14) QJ - Q+ = sin {6 - br + coyt - i/^J P(t), where ^py = /5w/ and P(t) is given by (11.09). REFERENCES 1. Y. W. Lee, Synthesis of Electric Networks by Means of the Fourier Trans- forms of Laguere's Functions, J. Math. & Phys., June, 1932. 2. H. W. Bode, Network Analytiis and Negative Feedback Amplifier Design, D. Van Nostrand Book Company, 1945. 3. H. Nyquist, Certain Topics in Telegraph Transmission Theory, A.I.E.E. Trans., April, 1928. 4. H. Nyquist and K. W. Pfleger, Effect of Quadrature Component in Single Sideband Transmission, B.S.T.J., Jan., 1940. 5. H. A. Wheeler, The Interpretation of Amplitude and Phase Distortion in Terms of Paired Echoes, Proc. I.R.E., June, 1939. 6. C. R. Burrows, Discussion of 5 above, Proc. I.R.E., June, 1939. 7. G. N. Watson, Theory of Bessel Functions, Cambridge University Press, 1944. 8. E. Jahnke and F. Emde, Funktionentafeln, 1928, p. 149. 9. E. A. Guillemin, Communication Networks, John Wiley & Sons, Inc., 1935. 10. W. R. Bennett, Time Division Multiplex Systems, B.S.T.J., April, 1941. 11. S. Goldman, Frequency Analysis, Modulation and Noise, McGraw-Hill Book Co., 1948. 12. C. E. Shannon, A Mathematical Theory of Communication, B.S.T.J., October, 1948. 13. B. M. Oliver, J. R. Pierce and C. E. Shannon, The Philosophy of PCM, Proc. I.R.E., Nov., 1948. Bell System Technical Papers Not Published in this Journal AiKENS, A. J./ and C. S. Tiiaeler.^ Noise and Crosstalk Control on Nl Carrier Systems, Elec. Eng., 72, pp. 1075-1080, Dec, 1953. Alley, R. E., Jr.,i and F. J. Schnettler.^ Effect of Cross-Section Area and Compression Upon the Relaxation in Permeability for Toroidal Samples of Ferrites, Letter to the Edi- tor, J. Appl. Phys., 24, pp. 1524-1525, Dec, 1953. Allls, W. P.,1 and D. J. Rose.^ The Transition From Free to Ambipolar Diffusion, Phj^s. Rev., 93, p. 84 93, Jan. 1, 1954. Anderson, O. L.,i and D. A. Stuart.* Statistical Theories as AppHed to the Glassy State, Ind. Eng. Chem., 46, pp. 154-160, Jan., 1954. Arnold, S. M., see S. E. Koonce. Barstow, J. M.,^ and H. N. Chrlstopher.^ The Measurement of Random Monochrome Video Interference, A.I.E.E., Commun. and Electronics, pp. 735-741, Jan., 1954. ^ Bell Telephone Laboratories, Inc. ^ American Telephone and Telegraph Company. * Cornell University. 790 THE BELL SYSTEM TECHNICAL JOUKNAL, MAY 1954 Bashkow, T. R.i Stability Analysis of a Basic Transistor Switching Circuit, Proc. Xational Electronics Conference, 9, p. 748, Feb. 15, 1954. Blecher, F. H.^ Automatic Gain Control of Junction Transistor Amplifiers, Proc. Na- tional Electronics Conference, 9, p. 731, Feb. 15, 1954. Bogert, B. P.^ Erratum: On the Band Width of Vowel-Formants, [published in J* Acous. Soc. Am., 25, p. 791, (1953)J, J. Acous. Soc. Am., 25, p . 1203' Nov., 1953. The statement in the abstract which reads "The mean values for bars 1, 2, and 3 were 130, 100, and 185 cps, respectively," should be corrected to read "The median values for bars 1, 2, and 3, were 130, 150, and 185 cps, respectively." Brown, W. L.,^ R. C. Fletcher,' and K. A. Wright.^ Annealing of Bombardment Damage in Germanium — Experimental, Phys. Rev., 92, pp. 591-596, Nov. 1, 1953. Brown, W. L., see R. C. Fletcher. Burton, J. A.,^ G. W. Hull,' F. J. Morin,^ and J. C. Severiens.^ Effect of Nickel and Copper Impurities on the Recombination of Holes and Electrons in Germanium, J. Ph3^s. Chem., 57, pp. 853-859, Nov., 1953. Burton, J. A.,' R. C. Prim,' and W. P. Slighter.^ Distribution of Solute in Crystals Grown from the Melt — Theoreti- cal, J. Chem. Phys., 21, pp. 1987-1991, Nov., 1953. Burton, J. A.,' E. D. Kolb,i W. P. Slighter,^ and J. D. Struthers.' Distribution of Solute in Crystals Grown from the Melt — Experi- mental, J. Chem. Phys., 21, pp. 1991-1996, Nov., 1953. Campbell, M. E., see C. L. Luke. ' Bell Telephone Laboratories, Inc. * Massachusetts Institute of Technology. TIOCIIN'H AI, PAPKRS 791 Ca.sk, K. J../ and Iden Kerney.^ Program Transmission Over Type-N Carrier Telephone, A.I.E.E.. C'omimiii. aiul Klec'tronics, pp. 791-71)"), Jan., 1954. CiiRi'STOPiiER, H. N., see J. M. Barstow. Clark, M. A.^ An Acoustic Lens as a Directional Microphone, J. Acous. Soc. Am., 25, pp. 1152-1153, Nov., 1953. CoREXZwiT, E., see S. Geller. Coy, J. A} Heat Dissipation from Toll Transmission Equipment, A.I.E.E., Com- mun. and Electronics, pp. 762-768, Jan., 1954. Dunn, H. K.i Remarks on a Paper Entitled "Multiple Helmholtz Resonators," Letter to the Ediit)r, J. Acous. 8oc. Am., 26, p. 103, Jan., 1954. Fletcher, R. C.,^ and W. L. Brown. ^ Annealing of Bombardment Damage in a Diamond-Type Lattice — Theoretical, Phys. Kev., 92, ])p. 585-590, Nov. 1, 1954. Fletcher, R. C, see W. L. Brown. Fracassi, R. D.,^ and H. Kahl.^ Type ON Carrier Telephone, A.I.E.E., Commuii and Electronics, pp. 713-721, Jan., 1954. Fuller, C. S., see J. R. Severiens, Geller, S.,^ and p]. Corenzwit.^ Hafnium Oxide, HfO: (Monoclinic), Anal. Chem., 25, ]). 1774, Nov., 1953. l^oll Tcloiilionp T.al)or;ttoii(>s, Inc. 792 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 GiBBS, W. B.3 Investment Cast Artificial Larynx Eases Fabrication Difficulties, Precision Metal Molding, pp. 34, 35, 75, Dec, 1953. Green, F. 0.» Cost Control — Bonus from Centralized Maintenance, Plant En- gineering, pp. 88-91, Jan., 1954. Hagstrum, H. D.^ Instrumentation and Experimental Procedure for Studies of Electron Ejection by Ions and Ionization by Electron Impact, Rev. Sci. Instr., 24, pp. 1122-1142, Dec, 1953. Hanson, A. N.^ Automatic Testing of Wired Relay Circuits, A.I.E.E., Commun. and Electronics, pp. 805-857, Jan., 1954. Hull, G. W., see J. A. Burton. Karl, H., see R. D. Fracassi. Kerney, Iden, see R. L. Case. Koch, W. E.^ Use of the Sound Spectrograph for Appraising the Relative Quality of Musical Instruments, Letter to the Editor, J. Acous. Soc Am., 26, p. 105, Jan., 1954. KoLB, E. G., see J. A. Burton. Koonce, S. E.^ and S. M. Arnold. ^ Metal Whiskers, Letter to the Editor, J. Appl. Phys., 25, pp. 134- 135, Jan., 1954. 1 Bell Telephone Laboratories, Inc. ' Western Electric Company, Inc. TECHNICAL PAPERS 793 Kretzmer, E. R.^ An Amplitude-Stabilized Transistor Oscillator, Proc. National Elec- tronics Conference, p. 756, Feb. 15, 1954. Li:wi8, II. W.' Search for the Hall Efifect in a Super-Conductor — Experiment, Phys. Rev., 92, pp. 1149-1151, Dec, 1953. LiNVILLE, J. G.' A New RC Filter Employing Active Elements, Proc. National Elec- tronics Conference, 9, p. 342, Feb. 15, 1954. Luke, C. L.,^ and M. E. Campbell.^ Determination of Impurities in Germanium and Silicon, Anal. Chem., 25, pp. 1588-1593, Nov., 1953. AIahoxey, J. J., see E. H. Perkins. May, J. E.i Characteristics of Ultrasonic Delay Lines Using Quartz and Barium Titanite Ceramic Transducer, Proc. National Electronics Conference, 9, p. 2G4, Feb. 15, 1954. MoRix, F. J.i Lattice Scattering Mobility in Germanium, Phys. Rev., 93, pp. 62-63, Jan. 1, 1954. MoRiN, F. J., see J. A. Burton. Murphy, E. J.^ Surface Migration of Water Molecules in Ice, J. Chem. Phys., 21, pp. 1831-1835, Oct., 1953. Pexnell, E. S.' A Temperature Controlled Ultrasonic Solid Delay Line, Proc. Na- tional Electronics Conference, 9, p. 255, Feb. 15, 1954. * Bell Telephone Laboratories Inc. 794 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 Perkins, E. H.,^ and J. J. Mahoney.^ Type-N Carrier Telephone Deviation Regulator, A.I.E.E., Commun. and Electronics, pp. 757-762, Jan., 1954. Peterson, G. E.,^ and Gordon Raisbeck.^ The Measurement of Noise with the Sound Spectrograph, J. Acous. Soc. Ain., 25, p. 1157, Nov., 1953. Prim, R. C, see J. A. Burton. Prince, M. B} Drift Mobilities in Semi-Conductors. I — Germanium, Phys. Rev., 92, pp. 681-687, Nov. 1, 1953. Raisbeck, Gordon, see G. E. Peterson. Rea, W. W.3 Oscilloscope Reduces Cost of Jig Grinding Operations, INIachiiiery, pp. 201-203, Dec, 1953. Remeika, J. P. Method for Growing Barium Titanate Single Crystals. J. Am. Chem. Soc, 76, pp. 940-941, Feb. 5, 1954. Rose, D. J., see W. P. Allis. Schlaack, N. F.i Development of the LD Radio System, Trans. I.R.E., Professional Group on Communication Systems, pp. 29-38, Jan., 1954. Schnettler, F. J., see R. E. Alley, Jr. Severiens, J. C., see J. A. Burton. 1 Bell Telephone Laboratories, Inc. ^ Western Electric Company, Inc. ^ University of Michigan. TKCIIXICAL PAPERS 795 Severiens, J. R./ and C. S. Fuller.^ Mobility of Impurity Ions in Germanium and Silicon, Tjcltcr to llie Editor, Pliy«. Wvv., 92, pp. 1322, Dec, 1953. Shockley, W.^ Transistor Physics, Am. Scientist, 42, pp. 41-72, Jan., 1954. Shockley, W.^ Some Predicted Effects of Temperature Gradient on Diffusion in Crystals, Letter to the Editor, Pliys. Rev., 93, pp. 345-346, Jan. 15, 1954. Slighter, E. D., see J. A. Burton. SXOREK, F.^ Cut Tool Costs with Precision Casting, Iron Age, pp. 136-139, Feb. 11, 1954. Stiles, K. P.^ Overseas Radio Telephone Services of A. T. and T. Co., Trans. I.R.E., Professional Group Communications Systems, pp. 39-44, Jan., 1954. Struthers, J. D., see J. A. Burton, and C. D. Thurmond. Stuart, D. A., see O. L. Anderson. Thaeler, C. S., see A. J. Aikens. Thurmond, C. D.^ Equilibrium Thermochemistry of Solid and Liquid Alloys of Ger- manium and Silicon — The Solubility of Ge and Si in Elements of Group III, IV and V, J. Phys. Chem., 57, pp. 827-830, Nov., 1953. Thurmond, C. D.,i and J. D. Struthers.^ EquiHbrium Thermochemistry of Solid and Liquid Alloys of Ger- 1 Bell Telephone Laboratories, Inc. ^ American Telephone and Telegraph Company. ^ Western Electric Company, Inc. 796 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954: manium and of Silicon — The Retrograde Solid Solubilities of Sb in Ge, Cu in Ge, and Cu in Si, J. Phys. Chem., 57, pp. 831-834, Nov., 1953. Walker, L. R} Dispersion Formula for Plasma Waves, Letter to the Editor, J, Appl. Phys., 25, pp. 131-132, Jan., 1954. Wolff, P. A.i Theory of Plasma Waves in Metals, Phys. Rev., 92, pp. 18-23, Oct. 1, 1953. Wright, K. A., see W. L. Brown. ^ Bell Telephone Laboratories, Inc. Contributors to this Issue M. 'SI. Atalla, B.S., Cairo University, 1945; M.S., Purdue University, 1947; Ph.D., Purdue University, 1949; Studies at Purdue undertaken as the result of a scholarship from Cairo University for four years of gradu- ate work. Bell Telephone Laboratories, 1950-. For the past three years he has been a member of the Switching Apparatus Development De- partment, in which he is super\'ising a group doing fundamental research work on contact physics and engineering. Current projects include fundamental studies of gas discharge phenomena between contacts, their mechanisms, and their physical effects on contact behavior; also fundamental studies of contact opens and resistance. In 1950, an article bj^ him was awarded first prize in the junior member category of the A.S.INI.E. He is a member of Sigma Xi, Sigma Pi Sigma, Pi Tau Sigma, the American Physical Society, and an associate member of the A.S.M.E. \ James M. Early, B.S., cum laude. New York State College of Fores- try, 1943; M.S. and Ph.D. Ohio State University, 1948 and 1951. Bell Telephone Laboratories 1951-. After teaching Electrical Engineering at Ohio State L'^niversity for five years while studying for his Master's and Ph.D. degrees. Dr. Early joined an electronic apparatus development group, participating in the development of the junction transistor. At present he is doing general theoretical work as well as development work on high frequency junction transistors. Member of the I.R.E. and Eta Kappa Nu. Associate of Sigma Xi. Walter T. Eppler, B.S., in E.E., Tufts College, 1927; Western Electric Company, purchase of special machinery and development of coil manufacture, 1927-1932; Crystal Golf Ball Company, Superin- tendent of Manufacture, 1933-1936; New Haven Clock Company, 1936-1937; Western Electric Company, 1937-. In 1941, he engineered a conveyorized dispatch control system for key assembly at the Kearny plant. For the past six years, he has been engaged in cable sheath en- gineering on Alpeth and Stalpeth cable. Member of New Jersey Society of Professional Engineers. 797 798 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 Stewart E. Miller, University of Wisconsin, 1936-39; B.S. and M.S., Massachusetts Institute of Technology, 1941. Bell Telephone Laboratories, 1941-. Except for World War II work on airborne radar systems, Mr. Miller's first eight years at the Laboratories were con- cerned with studies on coaxial carrier transmissions systems. A member of the radio research group, he is currently in charge of research on guided systems and associated millimeter and microwave techniques at Holmdel. Member of the I.R.E., Eta Kappa Nu, Tau Beta Pi, and Sigma Xi. Harry Suhl, B.Sc, University of Wales, 1943; Ph.D., Oriel College, University of Oxford, 1948. Admiralty Signal Establishment, 1943-46; Bell Telephone Laboratories, 1948-. Dr. Suhl conducted research on the properties of germanium until 1950 when he became concerned with electron dynamics and solid state physics research. His current work is in the applied physics of solids. Member of the American Institute of Physics and Fellow of the American Physical Society. Erling D. Sunde, E.E., Technische Hochschule, Darmstadt, Ger- many, 1926. Brooklyn Edison Company, 1927; American Telephone and Telegraph Company, 1927-1934; Bell Telephone Laboratories, 1934-. Mr. Sunde's work has been centered on theoretical and experimental studies of inductive interference from railway and power sj^stems, light- ning protection of the telephone plant, and fundamental transmission studies in connection with the use of pulse modulation systems. Author of Earth Conduction Effects in Transmission Systems, a Bell Laboratories Series Book. Member of the A.I.E.E., the American Mathematical So- ciety, and the American Association for the Advancement of Science. Laurence R. Walker, B.Sc. and Ph.D., McGill I^niversity, 1935 and 1939; University of California, 1939-41. Radiation Laboratory, Mass- achusetts Institute of Technology, 1941-1945; Bell Telephone Labora- tories, 1945-. Dr. Walker has been primarily'- engaged in research on microwave oscillators and amplifiers. At present he is a member of the physical research group concerned with the applied phj^sics of solids. Fellow of the American Physical Society. rHE BELL SYS n M / >EVOTED TO THE SCIENTIFIC mcai lournai 1^^^ AN ND ENGINEERING .SPECTS OF ELECTRICAL COMMUNICATION OLUME XXXIII JULY 19 54 NUMBER 4 KANSAS CIJY, MO. Negative Resistance Arising from Tr^4illl-.^Me imS6ni^^¥uctor Diodes w, shockley 799 JtiL -7 i^;>4 Transistor and Junction Diodes in Telepnohe I*bwerTP'Iants F. H. CHASE, B. H. HAMILTON AND D, H. SMITH 827 Wire Straightening and Molding for Wire Spring Relays A. J. BRUNNER, ^. E. COSSON AND R. W. STRICKLAND 859 Some Fundamental Problems in Percussive Welding e. e. stjmner 885 Automatic Contact Welding in Wire Spring Relay Manufacture A. L. quinlan 897 Electronic Relay Tester T. E. DAVIS AND A. L. BLAHA 925 Topics in Guided Wave Propagation Through Gyromagnetic Media Part II — Transverse Magnetization and Non-Reciprocal Helix H. SUHL AND L. R. WALKER 939 Theoretical Fundamentals of Pulse Transmission — II E. D. SUNDE 987 Bell System Technical Papers Not Published in this Journal 1011 I Recent Bell System Monographs 1017 Contributors to this Issue 1019 COPTRIGHT 1954 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD S. BRACKEN, Chairman of the Board, Western Electric Company F. R. K A P P E L, President, Western Electric Company M. J. KELLY, President, Bell Telephone Laboratories E. J. M c N E E L Y, Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE W. H. D O H E R T Y, Chairman F. R. LACK A. J. B U S C H W. H. N U N N G.D.EDWARDS H. I. R O M N E S J. B. FISK H. V. SCHMIDT E. I. GREEN G. N. THAYER R. K. H O N A M A N J. R. W I L S O N EDITORIAL STAFF J. D. TE B O, Editor M. E. STRIEBY, Managing Editor R. L. SHEPHERD, Production Editor THE BELL SYSTEM TECHNICAL JOURNAL is published six times a year by the Americ£in Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. Cleo F. Craig, President; S. Whitney Landon, Secretary; John J. Scanlon, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed in U. S. A. THE BELL SYSTEM TECHNICAL JOURNAL voLUMEXXxiii JULY 1954 number4 Copyright, 1964, American Telephone and Telegraph Company Negative Resistance Arising from Transit Time in Semiconductor Diodes By W. SHOCKLEY (Manuscript received January 22, 1954) The structural simplicity of two-terminal compared to three-terminal devices indicates the potential importance of two terminal devices employing semicondvctors and having negative resistance at frequencies properly re- lated to the transit tiryie of carriers through them. Such negative resistances may he combined with im symmetrically transmitting components, such as gyrators or Hall effect plates, to form dissected amplifiers that may he made to simulate conventional three-terminal amplifiers and operate at high fre- quencies. The characteristics of several structures are analyzed on the hasis of theory and it is found that negative resistances are possible for properly designed structures. 1. NEGATIVE RESISTANCE AND DISSECTED AMPLIFIERS Because the drift velocities of current carriers in semiconductors are smaller than the velocities attainable in vacuum tubes, transistor struc- tures must be smaller to achieve comparable frequencies. In principle it is possible, of course, to make compositional structures (i.e., distribu- tions of donors and acceptors) in semiconductor crystals on a scale much smaller than is possible for vacuum tubes. At present, however, the available techniques are limited and it may require many years before the ultimate potentialities are approached. 799 800 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 It is instructive, however, to speculate on some of these ultimate po- tentialities. For example a grain boundary formed of edge type disloca- tions is in a sense an analogue of a grid. Possibly it can be made into a grid by acting as a locus for an atmosphere of donors or acceptors. Evidently such a grid ^^'ill approach the smallest spacing that can be achieved with any kno^vn form of matter. If the spacings perpendicular to the grid are made comparable to a mean free path of the carriers used, the device ^^ill operate like a vacuum tube with carrier velocities con- trolled by inertia rather than by mobility. It is not easy to conceive of a structure having the potentialitj^ of operating at higher frequencies. It is evident that the difficulty of making small structures increases with the number of electrodes. For example, it is now possible to make diodes which give usable rectification at frequencies above 10^° cps. In these the "working volume" is a very thin layer under the metal point. The thickness of this layer is controlled by surface treatments and the applied voltages. The diameter of the point, which is the minimum di- mension mechanically controlled, is much larger than this thickness of the layer. In order to make a transistor of comparable frequency, it would be necessary to make structural elements having dimensions comparable to the thickness of the layer and this would be a much more exacting task than making the diode. These considerations point out the importance of giving serious con- sideration to two-terminal structures as amplifying elements. It is pos- sible, in principle at least, to have structures which are much smaller in one dimension than the other two and which exhibit negative re- sistance and thus give ac power at frequencies comparable to the reciprocal of the transit time across the small dimension. The attractiveness of such negative resistance diodes for amplification is enhanced by the possibility of using them in dissected amplifiers ' in combination with nonreciprocal elements such as gyrators or Hall effect plates. Combinations of negative resistance elements and nonreciprocal elements can lead to structures having gain and unsymmetrical trans- mission that simulate conventional amphfiers. The adjective dissected has been suggested for them since elements giving power gain are physically separated from those giving one-way transmission. In this article we shall not consider the possible forms of dissected amplifiers, of which there are a wide variety. Instead we shall give an introductory treatment of some forms of negative resistance that may arise from transit time effects. In some cases the most instructive way of treating the structure is by way of the "impulsive impedance" and we devote most of the next section to considering this method. NEGATIVi: ItESISTANCE IN SEMICONDUCTOR DIODES 801 2. THE IMPULSIVE IMPEDANCE AND NEGATIVE RESISTANCE The impulsive impedance D{t) for a two terminal (knice is defined in terms of its transient response to an impulse of current. Thus if the current through the de\ice is J{t) = J + jit), where J is the dc current and m = 0 except very near t = 0 and j j{t)dt = bQ, then the voltage is F(0 = F + v{t), where V is the dc voltage and v{t) = 8Q D{t). In other words, if in addition to the dc biasing current, a charge 8Q is instantaneously forced through the circuit at time t = 0, the added voltage is D(t) per unit charge. These equations also serve to introduce the notation used in this article: Notation. In general, cjuantities that are functions of time or position will haxe the functional dependence explicitly indicated. In Sections 4 and 5, however, the symbol 5 will be used to distinguish the transient parts bE and 6p from the dc parts of the electric field and charge density. Capital V{t) and J{t) stand for total voltages and current. Without functional dependence upon {t) they are the dc parts. Similarly v{t) and j{t) are the ac or transient parts. A sinusoidal disturbance is represented by v{t) = V exp iwt, j(t) = j exp iut. where v and j are not functions of time. Where it is necessary to distin- guish the displacement current at a particular location from the con- duction current, as in the next section, we shall write J(D, S2,t), 802 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 meaning the displacement current across space charge region S^ as a function of time. In this section we shall treat J and j as circuit currents. In subsequent sections, we shall be concerned with current densities and shall use the same symbols. The complex impedance of the device is evidently Z(o:) = v/j, where v and j are the coefficients in the sinusoidal case. In terms of the system of notation introduced above, Z(co) may also be expressed in terms of D(t) by expressing j exp icjot in terms of incre- ments of charge dQ = ye'"' dt, and summing over all increments up to time t. This leads to Z{co) = I D{t) exp i-wt) dt. Jo A negative resistance will occur if 0 > / Dit) cos wtdt = (-l/oj) / D\t) sin cot dt, Jo Jo the latter form coming from integration by parts for the case of Z)(oo) = 0, the only situation treated in this article. The use of D(t) in analysing the potential merits of diode structures from the point of view of negative resistance is illustrated in Fig. 2.1. Here three cases of D(t) together with certain cosine waves are shown. It is seen for case (a) that a negative real part of Z will be obtained. For case (b), the real part of Z is zero for the frequency shown; this represents a limit ; for other frequencies, a positive real part will be ob- ^a) \ (b) V (c) Jl D(t) \ \ 0 \ \ / 0 \ \ V- 0 \ \ / c 1 1 / c / ( \ / / / Fig. 2.1 — Some hypothetical D{t) characteristics. NEGATIVE RESISTANCE IX SEMICONDUCTOR DIODES 803 iMined. Case (c) represents an exponential fall such as might occur for a capacitor and resistor in parallel. We shall discuss this example below. Tlie conclusions regarding (a) and (b) may be somewhat more easily seen from the corresponding —D'(t) plots shown in Fig. 2.2. From part (a) it can be seen that the negative maximum in the sine wave at the end of the rectangular D(t) plot is particularly favorable. From part (b) it is seen that no choice of co will result in more negative area of sine wa^'e than positive. For (c) it is evident that each positive half cycle of the sine wave gives a larger contribution than the following negative half cycle and hence that a positive resistance will be obtained. For case (c), it is instructive to obtain the value of Z analytically by using D(t) = Cr' exp (-t/RC). This leads correctly to Z(a;) = (R -' + io,C)~\ For small values of uRC, Z reduces to R; furthermore, for this case, the decay of D(t) occurs while cos cot = I. Under these conditions Z(c.) = f Jo D{t) dt. This result is useful for estimating the effect of quickly decajdng contri- l)utions to Dit). These evidently contribute a positive resistance to Z equal to the area under the D(t) curve. From these considerations it follows that an upward deviation from the linear fall in Fig. 2.1(b) towards Fig. 2.1(a) will result in negative resistance. In Sections 4 and 5 we shall see how particular structures may lead to such favorable, con vex-upwards characteristics for D(t). (b) r\ / 1 D \ / Fig. 2.2 — The —D'(t) characteristic corresponding to fig. 2.1. 804 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 3. MINORITY CARRIER DELAY DIODE As a first example we shall consider the behavior of the device shown in Fig. 3.1. We have chosen a p-n-p structure rather than an n-p-n so as to deal with positively charged carriers and thus avoid numerous minus signs in the ecjuations. In this figure we have used capital letters P and A^ to designate specific regions, reserving the small letters to indicate carrier densities and conductivity types. Several features that simplify the theoretical treatment should be noted : (a) The PiN junction is 100-fold more heavily doped on the Pi-side. (b) The doping in the layer N varies exponentially ^^•ith distance by a factor of 10 across the layer. —*\ ^ P, 1 S, 1 N ^2 1 Pa ^ 10'9 10'8 10'^ - 10'6 - ID'S 10'3 I0'9 I0'6 I0'7 1016 lO'S Na - - ^__j:Jd^ - Na 1 Fig. 3.1 — Constitution of minority carrier delay diode. NEGATIVE RESISTANCE IN SEMICONDUCTOR DIODES 805 (c) Throughout A' the coucentratiou of holes is less by a factor of 10 than the electron concentration. (d) The thickness of X is large compared to the (lei)th of space charge penetration into it. (e) The ^•oltage di-op across the space charge region *S2 is large com- pared to the other \'oltage drops. The conditions lead at the operating frequency to the folhnving conse- ([uences: (1) The current across the first junction is carried preponderantly by holes. (2) The hole drift in .V is substantially unaffected Ijy the ac field and thus represents the delayed diffusing and drifting current injected across the first junction. (3) The ac voltage drop occurs chiefly across So . We shall show below (Section 3.2) how (a) to (e) lead to (1) to (3), but we shall first bring out the importance of (1) to (3) by using them to give a simple treatment of the impedance of the diode. 3.1. Calculation of Impedance. If the total current is J{t) = J ^je'"', "(3.1) then the ac hole current across *Si is also in the notation discussed in Section 2 with the addition of the sj^mbol p to indicate holes j(p,S„t) =.7V"'. (3.2) This current flows through the n-layer unaffected by the ac field and arrives at S2 delayed and attenuated by a complex factor /3 = \0\exp(-id). (3.3) Because of the high field in So , the transit time there is negligible so that the hole current arriving at Po is j(p, S.;)e''" = ^je'"'. (3.4) In addition to this current, there is a dielectric displacement current in So which is converted to hole conduction cm-rent in Po . If the voltage drop across »S2 is V(S2 , t) = V{So) + v{S,)e'''\ (3.5) 806 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 then the ac displacement current is j{D, ,S'2)f''"' = ia)C,i;(*S:,)e'"'. (3.0) Now the total current is constant through the device, hence J = Jip, Sd + j(D, S2) (3.7) which leads to j = io:C2v{S2)/{l - 13). (3.8) If ^(^2) is substantially eciual to the ac voltage across the luiit, then the impedance is (3.9) = (l/icoCo) + (I/C.C2) I ^ 1 exp t[{7r/2) - d]. Evidently ii 6 > tt and 6 < 2ir, the second term will have a negative real part so that the diode will act as a power source. If we neglect the ac electric field in N then |3 may be calculated in terms of the thickness L = X2 — rci of the layer and the potential drop across the layer. This latter arises from the concentration ratio Ndi/Nd2 between the two sides of N. Since the donor charge density is neutralized substantially entirely by electrons, and since almost no electron current flows, the electron concentration difference must result from a Boltzman factor (at lO^Vcna^ Fermi-Dirac statistics are not needed) and this leads to AVn = (kT/q)hi(Ndi/Nd2) (3.10) for the potential drop across A^. In A^ the electric field is thus E = AVn/L. (3.11) The differential equation for hole concentration for a disturbance of frequency co is p = i^p = -^pE ^ + i), g . (3.12) This linear differential equation has two linearly independent solutions. These must satisfy at .r2 , the left edge of the space charge layer S2 , the boundary condition that the hole density is practically zero. The appropriate solution is p = e'"' [e"'^''-''^ - e"'^'"'''), (3.13) NEGATIVE RESlSTAiNCE IN SEMICONDUCTOR DIODES 807 where Lh ^ (.To - .ri)A-i = «[1 + (1 + iyY"], (3.14) LA-o ^ (x, - .ri)A-2 = «[1 - (1 + iyY"] (3.15) where a = qAV/2kT, (3.16) 7 = 4coDpAr', (3.17) ?< = UpE = MpAF/L. (3.18) The current is j(p, X, t) = q(up - Dpdp/dx) (3.19) and the ratio of currents at .Ti and xo , which is /3 by definition, is l3 = j{p, .1-2 , t)/{j(p, Xi , t), ^ Lh - Lk. Kh exp (-LA:,) - KA;2 exp {-Lh) (3.20) ^ 2(1 + iyye" [1 + (1 +iyy'^expa{l+iyy'^ - [1 - (1 +*7)''']exp - cx(l-\-iyy'' ' The phase lag in jS must exceed 180° or tt in order to give negative re- sistance. It can be seen that this phase factor must result from the first exponential in the denominator by the line of reasoning suggested below: The real part of the exponent is larger than the imaginary part. Hence the absolute ratio of the two exponentials is at least 27r. For this condi- tion the second term in the denominator is negligible compared to the first. Hence the phase of (3.20) is determined largely by the first ex- ponential. As a helpful approximation we may neglect the second term and write . ^ 2(1 + n)"'exp [g - c. (1 + iyY"] . . ^ 1 + (1 + iyy^ • ^^-^^^ Two limiting cases are worthy of special note: (I) a ^ 0, uniform n-layer, y — > so . ^ = 2 exp -aiiyY" = 2 exp -(1 + i)(o:/2Dy" L. (3.22) (II) a -^ 00 , qAV/kT » 1 , 7 ^ 0. /3 = exp [-za7/2 - ay^/S\, (3 23) - exp [-i{c^L/u) - (DL/u)/(u/o:f]. 808 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 These expressions may be interpreted as follows: In case (I), flow is by diffusion and the propagation factors A'l and A'2 take the form ±(a/L){iyY'^. For this case attenuation and delay terms in the ex- ponential are equal, and the largest negative term occurs in Z when the phase angle is 57r/4 (as may be verified by differentiation.) This leads to ^ = 2(-l + i)2~"^exp (-57r/4) = 0.028(-l + i), (3.24) which gives Z = (l/coC2)(- 0.028 -i 1.028), (3.25) the impedance of a condenser \\\i\i a negative Q of 37. In order to make an oscillator by coupling this to an inductance, an inductance with a Q of more than 37 must be used. It is obviously advantageous to reduce the magnitude of the negative Q. For case (II) in its ideal form, the ac current simply drifts through the n-layer without attenuation. This produces a phase lag of co times the transit time L/u. If this were the only effect involved, a capacitor with a negative Q of less than unity could be produced. In addition, however, there is attenuation due to spreading by diffusion. This effect is depend- ent upon the ratio of the spread by diffusion (DL/uY''^ to the separation of planes of equal phase in the drifting hole current. This latter separa- tion is 2Tru/o). The square of this ratio appears in the attenuation term in the second form of (3. We shall estimate the effect of the attenuation term by taking ay/2 = 37r/2, (3.26) so that the desired phase shift is obtained. The attenuation term is 7/4 smaller than this so that if 7/4 is considerably less than one, the attenuation in (3 will be small while the phase shift is correct. If we take 37r/2 for the value of ay/2, then the value of 7 becomes 7 = 4:a>D/u = 6TkT/qAV. (3.27) Thus the approximation on which (II) is based fails unless qAV/kT > 18, a value which implies an enormous range of concentration in the n-layer. We must, therefore, investigate the case of gradients in the n-layer by more complete algebraic procedures. We shall denote by — di the phase shift in jS due to the exponential in equation (3.21). The total phase shift 6 is somewhat less since the alge- braic expressions give a small positive phase shift of at most about 15°, which vanishes for large and small values of 7. Similarly the attenuation of ;S arises chiefly from the real part of the exponential since the absolute NEGATIVE RESISTANCE IN SEMICONDUCTOR DIODES 809 value of the algebraic expressions lies between 2 for 7 = 0 and 1 for Y = 00 . It is instructi^'e to express the real part of tlie ex})oiieiit in terms of a and ^1 . This is done as follows: a - a(l + iyf- = --q - id^. (3.28) This can be solved for r] and (1 + ^7)"': 7/ = («- + diY' - a, (3.29) (1 + iyf" = [(a- + d,Y' + ie,]/a. (3.30) From there it is seen that for a fixed value of ^1 , the attenuation can be greatl}^ reduced by increasing a. Unfortunately, this requires very large changes in concentration. For example with ^1 = 37r/2 and a = ^1 , the value of 7} is reduced to r; = 0.414 di . However, the value of potential difference is qAV/kT = St, (3.31) giving N,,/N,2 = 10'. (3.32) For the case shown in Fig. 3.1, a = 1.15 and for 9i = 57r/4 = 3.9 we obtain n = 2.95 = Ine 19.5 (3.33) This is an improvement of about 1 factor of e in the exponential com- pared to having AT' = 0. The value of (8 is /3 = 0.082 X exp (-i 218°), (3.34) and this leads to Z = (C0C2)"' (-0.05 - i 1.0G5). (3.35) Thus at the operating frequency, the diode appears to be a capacitor with a negative Q of 21. Increasing the concentration change to a factor of 100, so that a = 2.3, gives ri = 2.25, (3.36) /3 = 0.18 < - 220°, (3.37) Z = (cuCs)"' (-0.11G - f 1.14), (3.38) 810 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Q = -10. (3.39) The calculations indicate that attenuation can be controlled to a considerable degree while maintaining the desired phase shift. 3.2. Justification of Consequences (1), (2) and (3) In germanium at room temperature the product np is about 10 under equilibrium conditions. At the first junction of Fig. 3.1 it is 10 ", implying a forward bias^ " of (kT/q) 2.3 X 5. In order to maintain this forward bias a flow of electrons must be furnished to A^. There are sev- eral ways of accomplishing this. In the first place, the reverse bias across Si drawls a reverse current of thermally generated electrons from P2 . This current can be controlled by controUing the lifetime and temperature in the P2 region. Alternatively, electrons may be injected into P2 ; some of these will diffuse to S2 and arrive at N. Still another means of controlling the bias across *Si is to make contact to .V itself. Since only the dc bias need be controlled, the series resistance across A^ itself is unimportant; the source should be of high impedance. The decrease in density of 10 across the junction in carrier concentra- tion implies a potential difference of 9.2 {kT/q). Most of this potential difference occurs where the carrier concentration is negligible. Hence the space charge theory may be applied. Furthermore, the acceptor con- centration is much higher than the donor concentration. Hence the space charge extends chiefly into the donor region and we may write A7i = {2TrqNd/K)W^ (3.40) for the relationship between width W of the space charge region and voltage drop AF. If this voltage drop has an ac component, then a charging current will be required to change W. This current is determined by the admittance coC = w/c/47rTF (3.41) of the space charge region. At the same time injected hole and electron currents flow across the junction. The admittance associated with the hole current is approxi- mately A = {iw/Df" cTpi , (3.42) where o-pi is the hole conductivity just inside the n-layer. ' Actually, as discussed below, the admittance is somewhat higher. NEGATIVE RESISTANCE IN SKMirONDrCTOIt DIODES 811 The ratio of the admittances is 9~l Air coo-pi 4^17' (3.43) 47ro-pi qAV a pi For our example this expression is much greater than 1 as may be seen as follows : The first fraction is the ratio of the dielectric decay constant to oj. This is 10 or more larger than co need be. The next term is about 10 and the last term is the ratio of hole to electron density at Xi and is about 10~ . Hence the ratio of impedances is about 10: 1. We shall next consider why the expression for A for holes must be examined more closely. The admittance formula used above applies to the case of zero field to the right of the junction. The aiding field will increase the flow of holes into the n-layer and raise the admittance somewhat. Correcting for this will increase A in respect to wC and \\dll thus strengthen rather than weaken the argument. Also in the expression for A, no account was taken of the transit time across the region W. If we assume a uniform field in this region for purposes of making estimates, then the solution of equation (3.20) may be applied. Since now m corresponds to drift velocity due to 9kT/q of voltage drop across W which is much less than L in length, it is evident that 7 will be less by a large factor in this region compared to its value in .V. This leads to the conclusion that phase lags will be unimportant in this region. We have neglected the effect of electron injection into Pi . By the customary arguments for unsymmetrically doped junctions, it follows that this current is very small compared to the hole current. This justifies consequence (1). Consequence (2) may be justified as follows: At Xi all the ac current j exp (lot) is carried by holes. If a pure drift case occurred, the hole cur- rent might be reversed at some point in the n-Iayer and be —j exp (iwt). Under these conditions the electron current would have to be 2j exp (io)t). Under no conditions, however, \xi\\ the electron current be larger than this. This maximum possible electron current will require an electric field and this field \x\\\ also affect the hole flow. Since the electron con- ductivity is at least 10 times larger than the hole conductivity, the hole current due to the ac field will only be about }/{q of 7 at most. Thus the hole current is only slightly affected by the ac field. Consequence (3) follows from the fact that the reverse biased junction 812 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 >S2 has much higher impedance than Si . Si has higher impedance than that for hole injection into N. The impedance for hole injection into A^ corresponds to the hole conductivity in the n-layer over a distance com- parable with the thickness of .V. However, the impedance of A^ itself is that due to the much larger number of electrons in it and is thus much less than the impedance of *Si . Thus it follows that the impedances across Si and across A^ are much less than across S2 . This conclusion is not affected by the modification of impedance of S2 due to hole flow across it. 3.3. Modifications The treatment presented abo\'e has been based upon the conditions (a) to (e). Some of these are advantageous from the point of ^'iew of operation but others have been introduced to simplify the treatment. Among the latter is the condition that the current across Si is carried chiefly b}^ holes. If the current were chiefly capacitative at this junction, then the voltage would lag 90° behind the current. This adds a desirable phase lag in the hole injection across >Si and thus requires less phase shift in the n-layer. By suitably'' adjusting the ratio of capacitative and in- ductive admittances, a net improvement in Q may be obtained. 4. THE TRANSIENT RESPONSE IN A UNIPOLAR STRUCTURE In the previous section the electric field produced by the injected holes had a neghgible influence on the motion of the injected holes. In effect i Pi 1 N 1 P2 / (a) (b) Hg. 4.1 — Space charge limited hole flow. NEGATIVE RESISTANCE IN SEMICONDUCTOR DIODES 813 this \\as due to the bipolar nature of the mode of operation considered, the majority cari-icrs in the region N acting to yhi(>l(l th(^ minority car- riers from their own space charge. In this section we shall deal with unipohir diodes in which only one type of carrier is present in sufficient number to have a major effect. In these the influence of the space charge of the carriers upon their motion plays an important role. Fig. 4.1 illustrates one example of the type of structure covered by the theory of this section. It is again a p-n-p structure like that considered in Section 3. However, in this case the dimensions, the donor density and the applied potential are such that the space charge "punches through" the device. ' Under these circumstances a condition of space charge limited emission is set up so that holes are injected from the positive region Pi to just such an extent that their flow is limited by their own space charge. This limitation is associated with the maximum of potential just inside A^". The potential maximum is evidently a "hook" for electrons generated thermally in Po and in A^. Under some circumstances electrons may accumulate and form a layer in which there is no electron flow and hole flow is carried equally by diffusion and drift. Such stagnant regions will tend to be suppressed if Pi is made of short lifetime material, so that electrons are siphoned out of A^, or if p at the maximum is larger than p for intrinsic material and the lifetime is locally low. We shall treat the transient response of this structure of Fig. 4.1 from the point of view of the impulsive impedance discussed in Section 2. Accordingly we suppose that a steady current J flows per unit area. At ^ = 0 an added pulse of current occurs carrying a total charge of 8Qi per unit area, the subscript "i" signifying initial condition. Our problem is to determine how this added charge is carried by a transient disturbance in the hole flow and what is the resultant dependence of voltage upon time ; by definition the added voltage across the device is v(t) = dQMt)- (4.1) Since we are dealing with a planar model, we shall suppose that the initial condition at f = 0 corresponds to added charges 8Qi and — SQ,- on the metal plates on the P-regions. These charges set up an added field 8Ei = 8Qi/K, (4.2) where K = Keo (4.3) 814 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 ill MKS units. The initial value v{0) is then simply 8Ei times the total width of the structure. The first effect, which takes place in a neghgible time in respect to the frequencies involved, is the dielectric relaxation of the field in Pi and P2 . The added current due to dEi leads to an exponential decay of dE in these regions with a transfer of 8Qi and —8Qi to the two boundaries of N. If Pi and P2 are thin compared to N, the resulting drop in v(t) is small. In any event it can be shown by the reasoning at the end of Sec- tion 2 that this contribution to D{t) adds simply the series resistance of Pi and P2 to the impedance. The next effect is the transport of 8Qi on left side into N by hole flow over the potential maximum. It will be easier, however, to discuss this process after the treatment of the transient effects that occur in A'^ itself. Consequently, we shall at this point assume that after a time, short compared with the important relaxation time in the structure, the disturbance of hole density is as shown in Fig. 4.2(a). Fig. 4.2(a) shows added charges -{-8Qi and —8Qi produced by a disturbance denoted as 679 in the hole density. The charge —8Qi on the right side is produced by an increased penetration of the space charge into P2 ; it is similar to that produced by increasing reverse bias on a p-n junction. Fig. 4.2(b) shoAvs the corresponding disturbance in electric field. This disturbance is denoted by 8E which is a function of x and t. Evidently vit) f Jo 8E{x, t) dx. (4.4) and this is the area under the 8E curve. The other parts of the figure indicate qualitatively a subsequent stage in the motion and decay of 8p and 8E. Our problem is to formulate mathe- matically this decay process. We shall treat the decay process in terms tfp + 6Q, (a) -dQi (0 ^^=^ dE (b) cfE H (d) Fig. 4.2 — The initial stage and a subsequent stage of the transient. XEGATIVK RESISTANCE IX SEMICOXDrcTOU DIODES SI.") of the effect of drift in the electric lield and neglect the effects of {liffu- sioii. This })r()cedure can he justified hy the fact that as soon as a hole had reached a point where the potential has fallen hy A'/'A/ helow the niaxinunii, its flow is iioxcrned hy diifl rather than diffusion and the |)redominance of drift continues to increase towai'ds the right.'' " If diift in the held is the predominant cau.se of hole flow, then the equations go\-ei'ning the situation in .V are J + bJ = (p + 8p)(n + 8u), (4.5) where the terms with 8 represent the transient effects and those without represent the steady state solution, p = qp is the charge density of the holes and u their drift velocity. The equation for the change of E with distance is K(d/dx)(E + 8E) = pj.^ p^ 8p, (4.6) where p, is the fixed charge density due to donors and acceptors. (We neglect any effect of traps.) The steady state equation for E is thus K(dE/dx) = pf + J/u. (4.7) In a region where pf is independent of .r, this equation may be reduced to quach-atures by writing K dE/(pf + J/u) = dx; (4.8) the left side is then a known function of E through the dependence of u upon E. It is convenient to introduce a time-hke variable s which is the transit time for the dc solution. Evidently ds = dx/u = KdE/(pfU + J). (4.9) For the case of space charge limited current, s may be conveniently measured from the potential maximum. Even though the solution is invalid at that point, the integrals converge and the contribution from the region within kT/q of the maximum is small. We shall assume that the equations for the steady state case have been solved and that the functional relationships are known between E, x, v and s. The differential equation for 8E maj' then Ijc obtained as follows: To the left of the pulse in 8p in Fig. 4.2(a), 8E is zero. From equation (4.6) we have KdE(x)/dx = dp. (4.10) 816 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Integrating this from the region where E is zero gives KdE(x, t) = / dp{x, t) dx. Jo (4.11) Equation (4.11) states that the dielectric displacement at x is equal to the excess charge between the potential maximum and x. Evidently during the transient following Fig. 4.2(a), the rate of change of this extra charge is —8J(x, t) since the dc current is flowing in at the left and an excess current 5 J flows out at the right. Hence we have KdhE/dt - -bJ, = —{bpu + pbn). For the change in drift velocity we may write hu = {du/dE) 8E = iM*dE. (4.12) (4.13) For high electric fields u increases less rapidly than linearly with E and /x* is less than the low- field mobility. For very high fields fx* is nearly zero and there are theoretical reasons for thinking that there may be a range in which fi* is negative. We shall return to this point in the next section. In Fig. 4.3 we show a diagrammatic representation of the transient so- r (b) 0 f dp ft 0 . 3 Fig. 4.3 — Graphical representation of the dependence of SE upon time. (4.15) NEGATIVE RESISTANCE IN SEMICONDUCTOR DIODES 817 lilt ion. Each of the dashed lines represents the decay of 8E as measured in a moving coordinate system: Thus wo consider 8E measiu'ed at a position x(so + 0; this is a position that moves with the dc velocity 7/. This BE is evidently expressed in terms of dE(x, t) by writing x = x(so -\- /): dE in moving system = 8E„Xso , t) = dE[x(so + t), t]. (4.14) The differential equation for SE^ is (d/dt) 8Em = (d8E/dt), + (d8E/dx)tdx/dt, = {d8E/dt), + {d8E/dx)tU, = - {u8p + p8u)/K + (8p/K)u, = -{pn*/K)8E = -v8E, where the quantity V = pn*/m (4.16) is an effective dielectric relaxation constant being the differential con- ductivity pn* divided by the permittivity K. E\'idently j/ is a function of position x only and may be expressed as v(s) through the dependence of x upon s. Thus we may write (d/dt)8EUso , t) = -v(so + t)8E^(so , t) (4.17) which has a solution 8E^iso, t) = 5^„(.So, 0) exp [-g(so + 0 + ^(^o)], (4.18) where g{so + t) = [ v{s) ds. (4.19) Js' The lower Hmit s' is chosen for convenience so as to avoid singularities in g{s). This integration shows that 8Em decays exponentially as the elec- trical field would decay in a material whose dielectric relaxation constant changed -snth time just as v changes as observed on the moving plane. Fig. 4.3 shows on the dashed lines the decay of 8Em on the moving planes. Since 8Em is zero to the left of the initial pulse in Fig. 4.2(a), it remains zero on all moving planes which follow the pulse of 8Qo . This justifies the statement made earlier. The solid curves labelled /i , t-i etc. show the spatial dependence of 8E for times ti , ^2 , etc. after the charge 8Q1 is added. The values of the transient voltage v(t) at time ti , for example, is the 818 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 integral under the curve ti . This curve is zero for x < x(ti) and for X > x(ti) it is 8E(x, h) = (SQi/K) exp [- g(s, + ^0 + g(so)], (4.20) where X = x(so + t). (4.21) If the total transit time across A^ is *S so that x(S) = L, (4.22) then v(h) = f 8E(x, h) dx. (4.23) From this expression we can derive the desired formula for D(t). For this purpose the integral over dx is replaced by an integral over s. At time t the range of s is evidently from t to S and dx = u{s) ds. From this we obtain: D(t) = vit)/5Qi , (4.24) = (1//0 [ exp [- g{s) + g{s - tMs) Js ds. From Fig. 4.3 we can see that there are competing tendencies in the decay of D(t) some of which tend to produce the desired convex shape discussed in Section 2 and others the concave shape. The effect of the dielectric relaxation constant is adverse and tends to produce an ex- ponential decay. On the other hand the advance of the pulse of holes from left to right in Fig. 4.2 proceeds in an accelerated fashion with the result that the range of x over which 8E is not zero decreases at an ac- celerated rate. If the dielectric relaxation Avere zero, this would result in the desired convex upwards shape. The resultant shape of the D{t) curve is thus sensitive to the exact relationship of the transit time and dielectric relaxation. This can be illustrated by giving the results of analysis for a p-n-p structure, neglect- ing diffusion and considering n to be constant. The solutions of the dc equations are readily obtained for this case and have been published. For convenience we repeat them here: E = (J/m)(e"' - 1), (4.25) x(s) = iiJKinpf)-' (c"' - at - 1), (4.26) NEGATIVE RESISTANCE IX HEMI('()NDTT(T< )1{ DIODES 819 L = x{s) = {JK/f,p/Xe' - /? - 1), jS = as. From these it is found that This leads to Ingis) = (1 - e-^. D(t) = (J/tJLpi) [/ + e"' (at - = (J//XP/) Di^, at). - 1)] (4.27) (4.29) (4.30) For t = 0 this reduces correctly to L/K. Figure 4.4 shows the resulting shape of the D curves with /3 as a parameter. Large values of (3 correspond to cases in which the hole charge density is small compared to p/ and to relatively long relaxation constants. For them the desired convex upward shape results. Figure 4.5(a) and 4.5(b) show the real and imaginary parts of the impedance expressed in terms of Z{^, 6) : Jo 'D(t) dt = {KJ/^.Vf)Z{^, d), = coT. (4.31) (4.32) 0.8 Q 0.6 s Q 0.4 lyi^^ ^ /3=\o\j \^\3 ^ V ^ ^ N ^ k 0 0.2 0 4 0 6 0.8 10 at//3 Fig. 4.4 — Impulsive impedance lor various values of /3 in p-n-p structure. 820 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 It is seen that values of — Q as small as about 10 can be obtained for 13 ^ 3. In the next section we shall consider modifications which may result from variations in n* and for changes in geometry. We must return to the question of how the charge +5Qi passes the potential maximum. In order that the theory given above apply, it is necessary that the time required for 8Qi to enter the drift region be short compared to the transit time. At the potential maximum the charge density may be estimated by the methods previously dealt "with in the theory of space charge limited emission. Initially -\-8Qi appears to the left of the maximum and the field at the maximum is 8Ei . This field will 10-2 /3=5 \ — ^ NEGATIVE VALUES 4 \ \ \ ^=5 3 S \ / \ \ r \ \ \ / ^ «y > \i y- S^i \ /3=5 \i 1 \l ^ — ^ \ f "^ \ \ 1 Y \r A '/- \l 4- / 1 » / \l Ni \ ^ \ f ^^-^ ^ 1 s \ 1 i 1 P3 2 J 1 ' 100 200 300 400 500 600 700 800 6 IN DEGREES Fig. 4.5 — Impedance of a p-n-p structure, (c) Real part of impedance. NEGATIVP: resistance in SEMirONDUCTOIt DIODES 821 then relax with a relaxation constant of about iJ.p(rciax)/K where p(max) is the hole charge density. Actually the relaxation may be somewhat (juicker because the concentration gradient of the added lioles also contributes to the flow over the maximum. Since the charge density at k'T/q below the maximum is comparable to that at the maximum the entire relaxation process will proceed at about this rate. Thus a criterion for the applicability of the theory is that /C/^ip(max) be much less than S, the transit time or total decaj' time for D(t). 5. MOBILITY AND GEOMETRY EFFECTS 5.1. The Effect of a Region of Negative n* In very high electric fields holes may be expected on the basis of theory to exhibit a negative value of /x*. This theory^ ^ is founded on the idea that a hole can lose energy to phonons at a certain maximum aver- 100 200 300 400 500 600 700 800 e IN DEGREES Fig. 4.5 — Impedance of a p-n-p structure, (h) Imaginary part of impedance. 822 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 age rate Pmax • (-Pmax is the staukonstante of Kromer.) The holes probably achieve this rate when their energy is near the middle of the valence band. Under these conditions the power input from the electric field must be no greater than Fmax : From this it follows that qEu ^ Pr, u ^ PmaJqE, (5.1) (5.2) so that the drift velocity Avill decrease mth increasing field at sufficiently high fields. Furthermore, if the \\idth of the valence band is less than the energy gap, then a hole cannot acquire enough energy to produce hole electron pairs. Thus in such a case, the negative resistance range should be reached before breakdown effects occur. In Fig. 5.1 we illustrate the general trends of the u versus E curve, to be expected if the staueffekt occurs. As is indicated, the maximum drift velocity will be referred to as iim ■ It occurs at a field Em . Since we are here concerned A\dth principles rather than details, no attempt has been made to indicate the square root range in which u is proportional to E^^^. This range has been observed by E. J. Ryder ' and shown by G. C. Dacey^'^ to control hole flow in space charge limited hole currents in germanium and has been treated theoretically. ' Dacey has also in- vestigated the effect of the square root law upon the D(t) curves for the p-n-p structure of Section 4 and reports that the effects are so unfavor- able that no negative resistance is to be expected. The staueffekt opens the attractive possibility of making negative resistance devices in which the current decreases with increased dc voltage so that negative resistance will be exhibited over a wide fre- quency range. Unfortunately, when the boundary conditions are taken Fig. 5.1 • — Qualitative representation of drift velocity versus field as affected by "staueffekt." NEGATIVE RESISTANCE 1\ SEMK '<)\Dl( TOH DIODES 823 into account, it is found that a device in wliicli most of the curreiit flow- occurs in a ne will require a distance Ax = KE„,/pa ■ (5.7) If this value is much smaller than L, then the situation represented in Fig. 5.2(c) and (d) will occur. On the other hand if L is smaller than Ax, the region of space charge and high field will extend throughout most of the structure. In any event eciuation (5.4) leads to positive resistance. This can be seen from the fact that increasing J always means a decrease in x for the same value of E and hence an increase in E at all values of x and thus an increase in voltage at any fixed value of :r. The above conclusion that a positive dc resistance will be exhibited by a structure like that discussed above may also be reached by considering the transient response. The theory of Section 4 may be once at be apphed to this case by simply taking account of the fact that v is negative for part of the structure and thus that 8Em increases with increasing s. In Fig. 5.3 we illustrate a stmcture to which these considerations may be relatively simply apphed, at least in a limiting case. It consists of four layers, the two outer being p^ as before. Space-charge limited emis- sion then enters the intrinsic layer which is of such a width that at its right hand boundary the electric field has a value E^ that exceeds E^ ■ At this point the hole space charge is P.3 = J/lh (5.8) p + L P P + Fig. 5.3 ductance. A structure having a region of uniform negative differential con- NEGATIVE RESISTANCE IN SEMICONDUCTOIl DIODES 825 where Ws is u{Ez). In the P-region this space charge is compensated by acceptors to produce a region of uniform field in which ^l* is negative. If the F-region is wide compared to the /-region, then the transit time through it will also be relatively large. As a consecjuence bQ will be transferred quickly into the P-region. From that time on bEm. curves, like those of Fig. 4.3, will show an exponential increase with time and also with distance since for this case of constant u in the P-region, time and distance are linearly related. This will lead to a D{t) of the form D{t) = {u,/K)(S- t)exp\^.*p,/K\t, (5.9) where the absolute value signs emphasize that for this case of negative n* there is a build-up in time. This form of D is always convex upwards and, in fact, if ,S|M*P3/i^| > 1, (5.10) it starts ^^^th a positive slope at s = 0 so that the transient voltage actually builds up initially with time. Even an initially growing D(t) does not give a negative resistance at low frequencies, however. As shown in Section 2, the dc resistance is simply the integral under the D{t) curve and thus \nll still have a posi- tive value. 5.2. Convergent Geometry It is possible to obtain marked improvement of the D(t) curves without the aid of the negative values of m*- This possibility is based upon con- vergent geometry. A possible case is illustrated in Fig. 5.4. In this case it is supposed that the field in the inner P-region is so large that a sub- stantial reduction in fx* has occurred. As a consequence, the decay of field in this region is relatively slow. Furthermore, since both the dc and Fig. 5.4 — A convergent flow structure. 826 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 transient fields are high in this region, essentially because of the inverse square law, the principal contribution to D{t) comes from this region. These two factors — relatively slow dielectric relaxation near the center and principal contribution to D(t) from near the center — combine to give a D{t) characteristic which holds up well until the pulse of injected holes reaches the inner region. This may result in a favorable convex upwards D(t) characteristic. ACKNOWLEDGEMENTS The writer is indebted to a number of his colleagues for helpful dis- cussions and to W. van Roosbroeck for the calculations for Fig. 4.4, and to R. C Prim for Fig. 4.5. REFERENCES 1.1. W. Shockley and W. P. Mason, J. of Appl. Phys., 25, No. 5, p. 677, 1954^ 3.1. W. Shockley, Electrons and Holes in Semiconductors, D. van Nostrand, N. Y., 1950, p. 312. 3 2 ScG RsfGrGHCG 3.1. 3^3'. W. Shockley, B.'s. T. J., 28, p. 435, 1949, Section 2.4. 3.4. See References 3.1 or 3.3. 4.1. W. Shockley and R. C. Prim, Phys. Rev., 90, pp. 753-758, 1953. See also G. C. Dacey, Phys. Rev., 90, pp. 759-763, 1953, and W. Shockley, Proc. I.R.E., 40, pp. 1289-1314, 1952. 4.2. W. Shockley and R. C. Prim, Reference 4.1. 4.3. E. J. Ryder, Phys. Rev., 90, pp. 766-769, 1953, references. 4.4. See Shockley and Prim, Reference 4.1. 5.1. This theory of mobility in higher fields has been published by H. Kromer, Zeits f. Physik 134, pp. 435-450, 1953. Kromer considers the theory in con- nection with the values of a in point contact transistors but does not ex- plore it as a power source. The present writer derived the same result in a more primitive form in 1948 as a potential means of obtaining high fre- quency power. However, experimental results by E. J. Ryder did not show evidence of this effect. The effect may possibly have occurred without being recognized because of the absence of an adequate appreciation of the im- portance of the boundary conditions discussed in this section. 5.2. See Reference 4.3. 5.3. G. C. Dacey, Phys. Rev., 90, pp. 759-763, 1953. 5.4. W. Shockley, B. S. T. J., 30, pp. 990-1043, 1951. 5.5. Personal communication. Transistors and Junction Diodes in Telephone Power Plants By F. H. CHASE, B. H. HAMILTON and D. H. SMITH (Manuscript received November 30, 1953) This paper describes the use of junction diodes, reference voltage diodes, and junction transistors in regidated rectifiers for telephone power plants. It discusses the pertinent characteristics of these semiconductor devices, together ivith illustrative circuits in which they are used to control the flow of direct current power. 1. INTRODUCTION Recent articles in the literature have treated the theory and proper- ties of semiconductor de\dces. In particular, papers by Messrs. Shockley, Ryder, Wallace and others have emphasized the theoretical aspects of the new devices; their reliability, reproducibility and performance at high frequencies to name only a few.^' ^' ^- ^' ^ In addition many papers have been published concerning their applications in the transmission and computer fields. There is also a field of application for these devices in the conversion and control of power, and this paper discusses some of these power applications. 1.1. Scope The first three groups of sections in this discussion review the perti- nent characteristics and practical engineering aspects of junction recti- fier diodes (Section 2.1), reference voltage diodes (Section 2.2) and junc- tion transistors (Section 2.3). The second three groups of sections concern respectively shunt transistor regulators (Section 3.1), series tran- sistor regulators, (Section 3.2) and power regulating circuits employing magnetic amplifiers in combination with transistors and junction diodes (Section 3.3). The last two groups of sections treat specific appUcations (Sections 4.1 through 4.3). 827 828 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 2. DEVICE CHARACTERISTICS 2.1. Junction Rectifiers A junction rectifier is made from a wafer cut from a single crystal of semiconductor material. The materials now being used for this purpose are germanium and silicon, but to date the use of germanium is more common than silicon. Pure germanium in its undisturbed or intrinsic state is a poor conductor; but its conductivity can be increased by dis- turbances such as cosmic rays, photons of light, external potentials, or by the addition of very small amounts of selected impurities. We are concerned here only with the addition of impurities. There are two classes of these impurities, called "donors" and "acceptors." The physical mechanism by which pure germanium becomes conductive depends on which of these two classes of impurities are present. Donor impurities result in a surplus of free electrons which can conduct current by nega- tive charges passing through the germanium crystal. Thus the addition of donor impurities to pure germanium creates "n" type material. Presence of acceptor impurities results in a shortage of electrons creating "holes," which have positive charges. These holes are mobile and they can conduct current through the crystal.'^ Thus the addition of acceptor impurities to pure germanium creates "p" material. When an abrupt change is made from p to n type material inside the crystal a rectifying junction exists at the boundary between the two materials. This p-n junction exhibits rectifier action in that it \\\\\ con- duct current every easily from p toward n; but, in its rectifier operating range, only minute currents can be made to flow from n toward p. We say that this junction has a low forward resistance and a high reverse resistance. All rectifiers have these characteristics to a greater or lesser degree and the p-n junction rectifier characteristics have been compared elsewhere to other rectifier de vices. ^ There are two methods of producing the junction inside the crystal. It can be obtained by growing part of the crystal from p type material and part from n type. This is called a "grown" junction. It can also be obtained by diffusing impurities into the crystal after it has been grown. This has been called an "alloy" process, a "fused junction" process, or a "diffused junction" process. 2.11. Junction Rectifier Terminology Before discussing the characteristics of junction diodes, it may be helpful for the reader to consider the terminology employed. As in other TRANSISTORS AND JUNCTION DIODES 829 rectifying cells, there are two directions of current flow, forward and reverse. Each diode has a positive and a negative terminal, and we de- tine the positive terminal as that terminal towards which forward current flows within the diode. LikeA\ise, the negative terminal is that terminal towards which reverse current flows within the diode. In Fig. 1(a), terminal 1 is the negative terminal and terminal 2 the positive. The circuit convention for the diode is a shorthand method of indicating the polarity of the diode to the engineer. If a battery is connected to a diode as shown in Fig. 1(b), forward current A\ill flow, and if connected per Fig. 1(c), reverse current \\\\\ flow. If the battery is replaced by a source of alternating current, forward current will flow through the diode during the half cycle that terminal 1 is positive, and reverse current \\\\\ flow during the half cycle that terminal 2 is positive. The rectifier is said to "conduct" during the first half cycle and to "block" during the second half cycle, for the resistance in the conducting direction is very much less than the resistance in the blocking direction. The figure of merit of a diode is a measure of this ease of conduction and the effectiveness of the blocking action. The ease of conduction can readil}' be determined on a static basis by applying a dc voltage to the diode as shown in Fig. 1(b) and plotting forward current through the diode as a function of applied voltage. Like^^^se, the blocking charac- teristic can be determined if a circuit per Fig. 1(c) is employed. 2.12. Typical Junction Rectifiers Fig. 2 is a photograph of several sizes of typical junction diodes. The diodes sho^^^l have a range of forward current from several milliamperes (Diode I) to hundreds of amperes (Diode IV). Diode I is made from a crystal of silicon and the balance are made from germanium. Most rec- tifying diodes have a particular field of use dictated mainly by their power handling capacity in the forward direction of current flow, al- though Diode I is of interest because of its unusual reverse or blocking characteristic, as will be pointed out later in this paper. (-)o. 2, DIFFICULT DIRECTION OF CURRENT FLOW (a) FORWARD CURRENT FLOW (b) Y'lg. 1 — Rectifier terminology. REVERSE CURRENT FLOW (c) 830 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 DIODE m DIODE m Fig. 2 — Typical junction rectifiers. Diodes II, III and IV, courtesy of the General Electric Company. TKAXS16TOKS AND JUNCTION DIODES 831 Fig. 3 is a plot of the static forward aiul ro verse eharactiMistics of the four diodes shown in Fig. 2. Tlie eharacteristies were obtained using the circuits in Figs. 1(b) and 1(e), respectively, measurements being made in still air at room temperature. The curves in the first (luadrant, ( + /i' + /) are the forward characteristics and the cur\'(>s in the third (juadrant ( — E —I) are the re\'erse characteristics. Notice that the scales are different in these (juadrants. In general, at any other temperature the curves would shift their positions with ies{)ect to the reference axes. This must be taken into account by the ciicuit designer. 2.1.3. ,1 nnction Temperature We will limit further discussion of general characteristics to those of Diode IV, for in many respects this is the most interesting rectifier for :aj 60 i| < 40 ;^ :5 20 iZ 0 : in 'ft 2.5 ir Q. 31 in _j Si ?, ^ 7.5 / REVERSE CURRENT FORWARD CURRENT IN MILLIAMPERES IN AMPERES OOOOOO OOO 1 DIODE N0.1 / / DIODE NO. 4 1 / J — y / / t J 40 30 20 to REVERSE VOLTS 0.5 1.0 t.5 2.0 FORWARD VOLTS 160 120 80 40 0 0.2 0.4 0.6 0.8 REVERSE VOLTS FORWARD VOLTS Z 3 O CE 5 u. 3 < iu5 ; tr < )(r_i DIODE NO. 2 1 / ^ / . — * ^^ yy ^ 5Z 0.5 Z OJ 31 10 IT) _| Sz DIODE N0.3 / / ^ r -^ ^ \^' 400 300 200 100 REVERSE VOLTS 0.5 1.0 1.5 2.0 FORWARD VOLTS 20 15 10 5 REVERSE VOLTS 0.2 0.4 0.6 0.8 FORWARD VOLTS Fig. 3 — Junction rectifier static characteristics. 832 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 power applications. Laboratory experience indicates that it is not de- sirable to operate the junction of this diode above 65° to 70° centigrade. The value of this critical temperature is not accurately kno^^^l on ac- count of the difficulty in measuring the junction temperature inside of the crystal. However, below the critical temperature, those changes in characteristics which are associated with changes in junction tempera- ture are reversible, that is, if the temperature is raised and then reduced, the characteristics will shift back to values previously experienced at the reduced temperature. Beyond the critical junction temperature any change in the reverse characteristics is permanent and has the effect of reducing the reverse resistance. In an operating circuit, this effect leads to progressively greater permanent damage to the diode. Lowered re- verse resistance allows more reverse current to flow, increases the re- verse power dissipation and elevates the temperature of the junction causing further reduction of the reverse resistance, and so on until the diode no longer blocks. Thermal damage to the junction can be prevented by removing heat. This method is employed "\Aith the diode under discussion by forcing air through the cooling fins at a high velocity. The quantity of air needed depends on the amount of heat generated in the junction, the efficiency of the cooling fins and the temperature of the air employed for cooling. In most Bell System applications, the maximum temperature of the <^ 100 ^ ^^ ^ y ^ / / ; 1 1 r : + /I 2,4, LINE ' 1 1 / / LOAD VOLTAGE CONSTANT AT 50 VOLTS Fig, D 800 1200 )600 2000 2400 28 LINEAR VELOCITY OF AIR FLOW IN FEET PER MINUTE (TO KEEP HOTTEST JUNCTION AT 65°C) 4 — Junction rectifier forced cooling characteristics. TRANSISTORS AND JUNCTION DIODES 833 ambient air is 40°C, which permits the junction temperature to be 25 to 30°C above the air temperature before the critical vakie is reached. ^ ^A typical load-current versus air velocity curve is shown in Fig. 4. The curve is based on a G5°C junction temperature measured by thermo- couples attached to the radiating structure near the junction and 40°C ambient air. Notice that the curve is taken with a working circuit com- posed of six diodes in a three-phase full wave bridge arrangement. In general, engineers developing rectifier circuits find that curves showing the properties of combinations of rectifying diodes an; more useful than single diode characteristics, except where the properties of the diode are such as to make it useful as a valve, or as a reference standard, as is the case of Diode I in Fig. 2. This leads directly to a more detailed considera- tion of the blocking or reverse characteristics of junction rectifiers. 2.2. Reference Voltage Diodes 2.21. General In the case of sihcon junction diodes it has been possible to reduce the reverse current to a very low value for reverse voltages up to a value called the "saturation voltage." When the saturation voltage is reached the electrons and/or holes which comprise the leakage current are given sufficient energy to create other electron -hole pairs which add to the original reverse current. This process is cumulative and leads to large in- creases in current for small further increases in voltage. The effect is illustrated by the reverse voltage-current characteristic for Diode I in Fig. 3. This curve shows the reverse current to be quite low for volt- ages less than 22 volts. This portion of the characteristic is called the "high resistance region." As voltage is further increased the curve goes through a "transition region" to the "saturation voltage region" at 23 volts where voltage is nearly constant over a wide range of current. The voltage saturation characteristic makes the diode suitable for use as a source of reference potential in the control of power. Those readers who ^\ish to study the basis of these properties will find the theory cov- ered elsewhere in the literature.^ -^ The rectifier selected for study in this Section is Diode I. This is a p-n junction rectifier made from silicon. It has been constructed to obtain a reasonably constant saturation voltage as shown in Fig. 5. In order to show the wide range of current values where this voltage is sub- stantially constant, Fig. 5 is plotted to a logarithmic scale. In this con- nection it is interesting to note that the saturation voltage can be con- trolled in manufacture from a few volts to several hundred volts. This 834 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 ^-HIGH RESISTANCE REGION ->--*« SATURATION VOLTAGE REGION- 1 1 1 TRANSITION 1 / REGION r 1 \\ 10"^ 10" = CURRENT IN AMPERES Fig. 5- — Reverse characteristics of a reference voltage diode. range can be compared to the 60 to 150 volts range of cold cathode volt- age regulator tubes which are also used as sources of reference potential. 2.22. Saturation Voltage Utilized in Regulating Circuits In all check back (feedback) regulating circuits the potential to be regulated is compared to a reference potential. This comparison is a form of subtracting the two values so that the changes in the potential to be regulated produce a large percentage change in the difference or error voltage. The methods by which this is accomplished in direct ciu'rent circuits are illustrated in Section 3. A stable source of reference potential is required for this type of regulation. When the saturation voltage of a silicon junction diode is used for this piu'pose, we have called the device a "reference voltage diode." 2.23. Effect of Temperature on Saturation Voltage In order to evaluate the stability of Diode I in its saturation voltage region a small section of Fig. 5 has been redrawn in Fig. 6 using a linear scale. Additional curves are included in Fig. 6 to show the change of voltage with ambient temperature variations. The slope of the 30 degree curve in Fig. 6 is equivalent to a resistance of 200 ohms in series with a 23-volt battery with current flowing through this combination from an external source. The change of potential with ambient temperature is equivalent to a 0.07 per cent change per degree C. It should not be in- ferred that these are limiting values, for diodes hiive been tested which exhibit slopes of less than 10 ohms and temperature coefficients of less TKANSISTOKS AND JUNCTION DIODES 835 tluiii 0.01 per {'(Mit per de<>;ree C The specific applications covcM'ed later ill this discussion show methods to compensate lor slope and temperature variation when necessary. 2.3. Junction 'J'ransiir Action 2.31. Tiro-Rectijicr Anah/sis In junction transistors tiiere are two p-n junction rectiliers contained in the semiconductor material. Of the materials now in use germanium is the more pre\'alent. llememl)ering the results of adding donor and ac- ceptor impurities to obtain n and p type materials covered in section 2.1 these two rectifiers are obtained by interposing a layer of p type material between two layers of n type making an 7i-p-n transistor or interposing a la^'er of n tj^pe material between two layers of p type making a p-n-p t7-ansistor. The electrical connections are designated as the collector terminal, the emitter terminal and the base terminal. Both types of transistors (n-p-n and p-n-p) have a rectifying junction between the collector and base terminals and another rectifying junction between the emitter and base terminals. The polarity of the collector and emitter rectifying junctions determines whether the transistor is n-p-n or p-n-p. Figs. 7(a) and 7(b) are simplified diagrams illustrating respectively the internal circuits of n-p-n and p-n-p transistors. The figures show the characterizations of transistors by means of a two-rectifier analogy. Although a transistor may be somewhat over-simplified by this method of characterization, the analogy permits the power engineer to approxi- mate the operation of transistors in familiar terms. Experience in the development of the circuits described later in this article has proven that the analogy is valid under circumstances where the operation of the transistor as a dc amplifier is of interest. 26 25 z24 23 22 U^ ^ +50^ "^^ -^ .-^ ^ 1 -^^ "+30° -30°C ^ "^ r Fig, 0 1 23456789 10 REVERSE CURRENT IN MILLIAMPERES 6 — Saturation voltage characteristics of a reference voH-age diode. 836 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 In an n-p-n transistor the collector and emitter terminals are the posi- tive electrodes of the rectifiers, see Fig. 7(a), and in a p-n-p transistor the collector and emitter terminals are the negative electrodes of the rectifiers, see Fig. 7(b). The base terminal is the common point of the two rectifiers. In a given transistor each rectifier has a satin'ation voltage, usually stated in the characteristics, which must not be exceeded in normal operation. Thus, the saturation voltage of the collector rectifier determines the maximum instantaneous collector potential. The emitter rectifier also has a saturation voltage which determines the maximum potential which can be applied between the base and the emitter. The saturation voltage of the collector rectifier usually differs from the saturation voltage of the emitter rectifier. 2.32. Transistor Action If a source of potential, Ece in Figs. 7(a) and 7(b) is connected between the collector and emitter terminals, the resulting current will flow in series through the collector rectifier in its reverse direction and through the emitter rectifier in its forward direction. This is the direction of current flow for transistor action to take place. In Fig. 7 the reverse resistances of the collector rectifiers are sho^vn and the forward resist- ances of the emitter rectifiers are also shown. Ece -^- COLLECTOR TERMINAL EMITTER TERMINAL COLLECTOR RECTIFIER EMITTER RECTIFIER FORWARD RESISTANCE . OF EMITTER. REVERSE RESISTANCE OF COLLECTOR BASE TERMINAL AAA ? FORWARD RESISTANCE , OF EMITTER, H EMITTER RECTIFIER -' RESISTANCE OF BASE EMITTER TERMINAL COLLECTOR RECTIFIER ''1 REVERSE RESISTANCE OF COLLECTOR COLLECTOR TERMINAL (a) n-p-n transistor (b) p-n-p transistor Fig. 7 — Junction transistor analogj'. 1 TRANSISTORS AND JUNCTION DIODES 837 2.33. Current Gain Now when a second relatively small potential is connected between the base and emitter rectifier (Eb in the sketches) additional current, !« will flow through the emitter rectifier in the forward direction and Tc w ill also increase. This increase in /^ caused by the increase in le is transistor action. The increase in /^ is related to the increase in /« by the factor alpha (a) as written below: Ale = (xAle. (1) llie application of lvirchoff''s current law to the sketches in Fig. 7 gives the change in lb as follows Ah = Ale - Ale . (2) By combining equations (1) and (2), Ale can be written as a function of Ah only ^h=j.r^Ah. (3) (1 - a) The usual value of a for junction transistors is near but slightly less than unity. In a typical case a might be 0.98. This value when substituted in equation (3) shows the current gain of the transistor, Ah/ Ah to be 49. Most of the circuits discussed in this paper are based on equation (3). It has been shoAvn how a small change in base to emitter potential with a small change in base current effects a large increase in collector current at a higher voltage. This explains how large power gains, of the order of 60 db, can be obtained from the junction transistor. The sketches in Fig. 7 do not show why this transistor action takes place. The reasons for it involve the use of such solid state physics terms as the migration of electrons and holes through a crystal lattice, and the interposition of junction barriers. The "why" for transistor action is very important in the manufacture of transistors, and it has been thor- oughly covered in the literature.^' '^ For present purposes it is only necessary to examine the static characteristics of an n-p-n transistor as sho^\^l in Fig. 8. This figure presents transistor characteristics in a manner which simpUfies the explanation of the operation of the transistor control circuits covered later in this paper. Referring to the curves in Fig. 8, it will be seen that in the straight portion of the 1.5- volt curve, a change of 50 microamperes in the base current will result in a change of about 2 miUiamperes in the collector current. This illustrates the current amplification of transistors and the 838 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 tr 4.5 O UJ 4.0 O O 3.0 ^ 2.5 2.0 5 0.5 \ A J / COLLECTOR TO _ ,p^/ EMITTER VOLTS ~ '"^ / / / / 6\y / / ' / 0 '^SV / / '/ / / ^ r //, y ^ 50 MILLIWATT n-p-n TRANSISTOR ^ y 30 40 50 60 70 80 90 100 MICROAMPERES FLOWING INTO THE BASE 110 120 130 Fig. 8 — Junction transistor static characteristics. current gain of this transistor is equal to 40. The measured a for this transistor was 0.976. Substituting in the current gain formula, equation (3) above, the calculated current gain is 40.6 which agrees with Fig. 8 within the accuracy of the measurements. 2.34. how Voltage Characteristics Again referring to the curves in Fig. 8, it will be seen that transistors operate at low collector to emitter potentials. The 1.5-volt curve is not the minimum potential at which this transistor ^\dll operate. Some transistors have good current amplification at potentials as low as two- tenths of a volt. When the base current is reversed, the characteristics in Fig. 8 can be extended to smaller collector current values. One might assume that the collector current can be reduced to zero by causing enough current to flow out of the base. This is not true. There is a mini- mum collector current, called the saturation current, and increasing current flow out of the base will not decrease the collector current below this value. This saturation current is assigned the symbol Ico ■ This Ico current is usually a few microamperes but it increases at the rate of 7 or 8 per cent per degree Centigrade increase in temperature of the col- lector junction. Transistors also have a critical junction temperature TUANSISTOUS AND JUNCTION DIODES 839 \\hich should not be exceeded under any operating conditions, and this must be kejit in mind during the design of the reguhiting circuits. 2.35. Equivalent Circuit of a T7'ansistor Ryder and Kircher^ have shown that it is possible to convert the sketches shown in Fig. 7 into a small signal etiuix'alent circuit using alpha and the three characteristic resistances of the transistor. These resistances are the emitter resistance Vc , the base resistance fh and the collector resistance i\. . Two forms of equivalent circuit are shown in Figs. 9(a) and 9(b). In the equivalent circuit in P'ig. 9(a) the active portion of the transistor is characterized as a current generator. This cciuivalent circuit is more directly related to the physical piocesses occurring inside the transistor than the eciuivalent circuit in Fig. 9(b) which characterizes the active portion of the transistor as a voltage generator. Although both equivalent circuits are useful the one in Fig. 9(a) is preferred in power work because t'c is much larger than the load resistance in many cases and can be neglected. Typical ^'alues for the eciuivalent circuit param- eters are given in the caption of Fig. 9. The use of the equivalent circuits are further discussed in some of the articles listed at the end of this paper. The article^ by R. L. Wallace Jr. and W. J. Pietenpol is of particular interest in this connection. 2.36. Typical Junction Transistors Fig. 10 is a photograph of two Bell System junction transistors made from germanium. The smaller one will dissipate 50 milliwatts, and the larger one is an exploratory model that will dissipate 2 watts when it is attached to a suitable heat sink. These transistors are hermetically sealed to protect them from the infiltration of moisture. The characteris- tics shown in Fig. 8 were measured using the smaller unit. Veb i T o -'/.A- l = aU (a) Vcb i t Veb 1 1^ c \AAr ,4 s i v=rml +_ arc) 1 Vcb \ -O T fb) Fig. 9 — Junction transistor equivalent circuits. Typical values for a .SO-mil- liwatt transistor: /•„ , 25 ohms; n , 500 ohms; ;> , 5 megohms; a, 0.98; and /„, , 4.9 megohms. 840 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 50 MILLIWATT 2 WATT Fig. 10 — Typical junction transistors. Thus, note that there are two kinds of transistors with respect to the polarity of the electrodes. The n-p-n transistor operates with positive collector potential and the p-n-p requires negative potential on the col- lector. Both will amplify current changes in the base circuit into much larger current changes in the collector circuit. The transistors have simi- lar equivalent circuits and parameters but all of their operating poten- tials and currents are reversed. It is also significant that the normal direction of current flow is out of the base terminal of the p-n-p transistor and that the normal direction of current flow is into the collector ter- minal of the n-p-n transistor. Likewise, the normal direction of current flow is out of the collector terminal of the p-n-p transistor and into the base terminal of the n-p-n transistor. This relationship between direction of current flow in n-p-n and p-n-p transistors is cafled reversed or com- plementary symmetry, and enables the circuit designer to cascade direct coupled transistors, alternating n-p-n and p-n-p. This is not possible with vacuum tubes because there is no tube that will operate \\ath negative plate potential. It ^\nll be sho^\^l how this complementary sym- metry can be used to advantage in multistage direct current amplifier circuits. 3. TYPICAL REGULATING CIRCUITS 3.1. Shunt Regulators 3.11. Simple Diode Regulator If a load is connected to a source of power, the current through the load and thus the voltage drop across the load will depend on the po- tential of the source of power, the internal impedance of the power supply and the load impedance. The voltage drop across the load can be made very nearly independent of these three parameters by employing a cir- cuit known as a shunt regulator. TRANSISTORS AND JUNCTION DIODES 841 A shunt regulator is a variable current device, connected in parallel with the load. Both the load and th(> shunt rept that the base to emitter potential decreases, the collector current decreases, the ^'oltage drop across the regulating re- sistor decreases and the load \'oltage rises to the n^gulated value. The value of the regulated output voltage is determined by the ad- justment of the potentiometer. Of course in a practical shunt regulator circuit, the adjustable range of the potentiometer would have to be limited to correspond with the operating range of the transistor. The maximum allowable positive potential between the base and the emitter is limited l\v the safe value of the maximum collector current. The maxi- mum allowable negative potential between the base and the emitter is limited by the saturation voltage of the emitter rectifier. The accuracy of this shunt regulator circuit is restricted by the slope of the characteristic curves for the reference voltage diode. All of the changes in base and collector currents required for regulation flow through this diode and cause changes in the saturation voltage. The ad- dition of the transistor does not increase the accuracy of regulation but only allows adjustment of the regulated output potential to a value which is greater than the standard potential. However additional stages of transistor current amplification minimize the reference potential changes b}' restricting the range of current excursions through the diode. An example of a multistage shunt regulating circuit is given in Fig. 13. REGULATING RESISTOR : — VA- UNREGULATED INPUT VOLTS p-n-p TRANSISTOR SECOND CURRENT AMPLIFIER (INTERMEDIATE SIZE) n-p-n TRANSISTOR SHUNT AMPLIFIER (large Size) n-p-n TRANSISTOR! FIRST CURRENT AMPLIFIER (Small size) Fig. 13 — Transistor sliunt regulator using three transistors. 844 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 3.13. Multistage Transistor Shunt Regulator In Fig. 13 two additional transistors have been added to the simple shunt regulator of Fig. 12 in order to increase the accuracy of regulation. The first stage (subscript 1 is used for the transistor currents in this stage) compares the output potential to the reference voltage, drives the second stage (subscript 2) which in turn drives the third stage (subscript 3). The first stage transistor operates in a similar fashion to the transistor in Fig. 12, except that its collector current now is the base current of the second transistor. The collector current of the second transistor is the base current of the third' transistor. The second and third stage tran- sistors amplify the collector current of the first transistor. The shunt regulating current is the sum of the currents in all three transistors. An examination of Fig. 13 mil reveal that the first transistor is an n-p-n, the second transistor is a p-n-p, and the third transistor is an n-p-n and that no coupling networks are used. This illustrates the advantages of complementary symmetry. In Fig. 13 different sizes are specified for the three transistors. The transistor shown for the first current amplifier might be a 50-milliwatt transistor operating at a collector potential of about 10 volts. Then the maximum base current of the second stage p-n-p transistor should not exceed 5 milliamperes and, with an assumed current amplification of 20 times, the maximum collector current of the second stage could be 100 milliamperes Such a transistor has been developed. With 100 milli- amperes flomng into the base of the large n-p-n transistor and an assumed current amplification of 20 times the maximum shunt regulator current would be about 2 amperes which would compensate for consider- able load current variations. Large size transistors such as would be necessary in the third stage are now under exploratory development within the industry.^ The circuit in Fig. 12 can be modified to use a p-n-p transistor and several other modifications can be made. Similar modifications can be made in the circuit sho^vn in Fig. 13. It is not mthin the scope of this article, however, to show all the permutations and combinations of transistor regulator circuits that are usable. Section 3.2 below covers some typical transistor series regulator circuits. 3.2. Series Regulators Precise voltage control can be obtained with shunt regulators but series regulator circuits are usually more efficient. This comes about because the shunt regulator wastes the shunt current plus the voltage TRANSISTORS AND JUNCTION DIODES 845 drop across the regulating resistance whereas the series regulator wastes only the voltage drop across the series device. At light load the powoi- dissipated in the shunt current is usually greater than the power dissi- pated in the series circuit. With a transistor used as the series regulator device this difference in efficiency is more pronounced because of the small collector voltage that can be used for the full load ciu-rent. This collector voltage is the voltage drop across the series transistor as shown in Fig. 14. 3.21. Simple Series Regulator P'ig. 14 shows a simple transistor series regulator circuit. A p-n-p transistor is shown connected so that all the load current must pass through it. The comparison of the output voltage to the reference po- tential in the current amplifier of Fig. 14 is accomplished by holding the emitter at a constant potential with respect to the positive output ter- minal. Note the difference between this method and that covered in the previous section on the shunt regulator 3.12, where the emitter was held at a constant potential ^^ith respect to the negative output terminal. Now, when the output potential increases by an amount AE, the base voltage becomes more negative A\dth respect to the positive terminal by the proportion of AE developed across points 2 and 3 of the potentiometer. Since the emitter cannot change with respect to the point of reference (the positive terminal), the net effect is to decrease the base to emitter potential and the collector current for an increase in output voltage. The collector current decrease is amplified by the current gain of the p-n-p series transistor to decrease the load current, reducing the output UNREGULATED INPUT VOLTS P-n-p TRANSISTOR SERIES c AMPLIFIER (intermediate size) REFERENCE k VOLTAGE DIODE Ic ADJUSTABLE POTENTIOMETER n-p-n " TRANSISTOR CURRENT AMPLIFIER WITH PHASE REVERSAL (SWALL size) I_ REGULATED OUTPUT VOLTS Fig. 14 — Transistor series regulator constant voltage regulation. 846 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 voltage, and thus regulating it. The value of the regulated output voltage is again determined by the adjustment of the potentiometer. The ohmic value of the R^ resistor in Fig. 14 is selected to keep the cur- rent flowing through the reference voltage diode in its saturation voltage region. Fig. 14 is the simplest form of a transistor series regulator circuit. It requires two transistors whereas the most simple form of a transistor shunt regulator shown (Fig. 12) reciuires only one transistor. But the added current gain of the second transistor in Fig. 14 results in better regulation than can be obtained with Fig. 12. If desired the circuit in Fig. 14 can be modified to change the series transistor to the negative output lead by using the complementary p-n-p first current amplifier and an n-p-n series transistor. This illustrates another advantage of the complementary symmetry of the two types of transistors. Also, if more gain is rec^uired, additional transistor stages can be used employing the principles outlined above. 3.22. Series Current Regulator The circuits covered so far regulate for constant output voltage. Similar transistor regulator circuits can be developed which will regulate for constant output current. One of these is shown in Fig. 15. In this circuit the load current produces a voltage drop across the regulating resistance and, in the n-p-n transistor, this voltage drop is compared to the reference voltage. The difference between these two potentials controls the n-p-n transistor base current and this base current is am- plified by the current gain of both transistors to control the load current. REGULATING RESISTOR I LOAD REGULATED OUTPUT CURRENT Fig. 15 — Transistor series regulator constant current regulation. TI{A\SIST()i;S AXD JUXCTIOX DIODKS 847 'lliis ciicuit is phased so thai tlie load cunciit will he increased when it is too sinall and decreased when it is too large. Tlie values of the regulat- ing resistor and the r(^fer(Mic(> \'oltage deteiinine the value of the regulated load curi'ent. Additional curKMit amplifier stages can be included or the circuit can l)e modified to chaiig(> the series transistor to the miiuis lead as covered above. 3.3. Transislors Combined With Magnet ic Amplifiers 3.31. General Transistors can be used to control directly the flow of power to a load as pointed out in the sections on series and shunt regulators. However, their direct use is limited to moderate voltages (below 100 volts) or moderate currents (up to 1 ampere) with transistors now contemplated. 3.32. T7-ansisfors as DC Preamplifiers In cases where regulation of higher power is retjuired, it is expedient to comliine transistor circuits with other devices having higher power- handling capacity. One type of combination is shown in Fig. 16, where a transistor is used to amplify weak dc error signals to a magnitude suf- ficient for driving a magnetic power amplifier. In Fig. 1(3, emitter (e) of the n-p-n transistor is held at a fixed negative \oltage with respect to the positive output of the power supply by the reference voltage diode ("*S"')- Another negative voltage derived from the output voltage of the power supply through potentiometer (P) is applied AC LINE SATURATION CURRENT LINE WINDING SATURATION WINDING 'V-T' LINE WINDING — ^m- — POWER RECTIFIER MAGNETIC AMPLIFIER REFERENCE VOLTAGE DIODE ADJUSTABLE POTENTIOMETER lb " n-p-n TRANSISTOR CURRENT AMPLIFIER WITH PHASE REVERSAL T REGULATED OUTPUT VOLTS Fig. 16 — Transistor coiitiol circuit tor a magnetic amplifier regulated rectifier with constant voltage regulation. 848 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 to base (6) of the transistor. This latter voltage is made a little smaller than the emitter voltage so that the base (6) is positive with respect to the emitter. Now assume that the load voltage increases for some reason such as an increase in the line voltage or a decrease in the load current. A portion of the increased load voltage appears across points 2 and 3 of potentiometer (P), and tends to make the base voltage more negative. Since the base is slightly positive with respect to the emitter, the net effect of making the base more negative is to decrease the base-to-emitter voltage. Through transistor action, the collector current, which is also the saturation current of magnetic amplifier, decreases and the ac im- pedance of the line windings rises. The line windings absorb more input voltage and the output voltage is brought back very nearly to the original value before the change. The circuit of Fig. 16 is of interest because it can control larger amounts of power than can be handled by transistors alone and, in ad- dition, it is capable of faster regulating action than an all-magnetic regulating circuit with the same loop gain. The use of the transistor in this circuit eliminates the need for one or more stages of milliwatt-size magnetic preamplifiers. 3.33. Increased Gain in Voltage Regulators Additional amplification to improve the regulation can be added to Fig. 16 in two ways. Several stages of transistor current amplification can be added or more magnetic amplifier stages can be used. Of course AC LINE SATURATION CURRENT LINE WINDING SATURATION WINDING LINE WINDING POWER RECTIFIER MAGNETIC AMPLIFIER ^ REGULATING RESISTANCE — vw — REFERENCE VOLTAGE DIODE I LOAD TRANSISTOR CURRENT AMPLIFIER REGULATED OUTPUT CURRENT Fig. 17 — Transistor control circuit for a magnetic amplifier regulated rectifier with constant current regulation. TRANSISTORS AND JUNCTION DIODES 849 a combination of the two methods is also feasible. Additional magnetic amplifiers have the disadvantage of adding time delay. Transistor action likewise is not instantaneous because it takes a finite amount of time to move the charge over a finite distance in the crystal lattice. However transistor action is much faster than the time required to change the current in practical magnetic amplifiers. 3.34. Current Regulators Fig. 17 shows a simple transistor control circuit to obtain constant ciu"rent regulation with a magnetic amplifier regulated rectifier. The operation of this circuit is similar to Fig. 15 and its description will not be repeated. 3.35. Temperature Effects One limitation of the foregoing transistor regulating circuits is the sensitivity of collector current to ambient temperature variations. The collector current increases ^^^th increasing temperature even if the base- to-emitter bias is held constant. This is the result of three factors. (1) 7co , the uncontrolled portion of Ic increases greatly as covered in Section 2.34; (2) the emitter resistance (r^) decreases causing /& to increase, and (3) alpha changes. The effect of the temperature sensitivity of the col- lector can be greatly reduced by using a differential or push-pull circuit of the type illustrated in Fig. 18. 3.36. "Push-Pull" DC Amplifier The pu.sh-puU circuit uses two emitter-coupled n-p-n transistors and is in many respects similar to a cathode-coupled vacuum tube amplifier. AC LINE WINDING MAGNETIC AMPLIFIER I MINUS Fig. 18 — Push-pull transistor control circuit for a magnetic amplifier regulated rectifier with constant voltage regulation. 850 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 A voltage proportional to the regulated output is connected to the base of transistor {Tl). Fixed resistors are sho\Mi in the base circuit of {Tl) in Fig. 18, but a variable potentiometer could be used. The reference voltage diode "s" applies a constant reference voltage to the base of transistor {T2). If the output \'oltage tends to increase, more collector and emitter current flows in transistor {Tl) due to the increase in its base-to-emitter voltage. This increase in emitter current of transistor (Tl) flows through resistor (Rl) and tends to raise the emitter voltage of transistor {T2). Since the base potential of transistor {T2) is fixed, the effect is to decrease the base-to-emitter voltage of {T2) and its collector and emitter currents decrease. The result is an increase in Id and an almost equal decrease in Ic2 • If the two saturation A\indings on the mag- netic amplifier are oppositely poled, the changes in Id and /^o represent a net decrease in the control ampere turn input to the magnetic amplifier. As before, the magnetic amplifier responds by absorbing more voltage. If, however. Id and 7^2 both increase equally due to an increase in am- bient temperature, no net change is made in the control ampere turn input to the magnetic amplifier. Thus if the two transistors are perfectly matched, and the reference voltage diode has a low temperature co- efficient, temperature changes will have little effect on the output regu- lated voltages. A further advantage is the reduced variations in the current through the reference voltage diode. As in the case of the other circuits additional stages of transistor or magnetic amplification can be added to increase the loop gain and the precision of regulation. 4. APPLICATIONS 4.1. General The last sections of this discussion cover some specific applications of the principles discussed above. Section 4.21 covers a one-stage transistor shunt regulated rectifier as a grid battery eliminator for phase controlled thyratron tube rectifiers. Section 4.22 covers a transistor voltage am- plifier circuit as a grid battery eliminator for magnitude controlled thyratron tube rectifiers. A two-volt, three-ampere regulated rectifier covered in Section 4.31 illustrates how a low voltage, high current, regulated rectifier with a transistor and magnetic amplifier control circuit can be obtained. Section 4.32 covers a 65-volt, 200-ampere regu- lated rectifier for telephone central office battery charging. It uses the p-n junction rectifier devices covered in Section 2.1 and a modification TRANSISTORS AND ,11 "\-coiit rolled thyratron rectifier. 852 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 to 115 per cent of its normal value. In the thyratron tube rectifiers, the circuit of Fig. 19 operates into a constant resistance load of several megohms. With such a high value of load resistance, the addition of the compounding resistor on the load side of the regulating resistor does not cause appreciable error. In fact, laboratory measurements on an ex- perimental unit show that the compounding can be adjusted to obtain improved regulation of the thyratron tube rectifier when the grid battery eliminator is used in place of the normal grid battery. This is because the grid battery eliminator can be adjusted to over-correct for line voltage variations and thus compensate for the slight amount of residual line regulation error in the thyratron circuit. The thermistor in Fig. 19 is a shunt element across one of the resitors in the potentiometer and a change of its resistance is equivalent to chang- ing the potentiometer adjustment. The thermistor decreases its resistance with an increase of ambient temperature so it will change the output voltage when the temperature is changed. This output voltage change is opposed to the voltage changes resulting from the temperature effects in the reference voltage diode and the transistor. By selecting the proper thermistor and the proper ohmic values for the potentiometer resistors, these temperature variations will nearly cancel and the regulated output voltage will be temperature compensated. .THYRATRON TO OTHER THYRATRON REGULATED OUTPUT MINUS 1 Fig. 20 — Grid-battery eliminator for magnitude controlled thyratron rectifier. I TRANSISTORS AND JUNCTION DIODES SoS 4.22. Magnitude Controlled Thyratron Tube Rectifiers The grid battery eliminator covered in Section 4.21 is also usable in magnitude controlletl thyratron tuV)e regulated rectifiers but a simj)le, less expensive circuit can be used for this aijplication. It is illustrated in Fig. 20. A simplified schematic of the thyratron rectifier is also shown in Fig. 20 and the grid battery eliminator is the portion of the circuit enclosed by the dotted line. It is actually a transistor voltage amplifier circuit. This type of circuit has not been covered previously in this dis- cussion so its operation is described in some detail below. Referring to Fig. 20 a portion of the output potential is compared to the reference potential by the base and emitter connections to the tran- sistor. The chfference between these two potentials causes the base cur- rent h to flow. This base current is amplified by the current gain of the transistor and it results in flow of collector current Ic , through the Re collector resistance. The voltage drop across Re is the negative grid potential applied to the thyratron tube. Now when the output potential is increased the base current is increased, the collector current is in- creased, the voltage drop across the Re resistor is increased and the negative grid potential at the thyratron tube is increased. This ^^dll dela}'^ the firing of the thyratron and thus reduce the output potential. If the ohmic value of the Re resistor in Fig. 20 is zero the voltage amplification of this transistor circuit will be about 10, or a small change in the output potential will result in about 10 times this change in the thyratron grid potential. This is ^'oltage amplification added to the circuit by the grid battery eliminator and a voltage gain of 10 is more than present circuits can use. The emitter resistance Re reduces the voltage amplification of the grid battery eliminator to reasonable pro- portions. The Rt resistance in the potentiometer circuit of Fig. 20 is wound wdth nickel resistance Anre. Its positive temperature coefficient of resistance compensates the grid battery eliminator circuit for the temperature effects in the reference voltage diode and the transistor. This nickel wire resistance accomplishes the same result as the thermistor in Fig. 19. This is another method of compensating transistor regulator circuits for ambient temperature variations. The "Adjust Output Volts" potentiometer and the auxiliary rectifier shown in Fig. 20 are part of the present magnitude controlled thyratron tube rectifiers. The auxiliary rectifier adds some ac line voltage com- pounding to the rectifier regulation. It is also used with a time delay relay circuit, not shown, to bias the grid potential of the thyratron tubes 854 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195-4 SO that they will not fire during the required rectifier starting time in- terval. 4.3. Magnetic Amplifier Regulated Rectifiers 4.31. 2-Volt, 3- Ampere Regulated Rectifier The control of direct current at low voltage levels has been complicated by the lack of inexpensive low voltage reference standards, and b}^ the very poor efficiencies of most rectifiers at low voltage. The new semi- conductor devices have made important contributions in this field. Germanium diodes with their relatively low forward resistance seem naturally suited for use at low voltages, and the high sensitivity of junction transistors likewise makes them an almost ideal amplifier of small dc potentials. Fig. 21 illustrates the use of junction diodes, junction transistors and a magnetic amplifier combined to furnish a regulated 2-volt, 3-ampere dc power supply. It will be noticed that the circuit of Fig. 21 is very GND Fig. 21 — Two-volt three-ampere magnetic amplifier regulated rectifier. TKANSISTOKS AM) .lUNCTION DIODKS 55o similar to that in Fi.u;. 18, except that an adchtioiial sta<>;(' oi' cunciit anij)h{i('atioii has been added to the basic push-pull ciicuit. Briefly, the reiiulating action is as follows. (1) 'I'he currents /i and 7u r(\spoud in push-pull I'ashion to changes in output \-oltage VI as covei-ed in Section 3. 31), (2) currents /i and h are amplified by the 2-watt n-p-n transistors {T^) and (7^4), (3) the amplified currents (I3) and (74) flow in control windings (Ci) and (C2) of the magnetic amplifier to con- trol the voltage absorbed by the power winding (LI), (4) this action regulates the average value of the voltage rectified by the germanium diode (Dl), thus completing the feedback loop. Tests show that this circuit is capable of ±1 per cent accuracy of the output voltage with a zbl5 per cent change in the line voltage and with load current variations of from 10 to 100 per cent of rated output current. 4.32. 60 -Volt, 200-Am'pere Germanium Rectifier Fig. 22 is a circuit sketch of a 65-volt, 200-ampere regulated rectifier suitable for charging and floating central office storage batteries. This rectifier employs six of the power rectifying cells with forced air cooling FILTER REGULATING REACTORS TO 3 PHASE 60 CYCLE LINE HIGH-LEVEL LOW-LEVEL (2 WATTS) (50 MILLIWATTS) Fig. 22 rectifier. Si.xty-five voH two-hundred amjjore nuigiietic amplifier regulat(!d 856 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 described earlier (Diode IV, Fig. 2), a reference voltage diode (Diode I, Fig. 2), two 50-milli\vatt, and two 2-watt junction transistors. The dc output voltage of the rectifier is controlled by a high gain self saturating magnetic amplifier. High gain in the magnetic amplifier is achieved by using tapewound gapless nickel-iron cores having rectangu- lar hysteresis loops. The control current for the magnetic amplifier is provided by 2, 2-watt n-p-n transistors acting in push-pull. The 2-watt transistors are driven by 2, 50-milliwatt p-n-p transistors also acting in push-pull. The circuit is similar to Fig. 21. Again, the reference potential is furnished by a reference voltage diode. Where the rectifier is connected to storage batteries an additional feature kno^vn as "current droop" is needed to protect the rectifier. The output characteristic of the rectifier ^^dth current droop is sho^Mi in Fig. 23. This characteristic is obtained by coupling a signal proportional to load into the first stage transistor amplifier through a gating circuit. This signal is provided by a dc current transformer which is another form of magnetic amplifier. At currents below the "droop" value the current signal is blocked from the amplifier. At full load the gating cir- cuit allows the current signal to take over and hold the output current constant over a wide range of output voltage. In Fig. 23, the performance 1 AMBIENT TEMPERATURE = 0°C 40°C 60 80 100 120 140 160 180 200 220 240 LOAD CURRENT IN AMPERES Fig. 23 — Output characteristics of experimental 65-volt 200-ampere ger- manium rectifier. TRANSISTORS AND JUNCTION DIODES 857 is shown over the range of ambient temperatures normally (Micouiitered in central offices. 5. CONCLUSIONS It is seen from the above discussion that semiconductor junction diodes and transistors have a wide field of application in power con- \'ersion and control. Certain difficulties remain to be overcome, among which the \'ariation of the device characteristics with ambient tempera- ture appears to be the most troublesome at the present time. It has l)een shown that these variations with temperature can be minimized by two methods. First through the use of thermistors (negative temperature coefficient) or nickel-wire resistors (positive temperature coefficient) and second, through the employment of circuits in which the tempera- ture variations of one element are balanced out by similar temperature variations in a complementary element. Thus, errors due to temperature changes can be minimized by further reduction of the sensiti\nty of the device characteristics to ambient temperature changes and by improved uniformit}' of the devices. Another important aspect of the circuits covered in this paper is their freedom from dependence on auxiliary sources of dc potential. In most cases it is possible to power the regulating circuit directly from the regu- lated output, thereby eliminating the necessity for the transformers, rectifiers and filters usually needed to furnish plate potential for the regulating tubes and voltage standards. The regulating circuits discussed in this paper are of the checkback type. In all of them, there must first be an error in the load voltage to start and maintain the regulating action. The load voltage Avill only return to precisely the original value if the regulating amplifier has infinite gain. These effects, however, are common to all closed-loop feed- back regulating systems. Transistors and junction diodes, at their present stage of development seem well suited for use in checkback circuits having a high quality reference potential, for the feedback principle helps to minimize residual errors due to changes in the device characteristics with, changes in ambient temperature. Of course, the small size, long life and high efficiency of these semi- conductor junction devices will also be very gratifying to the design engineers. 6. ACKNOWLEDGMENTS The authors wish to acknowledge the assistance of C. W. Van Duyne of Bell Telephone Laboratories and the personnel in his group. 858 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 who were instrumental in obtaining the experimental data upon which this paper is based. 7. REFERENCES 1. W. Shocklev, The Thcorv of p-n Junction in Bemiconductois and p n Junction Transistors, B. S.T.J. ,"28, p. 435, 1949. 2. C. L. Rouault and G. N. Hall, A High-Voltage, Medium-Power Rectifier, I.R.E. Proc, 40, p. 1519, Nov., 1952. 3. G. L. Pearson and B. Sawyer, Silicon p-n Junction Alloy Diodes, I.R.E. Proc, 40, p. 1348, Nov., 1952." 4. R. M. Ryder and R. J. Kircher, Some Circuit Aspects of the Transistor, B. S.T.J. 28, p." 367, 1949. 5. R. L. Wallace, Jr., and W. J. Pietenpol, Some Circuit Properties and Applica- tions of n-p-n Transistors, B.S.T.J., 30, p. 530, 1951. 6. R. N. Hall, Power Rectifiers and Transistors, Proceedings of the I.R.E. Proc, 40, p. 1512, Nov., 1952. 7. W. Shockley, Holes and Electrons in Setniconductors, D. Van Nostrand, 1950. 8. K. G. McKay, Avalanche Breakdown in Silicon, Phys. Rev., 94, p. 877, Mav 15, 1954. Wire Straigliteiiing and Molding for Wire Spring Relays By A. J. BRUNNER, H. E. COSSON and R. W. STRICKLAND (Manuscript received Januai-y 19, 1954) The basic design of the wire spring relay departs from c&nventional relay design in many ways. Translation of some of these design departures into commercial relay manufacture has necessitated the development of new machines and new methods because those available were incapable of pro- ducing to the new design requirements. Two developments in this category involved the straightening of large quantities of small diameter wire and the molding of a multiplicity of straightened wire inserts into phenolic resin blocks. The manner in which these developments were reduced from prob- lems to practice is the subject of this paper. Part I — Automatic Wire Straightening Ordinarily wire is received from suppliers on spools or reels. In the spooling operation a spiral bend is placed in the wire which persists when it is iinspooled. For use as a wire spring in the wire spring relay this spooling bend must be removed if the wire is to be positioned with tlie precision required for the desired functioning of the relay. It is necessary', also, to have the wire free of bends if automatic manufactur- ing methods are to be employed. For these reasons, it is important that the nickel silver and silicon copper wire used in the Avire spring relay be straightened as the initial operation in the manufactiu'e of wire block assemblies or "combs" for these relays. Wire straightening can be accomplished by cold working the wire under controlled conditions until sufficient stress has been l)uilt up, particularly at the surface, to make the wire resist bending efforts. The degree of straightness reciuired is governed, of course, by the de- sired performance of the comb in the operation of the wire spring relay. For the 0.022()-inch nickel-silver wire used in the twin wire comb this has been established, for example, as a deviation not exceeding 0.010- inch from absolute straightness measured at the contact end of the comb, 859 860 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 i.e. 2%-inches from its anchorage point in the phenol resin block. This degree of straightness is satisfactory also for the automatic manufacture of relay combs in which a multiplicity of straightened wires are guided into a molding die and positioned so accurately that they can be per- manently imbedded in phenolic resin to the close dimensional limits necessary for ultimate assembly into relays. EXPERIMENTAL WORK Wire Straightening The original experimental work on wire straightening was done at the Bell Telephone Laboratories to aid in establishing the feasibility of a wire spring relay design. After eliminating other approaches it was decided to straighten the wdre in a motor-driven machine by pushing the wire through carefully oriented dies in a rotating head. The mre pro- duced in this manner was known to have a twist but was adequate for making model parts. Subsequently, Western Electric development engineers made a survey of available commercial \\dre straightening ma- chines. A machine was purchased which, while not intended for straight- ening \vire of the small diameters used in wire spring relays, was capable of modification. Among the important things learned from the operation of this machine were first, it is preferable to push instead of pull wire through the rotating die head because of interference at the puller due to twist in the straightened wire; second, much of the twist can be re- moved from the straightened mre by spinning the spool of raw mre counter to the direction of the driven die head; and third, it appeared that a simpler approach than spinning the. spool of raw wire would be to pass the ^vire through a second straightening head rotated in the op- posite direction from the first. On the basis of these observations, a Hawthorne-designed experimental straightening machine was con- structed. This machine featured two die heads independent of each other and counter rotating in operation. Five individual sets of die blocks, wth provision for spacing adjustment as found on the rotary die holder of the commercial straightener, were retained in each head. Subsequently, this experimental machine was used for an extended series of tests to determine such things as the optimum spacing between individual dies, the proper offset from the center line of the head for each die, the best ratio of opposing head speeds, and the maximum rate of mre feed with respect to the rotational speed of the die holder head required to produce straight wire in the diameters employed in the mre spring relay. Since it had become evident by the time this study was well advanced WIRE STRAIGHTENING AND MOLDING FOR RELAYS 861 that a multiple head machine would bo rcHiuiicd, a second experimental machine was built. This machin(% V'l^. I, (Unsigned with the driving mechanism and spacing allowances considered ncn-essary for an automatic multiple head straightener, had onl_v one double head capable of straight- ening a single wire. A major change, to be tliscussed later, was rei)lac(^- m(Mit of the five adjustabh" die blocks in each head l)y a pair of opposing die l)lades contoured to provide a wire passage space between them identical to the predetermined path previously forced upon the ^^^rc by the fi\'e die sets. These die blades were retained by a spindle keyed to the drive mechanism. To accommodate the double head featuie, a rotating unit consisting of two spindles coupled together was employed. Much of the remaining experimental work, such as optimum rotational speed of the spindles, the effect of different configurations of the die blade wire path surfaces, rate of ^\ire feed, etc. was performed with, this ma- chine. Except for minor changes it became the prototype for the auto- matic multiple head machines constructed later. Straightened Wire Storage In contrast to the manual operations needed to assemble the springs and phenol fibre insulators of U- and Y-type relay spring pile-ups, it was planned from the beginning to mold straightened ^\^re into phenolic resin by automatic means so that imit assemblies would be obtained for Fig. 1 — Experimental machine with one double head, prototype of 24 and 30 double head wire straightening machines. 802 THE BELL SYSTEM TECHNICAL JOUKXAL, JULY 1954 the wire spring relay. The labor economy of the latter procedure is obvious. To implement this plan it was necessary that straightened wire be available at the molding press in the (juantities required to prevent loss of molding time. To be successful it was important that interrup- tions to the regular reciu"rence of molding cycles, such as rethreading the multiplicity of wires into molding dies, he kept to a minimum. The original plainnng envisioned a battery of single strand wire straighteners operating continuously to make relatively long lengths of straightened wire. How to store this wire between the wire straightener and the molding press presented the real problem. An early attempt toward a solution involved winding straightened \\'ire on 3()-inch diame- ter reels until sufficient length had been accumulated for eight hours' molding time. These reels would be mounted ahead of the molding press as shown in Fig. 2, and changed at the end of each eight hour shift. Fig. 2 — Sketch showing handling of straightened wire on storage reels. WIKK STUAICIITKNINC AND AlULDINti FOR ]{ELAYS 803 Initial efforts indicated that this procedure was practicable. However, a new shipment of nickel silver wire revealed that, while not detectably different from pre\'ioiis shipments, the new wire took a permanent set on the 3()-inch reels therel)y making it unusable at tlie moUHiis piess. I'rincipally because of the incipient possil)ility of straightened wire accjuiring a set when not stored on flat siufaces, this reel approach was abandoned. Another effort consisted of providing a multiplicity of straight storage tubes, of either metallic or plastic material, into one end of each of which an eight hour supply of a single strand of wire was pushed by tlic wire straightening machine and from the other end of which a molding press would withdraw its requirement of wire (Fig. 3). This was found unworkable l)ecause often the wire straighteners were unable to push the recjuired length of wire into the tubes due to the lead end becoming snarled from twist in the wire. It was decided, finally, to discard the idea of continuously straightening and storing wiic in favor of placing multiple head machines adjacent to the molding presses and operating them only as recjuired. This meant increasing the straightening machine investment because intermittent operation of the straighteners necessitated more wire straightening facilities. A compensating factor was the elimination of investment in storage facilities. It was found that interrupting the continuous operation of the straightening heads had no detectable effect on the characteristics of the straightened wire. Accord- ingly, multiple head automatic wire straighteners are now placed adjacent to the molding press and operated at a speed slightly greater than the wire consumption of the molding dies. Automatic control of the length of a partial loop of wire extending from the wire straightener assures an adequate supply of wire at the molding press. The ultimate length of continuous .straightened wire available to the molding press by this arrangement is governed only by the length of raw wii-e on the spool and is sufficient for many operating shifts. ArTOMATIC MULTIPLE HEAD WIRE STRAIGHTENER Both 24- and 30-doui:)le head automatic wire straighteners have been built by the Western Electric Company. The 24-d()uble head straight- eners are used in making comV)s for the AF, AG and A.I type general purpose wire spring relays and the 30-double head machines for the 286, 287 and 288 type multi-contact relays. In practice the phenol resin molding operation is accomplished in four-cavity dies with the cavities arranged symmetrically about the center of th(^ di(\ Thus two forward cavities face the \nre straightener with the remaining two cavities in tandem. When making the twelve wire single comb of general purpose 864 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Fig. 3 — Sketch showing method of handling straightened wire from storage tubes. relays, half of the wh^es from the 24-double head straightener are guided into each forward cavity. When the twenty-four wire twin wire comb is being made, on the other hand, the entire production of a 24-double head straightener is guided into each forward cavity and two wire straighteners are necessary for each molding press. The 30-double head straighteners are arranged similarly when the fifteen wire single wire comb and the thirty "wdre t^^'in ^\^re comb of the multi-contact relay are being molded. WIRE STKAICHTKNING AXD Mt)LDINr, FOR RELAYS 865 WIRE SUPPLY One spool of raw wire is cradled in the mre straigh tenor, Fig. 4, for each wire required in the molded comb. There are three sizes of wire straightened, 0.0200 and 0.0226-inch diameter wire for the twin wire comb of multi-contact and general purpose relays, respectively, and 0.0400-inch diameter wire for the single wire comb of both relays. The smaller Avires are nickel silver while the 0.0400-inch AAdre is a silicon- copper alloy. All three Avires are in the hard temper range. Originally the wire was pulled from the spools by the drive (pusher) roll of the wire straigh tener. However, the pulling force required varied A\idely from spool to spool. The result was an unequal rate of wire feed through the straightening heads. To avoid this, a capstan with an in- dividual pulley adjustment for each wire was added to the machine. This capstan, in addition to pulling the wire from the supply spools, meters the amount of wire fed into the straightener. An occasional ad- justment of individual capstan pulleys is all that is necessary now to assure production of straightened wire at a uniform rate from every head. WIRE STRAIGHTENING MECHANISM Fig. 5 shows the wire straightening mechanism. Some of the spools of raw wire are visible to the right below the 24 wres, in this instance, being pulled from the capstan pulleys by the grooved shaft mounted just inside the machine housing. This shaft has 24 grooves, one for each \nre, which mate with twelve spring tensioned tA\in grooved wobble rolls underneath to provide the means for pushing the wires through the straightener heads. Both the grooved shaft and the twelve mating rolls are power driven. The wires are pushed through the tubes to the left of the grooved shaft which guide them to the spindles in the straightening heads. These heads, arranged in two vertical rows, make it possible for a common spiral geared drive shaft to rotate all 24 heads at identical speed. This arrangement, however, causes twelve of the heads to rotate clock\nse and twelve to rotate counterclockwise. The opposite twists produced in the upper and lower wires under this circumstance are cor- rected for by the double head arrangement in which the second set of heads rotate counter to the first set. DIE BLADES AND SPINDLES Inside each head is a removable spindle for retaining a pair of con- toured wire straightening die blades. The spindles are suitably keyed 86() THE BELL SYSTEM TErHNICAL JOURNAL, JULY 195-1: Fig. 4 — 24 double head wire straightening machine. to the heads to assure rotation. To conform to the double head design two spindles, joined by a loose coupling, are used. This is illustrated in Fig. 6 which also pictures two sets of die blades removed from the spindle slot. The space between each pair of contoured die blades as positioned for the photograph shows clearly the path of the ^vire duiing its transit of the rotating heads. The die blades maintain the same spacing, offset and length that con- stituted the desired wire path through the five individually adjusted sets of die blocks of the early straightening machines. The continuity of a die blade is accomplished simply by bridging what had been air spaces between the individual die elements and removing enough metal to prevent wire contact in the bi'idging sections. WIRE STH AKJH'I'KNINC; AX1> MOLDINO FOR RELAYS 807 'I'hc (lie lihulcs arc used not (»iily to coii.sc'rN'c space l)iit also to iiiiiiiinizc (lie costs. The hitter is acconii)hsli(>(l l)y making them from inexpensive sheet metal on a puncii press and discarding- them as soon as wear has dc'stroyed their us(>fuln(\ss for wire straightening. I'lilike the indix'idual die blocks used in the previous rotary head straighteners, it is not neces- sary to groove these die blades to direct the flow of wire through the head. There is a slot milled into each spindle to hold the die blades as shown in Fig. (i. The walls of these slots guide the wire and limit its sideways movement in much the same manner as the groo\'es in the indi\-idual die l)locks. The actual thickness of the die l)lade was estab- lished as slightly more than that of the diameter of the largest wire to be straightened for wire spring relay combs. Thus, one slot of uniform width is milled into each spindle allowing interchangeability of spindles regard- less of the diameter of the wire to be straightened. Wire in its transit through the die blades is flexed and burnished to the extent required to produce the desired degree of straightness. It is not rotated during the straightening operation but may acquire twist and even a spiral threadlike burnished appearance from rotation of the die blade surfaces. Fig. 5 — Straightening nicrhanisin ot Ul douhlc head wire si raigtiicniiig niachiiic. 868 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 OBSERVATIONS ON WIRE STRAIGHTENING The straightening operation affects some physical properties of the wire. Tensile strength is reduced about 10 per cent while elongation is increased around 50 per cent. The diameter of the straightened Avire is usually from 0.5 to 1.0 per cent greater than that of the raw wire AAith commensurate loss in wire length. Both straightness and twist appear to be dependent in large part upon the contours of the die blades. Thus far these contours have been determined by trial and error on the ad- justable die block straightener, although .general relationships, especially with respect to wire size, are becoming evident. It is expected that fur- ther study and experience wi\\ establish bases on which contours can be calculated with accuracy. Twist imparted to the mre by the rotating action of the spindle has been found difficult to measure. What is referred to as twist is actually Fig. 6 — Double spindle showing slots and conipU-uuni oi l wo sets of die blades. radial distortion of the wire about its longitudinal axis resulting from partial release of internal stresses remaining in the "wire after straighten- ing. Further release of internal stresses may occur when the wire ends of the twin wire comb are formed before welding, in which event misloca- tion of contacts will result. Fig. 7. This is objectionable from the stand- point both of subsequent manufacturing operations and of relay per- formance. The internal stresses are caused by the crank action applied to the wire surface while it is passing between the die blades in the rotating spindles. Internal stress which is not apparent until after its release, as by forming, has been designated as "residual twist". A rough approximation of the amount of residual twist in wire can be obtained by measuring w^hat has been termed "apparent tmst". Apparent twist is the amount of visible rotation at the end of a wire after leaving the straightener. It can be measured in degrees of rotation per foot of wire straightened. When the apparent twist is Ioav, usually the residual twist also is low. A working range for permissible apparent WIRE STRAIGHTENING AND MOLDING FOR RELAYS 869 Fig. 7 — Photograph showing the affect of residual twist in the upper set of wires as compared to freedom from residual twist in the lower wires. twist has been established which has been successful generally in main- taining acceptable limits on residual twist. There are three variables which largely control the (luality of straight- ened wire. The first is the physical properties of the wire itself. Although a shipment of wire may, on the basis of the sampling method employed, meet specification requirements limiting physical and chemical char- acteristics, an occasional spool or part spool of \\ire can be expected which will be enough outside limits to cause unsatisfactory straightness and unmanageable residual twist. The second variable affecting straightness and twist in wire processed on multiple head machines lies in small differences between the rotating head assemblies. While all critical dimensions of the die blades, spindles, and spindle housings are held to close tolerances, it is possible to obtain an accumulation of dimensional deviations in some spindle assemV)lies of such magnitude as to cause appreciable difference in ^^ire t^\ist and sometimes in wire straightness. It has been necessary, therefore, to provide means for balancing such dimensional variations. The third variable is wear on working surfaces of the die blades. Con- tinued sliding of wire over die blade surfaces eventually produces grooves which decrease offset and increase clearance in the wire path. It has been found in general that, as the die blades wear, t\dst decreases until it eventually reverses direction. Simultaneously, straightness may improve to a critical point from which it rapidly deteriorates. Accordingly, re- placement of die blades must be made before wear has rendered them ineffective. CONCLUSIONS Satisfactory performance of the multi-head wire straightening ma- chines described has been demonstrated dining the pilot plant period of wire spring relay manufacture. Further refinements in the means 870 THE BKhh SYSTEM TECHNICAL JOURXAL, JULY 195-1: for controlling known variables must be made, however, to assure the reliability demanded of heavy duty mass production machines. Part II — Automatic Molding of Wire Spring Relay Block Assemblies Parallel with the effort directed toward development of wire straight- ening facilities, an investigation w'as undertaken by Western to develop automatic facilities for molding an array of straightened wires into small plastic blocks spaced at specified intervals. These blocks were designed not only to hold the ^^ires securely and to locate them accurately but also to insulate them from each other electrically. The design engineers at Bell Telephone Laboratories had decided, after evaluation of the physical properties of available plastic molding materials, that a thermo- setting phenolic type resin would best provide the characteristics needed for wire spring relay block assemblies. Proceeding on this information. Western Electric development engineers reviewed the merits of molding methods adaptable to embedding a multiplicity of inserts, wires in this instance, into phenolic resin. Such economic factors as molding time and material cost were balanced against molding problems like shrink- age and flow characteristics. It appeared from this review that transfer molding offered the most favorable possibilities. It appeared also that the shortest practicable molding cycle might be achieved by preforming the phenolic resin material, preheating these preforms elect ronicalh^ and automatically feeding them into the molding die. Further study of the molding problems indicated that molding presses for this purpose would have to be specially designed, particularly if multica\'ity dies were to be used. The special design features, such as wider spacing be- tween the tie rods, pro\asion of an electronic preform heater, and micro- timing devices, are discussed later as they become pertinent to the description of the machines finally adopted. automatic molding machine Hydraulic molding presses appeared to offer the most ad\antages for this job. Essentialh% these consist of two opposed hj'draulic rams mounted \'ertically; the lower and more powerful ram providing the force required to close and hold closed the split die employed and the upper ram providing the force needed to transfer the phenolic resin in a softened or plastic state into the die cavities. X'nusually wide spacing between the tie rods of the press had to be pro\ided to accommodate the complex progressive die required to make WIRE STHAKiHTENlNG AND MOT.DIXC FoU UKLAV 871 the molding operation automatic. This die liad lo he d(\si;. 15. Aho\-e the transfer area a cyhnchical opening extends \-ei t ic;illy throutih the ujjper die hah" to permit i)assa of molded assemblies is adjustably mounted on the base phiti> of the lower die half, Fig. 17. When the die is closed, the upper details butt against the upper (he plate thereby forcing the cutting shears upon the wire. To reduce the forc(> lecjuired, the cutting blades are so tapered that they cut each wire in succession. Upon termination of the cutting operation, the parts fall free (jf the guide rails into chutes leading to the fiont of the pre.ss, thus comjjleting the molding operation. CONCLUSION The original objective of embedding a multiplicity of straightened small diameter wires in phenolic resin blocks (Fig. 18) on a commercial basis has been accomplished. These wire spring relay parts are being pro- duced at low cost to the required dimensional accuracy in automatic molding machines. P"ig. lb — Wile l)l()ck asscinhlies ;is iii;inut;ictui(' k 'x' 4 6 8 10 12 SEPARATION IN INCHES 16 XlO-3 Fig. 2 — Three sigma limits of Ijreakdovvu voltages between a plane surface and (a) a flat end and (b) pointed end of a round cross-section wire. excess of that which one woukl compute as average heating. As a result \'er3' high temperatures occur in the arc region and material is evapo- rated. This somewhat increases the arc length and the arc moves on to a new spot. As a result of this mechanism some material is burned off in the arcing period and the arc duration is therefore longer than the initiation separation divided by the approach velocity. It is quite diffi- cult to predict the amount of material evaporated. In lieu of more ex- tensive experiments it can be stipulated that the evaporated material should be proportional to the energy input minus the energy directly radiated to the surroundings. It is of interest, however, to note that the phenomenon of evaporation tends to increase the spread of arc duration as computed from the initia- tion separation alone. This is simply due to the fact that a large initiation 890 THE BELL SYSTEM TECHNICAL JOUKNAL, JULY 1954 separation means a longer arc duration, pi-oviding a larger energy in- put and hence increased evaporation. This means that the mo^'ing part has to tra\'el further before the arc is extinguished thus further increasing the arc din'ation. C. Bridging Early experiments indicated a variation in arc duration greater than that which could be explained on the basis of variation of initiation sepa- ration and l)urn-off alone. It was realized that there was a third phenome- non involved. This phenomenon in which metal filaments form and extinguish the arc prematurely will be called bridging. As a demonstra- tion of bridging the two surfaces to be welded were spaced 0.002 inches apart and voltage applied between them. The resultant arc produced a molten filament between them and a weld was formed.* With the ma- terials used the welds produced were somewhat porous and not too strong but tests were much too fragmentary to properly evaluate this process. Electrostatic forces are too small to account for the bridging phenome- non. A speculative explanation on the basis of magnetic forces may be attempted. Let us first examine a luiiform liquid filament carrying a current /. Application of the electromagnetic stress tensor demonstrates the presence of radially compressive pressures equal to: t2 2 p = !?4;, (1) STT-^a" where / equals total current carried by filament. r eciuals radial distance from center of filament. Mo equals permeability of filament material. a equals radius of filament. In the presence of these compressive stresses the filament will tend to elongate. Consider now the two surfaces of the parts to be welded covered with a thin film of molten material. If due to the turbulence caused by the arc a small filament forms on one surface it may tend to elongate in the presence of magnetic forces and bridge the gap between the surfaces. A detailed dynamical analysis of the formation and stability of these metal bridges is quite difficult. The following treatment is a very rough * This may actually be an alternate method of welding with the advantage of offering excellent dimensional control. PKHCrSSlVK WKLDIXC 891 model which will f^how that coiisidcratioii of the maj;iietic forces does yield an explanation \-erifyinji; tlie order of mafj;nitude of the bridf>;in}2; ol)ser\('d ill experiments. The model simulates th(> I)ridi2;iiijj; phenomenon by the translational motion of a small cylindrical filament across the f>;ap between the two surfaces. It is argued that the magnetic energy originally stored in the arc should be comparable to the kinetic energy of the mo\ing filament. The magnetic energy n^siding within a small cylindi'ical filament of diameter d, length (, carrying a current / is: IGtt (2) If such a filament moves at constant velocity through a distance /' in time / its kinetic energy will be 8, = XiM^ = -^ , (3) wh(>r(> p is the density of the filament material. If the two energies are comparable then the bridging time is roughly. ( ~ ^'- 4/i (4) If we assume the following as reasonable numl)ers: / = 1,000 amperes, ( = 3 mils, d - 20 mils, p = 10 gm cm^ then equation (4) gives a transition time of 15 X 10~^ seconds. This figure is of the same order of magnitude as bridging times observed during experiments. The rough model used here is only one stage more refined than purely dimensional analysis but seems to give reasonable agreement with ex- periments. Of importance is the design guide offered by equation (4). By means of proper choice of the welding circuit it is possible to select an arbitrary current versus time relationship. In order to avoid bridging effects, which are, of course, very erratic, the bridging time / should be maximized. By ecjuation (4) the ratio of current to separation should, therefore, be minimized. Stated in words this means that if large cur- rents are necessary for the process (this will be shown to be desiiable 892 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 in a later section) they should be confined to a period when the separa- tion between the surfaces to be welded is quite large. The ciu-rent should be sharply decreased as the surfaces approach each other. DESIGN OF IDEAL WELDING CIRCUIT In the previous section it has been shown that arc duration \\-ill vary over a wide range. This suggests that the system be designed in such a way as to be independent of arc duration. The procedure ^^'ill be to start with the desired temperature versus time relationship of the two opposing surfaces. From this the correspond- ing current versus time relationship can be found and finally the cir- cuit giving such a current distribution selected. Obviously, the "safest" temperature-time relationship is one where the temperature is kept constant at the desired level T. The corresponding current distribution will be derived on the basis of one-dimensional heat flow. Let u = temperature T = desired temperature at surface a^ = diffusivity of material X = distance in direction of heat flow A = cross-sectional area over which heat flow occurs K = heat conductivity of material Vm = voltage across arc i = transient current Start mth the differential equation for one-dimensional heat flow: For the boundary conditions: u = 0 t <0 (6) u L=o = T t> 0 The solution is : (2 rxliaVt ., \ (7) In order to determine the heat input required we must find the gradient at the surface. Expanding equation (7) : PERCUSSIVE AV ELDING 893 du dx du 2T ~ 1 _2avT ~ X^ X* dx Vtt T x=o ay/ir {2aV~ty'^2l{2aV'ty"'\ (8) (9) The required heat input mu.st now !)(' set e(iual (o that suppUed by the weUUn"- circuit:* or 1/17 ^^^^ 2KA T aVm^/T y/t (10) (11) The current time relationship represented by equation (11) is that due to a capacitati^'e transmission line working into a short circuit. Since the arc ^•oltage is considerably low^er than the voltage to which the line is charged, it is substantially a short circuit. - (a) IDEAL CIRCUIT fCAPACITATIVF 1 INF1 - (b) (C) SINGLE SECTION RC NETWORKS \ t 1 1 \ \ \ \ \ \ \ \ \\ \\ \» - Si? - — lf^^^ s. \ "V ^^ ""*— - ^- -i£l '"•-^ \ — "-- t=^ \ \ 1 ■= Fig. 3- — Current time characteristic, (a) Ideal circuit (capacitative line), (b) and (c) single section RC networks. * It will be assumed tliat the energy of the arc is divided {>((ual]y l)y the two opposing surfaces. 894 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 CHARGING CIRCUIT : 80MF IOO>WF Fig. 4 — Practical welding circuit. SELECTION OF A PRACTICAL CIRCUIT It is probably not practical to use a distributed constant capacitative line having a current time relationship as plotted in Fig. 3. The line can, however, be approximated to the desired degree of accuracy by means of a series of r-c sections. The current discharge curve for a single r-c section plotted on the semilog graph of Fig. 2 A\ill be a straight line. Clearly this is not a very good approximation of the ideal curve. If the constants are adjusted such that the desired initial high currents are met, the current will decay too fast allowing the surface to cool prematurely for long arc durations. If the constants are adjusted to match the desired curve for long arc times, the initial heating will be insufficient and short arc durations will produce poor welds. A fairly good approximation of Fig. 2 can be obtained by as little as two r-c sections in parallel (Fig. 4).* Use of this circuit has resulted in considerable improvement not only in the uniformity of the welds ob- tained but curiously enough in the control of arc duration. The reason for the latter phenomenon is that with the multiple section circuit the desired bridging characteristics can be met much more closely. THE MECHANICAL STRUCTURE Relatively little is known about the effect of the mechanical design of the welding apparatus on the process. Basically, the mechanical constants of interest are the mass and velocity of the gun when the arc is being extinguished and the forces propelling the gun. The gun contains kinetic energy part of which is absorbed during the impact of the two parts to be welded. The remaining part will tend to produce rebounding of the gun. Clearly the weld must have cooled sufficiently when the gun draws back such that it can withstand the forces tending to pull it apart. The time allowed for cooling is then roughly one-half the period deter- * The circuit configuration shown is ecjuivalent to two L sections of a lumped constant line as usually depicted. PERCUSSIVE WELDING 895 mined ])y the mass of the gun and the stillness of the statio!iarv part to he welded. The effeets on weld (niality of that hammer blow jjioduced by the gun are not well understood but there is some evidence that this l)i()w may be advantageous in producing intimate mixing. SUMMARY The basic processes of percussive welding haxe been discussed. Large variations in arc duration are caused by the spread in the initiation separation, l^ridging phenomena, and the amplifjdng effect of evapora- tion. The relative spread of initiation separation is minimized by working at high \oltages, in excess of 1,000 volts. Bridging, which causes pre- mature extinguishing of the arc, is minimized by maintaining the ratio of current to separation at a minimum. A welding circuit offering independence from arc duration variations has been developed on the basis of one-dimensional heat flow. The analy- sis presented suggests a capacitative transmission line, which can how- ever be approximated by two or more r-c sections. Greatly improved process control has been effected with this circuit. ACKXOAVLEDGMENT The author gratefully acknowledges the assistance of J. J. Madden in all phases of experimentation connected with this project. Miss L. Mitchell performed the study of breakdown voltage. S. P. Morgan sug- gested the model of bridging time. Automatic Contact Welding in Wire Spring Relay Manufacture By A. L. QUINLAN (Manuscript received January 19, 1954) Welding of precious metal contacts to the new wire spring relay has pre- sented some unusual manufacturing problems. As the name implies, the springs of these relays are wires. The contacts through which electrical cir- cuits are established consist of small blocks of palladium accurately and securely welded to one end of these wires. The wires, arranged in a parallel array, are imbedded for part of their length in molded phenol plastic to form parts which will be designated as "combs" in this paper. There are two kinds of combs, those with the wires arranged in pairs called twin wire combs, aiid single wire combs. Different welding techniques are required, each of which will be described separately. INTRODUCTION The ^^dre spring relay, Fig. 1, was designed wdth such advantages over U and Y t3'pe relays as higher operating speed, longer life, lower power consumption and lower cost, as described in a recent article in this Jour- nal.* The lower cost will be achieved largely by reduction of assembly labor time, by reduction of adjustment effort after assembly due to greater precision in the manufacture of component parts and by exten- sive use of automatic manufacturing processes. To attain these goals the closest cooperation has been necessary, particularly during the design stage, between Bell Telephone Laboratories relay engineers and Western Electric development engineers. Small lots of wire spring relays of several of the early designs were manufactured by the Western to furnish the Laboratories \vith relays for testing. These operations provided Western development engineers with valuable manufacturing experience. The present design of relay has been in production on a pilot plant basis. The contact welders have operated as individual units during * A. C. Keller, A New General Purpose Relaj' for Telephone Switching Systems, B.S.T.J., Nov., 1952. 897 898 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Wire spring relay. that period as contrasted to their inclusion as components in automatic welding and wire forming lines now being placed in operation. The per- formance reported herein is based on individual operation. Part I — Automatic Multiple Resistance Welding of Twin Wire Combs The problem to be solved by Western Electric development engineers was that of welding small blocks of palladium, a precious metal, to flat- tened ends of 0.0226-inch diameter nickel silver wires molded in twin wire combs. Fig. 2. In addition to a secure weld, close limits had to be Fig. 2 — Twin wire comb. CONTACT WELDING IN WIKE ttl'KlNC KELAYS 899 mot on the location of the pnM-ious metal. For example, Fiji;. 5, the upper contact surfaces had to be located with a precision of 0.004 inch. The con- tact itself had to he centered lateially with respect to the straight portion of the wire within 0.004 inch. X'ariahles in the \vir(^ materials, such as elasticit}', make such limits difficult to meet. The wires in the twin combs are oriented in pairs to pro^'ide bifurcated contacts, i.e., the con- tacts on both wires of any pair mate with one stationary contact of the single wire comb to furnish two current paths. Any number of pairs up to a maximum of twelve may be required })j^ a code of relays. To conser^'e precious metal, contacts are welded only to those wires needed for a particular comb. The wires not required are clipped from the comb l)y a hydraulic press operated die in the automatic welding and wire forming line. The 24-circuit capacitor discharge resistance welders, one of which is shown in Fig. 3, were designed and constructed for welding twin wire contacts. As stated before, this welder is one of the units in a welding and wire forming line. Combs from the molding operation are delivered to the beginning of the line in magazines. By fully automatic means they are removed from the magazine, carried by a reciprocating conveyor through each machine luiit in the line and, when fully processed, placed in another magazine. The line of machine units, Fig. 4, forms the con- tact end of the ^^■ires, degreases the wires, welds the contacts to the \nres, coins the contacts to the specified dimensions, forms the termi- nals, clips the ends to length, tension bends the wres, removes unneces- sary ^Wres and tins the terminal ends. REASONS FOR SELECTION OF PROJECTION TYPE RESISTANCE WELDING A capacitor discharge resistance projection welder was chosen for this job for the following reasons: 1. It is capable of welding automatically in a line of other machines. 2. It can pro\'ide the fast rate of temperature build-up required to prevent excessive heating of the small nickel silver wire ends. 3. A capacitor, charged to a fixed voltage, offers a good means of con- trolling weld energy within narrow limits. 4. Resistance projection welding can concentrate heat at the point required. The use of a projection at the welding interface lowers the electrical current needed to a value which can be handled by electrodes without excessive heating. By plachig a projection on the metal with the lower electrical resistance, in this instance the palladium, the temperature of the palladium can be increased in the weld zone as compared to that of the nickel silver wire, thus securing a better heat balance at the joint 900 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 CONTACT WELDING IN WIRE SPRING RELAYS 901 902 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 COXTACT WELDING IX WIKE 81'1{1N(J I{ELAYS 1)03 The projection also confines the location of the weld to a central area of the wire end. When the weld nugget is confined to a central area, cold surrounding metal helps prevent flashing out of hot metal and results in stronger welds. To further centralize the weld nugget and to limit the projection area in contact with the nickel silver wii-e, the wire siu*- face is preformed to a large radius at right angles to the weld bead. WELDER HEAD OPERATION Welding is done in two stages. Head No. 1 welds the odd numbered \\ires of each pair, after which the comb is advanced and head No. 2 welds the even number wires. This is necessary because of the small distance between the wire centers. The electrodes in each head are spaced to match the interval between odd or even numbered ^^^re centers. Fig. 5 shows an isometric sketch of a welding head. The sequence of operations for either head is as follows: 1. The tAnn A\'ire comb is advanced to a locating nest and lowered into position with the reference holes in the plastic engaged on pilot pins. 2. As the comb is lowered, the contact wires enter guide slots of a rake at a point adjacent to the plastic. 3. This rake is moved toward the ends of the wires, thereby spacing and positioning them as shown in Fig. 5. A plastic spacing member is incorporated in the relay for aligning the contacts in the relay assembly. 4. The palladium contact metal, in tape form, is advanced the proper distance to pro\'ide a contact of the specified length. These tapes, parallel \Wth the \Aires, extend from guide slots for a distance slightly greater than a contact length and project over the \\are ends. The tape guide slots, incorporated in a split holder, consist of steel inserts embedded in phenol fibre so designed that when the upper section is shifted laterally with respect to the lower section, the steel inserts clamp or release the tapes. Since the rake is located A\ith reference to the tape guide, accurate spacing of contacts in the relay assembly is secured regardless of minor deviations of position of the wire ends at the welding machine. The phenol fibre mounting electrically insulates the tapes from one another to prevent part of the weld current from passing through adjacent tapes. 5. The upper electrodes, on the end of cantilever arms, are brought do^vn on the palladium-nickel silver wire assembly. 6. The capacitors are discharged and the welds are made. 7. The upper electrodes are pivoted upward out of the way. 8. The ^^^re guide rake is returned to a position near the plastic. 9. The upper shear blade is lowered onto the contact ends clamping them against the loAver electrode. 904 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 10. The lower shear blade is moved upward, shearing off the tapes. 11. The upper shear blade is raised and the welded comb is transferred either to the second similar welder head or to the next operation. The upper electrode tips are pins of special electrode material fixed in the ends of the supporting cantilever spring members by tapered joints, Fig. 5. Each electrode member is insulated electrically from the others and from ground by a coating of Teflon and a slotted phenol fibre guide block (not shown in Fig. 5) at the end adjacent to the elec- trode tips. Teflon furnishes the necessary insulation in a minimum of space and is slippery enough to allow free movement of any member. The cantilever construction makes a very light electrode assembly ^^^th the quick follow through necessary to maintain pressure on the weld area as the projection on the palladium tape is melted during the short weld cycle. A deflection of one-sixteenth inch at the tips vdW produce a force of about ten pounds, which is ample for this welding job. The working surfaces of the electrode tips are dressed approximately every 20,000 parts by an abrasive wheel mounted on an arbor and rotated back and forth by hand with the electrode tips pressed against the fiat side of the wheel. In this manner all electrodes are dressed to a uniform length and angle simultaneously. After repeated dressings have reduced the tip length by approximately one-eighth inch, new tips are inserted. MECHANICS OF THE WELDER The w'elder is driven by an electric motor connected to the main cam shaft through a suitable reduction gear. This cam shaft is located just under the table top and extends the length of the machine. A me- chanical overload clutch is provided on the output shaft of the reduction gear. All cams for the head movements are located on the main shaft. A solenoid operated clutch stops the main cam shaft when necessary. When no comb is in the weld position the weld control circuit is held open and solenoids shift either of the tape feed cams axially out of line with their followers to prevent the feeding of tape. This prevents burning the electrodes, saves precious metal and avoids loose contact pieces which might interfere \^^th succeeding welds. A reciprocating conveyor, extending through the machine about four inches above the table top, carries the combs through the two weld positions. Excessive load, as when a part jams, mil trip a microswitch and stop the machine. TAPE SUPPLY The contact tapes are advanced the amount required to provide the exact contact lengths by means of opposing rubber-faced feed rolls in COXTACT WKLDIXCi 1 X \MKK Sl'KINC KKLAV; U05 ^^ ^ 0.058 _i a 0.057 < H lij < Si BEFORE /'""^L. ADJUSTING/ r\ r \ "> r" \ c L »,FTEP T H-4 y / ADJUSIINt >^ ) ^I^-^ 10 12 14 16 WELD POSITIONS Fig. 6 — Contact length before and after adjusting hardness of rubber feed rolls and omitting gears between them. each head. Fig. 6 shows the results of an early test on feed accuracy. Impro^•ed accuracy was obtained by providing free running tape reels. All portions of the tapes and reels are insulated from each other to prevent loss of weld energy. ELECTRICAL FUNCTIONS Electrically the welder has 24 separate weld circuits. Each circuit, sho^^^l schematically in Fig. 7, includes a capacitor w^hich is charged through its own thja-atron to a predetermined high voltage maintained by a voltage regulating transformer. The capacitor discharges through a thyratron and a transformer to produce the customary low \-oltage- high current weld energy. The transformers are located under the table near the welding heads. The electronic equipment is mounted in cabinets to the right and left of the welder proper, as shown in Fig. 3. The capacitors are located in low cabinets, on either side, together with their solenoid controlled safety short-circuiting system. The cabinet above the welding heads contains control switches and relays. Toggle s^^^tches are provided for cutting off the weld energ}^ for each of the 24 circuits. - — VW WELD TRANSFORMER WELD ELECTRODES Fig. 7 — Schematic of weld circuit. 906 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1 954 a. V) ULi Q. 5^ 34 32 A BEFORE / \ ADJUSTING \ \ k- > / \ A / VA \ J 30 28 26 ?4 PT^ 1 N rd k L-^ ^ ?^ Y\ 1 1 1 / >^ ^ 7 AFTER ADJUSTING 8 18 20 22 24 26 Fig. 10 12 14 16 WELD POSITIONS Weld strength versus resistance adjustment. The welds occur in rapid succession in each head thereby preventing electrical interference between the many circuits involved. When combs which require less than a full complement of contacts are to be welded, the tapes in the not wanted positions are removed from the feed rolls and the toggle switches on the control cabinet are set to direct the weld energy in the sequence desired. To aid in obtaining uniform weld strength from the 24 circuits, all secondary leads from the transformers to the upper welding electrodes are of the same length. To further aid in effect- ing uniform weld strength, rheostats are provided in the high voltage side of the discharge circuit. Fig. 8 shows the result of adjusting these rheostats to balance the weld strengths produced by all circuits. A longer pulse would give the heat produced more time to be conducted away from the weld zone and would tend to heat more of the wire end, to the point, perhaps, of melting it completely. The three millisecond welding time has proven to be satisfactory for this job. However, the shorter the duration of the weld the higher will l^e the current peak re- quired to obtain the necessary heat; the higher the current peak, the greater will be the likelihood of burning the electrodes and shortening electrode life. Adjustment of the pressure on the electrodes can be em- ployed as a compensating factor. As the pressure is increased the tend- ency to burn the contacting surfaces is decreased, but the weld pro- jection on the palladium is flattened correspondingly and the tempera- ture of the weld, for a given current, is lowered accordingly. These variables can be adjusted to maintain optimum balance between uni- formity of weld strength and electrode life. CONCLUSIONS More than ten million welds from this automatic machine have offered convincing proof of its capabilities. Fig. 9 indicates that good weld CONTACT WELDlACi IN WTKE ttPKlNU KELAYS •JUT strengths are being obtaiiuxl. These vahies are shear strengths obtained on a dial indicator type of gage, reading directly in pounds, when the contact is sheared from the wre along the wire axis. A minimum test requirement of ten pounds has been established. Part II — Automatic Percussion Welding of Single Wire Combs Percussion welding is not new. Early work goes back beyond the l)e- ginning of the c(Mitury, but little application has been made of it and only a meager amount of literatiu'e is a\'ailable. However, this method has a real field of usefulness as the application described in this paper will show. The original Vang process, wherein a capacitor charg(Hl to a high potential, often several thousand volts, is discharged across the gap be- tween parts as they approach each other under a propelling force, is a good general description of the method used. The arc so produced heats the abutting surfaces before they collide so that a very thin layer of metal is brought to welding temperature. The propelling force, con- tinuing to act, brings the parts together percussively and the weld is made. Little metal is heated and little heat penetrates the adjoining metal; therefore, the heat balance problem is greatly minimized and different metals weld together with little trouble. There is, however, the problem of protecting personnel from high voltage. Also, the two surfaces being welded must be insulated electrically from each other. This excludes the use of this process for joining the ends of the same piece of metal as in making a ring. The project undertaken by Western Electric development engineers in the case of the single wire combs was the development of a machine for MINIMUM LIMIT " "" " V- \ 1 > 1.:,:^-- 1 _ ■■:■:■; 6 8 10 12 14 16 18 20 22 WELD STRENGTH IN POUNDS SHEAR Fig. 9 — Weld strength distribution. 24 26 28 30 908 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 the automatic multiple percussion welding of contact blocks to the ends of an array of small wires extending less than a quarter of an inch from molded phenolic plastic. This array, fixed in plastic, forms a comb which is illustrated in Fig. 10. WTien completed this comb becomes the station- ary contact member of the ^^'ire spring relsiy. The small blocks of metal on the ends of the wires are cut from a composite tape of which a small portion near the top and/or the bottom surface is palladium. There is a family of combs to be welded, depending on the number and type of contacts required by the code of relays into which each comb is assem- bled. Unlike the twin wire combs, wires which do not require contacts are left in the single wire combs, primarily to facilitate reading terminal locations during wiring into equipment. All top palladium contact surfaces must be located in the same plane across the 12 wire positions of the comb within a tolerance of dzO.002 inch to meet design reciuire- ments. In addition, other dimensions for locating the precious metal must be held to close limits for reasons of precious metal economy. The contact blocks for the wire comb are welded to the wire ends by the automatic percussion welder. Fig. 11, which is a unit in an automatic welding and forming line, Fig. 12, similar to but not identical with the FORMED TERMINALS THE FOUR CONTACT CONDITIONS (l) NO CONTACT (2) MAKE- PA CAP [3) BREAK -PA CAP DOWN (4) TRANSFER -PALLADIUM CAP UP S. DOWN 0.042"-: 0.010" O.IOI" K- I C f^ ^ ^ I I f^- LLADIUM f>r:^ OWN y/ 0.040" Fig. 10 • — Single-wire coml) with percussion welded contacts. Also view of wire spring relay. CONTACT ^VKLDl\(i 1 X WIK'K SI'IM\(! HKLAYS OOO liiu> used I'oi' twin wire coinh inainit'acluic. 'riicsc lines, hy bcin*; fully autonialir in operation, arc iiitcrcsl in^ examples of auton\alion. UKASONS FOU SELIOCTION OF PERCUSSION WELDING Percussion \v(^l(lini>; was selected for tliis proc(\ss for tiie followius reasons: 1. 'Hie electrodes may be placed well away from tlie weld zone. The lesser cuirent lequired for arc wielding as compared to resistance welding makes it possible to conduct this current through the wires without heat- ing them appreciably. The electrodes must be placed away from the wire ends so the clamping force will not deflect the wires from their normal position, thus causing a misalignmerit of contact surfaces. 2. A suitable heat balance in the weld zone can be obtained readily. If the slower butt wielding method were used this would be more difficult because of the unequal size and differing electrical and heat conductivi- ties of the abutting parts. Fig. 11 — Percussion welder lui (unutcio on siugle-vviie cum bO 910 CONTACT WELDING IN WIRE SPHIXG RELAYS 911 Fig. 13 — Experimental percussive welder for welding contacts to a molded comb on a "one at a time" basis. 3. The fast welding time recommends its use in high speed automatic welding machines. EXPERIMENTAL WORK The earliest experimental work was performed on a simple welding fixture with a spring loaded sliding jaw for retaining a contact sized piece of metal and a clamp for holding the wire. Some welds were pro- duced but they varied widely in strength. The next fixture built, Fig. 13, was designed to weld contacts to a molded comb on a one at a time basis. A lightweight spring jawed slider held the contact and guided its travel along a fixed path. This slider was propelled by a lever actuated by a spring and controlled by a cam. The cam was powered by a variable speed drive. An extended study failed to show good weld results except at high speeds when the lever was unable to follow the cam surface and moved 912 THE BELL SYSTEM TECHNICAL JOURNAL JULY 1 954 • \ \\ V \ \ \ \ \ 1 k\ \ \\ d\ N \ V □ ^ / / / / 700V 0.01 Q1 1 5 10 20 30 40 50 60 70 80 90 95 CUMULATIVE PER CENT OF CONTACTS WITH WELD STRENGTHS; 99 99.9 VALUE SHOWN Fig. 15 — Effect of voltage variation on weld strength. 914 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 (xoyddv) SQNnod ni SHisNayis a"i3M roXTACT AVF.T.DlXr; IX WIRE SPRIXn KKLAYS 915 ? 0.08 DEFLECTION r- UNDER IMPACT " OF GUN = 0.013" JL —a/ ^ °!r^K^ trocJ >cicr' 0.020 5:^ -0.8 -0.4 0 0.4 0.8 ARC TIME IN MILLISECONDS 0.010 0 -0.010 -0.020 5x Fig. 17 — Motion of contact during percussion welding plotted from measure- ments on film. evidence of what occurred during the welding operation as could be re- vealed \iy high speed motion pictures. Fig. 17 shows the kind of informa- tion obtained from these films; notably the speed of approach, the amount of burn-off, and the deflection of the comb ^^-ires back of the front plastic block upon impact by the welding gun. In connection ^\ith a study directed toward a better understanding of the mechanism of percussion welding undertaken by E. E. Sumner of Bell Telephone Laboratories tests were made with an electro-capacitive transducer, oscilloscope, and polaroid camera arrangement for the purpose of recording variations in the velocity of the welding gun. The characteristics of the current discharge during welding was recorded bj^ another oscilloscope-camera setup. Burn-off was measured and broken weld surfaces on the contacts were examined and photographed. These tests pointed to the value of a parallel capacitor discharge circuit as a solution to the arc duration variability problem. A commercial trial of parallel capacitor circuits at Western Electric demonstrated the merit of this type of circuit, which will be described in more detail later. Calculations from the transducer traces indicated that the striking force of the contact and contact holder upon impact with the comb \\ire is less than seventy-five pounds. The welding gun was operated on a reed mounting during Mr. Sumner's study. Early experimental work on percussion welding showed that small gas pockets in the weld zone were causing weak welds. Nickel silver was used at that time. Its component of easily volatilized zinc was sus- pected of causing the trouble. Then various metals and alloys were tested and silicon-copper was chosen for the A\ires in the comb. This alloy. 916 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 however, was not suited for use as the base metal of the contact block because it was difficult to weld in the roll welding process used for fabri- cation of the contact tape. A 70-30 per cent cupro-nickel was selected for this base metal because it approximated the resistance of the pal- ladium, a necessary condition for roil welding, and it did not have an easily volatile component to weaken the percussion welds. Another subject investigated was the speed of approach of the parts during the percussion welding operation. The comb and its array of SELECTIVE HITCH FEED TO DRAW TAPE FROM 1 OF 3 REELS AND PUSH THROUGH DIES -."STEPPED ALONG" AS CONTACTS ARE WELDED -0.040" DIA COMB WIRES Fig. 18 — Schematic of the percussion welder. CONTACT W KLOIXC I\ WIKK SPIMXC UKLAYS 017 wires is held stationary and the contact block is moved to the wires to make the weld. The speed of approach is an important factor in con- trolling the arc duration and the impact force. \'arious speeds of the con- tact block carriage from approximately 1 to 80 inches per second were tried. The best welds were obtained by a spring actuated gun which attained a \(^locity at impact of approximately 40 inches per second. AITOMATIC WELDEK OPKHATION The automatic percussion welder contains duplicate welder heads which are miri'or images of each other. Two contacts are welded at a time, one on each half of the comb, Fig. 18. After each welding opera- tion or one cycle, the comb is indexed to the next pair of wire centers. Six cycles complete the welding of tweh'e contacts, or a lessei- number if rf^quired. The guns do not weld at exactly the same time. There is an interval of one degree of revolution of the main shaft between the firing l)oints to prevent electrical or mechanical interference. The welder was designed to select the type of contact, cut it from the tape and weld it in one eycle, to avoid the handling problems associated ^^^th precutting and magazining contact blocks. TAPE SUPPLY AND SELECTOR One of four contact conditions ^\^ll apply for each ^^'ire; (1) no contact, (2) contact ^^'ith palladium cap up, (3) palladium cap down, or (4) pal- ladium cap both up and down. This requires three reels of tape on the right for one head and three on the left for the other. Adjustable knobs on both right and left tape feed cams are set for one of the four tape feed conditions for each ^\•ire position of the comb. Thus, any combination of contact conditions can be set up to make parts for any code of relay. CONTACT SHEAR AND TRANSFER The three tapes enter the shearing die through nidnidual openings. However, only that tape selected by the tape selector is fed into the die and subsequently sheared by one punch stroke. Fig. 18. The tape is punched from such a direction that the base metal is not dragged over the boundary line into the palladium zone. This avoids contamination of the palladium. As the contact is blanked out, the walls of a notch in the punch confine it on the precious metal sides to prevent (Hstortion. The punch delivers the contact to a transfer position at the end of the shearing stroke. There a transfer finger pushes the contact out of the punch notch, through a guide channel and into the waiting gun jaws. 918 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 WELDING GUNS The welding gun is a light reciprocating member that carries two opposing steel fingers or jaws to receive the contact block from the transfer finger mentioned above. The jaw opening is a few thousandths of an inch less than the nominal height of the contact, however, the edges are beveled so that when a contact is pressed against them they spring open, the contact enters and is held securely in place. After welding the jaws are pulled off the contact. At the extreme return travel of the gini any contact which might remain in the jaws because it was not properlj^ welded is removed by an ejector blade. When in the loading position, a portion of the blade stops the travel of the contact through the jaw opening so it is held in a uniform position and A\ill be located on the Anre with precision. GUN MASS CONSIDERATION During welding the comb must be supported accurately to meet the close contact location requirements. It must be supported securely to ^\^thstand the impact of the guns, or weak welds may result. The mass of the gun is important. Evidence indicates that more uniform and higher weld strengths are obtained with lightweight guns. A magnesium gun weighing about 60 grams produced better welds than did the original steel gun weighing about 130 grams. A newly installed steel gun weighing about 30 grams appears to be even more satisfactory. Other guns are being designed and tested to further check various features. A striking force of approximately 75 pounds tends to loosen the ^^dres in the plastic and to produce weak welds. The 60 and 30 gram guns propelled at about 40 inches per second at impact produce less than half this force. The velocity during the arcing period is important to control the amount of heating. One-half cycle of vibration of jaw and ^^dre after impact is the time available for the weld to freeze before a tension strain is placed on it. This time has been measured in the laboratory by the use of a trans- ducer. Weak welds resulted when the time was less than 1 millisecond. ELECTRICAL FUNCTIONS Fundamentally, each weld circuit includes a capacitor which is charged during a small portion of each cycle and subsequently discharged through a resistance in series with the weld. During the charging period, which is controlled by a cam and microswitch, the contact and the wire end of the comb are separated electrically at the weld point. A mul- CONTACT WELDING IN WIRE SPRING RELAYS 919 tiple leaf brush under considerable pressure connects the one side of the circuit to the terminal end of the individual wire beiiifi; welded. Tiie ^un, and through it the contact block, is connected to the other side of the circuit. After a cam frees the gun, the spring propells it toward the wire end and an electrical arc is established by the high potential (900 to SILICON-COPPER WIRE »*»-ili^*!8SaC CUPRO- NICKEL CONTACT . Fig. 19 — Section of a percussion weld. Original amplification 100 X. STRENGTH ♦ --PROPORTIONAL—*! TO ARC DURATION SUITABLE MIN ARC DURATION r*'"' • 1 1 \ : « • 1 • < • • • < < 1 / < 1 • / f / /( 1050V -7^ POUNDS -0. 150" COMBS 21-30, (12-1-52) /■ 1 20 40 60 80 100 120 140 160 180 200 220 240 ARC DURATION IN MICROSECONDS P'ig. 20 — Weld strength versus arc duration. 920 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 1800 volts) just before the parts touch. The arc initiates itself when the gap has been reduced to a few thousandths of an inch. A portion of the abutting surfaces of both the wire and the contact base metal (normally 0.005 inch to 0.010 inch) are melted and expelled in liquid and gaseous states before the molten surfaces are forced together. The arc is extinguished when it can no longer melt and expel metal to maintain a gap. Under good operating conditions it persists from 0.1 to 0.4 milli- seconds. Nearly all of the heated metal is expelled from the joint during the welding operation as illustrated by Fig. 19. This micrograph of a typical sectioned percussion weld shows only a 0.001 inch to 0.002 inch thick layer which was melted or heated sufficiently to change the struc- ture. A small stream of compressed air is directed into the weld area to remove gaseous, possibly ionized, arc products so that they will not Fig. 21 — Schematic of percussion welder circuit. CONTACT WELDING IN WIHK SPUING HELAYS 921 interfere with the mitiation of the arc during- the next weld cycle. Limited tests made with nitrogen and iieliuin atmospheres gave no indication of improvement in weld ciuality. In the course of the many studies employing diricrcnt kinds of in- strumentation, an electronic counter was used to measiu'e successive arc durations. Variations from 20 to 230 microseconds were observed for the prevailing conditions during early tests. The contacts welded were measured for strength of weld and a correlation found between short arc durations (up to 65 microseconds) and weak welds, Fig. 20. Circuit impedance was found to be important in producing uniform weld strength. The assistance provided by the Laboratories in the early work led to better control of arc duration. Hawthorne continued this study and arrived at the circuit, Fig. 21 which greatly improved the uniformity of welds. The improved circuit resulted in less arcing in th(^ jaws, thus prolonging their life and precision. Uniformity of burn-off and of weld strength both showed marked improvement. Fig. 22 shows a typical weld strength distribution for this circuit based on standard PER CENT OF SAMPLE 3500 3000 3 2000 O 500 in 00 — (\J (0 -' GAUGE TOLERANCES -rE7=0 P'ig. 1 — Schematic diagram of electronic relay tester. ELECTRONIC RELAY TESTER 927 cient gain to provide adequate deflection voltage to the horizontal plates of the scope. The oscillator comprises a 6AU6 pentode tube and an associated tank circuit. The tube is connected as an electron coupled oscillator with the energy for the oscillations being supplied by the screen circuit. The plate output is a modulated current that passes through the screen and sup- pressor grids. Considerable isolation of the coupling circuit from the oscillator is obtained by this arrangement. An antiresonance circuit similar to the tank circuit is used to couple the plate of the oscillator tube to the rectifier. Since the electron coupled oscillator acts as a high impedance generator or constant current source the voltage across the coupling impedance is proportional to the value of the impedance. Hence as the oscillator frequency is shifted by a change ill the position of the armature a corresponding shift occurs in the ac voltage across the coupling impedance and rectifier tube (6AL5). The output voltage of the rectifier also shifts with the input to the rectifier since the changes are slow compared with the time constant of the out- put circuit of the rectifier. In normal operation the output of the rectifier may vary from G volts 928 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 £ VW ELECTRONIC RELAY TESTER 929 to 10 volts with 10 volts appearing when the armature is operated. A typical voltage-displacement curve is shown on Fig. 4. This output is applied to the grid of the "zero set" cathode follower and its bias is set so the output will be from approximately —2 volts to +2 volts. This voltage is applied to the horizontal dc amplifier of the scope. Theory of Operation of Gauge Since the capacitance probe is in the tank circuit the frequency of the oscillator changes with variations in the separation of the probe elec- trodes. Hence a displacement of the moving electrode causes a change in the frequency but no change in the amplitude of the alternating current through the pentode plate. The radio frequency voltage that appears across the coupling impedance is the product of the plate cur- rent and the coupling impedance. Since the current is constant in amplitude: E = hZ, where E = rf voltage applied to rectifier, /o = rf plate current from os- cillator, and Z = coupling impedance. Since a change in E is due to a change in the oscillator frequency and the resultant change in Z; Let / = K- = oscillator frequency, 2ir /2 = ;r— = resonance frequency of coupling circuit. 2 T Q = and 002jL'2 f 7=1--. J2 where L2 = inductance of coupling circuit, and R = resistance of coupling circuit. The coupling impedance can be written : At resonance Zmax. = ^iL-iQ. 930 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Sin(!0 the maximum value of E is obtained when Z is at a maximum: Z_ = ^ = ^ (2) provided co/^co2 is near unity. Probe and Grid Circuit of Oscillator The oscillator frequency is determined by the resonant frequency of the tank circuit. Fig. 3 shows that the tank comprises the capacitance of the probe plus a fixed capacitance and an inductance. The probe which is assumed to be a parallel plate condenser has a capacitance which is inversely proportional to the plate separation. Let: d be plate separation of probe, di be separation at resonance (where / = /a), d di Then the variable capacitance of probe is given by: n _ fi di _ Ci l^p — <- 1 -^ — d X and the total capacitance of the tank by: C Cq -{- Cp = Co + — , X where Co = fixed capacitance of tank including fixed part of probe capacitance. Let: then (-9 and 1 LiCc LiCo(l + K) where w and 0)2 are radian frequencies of the oscillator at probe separa- tions d and di respectively and Li is the effective inductance of the tank circuit. It follows that: KLECTKOXIC Ki:i>.\Y TlOSTKlt 931 C02" 1 + K X and tlierefore (3) This expression for Y may be substituted in (2) giving: E 1 1 Em.. VQ'Y^ + 1 For .v/K ^ 1 (4) may be written: E 1 + Q' (4) K ^l + K^Q.(lZ^' (5) Equation (5) gives the ^'oltage across the rectifier as a function of probe separation and the product KQ. It can be used to determine the optimum values of: (1) K the ratio of variable to fixed oscillator capaci- tance, (2) the Q of the coupling impedance, and (3) the range of probe 0 10 20 30 40 50 60 70 80 90 ARMATURE DISPLACEMENT AT STOP PINS IN MILS Fig. 4 — Typical out])ut voltage versus disiilaccmcnt curvo of tlio electrostatic gauge. 932 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 separations over which a prescribed degree of linearity can be obtained. It can be shown that the maximum range of linearity is obtained with {KQY = 2 and that the center of this range is at a: = 3^. Fig. 5 is a graph of (5) against x with {KQY set at 2, SCANNING CIRCUIT The scanning circuit is a high speed electronic switching and detecting device which lapidly selects successive contacts, checks whether the contact is opened or closed, and displays this information on the vertical plates of the scope. Each contact is assigned a particular vertical refer- ence level. When gauging, each level appears as a horizontal line with a vertical step on the line indicating the armature position for the contact operation. Electron tubes designated "C" on Fig. 6 serve as switches to connect one contact at a time to the vertical plates of the scope. Each contact is connected to the grid circuit of a C tube. In order to test a contact its 5 UJ 0.5 0.3 0.2 / ^ ~^ \ s / y /[mf= 2 / / ' / / 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 X=d/d' Fig. 5 — Curve showing E IE max versus x with (KQ)^ = 2 computed from equa- tion (5). ELECTRONIC RELAY TESTER 933 associated C tube is made to conduct by shifting the grid voltage. All of the C tube cathodes are connected together to a common cathode resistor which couples the \'oltages from the contacts to the vertical amplifier of the scope. The plate current that flows when a C tube is fired causes a voltage drop through the common cathode resistor which deflects the scope beam to the proper vertical level. The level is set to the desired value by adjusting the plate circuit resistance. Since the contact under test is connected to the grid circuit, the grid voltage and the cathode voltage are shifted a small amount by opening or closing the contact. That is, when a C tube is fired the vertical plate voltage jumps to one value when the contact is open and it jumps to another slightly different value when the contact is closed. The f 6 C tubes corresponding to the 16 contacts under test are fired one at a time in succession by 16 associated multivibrator stages. The multivibrators are connected in a ring so that each stage is fired by the preceding stage. When a stage is fired it holds its C tube in a conducting condition for two microseconds. Fig. 6 is a detailed schematic of the single stage multivibrator (tubes A and B) with an associated modulator (C) tube. The multivibrators are normally in a stable waiting state and go into a temporary unstable state onlj^ when a transient is applied. Normally the A tube is cut off and the B tube is conducting. When a pulse is applied to the A tube grid the A tube conducts and the B tube is cut off. After two microseconds the A tube reverts back to its waiting state and sends a pulse to the next stage. When the B tube is cut off it provides a flat two microsecond pulse through the coupling circuit to the associated C tube. Each multivibrator stage consisting of the A and B tubes with other circuit elements is mounted on a plug-in turret which may easily be changed in case of trouble. BRIGHTNESS CONTROL CIRCUIT When pulsing a relay the armature remains on the operated and re- leased positions for a relatively long time interval. If the intensity of the scope beam is allowed to remain constant the spots at the ends of the traces are bright enough to fog the entire scope face. This makes it difficult to see the relatively weak lines that are caused by the motion of the armature. Therefore a control circuit is pro^'ided to brighten the scope trace only when the relay armature is in motion. The circuit shown on Fig. 8 provides the voltage required to intensify the cathode beam of the scope. This voltage is obtained by differentiating the output voltage of the electrostatic gauge. As the latter is proportional to the 934 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 +165 V Fig. 6 — Schematic circuit of two switching stages showing modulator tubes (C) and their associated multivibrator tubes (A and B). ELECTRONIC RELAY TESTER 935 armature position, the clifferentiatod voltage is proportional to the armature velocity. After amplification the voltage from the gauge is applied to a tube with a transformer in the plate circuit. The voltage across the secondary coil of the transformer is the differentiated voltage, proportional to the armature velocity. It is applied to a germanium diode bridge, which may be connected as a half-wave or full-wave rectifier by the selector switch. The rectified voltage is amplified and clipped and then applied to the Z axis terminals of the scope. The scope trace is brightened during operate, during release, or during both in accordance with the setting of the bridge selector switch. GAUGING A RELAY The rela}' is plugged into a holding fixture with a jack for the coil and contact terminals. The jack connects the relay coil to the circuit which provides power for operating the relay. It also connects the relay contact terminals to the scanning circuit. The electrostatic transducer electrode is mounted on a bracket which is attached to the front end of the fixture by means of a hinged bracket. After the relay is plugged into the jack, it is clamped and the gauge electrode is rotated into position near the armature. Then the "zero set" potentiometer on the dc amplifier is used to align the operated armature position with the zero marking on the calibrated horizontal scale of the scope. The contact selector switch is used to select the combination of con- tacts to be scanned such as: all contacts, all breaks, or all makes. For convenience in checking relays with 8 contacts or less a switch on the scanning circuit is used to connect 8 of the multivibrators in a ring instead of 16 so that onh^ 8 lines appear on the scope. These 8 lines may be shifted to obtain greater spacing for easier reading. A typical gauging pattern on an oscilloscope screen is shown on Fig. 7. This also improves the gauging accuracy which depends upon the amount of armature motion between successive scanning dots. For 16 horizontal lines on the screen the time interval between successive dots is 32 microseconds. For an armature velocity of 30 inches per second the corresponding distance between successive dots on one line is about 1 mil-inch. If 8 lines are used this distance is about 0.5 mil-inches. The gauging error resulting from this dot definition is actually less than indicated because successive operations of the armature give a small random variation in the dot locations. 936 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Fig. 7 — Photograph of the 7-inch oscilloscope with a typical gauging pattern on the screen. The brightness control switch is used to compare gauging when the relay is operating with the corresponding gauging when it is releasing. An apparent difference of 2 to 3 mil-inches is noticed between the operat- ing gauge point of a contact and the releasing gauge point. This measure- ment change is ascribed to the tip flexure or follow of the contact wire. Contact closure occurs when the contacts first touch, in advance of the short follow travel during which the tension of the contact wire is transferred from the card to the contact. In dynamic operation, the contact opens when it is first struck by the card, before the tension would be transferred statically from the contact to the card. Thus the dynamic contact closure points agree with the static gauging, and differ from the dynamic opening point by the amount of contact follow. Good ELECTRONIC RELAY TESTER 937 correlation is obtained between the static and dynamic gauge measure- ments if make contacts are gauged on operate and break contacts on release. OTHER RELAY CHARACTERISTICS Switches are provided to permit other relay charcateristics to be ob- served on the scope such as armature motion, contact chatter, and the contact operate and release times. The contact gauging is obtained with the horizontal plates connected to the electrostatic gauge. With this setting, the time for a dot to move across the screen may be less than 3 milliseconds (depending on the armature transit time) and contact chatter may be observed during this time. With a time base sweep on the horizontal plates, the armature motion -H65V 119E oJ REPEATER «=5 COIL -^ lre practically plane waves are almost always employed in this connection, the non-reciprocal nature of this effect is so familiar that it haixlly requires restatement here. 939 940 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 By contrast, Part II of this paper deals with devices whose non- reciprocal operation depends in principle as well as in numerical detail on the disposition of the boundary, or, more generally, on geometrical configuration. These devices employ magnetizing fields transverse to the propagation direction. Some electromagnetic field configurations are unaffected by such a dc field, but, whenever the rf magnetic field in the case of a ferrite (or the rf electric field for a plasma) has a component normal to the dc magnetic field, this is no longer the case. For, now, the magnetization (or the charges) will be caused to precess about the dc field, giving extra terms in Maxwell's equations and a resultant change in the propagation. This change may be simply an alteration in phase velocity, the propagation remaining reciprocal. This is the case, for example, for the propagation of plane waves in an infinitely extended medium [Cotton-Mouton effect]. Here, since every direction of propaga- tion normal to the dc field is physically equivalent to any other and, in particular, to the opposite direction, no non-reciprocity can arise. For reciprocity to be preserved in the presence of the dc magnetic field is, however, exceptional and requires a certain amount of geo- metrical symmetry in the system. That non-reciprocity may be expected in asymmetrical systems may be foreseen if we consider a system, typical of those to be treated in this paper, in which all the rf fields are inde- pendent of the coordinate along which the dc magnetic field is pointing. The relevant conducting boundaries and any interfaces between ferrite (plasma) and air are all surfaces parallel to the direction of propagation and lying in the dc magnetic field direction. Suppose the system to be divided into two parts by another surface of a similar kind and examine the surface impedance of one of the parts (which should contain some gyromagnetic material). If the propagation direction be reversed it is necessary to reverse the magnetic field to retrieve a situation in the part considered geometrically equivalent to the original. But, since the precession of the magnetization (or charges) about the dc magnetic field has a definite sense, the magnetic or electric current associated with this precession will be reversed when the dc field is reversed. Thus, the properties of the medium are altered and the surface impedance will be different for the two directions of propagation. In general, the surface impedance of the other part of the system will not compensate for this distinction between the two directions and we shall find different propa- gation constants for opposing directions. An exception will occur if the system contains a surface about which it has geometrical symmetry, for, then, compensation clearly takes place about this surface and the system is reciprocal. GUIDED WAVE PROPAGATION TIlKuUCill GYllOMAGNETIC MEDIA. II 941 An example of a simple non-reciprocal system is indicated in Fig. 1(a). Here a slab of ferrite is inserted into a rectangular waveguide parallel to the narrow walls and closer to one of them. Several workers have demonstrated that this arrangement and a similar arrangement in a circular waveguide are non-reciprocal for what is essentially the dom- inant mode. ' ' When the slab is centered in the guide we have a plane of symmetry and the non-reciprocity vanishes. Another configuration of the transverse field type is represented by the system shown in Fig. 1(b). Here a hollow ferrite cylinder is mag- netized circumferentially and propagates a TEon-mode. It is clear that any arrangement of this sort, which might, in principle, include conduct- ing sheaths, internally or externally, or might have the ferrite extending to an indefinitely large or small radius, cannot have any symmetry MAGNETIZING FIELD Ho WAVEGUIDE ~ WINDINGS FOR MAGNETIZING--, CURRENT PROPAGATION DIRECTION CIRCUMFERENTIAL,-' MAGNETIC FIELD (b) Fig. 1 — (a) Rectangular waveguide and ferrite slab, (b) Circumferentially magnetized ferrite cylinder. * It is expected that an article on this subject by S. E. Miller, A. G. Fox and M. T. Weiss will appear in a forthcoming issue of the Journal. 942 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 about a cylinder coaxial with the ferrite. Thus, the internal and external impedances of such a system at any coaxial cylinder can only com- pensate accidentally (perhaps at a single frequency) and non-reciprocity is the rule. It is freciuently asserted without ciualification that for non-reciprocity a further condition upon the relevant rf field is that its projection upon a plane normal to the magnetizing field be elliptically or circularly polar- ized in the limit of vanishing magnetization. The argument is based on the consideration, in itself correct, that the effective material constants are different for right- and left-circularly polarized field vectors. Suppose that the magnetization direction is y. Then the tensor relating B to H is (see Part I, Section 2.1): Jii 0 Mo 0 ■JK For right- and left-circular fields with H^ = ij/Zx , therefore, the medium is isotropic in the plane transverse to the dc field with permeabilities Iji -\- K, iJL — K respectively. Since opposite circular polarizations accom- pany opposite propagation directions, (see for example. Fig. 2) the per- meabilities, and hence the propagation constants, are different for oppo- site propagation directions. It is then argued that the field must already be circularly, or at least elliptically polarized to start with, if non- reciprocal effects are to result from application of the magnetization. However, the argument is true only for effects of first order in the mag- netization. For general values of magnetization, the rf field, even if linearly polarized to begin with, will become elliptically polarized, and ■//////////////////////////////////////////////////////////////////////////, l;l; { " ^ \\\\ III C"~^~"~^ ' ! l! I M I V /' I M I /Ml ■• /III III I M III I I I I ' I I ' ^///'/■///'A^//// I I I I I I I I I I I ' I nil III', 1 I I ( I II I III ImI Ml, ■/ --//// ^ // - '/■■' ■■///-. ^//////^/. ^r^ Fig. 2 - waveguide. Magnetic lines of force parallel to the broad side of a rectangular GUIDED WAVK 1'U<)1'A(;AT1()\ TllUOl (;II <;^ i;()MA(i.\HTI(' MKDIA. 11 943 noii-i'cciprocity will occur. It is understandable that this double function of the magnetization (conversion to elliptic polarization plus creation of (lit'fereiices in permeal)ility) leads to higher order effects. Ilowev'er, inasmuch as for tlie ferrite and plasma the magnetization can produce large changes, the re(iuirement of elliptic polarization in zero magnetic field cannot l)e regarded as essential in practice. These considerations are demonstrated by a simple example in Section 2.2. The (Unices considered in this paper actually are such that the electric or the magnetic held vector in the plane normal to the held is ellipti(;ally polarized even in the absence of the gyromagnetic medium. For example, the magnetic lines of force of the TEio mode in a rectangular waveguide form two sets of closed loops in a plane parallel to the wide sides (see Fig. 2) and repeating every wavelength. This pattern moves bodily down the guide with the phase velocity of the mode, so that an observer stationary at any point not at the center or at the narrow walls of the guide sees a magnetic field rotating at the signal frequency and tracing out a generally elliptic path. The sense of the rotation is opposite on opposite sides of the center plane, and depends on the propagation direc- tion. The conditions outlined in the previous paragraph are therefore satisfied; introduction of a ferrite slab magnetized as in Fig. 1(a) will yield first order non-reciprocal effects. The problems considered here are such that the electromagnetic fields do not ^'ary in the direction of magnetization. Under these condi- tions the field can be split into a TE and TM field satisfying different wave equations. In general, the two fields are coupled through the bound- ary conditions. Most of the paper is devoted to the analysis of the non- reciprocal helix, a problem that has recently gained importance in connection with high power traveling- wave-amplifiers. The conventional amplifier suffers from a limitation on its maximum useful gain; waves reflected from the output end will make the tube "sing" above a certain critical gain. Ferrites offer the possibility of preferentially suppressing these ])ackward waves and so of increasing the permissible gain by a large amount.* In section 2.3 the "fiat" helix (one of infinite radius) is considered. For the slow^ waves employed in practice a rather complete treatment is possible in this case of planar geometry. In Section 3 the cylindrical helix is treated. Inasmuch as the solutions involve functions for which no extensive tables exist, the treatment has to be more sketchy. * ]\Iore speeificallj-, in high-power traveling-wave tubes the large beam cur- rent employed may t)e above the critical value required forlnickward wave os- cillations due to sjiatial harmonics of the helix structure. In such cases the larger attenuation of tjackward waves will permit a higher beam current and therefore stable amjjlification to higher power levels. 944 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Thus the loss is neglected and only the non-reciprocal phase charac- teristic is considered. The losses have then to be determined approxi- mately by differentiation of the phase characteristics. A few further problems were considered as illustrations of the general principles. One case, that of the plasma filling the space above an im- pedance sheet can actually be solved analytically and provides a par- ticularly clear demonstration of the non-reciprocity. The case of the rectangular waveguide with a ferrite slab has already been considered extensively elsewhere, and only the results for a thin slab are given here. A problem with cylindrical symmetry is taken up in Section 3.3: a cyUndrical waveguide fitted with a circumferentially magnetized cylinder Ph 1 -3.5 -4.0 , o // A J 7 7 7 CM o" / /; / / / I =s ^ X y y y f J — — ^ ^ ^2__ ^ siF ^m ^ = ^ ^ ffl / 0.8 1 0.6 0.4 S /It 7 1 1 L ' sfcfeb rvj d m WW 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09 1.0 I.I 1.2 1.3 1.4 I. cr Fig. 3 — pff versus a for various p. GUIDED WAVE PKOPAGATION THROUGH GYROMAGNETIC MEDIA. II 945 of ferrite. Again the discussion had to be sketchy in ^•io^v of the scarcity of information on the functions that solve the problem. 2. PLANAR GEOMETRY 2.1. Fields and Impedances In this section we consider planar transverse field problems which are characterized by the following conditions. The dc magnetic field is of uniform strength Ho within the gyromagnetic medium and points along the y-axis. All rf field components are independent of the y coordinate. We discuss the ferrite case first, then indicate how the results are to be translated for the plasma. For the orientation of the dc magnetic field which has been chosen the permeability matri.x is of the form: 0 0 no 0 -JK 0 fx and K are, in general, even and odd functions of Ho; the permeability of unmagnetized ferrite is taken to be no as in free space. Following the procedure of Part I we shall assume specifically for n and k the formulae given by Polder's treatment of the dynamics of the medium. Thus, we have the expressions (for the case of no loss): Mo K Mo 1 — pa — 1 - 1 - (72 , and (1) _ K _ p fJL \ — pa — a^ It I where a is the ratio of the precession frequency, '—— Ho , to the signal It I frequency and p is the ratio of a frequency, -~ Mo/mo , associated with the saturation magnetization, Mo , to the signal frequency. It should be noted that p and a have always the same sign. The behavior of fj. and k as functions of a- was shown in Fig. 1(a) and (b) of Part I. pf is shown as a function of a in Fig. 3. The dielectric constant of the ferrite is taken to be e. For reasons given in Part I, \p 1 is assumed less than unity. 946 THE BELL SYSTEM TECHNICAL JOUKNAL, JULY 1954 When the condition, d/(dij) = 0, is put into Maxwell's equations the latter are found to be separable into two sets: -^-^^jo^eE., (2a) oz ^' = >e£., (2b) dx dz dx — jc^noHy , (2c) and - — -^ = - jco[/x//x - jkH,] , (3a) dz ^^= -jc.[>//.+ m//J, (3b) dx dHx dHz J-, frf\ dz dx It is to be stressed that such a separability is possible only when the rf fields do not vary along the dc magnetic field. The sets of equations (2) and (3) correspond to the separate equations for H, and E, which arise from (13) of Part I when /3 is there set equal to zero. The first set de- scribes a TM field of the familiar type, whose propagation through the medium is unaffected by the magnetic field. The second set describes a TE field whose components, because of the presence of k, are con- nected by different relations from those which exist in an unmagnetized medium. The separability of the two fields is equivalent to saying that they are not coupled by the medium itself, but they may, of course, be coupled at the boundaries. We may write (3a) and (3b) in the form • / 2 •2\TT ■ ^-^y ^-^y f 1 N - ja)()U - K )Hx = JK—- - ^x—- , (4a) dx dz - JC0(m - K )Hz = /X-— + JK—- , (4b) dx dz and upon eliminating H^ and H^, , the wave equation for Ey is found to be: d'Ey d Ey 2 f^' — K' „ ,.. -V^ + -^^ + ^ t Ey = 0, (o) dx- dZ' ju where Ey (and also the H's) are evidently propagated in the ferrite as (aiDKD WAVK PHOl'AGATIOX TlinorCII (! VHOMAGNETIC MEDIA. II 947 3.0 2.5 ^0 0 \ \ A \ VJ '^ ^^pl = 0.8 X — 2£_ ^ N \ \ V \ \lpl = o.s \ 0 0.2 0.4 0.6 0.6 1.0 1.2 1.4 1.6 1.8 2.0 \ 0) even when n — k is negative, as ^dll happen for Vl + pV4 - \p\/2 < (7 < 1. Since the medium itself has no non-reciprocal properties it is clear that if the latter are to arise they must do so as a result of interaction between the medium and its surroundings. The boundaries of the ferrite GUIDED WAVE PKOPACATIOX THROUGH GYROMAGNETIC MEDIA. II 949 at which matching will be necessary are surfaces parallel to the 7/-axis and here we need an expression for Hx^nJEy where i/tane is a tangential magnetic field at the surface. For the moment we will consider the admittance looking into the ferrite and take tangential components in the counterclock^^^se sense. From equation (4), then, ■^tang _ Ey dv _ Ey da (7) where d/(dv) is a normal derivative (outward) and 8/ (da), a tangential derivative. It is possible, although by no means essential, to interpret the terms of (7) in the following fashion: the first term is just the admit- tance of a normal TE mode propagating in the interior of the ferrite (which is to have the permeability, ii^s) ; the second term is to be ascribed to an independent surface current, da )(m- — K-) Using this picture one may see how non-reciprocity arises in a simple case. If the ferrite be bounded by the planes x = Xi and a; = X2 , and the 2-variation is of the form e~^^^, the admittances due to the surface cur- rents are -\-JK(3/ 0 and bounded at .r = 0 by a sheet of constant impedance. This impedance is to depend upon frecjuency but not on the propagation constant. It will be written as j\^iJLo/eoZ{oo) where no and eo are free space values. A practical realization of such a sheet might consist of a very large number of similar fins of negligible thickness and separation, parallel to the ?/-axis and attached normally to a conducting plane x = constant. The fields between separate fins are uncoupled and E, is uni- form between fins. For such an arrangement, Z(co) = tan o^y/eofj-'oXo > * See Section 2.2 of Part I. (aiDKD WAVE PUOPACATIOX Til 1!( )r( J 1 1 ( J^' liOM AdNKTIC MKDIA. 11 \)~A where .Co is the depth of the fiius. If the ,c-\ari;i(i()ii of llic lichls is e~^ ~ and the waves are guided, the .r-depeiideiice in the plasma must he as e.\p — V/S- — w-)U(i€eff •<■• From e(|uati()ii (81)) ;^ hitching at .r = 0 gives ■ni^ — e ViS^ — wVoCeff _ „• , /Mo ^/ N This vields /i - r,co 4/ ^ Z(co^ PO 2, 2 I 2 A^U 2,^2/ \ CO (xije + CO — € Z (oj; Co or —7= = ^ Z(co) ± 4 A + "1, Z'^(co) . (11) CO VMo€o fO K €0 €()- Tlie non-reciprocity is clearly exhibited, since the two values of (3 are not equal and opposite. The solution (11) is valid only if IS" > co>oepff , corresponding to guided waves. In the second example we assume that the region between conducting planes at .r = 0 and x = xo is filled by a plasma. When no magnetic field is present E, is supposed to vanish and E^, is uniform across the gap. The unperturbed field is then plane polarized (TEM). The magnetic field is now applied parallel to the y axis to that part of the gap between .r = 0 and x = Xi . E, in the magnetized plasma is now given by E, = £"0 sin Vco-Moeeff — l^-x since it must vanish at x = 0. The z variation is again exp — jl3z. Hy may be found from equation (8b) and is ^y = .. "^^ " oi \-'^^ ^"^ Vco-Moeeff - |S- X C0-)UO€ — P^ - e Vc«j2^o€eff - /3- cos Vw-Mi.eeff - /S^ .t]. The admittance of the magnetized section at x = Xi , is thus, T7^ = ., "^^ — T, hiS - e Vco'-Mi)«eff - /3- eot Vco-M(.feff - /3- a'l], and, analogously, that of the uiimagiiet izcd pai't is ;, [-61 VcoVoCi - iS" cot \/co-Mo€i - |S^ (:^o - Xi)], Hy joi Ez co-/io«i — jS 952 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 where ei = eo(l — (f)- Suppose now that the applied field is weak, so that CO ^loe "^ CO Moci '^ /8 . Then, the cotangents may be expanded and by equating admittances one obtains J^ r?/3 - - J^^i (co-Moei - (3-)(a;o - .Ti) or ^ = 1 + — ' - " ^ ^"^^ -ei + ( '7(8 - — ) (a:o - Xi) (12) Since e — ei is of the second order in a this equation may be solved by substituting the unperturbed value of /?, ± co\/juo€j, in the right hand side. It is clear that although the system is non-reciprocal, it will be so only to third order in a. This system, therefore, illustrates the fact pointed out in Section 1.1 that even when the fields are plane polarized in the absence of a dc magnetic field, non-reciprocity may arise, although it may be very small in weak fields. The third example to be considered is one which has been referred to in Section 1.1, namely that in which a strip of ferrite is placed across the short dimension of rectangular wave guide, see Fig. 1(a). In view of the fact that this problem has been discussed with great thoroughness by Lax, Button and Roth, we shall, after deriving the characteristic equation, consider only the case of a very thin strip. Let the thickness of the strip be 2xo and the distance from its two faces to the nearest guide wall be Xi and x^. respectively. The admittances at the two faces are then — cot ViSo- - jS' xi and -J 2 — cot V/3o'- - /S^ X2 COjUo 1-, = -[<^W - /?'] = -(/3/ - ^'), respectively, where /So = co juoco . Inside the ferrite and 2n -rr dEv ax GUIDED WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA. II 953 Thus, immediately within the ferrite, 1 dEy . ///A ■Liy OX \-/v J/ /external It' the two faces of the ferrite are x = —.To and x = .To we tlieu have (1 f^) = ^ cot V^T^T^. .,, - ,,^ = A, \Jby ax /x=-Xq fXO and (^ ^-f) = -^ cot V^^^-^ X, - p.^ = B. \hy dx A=xo Mo If we write 1 dE ^. „ = Vl3r - /3" tan (V/3r -/32 a;), iiy do; and make use of the boundary conditions we obtain tan2V^V^=..^Vg3J;U-^.) (13) The non-reciprocity is clearly contained in the odd power of /3 in A and B. For small thickness we replace the tangent by its argument, and, sub- stituting for A and B on one side of the equation, obtain ^ [cot V/3o- - ^- ^1 + cot V/3o' - iS^ rcs] = 2.ro[iS/' - /3' - AB], Mo or sin ViSo^ - /32 (xi + a^s) (14) = 2a;o — sin V)8o' - /3^ Xi sin V/3o'' - jS^ rczLS/ - jS' - AB]. Meff Since the guide is almost empty, we may write ViSo' - /32(a;i + 0:2) = TT - 5, where 5 is small. Or, 27r5 ^ = '^i + FT 1—^2 ' i3i(a;i -f 0:2)2 where ft^ = /3o^ - 7rV(a;i + X2)\ 954 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Writing 0 = fSi m the right hand side of eriiiation (14) and noting that if then \/(3(r — (3{^ ^2 = w — 6, we have 8 = 2a;o — sin Meff IS/ _ ^^-^ _ ( ^ cot ^ - p„^i ) ^ cot d - p«/3i Mo /\ Mo =^2.^0^ hi - (1 + PH')liA sin- d - r^j COS- ^ (15) Meff L + 2 -^-^ pjy/3i cos 0 sin 6 Mo The non-reciprocal part of (3 is thus 47r.ro(a-i + x^Y'ph sin 20. This has a maximum value for d = x/4 or 37r/4 and, hence, Xi = (xi + .T2)/4 or 3(.Ti + X2}/4:. This result may be understood ciuahtatively by con- sidering the fields in the guide before the ferrite is inserted. We then have Ey = Ed sin {■Kx/a) where a = Xi + .^2 and consequently, . IT J — jj fi . TTX , jj. a irx Hx = — — sm — and Hz = — cos — ■ . co^t a cofj. a The amplitudes of the left- and right-handed components of circular polarization at a point are then proportional to , _, . TT.T , TT TTX and — /3 sni — + - cos — . a a a The difference in the squares of these amplitudes is 2/5(7r/a) sin (irx/a) cos (Tx/a) and this is proportional to the difference between the energy stored at x in the left-handed wave and in the right-handed wave. It is plausible that this should be a measure of the non-reciprocal effect produced by a thin piece of ferrite at x. 2.3. The Plane Helix In dealing with transverse field problems with cylindrical geometry we shall consider non-reciprocal propagation along a helix which is surrounded by circumferentially magnetized ferrite. The analysis of this problem is rather cumbersome and it is advantageous to study first an analogous plane problem. The simplicity thus gained allows us to examine somewhat more complicated problems. The ''plane hehx," n ■ TT-^ IT irx — ^sni — - - - cos — a a a GUIDED WAVK rii( )1'A(;ATI( )X TllHorciI CVKOMACXKTK' Ml.DI A. II *);>;) to l)e treated here, is a sheet of neghgible thickness, lying in the plane X = 0, which conducts only in a single direction making an angle \p with the //-axis. In this direction it will l)e supposed lossless. In addition we assume that the i-egions, x < 0 and 0 < x < .ro are empty, while the space .ri > .ro is hlled with ferrite. As usual the magnetic field is along the 2/-axis and the fields are independent of //. The problem is clearly the limit for very large radius of that of an empty, helically-conducting c^'linder, surrounded by an infinitely thick shell of ferrite, circumfercn- tially magnetized, the whole system carrying fields with no angular \ariation. We first consider the boundary conditions for the plane helix, after noting that it is evident that both TE and TM fields ^\'ill be required. The tangential electric field on either side of the sheet must necessarily be at right angles to the direction of conduction since the conductivity is infinite. Further, the tangential electric field must be continuous through the sheet. Hence, if the field normal to the direction of conduction is £"0 (omitting here and elsewhere the factor e~^^^), we have ^/ = Er = Eo cos lA, Ey'^ = Ey" = -Eosin^l/, where the symbols -j- and — refer to x > 0 and x < 0 respectively. Again, since current cannot flow normal to the direction of conduction, the tangential magnetic field along the latter must be continuous through the sheet or (F,+ - H~) sin rp + (/// - Hy') cos lA = 0. The boundary conditions may be combined into a single equation, by introducing admittances, in the form Hy _ Hy j:.+ E, (Hi _Hr\ \Ey+ Ey-j ' 77-T - VT- I sill' "A = cos' ^. (16) The left-hand side refers to the TE fields and the right to TM fields. In the empt}^ regions surrounchng the sheet ^-, - (^^ - w-eoMo) H,. = dx' Hy = 0, E. = jweo dx and 956 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 h - ''' - ^»^' Ey = 0, H.= -1 dE. joijiQ dx with /3o^ = co^eoMo • If the waves are to be guided, /3^ > ^o, and, then, for re < 0, we have d/dx = V/S^ - iSo^. Thus Hy~ _ ytoeo Er ~ V^' - /3o^ ' and ^ = _ V<3^ - i8o^ If the admittances at the surface of the ferrite are Hy^/E/ and U^ /Ey\ then Ey'^ /Ez" and U^ /Ey^ are given by the impedance transformation : jcoeo and ^, ^j:^f^'-tanhV^-r^W^.o 1 - V£EW f/ .tanh VW^^JC- X, ^+ JWCO 1 - — jWMo /Z"^ _V/3^ - /3o^ ^/_ •tanh V/S" - iSo" a:o Within the ferrite the TM-fields fall off as exp — \//3- — w^eMo^^ and the TE-fields as exp — V/S^ — co^e/ieff^- We then have and i7/_ — ycoe £.^ V^=^ - oj^cMo' J'<^Meff ViS^ - CoVeff - /Jk/m " These results may now be collected and substituted in equations (IG). The equation of condition so obtained is A + 1 (r - /3o^) A + tanh V/S^ - ^x^ 2 _ .2 1 T> = jSo" cot' i/' ;— -T — / , , (17) ^ I - B tanh ViS^ - j8o'a;o GUIDED WAVE PKOPAGATIOX THROUGH GYKUMA(;\ETK' M1:DI A. 11 957 where and 5 = - €o V^2 _ ^^'. 6Mo Wo shall assume that we are dealing with slow waves (/3 large), 'i'liis is the case of greatest practical interest and is usually ensured by making cot yp large. Assuming that the waves are slow we simplify the equation (12) and find certain values for ^3. We may then ascertain in what ranges of a and p the simplifying assumptions and the results are consistent. In equation (17), then, we replace all square roots by | |8 | and /3 — /3o by /3', obtaining .2 .2 , A + tanh \^\xo 1 - B 00 cot xf/ - with and A -\- 1 1 - B tanh | ^ | rco ^ Meff/Mo M + '^ sgn /3 1 — Ph sgn /3 Mo B= -el fo where sgn /3 = 1 for jS > 0 and sgn ,3 = — 1 for jS < 0. Taking first the simplest case of no separation (.ro = 0), this becomes 2 ^ |8o"(l + e/eo) cot' ^ ^ i3o'(l + e/ep) cot" i/' 1 _(_ 1 - Pg Sgl^ /3 -^ _j Mo Meff/Mo M + '^' Sgn /3 Substituting the expressions (G) and (1) for Meff/Mo and pn we arc led to* ^+ = /3o cot v^. /^l (1 + e/eo) //^^pyq=^ ' ^^^^^ ^_ = -/3o cot ^. /^\ (1 + .Ao) ///_~//^^^ (18b) * Since reversal of the magnetization is equivalent to inlerchanji;e of the propagation directions, we are at liberty to consider a and p always i)ositive, and to deal with the two cases /3 >0 and jS < 0 separately. 958 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 (T + I + p , The l3+ mo(l(> ni-opjiaates for all a, , , , t: decliiiiiiQ; steadily trom 0- + 1 + p/2 I + p ; — , — to unity. The /3_ mode on the other hand is cut ol'f, with B' = 0 1 + p/2 at a = I — p and then reappears with (S" = x at a ^ 1 — p/2. The behavior of the two modes is shown in Fig. 5. Self-consistency requires that /S" » oj'e/io or oo'e \ Heft \, whichever is the greater. The condition /3^ y> co'e I Meff I fails to be met near when (7 = (To = (To 9 r + 1 ^ 2 f + i-f- < P JV , 1 ^ /? _ i\ 0 + ^Ao) cotiA - 1 The condition fi' » w'e/jo will fail for j8_ near p=0.6 p=0.6 p^o.aV V^ 4 P = 0.8 \p=0.6 p=1.\ ^ p=1.0~' 0.5 1.0 ff Fig. 5 — The non-reciprocal propagation constants of the flat helix. The dotted curves represent /3+ , the solid curves /3_ . (Loss free case). produce an appearance of sharpness in the variation of the loss at higher frequencies. If the slow wave assumption be made again it is possible to obtain solutions for the case in which the ferrite does not touch the helix and the latter is lossless. With the slow wave approximation, (17) becomes ^±^ 1 + M + K sgn |8 Mo + tanh I ,8 I x^ /3o2 C0t2 ^ e , , 1 + — tanh I (8 I rco Co M + K sgn /3 Mo + 1 If we write ] /3 [ a'o = t<, then the above equation (expresses /3± in terms of u. At the same time .To = \i/\ 0±(u) \ and we evidently have a para- metric representation of ^^ and Xo . The results of such computations 960 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 /3o -fTT^Tfo COT 5^ Fig. 6 — Real and imaginary parts of the reverse propagation constant /3_ of a fiat helix versus a for various p and for a = 0.1 (the parameter a is marked along the curves). The forward propagation constant has a ver\- small imaginary part which hardly varies with a. are shown in Figs. 9(a), (b) and (c), where /3 is plotted against Xq for various fixed a for two vahies of p. It is to be noted that the introduction of any characteristic length or scale into the problem, such as is provided here by the distance .to immediately produces a great complication in the mode spectrum. The plane helix with the ferrite in contact may be thought of as a highly degenerate problem. To carry out loss calculations using the appropriate expressions for IX -\- K sgn (3 would be very tedious in the separated case. However, it was pointed out in Part 1 that to order a the expressions for /x and k are given correctly if we put a + ja in place of a in the lossless formulae and that, in consequence, for small a, the imaginary part of the propa- gation constant is approximately given by a — • da In Figs. 10(a) and 10(b) the loss calculated in this way for the cases considered in Figs. 9 is shown. GUIDED WAVE PUOPAGATION THHOUGH GYUOMAGNETIC MEDIA. 11 U(i 1 3. CYLINDRICAL GEOMETRY 3.1. Impedances In this section we consider systems witli cylindrical symmetry about the propagation (Urection. All boundaries, those of the circuit as those of the medium, are coaxial circular cylinders, and the medium is as- sumed to be magnetized circumferentially. The practical means for bringing about such a magnetization — for example, thin wires threaded through a C3'linder of ferrite and carrying a dc current, Fig. 1 (b) — are assumed to effect the electromagnetic field to a negligible extent. As in the case of planar geometry, we restrict ourselves to fields which have no \-ariation along the magnetizing field; that is, in the azimuthal direc- tion. Only the ferrite is considered here; the results for a plasma are obvious corollaries. The magnetizing field and the dc magnetization 1.95 , 1.9 — y X^ •^ k. J = 2.0 \ Z--W \ ±p+i-j|pia V±p+2-j|p|a P= SATURATION MAGNETIZATION PARAMETER MARKED ALONG CURVES a= DAMPING CONSTANT Z=/3/(V'+-^/>oCOTi^) \ =0.1 > Vl.6 \ bi.5 1.75 ^ ^ > ^. = 0.3 1.41 P= 2.0 \ r BACKWARD WAVE ^ i.5 ^' 0.5 \ \ P = V = 2.0 \ \ t.2 v° '•°l r < 1 / J 1 / 0.6rf J ^0.8 0^ / p=o O.^J^J^. 5 a ^B 0.2 FORWARD WAVE VARIATION IN PHASE AND LOSS OF FORWARD WAVE IS SMALL -0.6 RJZ,Z Fig. 7 — Real and imaginary parts of /3+ and /3_ for a flat helix at the veiy .small magnetizing field recjuired to saturate the ferrite. 962 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 ^ 1 1 1 1 1 A /^ / X \\ 1.0 1 \ L,, / / , A k / / / a\ ^2.0 / / / A/ a\* / j 1 A w / J J /^ \\\ y. :::: '^ yy X,,^>s>s ^^ _L — g; a- u)^ Fig. 8 • — Attenuation of reverse wave versus frequency at a fixed magnetic field and saturation magnetization for various values of wjj/com = {n!iH)/M. The curves for 0.5 and 1 are discontinued when p reaches unity. are supposed to be independent of radial distance from the cylinder axis. For the geometries employed in practice, this \\\\\ be a reasonably good assumption. Thus it is possible to relate the components of B and H in cyHndrical coordinates (r, ^, z) through the tensor M 0 -JK 0 Mo 0 i« 0 M where m and k are given by the Polder relations (1) and are independent of position. Written in cylindrical coordinates, Maxwell's equations in the ferrite are therefore GUIDED AVAVE PKOPAf! ATIOX TIIKOT^GII CYHOMAGNETIC MEDIA. IT 963 dr 1 a_ r dr -pEr dr = -jo)noH^, 1 '\ - T- rE^ = - juiJLijpgHr + Hz). r dr (20a) (20b) (20c) (20d) (20e) (20f) Fig. 9 — Propagation constants for a fiat helix separated from tlie ferrite by a distance Xo for various values of a. (Loss-free case) (a) /3_ and/3f for p = 0.2, (b) (i- for p = 0.8, (c) 0+ for p = 0.8. In (a), above, the dotted lines bound all o- values. 964 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.^ Xo/3oCOT J^ Fig. 9(b) — See Fig. 9. 1.0 1.1 1.2 1.3 1.4 1.5 As in the case of planar geometry, the field-components can be grouped into TE and T]\I parts; only the TE part will depend explicitly on the gyromagnetic properties of the medium. Equations (20a, 20c) and (20e) determine the TM field. Er and H^ can be eliminated from them, yield- ing the familiar wave equation (JUII)ED WAVE PKOPACATIOX TIIROUCII C YliOMACNETIC MEDIA. II 005 ^ 2.5 2.4 2,3 2.2 2.1 2.0 1.9 1.8 1.7 1 1^ = 10 P = 0.8 \ \ \ \ \ s. \ Nt"' ,,-- CLOSE TO THIS LINE FOR ALL CT N \ .. — 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Xo/3oCOT}^ Fig. 9(c) — See Fig. 9. whose solutions are zero order Bessel functions (or linear combinations thereof) of a kind depending on the region under consideration. Thus if jS > (Si , and the region occupied by the medium includes the cylinder a.xis, the modified Bessel function /o {r\^f3- — /3i^) has to be chosen if the field is to be finite at r — 0; if the medium extends from a finite r to infinity, the function A'o (r\/j3'^ — j3i'^), regular at infinity, is se- lected. Correspondingly, if /3 < |Si , the Bessel and Hankcl function Ju (r\^l3i- — I3-) and Ho^' ' (r -x/jSi^ — I3-) replace h and /vo respectively. In terms of the appropriate solution of (21), the remaining field com- ponents are iS dE^ _ „ juei dE^ Er = -j H^ — — - 0{' - I3'' dr ' "" 01^ - /3- dr The tangential admittance for the T^M field is thus J'^TM — E. 'jSi^ - )32 dr log E, (22) 966 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 ^1 toko 0.5 0.4 0.3 0.2 O.t 0 (a) d =Xo/3oCot;Z' N. "^ == 0.2 0.1 I (b) d =/9oXoCOT^ 1 \ ^ k \ ^ „0^>0.8 0< 0-< 0.15 '-'"^^ 1 1 ^ IT — " ==, ===„ 0.2 0.4 0.6 0.8 1.0 d 1.4 1.8 2.0 Fig. 10 — Ratios of reverse attenuation for a flat helix separated from the ferrite by a distance Xo to the reverse attenuation at Xq = 0. Computations made from the approximate slope-formuhi for the loss, where applicable, (a) p = 0.2. (b) p = 0.8. In (a) all applicable o- values lie in the shaded region. For example, if /3 > /3i , and the medium occupies all space from a finite r to infinity jcoei K o(air) —jcoeiKi(air) i^TM — ai Ko(air) aiKo(air) (23) where ai = V(8- - /3r. The field components of the TE field are determined by equations (20b), (20d) and (20f). Elimination of E^ and Hr from these gives 1 d dH, , / 2 2 PHd\ r dr dr \ r 0, (24) where ^/ = wVei (1 - ph). The term /?/ - jS" in the bracket is already GLIDKI) WAVK TROPAGATION TIIKOUGII GYEOMAGNETIC MEDIA. II 967 familiar from the planar case; it depends on the magnetic; field and on the propagation direction in a pin-ely reciprocal wny. The term pn^/r, however, reverses sign when either magnetizing field or propagation constant changes sign. The solntions of e({iiation (24) are therefore different for opposite propagation directions (or opposite magnetiza- tions). Thus, in contrast to the planar case, where non-reciprocity arose only through the ])0undary conditions, the cylindi'ical case is inherently non-reciprocal. In the absence of the last term in the bracket, equation (24) would be solved by zero order Bessel fimctions, just like equation (21). In the presence of this term, the solutions become confluent hypergeometric functions. Different changes of ^'ariable bring these functions into forms known by different names and notations. One such change leads to Laguerre functions, another to AVhittaker functions. We shall choose the latter representation, since it is closely related to Bessel functions, and numerical tables seem about equally scarce for all representations. In equation (24), let 13" — /3/ = a2 , and let a2r = y. Then it becomes l^/"'-U^^Ji)H. = 0, (25) ydydy \ y / where x = — /Spi//2a;2 . Further, let y = x/2, and H^(y) = h(x)/\/x; then equation (25) becomes which is the standard form of the equation for zero order Whittaker functions. It is a special case of the equation for ju order Whittaker functions : /I 2 \ (27) The solution of equation (27) which is regular near zero is denoted in the literature by MxA^)', it is proportional, in the limit x = 0, to the Bessel function I^(x/2). The solution of equation (27) regular at infinity is denoted by W^.^ix), and in the limit x = 0, is proportional to K^{x/2). The factors of proportionality are found in Appendix I, In this notation, the solutions for Hz are thus M^fiCIaiv) Wxfi{2a2r) * If /3 < /32 , ])oth argument and suffix x are imaginary. These functions are then related to ./o and //o respectivelj'. 968 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Once the appropriate H^ is determined, Hr and E^ are given by (28) /3- — uryLii \ dr (f - -^-0 ^2 - coV^i The tangential impedance for TE-fields is thus ^^- "'^ ^^log//. -pH/3). (29) H, ji0-' - co'uLe,) \dr The reader will recall that for isotropic media (pn = 0) the numerator of the right hand side of equation (29) can always be expressed as the ratio of first order to zero order Bessel functions by virtue of relations Hke Ko (x) = —Kiix), V(x) = h(x) and so on. Analogous results hold true in the present case. Suppose, for example, that we are dealing with a geometry such that the correct H, is Hz = — AC — = Rxo{2a2r), say Then and Hz or 2a2C0n RxO + xR xo i((S2 — co-fiei) R^o It is shown in Appendix I that ^xo' + X^xo = (x - ^) Rxi ■ Therefore ^ _ a^M«o(2x - 1) Rxii2a.r) j{l3~ - o}-fxei) Rxo{2a2r) ' ^ co/ia2(2x — 1) W^,i(2a2r) i(^2 _ ^2^,^) Wx.oi2a.yr) " A similar difference relation shows that if the region is such that Hz = Mx,o{2a2r)/V2a2r, (30) (33) GUIDED WAVK rUorAGATION THROUGH GYHOMAGNKTH' MKDIA. 11 969 then In the uumagnetized case equations (30) and (31) reduce to _ ^ii-MCi\ Kijair) ^ _cojXoKi{ocir) , -,. j(|8"^ - «-Mo€i) Ko{air) jaiKo{air) ' and to upo Iijair) jax Io{air) ' One might be led to believe that the search for sohitions of Maxwell's equations Anth angular dependence e^'"^ will lead to ri'^ order Whittaker functions (just as in the isotropic case this dependence leads to ri'^ order Bessel functions). Such is not the case, however. Unless n = 0, one is led to two simultaneous second order equations for E^ and H^ , and the character of the problem is changed completely. 3.2. The Cylindrical Helix We are now in a position to derive the characteristic equation for a close wound cylindrical helix and approximated by a helically conducting sheet surrounded by ferrite. We confine the discussion to the case in which the ferrite is in actual contact with the helix, P'ig. 1 1 ; the case of finite separation discussed for the planar hehx (Section 2.3) would be too cumbersome here. Losses are neglected. If they are not excessive, they can be deduced from the curves for the propagation constant in the loss free sample by differentiation, as outlined in Sections 2.1 and 4.15, Part I. The boundary' conditions are just the same as in the planar case. In Section 2.3 they were stated in terms of admittances, and it is only necessary to substitute for these the admittances just derived for cy- lindrical geometry. Thus for Hy/Ez we substitute Ftm , and for HJE^ we write Fte = 1/^te • If superfixes i and c refer to the inside and outside of the helix (in Section 2.3 on the plane helix i and e were denoted by " — ", and " + "), the characteristic equation is {F™^" - FTM^^^lr=.oCOtV = (FtE^" - FTE^'^^ro, (34) where ro is the radius of the helix. * The ai)peararice of difTerent factors (2x — 1) and (3^^ — x^) is .siini)ly duo to the way the functions H , M are normalized in the literature. 970 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 For waves bound to the helix, Fte * is to be derived from that solu- tion of (24) which tends to zero ar r — > oo. This solution is TFx,o(2a:2r)/ \/2air, and so Fte^*' is given by equation (30) 1 TV. 1 _ i /3^ — coVei W^,(,{2a2r) Zte Wju q;2(2x — 1) TFx,i(2Q;2r) * Similarly, from equation (23), we have, for bound waves, with E^^ Kq , FERRITE DIRECTION OF CONDUCTION CIRCUMFERENTIALLY MAGNETIZED Fig. 11 — (a) Cylindrical helix surrounded by ferrite, (b) Magnetic Reld lines projected onto a plane containing the axis of the helix. GUIDED WAVE PROPAGATION' TIIliOT'GII GYUOMAGNKTIC MKDI A . 1 1 <)7 I (e) j(^ei Kiicxir) Y ai /vo(air) Inside the helix, we require the solution for an isotropic region which remains regular as r -^ 0. Accordingly ^ (i) _ joco hjaor) . /— — w/xo Iiiocor) ' ?o = w juoeo and „ (i) _ . weo /i(aoro) -» TM —J ao /o(aoro) ' where eo , mo are the dielectric constant, and permeability of vacuum. Combining these expressions in (34), we obtain after slight rearrange- ment hiaoVo) . ooei Kiiain) t ^ .. \ a (35) tan ^ /o(aoro) aieoKo{airo) ]_ Iiiaon) ixa^ (2% - 1) TFx,i(2a2r)_ which determines /3. A complete solution of equation (35) is out of the question. However, as in the planar case, for the slow waves used in travehng wave tube work, the equation may be simpUfied so that solutions may be com- puted rather easily. For electron velocities usually employed the result- ant jS must be about 10/3o . Therefore in equation (35) it wU be permis- sible to neglect all the quantities ^q, ^1 , ^2", w nei , in comparison with 13^, except in the narrow ranges of magnetic field such that n or fi(l — pb) becomes very large. This will occur near dzco where o-q = — p/2 -f ■\/p-/4: -{- 1, and near o- = 1. A solution obtained by as- suming a large ^ must be self-consistent; that is, it can be credited only in regions where it does, in fact, predict large |S. However, in Section 2.3 it was shown for the plane hehx that in any practical case the ranges of magnetic fields so excluded are very narrow, even in the loss-free case, and one may suppose that this is true also in the cylindrical case. For slow waves, each of the a's reduces to ] jS | ; the absolute value sign derives from the fact that the positive square root is implied in the definition of the a's. Therefore Now the suffix x of the "Whittaker functions no longer depends on the 972 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 magnitude of j8, and it is chiefly for this reason that further progress is possible. For large /3, equation (35) can be written iS' = /3o' cot' ^ /i(| /3 I ro) 61 Ki(| 13 I ro) ToOlko) 6oKo(|^|ro) /o(| ^ I ro) 1 PF,.o(2|^|n)) • (37) /i(l /3 ko) M. .2^ _ j^ T^x.o(2 I /3 I ro) Mo where x is now given by equation (36). Equation (37) is now solved by -7 -6 -5 O ^5. -3-r: -2 •^ 1 O u o Q. 2 \ \ / p=0.2 r / 0- = 2^94 _ "5 d II in 00 d f ^ ^ 0.95 1.0 M c^--"^ == 1.2 V. rl.5,r0.1 6.?-^ / ^=^ 0.6 0.75 i 0.79 *> ^'•*«.. '**««= ===== _02_=1X)_ O.T' ^^^« =^«- 0.2 0.4 1.0 1.2 1.4 1.6 0.6 0.8 ToyGo COT Tp Fig. 12 ■ — Reduced forward and reverse propagation constants versus reduced radius of a cylindrical heli.x (loss-free case) for various a and p. The range en < (7 < (To where = /jA+ pz and /' + M contains an infinity of shape resonances and is not shown here, (a), above, p = 0.2. (b) p = 0.6. (c) V = 1.0. (UHDED AVAVE P1U)PA(..\TI()N THHOICII (!YU()MA(iNKTl(' MEDIA. II 973 ^ -4 Q. -3 <^ 1 1, P=0.6 1 ^ ^ r^b^ rA4 1 in d o d CD d U3 d / o d 1 . / f ::;=: == ^=0.85 0.90 d \ k i /// . ^ 1.5 ^ m i ^S— - ^^ ■ 0.2 0.3 0.39 N ** — — . ^---r----. _£r = 1.0 0~ 0.4 0.6 0.8 1.0 ToPoCOT^ Fig. 12(b) — See?Fig. 12. 1.2 the following procedure: Introduce the parameter = I /3 I ro . (38) For a given pu and m (or a and p), and a given sign of /3, each value of w determines /3 through equation (37), and then ro through equation (38). Thus /3 can be plotted versus ro . The procedure is repeated for the op- pasite sign of /3 (and therefore the opposite sign of x)- A different curve of jS versus ro is then obtained. Thus for a given value of ro , the "forward" and "backward" propagation constants are different in magnitude. The results (computed for a typical ratio ci/to = 10) are conveniently stated in terms of |8 = /3/(/3o cot i/') and fo = ro/3o cot i/' and are shown in Fig. 12(a) to (c), and again, for fixed fo , in Fig. 13(a) to (e). We note that for fo in excess of about 1 .5, the results are almost the same as those 974 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 -7 -6 ■a. C^ -3 -2 p=i.o 1 W c o r X ^ 3^, ^x.o , and therefore Z^ , has a zero which increases from w = 0atx = j^tow= latx =^'^- Accordingly Z^iu) in ^ > ,f<^ V \ T^ >\' V \ \ \ \\ \ Fig. 15 — The function Zx(u) versus x for various w, in the range — % 3^ starts from 0 at u = 0 with a downward-directed vertical tangent, achieves a negative minimum, then increases, through its zero, to the asymptotic value unity as w -^ co. The minimum becomes deeper as X approaches %. At x = /4, ^x.i('^) developes a zero at u = 0, which steadily moves to larger u as x increases further towards %. At the same time the zero of TF^.o already discussed moves from 1 to 2 + \/2, and a new zero arises at m = 0, x = %, which increases to 2 — ■\/2 as X approaches ^^, but which lags behind the zero of Wx,i(u). The function Zy.(u) now has a pole and two zeros, and behaves as shown in Fig. 14. This process continues; each time x passes (2n + l)/2, a new zero and a new pole appear. (For a detailed list of poles and zeros the reader is referred to Appendix I). To apply these results, we first resort to the Polder relations. In terms of a, p, we have, for /3 negative and 1 x = M p 1 — p(T — a^ 1 - <7 ^(2x-l) ^^-1 + ^ Mo = A, say. 978 THE BELL SYSTEM TECHXICAL JOURNAL, JULY 1954 The characteristic equation is /i(u/2) 61 lUun) Io(u/2) "^ Co Ko(u/2) hiu/2) h{u/2) I /S I fo = -u/2 - AZ^{u) (37) and can now be discussed in terms of o- at a fixed p. Suppose that p < 1. Then A is negative for o- < 1 — p- In the same range, x < M, so that Z behaves essentially as Ko/Ki . Therefore both numerator and denomi- nator are positive for all u, and the ratio tends to the planar result 1 + €0 1 -A asw-^ oo.Asw— >0, ^ tends to zero along the vertical, as can be shown by an examination of the various functions near u = 0. For a < 1 — p, the course of the 0^ versus ^/-curves is as shown schematically in Fig. 16(a), and it is easily seen that the ^ versus fo curves run in essentially the same way, Fig. 12(a) to (c). However, as a approaches 1 — p, the ^ versus u curves steadily fall, until at a- = 1 — p, j8^ = 0 for all finite u, since A = — oo . As 0- passes 1 — p, A changes sign and at the same time x passes 3^ so that Zx acquires a zero. As a varies from 1 — p to 1 — p/2, A decreases from + oo to unity. Therefore, while u < Ui , the zero of Z^ , 1 — AZ^ is positive; however as u increases beyond Ui , {Iq{ii/2)/Ii{u/2)] — A2iy{u) eventually passes zero, since ZJ^u) and 7o(w/2)//i(w/2) both approach unity. On the other hand the numerator of equation (37) is ZERO OF D = t X<4r (a) (b) (c) Fig. 16 — Schematic variation of /3_ with m. a) x < K; b) ]4. < x < H',c) ^ < X < %. GUIDKD WAVK PROP AOATIOX TIIIJOUOII (nHOMAGNKTIf MEDIA. II 979 always positive; therefore jS" approaches infinity at h{u/2)/Ii{u/2) — AZ^{u) = 0, and no real vakies of ^ exist thereafter (see Fig. 1Gb). Since this "cut-off" occurs at a finite \'alue of u, the corresponding value of To is zero. This explains the bulging of the corresponding j8 — 7"o curves in Fig. 12(a) to 12(c). The next major change in the curves occurs when x exceeds ^, (that is, a exceeds (Tl V^+M-) For p < 2, cTi is still less than 1 — (p/2), so that, initially at anj'' rate, A is still greater than unity. In addition to the infinity of /3^ just dis- cussed, a further infinity arises between u = 0, and the pole of Z^{u), as is seen from Fig. 16(c). jS increases from zero at w = 0 to this infinity, thereafter it is negative, until the pole of ZJ^u) is reached. Thereupon it resumes at |8" = 0 and approaches infinity at the zero of the denomina- tor [/o(i//2)/7i(w/2)] — AZ^{u) already discussed. Thus there are now two branches of the ^^ — u curve; their corresponding traces in the j8 — f 0 plane are shown schematically in Fig. 17. (The computations on which Fig. 12(a) to 12(c) were based were broken off at a = o-i .) A further branch is added each time x increases beyond a number of the form (2n + l)/2 (o- increases beyond 2 4A + V 2n + 1 These all resemble the two branches just discussed, until n > Fig. 17 — Schematic variation of /3_ with ro for K < X < %• 980 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 1 (a) ANCHES 1 ER BF /3pLANE HELIX / FURTH - BRANCH^ y^ TO / y^ /^PLANE HELIX Fig. 18 — (a) &- versus u when 1 — (p/2) < o- < ao . (b) /3_ versus ro under the same circumstances. }/>{- — 1). When this occurs, o-„ > 1 — (p/2), so that there will be a V value 0- between cr„_i and o-„ beyond which the denominator no longer decreases through zero as li -^ qo , but approaches a finite positive value, Fig. 18(a). Accordingly /S" approaches a finite positive value, and cut-off of the extreme right hand branch. Fig. 16(c), no longer occurs. The cor- responding j8 versus fo branch is as in Fig. 18(b). As n — >• 00 (o-„ -^ o-Q = \/l + (pV-l) ~ p/2) the number of branches increases to infinity. This situation resembles that in the completely filled waveguide (Part I), where we found an infinity of modes ("Shape- resonances") in the range o-q < tr < 1. In the present case, however, they are to be found in the range 1 — p < a < o-q • When cr = (To + 0, X is infinite and negative. The function Z^{^ is then constant and equal to unity. A is less than unity, and the denomi- nator of equation (27) has no zeros. The ^ versus fo curve is now "nor- mal" again, see Fig. 12(a). As a increases further, the curve falls (since A decreases steadily to — 1 as o- -^ =o), and no more quaUtative changes occur. 3.3 Cylindrical Waveguides As pointed out before, the fact that the propagation problem in the cylindrical case can always be integrated in terms of Whittaker functions when the fields show no angular variation is an accident, and in view of the lack of numerical tables, not a particularly fortunate one. Only in special cases (like that of the slow-wave helix) is the text-book informa- tion on these functions of any great utility. In general, it will be more convenient to solve the differential equations numerically. However, for completeness, we shall state some of the formal results for a cylin- drical waveguide containing a cylinder of circumferentially magnetized ferrite, and propagating a TEo mode. GUIDED WAVE PROPAGATION TIIHorOH G VHOMAGNETIC MEDIA. II 081 First Ave consider a waveguide, radius /n , into which is fitted a hollow cylinder of ferrite, outer radius /o , inner radius ri . In that cylinder, the magnetic field //; maj^ be taken to be a superposition /l*S'xo(2Q:2r) + BR^o(2air) where, as before. a2 = /3" - co-€i/x(l - Ph); S{x) = --^ ; R{x) = — /^ 0Ph 0:2 may be either real or positive imaginary. [In choosing this combination we depart from the usual practice of taking a superposition of Jo and Nq in the isotropic case. Were we to follow this practice, it would be necessary to define a new function R^f.CIjx^e^'''"'''^ + Ry^^{ — 2jx)e~^'''^ to correspond to N^(x). Our choice corresponds to taking a combination of Joix) and one of the Hankel functions Ho(x) in the isotropic case. Since the functions H, J, N are linearly dependent, this will not affect the results.] In view of the difference relations, equation (39) in Appendix I, and of equation (29) we obtain for the impedance in the ferrite E^ ^ >M«2 U(i^ - x')S^ii:2a-2r) + B{2x - l)i?,i(2ct2r)] H, (J3'' - coVei) [AS^o{2a2r) + BRA2a2r)] A and B must be adjusted so that this quantity vanishes at ro , the guide wall. This gives = -(2x - mi - xO ]r,.i(2«2ro).¥,,i(2«2r) - J/,.i(2a2ro)Tf,,i(2c.2r) (2,_i)TF,,i(2«2ro)ilf„o(2a2r) - (i^ - x')M^A2a,ro)WM^a.^) ' In the vacuum, between r = 0 and r = ri , //j is /o(ao?') and the im- pedance is EJH^ = —j(u:ixo/ao)Ii(aor)/Io(aor) where ao" = jS^ — coVeo • At r = ri , the ferrite-vacuum interface, the two impedances must be equal. Thus we obtain the characteristic equation Mo /i(ao'*i) A-, iN/i/ 2n a-M = (2x - l)(M - X ) ao loiaoTi) (3- — w-yuci ir,.i(2a2ro)Mx.i(2a2ri) - M ^ A2cc2ro)W ^ A'^a^n) (2x - m\A2a,r,)M^,o(2a,n) - (M - x')^1/.,i(2a2rn)Tr,,n(2a2/-i) ' (It is understood that for "normal" waveguide propagation a^ will be imaginary, and the / will be replaced by ./). As a simple illustration we 982 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 consider the case in which the ferrite cylinder fills the waveguide com- pletely. E^ is then proportional to Tlf xi(2«2^), and so ^ is determined from the condition Mxi(2a2ro) = 0. When "normal" propagation prevails (l3 less than the natural propaga- tion constant of the medium co\/ei/x(l — pa^)) both 02 and x are imagi- nary, equal to jai' and jx', say. Under these circumstances little is known about the zeros of M. However, it is possible to say something about the solution for large radial mode numbers. It follows from Erdelyi et al.^ 1, p. 278, formula (2), that for large argument MjY.ii^M'n) = const •sin[a2'ro + x'log a2% + x'log 2 + $ (xO - 7r/4]. where $(x') = arg r(% -f jx')- The zeros of this expression are at ^2 To + X log Q!2 ro = — - — X log 2 — $(x ). n = a large integer This equation may be solved graphically by setting 0:2^0 = u, assigning values to ^, ps , u (and hence to x', 0C2). From a solution u one then finds n = u/a2. M also has zeros for real a^ , x, if X is large enough. Thus the wave- guide Mdll support waves with a 13^ greater than coVei(l — p/)- It is shown in Reference 2 (1, p. 289) that when x is between ^ and %, M has one zero, when x is between % and J^, M has two zeros and so on. Suppose that pn is negative, = — \ ph\ . Then ^ /3|ph| For real positive j8, this equation has a solution for 0 ii \ ph \ < x' Vx' - Ph' ' ^^ % < X < /4i M will have a zero ?/(x) depending on the value of x- Thus the equations Vx^ - PH^ V/5^ - iS2^ I Ph I 182 solve the propagation problem parametrically. Similarly when x is between % and J^, there are two zeros of M given by two functions GUIDED WAVE PROPAGATION' TIIHOTTGII G VHoAIAGNETIC MEDI.V. II US.S ?/i(x) and W2(x)- There are now two possible modes, with the same restrictions on p,i . An additional mode arises each time x is allowed to patss a number of the form (2n + l)/2. It is to be noted that those modes are not confined to the resonance range. For /3 positive, they can exist in the range o- > ao and in the range — o-q < o- < 0. Appendix I. Some Properties of Wiiittakek T^tnctions Used in This Paper I. relation to bessel functions if/-!^i^ = 2'«r(M + DUr), Lit rp— — / — , x-»o v2a; Vx y ^ X^A^ -^1^) V TT -M+lrr (1)/ \ j V2jx V -2jx J Lt x-o L II. difference relations The foUoAving results can be obtained either by reference to Erd^lyi, , 1 pp. 258, 254, by differentiation and subsequent integration by parts of integrals such as u+l/2 -x/2 .CO r(M + H - x) •'o or by observing that combinations of the form satisfy Whittaker's equation with m "= 1- In the last mentioned method, the required constant multiphnng the first order Whittaker function can be obtained by reference to the limiting behavior for small x. If ■^x^ = ^^xA^)/y/x and *SxM = My^,^{x)/\/x the results are i^xo' + X^xo = (X - yz)Rxl , >Sxo' + x5,o -V2{K- X) >Sxi . 984 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 III. ZEROS When X = (2n + l)/2, n = 1, 2 , W^Jx"'"'^ reduces to a polynomial times a function of x- This may be inferred from the asymp- totic expansion, 1+ Z [m' - (x - y2)V - (x - Vz)'] •■■in'- ix-n+ y^f] nix"" which terminates if /x = 1 and x = ^i + 3^(w= 1,2...), or from the fact that for these values of the suffi.xes, W reduces to the generalized Laguerre polynomial Similarly when x = (2/i + l)/2, ?i = 0, 1, 2 . . . ., W-^^ reduces to the Laguerre polynomial Lx-(M2){x). The zeros at the critical values of x are given in the following table X Zeros of W-^,q Zeros of ]{\, i V2 0 0; 1 0; 0.586; 3.414 0; 0.416; 2.294; 6.290 0; 0.323; 1.746; 4.537; 9.395 0; 0.26356; 1.413; 3.596; 7.086; 12.641 None 0 0;3 0;6;2 0; 1.517; 4.312; 9.171 0; 1.227; 3.413; 6.903; 12.458 Betw^een n -\- }/2 and w + M, TF^.o has n + 1 zeros (n = 0, 1, 2 . . .) and TFx.i has n zeros {n = 1, 2 . . . .). The zeros of M^^^ coincide with those of ir^.i when x = n + 3^^, and at those values of x only. The functions il/x.i and IF^.i then are propor- tional to each other.* IV. THE RICCATI-EQUATION FOR THE IMPEDANCE FUNCTION TF^.o/^ x.l- The computations concerning the cylindrical helix required a study of the function Z(«) = ^ . PFx.i(m) * At these critical values of x, the solution to the problem of the hollow cyl- inder of ferrite in the waveguide breaks down, since M and W are then not in- dependent. A further independent solution must then be constructed. GUIDED WAVE PROPAGATION Til HOUGH GYKOMAGNETIC MEDIA. II 985 We Avill show that Z^(u) satisfies a non-linear first order differential equation of Riccati-type. From the difference relations, equation (36), we have say and from Whittaker's equation Therefore or ;;^ ^ + -, + M i - 2x ) + (x^ - K) = 0, dug g- g \u l = ^ + S-2^)^+(^'->4)^' Finally, let Then dZ_ du z= ix- K) giu) = 5^) (x - 3-^) + (^ - 2x)^ + (x + M) Zl Since this equation is satisfied by Mx.oiu)/Mx,i(''^)) as well as by Wx,o{u)/Wx,i(u), a selection has to be made from all the possible solu- tions of this equation. We require the one which for large u approaches unity. But for large u the equation is ^ = {I - z) [x - y2 - (x + y2) z], whose integral is ^ ^ (x - M)^g" - 1 (x + 3^)^e" - 1 For large u, therefore, the solution is either unity, when A = 0, or else (x — K)/(x + 3^), A z|z 0. The solution with A i^= 0 corresponds to the M functions; that A\ith A = 0 to the IF-functions. 986 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 The case A = 0 was integrated on an analogue-computer, and the results are shown in Fig. 14(b). The computation was restricted to the range | x I < /^- Beyond these values, the heUx-problem was discussed only qualitatively. REFERENCES 1. M. L. Kales, H. N. Chait, and N. G. Sakiotis, Letter to the Editor, J. Appl. Phys., 24, No. 6. 2. Erd^ivi, Magnus, Oberhettinger and Tricomi, Higher Transcendental Func- tions, I, McGraw-Hill, 1953. 3. E. H. Turner, I. R. E. Proc, 41, p. 937, July, 1953. 4. J. S. Cook, R. Kompfner, and H. Suhl, Non-Reciprocal Loss in Traveling Wave Ferrite Attenuators, Letter to Editor, I. R. E. Proc, to be published. Theoretical Fundamentals of Pulse Transmission — TI By E. D. SUNDE (Manuscript received September 23, 1953) Part II. 12. Impulse Characteristics and Pulse Train Envelopes 987 13. Transmission Limitations in S3'mmetrical Systems 991 14. Transmission Limitations in Asymmetrical Sideband Sj^stems 996 15. Double vs. Vestigial Sideband Systems , 1004 16. Limitation on Channel Capacity by Characteristic Distortion 1007 Acknowledgements 1010 References 1010 Part I of this paper dealt with various idealized transmission characteris- tics and with methods of evaluating pulse distortion resulting from various system imperfections. In Part II the resultant transmission impairments or limitations on pidse transmission rates are discussed for systems with low- pass, symmetrical hand-pass and asymmetrical hand-pass characteristics, and a comparison made of the transmission performance of double and vestigial sideband systems. The limitation on channel capacity imposed by random imperfections in the transmission-frequency characteristic, as com- pared to random noise, is also discussed. 12. IMPULSE CHARACTERISTICS AND PULSE TRAIN ENVELOPES In pulse modulation systems pulses are transmitted in various com- binations to form pulse trains, and at the receiving end the envelope of the pulse train is sampled at regular intervals to determine the ampli- tudes of the transmitted pulses. As a result of pulse overlaps there may be appreciable distortion of the pulse train envelope, which may cause errors in reception or noise, depending on the type of system. To evalu- ate transmission impairments, or limitations imposed on transmission capacity to avoid excessive transmission impairments from pulse dis- tortion, it is necessary to establish basic relations between the impulse characteristic of the system and the envelope of the received pulse train. In Fig. 42 are shown three transmitted pulses of different peak ampU- tudes, A-i , Ao and Ai , transmitted at intervals r with the first and third 987 988 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 pulse overlapping into the middle pulse. The instantaneous amplitude of the received train at a time to referred to the peak amplitude of the middle pulse is W{U) = A.,Pito - r) + AoPito) + AiP{to + r), 1 I n=-l = tl AnPito + nr). (12.01) When the sequence of pulses transmitted at uniform intervals r extends between n = — and , the instantaneous amplitude of the pulse train at time to is TF(/o) = Z AnPHo + nr). (12.02) The above equation gives the instantaneous value W(to) for any se- lected combination of transmitted pulses. The transmitted pulses may have any value within certain limits, as when they represent signal samples in a pulse amplitude modulation system, or may assume two or AMPLITUDES OF TRANSMITTED PULSES Ao Ai ])tained b^' taking the maximum positixe and negative values of the summation term in (13.01). As discussed in prcxious sections, certain types of trans- mission system imperfections gi\e rise to pulse distortion extending over long time intervals, such as fine structure (le\-iations over the transmis- sion band, a low-frequency cut-off and pi'onounced band-edge phase deviations. Evaluation of peak intersymbol interference is then rather difficult, and a more convenient approximate method is to evaluate rms inters3'ml)()l interference, which can be related to vms deviation in the transmission fre(iuency characteristic by methods discussed previ- ously. Peak inters3'mbol interference may then be estimated by applying a peak factor between 3 and 4, depending on the type of transmission distortion. If Poinr) designates an ideal impulse characteristic, which is zero for 71 = ±1, ±2 etc., the deviation from the ideal envelope of a pulse 5 40 \ \ N \ ^^ ^, N \ \ \ DELAY DISTORTION \ \ Umax > / \ V 1 \ \ \ 1.0 1.5 2.0 2.5 3.0 dMAX'MAX = 2d^lAyT| •"MAX I WAX Fig. 43 — Margin against exoessivo jjcak interference in systems emploj'ing two ))ulse amplitudes with intervals between i)ulses r = ti = l/2/i = 1/fmax. for impulse transmission characteristic as shown in Fig. 23. (13.09) (13.10) 994 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 train may be written 00 AT^(O) = W{0) - WoiO) = X; AnlPM - PoMl (13.08) n=— 00 The rms deviation becomes, with APinr) = P{nT) — Poinr) ATF(O) = a(j: [APinr)fY, \ n=— 00 / ^ 4 (^ /] [^Pit)T dtj', = AP(0)U. (13.11) A is the rms ampUtude of the transmitted pulses and U the rms inter- symbol interference referred to unit amplitude of the received pulses. Expressions for U applying to fine structure imperfections in the trans- mission frequency characteristic were given in Section 8, for a low-fre- quency cut-off in Section 9 and for band-edge phase deviations in Sec- tion 10. For balanced pulse systems employing positive and negative pulses, rms intersymbol interference in the positive and negative directions will be equal. For such systems the maximum value of the summation in (13.02) becomes A"^(0) and in (13.03) -kWiO), where k is the peak factor. Equation (13.04) is then replaced by M = — A ■ q- 1 = 2Am^P{0) P(0) - 2kAP{0) U, Lq -i-^ - fc[/(4Mmax) (13.12) when Amin = — ^max • In a balanced pulse system employing q pulse amplitudes, i.e., q/2 positive and q/2 negative amplitudes, with equal steps 2^1 max/ (? — 1) between pulse amplitudes, the following relation applies if all ampli- tudes have equal probability. 1/2 .A/Amax — g+ 1 L3(g - 1)J Hence, M = 2Amax-P(0) q- 1 1 - k q' - 1^^'^ U (13.13) (13.14) As mentioned before, the factor A; may be as high as 4, in which case the THEOKKTU'AL FUXDAMKNTALS OF PULSE Tl< A NSM ISSK )N !)<).") maximum t(ilorable rms iiiters^niibol interference U referred to unit peak amplitude of the rec'ei\'ed pulses heeomes for M = 0: Q = 1 U = 0.25 4 0.112 0.054 In (13.14) and in the above table, U is the maximum tolerable rms intersymbol interference from all sources, such as fine structure imper- fections over the transmission band, band-edge phase distortion and a low-frequency cut-off. Interference from these various sources may be combined on a root-sum-square basis. In the above evaluation of rms intersymbol interference a balanced pulse system was assumed. An unbalanced system can be obtained by superposing on a balanced system an infinite sequence of pulses of equal amplitude and polarity at uniform intervals as indicated in Fig. 44. This superposed sj-stem will give rise to a fixed intersymbol interference or displacement of the received pulse train, which does not alter the margin for distinction between pulse amplitudes and Avhich can be corrected by a fixed bias at the receiving end if necessary. For this reason, in the case n u n (a) BALANCED PULSE TRAIN WITH EQUAL MAXIMUM AMPLITUDES OF POSITIVE AND NEGATIVE PULSES Q Q (b) SUPERPOSED INFINITE PULSE TRAIN TJ n (c) UNBALANCED PULSE TRAIN (a) + (b) Fig. 44 Derivation of an unbalanced from a balanced pulse train by super- position of an infinite train of pulses of equal amplitude. 990 THE BKLL SYSTEM TECHNICAL JOURNAL, JULY 1954 of an unbalanced system, only the balanced component need to be con- sidered in evaluating rms intersymbol interference, which will thus be the same whether or not the system is balanced. As shown previously, peak intersymbol interference, or the margin for distinction between pulse amphtudes, depends only on the peak to peak pulse excursion and is thus the same for unbalanced as for balanced systems. It may be noted here that for a balanced system the transmitted power is a mini- mum for a gi^^en margin in pulse reception, as is the interference in other systems that may be caused by the transmitted pulses. For a symmetrical band-pass system, rather than a low-pass system as discussed above, Qinr) = 0 in (12.08). The envelope of the pulse train then becomes 00 TF(0) = E AnRM, (13.15) 71= — 00 where Rinr) = R-inr) -\- R+inr) = 2R+(nT), with R^ and R+ given by (2.10). Since (13.15) is of the same form as (13.01), the relationships estab- lished above for low-pass systems also apply to symmetrical band-pass systems, with Ri^nr) replacing Pinr). Rinr) will have the same shape as Pint), but will be greater by a factor 2, which will appear as a multiplier in the various expressions and hence not alter the requirements on toler- able pulse distortion or intersymbol interference. 14. TRANSMISSION LIMITATIONS IN ASYMMETRICAL SIDEBAND SYSTEMS The formulation of transmission limitations imposed by pulse distor- tion in asjrmmetrical sideband systems is complicated by the presence of the quadrature component in the impulse transmission characteristic. Of particular interest are the transmission limitations with vestigial sideband as compared with double sideband transmission, assuming the same bandpass characteristic in both cases, a question which has been dealt with in literature for systems with a linear phase characteristic .' Relationships (2.18) and (2.19) facihtate a comparison also for systems with phase distortion, as shown in the following. If the envelope of the impulse characteristic with double sideband transmission is P{t), the in-phase and quadrature components with vestigial sideband transmission are given by (2.19), with coy = cos or R = R^ -{- R+ = cos (i^st - rps) P(t), (14.01) Q = Q_ - Q+ = sin (o^st - ^s) P{t). THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 997 If t is so chosen that a\4 — \p, ^ 0, and the time wilh respect to this \-aUie of t is designated /o , then A'(/o) = cos c^sUPih), Q(fo) = sin co./oP(/o). (14.02) An apphcation of this method to the impulse cluiracteristic shown in Fig. 23 for 6 = 15 radians is ilkistrated in Fig. 4."). In order to compare vestigial with double si(lel)and transmission, it suffices to evaluate the in-phase and ([uadrature comi)onents at the sampling instants. With r = 7r/2co,, =1/4/.. , the in-phase and quadrature components at times rnr, for m = 0 ±1, ±2, etc., will be as illustrated in Fiff. 46. Fig. 45 — Determination of in-phase and quadrature components of impulse characteristic for vestigial side-band transmission. 998 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 With double sideband transmission, pulses would be transmitted at the points m = 0, ±2, ±4, etc. At these points the quadrature com- ponents vanish, as indicated in the above figure, and the in-phase com- ponents are the same in amplitude as with double sideband transmission. Thus, if pulses were transmitted at the same rate as with double side- band transmission, the sum of the absolute values of the in-phase com- ponents at the sampling points would be identical with the sum of the absolute values of the envelope with double sideband transmission. It follows from the criteria established in Section 13 that for this particular pulse transmission rate the effect of pulse distortion would be the same with both transmission methods. With an ideal transmission frequency TRANSMISSION FREQUENCY CHARACTERISTIC P(t) = ENVELOPE OF IMPULSE CHARACTERISTIC R = IN-PHASE COMPONENTS AT SAMPLING INSTANTS Q = QUADRATURE COMPONENTS AT SAMPLING INSTANTS r = 1/0)3 = PULSE INTERVALS WITH DOUBLE SIDE BAND TRANSMISSION Fig. 46 — In-phase and quadrature components of impulse characteristic with vestigial side-band transmission. THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 090 characteristic ha^^lng a linear phase shift, there would be no intcrsjTnbol interference with cither method for the above rate of pulse transmission. Assume next that the pulse transmission rate is doubled and that the quadrature component is eliminated. This is possible if the carrier fre- quency is transmitted and is deri\-ed at the receiving end with the aid of filters and applied in proper phase to a product demodulator, a method known as homodyne detection. At the points m = 1, 3, 5, etc., there would then be no quadrature components and no in-phase components. The sum of the absolute values of the in-phase components at the other sampling points, m = 2, 4, etc., would be the same as with double side- band transmission. It follows that the transmission capacity (pulsing rate) can be doubled by vestigial sideband transmission if the quadrature component is eliminated by homodyne detection, for the same margin against excessive intersjonbol interference as \\ith double sideband transmission. An increase in transmission capacity can be realized mth vestigial sideband transmission without elimination of the quadrature component by homod3Tie detection, although a two-fold increase is then possible only if the phase characteristic is linear, as discussed below. Vestigial sideband transmission can be employed A\ithout transmission of the carrier, or with a fixed level of carrier in the absence of pulses and a higher level in the presence of pulses. The latter method is equivalent to the transmission of two or more pulse amplitudes, with the minimum ampUtude greater than zero. With this method the effect of the quadra- ture component on the envelope of a pulse train can be reduced, and even eliminated provided the phase characteristic is linear. In the follow- ing, vestigial sideband transmission with two pulse ampUtudes at twice the double sideband pulsing rate is discussed, for the case in which the minimum pulse amplitude is finite rather than zero. With pulses transmitted at twice the double sideband rate, i.e., -with the interval bet^veen pulses equal to t = ir/2ws , equation (12.08) for the envelope becomes in view of (14.02) W(0) ^ An cos COsnrP(?lT) + CO _ -|2\ 1/2 2 An sin cosnrPinT) ) (14.03) At the even sampling points, i.e., n = 0, 2, 4 • • • , cos co.nr = ±1 and the in-phase components may be written R(±2mT) = ±P(±2mT), m = 0, 1, 2 • • • At the odd sampUng points, i.e., n = 1, 3, 5 • • • , sin co.nr = ± 1 and 1000 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 the quadrature components may be written Q[±{2m - 1)t] = ±P[±(2m - 1)t], m = 1, 2, 3 Let m=l 00 J^R~ = Z [^~(2mr) + R'{-2mr)], 7n^ 1 EQ"" = £ ^""[(2^ - l)r] + Qn-(2m - l)r], (14.04) EQ" = Z Q"[(2w - l)r] + Q-[ -(2m - l)r], where i?""", Q"*" designate positive values and R~, Q~ the absolute values of negative amplitudes of the in-phase and quadrature components. Let it be assumed that two pulse amplitudes are employed, Am in and A max • When the minimum amplitude is transmitted, the maximum value of the envelope is obtained by considering the maximum positive over- laps of the in-phase components in conjunction with the maximum value of the quadrature component. The value thus obtained is Tfmax = [(^min RiO) + ^ma. Z^^)' It is assumed that ZQ" > ZQ^> otherwise Q~ and Q^ would be inter- changed in the last term. When the maximum amplitude is transmitted, the minimum value of the envelope is obtained by considering the maximum negative overlaps of the in-phase components, in conjunction with the minimum value of the quadrature component, which gives Wmin = [(Amax i^(0) " Amax Z^")' + ^^'min (ZQ" The margin for distinction between Am in and A max is M = TFmin — Wma.x and becomes M = Amax [{R(o) - ZRy + ahq"- - ZQ-yf" (14.07) - Amax [{HR(0) + Z^^)' + (ZQ" - f^ZQyf", where THEORETICAL FU.NDAMENTALS OF I'ULSE TRANSMISSION 1001 The margin for a unit dilference A^ax — -Imi,, , i-c. Mi = M/(A, ■ -I mill) becomes: 1 — M (14.08) The s{)ec'ial case of an ideal transmission characteristic as sliown in Fig. 47 will be considered first. In this case 72(0) = 1 R{2t) = 0 R{-2t) = 0 R{4.t) = 0 A'(-4r) = 0 Qir) = 0.5 Q(-r) = -0.5 (KSt) = 0 Q(-3t) = 0 so that: 1:^+ = 0 X^~ = 0 Z^"" = 0.5 DQ- = 0.5 Equation (14.08) in this case simplifies to Ml 1 - 1 - m' + ^ (1 - m)' ni/2 (14.09) For various values of |U = ^4niinA'lmax the margin for luiit (hffer ence m ENVELOPE Pit) ^ 1 ~ -A P(o) = 1.0 P(rj = 0.5 P{2T) = 0 P(37-) = 0 Fig. 47 — Envelope of idealized impulse characteristic. 1002 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 pulse amplitudes becomes: Ml. 0 0.2 0.3 0.5 0.8 0.9 1.0 0.50 0.69 0.77 0.87 0.96 0.998 1.0 Thus, for an ideal impulse characteristic as assumed above, the quadrature component gives rise to 50 per cent maximum intersymbol with fji = 0, and to negligible intersymbol interference when /x = 0.8 or greater. By way of comparison, the margin would be zero with double sideband transmission at the rate assumed above, i.e., twice the normal double sideband rate. This follows from (13.07) when it is considered that P{zkT) = }4 P(0), P(±2t) = 0, so that the sum of the absolute values of the impulse characteristic at the sampling points is equal to P(0) and thus M/ikfmax = 0. Elimination of the effect of the quadrature component by the above method is contingent on a symmetrical impulse characteristic, i.e., P{nT) = P{ — nT), a condition which can be realized only with a linear phase shift. Furthermore, the in-phase components must vanish at the sampling points, which entails an ideal amplitude characteristic. In the presence of phase distortion the effect of the quadrature component cannot be eliminated but may be reduced by proper choice of the ratio ju, as discussed below for a transmission characteristic with moderate phase distortion. As an example consider an impulse characteristic as shown in Fig. 23 for h = 5 radians. The in-phase and quadrature components at the various sampling points are in this case R{0) = 0.97 R{-2t) = -0.09 R(2r) = 0.13 P(-4r)^0 P(4r)^0 Q(-t) = -0.54 Q(t) = 0.44 Q(-3r) ^0 Q(Zt) = -0.03 Q(-5t)^0 Q(5r)^0 Hence 2;P+ = 0.13 ^R~ = 0.09 SQ"^ = 0.44 ^Q" = 0.57 Equation (14.08) in this case becomes ilfi = -J— ((0.88' + 0.13V)'" - [(0.97m + 0.13)' 1 — M + (0.57 - 0.44m) Y''). TllEOHKTlCAL FUXDAMEM'ALtJ OF I'LLSE TKAAfciMlttaiON 1003 For various values oi n ^ Amin/Am^^^ the margin for unit difference in pulse amplitudes becomes fjL 0 0.2 0.3 O.l 0.-) 0.0 0.7 0.75 ,1/1 0.30 0.375 0.40 0.375 0.34 0.25 0.13 0 The optimum condition is thus in the aboxe j)articular case ol)tained with 11 = 0.3, with a comparatively small variation in transmission per- formance for any value of n between 0 and 0.5. In the above discussion of vestigial sideband transmission, modulation of a carrier was assumed, with elimination of one sideband except for the wanted vestige. The equivalent performance can be secured by applica- tion of impulses to a band-pass transmission characteristic with the proper interval between pulses in relation to the midband frequency, as discussed below: When equation (12.03) is written with respect to the midband fre- quency, Ur = oom , and a sjnnmetrical amplitude characteristic is assumed so that Q = 0, the following relation obtained. W{to) = cos (jOmto ^ Ar, cos co„,nTR(to 4- nr) (14.10) — sin Umfo X^ --In sill comiiTRito + nr), n= — 00 in which R may be replaced by P, the envelope of the impulse charac- teristic. Let it be assumed that t is so chosen that cos Umtir = cos n7r/2 in which case sin a)„nr = sin n7r/2. The above equation then becomes 00 W(fo) = cos o^mh X) AnPih + nr) cos mr/2 — sin (jimh ^ AnP{U) + nr) sin mr/2. n=— 00 The in-phase and quadrature components of the envelope at the sampling instant ^0 = 0 are accordingly 00 i^(0) = Z) AnPinr) cos mr/2, (14.12) 00 Q(0) = X) AnVim) sin mr/2. Pulses in even positions, i.e., A^ , A2 , Ai , etc., will thus contribute an in-phase but no quadrature component while pulses in odd positions 1004 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Ai, Ai , A5 , etc., will contribute a quadrature but no in-phase com- ponent. It will be recognized from Fig. 42 that this is the same condition as encountered in vestigial sideband transmission with pulses in the latter case transmitted at intervals r = 7r/2cos = 1/4 /« . To realize the above condition ^nth pulses applied to a band-pass filter, it is necessary that in (14.10) co^nr = n7r(>^ + N), (14.13) where N is an integer, or that ^ = ^ = —77 — • (14.14) 2com 4/„ The interval between pulses must thus be an integral number of half- cycles plus one quarter cycle of the midband frequency /„ , as illustrated for a particular case in Fig. 48. When /,„ is large in relation to the side- band frequency this condition can be achieved with substantially the same pulse spacing as ^^'ith vestigial sideband transmission. To secure exactly the same rate of pulse transmission it is necessary that r = 1/4/., which, in conjunction with (14.14), gives N = h (fm/fs - 1). (14.15) Thus, if /,„ = 5fs , N = 2 and the interval r between pulses as obtained from (14.14) is 1.25 cycles of fm . If fm = 10/^ , A^ = 4.5 and it is not possible to have exactly the same pulsing rate as with vestigial sideband transmission, since N must be an integer. It is then necessary to take A^ = 4 or 5. With A" = 4 equation (14.14) gives r = 9/40/^ and with A^ = 5, T = 11/40/s . This compares with r = 1/4/, = 10/40/, with vestigial sideband transmission, so that there is a minor difference in pulse intervals with the two methods. 15. DOUBLE VERSUS VESTIGIAL SIDEBAND SYSTEMS From the preceding discussion it follows that, for the same bandwidth and margin against interference from characteristic distortion, a two- fold increase in transmission capacity can be approached with vestigial over double sideband transmission. This assumes that the carrier is transmitted at the proper level and that the phase characteristic is linear, or that otherwise homodjnie detection is used to cancel the effect of the quadrature component. TIIKOKKTK'AL Fl" NDAM KXTALS OF ITLSK TUAXSMISSIOX ion.-) For the same bandwidth, the same transmission capacity can be realized with a double sideband sj^stem employing fom- jiulse amplitudes as with a vestigial sideband system Avith two pulse amplitutles. However, the latter type of system will have a greater tolerance to interference from characteristic distortion than the former. This follows when it is considered that in a quaternary system the maximum tolerable inter- ference is ^ the maximum pulse amplitude, as compared to 1^2 the maxi- mum pulse amplitude in a binary system. With )u = ^minMmax = 0, the quadrature component reduces the margin by a factor of 0.5, so that the maximum tolerable interference in relation to the maximum pulse PULSE SPACING (1.5 tO.25) cycle PULSE SPACING (1.5 +0.25) CYCLE PULSE TRANSMITTED IN POSITION n = -1 PJLSE TRANSMITTED IN POSITION n = 1 RESULTANT ENVELOPE Fig. 48 — Impulse exitation of l)aiKl-pass .sA-stem with pulse spacing selected to provide equivalent of vestigial side-band transmission. 1006 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 amplitude is ^ as compared to 14 for a quaternary system. If the phase characteristic is linear and the carrier is transmitted at the optimum level, or if homodyne detection is used, the effect of the quadrature com- ponent is cancelled. The maximum tolerable interference is then l^ as compared with ^^ for a quarternary double sideband system. In the presence of phase distortion, a substantial advantage can also be realized with a binary vestigial system, which can be illustrated by considering the example in Section 14. For the optimum condition fx = 0.4, the margin is reduced by a factor 0.4 and is thus 0.2. For a quater- nary double sideband system the factor by which the margin is reduced is given by (13.07), with g = 4 and with ^|:|P(n.)| + |F(-n.)|=^Efi-+E«-), where R{0) = 0.97, 2^^ = 0.13 and X^~ = 0.09, as in the example in Section 14. The reduction in margin thus obtained is M/Mmax = 0.32. Hence the maximum tolerable interference for a quarternary double sideband system is 0.32/6 = 0.053 as compared with 0.20 for a binary vestigial sideband system under the optimum condition ^ = 0.4. For the same transmission capacity and same number of pulse ampli- tudes, a substantial transmission advantage may be realized with ves- tigial over double sideband transmission in circuits with pronounced phase distortion, owing to the circumstance that a two-fold reduction in bandwidth with vestigial sideband transmission may afford a sub- --^-AMPLITUDE CHARACTERISTICS FREQUENCY DELAY DISTORTION JMAX I MAX - 0.15 CJmAX fwAX Fig. 49 — Comparison of double and vestigial side-band transmission in the presence of delay distortion. THEORETICAL FUNDAMENTALS OF Pl^LSE TRANSMISSION 1007 stantial reduction in delay distortion over the transmission band. This is illustrated in Fig. 49, Avhere a co.sine variation in transmission delay is assumed. With a two-fold reduction in bandwidth, the product d'max/'max for vcstigial sideband transmission is about 15 per cent of the product c/max/inax for doublc sideband transmission. Thus, with c?max/max = 8.3, d'lnaxf max = 1-25, Corresponding to 6 = 5 radians, as assumed in the example in Section 14. Vestigial sideband transmission is in this case feasible with an adecjuate margin, about 40 per cent of the maximum margin in the absence of phase distortion. Double sideband transmission would not be possible, as is evident from Fig. 43, since it would be neces- sary to have c^max/max Icss thau 4, as compared with 8.3 in the above case. The above discussion of vestigial vs double sideband transmission pertains to the effects of characteristic distortion rather than noise, and the relative complexity of terminal equipment was disregarded. Because of the simpler terminal equipment with double sideband transmission, this method is ordinarily used w^here bandwidth is not a primary con- sideration, as for example in providing a number of telegraph' channels over a \'oice freciuency circuit. 16. LIMITATION ON CHANNEL CAPACITY BY CHARACTERISTIC DISTORTION For an ideahzed channel of bandwidth /i with a transmission-frequency characteristic as shown in Fig. 7, the transmission capacity in bits per second for a signal of average power P in the presence of random noise of average power N can with sufficiently complicated encoding methods approach the limiting value given by Shannon :^^ C = /i log2 (1 -f P/N). (16.01) The above expression also appHes to certain other ideahzed channels with a linear phase characteristic, w^hen /i is defined as in Fig. 10. In all of these cases the integral of the area under the amplitude characteristic, or the equivalent bandwidth, is /i . By way of comparison, for pulse code modulation systems the channel capacity is of the same basic form as (16.01), namely : C = /:log2(n-^), (16.02) where K = 8. Thus about an 8-fold increase in signal power is required to attain the same channel capacity as with the idealized but imprac- ticable encoding system underlying (16.01). The above expressions give the limitation on channel capacity imposed by random noise. From the discussion in Sections 13 and 14 it follows that a limitation is placed on channel capacity by characteristic distor- 1008 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 tion, in the absence of noise. In idealized communication theory, charac- teristic distortion has been disregarded in determining channel capacity on the premise that unlike random noise it is predictable and can there- fore be corrected, at least in principle. In actual systems, however, com- plete elimination though possible in principle caimot be accomplished in practice. The resultant limitation on transmission capacity may be as important as that imposed by the maximum signal power that can actually be provided to override noise. In the following it will be assumed that correction of amplitude and phase deviations is made by equalization, so that the amplitude and phase characteristics are as assumed for an ideal channel, except for small fine structure residual deviations as illustrated in Fig. 30. These small fine structure deviations may be regarded as of random nature in the sense that they differ among channels and cannot be predicted, although for a given system they would remain fixed in the absence of temperature variations or changes in amplifiers -with age. From equation (13.12) it follows that the maximum number of pulse amplitudes or quantizing levels as limited by characteristic distortion is obtained from the relation 1 = kUA/An.^ , (16.03) or 9=1 + ^ Axnax/4- (16.04) In the absence of characteristic distortion, the maximum number of pulse amplitudes as limited by an rms noise amplitude An or a peak noise amplitude kAn is obtained from the following relation for a bal- anced pulse system. ^"''" kAn , (16.05) q- 1 or g = 1 + -^ ^max/4. (16.06) KiAn Comparison of (16.04) and (16.06) shows the following equivalence between intersymbol interference and noise from the standpoint of limitation on the permissible number of pulse amplitudes U = A„/A, (16.07) THEORETICAL FUNDAMENTALS OF PULSE TRANSMLSSION 1009 or U'- = J) = N/P. (10.08) This means that random characteristic cHstortion has the same effect as a random noise power N = DP, where Z) is a distortion factor. In view of the above equivalence, the channel capacity of a PCM system in the presence of random characteristic distortion, but without noise, as obtained by substitution of (1G.08) in (16.02) becomes C = /i log, (l + ]^) • (10.09) With random interference from both characteristic distortion and noise, the interfering powers add directly, so that for a PCM system ''-f^'A' + mTNjp))- ^'"'"^ The equivalence (16.08) was established above on the liasis of discrete pulse amplitudes, but it is independent of q and would thus apply also for continuous signals. On this basis it would apply for an^^ method of modulation or of encoding signals and the maximum channel capacity as given by (16.01) would be modified to It follows from the above that for any modulation method the toler- able distortion factor is directly related to the average signal-to-noise ratio. Thus two modulation methods which are equivalent from the standpoint of signal-to-noise ratio are also equi\'alent from the standpoint of tolerable rms distortion, provided faithful reproduction of the trans- mitted signal is required, as assumed here. From (8.14) the following relation is obtained between the distortion factor D = U^ and small rms deviations a (nepers) and b (radians) in the amplitude and phase characteristics D = or -\- b\ (16.12) In order that characteristic distortion may be disregarded in compari- son with noise, it is necessary that D <5C N/P or a' + &' « N/P. (16.13) For example, in communication systems employing the same band- width as the original signal, such as a pulse amplitude modulation sys- tem, a representative signal-to-noise ratio would be about 40 db, or N'/P = 10~*. In order that characteristic distoition may ])C disregarded in this case, it would be necessary for both a and b to be substantially 1010 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 less than 10"^ nepers and radians respectively. This would correspond to an rms gain deviation over the transmission band well below 0.08 db and an rms deviation from a linear phase characteristic well below 0.6 degrees. Since these tolerances are difficult to realize in actual systems, at least for wire circuits, characteristic distortion rather than noise may impose a limitation on channel capacity of systems employing about the same bandwidth as the original signal. In accordance with (16.01), the bandwidth can in principle be halved without change in channel capacity if the signal-to-noise ratio is squared, i.e., if N/P = 10"^ rather than 10~* in the previous example. The toler- able rms amplitude and phase deviations would then be ^ + ^ « 10~^ Thus both a and h would have to be substantially smaller than about 10"*, which would preclude a substantial bandwidth saving in practical systems from the standpoint of characteristic distortion, even if it were feasible from the standpoint of signal power required to override noise. The above considerations apply when faithful reproduction of the transmitted signal is required, as for example in data transmission. In speech transmission considerable distortion can be tolerated, a circum- stance which permits appreciable phase distortion in the usual frequency division system without noticeable impairment of intelligibility, but which cannot be taken advantage of in time division pulse systems be- cause of the resultant intersjonbol interference. The characteristics of speech sounds also permit a reduction in the bandwidth of the original transmitted signal, by such devices as vocoders or frequency compandors, without excessive impairment of intelligibility. ACKNOWLEDGMENTS This paper is based on studies in connection with the application of pulse systems to ware circuits, suggested by M. L. Almquist and J. T. DLxon and carried out under their direction. Some of these studies were made on behalf of the Signal Corps, under contract W-36-039-SC-38115. The Signal Corps Engineering Laboratories have consented to the pub- Hcation of results obtained in these studies. The writer had the benefit in these studies of discussions with C. B. Feldman and W. R. Bennett, and of some of their memoranda on pulse modulation systems. He is also thankful for comments and advice from F. B. Llewellyn and others in connection with preparation of the paper. REFERENCES See Part I. Bell System Technical Papers Not Published in this Journal. 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C/ and Fageant, J} Orientation Relationships in Cast Germanium, J. Metals, 6, Pt. 2, pp. 291-294, Feb., 1954. Ellis, W. C.^ and Greiner, E. S.^ Production of Acceptor Centers in Germanium and Silicon by Plastic Deformation, Letter to the Editor, Phys. Pie v., 92, pp. 1061-1062, Nov. 15, 1953. Fageant, J., see Ellis, W. C. Fine, M. E.,^ Van Duyne, H.,^ and Kenney, Nancy T.^ Low-Temperature Internal Friction and Elasticity Effects in Vitreous SiUca, J. Appl. Phys., 25, pp. 402-405, March, 1954. Fuller, C. S., see Pearson, G. L. Fuller, C. S., see Severiens, J. C. Galt, J. K.,^ Yager, W. A.,^ and Merritt, F. R.^ Temperature Dependence of Ferromagnetic Resonance Fine Width in a Nickel Iron Ferrite — A New Loss Mechanism, Phys. Rev., 93, pp. 1119-1120, March, 1954. Germer, L. H. Arcing at Electrical Contacts on Closure. Part IV — Activation of Contacts by Organic Vapor, J. Appl. Phys. 25, pp. 332-335, March, 1954. GoHN, G. R.,^ GuERARD, J. P., and Herbert, G. J. The Mechanical Properties of Some Nickel Silver Alloy Strips, Proc. A.S.T.M., 54, Jan., 1954. Greiner, E. S., see Ellis, W. C. GuERARD, J. P., see GoHN, G. R. ^ Bell Telephone Laboratories. TECHNICAL PAPERS 1013 Hagstrum, H. D/ Reflection of Ions as Ions or as Metastable Atoms at a Metal Surface, riiys. Kev., 93, p. ()52, Feb., l\)P)[. Absinicl of paper proseulcd at Gaseous Electronics Conference Oct. 22-24, 1953. Heffner, H., see Clogston, A. M. Herbert, G. J., see Gohn, G. R. Johnson, J. B.,^ and McKay, K. G.^ Secondary Electron Emission from Germanium, Phys. Rev., 93, pp. 668-672, Feb. 15, 1954. IvAHN, A. H., see Tessman, J. R. Kenney, Nancy, see Fine, M. E. KocK, W. E.' Use of the Sound Spectrograph for Appraising the Relative QuaUty of Musical Instruments, Letter to the Editor, Acous. Soc. Am., J., 26, pp. 105-lOG, Jan., 1954. Kretzmer, E. R.^ An Amplitude StabiUzed Transistor Oscillator, Proc. I.R.E., 42, pp. 391-401, Feb., 1954. Lewis, H. W.' Search for the Hall Effect in a Superconductor — Experiment, Phys. Pvev., 92, pp. 1149-1151, Dec. 1, 1953. LiNVILL, J. G.^ RC Active Filters, Proc. I.R.E., 42, pp. 555-564, March, 1954. MacColl, L. A.^ Geometrical Properties of Two-Dimensional Wave Motion. Am. .Math. Monthly, 61, pp. 90-103, Feb., 1954. '■ Bell Telephone Laboratories. 1014 the bell system technical journal, july 1954 Machlup, S/ Noise in Semiconductors: Spectrum of a Two-Parameter Random Signal, J. Appl. Phys., 25, March, 1954, Matthias, B. T/ Transition Temperatures of Superconductors, Phys. Rev., 92, pp. 874-876, Nov. 15, 1953. McKay, K. G.. see Johnson, J. B. Merritt, F, R., see Anderson, P. W. Merritt, F. R., see Galt, J. K. Morin, F. J/ Lattice-Scattering Mobility in Germanium, Phys. Rev., 93, pp. 62-63, Jan. 1, 1954. Morin, F. J., see Pearson, G. L. Pearson, G. L.,^ and Fuller, C. S.^ Silicon p-n Junction Power Rectifiers and Lightning Protectors, Proc. I.R.E., 42, pp. 760, April, 1954. Pearson, G. L.,^ Read, W. T., Jr.,^ and Morin, F. J.^ Dislocations in Plastically Deformed Germanium, Phys. Rev., 93, pp. 666-667, Feb. 15, 1954. Ppann, W. J.' Comment on Paper by Tiller, Jackson, Rutter and Chahners, Letter to the Editor, Acta Metallurgica, 1: pp. 763-764, Nov., 1953. Pfann, W. G.^ Redistribution of Solutes by Formation and Solidification of a Molten Zone, J. Metals, 6, Pt. 2, pp. 294-297, Feb., 1954. 1 Bell Telephone Laboratories. TECHNICAL PAPERS 1015 PlERCi:, J. "R.' Coupling of Modes of Propagation, J. Ajipl. Pltys., 25, pp. 179-183, Fel)., 1954. Read, W. T., .hi} Dislocations or What Makes Metals So Weak? Aletul l^()ij;iess, 65, pp. 101 UUl, KiS, 170, 172, F(O)., 1954. Read, W. T., Jr., see Pearson, G. L. Remeiic\, J. P., see Anderson, P. W. Rose, D. J.,' and Allis, W. P.^ Transition from Free to Ambipolar Diffusion, Phys. Rev., 93, pp. 84-93, Jan. 1, 1954. Ryder, R. ^l.} and Sittner, W. R.^ Transistor Reliability Studies, Proc. I.R.E., 42, pp. 414-419, Feb., 1954. Schlaack, N. F.^ Development of the LD Radio System, I.R.E., Trans., P.G.C.S., 2, pp. 29-38, Jan., 1954. Severiens, J. C.,^ and Fuller, C. S. Mobility of Impurity Ions in Germanium and Silicon, Letter to the Editor, Phys. Rev., 92, pp. 1322-1323, Dec. 1, 1953. Shive, J. N., see Slocum, A. Shockley, W.,^ see Tessman, J. R. SiTTNER, W. R., see Ryder, R. M. Slocum, A.,^ and Shive, J. N.^ Shot Dependence of p-n Junction Phototransistor Noise, Letter to the Editor, J. Appl. Phys., 25. p. 40(3, March, 1954. Bell Telephone Labaratories. 1016 the bell system technical journal, july 1954 Smith, C. S.' Piezoresistance Effect in Germanium and Silicon, Phys. Re\-., 94, pp. 42-49, April 1, 1954. Stiles, K. P.' Overseas Radiotelephone Services of A. T. & T. Co., I.R.E., Trans , P.G.C.S., 2, pp. 39-44, Jan., 1954. Stubbs, R. R.^ Telephone Service is a Big Bargain, Telephonj^, 146, pp. 20-21, 43, March 13, 1954. Tessman, J. R.,^ IL^hn, a. H.,^ and Shockley, W.^ Electronic Polarizabilities of Ions in Crystals, Phys. Rev., 92, pp. 890-895, Nov. 15, 1953. Thomas, D. E.' Single Transistor FM Transmitter, Electronics, 27, pp. 130-133, Feb., 1954. Van Duyne, H., see Fine, M. E. Valdes, L. B.^ Resistivity Measurements on Germanium for Transistors, Proc. I.R.E., 42, pp. 420-427, Feb., 1954. Wilfong, J. C, Jr.''' The Telephone, an Instrument of Culture, Telephony, 146, pp. 22-23, 45, March 20, 1954. WOJCIECHOWSKI, B. M.^ Capacitance Gage Checks Cable Sheath Thickness, Electronics, 27, pp. 134-137, April, 1954. Yager, W. A., see Galt, J. K. 1 Bell Telephone Laboratories, Inc. 2 American Telephone and Telegraph Compan3^ 3 Western Electric Company, Inc. * Southern Bell Telephone Company. ^ Department of Physics, University of California, Berkeley, Calif. " Chesapeake and Potomac Telephone Company-. lecent Monographs of Bell System Technical Papers Not Published in This Journal* BiKLixG, C. A., see Edelson, D. Bowx, R. Vitality of a Research Institution and How to Maintain It, jNIonograph 2207. Campbell, W. H. Current Status of Fretting Corrosion, Monograph 2160. COXWELL, E. M. High Field Mobihty in Germanium with Impurity Scattering Dom- inant, Monograph 2158. Dacey, G. C. Space-Charge Limited Hole Current in Germanium, jMonograph 2157. Edelsox, D., Bip:lixg, C. A., and Kohman, G. T. Electrical Decomposition of Sulfur Hexafiuoride, Monograph 2175. Gray, Marion C. Legendre Functions of Fractional Order, Alouograph 2104. Grisdale, R. O. The Formation of Black Carbon, Monograph 2161. * Copies of these monographs may be obtained on request to the Publication Department, Bell Telephone Laboratories, Inc., 463 West Street, New York 14, X. Y. The numbers of the monographs should be given in all requests. 1017 1018 the bell system technical journal, july 1954 Groth, W. D., and Slade, F. D. Principles of Tape-to-Card Conversion in the AMA System and Mechanized Billing of AMA Toll Messages, Monograph 2190. KoHMAN, G. T., see Edelson, D. Pierce, J. R., and Walker, L. R. "Brillouin Flow" With Thermal Velocities, Monograph 2191. Prim, R. C., see Shockley, W. Read, W. T., Jr. Dislocations or What Makes Metals so Weak? Monograph 2211. Rose, D. J. The Transition from Free to Ambipolar Diffusion, Monograph 2205. Shockley, W., and Prim, R. C. Space-Charge Limited Emission in Semiconductors, Monograph 2156. Slade, F. D., see Groth, W. B. Thomas, D. E. Low-Drain Transistor Audio Oscillator, Monograph 2204. Thurmond, C. D. Equilibrium Thermochemistry of Solid and Liquid Alloys of Ger- manium and of Silicon, Monograph 2210. Walker, L. R., see Pierce, J, R. Wilkinson, R. I. Random Picture Spacing with Multiple Camera Installations, Mono- graph 2188. Contributors to this Issue Albert L. Blaha, B.S. in E.E., Polytechnic Institute of Brooklyn, 1950; Bell Telephone Laboratories, 193G-. In 1937 and 1938, Mr. Blaha was in the quartz crystal development shop. Since then he has been pri- marily concerned with the testing and development of relays. During World War II he worked on magnetostriction type sonar devices. A. J. Brunner, B.S. in M.E., Lewis Institute, 1934; Western Elec- tric Company, 1920-. During Mr. Brunner's early association with Western Electric he was included in an engmeering group developing special machines for the manufacture of telephone products. Later he worked on the development of die casting processes. For the past decade his assignments have been in the field of plastic molding. He did notable work in connection with molding 500-t3^pe handset parts and is presently engaged in engineering the facilities required to manufacture molded parts for the wire spring relay. He is the holder of numerous patents. F. Harold Chase, University of Illinois, 1921; Western Electric Company, 1917-1918; Bell Telephone Laboratories, 1921-. Until joining the Power Engineering Group in 1943 he was concerned with the design of carrier system equipment and maintenance practices on toll equip- ment. Since 1951, he has been developing new uses for transistors in the control of power equipment. H. E. CossoN, B.S. in M.E., Michigan College of Mining and Tech- nology, 1949; Allis Chalmers Manufacturing Compan}^, 1949-51; A. 0. Smith Corp., 1951; Western Electric Company, 1951-. Mr. Cosson served thirty-one months during World War II, fifteen of which w'ere on a naval aircraft carrier. Since joining the development engineering group at Western Electric Company, he has worked on problems associ- ated with straightening wire for the wire spring relay. Junior member A.S.M.E. Thomas E. Davis, B.S. in E.E., University of Arizona, 1928; Bell Telephone Laboratories, 1928-. He has been concerned \\ith apparatus development projects, including those related to microphones, handsets 1019 1020 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 and echo suppressors for long telephone lines. During World War II he worked on underwater sound systems for the Navy and was awarded the Naval Ordnance Development Award by the U. S. Navy. Since then he has been working with the wire-spring relay and Hne concentrator for the No. 5 crossbar system. Member American Institute of Electrical Engineers and Tau Beta Pi. B. H. Hamilton, B.S. in E.E., University of Kansas, 1949; Bell Tele- phone Laboratories 1950-. With the Laboratories he has worked on de- velopment of equipment to power the L3 carrier system and has been concerned with fundamental studies of new types of regulated rectifiers. Member American Institute of Electrical Engineers, Tau Beta Pi, Sigma Tau, Sigma Xi, Kappa Eta Kappa. A. L. QuiNLAN, B.S. in E.E., University of Kansas, 1921; Western Electric Company, 1921-. Prior to World War II, Mr. Quinlan worked extensively on the development of manufacturing methods and machines for loading coils. He was granted patents on loading coil case designs and on toroidal coil winding machines and was co-author of an article, Recent Improvements in Loading Apparatus for Telephone Cables, published in the A.I.E.E. Journal, Dec. 1947. During the war he had engineering as- signments on gun director, precision coil manufacture and vacuum tube projects. Since then he has developed manufacturing facilities and meth- ods for welding precious metal contacts to telephone switching apparatus. Notable among these are the roll welding of contact tape to crossbar switch multiples and the resistance and percussion welding of contacts to Avire spring relays. Member A.I.E.E. William Shockley, B.Sc, California Institute of Technology, 1932; Ph.D., Massachusetts Institute of Technology, 1936; Teaching Fellow, M.I.T., 1932-1936; Bell Telephone Laboratories 1936-1942; Director of Research, Antisubmarine Warfare Operations Research Group, Division of War Research, Columbia University, 1942-1944; Expert Consultant, Office of the Secretary of War, 1944-1945 ; Bell Telephone Laboratories 1945-. Appointed Director of Transistor Physics Research December 1, 1953, he had directed the group which invented the point-contact transis- tor. During the past six years he has made many contributions to solid state physics particularly in connection with the transistor. In addition to solid state physics and semiconductors, his work has also included vacuum tube and electron multiplier design, studies of various physical phenom- ena in alloys, radar development and magnetism. Medal for Merit, US. CONTRIBUTORS TO THIS ISSUE 1021 War Department, 194G; Air Force Association Citation of Honor, 1951; Morris Licbmann jNIemorial Prize, I.R.E., 1952; Oliver E. Buckley Solid State Physics Prize, American Physical Society, 1953; Certificate of Appreciation, Department of Army, 1953; Comstock Prize, National Academy of Sciences, 1954. Fellow of American Physical Societ}^; Senior Member Institute of Radio Engineers; Member of National Academy of Sciences, Tau Beta Pi, Sigma Xi. For the past few months he has been on leave to California Institute of Technology for teaching and study in the field of solid state physics. DoxALD H. Smith, B.S. in E.E., University of Minnesota, 1944; Bell Telephone Laboratories, 1947-. After working with the Systems Depart- ment of the Laboratories on trial installations, Mr. Smith was concerned with rectifiers and regulating systems in power development. He is cur- rently in charge of the group doing long-range engineering on power de- velopment. ^Member of A.I.E.E. and the Amateur Astronomers Associa- tion. R. W. Strickland, B.M.E., University of Florida, 1951; Western Electric Companj^, 1951^. Mr. Strickland served two and a half years in the U. S. Armed Forces prior to receiving his degree. Since coming to the Western Electric Company, he has been active in the development of equipment and processes for molding of plastic components of the wire spring relay. Junior member A.S.M.E. Harry Suhl, B.Sc, University of Wales, 1943; Ph.D., Oriel College, University of Oxford, 1948. Admiralty Signal Establishment, 1943-46; Bell Telephone Laboratories, 1948-. Dr. Suhl conducted research on the properties of germanium until 1950 when he became concerned with electron d>aiamics and solid state physics research. His current work is in the applied physics of solids. Member of the American Institute of Ph\'sics and Fellow of the American Physical Society. Eric E. Sumxer, B.M.E., Cooper Union, 1948; M.A. Degree in Physics, Columbia University, 1953; Instructor of Physics, Cooper Union, 1947-48; Non-resident instructor of Massachusetts Institute of Technology on Probability and Statistics — Applications to Sampling and Quality Control, summer, 1950; Bell Telephone Laboratories, 1948-. INlr. Sumner was given rotational assignments in apparatus, switching, and Television transmission development and switching research, and has worked on a number of projects, including the card translator, the mag- 1022 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 netic drum, video transmission e valuator, vibrating reed selector, de- velopment of wire-spring relay, trouble recording apparatus development, and transistor circuitry for a subscriber line concentrator. He is cur- rently engaged in developing small functional circuits for an electronic switching system. Member of Tau Beta Pi and Pi Tau Sigma. Erling D. Sunde, E.E., Technische Hochschule, Darmstadt, Ger- many, 1926. Brooklyn Edison Company, 1927 ; American Telephone and Telegraph Company, 1927-1934; Bell Telephone Laboratories, 1934-. Mr. Sunde 's work has been centered on theoretical and experimental studies of inductive interference from railway and power systems, light- ning protection of the telephone plant, and fundamental transmission studies in connection with the use of pulse modulation systems. Author of Earth Conduction Effects in Transmission Systems, a Bell Laboratories Series Book. Member of the A.I.E.E., the American Mathematical So- ciety, and the American Association for the Advancement of Science. Laurence R. Walker, B.Sc. and Ph.D., McGill University, 1935 and 1939; University of California, 1939-41. Radiation Laboratory, Mass- achusetts Institute of Technology, 1941-1945; Bell Telephone Labora- tories, 1945-. Dr. Walker has been primarily engaged in research on microwave oscillators and amplifiers. At present he is a member of the physical research group concerned with the applied physics of solids. Fellow of the American Physical Society. rHE BELL SYSTEM / echmcai lournal EVOTED TO THE SC I E NTI FIC^^^ AND ENGINEERING SPECTS OF ELECTRICAL COMMUNICATION OLUME XXXIII SEPTEMBER 1954 NUMBERS ' — KANSA.y ^i'VV, MO. — " PUi>LiC LIi>i direction of the 1023 1024 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 magnetization in such materials. In the ferromagnetic metals, it is well known that these losses ordinarily arise largely from the eddy currents which are induced by the motion of the domain walls. In the ferrites, however, the conductivity is so low that the contribution of eddy cur- rents to the losses is never overwhelming and is often negligible; the losses must therefore in large part arise from other sources not yet under- stood. It is the purpose of this paper to present some recent studies of these losses and to discuss their relevance to the losses in ferrites gen- erally. In any ordinary sample of a ferromagnetic material, a study of domain wall motion and the associated energy losses is complicated by the fact that the domain pattern is very complex. Any attempt to provide a theoretical explanation of data taken on such samples must invoh^e an averaging process over many domain walls of varying area, crystal orientation, etc. This makes it extremely difficult to describe the be- havior of such patterns uniquely and quantitatively, although some progress has been made. ' A method of avoiding this difficulty has been developed by Williams, Bozorth, Shockley, Kittel and Stewart '^ in working on silicon iron. This method consists in cutting a polygonal ring from a single crystal in such a way that each leg of the ring lies along one of the easy directions of magnetization in the crystal. In silicon iron this leads to a rectangular ring with each leg along a [100] crystal direction. In the ferrite which we use this technique to study here, the easy directions are [111] directions, and we use a diamond shaped sample as shown by the solid lines in Fig. 1. Each leg is along a [111] direction, and the major face is a (110) plane. If the sample is good enough, the domain pattern is that indicated by the dotted lines in Fig. 1. This pattern consists of four stationary walls, one at each corner, and one movable wall which goes all the way around the sample. The mag- netization thus travels around the sample in two paths, one clockwise and the other counter-clockwise, and the position of the movable wall therefore determines the net circumferential magnetization. In such samples we study quantitatively and in some detail the motion of an individual movable wall. Williams, Shockley and Kittef studied the motion of the movable wall on one of the rectangles cut from a single crystal of silicon iron. They found the motion to be viscously damped, as Sixtus and Tonks had in earlier experiments with more complicated domain walls. I^e- cause of the simplicity of their domain pattern, Williams, Shockley and Kittel were able to cak^ulate the eddy current losses in their experi- ments, and to show that they accounted for most of the observed damp- MOTION OF IXDIVIDIAL DOMAIN' WALLS 1025 ing as expected. There was an additional contiibution, however, which Kittel suggested was due to mechanisms of the sort which give rise to the width of ferromagnetic resonance hnes. These mechanisms of course are the controlhng ones in the ferrites, where eddy current losses are small. The motion of a domain wall damped by such effects and un- affected by eddy currents was first discussed in a classic paper by Landau and Lifshitz. The experiments reported in the present paper consist of measure- ments of the velocity of a movable wall as a function of applied magnetic field in a sample like that shown in Fig. 1 . The measurements are made by observing the voltage induced in a secondary winding on such a sample when a knoAvn field is applied by means of a pulse of current in a primary winding. The composition of the ferrites used in these studies is given by the approximate chemical formula (NiO)o.75(FeO)o.26- Fe203 . Data have been taken as a function of temperature on several samples. The large, perfect crj'stals of the ferrites which are essential to the success of these experiments have been obtained through Dr. POSITIONS OF ' DOMAIN WALLS Fig. 1 — Sample of fcrrite used in this study. 1026 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 G. W. Clark from the Liiide Air Products Company. Similar studies have been performed pre^'iously at, room temper-ature on Fe304 ' by the author, and in a preliminary way on (Xi())(i.9i (FeO)o.,,9 FeoOs ^ by the author in eollaboration with J. A. Andrus and H. G. Hopper. THE EXPERIMENTS Preparation of Samples The key to the success of experiments of this sort is of course obtain- ing the domain pattern shown in Fig. 1, so that we ha\'e only one mo\'able wall in the sample. The achie\'ement of this pattern, and the obser\'a- tion of it when achieved, depend in turn on success in producing a perfect or almost perfect sample. It therefore seems worth while to describe the process which has finally emerged as a satisfactory way of producing these samples. The rough crystal is first oriented by means of X-rays (Laue and X-ray goniometer techniques) to an accuracy of a few minutes while it is mounted in such a position that afterwards we can grind a flat on it w^hich is coincident with the (110) plane. This flat is ground simply with a belt grinder. A cut is then made with a diamond saw parallel to this flat, so that we have a disc whose faces are (110) crystal planes. This disc is usually made about 2}/^ to 3 mm thick. The second face is ground accurately parallel to the first in a paralleling block. A flat coincident with the (100) plane is ground on the edge of this disc for later use in orienting the sample in the plane of the disc. This flat is also ground with a belt grinder after orienting the disc wdth Laue and X-ray goniometer techniques. The disc is ground as smooth as possible with 3031.2 emery, and then very carefully polished. The polishing is done first on a lap surfaced with No. 0000 french emery paper, with Linde A abrasive loose on top of the emery paper. After this the lap is surfaced with a sheet of \'ery smooth paper and then Linde B abrasive is used. The polishing process takes four to eight hours per disc, and removes all pits ^'isible under 50 X magnification, except those inherent in the crystal. Only on such a smooth surface can the very small holes which sometimes occur in these crystals be seen. Sometimes, however, the polishing process conceals fine (Tacks. It also cannot reveal variations in chemical composition which sometimes occur from point to point in the crystals. Such com- position variations are presumably variations in the concentration of divalent iron from point to point in the crystal. In order to reveal these latter imperfections, the disc is etched for three hours by boiling it in MO'I'IOX OF l\I)IVinr\L DOMAIN WALLS 1027 50 per cent H2S()4 uiuler a reflux condenser. An asl)esto.s pad is placed between the flame and the bottom of the flask containing the II2SO4 ill order to pre\-ent sharp temperature fluctuations in the bath. The rate of the etching attack, and the ([uaUty of the surface it leaves are sharply- dependent upon temperature. It should be mentioned that if this etch is used on the discs before polishing, the surface remains rough or may c\ (Ml be made rougher, so that it is impossible to detect the imperfec- tions in the disc. The etch must start on a smooth surface. Once the disc is cut, polished, and etched, if it is found to be sufiiciciitly U-cc of imperfections, a sample is cut from it in such a way as to avoid those which there are, as they are revealed by the polishing and etching processes. First the diamond-shaped hole is cut by means of a jig whose rotational posiiion with respect to the disc is determined from the (100) flat on the edge of the disc. This jig is a piece of steel which is driven in \ibiation \ertically with a magnetostrictive drive.^ The surface of the disc is covered with a sIvht}^ of carborundum or diamond dust, and this abrasive is made to cut a hole in the disc as the vibrating jig is slowly lowered. With the hole cut in the proper orientation, the outer parts of the disc are ground down to form the legs of the sample. Another jig of the proper shape is used to hold the sample in position during this process. A hysteresis loop is taken as soon as the sample is cut. A relatively good loop taken on our best sample is shown in Fig. 2. All stich loops on these samples are taken on the Cioffi recording fluxmeter.'" This loop is obviously not yet in the form which we finally need. In order to scpiare the hysteresis loop, we anneal the sample for approximately an hour at ()00°C in a magnetic field of 10 to 20 oersteds. The field is pro- duced by running a current through a few turns of glass insulated wire wound on the sample. After such a heat treatment the hysteresis loop of this sample assumed the form shown in Fig. 3. Once the sample is prepared, the next problem is to observe the domain pattern and find if any important deviations from the pattern shown in Fig. 1 occur. The heat-treatment we give them corrodes the polished surfaces of the sample, and of course the faces exposed when the sample is cut fi'om the disc have not yet been polished. Consequently both the major (110) faces of the sample and the outer faces of the legs are pol- ished, and the sample is then etched in the same way as before. Usually a hysteresis loop is again taken at this point as a check. If the sample is good, it is not significantly different from the loop taken immediately after heat-treatment. The sample is l)rought to a demagnetized con- dition at this point so that the movable wall will be near the center of the sample where it can be observed. This completes the process of 1028 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 4000 ^ y^ ^^ 3000 / / ^ 1 / / B IN GAUSS ) C J ' 1 / } ' / / -4000 •0.8 -0.6 -0.4 -0.2 O 0.2 0.4 H IN OERSTEDS Fig. 2 — Hysteresis loop taken on sample before heat treatment in a magnetic field. preparing the samples for observing their domain pattern and for per- forming our experiments on them. Four samples have been prepared in this way for our studies on (NiO)o.75 (FeO)o.25 FesOs . Domain Pattern Observations The method used to observe the domain walls on these surfaces is the same as that used by Williams and his collaborators. We will there- fore not describe it in detail. It consists essentially of observing through a microscope the pattern formed by a magnetic colloid on the surface. The observation of domain patterns, even on these carefully pre- pared samples, is difficult. There is still some pitting on the surfaces. Also, many of the surfaces become rounded in the process of polishing. This produces surface spikes of the sort discussed by Williams, Bozorth and Shockley.^ The result is that on many surfaces the domain pattern of the sample as a whole has to be discerned in a substantial amount MOTION OF IXniVIDTAL DOMAIN WALLS 1029 of extraneous structure such as surface spikes and pits. It is therefore inipossiblo to show in one picture the whole pattern as diagrammed ill I'ig. 1. However, more detailed pictures of parts of the pattern do show that it is there. The essential features of the pattern on our best sample are shown in Figs. 4 and 5. Fig. 4 shows the stationary wall at one corner, and Fig. 5 shows a section of the movable wall on one leg. Differentiation of the domain walls in Figs. 4 and 5 from the many scratches is not very difficult after one has some experience in such obser\'ations. The variation in wall position along the leg shown in Fig. 5 is due to the effects of strains and other imperfections, present even in this care- fully prepared sample, in determining the position of the wall at rest. i ; I -0.2 0 0.2 H IN OERSTEDS Fig. 3 — Hysteresis loop taken on same sample as loop in Fig. 2 after annealing for one hour at 580°C in a magnetic field of approximately 20 oersteds. 1030 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 Fig. 4 — Picture of domain pattern at one corner of sample. The stationary wall is indicated by arrows. MOTION OF ixnivinrAi. domain walls 1031 Fig. 5 — Picture of a section of the movable wall along one leg of sample. The line has been slightly emphasized by retouching after the picture was taken. The edge of the leg can be seen at the bottom of the figure. It is unlikely that the wall is so distorted from a plane when in rapid motion, however, since then the driving force and the viscous damping resistance to motion are both much larger than the effects of these imperfections. The imperfections, of course, are primarily effective in tietermining the coercive force, as read from the hysteresis loop. The domain pattern as traced out on each of four samples is discussed below : Sample 1. Although they had spikes associated with them, the sta- tionary walls expected at the corners could be seen, at least in part. In addition, at one of the acute angle corners there was some rather ex- tensive domain wall structure. This structure had one form when the sample was magnetized in one direction, another when it was magnetized in the other. It was due to the presence of a small void at this corner whose magnetic energy was reduced by having domains of reversed magnetization around it. The existence of this void was established from stiiicture which {ippcared in the spots on an X-ray Laue photograph taken at this point. The process of magnetization in this sample con- sisted in (a) the growth of the wall from the nucleus around this void imtil it existed all around the ring, and (b) the motion of this wall to the other side of the sample, where it again shrank to a configuration 1032 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 which minimized the magnetic energy of the void. As a result of this state of affairs, the wall was not in equihbrium in the middle of the sample, but always shrank around the void so as to magnetize the sample in one direction or the other. It was impossible therefore to demagnetize the sample so that the wall was at the center of the legs and then have the wall stay there to be observed. The wall could be brought to the center, however, and held there using the technique mentioned by Williams and Shockley^ (see Fig. 8 in their paper). The legs of the sample were so small that it was difficult to do this, but the wall was found on three legs of the sample at different times in this way. The wall curved a good deal in traveling along the legs. This sample was etched repeatedly so that data could be obtained as a fuction of sample dimensions. The variations observed led to a viscous domain wall damping independent of dimensions if it was assumed that the domain wall was perpendicular to the major (110) face. Therefore it was to this (112) plane that the wall was brought for observation. Sample 2. Stationary walls at the corners w^ere rather patchy but visible. The movable wall was traced along the outside faces of two legs, which indicates that it lay in the (110) plane. The wall curved a good deal. There were also other walls which enclosed patches of surface. It is suspected that these patches were the bases of spike domains extending into the sample from strain patterns on the surface. The sample was therefore etched again. Unfortunately, the bath apparently became locally overheated, and this etch took off rather more material than expected. It also left a matte surface on which domain walls could not be observed. The data taken on this sample, however, check those on other samples if we assume a pattern in which the movable wall is in the (110) plane, as our observations lead us to suspect. Sample S. The stationary walls at the corners were seen, but only with difficulty. They were patchy. There was a good deal of structure all along the legs on the major (110) face of this sample, but no wall which ran around the sample could be seen on this face. On the outside of two of the legs, which are (112) faces, pitting and extraneous walls were so bad that the main wall could not be discerned. On a. third, the wall could be traced most of the way. On the fourth, however, there were two walls, one of which could be traced along the whole leg, the other of which went only three-fourths of the way along the leg. Both walls on this leg showed a good deal of curvature. It there- fore appears that the movable wall lies in the (110) plane, but that there is another wall big enough so that it may move and affect our data. This picture of a domain pattern with two movable walls was confirmed by checking the data obtained on this sample with those from others. MOTION OF IXniVIDlAL DOMAIN WALLS 1033 Sample 4- The stationaiy walls at the corners, although patchy, were seen. There was some extraneous domain wall structure on the major (110) faces, but nothing which looked at all like the main wall. On the outside (112) faces of the legs, however, the wall was traced almost all the way around. Only short sections were impossible to trace. The wall curved as usual, and there was some extraneous domain wall structure on these faces, but substantiallj'^ the whole of the ideal pattern shown in Fig. 1 was seen on this sample. The hysteresis loops shown in Figs. 2 and 3 and the domain pattern pictures shown in Figs. 4 and 5 were taken on this sample. Not only was the domain pattern on sample 4 the best and most com- plete, but the data taken on this sample was much the most reproducible. We shall therefore report the data taken on this sample in detail, and simply refer to the results on other samples as a check and to indicate the sort of variations which occurred from sample to sample. Measurements Our procedure in making the measurements is as follows. The sample is wound with a primary and a secondary winding. A square pulse of positive voltage is applied to the primary winding in series with a re- sistor which is large enough to keep the pulse rise time short. The rise time must be short compared to the time required for the field produced by the pulse to reverse the magnetization of the sample. On the other hand, since the pulse is applied for the purpose of reversing the magneti- zation of the sample, the length of the pulse must be at least comparable with the time required for the reversal to occur; if possible, it should be longer than this. The reversal time, of course, is the time required for the mobile domain wall to move from one side of the sample to the other under the field produced by the applied pulse. A second pulse, of nega- tive voltage, is applied to the primary during each cycle of the pulser in order to bring the wall back to its original position so that the phe- nomenon may be observed repetitively. By sjaichronizing an oscilloscope sweep with the pulser, the signal induced in the secondary winding is observed while the applied pulse is on the primary. Since this signal is proportional to the velocity of the wall, it is constant to a first approximation during the application of a constant field. Irregularities in the crystal may cause the velocity of the wall to vary somewhat as it moves across the sample, however, and this will cause the signal to vary too. In this case, the observer reads the average value. Fig. 6 shows an example of the signal induced in the secondary winding as seen on an oscilloscope. Sample 4 was used to obtain this picture. 1034 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 Fig. 6 — Oscilloscope trace of the induced voltage in the secondary winding while a sfiuare current pulse 8m sec long was applied to the primary. The voltage spike at the beginning is not completely understood, but is thought to be con- nected with the fact that the domain wall spikes associated with imperfections do not pull back on the domain wall until it has moved a little distance. In this sample, once this initial peak is over, the wall velocity comes to its steady state value and is not much disturbed by imperfections, so that the observer need do no averaging. Ih this case, the wall was moving with a velocity of 3590 cm/sec. At this velocity, the wall was unable to reverse the magnetization in 8/i sec, so the signal ends as the magnetic field goes to zero at the end of the pulse. After the pressure on the wall due to the magnetic field stops, the domain wall spikes asso- ciated with imperfections pull the wall back slightly, giving rise to the voltage spike in the opposite direction. The applied field due to the prunary pulse is deduced from the cur- rent in the primary winding (measured by observing the voltage across the series resistor) using the solenoid formula H = ^NI. To obtain the relation between wall velocity and induced \oltage per secondary turn we have: Volts/turn = {d^/dt) X 10~* = ^'K^L{^z/ M)w^-,n X 10~^ (l) where {Az/ M) is equal to the domain wall velocity v, and it\vau is the width of the wall between the boundaries of the sample in the direction perpendicular to the direction of magnetization. It is in deriving (1), of course, that we use our detailed knowledge of the domain pattern in the sample. We are thus able to obtain a ^•alue of the domain wall \-elocity v for each value of the applied field H. These data are the results of the experiment. The value of M^ used in (1) is the measured value (322.5 cgs units/cc) at room temperature. The values used at other temperatures ha^^e been deduced on the assumption that Me varies the same way with tempera- ture in this material as it does in magnetite as measured by Weiss and MOTION OF INDIVIDT^VL DOMAIN WALLS 1035 Ferrer. ^^ We have extrapolalcd tlicir data to got the variation up to our highest temperatures. Ill general, a plot of the data turns out to have the form shown in Fig. 7. This is the data taken on Sample 4 at 201°K. The wall does not move until the field exceeds the eoerci\-e force re(}uired to get it past various imperfections in the crystal. Its motion in fields higher than this is viscously damped. The wall velocity, v, therefore follows the relation: V = G{H - He), (2) where H^ is the coercive field and (r is the slope of the line drawn through the data. The \'alue of G is high if the losses are low, and \'iee versa, of course. RESULTS Data on Sample 4 of the sort shown in Fig. 7 have been taken at various temperatures. We show in Fig. 8 a plot of v/{H — He) as a function of temperature for this sample. Clearly, the outstanding fea- ture of the data is a tremendous increase in the viscous damping of the domain wall at low temperatures. Since the other samples were not as satisfactory, for reasons given above, we do not reproduce the data on them explicitly. Similar data ha\'e been taken on Samples 1 and 2, however, and they show the same l)eha^•ior within their accuracy except that the very sharp decrease in v/(H — He) seemed to occur at a somewhat higher temperature. This difference may be due to slight variations in composition among the a. 6 UJ 2 T = 201°K ^ V = 26,150(H-0.075) y y/X) ^ z^'' /^ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 HpRi IN OERSTEDS applied field. T\])ical plot of actual data for doiiiaiii wall velocity as a I'uiictioii of 1036 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 samples (Chemical analysis indicated that Sample 1 was (NiO)o.77(FeO)o.23Fe203 , as distinct from (NiO)o.7B(FeO)o.25Fe203 for Sample 4) but it is possible that other more subtle differences such as the arrangement of the divalent nickel and iron ions are involved. Volt- ages induced in the secondary winding on Sample 3 for various applied fields were much higher than for the other samples at room temperature. This confirms the presence of more than one wall as indicated by the domain pattern. Therefore no further data were taken on this sample. Each sample, after it had been cooled to the temperature of lic^uid 100 200 300 400 TEMPERATURE IN DEGREES KELVIN Fig. 8 — Plot of v/{H — He) for Sami)]e 4 as a function of temperature. MOTION OF INDIVIDI Al. DOMAIN" WALLS 1037 5 300 O 200 O • y T=77 "K y o O' ^^ y-'c) = 160 (H- 0.075) ^- ^ 1<^ 0.6 0. HpRI 8 1.0 1.2 1.4. 1.6 1.8 N OERSTEDS Fig. 9 — Plot of wall velocity as a function of applied field at 77°K. Note that at the higher fields the velocity' is no longer a linear function of the applied field. air once, gave the same value of vfijl — H^ as before. Samples 1 and 2, howe^'er, were cooled several times, and after the later runs they no longer did tliis. In both cases the value of v/{H — He) as deduced from the ideal domain pattern and (1) was lower by about one-half; we inter- pret this to mean that the domain pattern in these samples was changed by the repeated thermal shock. Direct confirmation of this interpreta- tion by observation was not possible, however, for reasons mdicated above in connection \\\t\i domain pattern observations on these samples. In view of the fact that only one point is plotted beyond the knee of the curve in Fig. 8, it should be emphasized that continuous qualitative observations made while the sample was cooling showed that the change was continuous and monotonic. On Sample 1, furthermore, the data at 201°K was somewhat doi\xn from the knee of the curve {v/{H — He) = 18000 cm/sec/oe] because of the fact that the knee occurred at higher temperature as mentioned above. Another feature of the data is indicated by Figs. 9 and 10. Data dis- cussed thus far have been taken at the lowest convenient velocities in order to minimize the possibility of wall distortion. However, when data were taken at higher fields, a non-linearity of the sort shown in Fig. 9 appeared. The average velocity increases more rapidly at the higher fields than our viscous damping coefficient would lead us to ex- pect. This effect was observed at all temperatures, but as comparison of Figs. 7 and 9 Anil show, it set in at lower velocities at low temperatures where the enlarged viscous damping appeared. Simultaneously with the appearance of this non-linearity in the apparent average velocity of the 1038 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 wall, the voltage induced in the secondary winding during the pulse of constant applied field is seen from the oscilloscope trace to distort with time. Fig. 10 shows a series of traces observed at room temperature on Sample 4 which show this distortion increasing from (a) to (d) as the applied field is increased. At the highest fields the trace forms a peak which is almost triangular in shape. We shall discuss the theoretical implications of these data in the next two sections. (O 11,600 — £ 27,000 62,000 — 5/U. SEC Fig. 10 — A series of oscilloscope traces showing the deviation of the wall ve- locity from a constant with time at the higher applied magnetic fields and there- fore at the higher velocities. These pictures were taken at room temperature, hut similar phenomena occur at all other temperatures in the range of applied fields where the nonlinearity in velocity shown in Fig. 9 becomes apparent. The ve- locities, as shown, increase from a to d. MO'l'IO.X OF l\l)l\ll)r \l, DOMAIN WALLS \(Y.V.) THEORY A theoretical aiialj'.sis of the experimental results given in the last section dixicies itself rather naturally into three parts. First we chai- acterize the data in terms of an equation of motion for unit, aica of domain wall. This means we determine the constants of motion (viseous resistance and coercive force in our case) of unit area of wall. Secondly, we show the relation between the viscous resistance of unit area of domain wall, and the constants which characterize the ferromagnetic material in general (saturation magnetization, crystal anisotropy, etc.). This is essentially an application of the recent work of Becker ' and Kittel. Lastly, we calculate the magnitude of the damping from a rela.xation mechanism which accounts for the low temperature effect shown in Fig. 8 at least ciualitatively. Consider unit area of a 180° domain wall between two regions of satiu'ated material. Such a system has an equation of motion for small amplitudes of the applied magnetic field H which may be written: nu + f3z + az = 2-1/,//, (3) where z is the displacement of the domain wall along its normal, ni is its mass per unit area, |3 is a parameter measuring viscous resistance, and a is a stiffness parameter which has meaning only for small fields such as those used in initial permeability measurements. When fields larger than the coerci\e force are applied, as in our experiments, the term con- taining a disappears and the field effectix'e in moving the wall is less than the applied field by an amount e(iual to the coercive force; this is shown b}^ the data given previously in the section on results. This re- formulation of (3) is ({uite reasonable when one remembers the spikes which pull back on the wall, in the experiments of Williams and Shockley, for small wall motions and snap off entirely if the wall mo\'es a large distance. Furthermore, since the velocity of the domain wall rises to its stead^^ value in a negligible time in the.se experiments (Fig. 6) the initial term in (3) is also negligible. As these remarks indicate, under the conditions of the experiment in wall velocity, (2) takes the form: ^k = 2il/.(//app - He). (4) This relation obviously fits the data given previously. The second step in the theoretical analysis starts from an equation of motion for the magnetization M in a small \'olume which was first used by Landau and Lifshitz, and takes ad\-antage of more recent woik of Becker'^ and Kittel." If we consider a volume small enough so that the 1040 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 magnetization in it is everywhere uniform even if we are inside the domain wall, our equation of motion is: "^ = y[M XH] - \/MTM x(M X H)l (5) at Here 7 is the gyromagnetic ratio {ge/2 mc), and X is a parameter, as- sumed to be characteristic of a given ferromagnetic material, which is determined by the magnitude of the damping effects in the motion of M. The magnitude of the last term on the right in (5) is thus determined by the amount of the damping losses. The rate of dissipation of energy in the small volume is H-(dM/dt), where AI is the magnetic moment of the volume, and H is the total magnetic field in the volume. The value of H requires some discussion. Outside the wall H = Ho where Ho is the applied field, which is parallel to the wall. When the wall is moving, however, there is an additional field He inside it. This field, which is normal to the wall, is a demag- netizing field which arises from the tendency of M in the moving wall to have a component normal to the wall. This field has just such a value that the magnetization in the moving wall precesses about it with the Larmor frequency. Its value is : He = -$/y){de/dz), (6) as Becker^^ has shown. Here v is the velocity of the w^all, 2 is a distance coordinate normal to the wall, and 6 is the rotational angle of the mag- netization as we pass through the wall along z. Inside the wall, He is much larger than Ho , but in any case H = He -\- Ho . From (5) we find : H-dM/dt^\He, (7) as Kittel" first showed. In the theory of the domain wall it is shown that 86/ dz in the wall is equal to the square root of the ratio of the in- crease in anisotropy energy as the magnetization turns away from the easy direction of magnetization, to the exchange energy constant. That • 14 IS : ^ = {W) - g{do)]/Ay^\ (8) oz The exchange energy constant A is defined by the following expression for the exchange energy per unit volume due to gradients in the direction MOTION OF IXniVIDTAL DOMAIN "WALLS 1041 of magnetization : Exchange energy/unit vol = A[(VaiY + (Vao)' + (Vas)^, (9) where ai , ao , and 0:3 are the direction cosines of the magnetization. g(d) is the anisotropy energy: g(d) = Ki (ai'a2 + a-ia^ + 0:3 «i ), (10) expressed in terms of 6, and g{9o) is the anisotropy energy along the (Urection of easy magnetization. Note that [g(6) — g(6o)] is always posi- tive. Ki is the first order anisotropy constant. If we use (6) and (8) in (7), and integrate over z along a cylinder of unit cross-section normal to the wall to get the rate of energy dissipa- tion for unit area of moving wall, we have: r H- ^ dz = iXv'/y'A"') r [g{d) - g(do)f' dd = 2HMsV, (11) J-00 dt Jbi where di and 62 are the angular positions of M on the two sides of the wall. 62 — di = T, of course, since we are considering a 180° domain wall. In order to obtain (11), we have used (8) to transform from in- tegration over z to integration over 6 as well as to evaluate (7). We set our result equal to 2MsH (pressure on the wall) times v since this is the rate at which the wall, considered phenomenologically, does work. We may now write : 2Msy'A'^' ^ X W) - g{e,)r dd ^'--^ This is the desired relation between wall velocity and applied field which is to be compared with (4). In this way we find: ^ = (\/y'A"') t [g{d) - g{9,)\"' dd. (13) We have thus shown the relation between the wall parameter jS and the parameter X which measures in general the losses associated with motions of the magnetization. The third part of our theoretical analysis is concerned with a cal- culation of the damping parameter X, or rather the relation between V and H itself, from an explicit physical mechanism. Such a calculation has not been made in the past since the appropriate mechanism on which to base it has remained obscure. Verwey and his co-workers have explained the well known transition at about 115°K in Fe304 as 1042 THE HELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 an order-disorder transition in the arrangement of the divalent and trivalent iron ions. M. Fine of Bell Telephone Laboratories has found a remnant of this transition in crystals of the same composition as those studied in the present research, by means of ultrasonic measurements of elastic constants/ We propose that the mechanism which causes the sharp rise in the damping of domain wall motion at low temperatures is a relaxation associated with this transition. Wijn and van der Heide" have explained in this way observations of theirs of losses associated with initial permeability at low frequencies in certain other polycrystal- line ferrites. The time associated with this relaxation should be short because this rearrangement of ions involves only the motion of electrons from one site to another. It should be of the order of the relaxation time associated with the electrical conductivitj^ of Fe304 . Snoek^^ has suggested some time ago that losses in the ferrites were due to an after- effect (relaxation) which, because of the short time constant involved, must be associated with electron migrations. It is extremely useful to compare our data with a theory of the damp- ing based on the above relaxation mechanism no matter what assump- tions we make in detail about what it is that relaxes. We shall see that we are led quite generally to the result that v/H '~ 1/r where r is the relaxation time for the process. However, in order to perform this cal- culation explicitly we must make more detailed assumptions about exactly what (quantity relaxes with the relaxation time r. Changes in the direction of the magnetization cause changes in stress in the sample because of magnetostriction. One possible assumption is that the re- sulting strain would lag behind this stress and mechanical energy would be dissipated in the crystal. This mechanism, however, cannot act in our case. The magnetization in the two domains on each side of the wall points in opposite directions, but causes the same strain in both, and they have such a large stifTness that the thin region occupied by the domain wall assumes this same strain even though the direction of the magnetization is different there. Thus regardless of the stresses produced in the wall, the strains remain the same, inside and outside the wall, whether it is moving or not; under these conditions no work is done on the lattice, and no energy can be lost in this way by the mo\'ing wall. A calculation has been made by the author on the assumption that it is the magnetization itself which relaxes with relaxation time r. This assumption leads to the result that wall velocity is not linearl}^ dependent upon {H — II r) ; it is therefore not correct, since the data shows such a linear dependence. A similar result is to be expected if we assume that the dielectric polarization relaxes. MOTKJN OF INDIVIOIAL DOMAIN WALLS ]{)\'.i What seems at present likely to })e the approximate nature of tlK^ mechanism, and what we will assume is tlie nature of tlie meclianism for the purposes of an i 1 lust rati \-e calculation is as follows. As the domain wall passes a [)oint in space, and the direction of .1/ chanties, the electrons on tiie (hxalent and ti'ixalent ii'on ions lend to i-eai'rani>;e tiiemselves so as to minimize tlie maj;netocrystalline anistropy energy."" If .1/ changes slowly this anistropy en(M-gv is lu^ar tlie minimum possible value (the rwcrsible value) at all times, and the piocess is almost isothermal. As a result of the fact that the process deviates irre\-ersibly from e(|uilihrium, however, net work is done in bringing about the change. If, on the other hand, the direction of .1/ changes so suddenly that the electrons have no time to rearrange, the process is adiabatic, and the magnetocrystalline anistropy energ}^ varies more \vid(4y with the angular position of M. Since our data is taken at low \-elocities and extrai)olated to zero velocity, it seems most appropriate for us to make a calculation of the losses on the assumption that as we increase the veloeity of the wall we are de\'iatiiig from the isothermal condition. Let us define as a ther- modynamical system the part of the magnetic lattice which lies in a small \olnme fixed in space. This volume is a sheet of unit cross-section in which the magnetization is uniform and which is part of the cylinder of unit cross-section normal to the wall mentioned in connection with (11). From the first law of thermodynamics, as the wall passes the small \'olume, we have: (hv = dU - clQ - dg, (14) where dQ is heat added to the system, dV is a change in internal energy, dw is work done on the system, and g is the anisotropy energy as- sociated with our rearranging electrons. Note that nerg3\ We find: Je 1 2M^ d^dd,/''- (31) hi dd'^ dz The derivatix'es of d with respect to z in (29), (30), and (31) are to l)e e\"aluated by means of (8) and (10). In using these equations, of course, we are assuming that the wall is mo\'ing slowly enough so that its shape remahis that of the wall at rest. Equation (31) shows that v -^ H/t as we mentioned earlier. Inspec- tion of Fig. 8 shows that this relationship explains very satisfactorily the sharp drop in v,'(H — H,) at low temperatures if we remember that T depends on temperature as follows: (32) where e is an activation energy. Finally, the right hand side of (30) may be set ecjual to: / {H-dM/dt)dz, J—r^ as calculated from the Landau-Lifshitz equation, see (11), to obtain a value for X. We find: X = r ^ ^'^ ^^" ^^ . (33) [ [g{d) - g{do)f''dd Je, DISC rs.siox It is clear that (4) fits our experimental data at each temperature. By fitting our data to this expression we obtain values for 13, the para- meter characteristic of the material which measures the damping of the wall. \'alues of 13 ol)taiiH'd in this way are gixcii in Table I. .\. more 1048 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 correct value of Ms for Fe304 at room temperature (475 c.g.s. units) has been used in calculating the jS given in Table I than has been used in previous calculations. In Table I we give in addition to values of /3, values of X at room tem- perature obtained from (13). The value for Fe304 is calculated from the new value of /3 and also corrected for the error mentioned in Reference 14. Values given in References 7 and 8 are somewhat in error. The A'alues of j3 and X given for Fe304 in Table I are the remainders after the contri- bution due to eddy currents has been subtracted out by means of the low field calculation of Williams, Shockley, and Kittel,^ [see their Equa- tion (11)]. Table I — • Room Temperature Data on Fe304 and (NiO)o.75 (FeO)o.25 FezOs (c.g.s. units) P (corrected for eddy currents) X domain wall X ferromagnetic resonance Fe304. (NiO)o 75 "(FeO)o ■z^Fe^Oz.'.'''.'. 0.44 0.023 5.5 X 10^ 4.6 X 10' 9 X 108 10 X 10' Table II — -Data for (NiO)o.75 (FeO)o.25 FesOa* T(°K) Ma (c.g.s. units) Ki (ergs/cc) v/(H - He) (cm/sec/oe) X domain wall (c.g.s. units) 77 201 300 363 400 445 .341 335 322.5 309 298 281 -8.1 X 10* -5.4 X 10* -3.8 X 10* -3.1 X 10* -2.8 X 10* -2.4 X 10* 158 261.50 28500 32500-35300 31750 36850 6.3 X 109 4.4 X 10' 4.6 X 10' 4.0-4.3 X 10' 4.5 X 10' 3.9 X 10' * The value of Ms at 300°^" is measured from the hysteresis loop of Fig. 3. Other values were obtained by assuming that M^ varied with T in the same wa}- as observed by Weiss and Forrer'' in FcsOi . The evaluation of X from (13) is done as follows. Since the wall is in a (110) plane, we find from (10): g{e) - cAOo) = L|l! (2/ Vs - V3 sin^ ey, (34) where d is the angle between M and the [100] direction which Hes in the plane of the domain wall. 6 is cos~^(l/\/3) on one side of the wall, and MOTION' OF INDlViniAL DOMAIN' WALLS 1040 TT + COS l(/\/3) on the other. Then, r+cos-l(l/-v/3) / •'C( loie) - g%)]''dd = om VrKTl. (35) co.s-l(l/v'3) In pcrformino- this integration, rare must be taken to use the positive ^•alue of the square root over the whole interval. It should be noted that in using (8) and (10) to evaluate (13) we are assuming that the wall is moving slowly enough so that its shape is the same as that of the wall at rest. A is best evaluated from a fundamental relation derived by Herring and Kittel^^ between A and the Bloch constant: A = [SoM"'[k/l3.^C"\ (36) where k is Boltzmann's constant, C is Bloch's constant as used in the relation Ms = il/o(l — CT^'^), So is the atomic spin, and fi is the atomic volume. (*So/fi) is equal to the saturation magnetization at 0°K divided by twice the Bohr magneton. For Fe304 , we find C = 4 X 10""' by fitting the Bloch T"- law to the saturation magnetization measurements of Weiss and Forrer." From (3G), assuming Ms at O^iv is 505 c.g.s. units, we then find A — 1.24 X 10~^ Furthermore, 7^1 = -1.1 X lO' as given by Bickford,'' and 7 = (1.76 X 10')^/2 = 1.865 X lO', where we have used Bickford's^^ value (2.12) of g.^^ Now from (13) we find X = 5.5 X 10' in Fe304 at room temperature. For (NiO)o.75(FeO)o.25Fe203 , we assume that .!/« , while different from that for Fe304 , varies in the same way with temperature so that C = 4 X 10" . From our measurement of M^ at room temperature (322.5 c.g.s. units) and this assumption about the variation of Ms with T, we find that Ms at 0°K is 342 c.g.s. units. This leads by (36) to A = 1.09 X 10~ . Ferromagnetic resonance experiments" done by W. A. Yager and F. R. Merritt in collaboration with the author on spherical single crystals of the same ferrite material as that used in the present research give a g value of 2.14 at all the temperatures mentioned in Table II except 77°K, where g = 2.19. These values are used in determining the value of y[^ 1.76 X 10 g/2] in (13). The anisotropy values in Table II are also taken from the results of these ferromagnetic resonance experi- 1050 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 .4 ments. The Ki at room temperature is —3.8 X 10 . The room tempera- ture value of X domain wall given in Table I and the values at various temperatures given in Table II are obtained from (13) using these data. An independent value of X may be obtained from the ferromagnetic resonance line width. The relation between the observed Hne width 2 AH and X has been given elsewhere. Sample shape enters this relation, but not in a critical way, and we therefore ignore it except as it affects the value of the dc magnetic field at resonance, Hres • The relation is: X = AHyMs/H,,, . (37) 22 From Bickford's data on line width and (37) we find X = 9 X 10 for Fe304 at room temperature. From the ferromagnetic resonance data reported elsewhere on (NiO)o.7o(FeO)o.25Fe203 , the material used in the present research, we find X = 10 X 10' at room temperature. Table I compares the room temperature values of X obtained in the two ways on the two materials. The differences between the domain wall experiments and the fer- romagnetic resonance experiments lead to quite different behavior of X in the two cases at low temperatures. These differences can be under- stood in terms of the frequency dependence of X as given by an extension of the relaxation theory given in the third part of the theoretical dis- cussion. A discussion of these relationships must await the detailed report on the ferromagnetic resonance results which is now in prepara- tion, where such an extension will be given. As Table I shows, the room temperature values obtained in the two ways are of the same order of magnitude. Let us now turn to a discussion of the relation between (31) and (33) and the data shown in Fig. 8. Qualitatively, of course, the 1/t factor in v/Ho which (31) reveals, taken together with (32) explains most satis- factorily the sharp increase in viscous damping of the domain wall at low temperatures. Furthermore, it seems tjuite possible, although the author has not investigated it, that the higher order terms in (2-1) account for the nonlinearities in Figs. 9 and 10. It should be mentioned that an increase in relaxation time at low temperatures which is consistent with (32) has been deduced by Bloem- bergen and Wang and Healy from ferromagnetic resonance data taken by them. Quantitatively, we have inadeciuate data for a satisfactory compari- son between Fig. 8 and (31), and the assumptions of the theory should perhaps be investigated further before any such comparison is taken .MOTION' OF 1M)1\ iDlAI. DOMAIN WALLS 1051 seriously. Nevertheless, j)laiisil)le assumptions ran be made which make an instructive comparison possible, i/i^ in (31) is equal to the difference in the variation of the anisotropy energy when measured adiabatically and when measured isothermally. At this stage of our knowledge, many assumptions which still retain the symmetry of the crystal are possible concerning the form of (ji^id). For our present purposes we will assume that it is given to within a constant by (34), but we must introduce a minus sign to take account of the fact that the rearranging electrons must ha\e a posilirc anisotropy encM'gy associated with them. Since we differentiate before substituting in (31), we do not need to subtract out the constant. The amplitude of ^i^ is not given by | i^i | of course; we will call this amplitude \ Kr \. When we introduce the minus sign into (34), and calculate the integral in (31), we find: _ 1 4.0M, "^^!Am/S^"- (38) This relation points up the fact that the mechanism we are discussing here is characterized by two parameters, an amplitude factor | Kr \ and a time r. Our experiment tells us nothing about \ Kr\, so we will ar- l)itrarily assume it is about j^ of | T^i | at room temperature, or 20000 ergs/cc. I Kr \ can be measured by comparing \'alues of | i^i 1 determined isothermally, say by measuring the torque on a disc, and values of I Ki I determined adiabatically, say by ferromagnetic resonance ex- periments, but this has not been done as yet. To determine r at 77°K, assume that the maximum frequency of rotation of dipoles in the wall is such that ccT = 0.1 when the non-linearity shown in Fig. 9 first occurs (at V = 150 cm/sec). We calculate the thickness of the wall to be 4 X 10~^ cm from standard formulae (see Kittel's review article. Reference 14) and find r = 0.5 X 10"^. These values of r and | Kr | , together ^vith I Ki I from Table II at 77°K and A as calculated above, gii^e v/Ho = 40 when in.serted into (38). This is to be compared \\'ith a measured v/(H — He) of 160 at 77°K. In view of the preliminary nature of (31), and the arbitrariness of some of our assumptions, this check is quite sat- isfactory. It would be naive to expect the theoretical result to be closer than an order of magnitude to the experimental one. Inspection of Fig. 8 suggests that the mechanism which gives the sharp increase in damp- ing at 77°K is submerged at room temperature in the effects of other mechanisms, perhaps other electronic rearrangements, perhaps exchange effects of the sort recently suggested in metals by Kado.'^ Equation (30) confirms this expectation when we use t = 10^'', as determined either 1052 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 from ferromagnetic resonance data' or from the conductivity of Fe304 . The above method of calculating r from domain wall data is less sat- isfactory at room temperature. The idea of associating losses in ferrites such as this one with changes of order in the divalent and trivalent iron ions explains the fact that the damping obser\^ed in domain walls at room temperatuie in Fe304 , where none of the divalent iron is replaced by nickel, is larger than that observed in (NiO)o.75(FeO)o.25Fe203 . Finally, as Wijn and van der Heide^' have pointed out, and as (31) and (33) show more analyti- call}^, this mechanism is a very satisfactory explanation of the sharp change in domain wall relaxation frequency observed by Gait, Matthias, and Remeika.^^ ACKNOWLEDGEMENTS The author wishes to express his gratitude to many friends and col- leagues for assistance with various aspects of this research. The crystals from which these samples were cut were obtained through Dr. G. W. Clark from the Linde Air Products Co. H. J. Williams has been of considerable help in observing the domain patterns. Most of the work of cutting the samples from bulk crystal and preparing them was done by J. A. Andrus, Mrs. M. R. Tiner, and R. E. Enz. The hysteresis loops were taken in collaboration with P. P. Cioffi and F. J. Dempsey. The special pulser used in obtaining the data was designed by H. R. Moore and built by H. G. Hopper. The chemical analyses were made by H. E. Johnson, J. F. Jensen, and J. P. Wright. Enlightening discussions were had with J. F. Dillon, Jr., C. Herring, A. N. Holden, B. T. Matthias, W, T. Read, H. J. Williams, and A. M. Clogston, who derived inde- pendently a result somewhat similar to that in (31). Useful comments were made on the manuscript by R. M. Bozorth, S. INIillman and P. W. Anderson. REFERENCES 1. Ratio, Wright and Emerson, Phvs. Rev., 80, p. 273, 1950. Rado, Wright, Emer- son and Terris, Phvs. Rev., 88, p. 909, 1952. G. T. Rado, Revs. Mod. Phvs., 25, p. 81, 1953. Welch, Nicks, Fairweather and Roberts, Phvs. Rev., 77, p. 403, 19.50. 2. D. Polder and J. Smit, Revs. Mod. Phvs., 25, p. 89, 1953. J. J. Went and H. P. J. Wijn, I'hvs. Rev., 82, }). 269, 1951. Wijn, Gevers and van der Burgt, Revs. Mod. Phvs., 25, p. 91 , 1953. H. P. J. Wijn and H. van der Heide, Revs. Mod. Phvs., 25, p. 98, 1953. H. P. J. Wijn, Thesis, Leiden, 1953. Separaat 2092, N. V. Philips Gloeilampenfabrieken, Eindhoven, Holland. 3. H. J. Williams, Phvs. Rev., 52, p. 7-47, 1937. H. J. Williams and W. Shockley, I'hys. Rev., 75, p. 178, 1949. Williams, Bozorth and Shockley, Phys. Rev., MOTION' OF INDIVIDVAL DOMAIN WALLS 1053 75, p. 155, 1940. Williams, Sliorklcv and Kittel, Phys. Hov., 80, j). 1()!K), llC)!). H. J. Williams, HoU Lahs. Kcconl 30, p. 385, 1952. 3a. K. H. Stewart, Vvov. Plus. Soc, 63A. p. TGI, 1950 and J. IMivs. ct Hadium, 12, p. 325, 1951. 4. K. J. Sixtus and L. Tonks, Phys. Rev., 42, p. 419, 1932. 5. W. A. Yager and R. M. Bozorth, Phys. Rev., 72, p. SO, 1947. 6. L. Landau and E. Lifshitz, Physik. Z. Sowjctunion, 8, i). 153, 1935. 7. J. K. Gait, Phys. Rev., 85, p. G64, 1952. 8. Gait, Andrus and Hopper, Revs. Mod. Phys., 25, ]). 93, 1953. 9. W. L. Bond, Phys. Rev., 78, p. 646, 1950, Abstract I 10. 10. P. P. Cioffi, Phys. Rev., 67, p. 200, 1945 and Rev. Sci. Instr., 21, p. 624, 1950. 11. P. Weiss and R. Forrer, Ann. Phys., Series 10, 12, p. 279, 1929. After the pres- ent work was completed, the author became aware of the extensive measure- ments of Pauthenet (Ann. Phys., Series 12, 7, j). 710, 1952) on Ms in nickel ferrite and magnetite, among other materials. Interpolation between Pau- thenet's values for these two materials suggests that the value of Ms we have used for (NiO)o.75 (FeO)o.25 Fe^Os is about 6% too high at 445°K and 3% too low at 77°K, with intermediate errors between these temperatures and room temperature. Since, however, the accuracy of our data is not significantly better than this, and since our conclusions would be unaffected by any such changes, no correction has been made to our data. 12. R. Becker, J. Phys. et Radium, 12, p. 332, 1951. 13. C. Kittel, Phys. Rev., 80, p. 918, 1950; J. Phys. et Radium, 12, p. 291, 1951. 14. C. Kittel, Revs. Mod. Phys., 21, p. 541, 1949. See Eciuation 3.3.9. Since a wall perpendicular to the [100] direction in a crystal with a positive anisotropy energy constant is discussed in this reference, g(5o) = 0 and is left out. The author is grateful to A. M. Clogston for pointing out the need for it in the present research. It was ignored in References 7 and 8 with the result that the values given for X in those references were somewhat in error. Correct val- ues are given in Table I. 15. E. J. W. Verwey and J. H. de Boer, Rec. Trav. Chim., 55, p. 531, 1936. E. J. W, Verwey and P. W. Haa^-mann, Physica, 8, p. 979, 1941. Verwey, Haaymann and Romeijn, J. Chem. Phys., 15, p. 181, 1947. 16. M. E. Fine and X. T. Kenny, paper to be published. 17. H. P. J. Wijn and H. van der Heide, Revs. Mod. Phys., 25, p. 99, 1953. H. P. J. Wijn, Thesis, Leiden, 1953. Separaat 2092, N. V. Philips Gloeilampenfabrie- ken, Eindhoven, Holland. 18. The author wishes to acknowledge a conversation with B. T. ^Matthias in which this mechanism was independently suggested in connection with the present research. 19. J. L. Snoek, New Developments in Ferromagnetic Materials, Ellsevier, 1947. 20. It should be mentioned that Xeel (J. Phys. et Radium 12, p. .3.39, 1951; 13, p. 249, 1952) has suggested that the after-effect losses discussed by Snoek'^ in connection with the diffusion of carbon and oxygen in iron arise from an anisotropy relaxation. Xeel's analysis, however, leads him to jjredict zero loss for large motions of a 180° domain wall, which is contrary to our ex- perimental results. We suggest that he is led to an erroneous result because he allows the anisotropy energy itself to follow a relaxation of the form of Equation (18), whereas kinetic lo.sses of this sort are due to the relaxation of one of two conjugate thermodynamical variables whose product is an energy. In our case these variables are the torque on the magnetization due to anisotropy, and the angle of rotation of the magnetization. It is this torque which satisfies the relaxation eciuation. 21. C. Herring and C. Kittel, Phvs. Rev., 81, p. 869, 1951. See i;^ ! ^^^^ V iX FIELD OF USE FOR 3DB OBJECTIVE o o o 1 00 oO X (\J G o o o (M OO (O X G 00 o *• O X O Ol 10 12 14 16 TRUNK LENGTH IN MILES 26 Fig. 2 — Illustrative field of use; toll connecting or tandem trunks. gauge. In general, it is the practice to select the smallest gauge which will give the required transmission loss. For short distances, the dif- ferential in cost between trunks of different gauges is not large but as trunk length increases it will be found that a smaller gauge trunk with a repeater will cost less than one of larger gauge without a repeater. For example, at 10 miles a 22-gauge circuit with a repeater will cost substantially less than a 19-gauge circuit with no repeater. Also, as transmission objectives are improved, there may be cases where a smaller gauge trunk with two repeaters will be cheaper than a larger gauge with one repeater. Furthermore, in cases where the costs are about equal or even where the larger gauge is slightly cheaper, it will be found that lower losses can be obtained with repeatered circuits and the engineer- ing choice would accordingly favor the smaller gauge. Construction costs differ considerably depending on local conditions; hence the differential cost between the various gauges of conductors will differ correspondingly. Howe\-er, for illustrative purposes Figs. 1 and 2 have been prepared which utilize average Bell System costs for both outside plant and telephone repeaters. In the case of the E23 repeaters, the experience so far has been somewhat limited so there is a greater degree of unccniainty as to the actual costs than for the older tyjie re- peater or the outside plant constru(;tion. Fig. 1 shows the field of use for various conductor gauges when used in interoffice trunks where an over-all transmission objective of 6 db for lOGO THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 the trunks has been assumed. In considering this chart it will be seen that non-loaded 24-gaiige conductors can be utilized for the shorter distances up to about two miles. The next step is to apply loading, which is indicated in the chart by the symbol H88 meaning 88 millihenry coils at 6,000 foot spacing. After reaching the limit of about 7 miles without repeaters on 22-gauge, the differential between 22- and 24-gauge is more than enough to pay for a repeater, the simple series repeater (E2) being used for distances up to 10 miles and then the improved E23 re- peater extending the use of 24-gauge to about 14 miles. Beyond this point the most economical combination is indicated. The general effect of the introduction of the negative impedance re- peater is to shift the average gauge distribution so that more small gauge cable can be economically utilized, accompanied in most cases by im- proved transmission. It will be noted from the figure that no 19-gauge cable will need to be added in the future to care for interoffice trunks up to distances as great as 25 miles between offices. Of course, in cases where 19-gauge is already available in plant it can be used to advantage despite the fact that for new construction a smaller gauge with the nega- tive impedance repeaters would be cheaper. It has also been assumed in making up Fig. 1 that supervision or pulsing requirements will not limit the use of the smaller gauges. In a specific case where some of the older type central offices are involved, signaling may have an important bearing and substantially distort the economic ranges indicated on the chart. Fig. 2 illustrates the field of use of toll connecting or tandem trunks. As a toll connecting trunk forms part of a multilink connection, and since there are always at least two (and often more) trunks in series on connections involving these trunks, the Bell System companies find it economical to plan for a maximum loss of 3 db for this type of trunk. Likewise for the tandem trunks, where two in series may be used in place of an interoffice trunk, 3 db is considered a reasonable ol:)jective. It will be noted again by referring to this chart that 19-gauge has very little future field of use and in all cases the new "series-shunt" repeater will be utilized rather than the earlier series type because of return loss considerations and also because of the greater gain required to reduce the trunk losses to the desired values. In the tandem and toll connecting case, signaling may again be a distorting factor though not to so great an extent as in the direct interoffice trunk case. To this extent, however, the curves are theoretical, as they haxe been made up without regard to this limitation which may apply in a few practical cases. NEGATIVE IMPEDANCE TELEPHONE REPEATERS 1061 SPECIFIC APPLICATIONS ^lany installations of the new rcpcatci-s have been ciijiiiiccrcd lor completion in 1954. In all eases economic stndies were made and re- sults of these studies broadly confirmed the hidications of the two charts discussed abo\'e. 'J'o bring out more clearly the effectiveness of the E23 repeater, it ma^' be worth while to consider a few specific cases invohing installations of the new repeaters being made this year. The first case shown on Fig. 3 is in the area of the Pacific Telephone and Telegraph Company. The figure shows th(^ two altcn-natives that were considered for providing additional tandem trunks between San Francisco proper and the East Bay Area, needed this year because of an extension of customer toll dialing arrangements. The engineering study for this project was somewhat more com- plicated than would ordinarily be the case because two possible routes of unequal length and with slightly different amounts of submarine cable were involved. The present route which touches Yerba Buena Island is subject to some hazard from dragging ship anchors but the circuit relief would have been cheaper here than on the other route shown, were it not for the new repeaters. The second route is consider- ably less hazardous and, in addition, is sufficiently removed from the present one to provide increased reliabilit}^ under disaster conditions. Without the repeaters, however, the second route would have l)een somewhat impracticable, since it does not allow easy installation and access to required loading points between the two shore lines. However, with two of the new repeaters on each pair of conductors, nonloaded 22-gauge cable gives substantially the same effective transmission loss as loaded 19-gauge conductors on the shorter route and over a future period will involve less annual cost per circuit. Furthermore, the use of 22-gauge cable permits more than twice as many pairs to be included in the same size sheath in the expensive submarine section. The initial installation of the new type repeaters on this cable will total about 1,300, divided between the Main Office in San Francisco and the jNIain Office on the East Ba}^ side. Individual repeater gains are 5 to 7 db at 1,000 cycles but the repeater gain characteristic is shaped to offset the increasing cable losses at the higher frequencies so that the over-all circuit has relatively uniform transmission in the voice range. The second case is one where the need for the repeaters results from the complex toll switching system at Chicago, Ilhnois. Here it has been found desirable, because of the great \'olume of toll traffic, to have several toll offices scattered throughout the city and suburbs. In general, toll calls coming into these offices from other cities are completed over toll 1062 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 a PQ W Eh NEGATIVE IMPEDANCE TELEPHONE REPEATERS 1063 connecting trunks direct to the called subscribers' central offices. It is not economical, however, to provide enough such trunks to handle peak loads. When all of these direct trunks are busy, an incoming toll call is switched to the subscriber's central office via a tandem office serving his general area. Since the losses of trunks between the tandem and local offices are of the same order as those of the direct trunks from toll to local offices, it would be desirable to operate the trunks between the toll and tandem offices at losses close to 0 db if this were practicable. In this way the same transmission objectives would be met on both rout- ings and no contrast in transmission would be evident to the same subscribers on calls completed over the different routes at different times. Fig. 4 illustrates a specific case of 36 trunks between toll office No. 3 • LOCAL OFFICES TOLL ROUTES Fig. 4 — Typical E23 repeater application; Chicago toll tandem trunks. 1004 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 and Belle Plaine tandem, a distance of about 16 miles. Without repeaters these trunks would have been operated at a loss of about 10 db but with the repeaters they are operated at about 2 db. The only alternative to using E repeaters in this specific case would have been to use the more expensive hybrid-type repeaters. In some cases similar to this one, it was found practicable to temporize by installing the series repeater first and operating it at limited gain until the shunt element became available. Later, when the production of the shunt element was started, the additional units Avere added and full advantage of the new series-shunt design utilized in reducing the equivalent of the trunks to the lowest value permitted by echo return loss considerations. The third example shown on Fig. 5 is in the city of Pittsburgh between Churchill tandem and the town of New Kensington, approximately 17 miles. Here the transmission on existing 19-gauge loaded cable without a repeater would have been about 8 db. The repeaters reduce this figure to between 2 and 3 db which should be satisfactory in this case. It should be noted, however, that 22-gauge with two repeaters would also provide a low enough equivalent, and should it become necessary to supple- ment the present cable at some future date, there will be an opportunity for further savings from the E23 repeaters. J NEW 'KENSINGTON 4 ^ Fig. 5 — Typical E23 repeater application; Pittsburgh toll connecting trunks. NEGATIVE IMPEDANCE TELEPHONE REPEATERS 100.-) Theory negative imi'kdaxc'e concept Both the E2 and E8 repeater elements contain an aniplilier ha\ing multiple feedback paths. The operation of an amplifier circuit of this type can be explained by classical feedback theory. However, experience with the El repeater over the past four years has shown the value of using a negative impedance concept in engineering such a device. Hence, in the explanation of operation given here, the repeater units will be treated simply as two-terminal networks which have negative impedance inputs over the freciuency band of interest. The effect of introducing these impedances into telephone circuits can then be computed by the same simple network theory used to determine the effects of passive impedances. THE E2 REPEATER UNIT The E2 repeater is essentially a two-terminal network the impedance of which has a magnitude | Z \ and a negati\-e phase angle that can vary with increasing frequency from minus 90 degrees, or less, through minus 180 degrees to at least minus 270 degrees. This type of negative im- pedance is shown in the diagram of Fig. 6(a). It has been known for many years as the series type because it could be produced by con- necting the output of an amplifier back in series with its input. More recently it has come to be known as an open circuit stable, negative -R R -R Fig. 6 — The two t\j)cs of negative impedance: (a) i)\w\\ Circuit Stahie and (!)) Short Circuit Stable. 1066 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 impedance, because it will not oscillate when its two terminals are open circuited. THE E3 REPEATER UNIT The E3 repeater is essentially a two-terminal network the impedance of which has a magnitude | Z \ and a positive phase angle that can vary with increasing frequency from plus 90 degrees, or less, through plus 180 degrees to at least plus 270 degrees. This type of negative impedance is shown in Fig. 6(b). It has been known as the shunt type because it could be produced by connecting the output of an amplifier back in shunt with its input. In more recent years it has become kno^\ai as the short circuit stable type because it will not oscillate when its two termi- nals are short circuited. THE NEGATIVE IMPEDANCE CONVERTER The amplifier circuits of both of these repeater units perform the same function: that of a negative impedance converter. The operation of such converters is illustrated in Fig. 7. Fig. 7(a) shows the converter as a four-terminal network having a ratio of transformation k and a shift of phase through a negative angle of approximately 180 degrees over the operating band of frequencies. If, as shown, an impedance Zn is connected to terminals 3 and 4 then the impedance seen at terminals 1 and 2 will be the impedance Z^ multi- plied by the ratio h and shifted in phase through a negative angle of 180 degrees. This impedance will (over the frequency range of zero to in- finity) fulfill the definition given for the impedance presented by the E2 repeater. Hence, Fig. 7(a) can represent the operation of the E2 repeater. IklAio^ZN c |k| /i80°: 1 (a) Fig. 7 — The Negative Impedance Converter: (a) E2, open circuit stable and (b) E3, short circuit stable. NEGATIVE IMPEDANCE TELEPHONE REPEATERS 10G7 Fig. 7(b) shows the same converter, but here the impedance Zx is connected to terminals 1 and 2. The impedance seen at terminals 3 and 4 (at least over the frequency l)and of inten^st) will be Z^ divided bj^ A: and shifted in phase through a positive angle of approximately 180 de- grees. This impedance will (if freciuencies from zero to infinity are con- sidered) fulfill the definition given above for the E3 repeater impedance. Thus Fig. 7(b) can represent the operation of the E3 repeater. From Fig. 7, it is apparent that the same converter circuit could have been used for both the E2 and the E3 repeaters. For practical reasons it w^as not. However, the ratio k and the phase shift in both the con- verter of the E2 and that of the E3 were made approximately the same. OPERATION IN TRANSMISSION LINES Within limitations, the E2 repeater can be represented by a negative impedance, —Z, and the E3 repeater can be represented by a negative admittance, — Y. With a negative impedance and a negative admittance available, losses of transmission lines can be reduced in the manner illustrated in Fig. 8. The transmission line is represented by two net- works as shown in Fig. 8(a). One of these (Network A) is in the form of a T network the series arms of which are represented by impedances Z; and the shunt arm, by an admittance F. This network has a propaga- tion constant ai + j/Si . The attenuation ai represents the major portion of the line attenuation, and the phase shift /Si is that just sufficient to make Network A realizable physically. This representation is necessary because Network A has image impedances each equal to the character- istic impedance (Zq) of the line. If the characteristic impedance of the line w^ere a pure resistance, then the phase shift through this network could be zero and I3i could be zero. But the characteristic impedances of actual lines are not pure resistance; thus the phase shift i8i must be in- cluded in Network A. The other network (Network B) is shown as a box. It has a propagation constant a2 + j02 • Here ^2 represents the remaining phase shift in the transmission line and 02 is an attenuation just sufficient to make Network B physically realizable in view of the image impedances which are both equal to Zo , the characteristic im- pedance of the line. Fig. 8(b) shows the addition to this line of a repeater consisting of a T network made up of negative impedances — Z in the series arms and a negative admittance — Y in the shunt arm. The arm — Z of the repeater adjacent to the line cancels Z of the line. The two admittances — Y and Y cancel and the other series arms — Z and Z also cancel. The result, as shown in Fig. 8(c), is that only the attenua- tion and phase shift of Network B remain. 1068 THK BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 In practice the amount of attenuation (ori) which can be canceled by the repeater depends on the uniformity, the loss and the type of line. The permissible magnitude is computed by conventional methods which are beyond the scope of this paper. Fig. 8 has shown how the combination of a series and a shunt repeater can annul a large part of the attenuation of a telephone line. Much may be accomplished also by a series negative impedance alone as illustrated in Fig. 9. In Fig. 9(a) a transmission line is again represented by two networks A and B. However, Network A now is shown as an L con- figuration having a series arm Z and a shunt admittance Y together with an ideal transformer of ratio 1:K where K exceeds unity. Network A of Fig. 9 is equivalent to Network A of Fig. 8, and Network B of Fig. 9 is the same as Network B of Fig. 8. In Fig. 9(b) the addition to this trans- mission line of a single —Z such as the E2 repeater is shown. This nega- tive impedance —Z cancels the series impedance Z of Network A. The result is shown in Fig. 9(c). NETWORK A z z NETWORK B Zo I «.i+J/3i \^ TRANSMISSION LINE *\ (a) Zo- (c] Fig. 8 — ()|)or;iti()ii of the "T" in a Iransinissioii line: (a) transmission line, 1)) repeater and transmission line, and (e) result of addition of repeater. NEGATIVE IMPEDANCE TELEPHONE KEPEATEHS IOC)!) Thus Fig. 9 .shows that when a .single — Z is added in .series with the conductors of a tran.smi.ssi()n line the attenuation is n^luced, hut the e(|ui\-alent of a shunt conductance }' together with an impedance trans- forniation is l(>ft. The impedance transformation ]:K could he cor- rected hy means of a transform(M- if it were not for the shunt (element )'. Thus an impedance irregularity is introduced, and this irregularity re- lUx'ts power back toward the source. When the reflected power becomes a significant part of the total power passing through the line, transmis- sion is unsatisfactory. Echo is excessive and therefore the use of the single series repeater is limited. THE BRIDGED T STRUCTURE The discussion of the T repeater, illustrated in P'ig. 8, was based on the use of tw^o series negative impedances and a shunt negative admittance. It is perfectly possible, and more economical, to obtain the same effect by using a single .series impedance in a bridged T structure and this NETWORK A Z NETWORK B UK Fig. 9 — Operation of —Z in a transmis.sion line: (a) transmission line, (b) repeater and transmission line, and (c) result of adding repeater. 1070 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 is, in fact, what is done with the E23 combination repeater. Fig. 10 shows how this is accompHshed by using a center tapped Hne coil for the E2 repeater with the E3 connected as a shunt element to the midpoints of the coil. If the coil is considered to be ideal this arrangement is equiva- lent to the configuration shown in Fig. 1 1 in which the coil provides the basic bridge structure: the E2 repeater is the series arm; and the E3, the shunt arm. Incidentally this arrangement of E2 and E3 repeaters is similar to G. Crisson's twin 21-type repeater. For a bridged-T structure, the image impedance equals the square root of the product of the series and shunt arms and the attenuation (in db) is as indicated on Fig. 11. If a network is to be inserted in an electrically long line without in- troducing an irregularity, its image impedance must match the char- acteristic impedance of the line. This would be the case for the bridged T network if Za were set equal to iVZo and Zb were set equal to Zo divided by N . Then the square root of the product of Za and Zb would be Zo . A network make up of negative impedances is designed to match a line in the same way and Fig. 12 is a representation of such a structure. Here, as a matter of convenience, the shunt arm is shown as an impedance E2 CONVERTER ^-vOMy-" LINE TRANSFORM )RMER I I E3 CONVERTER Fig. 10 — The bridged T repeater E23. NEGATIVE IMPEDANCE TELEPHONE REPEATERS 1071 with a +180° phase shift instead of an admittance as used previously. I'he letter A^ designates a numeric or i)i'op()rti()iiality constant. It will be observed that the product of the shunt and series impedances is a real or positive impedance and hence the image impedance is a positive impedance, Zo . The gain is d(>termined entirely by the value of N. Thus if the characteristic impedance of a transmission line is known, together with the gain that the line can support without risk of oscil- lation, then N is known and the repeater network can be adjusted to give the required gain. The advantage of the bridged T as compared to a single series nega- tive impedance such as the E2 can be demonstrated by comparing the relative transmission gains obtainable from the two arrangements. Fig. 13(b) shows the insertion loss of a single impedance Za connected in series with a transmission line having a characteristic impedance Zo . If Za is a negative impedance such as that produced by the E2 repeater then the repeater gain becomes a function of N as shown in Fig. 13(c). If N equals 2 the gain is infinite and the system will oscillate. Thus N must always be less than 2 where Za is a negative impedance of the series or open circuit stable type. Practically, the impedance of the transmission line is not a constant Zo but varies with termination, hue construction and temperature. Thus A^ should be decreased until the negative impedance is always less than the sum of the two line im- pedances in series with it taking into account all possible variations in these impedances. The same limitations on A^ apply to the bridged T repeater of Fig. 12 Zi = IMAGE IMPEDANCE Zi= fzTzE ATTENUATION Zi IN DECIBELS -20 LOG,, Fig. 11 — Schematic of the bridged T network. Zo = IMAGE IMPEDANCE GAIN IN DB = 20 LOG, -^ Fig. 12 — Schematic of the bridged T repeater. 1072 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 as apply to the single series repeater. If A'' is 2 the gain is infinite and the circuit will sing. This similarity goes further. Assume that a single negative impedance Za equal to A'Zo/lSO" is inserted in series in an electrically long line and N is adjusted for stability. If this series element is removed and the bridged T of Fig. 12 is inserted in the same place and adjusted by changing A^ until the system is stable it will be found that A^ will have the same value in the bridged T structure as it had for the single series negative impedance. Thus, if A^ is the same in the case of the bridged T as in the single series impedance, the gain advantage can be obtained by comparing formulas on Figs. 12 and 13 from w^hich it can be seen that the gain advantage of the bridged T is equal in db to 20 logio [1 + (A^/2)]. If a single series repeater can be used in a line to give an insertion gain of 6 db {N = 1) then a bridged T can be used to provide 20 logio (1 + 0.5) or 3.5 db additional. Thus, in this case the series repeater gives 6 db gain as compared to 6 + 3.5 or 9.5 db for the bridged T. These gains are theoretical; in actual lines with simply constructed repeaters the com- parison may not be c^uite so favorable to the bridged T. THE NEGATIVE IMPEDANCE CONVERTER So far the discussion of the E2 and E3 repeaters has been in terms of a "black box" which translates a positive impedance into a negative INSERTION LOSS _ o„ , „^ I; IN DECIBELS ""^O LOG,o — INSERTION GAIN _ -,„ , nr -^3 IN DECIBELS " ^0 LOG,o -^ = -20 LOGio- 20 LOGio (b) 2Zo (c) Fig. 1,3 — • Insertion gain of the E2 repeater. NWiATIVE IMPKDAXCK TKLKrHOX K UKl'KATKRS 1073 impedance tlirouiih a mult iplyiiiu; and pluise shift operation. It will ho inter(>stinj>; to ('\amin(> tlicsc boxes to see what faetors determine their characteristics. THE E2 CONVERTER The E2 negative impedance converter is the same as the El. As dis- cussed elsewhere^ it can be represented schematically as in Fig. 14(a) and also in terms of the eciuivalent circuit of Fig. 14(b) if the coils are assumed to be ideal. The conv(M-t(M- performs much like a transformer. An impedance seen through it is not only transformed in magnitude by the ratio of ] 1 — jU|8 1 to | 1 + m 1 it is also modified by the phase shift of this factor which over the operating band of freciuencies approximates 180 degrees. The symbol m stands for the voltage gain of the electron tube and /3 is the ratio of 1 to 1 + (1 juCR). If both C and R are large /3 approaches unity in magnitude and the ratio of conversion approaches 1— /xtol+M-IfMis large compared to unity then this conversion ratio approaches —1. This ratio of —1 is approximately realized in the E2 con\'erter, and therefore the conversion ratio is not changed appreciably b}^ small variations in /u. U. = VOLTAGE GAIN OF TUBE Ro = TUBE PLATE RESISTANCE jwCR Fig. 14 — E2 Converter; (a) schematic and (b) equivalent circuit. 1074 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 In addition to the transformation term there is also, as shown in Fig. 14(b), a series term, 2Rp divided by 1 + M- Here Rp is the plate resistance of the tube, and m is the voltage gain as mentioned before. The factor 2 results from the use of two tubes in push pull. If m is large compared to unit}' then this series term becomes approximately 2Rp/n. It is entirely dependent upon the characteristics of the electron tube. As the char- acteristics change from tube to tube with manufacturing variation or in the same tube over a period of time or with variation in batterj- supply potential, the term 2Rp/{l + n) will change accordingly. Percentage- wise this change may be large. This is the largest source of variation in the E2 converter. It can be minimized by operating the converter be- tween impedances much larger in magnitude than 2Rp/{\ + /x) so that variations in this term have relatively small effect. This has been done in the E2 repeater by stepping up the impedance of the transmission line by about 1:9 by means of the transformer shown in Fig. 14. THE E3 CONVERTER Theoreticallj^, the same converter used for the E2 and shown in Fig. 14 could have been used for the E3. Instead of connecting the line to AW REPEATER NETWORK AV^ (a) a c l+/i/32 l+/i/32 1+/Z./32 ■ ^VVv VV V"" NETWORK V V V b LINE 2 1+///32 4 a = RESISTANCE C = RESISTANCE b = OUTPUT IMPEDANCE OF AMPLIFIER jj, = RATIO OF OPEN CIRCUIT OUTPUT VOLTAGE TO INPUT VOLTAGE OF AMPLIFIER /3,= ^ a +b-i-c /32 = (b) a + b + c Fig. 15 — E3 Converter; (a) schematic and (b) equivalent circuit. NEGATIVE IMPEDANCE TELEPHONE REPEATERS 1075 the converter through terminals 1 and 2, terminals 3 and 4 would be used. However, because the E3 must be designed for connection across a transmission line a coil or transformer input is not practical since the coil would shunt the line at low frequencies and introduce excessive loss to dial pulsing and 20 cps ringing. Without a coil to step up the im- pedance of the line, variations in 27?p/(l + n) with standard triodes are too large to be neglected. For this reason another converter circuit was designed for the E3 repeater. This circuit is shown in schematic form in Fig. 15(a). It consists of two resistances, a and c, respectively, and an amplifier poled according to the plus and minus designation on Fig. 15(a). The output impedance of the amplifier has been designated as 6. If the input impedance of this amplifier is high compared to other circuit impedances. Fig. 15(a) can also be represented by the equivalent circuit of Fig. 15(b). Here is a conversion factor similar to that in the E2 converter and also a series impedance. The factor n is the ratio of the open circuit output voltage to the input voltage of the amplifier. In the E3 converter this voltage ratio n is quite high because the amplifier is a two-stage arrangement. In the design of the E3 both 0i and ^2 are approximately one half. Thus nl3i and m/32 are both large compared to unity so that the conversion ratio (1 — M/3i)/(l + f^^i) is approximately unity and relatively independent of variations in n. Furthermore, because ^(32 is large compared to 6, the series term in the converter circuit is relatively small and variations in this term have little effect on the operation of the converter. Circuit Description THE E2 telephone REPEATER The circuit function of the E2 telephone repeater can be divided into two parts: the electron tube (negative impedance) converter; and the adjustable two-terminal network associated with the converter. In order to reduce the effect of variations in the electron tubes to negligible proportions, and at the same time to operate the tubes with load impedances that will permit optimum energy transfer from tube to connected circuit, the impedance of the telephone line is stepped up by means of the input transformer. To insure adeciuate balance for use in the telephone lines, the low voltage side of the transformer is divided into two equal, balanced windings. Each winding is center-tapped and connected in series with a fine conductor. The circuit of the E2 repeater is shown in Fig. 16. In practice, it is advantageous to limit the conversion bandwidth so 1076 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 19o4 19 GAIN ADJUSTING NETWORK ELEMENTS SHOWN FOR TYPICAL ARRANGEMENT OF REPEATER BETWEEN TWO H88 TRUNKS 25 R13 14 R12 32(^M8 — Wv — 8»— \AAr- 390on 2000 n io«- 13I CI4 0.004/U.F 0.008/^F C12 0.001/^F C1I 0.0005/iF 7<| OJW^ 285n R 9 2000 n LI 285n FUSE ON POWER DISTRIBUTION PANEL ■^gi-;ain is hmited by rechicing l3 to a small value at low freciuencies by capacitors (^ and C'., of I'^ig. Ki in the plate-to-grid coupling network, and at high frefiuencies by the small capacitors (\ and ('2 across the grid resistors 7^5 and R& respectively. 'I'he conversion ratio is affected by small losses inherent in the elec- tron tube and transformer. These are balanced out by a fixed resistor R9 connected in series with the gain adjusting network to increase the amount of positive feedback. The final negative series impedance presented to the line is equal to approximately —O.IZn over the frequency band of 300 to 3,500 cycles. The impedance Z.v is determined by the configuration of the gain-ad- justing network comprising several inductors, capacitors, and resistors. These components may be arranged in any form to obtain the desired negati^'e impedance, which in turn, introduces the gain and frequency shaping characteristic desired for each type of line facility. The E2 repeater employs a Western Electric 407A twdn-triode elec- tron tube of the 9-pin miniature type. The tube heater circuit can be operated from 24- or 48-volt office battery. The heater current is 100 milliamperes for 20-volt operation, 50 milliamperes for 40 volt opera- tion and the plate current is 1 1 milliamperes. ^M-^ 17.8/1 j( ^AV Fig. 17 — Sclioinatic circuit of E3 repeater. NOT PART OF REPEATER T E2 REPEATER T LINE A - '.^'J 1 LINE B R 1 ""t ^1' R R28 17.8 n'^ C12B INPUT CONNECTIONS SHOWN TO E2 REPEATER ,o7_,o9- FOR AN E23 COMBINATION ^/p ' REPEATER ^ R18 3010 n CI3 20^iF R24 . 1000 n R14 3010 n R27 .17.8n C12A 1.07-1.09 PLT FUSE ON POWER DISTRIBUTION PANEL H||j 1 PIN JACKS FOR MONITORING AND TUBE CHECKS iv('(l Dcccinhcr 16, 1953) The solderless wrapped connection is initialhj held together by the hoop stress in the wire which enters the connection as a result of the tension put on the wire by the wrapping tool. Measurements made out to a time of l.o years at room temperature show that the tension has decreased to 70 per cent of the one day value (8000 lbs per square inch) in this period. Two methods of extrapolation are discussed, both of which indicate that at least half of the initial one day value will remain at the end of forty years at room tempera- ture. Another set of stresses enters the connection as a function of time, namely the dijfu.'iion forces produced by diffusion of the tin plating into the brass terminal and copper ivire. A number of experiments are discussed which show that the activation energy of diffusion is materially reduced by the shearing stresses in the connection. Measurements at two temperatures, which allow extrapolation to room temperature, indicaij that at the end of two years the force required to strip the wire from the terminal has increased by 5 per cent over the initial value and that at the end of forty years the increase will be 20 per cent. Support for these conclusions is furnished by tests on actual connections that have been in the field for one year and ten months, which show an increase of 5 per cent in the stripping force even though the relaxed hoop stress is only 68 per cent of the initial value. The increase, which is due to diffusion forces, can be made higher by using zinc, cadmium or aluminum plating, and the fusion occurs in a shorter time. IXTRODUCTIOX As discussed in a series of papers/ the solderless wrapped connection is an efficient and inexpensive method of connecting a wire to a terminal. All the tests made so far indicate that it is mechanically sound and sufficiently free from the effects of corrosion to have a trouble free life of at least forty years. Photoelastic and stress studies show that the » The Solderless Wrapped Connection, B. S. T. .]., 32, May 1953. 1093 1094 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 connection is initially held together by the hoop stress in the wire which enters the joint as a result of the tension put on the wire by the wrap- ping tool. As time goes on, another system of stresses are generated, namely, the diffusion stresses caused by the diffusion of one part of the connection into the other which eventually ehminate the surface between the wire and the terminal and in effect join the two together. It is the principal purpose of this paper to describe a number of ex- periments which show how these stresses develop, how large they are and how they can be increased by substitution of a different type of plating between the terminal and the wire. A comparison of the strength of a joint formed by a tin plated copper ware and a bare copper wire both wound on nickel silver or brass terminals shows that the diffusion forces develop more quickly when the wire is tin plated than when it is bare. Measurements made at different temperatures show that the activation energy of diffusion is decreased in proportion to the hoop stress in the wire indicating that the shearing stress at the contact sur- faces aids diffusion. This activation energy is considerably less for tin than for copper. The diffusion joint has a strength per unit area equal to the limiting shearing stress of the tin platmg. This is in the order of 3,000 pounds per square inch for tin but is considerably higher for other types of plating such as zinc, aluminum or cadmium. ^Measurements of the stripping force of connections made with bare and tinned copper wire on zinc, aluminum or cadmium plated terminals show that the stripping force increases by a factor of two as a function of time, and the time required for the diffusion forces to operate is considerably less with these types of plating. The combination of relaxation and diffusion stresses that are discussed later show that as the mechanical strength due to the hoop stress de- creases, the strength due to diffusion increases and at the end of forty j^ears the standard tin plated wire on nickel silver or brass terminals will be at least 20 per cent stronger than it is initially. The extensive corrosion tests described in Footnote 1 , taken together with the mechan- ical strength tests described here, show that the standard connection should not fail in the forty year period under consideration. RELAXATION OF HOOP STRESS AS A FUNCTION OF TIME The rate of relaxation of the hoop stress in the wrapping wire is an important quantity for the stability of the connections. This has been studied by wrapping 24 gauge wire with a constant tension of three pounds around a spring steel terminal 0.0124 inches thick and 0.062 inches wide. As shown by Figure 1 1 of Mallina's paper,^ this causes the STRESS SYSTEMS IN THE SOLDERLESS "VVI! APPED CONNECTION 1095 terminal to twist through an angle of about 25° wIumi 100 (urns are wrapped around the terminal. This twist is the result of a tor(iu(> wWn-h is caused by the fact that the wire has a hoop stress and the wire does not come back on itself but advances by the thickness of the wire for each turn. As shown previously,^ the total torc^ue is equal to Torque = (W.F.) ( -^ ) L .a + 6, (1) where (W.F.) is the wrapping force, 2a and 2h the thickness and width of th(> tei-minal and L the total length of the wrapped section. Hence, by calibrating the spring constant, the average hoop stress can be evaluated. In this manner, Mallina^ has found that after transient creep (defined 1.1 Ul in 1.0 UJ a: vX ALUMINUM^ < ^^. ^. ^^^ X. ^ ^ 01 2345678 HOOP STRESS IN THOUSANDS OF POUNDS PER SQ IN. Fig. 3 — Activation energj' for stress relaxation as a function of hoop stress. Measurements of the rate of stress relaxation for tinned copper wire at 200°C, 175°C, 150°C, and 100°C are shown by Fig. 2, and the activa- tion energies plotted against average hoop stress are shown b}^ Fig. 3. Values are given using 150°C to 200°C as the temperature range and 100° to 175° also. Both ranges give the same activation energies within the experimental errors and show that down to about 0.4 relaxation the activation energies satisfy an equation of the type H'= H - ^a (8) The curvature exhibited by the relaxation versus log t shown for all temperatures indicates that the activation energy must increase faster than (8) for low values of relaxation and the dotted line shows a hj'po- thetical curve ending up at the self diffusion activation energy for cop- per, 57 kilocalories per mole. To apply this method in general, one has to take account of anj^ trans- formation such as recrystallization in the temperature range of measure- ment. For example, Fig. 4, shows similar curves for aluminum wire. Recrystallization in aluminum is known to occur at temperatures above 150°C and this change is shown in the relaxation measurements by the lower values of relaxation that occur for long times. If one takes values of time above and below the recrystallization temperature, the activa- STRESS SYSTEMS IN THE SOLDERLESS WHAPPED CONNECTION 1099 tion energy will appear higher for this range than for a temperature range below the recrystallization temperatuiv. The agreement of the activation energy for copper for the tAvo temjierature ranges shown by Fig. 3, sliows that no transformation occurs from 25°C to 200°C and hence we can extrapolate the relaxation to room temperature taken as 25°C, with the result shown by the solid line lal)elled 25°C/. ""Jliis agrees well with the measured values and indicates a stress at the end of forty 3'ears equal to 5,200 pounds per square in. DIFFUSION STRESSES IN SOLDERLESS WRAPPED CONNECTIONS In addition to the hoop stresses, another set of stresses develops as a function of time, namely, the diffusion stresses caused by the diffusion of one part of the connection into the other. The first experiment that showed the presence of these stresses was the stripping force tests of Fig. 23 of the previous paper referred to in Footnote 3. These measure- ments were carried out on connections which had been held at 175°C for lengths of time up to ten days and it was found that the stripping forces did not decrease with time. A more careful set with twenty con- nections for each point have recently been run with the results shown by Fig. 5, solid line labelled 175°C. From this curve, it is seen that the stripping force decreases to 88 per cent of its initial value of 15.5 pounds average for six turns of 24 gauge tinned wire and then increases to 120 per cent of the initial value at the end of ten days. Similar increases are shown at 100°C over a longer period of time and recent tests of solder- less wrapped connections that have been in the field for one year and ten months show that the stripping force is about 5 per cent higher on the average than it was when the connection was foraied. 10-^ 10^ 10= 10° TIME IN SECONDS Fig. 4 — Relaxation curves for aluminum wire in solderless wrapped connec- tions. 1100 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 UJIU OO oo ILli. z? o< O AT 175° C A AT lOCC D AT ROOM TEMPERATURE Oj^jfd ^^^^ ^ - / ^ ^N ~"~~~-~. 1 HOUR 1 1 DAY 11 1 MONTH 1 1 YEAR 1 10 YEARS 10= 10" TIME IN SECONDS Fig. 5 — Ratio of measured stripping force to initial stripping force as a func- tion of time and temperature for tinned copper wire on nickel silver terminals. The dashed lines show the corresponding hoop stresses as a fraction of the initial hoop stress and hence it is evident that as the hoop stresses go down the stripping forces first decrease and then increase to higher values than were effective originally. We shall presently show that the difference between the measured stripping force and the proportionally relaxed hoop stress is due to a stress caused by diffusion of the tin in the plating into the wire and terminal of the solderless wrapped connection. One experiment which shows that the initial stripping force is caused bj^ the hoop stress in the wire is the experiment shown by Fig. 6. Here a terminal is made which has a slightly tapered tin plated pin in the middle and is cut back for some distance beyond the pin. The terminal is wound with five turns of No. 14 gauge (0.065 inch diameter) tinned copper wire. The initial stripping force to pull off the winding was determined and it was found to take 72 pounds force on the average to strip the wire off the terminal. A similar set of measurements was made on the force required to pull the pin out of the terminal and this averaged about 50 pounds or 70 per cent of the stripping force for the wire. Since the pin and terminal had the same coefficient of friction as the wire and ter- minal, and since the pin is required to support all the compressional stress in the terminal due to the hoop stress in the wire, it is evident that at least 70 per cent of the force required to strip the wire off the terminal is plain frictional force between the wire and the terminal. It is thought that the remainder of the force is due to the gouges cut in the terminal bj^ the winding process. When the wire is stripped off the terminal, these cuts gouge out parts of the wire and hence require a higher force. This effect is equivalent to friction for a rough surface, which is higher than that for a smooth surface. STRESS SYSTEMS INT THK SOLDKKLKSS ANKAPPKD CONNECTION 1101 Next the terminal was held at 175°C for various times as shown by Fig. 0, where the stripping forees and the forces to pull the pins are plotted. It is seen that the force re(iuired to pull the ])in decreases with time while the force required to strip the terminal increases with time, 'i'lu^ pin force duplicates tlu^ stress n^axation curve initially but departs from this curve more and more as time progresses. This gradual depar- ture appears to be due to some diffusion in the tin which makes partial contact with the terminal. The amount of diffusion between pin and terminal is less than that between wire and terminal for two reasons: (a) the contact area between pin and terminal is not as intimate as the contact area between wire and terminal due to excessive plastic flow in the latter but not the former case, and (b) the contact area between j)in and terminal is much greater than between wire and terminal. Nevertheless, the force recjuired to pull the pin is substantially the same as the frictional force holding the wire on the terminal due to the hoop stress in the wire. Hence, we can conclude from this experiment that the initial stripping force is due to the frictional force resulting from the hoop stress plus shearing forces required to gouge the wire. Several experiments have been undertaken to measure the effect of diffusion separate from friction. The most successful of these was the arrangement shown by Fig. 7. Here a wire was pressed against a double- toothed sharp edged block, and a constant pressure was maintained for various times at various temperatures. An attempt was made to produce an indentation of the same magnitude as that developed in the wrapped solderless connection, although, of course, the area of contact increased slightly with time. It was found that for aluminum or copper on nickel silver at room 10^ 10- TIME IN MINUTES Fig. 6 — Experiment showing difference between stripping force and frictional force due to hoop stress — temperature 175°C. 1102 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 temperature, no adhesion occurred even when the time of loading was very long. It was also found that if the load was removed, even mo- mentarily, before the sample was placed in the oven, no adhesion oc- curred. For a given temperature there is an induction period before the wire adheres to the block, and this induction period increased, in general, as the temperature decreased. This induction period was found to be a time of nucleation. This is shown by the fact that the period increases as the square of the contact dimensions. Since diffusion is a function of x/\/Dt where x is the distance, D the diffusion constant and t the time, this observation shows that nucleation starts at a given point and pro- ceeds for a certain fraction of the contacting surface before fusion strengths are observed. After the induction period, the fusion force — the force required to pull wire and block apart ■ — increased at a rapid rate for the cases of copper, tinned copper, and zinc wire on a nickel silver block. In order to account for the effect of fusion due to the increase of con- tact area, the ratio of the fusion force to the contact area was determined, yielding a shearing stress. The area of contact could be easily measured with a microscope since a bright surface was produced hy the shearing process. The shearing stress, Fig. 8, for the case of tinned copper wire on nickel silver is shown to approach a limit of about 3,000 psi which is approximate^ the limiting shearing stress for tin. Hence, it appears that tin diffuses into the copper wire and the nickel silver base. Further- TIME IN SECONDS Fig. 7 — Diffusion forces for tinned and bare copper and zinc wire on nickel silver base. STRESS SYSTEMS IX THE SOLDERLESS WRAPPKI) ( OX \ KCTK )\ IKK^ IT) 9^ 5 '-' < z / \ e ^>1 y ^ y / V X y /^ yy Yy r ^^ ,.-"' ^ ^ Kx r DAY n^y^' MONTH 1 YEAR 10-^ 10'' 10 = TIME IN SECONDS IQS Fig. 13 — Shearing strength of alumiiunn on nickel silver connections as function of time. 1108 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 copper wire on nickel silver terminals at 150°C and 100°C. The increase in strength does not occur as fast as that for tin plated copper, and hence the standard connection with a tin plated copper wire is better than one formed from })are copper wire. PROPERTIES OF SOLDERLESS WRAPPED CONNECTIONS FOR OTHER TYPES OF PLATING A study of the factors governing the diffusion strength of the solder- less wrapped connection suggests methods for increasing the rate of diffusion and the strength of the diffusing layer. These methods result 1 DAY 10 DAYS 1YR lOYRS lU UJ U O (£ a: OO u. 1.6 w (J) zz Q-CL Q-O. 1 1 1 1 ISO'C, /' / loa/: -cs f^ ^ ^ ^ ^ ^ "' ■'--, :^ >^^ - ^^ 150° C/ / y^o 25^^ 10"^ lO'^ 10'* 10^ 10" 10' 10° 10^ TIME IN SECONDS Fig. 14 — Stripping force and shear strength of bare copper on nickel silver connections as a function of time and temperature. in greater mechanical strength and a faster fusion of the wire and ter- minal. Experiments show that there are at least foiu" metals which diffuse faster than tin, but due to the economic problems and the brittleness of some of the alloys formed, none of them are being considered for solder- less wrapped connection. The reason the diffusion forces in the tin plated solderless wrapped connection do not cause an increase in strength of more than twenty per cent during the life of the connection is that the hmiting shearing stress in tin is only 3,000 pounds per square inch. If now we substitute for tin plating a plating with a larger limiting shear stress, the strength of the connection should increase by a larger factor. To be of use, how- ever, it is necessary that the diffusion forces shall develop rapidly. The data of Fig. 10 show that if we can produce a higher shearing stress on a layer of aluminum, the activation energy can be lowered and STRESS SYSTEMS IN THE SOLDEKLESS AVKAIM'ED COX X EC'l'lOX 1101) 10* 10^ 10^ lO' TIME IN SECONDS Fig. 15 — Stripping force and shear strength of coppcr-zinc-brass connections as a function of time and temperature. be made to approach the cold welding condition. This requires that aluminum be placed on a stronger material such as brass, nickel silver or copper in order that the area of indentation for a given hoop stress will decrease and the shear stress will increase. Hence, the connection should form in a very short time. Furthermore, since the limiting shear stress for aluminum is near 0,000 pounds per square inch, one should except that the strength of the connection will nearly double. Since aluminum is not easih^ electroplated this suggestion probably is not practical. Of all the other metals examined, the next most promising are silver, cadmium and zinc. The activation energy for diffusion of zinc into copper 01 2345678 HOOP STRESS IN THOUSANDS OF POUNDS PER SQUARE INCH Fig. 16 • — • Activation energy for a copper-ziric-hra.ss connection u.s a function of hoop stress. 1110 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 at no stress is given^ as 38 kilocalories per mole which is 4 kilocalories higher than that for aluminum and 3 kilocalories under that for tin. The activation energy of silver and cadmium into copper also have low values. Hence, if the effect of stress parallels that for tinned copper, zinc and cadmium should diffuse into copper and nickel silver faster than tin. Furthermore, the limiting shearing 5000 pounds per square inch. Zinc, silver and cadmium are readily plated on the terminals or the wire. Zinc plating has been tested experimentally bj- constructing a solder- less wrapped connection of bare copper wire on zinc plated nickel silver or brass terminals. The stripping force for a bare copper wire wrapped on a terminal plated with 0.001 inch thickness of zinc has been measured at 175°C and at 100°C as a function of time with the results shown by Fig. 15. The strength increases to 60 per cent over that found initially in a time of less than two hours at 175°C. At 100°C, the time required is a little over a day. If we subtract the relaxed force from the stripping force and divide by the area of the connection, the shear strength of the con- nection is as shown by Fig. 15, lower cur^'es, for the two temperatures. From these two curves, an activation energy versus hoop stress can be obtained with the results shown by Fig. 16. These values allow one to extend the time variation of the shear stress down to room temperature with the result shown by Fig. 15. Adding these values multiplied by the area of the connection to the relaxed hoop stress, the indicated stripping force at room temperature is shown by Fig. 15, room temperature curve. This force increases at such a fast rate that the strength can be observed to increase at room temperature and corresponding measurements are shown by the circles. Since corrosion cannot occur in a region of fusion, a criterion for the corrodability of a connection is the time required to complete half of the total fusion at room temperature. On this basis, the zinc plated connection has the lowest half fusion time of any of the materials tested. Although zinc diffuses more readily than other materials examined, it tends to form more brittle alloys and hence its use has not been seriously considered for solderless wTapped connections. 8 R. M. Barrer, Diffusion In and Through Solids, Cambridge University Press, 1941, Table 67, p. 275. I A New Multicoiitact Relay for Telephone Switching Systems By I. S. RAFUSE (Manuscript received April 8, 195-1) The trend in new telephone switching systems toward faster operation, longer life and lower cost, indicates a need for faster and more capable control circuits. This paper describes a new high speed wire spring ynulticontact relay designed primarily for these applications. The basic unit contains 30 make-contact pairs. Two variations of the new design provide relays of 60-contact capacity. They are mechanically and electrically interchangeable with all crossbar system muUicontact relays. INTRODUCTION In a modern dial telephone central office, many thousands of mo- mentary intraoffice control connections are made daily between the various parts of the switching equipment. For example, in the No. 5 crossbar S3^stem, seven major types of connectors are used to associate markers^ with other common control circuits, and with the switching frames, for brief intervals, to assist in setting up the talking connection. Connectors are required to simultaneously close a large number of circuit paths, as many as 240 in the trunk link connector. The earlier flat spring type multicoiitact relays* used for this purpose provide large blocks of contacts per relay and provide an economical means for common or multiple wiring. The trend in new improved switching systems toward longer life, faster operation, lower cost and reduced maintenance, indicates a need for faster and more rehable connector circuits. This paper describes a new multicontact relay, designed primarily for these apphcations. It is a wire spring relay incorporating the improved manufa('turing processes and many of the design features of the new general purpose wire spring relay.* The basic unit. Fig. 1, for use in new equipments, is a high speed 30-contact pair relay, with wiring terminals arranged for horizontal multiple connections. 1111 1112 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 Fig. 1 — New 30-contact relay with contact cover detached. This development also includes two modifications of the new wire spring relay for replacement use and for additions in existing crossbar equipments. They are 60-contact pair relays, each consisting of two 30-contact units attached to a common mounting bracket. They are completely interchangeable with existing crossbar system multicontact relays. When a new improved relay is developed, it is almost invariably necessary to continue in manufacture and carry in merchandise stock, small demand codes of the old relays for many years. In this case, how- ever, manufacture of the old multicontact relays will be discontinued and all future needs will be supplied by the new product. OBJECTIVES AND REQUIREMENTS At the start of the project, all requirements from the standpoint of operating performance, circuit design and equipment use, were prepared in detailed form. The principal design objectives are summarized as follows : Electrical and Mechanical 1. Operate and release times as fast as economically possible. 2. Forty year life, or 200 million operations, with no adjustment necessary during the first 100 million operations. Nin\ MlL'l'ICOXlAcr IIKLAY 1113 3. No contact chatt(M-. 4. No false actuation due to armature rebound. 5. No magnetic or xihrational inteit'erence. (■). 120-ohni and 27.")-olun coils, to work with e(iuipment ali-eady in use. Equipminl 1 . Lower nianufacturing costs. 2. [{educed mount in<>; s])a('e. 3. Terminal ai'i'aniiement tor multiple wirinjj; same as at present, or equivalent from a wiring; stand])oint. Maintenance 1. Contact failures due to dirt or insulating films should lie sub- stantially equal to and preferably less likely than in the present relay. 2. No contact locking due to contact erosion. 3. Contacts should be replaceable in the field. 4. Coil winding should be replaceable in the field. 5. Field adjustment should be reduced to a minimum. Replacement Relay The design objectives also included modification of the new relay, if possible, to replace multicontact relays in existing crossbar ecjuipments. Design History During the early stages of this development, considerable effort was directed toward improving the present flat spring multicontact relay. Later, many experimental models were constructed, to investigate other flat spring, and wire spring designs, and several contact actuating meth- ods. The most favorable designs of flat and wire spring multicontact re- lays were compared, and their differences were resolved by an analysis of manufacturing tolerances and their effect on performance and cost. Preliminary estimates of initial cost were only slightlj; in favor of the wire spring design. However, the wire spring relay offered significant advantages in (1) higher speed, longer life, less chatter, (2) better manu- facturing control of tolerances, (3) less maintenance, and (4) possibilities of future cost reductions as further improvements are made in mech- anized methods of manufacture. DP^SCRIPTION OF THE NEW RELAY General Stationary single wire and moving twin wire spring subassemblies are arranged in alternate layers attached to the core and mounting 1114 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 TWIN-WIRE ARMATURE BACK-STOP WINDING TERMINALS ARMATURE COIL \ TEST RETAINING V TERMINAL SPRING MOUNTING BRACKET ARMATURE HEEL- STOP HEEL PLATE Fig. 2 — Top view of rela}' showing location of parts. plate, as sho\\ai in Fig. 2. The wire spring assemblies form two rows, each containing 15 make contacts. Each contact pair consists of moving twin contacts on separate twin wires, associated with a single stationary- contact. The stationarj' springs are supported close to the contacts bj^ arms extending from the bracket. A detailed view of all parts and sub- assemblies for a 30-make contact relay is shown in Fig. 3. Moving con- tacts are pretensioned by relatively large pre-deflections as shown in Fig. 4. The method used in flat spring multicontact relays to obtain contact force is illustrated for comparison. It is apparent that contact force obtained by the "buckle" method depends on operating stud length and therefore is subject to change due to wear. The new preten- sioned wire springs are supported and actuated by a single molded phenolic card by the "card release" method as illustrated in Fig. 5. In the unoperated position, the card is held against the core by a re- storing spring, which also supplies the force to open all contacts. In the operated position, the armature supplies the force to move the card, releasing the twin wires and closing all contacts. This method of actua- tion has some important advantages: 1. Contact force is essentially independent of gauging and wear. 2. The effects of wear at points in the relay which affect gauging are compensating to some degree, and therefore tend to minimize changes in gauging. 3. Dimensional variations controlling contact separation and armature travel are reduced, making possible shorter annature travel, faster NEW MULTICONTACT JtELAY 1115 Fig. 3 — Parts of the new 30-contact relay. IIIG THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 operate and release times, less contact chatter and increased mechanical life. Normally, the armature is held against the card by the lightly pre- tensioned armature hinge spring. However, when the relay is released, the armature motion becomes quite complex. Overtravel at the front is limited by the core plate backstop, and motion at the back, or heel sec- tions, is limited by the heel stop studs. This freedom of movement is intentional, its purpose being to dissipate armature energy into the core plate and core rather than back into the card. Magnetic Circuit and Armature Analytical and experimental studies show that one per cent silicon iron, with its higher resistivity, relative freedom from aging, and lower eddy-current losses compared to ordinary magnetic iron, provides opti- mum speed in the new fast relay. The contact load, about twelve grams per twin contact pair, is about one-half the load required in the present relay, and this together with winding space and heating, largely deter- mines the size of the magnet. The magnetic structure is shown in Fig. 6. The core is a one piece "E"-shaped section. The armature is a flat mem- ber made of low carbon steel having specific magnetic characteristics. This material simplifies manufacture, resulting in a cost saving with PRE- DEFLECTION t Ql STATIONARY // ^'-CONTACT Si--_-_ '¥rEE POSITION OF TWIN I i-r , ~~~^------^__ ^^' WIRES BEFORE ASSEMBLY RMAL ^Ui .--.-^A,^ " NORMAL CONTACT MOTION '"POSITION AFTER ASSEMBLY CONTACT FORCES ARE CONTROLLED BY RELATIVELY LARGE PREDEFLECTIONS OF THE TWIN WIRES ON NEW WIRE SPRING RELAYS. NORMAL STATIONARY CONTACT MOTION CONTACT SPRING -=^ /' ^^^ ■-■MOVING CONTACT SPRING OPERATING I "-^OPERATED POSITION OF STUD ARMATURE MOVING CONTACT SPRING ON FLAT SPRING RELAYS, CONTACT FORCE IS OBTAINED BY "BUCKLING" THE MOVING CONTACT SPRING. Fig. 4 — Development of coiitaot forces in wire spring and fiat spring multi- contact relays. NEW Mli;rKO\TACT KKLAV 1117 no appreciable penalty in ])ertonnan('e. The armature has a(le([uate sections to carry tiie flux, optimum poleface area and lowest possible mass. Two small rectangular holes in the armature locate the base of the card in the horizontal direction only. The card is located xcitically by the restoring spring as illustrat(Ml in Fig. 1. Fast i-elease is obtained by a nonmagnetic separator strij), welded to the face of the armature. This stri]) also provides a smooth sup]iorting surface foi- the molded card. Xegligibh^ wear at this ci'itical point contributes materiall}' to long life and stal)le adjustment of this relay. .V cellulose acetate filled coif is asseml)l(Ml to the center leg of the core. A nonmagnetic core plate, illustrated in Fig. 7, is then forced over the three core legs to hold them in aUgnment. The center hole in the core plate also functions as an armature backstop and permits a certain amount of o^•ert ravel of the armature when the relay is released. Coil Assemblies For circuit reasons, r20-ohm and 275-ohm coil resistances used in the old relays are reciuired in the new relays. Nominal power savings which ordinarily would i-esult due to an improved magnet and reduced load, are therefore sacrificed in the new relays in fa\-or of increased speed. More than half of the new relays are expected to be used in circuits rec^uiring maximum speed of operation and will, therefore, have 120-ohm ,- RESTORING SPRING COMMON STATIONARY CONTACT '""'''"TiH TWIN CONTACTS (MAKE) TENSIONj ||_. tensionT iLfe ARMATURE'' nonmagnetic " separator TWIN WIRES ' (MAKE) REED HINGE UNOPERATED OPERATED Fig. 5 — Principal of contact operation. 1118 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 coils. In normal operation, these relays are not energized continuously, but operate only for short intervals while a talking circuit is established. The design provides for replacement of a coil winding in the field, but this is not expected to be necessary except in rare cases. Molded Wire Spring Subasseinhlies Wire spring subassemblies for a 30-contact relay are shown in Fig. 8. The two single wire subassemblies differ only in their terminal arrange- ment. The twin wire subassemblies are identical. Therefore only three basic molded parts are required which supply all needs for all new produc- tion relays. The wire spring sections are molded in continuous ladders^ as in the general purpose relay. Spring bending, contact welding and coining^ and terminal forming for solderless wrapped connections,^" are performed in automatic tools developed by Western Electric Com- pany engineers. A comparison of the wire spring parts used in two new 30 make relays and the corresponding parts of a 60-make flat spring relay, is shown in Fig. 9. This illustrates the reduction in parts and simplicity of wire springs compared with flat springs. Seven types of subassemblies are used in the twelve layers required for an equivalent flat spring relay. Terminals of the single wire subassemblies are formed for multiple wiring, and therefore differ in length and configuration. An improved CORE I y ARMATURE ARMATURE HEEL STOP Fig. 6 — Magnetic structure of the new relays. NEW MULTICONTACT RELAY HID form of open wire multiple strapping has been developed for use with this terminal arrangement. It consists of bare wires held together in ladder form by means of phenolic plastic blocks molded successively in a continuous process. Fig. 10 shows relaj^s with ladder type horizontal strapping soldered to the single wire terminals. The usual cable with new solderless wrapped connections is used on the twin wire terminals. Fig. 10 also illustrates the accessibility of multiple connections provided by locating them off to one side in the clear area between cable groups. Contacts All contacts in the new wire spring relays are palladium having a volume of contact material suitable for forty years life. This is equivalent to about 200 million operations for relays in high usage circuits. Con- tacts are easily visible, readily cleaned and may be insulated for test purposes. Contacts may be replaced in the field if necessary using Bell System field welding equipment." Suitable tools and electrodes have been de- veloped to permit use of this equipment on all wire spring relays. Assembly and Adjustment The relay pile-up is securely fastened by two high tensile screws and a spring compliance member. Laboratory tests show that a pile-up of COIL CORE CORE PLATE Fig. 7 — Core legs are held in alignment by the core plate, which is forced on the ends after the coil is assembled. 1120 THE RELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 Fig. 8 — Molded wire assemblies for a 30-contact relay. this design will remain tight under widely varying atmospheric con- ditions. In the design of the new relay considerable attention was given to manufacturing control of tolerances, with reduced assembly and ad- justment costs as an objective. The molding process provides dowels for aligning the four layers of contact springs, trunnion supports to locate the single wires, and control grooves in the single wire subas- semblies to align the twin wires. These features provide accurate registra- tion for mating contacts, for the location of wire subassemblies relative to other relay parts and for the pretensioned restoring spring which in turn locates the actuating card in relation to the operating contacts. The card determines contact separation of all twin moving contacts in relation to their respective single stationary contacts. Due to these controls, and because the pre-deflected twin wire springs require no adjustment of contact force, no factory adjustments are anticipated except on relays which fall outside acceptable limits for back tension or contact gauging (or follow) as assembled. If necessary, therefore, the restoring spring may be adjusted to control the armature back tension. Contact gauging may be controlled if required by independent mass adjustment of the single wire contact rows. Fig. 1 1 shows this operation being performed by bending the bracket arm using a tool designed for this purpose. The mass adjustment feature is expected to simplify field maintenance practice. XKAV MULTICOXTACT HKLAV 1121 i|i ill 'H IS Hi lii ill ili 'If in yiuuiy 111 III III III HI III ill IM III III imiiiiii BIB BIB Fig. 9 — A comparison of molded wire assemblies for a 60-foiitact replacement relaj' with the corresponding flat spring rela)' parts. 1122 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 ■11 ill Fig. 10 — New 30-contact relays showing multiple wiring by the new ladder type strapping method, and the new solderless wrapped cable connections. Fig. 11 — New 30-contact relay showing method of mass adjusting stationary contacts. NEW MULTICONTAOT 1{KL.\Y 1123 Replacement Type Relays Future replacements of flat spring multicontact relays in (existing crossbar equiimients for maintenance reasons re((nire a slightly modified form of the new relay, capable of complete interchangeal)ility. As these relays are commonly used in connector circuits, operate and release times must be comparable with okUn- type relays in order to avoid circuit interference. This was accomjjlished in the new wire spring relay with only minor changes in design and in the manufacturing process. As illustrated by Fig. 12, the interchangeable or replacement relay consists essentially of two 3()-contact units assembled on a common mounting bracket having the same vertical mounting centers as the 60- contact flat spring relay. Since horizontal mounting space required by the new relay has been reduced, the 60-contact wire spring unit may also be used to replace 30-, 40- and 50-contact flat spring multicontact relays. The model shown in Fig. 12 has terminals arranged for horizontal multiple wiring. In another variation of the replacement relay all termi- nals are arranged for cable wiring. Modifications necessarj^ to slow down the new relays for replacement use are (1) longer armature travel, requiring a different card; (2) an armature of larger mass ■ — although this is a different piece part it has the same contour as the armature for the fast relay and may be punched in the same tool setup; (3) a core of low carbon steel in place of the one per cent silicon iron used in the high speed relay; (4) a leak- age reluctance shunt element shown in Fig. 13 to by-pass a portion of the total magnetic flux; and (5) coil windings having resistances of 120 and 275 ohms as required for circuit reasons, but having the number of turns calculated for slower speed consistent with reliable operating capability. ' Relay Performance Measurements have been made on laboratory-built models carefull}^ prepared and adjusted to simulate extreme ranges of manufacturing tolerances, and more recently, on representative samples of pre-produc- tion relays. Although some of the performance characteristics studied will be determined accurately only after long-term use in the field, it has been possible by designed experiments and comparative tests, to obtain a fair appraisal of relay capability. These tests and measure- ments indicate that design objectives stated earlier in this paper have been substantially achieved. 1124 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 Fig. 12 — A 60-contact, replacement relay with one contact cover detached. XKW MrLTI('()\'l'\rT ItKLAV 1125 Load and Pull Characteristics Load and ])ull curves ai'e incasuicd under essentially static conditions. Spring load and ai'mature motion are both ol)served at the center line of the card. They are measured and simultaneously recorded in chart form in a modification of a (ensile testing machine. " A ty])ical load and pull chart is shown in ¥'ig. 14. The abscissa shows armature motion, as the armature moves the card and the spring load, through a distance of 0.030 inch to the operated position, and back again. The ordinate shows s])ring load on the armature on both operate and release, and also the magnetic pull which is (kn^eloped in the armature; for varions numbers of ampere turns in the winding. The load and pull chart provides a comprehensive picture of over-all relay performance. For example, starting from the released position, the force or back tension, holding the card against the core is about 140 grams. Following the upper curve, the spring load increases slowly as the armature mo\'es toward the core, until the first contacts make at a load of about 200 grams. The load increases rapidly as the remaining contacts are closed until the last contacts are closed at about 650 grams. Further travel of the armature to the operated position increases the spring load to a final value of about 700 grams. As the armature is allowed to return to the original position, the lower curve is traced. The area between the two curves is a measure of mechanical hysteresis, or friction, in the relay. This energy loss is a very small fraction of the spring load at all values of armature travel. The pull curves show ampere turns necessary to assure operation of LEAKAGE RELUCTANCE SHUNT Fig. 13 — Core assembly for replaceinent type relays showing leakage reluc- tance shunt. 1 1126 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 w \\ \\ PULL -350NI 800 \ \ \ \ \ \\ \ \ \ \\\ LATE ' \ \L<:- CONTACTS \ V (MAKE) \ \ 700 \\ \ \ \ H \ CRITICAL -••^-LOAD \ POINT \ 600 l\ \ \ \ \\\ \ \ 500 '^V ' y \ -250 \releasing1\ \ \ \ 400 M \^ \ \\ \ \ V ^ -200 300 h \\ W'load V W RELA \ UOPER/ TING Ny\ -160 200 nL > -150 N^ 100 \ CON \ (M RLV ^^ TACTS ->' ^ AKE) 1 ^ BACK 'TENSION ""''^'-^~-- ^- 1- -50 0 \\ OPERATED POSITION 0.01 0.02 0.03 ARMATURE TRAVEL f IN INCHES I RELEASED POSITION Fig. 14 — Typical load and pull characteristics of a 30-contact fast relay. the relay. The maximum ampere turns required are determined by the "critical load point." This occurs at 0.010 inch armature travel and about 650 grams. Under static conditions, therefore, 160 ampere turns would be required for complete operation. Circuit uses for these relays do not include nonoperate, hold, or release requirements. This informa- tion could however be obtained from the pull curves in a similar manner . Operate and Release Speed The new high speed multicontact relay operates two to three times as fast as its predecessor, the flat spring type relay. Operate and release NEW MULTICONTACT RELAY 1127 120 OHM 275 OHM HIGH SPEED WIRE SPRING RELAY FLAT SPRING RELAY Fig. 15 — Comparison of operate and release times of wire spring versus flat spring multicontact relays. times are shown in Fig. 15. The improved performance of the new relays is shown by nominal operate and release values, and also by greatly reduced spread, or difference between minimum and maximum values, as compared with corresponding data for flat spring relays. Minimum operate times of the replacement and existing flat spring relays are comparable. It will be noted however, that operate and re- lease time spreads are much less in the new relay. This generally should improve the operation of existing crossbar circuits as new relays are used for replacements or additions. Contact Performance As speed increases, rela3^s and other switching mechanisms become more susceptible to false operation of contacts. It is obvious also that faster operation adds to life requirements and therefore extends the period or increases the number of operations during whicli trouble-free contact performance must be provided. For these reasons, extensive studies were made of chatter, unprotected erosion, and locked contacts as applied to the new multicontact relay. Additional longer range tests 1128 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 are planned to study protected contact erosion and susceptibility to open contact failures. When contacts of the new high speed relay are operated, initial chatter does not normally exceed 0.1 millisecond. There is no shock chatter caused by spring vibration due to impact of the armature with the core or backstop. There is no chatter caused by hesitation of the armature in its travel at the point where it picks up the contact load. This is due largely to the low mass of 0.020 inch diameter twin wires, the low mass and short travel of the actuating mechanism, the type of card operation, and the rigid mounting of the relay structure. Electrical erosion of contact material is reduced in the new relay, because of the reduction in chatter. For this reason, the contact size provided is expected to be adequate for all normal use for the life of the relay, and contact maintenance should be greatly reduced. Locked contacts are substantially eliminated in the new relay by the single card release method of operation. Static and dynamic forces associated with the restoring spring and card system are powerful enough to break loose any random pair of locked contacts. Contact failures due to dirt or the formation of insulating films on the contacts are difficult to check in laboratory tests. Long-term ac- celerated tests are necessary, with, a large test sample, under carefully controlled dust conditions. Many precautions were taken in the design to minimize failures from this source, as follows: (1) a dust cover. Fig. 16, encloses the contacts, but does not enclose the coil; (2) the cover partially segregates the contacts in groups of three pairs of contacts, reducing air movement in the vicinity of the contacts; (3) palladium contact material is used on all relays; (4) twin contacts are coined- — the rounded surface reduces the area in contact, effectively restricts the area w^hich may trap lint or other foreign matter, and increases contact pressure; (5) card release actuation and wire springs with large pre- deflections insure that no appreciable loss of contact force will occur due to age or erosion; and (6) twin contacts are attached to completely independent wires. Rebound chatter is another form of false operation which occurs in the form of contact reclosures caused by rebound of the armature after striking the backstop when the relay is released. Fundamental studies were made of rebound behavior in relay structures and various models were constructed and measured. As a result there is no rebound chatter in the new high-speed wire spring relay within the range of normal adjustment. A comprehensive survey was made to determine the prob- ability of reclosures due to rebound, in relays having limiting adjust- NEW MULTICOXTACT RELAY 1129 Fig. 16 — Contact cover for the new relays showing compartments in which contacts are grouped. ments. The probability of reclosures exceeding one millisecond duration is estimated as one in 28,000 relays. This grade of performance is due primarily to: (1) an armature having low travel and the lowest possible mass; (2) an armature suspension designed to dissipate rebound energy into the core plate and core rather than into the actuating card; and (3) a stiff mounting bracket to reduce the natural amplitude of core \'ibra- tion due to armature impact on operate and on release. Life Less than 5 per cent of the new multicontact relays will l)e required to operate more than 100 million times in crossbar systems during an csti- 1130 THE BELL SYSTEM TECHNICAL JOURXAL, SEPTEMBER 1954 mated life term of forty years. Life tests show that no readjustment should be necessary during the first 100 million operations. The tests also indicate that not more than a small percentage of relays will re- quire readjustment prior to an estimated maximum life of 200 million operations. In extreme cases, still greater life may be obtained if re- quired, by replacing the molded card. This is an inexpensive part and replacement is easily accomplished. Stability When the new relays are exposed to extreme temperature and hu- midity cycles, the greatest change in contact separation is, in general, about 0.004 inch, and only a small percentage of relays are likely to be used in this manner. Tests indicate that changes of this magnitude leave adequate margin for 100 million operations before readjustment is necessary. For economy, most equipment is shop assembled and wired on a frame basis, and shipped complete, read}^ for installation as equipment units. It is important, therefore, that apparatus units should be capable of withstanding physical shock far in excess of normal usage. Design features in the new relay which provide an adequate margin of safety in this respect are: (1) a rigid mounting bracket; (2) the wire spring pile-up is attached securely to the bracket with two specially heat treated steel screws; (3) the cover is held in place by the bending moment of an embossed section of a spring clip with a force many times greater than the compressive force of a single spring; and (4) guard surfaces molded m the cover prevent twin wires from leaving their respective guide notches in the single wire combs. Excessive shocks during shipment have, at times, damaged flat spring relays by bending their brackets. The new relay's have been subjected to shocks of similar magnitude without damage. Magnetic Iiiierjerence Under certain marginal conditions, a relay may be affected by leak- age flux from adjacent relays entering its magnetic circuit, and changing its operate and release values. Tests show that interaction is negligible between the new relays and also between new and old type relaj's when they are used in adjacent positions. CONCLUSIONS The new 30-contact rela3^s provide faster operate and release times, longer life, improved contact performance, reduced maintenance, and NEW MULTICONTACT RELAY 1131 greater adaptability in new circuit and equipment units than previous multicontact relays. The new relays also require less vertical and hori- zontal space in new equipments. As a result of these improvements, substantial savings are expected when these relays are used. The design includes many features which permit the use of mechanized manufac- turing processes, which, in turn provide better control of tolerances. For these reasons, lower initial costs are expected as manufacturing and assembly methods continue to improve. The new 60-contact relaj^s are completely interchangeable with all codes of flat spring multicontact relays in existing crossbar equipments, and in addition, they provide superior performance, longer life and reduced maintenance. Therefore, manufacture of the flat spring multi- contact relays will be discontinued as soon as new relay production becomes adequate for all uses. ACKNOWLEDGEMENTS Many design problems required the cooperation, special knowledge and facilities of Materials, Chemical and Research Departments of Bell Telephone Laboratories. Some design features, particularly those involving new manufacturing processes, were developed in close co- operation with Western Electric Company engineers. These acknowledgements would not be complete without including the technical contributions and assistance of the many people in Switch- ing Apparatus Development Department who were directly and in- directly associated with the project, and to E. G. Walsh and T. H. Guet- tich for their assistance in the preparation of material and illustrations for this paper. REFERENCES 1. G. S. Bishop, Connectors for the No. 5 Crossbar System, Bell Labs. Record, 28, p. 56, Feb., 1950. 2. A. O. Adam, The No. 5 Crossbar Marker, Bell Labs. Record, 28, p. 502, Nov. 1950. 3. Bruce Freile, Multicontact Relay, Bell Labs. Record, 17, p. 301, May, 1939. 4. A. C. Keller, A New General Purpose Relay for Telephone Switching Systems, B. S. T. J., 31, p. 1023, Nov., 1952. 5. R. L. Peek, Jr. and H. N. Wagar, Magnetic Design of Relavs, B. S. T. J., 33, p. 23, Jan., 1954. 6. R. L. Peck, Jr., Internal Temperatures of Relay Windings, B. S. T. J., 30, p. 141, Jan., 1951. 7. C. Schneider, Cellulose Acetate Filled Coils, Bell Labs. Record, 29, )). 514, Nov., 1951. S. A. J. Brunner, H. E. Cosson and R. W. Strickland, Wire Straightejiing and Molding for Wire Spring Relays, B. S. T. J., 33, p. S59, July, 1954. 9. A. L. Quinlan, Automatic Contact Welding in Wire Spring Relay Manufacture, B. S. T. J., 33, p. 897, July, 1954. 1132 THE BELL SYSTEM TECHNICAL JOI'RNAL, SEPTEMBER 1954 10. Solderless Wrapped Connections, B.S.T.J., 32, May, 1953. Introduction, J. W. McRae, pp. 523 and 524. Part I ~ Structure and Tools, R. F. Mallina, pp. 525-556. Part II — Necessary Conditions for Obtaining a Permanent Con- nection, W. P. Mason and T. F. Osmer, pp. 557-590. Part III — Evaluation and Performance Tests, R. H. Van Horn, pp. 591-610. 11. W. T. Prichard, Relay Contact Welder, Bell Labs. Record, 22, p. 374, Apr., 1944. 12. M. A.Logan, Design of Optimum Windings, B. S. T. J., 33, p. 114, Jan., 1954. 13. E. G. Walsh, Continuously Recorded Relay Measurements, Bell Labs. Record, 32, p. 27. Jan., 1954 and H. X. Wagar, Relay Measuring Equipment, B. S. T. J., 33, p. 3, Jan., 1954. 14. E. E. Sumner, Relay Armature Rebound Analysis, B. S. T. J., 31, p. 172, Jan., 1952. Topics in Guided Wave Propagation Through Gyroniagnetic Media Part 111 — Perturbation Theory and Miscellaneous Results By H. SUIIL and 1.. R. WALKER Some prohlcnif^, complete discussion of which would be extremely difficult, are treated approxijnately by means of perturbation theory. Among these are the partially filled cylindrical waveguide, and the problem of multiple internal reflections in a sample of finite length filling the cross section of a cylindrical guide. Propagation in a ferrite between pai'allel planes, mag- netized along the propagation direction is discussed by the methods described in Part I. The paper concludes with an addendum to Part I — a numer- ical study of field patterns of the TEu-limit and TMn-limit mode for various dc magnetic fields. IXTRODUCTIOX Parts I and II of this paper were devoted to a number of specific propa- gation problems, whose solutions, though frec^uently quite compHcated, could be discussed with a reasonably modest investment of effort. Un- fortunately, not all of these problems pertain to situations met with in actual gyromagnetic devices. Actual devices frequently employ struc- tures whose performance could be predicted only as the result of lengthy computing programs. For example, the microwave gyrator using Faraday rotation usually employs a ferrite sample whose cross-section only partly fills that of the cylindrical waveguide. Although it is easy to formulate the corresponding equation for the propagation constant, the classifica- tion and survey, let alone the computation of solutions, would be very difficult to carry out. Thus, one must often be content with approximate results, and the bulk of the present paper is devoted to perturbation methods. These take as starting point a situation whose propagation problem is essen- tially solved. The small change in propagation constant due to a slight change in the original state of the system is then calculated. The small 1133 1134 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 change of state may be the weak magnetization of an originally un- magnetized specimen occupying a substantial part of the structure, or the introduction of a very small specimen with arbitrarily large mag- netization into the originally empty structure. Under the heading, "Small Magnetization — Arbitrary Sample-Size," we shall discuss the propagation constant for a pencil of ferrite of any radius, coaxial with a cylindrical waveguide, the space between guide wall and pencil being filled wth an isotropic medium whose dielectric constant equals that of the ferrite. This is discussed in preparation for the practically more important case of a ferrite pencil of any radius in an air-filled guide. Here the unperturbed state of the system, when the pencil is unmagnetized and therefore isotropic, is already rather complicated and recjuire some preliminary calculations. Under the heading "Small Sample-Size — Ar- bitrary Magnetization," we consider the case of a thin pencil of ferrite in an originally empty guide. Another topic, not easily treated except by perturbation methods, is that concerning end effects in samples of finite length. After a prelim- inary discussion of internal reflections in an extended slab of ferrite (a problem which can be treated rigorously), two cases are considered: a ferrite slug of arbitrary length, closely fitting a cylindrical guide, and a thin disc normal to the guide axis, of arbitrary size. In these cases in- terest centers around the effect of sample length on Faraday rotation, though for the ferrite slug a subsidiary effect, that of mode conversion, is also mentioned briefly. It should be emphasized that the perturbation methods employed here are not in themselves novel. They are standard to most linear eigenvalue problems of physics, and have been used in connection M'ith electromag- netic problems by many authors. The remainder of the paper is devoted to a discussion of a ferrite-filled "cable" in plane parallel form, using the methods of Part I. The treat- ment is kept in terms of saturation magnetization and magnetizing field, and is based on Polder's equations. The paper concludes with an adden- dum to Part I, which reports some calculations and graphs of field pat- terns in a cylindrical waveguide completely filled Avith ferrite. 1. PERTURBATION METHODS 1.1 General Method A number of authors have made applications of perturbation theory to the problems of propagation in gyromagnetic media and the exposi- tion w^hich follows is included mainly for completeness. We shall develop the subject in the follo^\ing fashion: it ^^^ll be supposed that the unper- GUIDED WAVE PROPAGATION THROUGH GYROMAGXETIC MEDIA. Ill 1135 turbed system is a wave guide containing a medium whose permeability and dielectric tensors are diagonal and isotropic, but ma}' \ary o\-er the cross section of the guide, although not in the 2-direction along tlie guide. For this system it will be assumed that a complete set of normal modes exists for which appropriate orthogonality relations are known. The perturbation of the system will then consist of changes in the permea- bility and dielectric tensors of the medium, including the addition of non-diagonal terms. If these changes are to be genuine perturbations, they must be of one of two kinds. Either, the variation in the properties of the medium is confined to a limited region, small in volume in some appropriate sense, in which case its magnitude may be large, or, we may have a small fractional change in the material properties exteiuUng over a considerable volume. The fields in the guide may be expanded in the normal modes and a system of equations is developed for the z-de- pendence of the amplitudes of these modes. These equations are then solved approximately, making use of the smallness of the perturbing terms. The results may then be specialized to the various situations of mterest. Let us suppose that the unperturbed permeability and dielectric con- stant are niix, y) and ti{x^ y) respectively and that the system is now altered so that it possesses a permeability tensor ii-iix, y) -jk{x, y) 0 jii{x, y) 0 and a dielectric tensor t2{x, y) M2(a;, y) 0 0 MsCa;, y) -jvix, y) 0 j-n{x, y) eoCr, ?/) 0 0 0 e^{x, y) Maxwell's equations for the perturbed system may be written, using the notation of Parts I and II, f in the form: dHt* dz dEt* dz — jC0€2Et — 0)7] Et* = 0, -f jwniHt + o^kHi* = 0, V-^,* - 3^€^E, = 0, V-Et*-^ jcofisH, = 0. (1) t We omit the vector signs from all transverse vectors, which are sufficiently labelled by the subscript "t." 1136 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 195-4 These may be rearranged as oZ f)F * V*E, - -^ + >Mi^^ = -Mt^2 - tii)Ht - c^kH,* = B, , (2) V-Ht* - icoeiK = Xes - €i)E^ = A, , where At , Bt , A^ and B^ are introduced as abbreviations for the terms on the right hand side of the equations. E^ and H^ may be ehminated by substituting in the first two equations the expressions E. = V-Ht* - A, and ^ ^ -V-Et*+ B. The two equations so obtained are JO) \ m / dz ju Ml and 1 V. ( ^^•) - ^^ + >..ff , = B, + A V* i- . (3) JO) \ €i / dz JOO €i We now suppose that Et and Ht can be expanded in the form Et = ^ an{z)Etn{x, y) n and //"< = 23 hn{z)Hin{x, Ij), n where E',„e-^''"^ and HtnC''^"' satisfy the unperturbed form of equations (3) and the boundary condi- tion that tangential E vanishes at the guide wall. These equations have solutions for certain values of j8„ only, but, if Em , Hm are solutions for )3„ = c > 0, then Etn , —Hu are solutions for fin = —c. We shall assume GinOED WAVE PROPAGATION TIinOT'GII GYROM AGXETlf MEDIA. Ill 1137 hereafter that Em and Hm pertain to positive j3„ ^'alues. For a given iinpei-turl)e(l mode it follows that -^, reverses sign when the direction of propagation reverses. Substituting these series for Kt and //, in ecjua- tions (3), one finds and E Z dz An -^ + J^nOn = -— Ct^ A„ (7a) (7b) Equations (7a) and (7b) are, so far, exact, but they involve, on the right hand side, the functions Et , Ht , E^ and Hz which are still unknown. We are interested in those cases where the integral terms are small, either as a consequence of the terms (62 — ei), 77 and so forth being small, or of their being finite only over a small region. In the first case the fields Et , Ht , E, and H^ may be replaced in the integrals by the values which they would have before the perturbation was made. In the second case this is not possible since a large change in the material constants of a GUIDED WAVE PROPAGATION THROUGH GYHOMAGNETIC MEDIA. Ill 113!) region alters the field substantially ^\■itllin that resion. TTore, then, we have a preliminary problem to solve, namel}- that ofdeterminina; the field in the perturbed region in a zero order approximation. Perturbation problems ma^^ i)e divided into two classes by another distinction. The changes in material properties may be independent of the ^-cooixhnate, so that the new prol)lem is to consider propagation in a uniform guide differing slightly from the original one. Typical of such problems is that of a waveguide containing a ferrite rod of infinite length parallel to the 2-axis; the perturbation consisting here of the change in the properties of the rod when it is magnetized. Clearly, in such cases, solutions for which all field components vary as exp — jl3z are still possible and the perturbation equations (7) become ecjuations to determine (3. On the other hand, there is a class of problems for which the perturbation is confined to a limited region in the ^-direction, and we are interested, perhaps, in the reflection and transmission coefficients for a wave inci- dent upon the obstacle. Here, for example, we might think of the case of a disc of ferrite across the guide. If we remain in the range for which perturbation theory is valid the changes in the amplitude of reflected and transmitted waves will be small, but the changes in phase may not be, if the perturbed region is sufficiently long. In the latter case, it would be possible, if the perturbation were uniform in z over the region in which it exists, to find solutions going as exp j^z, as described above, and to use these to fit the boundary conditions at the ends. It is also possible if the perturbed region is long, with slowly varying properties, to obtain suitable approximate solutions by the WKB method. Some of these cases will arise in the examples which we treat below\ 1.2. Perturbations Uniform in z We consider first the general case in Avhich the perturbation is uniform ill z. In the absence of the perturbation the m*^ mode is to be present. For the fields Et and H^ , in the perturbed region we write am{z)Etm(i{x, y) and am{z)Hzmo(x, y), respectively, where am(z) is the amplitude function for the ?n*'' mode. If the perturbed region is one in which, for no magneti- zation, the material properties differ only slightly from their unper- turbed values, we may justifiably identify Etmo ^vith Eim. and H^mo with H^rn ■ If the material properties are appreciably changed even in the absence of a magnetic field, Eimo and //jmo have to be calculated by an independent method. For a^ we put A^e^^ ^ where ^ = /3,„ + 6/3 and 5/3 is small. Similarly for Ht and E^ , we write 6„ Hi„,o and 6,„ Ez,„o , with bm -^ B^e"^^'. With such assumptions, the 7n"' set of equations (7) 1140 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 gives an eciiiation for (3, while any other set, with n ^ m, gives the excitation of the n^^ mode. Substituting in (7) we have + (/i3 — IJLl)H,moi^zm] dS ) .4„ = jLA, (8a) and + (63 - e,)E,m,E.„] dS Bm ^ jMB (8b) Ignoring squares and products of small ciuantities, one then has 5^ = 1^ (L + .1/). (9) The first example to be considered is the effect on the propagation in a circularly cylindrical waveguide, when a coaxial, magnetized pencil of ferrite of very small radius is introduced. The guide radius is n and that of the pencil is Vi . Before the ferrite is introduced, mi == /^o and ei = eo , where mo and eo are the free space values. The unperturbed fields are those of the usual TE and TM modes in round guide. It is necessary to calculate first the zero order electric and magnetic fields within the magnetized pencil ; it will be sufficient to work out the magnetic case and deduce the electric one by analogy. Since the cross-section of the pencil is very small and transverse propagation effects consequently negligible, the internal field may be calculated by solving a static problem. The transverse magnetic field before the pencil is inserted is Htm and it is assumed that the pencil is so small that over a circle with a few times the radius of the latter, Htm is essentially uniform. We must now solve Laplace's equation for a cylindrical rod immersed in a magnetic field which is to be uniform at large distances. Within the rod, Bt = iiHt — jiiHt*, and at its surface the usual boundary conditions pre^•ail. Hereafter we write m for jU2 . The fields are derivable from potentials $out , ^in , which are of the form 'I'in = (Htmo-r), ^ fu ^\ ^ (^■^) 'S'out = {Htm-r) + -— - , GUIDED WAVK PROPAGATION' TIlHorCII (!YHOM AGXKTIC MKDIA. Ill 11 11 where r = (.r, y), a is a constant vector and the coorchnate system has its origin at the centre of the rod. Continuity of tangonlial // at Iho sur- face of the rod requues H, H i,n + n' and then $out = Htm-r + 4 (^'"'O - Htm)-r. r- The normal derivative at the surface of the rod is or, externally, n 1 3$ , a$ - .(• — -r y — fi [_ dx dy_ [Htm-r — {Htmo - Htm)-r]. ^Matching the normal B's at the surface then gives jUo[2 Htm — Htmo] = tJiHtmO — j nH t. or n tmQ — 2mo[(/X + lJLf^)Him + JKHtm*] (n + Mo)- — K^ (10a) In a similar manner, one would find if the dielectric constant of the rod were e. EtmO — 2eoEtm e + Co (10b) The longitudinal fields E^m and H^m are unchanged wdthin the rod. Turning now to the expression (9) for 5/3, Ave have in the present-case. 80 = --^ [ dS ■^Za»« •'pencil '260(6 - 6o) e -\- €o 1 Etm I' + (6 - €o) I E, + 2mo M Mo Ht,n \' - JK 4iU0 HtJHt (m + Mo)- — K- ' ^"" ' ■' (m + Mo)^ — K^ where we have anticipated that Am is real, which we A'erify below. Since the integrand is constant the integral may be replaced by tt/'i times the value of the integrand at r = 0. We consider now a TE-mode 1142 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 with variation e"'"^, n ^ 0. We shall have, Elm = -jo)HO^*^m , Htm = -j^mV^m , where and ^m = Jn I Unm - j fi'"'^, J n{Unm) = 0 r, 2 2 Unm Pm = (^ eoMo — — r For the fields on the axis, one finds for n = dzl, nHr = -jH^ = -jn — — e' Zro and nE^ = jEr = jo)on -^ e'"*'. li \n \ 9^ I there is no first order perturbation. We now have 2 2A, But we have Am Trri e — eo 2 2 Unm~ , IJ-fiQm'Unm" jJL -\- UK — /UO Co — — w Mo — r + ^ j 1 e + €o r(/ To- fji -\- UK -\- Mo_ Jo = — 27r-w/xo/3« • ^ = —ZTW/JLoPm ?; \llnm — 1. and then, ''7171 1 /-f 2/3„, 7-0- Jn{nnmy-(u„m- — 1) 2 M + ^« — Mo I ^ 2 €l — Co Pm 1 j -t- PC , H + riK -jr Ho ei -\- €n_ . (11) GUIDED WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA. Ill For Tj\I modes \xc ha\'c Elm = —jl^m'^Xm Htm = icOeoV*Xm 1143 E = -i^^ V ■*-'3 ., An where and = Jn{j " n\Jnm) — vJ ^OT = CO €oMO — Jnm 1^ Proceeding as before, we find, 1 rx 1 6/3 = - 2/3. n' lo 2 M + ^« — Mo , ^ 2 € — eo "^ PO , , i- Pr. Jn'ijnmY \_ M + ^K + Mo e + Co. (12) A problem which is of some interest, although not of immediate prac- tical significance, is that of a ferrite pencil of arbitrary radius and infinite length in a round wave guide, with the remainder of the waveguide filled Mith a non-magnetic dielectric, whose dielectric constant, ei , is equal to that of the ferrite. The ferrite is supposed to be only weakly magnetized. For such a problem, we have. 5/3 = - 2A, / [(m - fio)Ht,n-Htm - JKHtm,*-fftm\dS. ''pencil Htm is the field of an unperturbed TE or TM mode in the dielectric-filled guide, n — Ho and k are supposed small, but 7\ , the radius of the pencil need no longer be small. For TE modes, we have as before (again excluding the case n == 0), Etm = -jo)fJLO^*%n , Htm = -jlSmV'^m , r ^m = Jn[ U and '■nm I 6 ?'0/ a 2 2 Unm Pm — o: eo/XQ — — ir r) 2 "^Inm Pi. ~ — T 1144 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 Hence, assuming the pencil coaxial with the guide. 2A. CO Jo L ^0^ \ ru/ , W r 2 / r + — Jn I Unm - r- \ To r (h- + -Ittk^J -2^/1 f " ./„ f //„„, -^ ./,/ (,(,„„ - \ dr, rn Jo \ ?■()/ \ n,/ = 27r/i„ (m — mo) / Jo xJ';{x) + /;(.r)" dx ' «nm (ri/ro) + 2m / Jn{x)Jn{x) dx Jo = 9 27r/3„ J? r/2/ (m -Mo)(.r./„(.r)./„'(.r)+.^J::(x) + — -^ ./«(.V) ) + K-7lJ\ I ?'« Making use of the \'alue of Am found in the preceeding paragraphs, we have [UnnT — l)Jn{Unm)~ L\MO / \ 2 + ""-^ Jlix) + n - J ,A Unm - jJ-Q ro (13) For TM modes. Ht,n = jo^eiV*\l/,„ , ./«./« To In this case. a - 2 Jnm ^ 2 Jnm fi,n = o) ei/io — — ^ = ^1 — — r 2x(a)ei) ( (/X — /io) / ~ Jn \ jnm - I \ Jo L^" V ^0/ + H J'n ijnn, -)] T dv + 2^/1 f ^^ ^^ ./„ (jn,n "^ /,/ 6»« -^ ^r) . /•o^ \ ro/J Jo ro \ ro/ V ro/ / GriDKD WAVK i'i{( )1'A(;ati()\ TiiKorcii (;yu()ma(;\ktic mkdia. hi The A'aluc of A,,, for this case is 2 The \"ahie of 8tS now hecoines 83 ^'' Pmjnm «/ n \3 nm) L\M './„(.r)./.'(.r) + . ./',;(.,■) - /-r=J„„,('-i/ro) Mil \ ^11 (14) We note that for a ferrite filled siiide \nth ri — tq , the nonreciprocal part of 5/3 \-anishes which confirms a result found in Part 1 of this paper for weak magnetization. The very high dielectric constant of the ferrites (about 10) puts rather sex'ere restrictions on the size of the pencils to which pertiu'bation theory is applicable, even for weak magnetization. This limitation would l)e substantially relaxed if we possessed exact solutions for rods of high dielectric constant inserted into round guide, which could be used as the basis for magnetic perturbation calculations. Unfortunately, the only extensive published calculations of this kind are for dielectric constants less than 3. However, at the suggestion of M. T. Weiss, a calculation of the propagation constant of the lowest mode varying as e^^^ in a wave guide containing a coaxial dielectric rod (ti = 10) has recently lieen made in the ^Mathematics Department, for varying rod diameter, but for a single \'alue of guide radius equal to 0.4 times the free space wavelength. With the aid of this information, which was made available to us, the magnetic perturbation calculation has been carried out. As before, the rachus of the guide is ro and that of the rod is rx . The dielectric con.stant of the rod is ci . We consider first the propagation in the unmagnetized case. Since we are considering only one mode, namely, that with an angular variation, e^'^, and of the lowest order radially, we need not identify the £"s and //'s by a label. We use a subscript "1" for fields in the dielectric and "0" for fields in the empty part of the guide. In general, we have a'Et = -j[c„jiV*H, + (3VE,], a'Ht = -j[^3VfI^ - o:eV*E,], where 2 2 ^2 a = 0} en — p , V'E, = -aE,. 1146 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 and €, fjL refer, for the time being, to the dielectric constant and perme- ability of the region considered. It will be convenient to put t — \/ — JJz,t ; \ Mo Mo m; /5 wVenMo CO V foMo 1 and to measure lengths in terms of ' — 7= . We shall continue to use coVMoeo V and V* with the understanding that they refer to the scaled units. We noAv have ja'Ht = )8 V^. - -eV*E, , _2 -_ 52 V £", = — a E, . At the surface of the rod E^ , H^ , E^ and H^ are continuous. We must have 1 \ dr /o and where "0- ro dr /o n dr 1 dr /i j'l Cl 61 = -, €0 cto" = 1 and - - - a^ Oil = ei — p . jix = 1, everywhere, if the uimiagnetized ferrite has the permeability of free space. These equations may be rearranged in the form: 1 (du, . ^ U^{r^ \ dr /o. + A. 1 r\ dH, 1 "o- «iV 1, «o- nJ: rJ.2 ri «!- L H^iri) \ dr /i £'.(ri) V dr Jo + ^ Ti 1 /dE, ai' L ^.(n) V dr )x_ jH.{r,), EM). (15) GUIDED WAA'K PROPAGATION THROT7GII GYK( )MA(;\1:T1C iMlODIA. lit I 1 17 Within the dielectric, since all fields are hounded at r = 0, both E, and If, are proportional to ./i(air) and, e\identl3', in the notation of Part I. (E^)o and {H^)q , similarly, Avill be those two Hnear combinations of Ji(aor) and Yi{aor), which, respectively, vanish and have zero normal deri^'ative at r = ro , in order to ensure the van- ishing of the tangential fields there. The functions r d(H.), , r d{E.)o and (^.-)o dr '" iE,)o dr Avill be called //(aor) and G(aor) respectively. Eliminating E~{ri) and H,(ri) from equations (15) we obtain the char- acteristic equation of the problem in the form an- ar/ \ a^- ao /\ «r "o" The perturbation in the present problem is that due to a mild mag- netization of the rod and referring again to equation (9) we have (in unsealed units) 5/3 = 5^ = -K^ f [(m - f^o)HfHt - JKH,*-ff,] dS, •'rod A„ = f E*-IJtdS. "guide ;m, t Jo L\Mo / Mo 2A ^^•ith 'guide Thus, in the scaled system, t*-Htrdr '0 JO (16) The evaluation of the normalizing integral in the denominator is an ex- ceedingly tedious business and it seems advisable to avoid it. This may be done in the following manner. The characteristic equation has been solved for numerous values of Vi and ^ may be considered to be a reliably dB known function of Vi . In particular the slope -p- is knoAvn. But we may dri dB also deduce -y- , by a perturbation calculation in ^\•hich we start "with a dri 1148 THE BELL SYSTEM TECHNICAL JOURXAL, SEPTEMBER 1954 rod of radius ri and increase the latter to /'i + di\ by changing eo to ci in the shell Vi < r < /'i + dri . For such a perturbation, since A\ and E^ are continuous at the boundary, they suffer no change when the boundary moves; Er however is discontinuous and (Er),, = -(7!-'r), where fAV), and (Er)i are the noimal fields just outside and just inside the rod. The per- turbation formula (unsealed), thus gives 6(3 = -f (ei - 6o) I E, r + \E^\' -\-^-^\ (Er), |- )-27rr,8n , or, scaled, with r and /'i representing scaled radii, 1 (ei - 1){\E,'C ^ lE^f ^ e^\{Er)if) 6/3 = r /■i5ri Ei*-Hrr dr The formula (16) for the magnetic perturbation may now be written 5^ = , Vi dvi L\Mo / Mo 61-1 1 EM) \' + I EM) P + ei I {Er)i Inside the rod, we may write H-Xr,) di and then H. = •_ 2 EAn) E, = jcE, jai Hr = jcfi dE, dr r ■}- - 7 dE- ,m'H,^j0^-E,-\- -er^ r or The integrals are readily carried out and are as follows: [ ' \Ht fr dr Jo E:(rr) (€i ^ c (3 )[ Fiain) + + I - Ic^ei f \Ht*-Ht)rdr J I) jE.\n) a,-' (ei + c /3 ) - 2ceil5 F(airi) + + — - Gl'IDKD WAVK PHOrAG ATION TIIHDrr.lI GYHOMAOXETK' MKDIA. Ill 111") The term in the (Icnoiniiialoi- may he cxaluated l)y iisinjz; ■ 2j. c ,, , -dE, r dr jai E^ = -jc + •- is, . dr /• We obtain I E.in) {' + 1 Ee(r,) f -^ h \ {Er), \' = E/in) 2 ;v.4 The pertui'baf ioii may now be wi-itten: 61 - 1 Amo / Mo rr«i4 + [^ - cF{a,r,)Y + €,[c - ^F(ain)]- where A = — + F{airi) + —y- and c may be obtained from equation 15, and the definition of c Gjaori) _ . Fjaifi) _ otQ^ 5i^ _ «! G(aori) — hoco FiaiTi) \ao- «! / with Ni(aoro)Ji{anri) — Jj(aoro)Ni{aori) ;i7) GCa-dri) = aori A^i(aoro)J'i(a:ori) — Ji(Q:oro)A^](anri) Fig. 1 shows the propagation constant as a function of the relati\'e diameter, ri/vo , of the dielectric rod in the unmagnetized case, with €i — 10. Fig. 2 shows the deri\'ative -r: — ;— : as a function of vi/rn . The a{ri/rQ) guide racHus is 0.4 times the free space wavelength. From (Hiuntion (17) 5^ may be written as 1150 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 1.2 0.8 ^ ^ / ^ / 1 f \ \ \ J 1 y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Fig. 1 — Propagation constant of a circular guide containing an unmagnetized coaxial, dielectric rod (e/eo = 10). ri is the radius of the rod and ro the radius of the guide, ro = 0.4Xo , where Xo is the free space wavelength. The computed values of Q and P — Q are shown in Fig. 3 as a function of relative rod radius. P — Q is plotted since P and Q are very nearly equal.* 1.3 Perturbations Non-Uniform in z So far we have been concerned only with structures indefinitely ex- tended in the direction of wave propagation. In practice the non-recipro- cal element is, of course, finite, but end effects can frequently be ne- glected, since the element is matched at its ends (by tapering of the finite * H. Seidcl and Miss M. J. Brannon, at the suggestion of M. T. Weiss, have i-e- cently calculated the dielectric loss for the guides containing a dielectric rod de- scribed above. By combining such information with that obtained here it is possi- ble to discuss figures of merit (degrees of rotation loss in dli) for various pencil radii. Such an analysis is being made by M. T. Weiss and will appear in an ar- ticle, by S. E. Miller, A. G. Fox and M. T. Weiss, in a forthcoming issue of the Journal. GUIDED AVAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA. Ill 1151 sample, for instance). The matching could be accomphshed in such a way that the transition region, whose characteristics would be very diffi- cult to compute, should contribute little to the overall non-reciprocal beha\'ior. Therefore, in many cases, the theory for the indefinitely ex- tended sample is adequate. For some special purposes, however, it is desirable to mismatch the sample deliberately. For instance, Howen' has suggested that the change in Faraday rotation, due to internal re- flections in an unmatched specimen, can offset to some extent the fre- quency dependence of the rotation which is implied by the Polder rela- tions, broadening thereby the useful band\\idth of the device. Consider an infinite slab of ferrite, magnetized in a direction normal to its two parallel plane bounding surfaces. A circularly polarized wave, normally incident on the slab, A\ill be partially transmitted, and, since for such a wave, the medium behaves as though it had an ordinary scalar permeability, the phase and amplitude of the transmitted portion are readily calculated. It is clear that, as the result of multiple internal re- flections, the phase of the emerging wave AAnll differ from the value ap- propriate to a single trip through the slab (such as would be obtained were the slab perfectly matched). Both amplitude and phase of the transmitted wave aWII depend on the electrical thickness of the slab and its dielectric constant, and on the effective permeability. The latter dc 0 r \ 1 \ i \ \ \ J 1 \ / ^ - — J 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 C= n/Po Fig. 2 7 — ^ versus — . a fe) 1152 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 n --- r^ 1 1 -1 ^ / \ / \P-Q // x ^^ — -^-> 7 ^ ^^i P-Q 0.3 0.4 0.5 0.6 0.7 0.6 nAo Fig. 3 — Q and P — Q versus n/ro . differs for right and left circular waves, so that a plane polarized signal (which is the sum of equal right and left circular components) will emerge, in general, elliptically polarized, with the major axis of the ellipse tilted from the polarization at incidence by an angle which differs from the single trip value as the result of internal reflections. It is clear that this change in rotation can be calculated in a very elementary way. When the sample is confined to a waveguide a similar effect occurs, but its calculation becomes extremely involved, at least for arbitrarily large magnetizations. The reason, which should be clear from Part I of this paper, is that the circularly polarized modes no longer have the same field configiu'ations inside and outside the sample.* Therefore any inci- dent mode excites all of the normal modes of the ferrite; these, in turn, give rise to all the mode patterns of the air-filled portion of the guide. Even if all but one of these are cut off, the excitation modifies the phase and amplitude of the reflected and transmitted portions of the propagat- ing mode. * This is due to the fact that there is now no ordinar}^ effective scalar perme- ability as for infinite geometry. GTIDKD WAVK I'HOPAGATIOX THROUGH GYHOMAGXKTIC MKDIA. HI I 1 .")8 Thus all modes have to be iucliuled in the problem, which conse- quently takes the form of an intinite system of linear equations for the mode amplitudes. This can be solved only to some approximation whose general \alidity it would be hard to establish. The problem could also be stated as an integral equation iiivoh-ing a complicated Green's func- tion, with no greater chance of complete solution. AVe are therefore forced to restrict the problem to the ranges of mag- netization, or of sample size, in which perturbation theory is applicable. However, we begin with a discussion of the infinite, plane, loss-free slab, a problem which can be solved completely, and which has some bearing on the perturbation problem for a slug of ferrite whose cross section completely fills the waveguide. Let the plane boundaries of the slab be normal to the 2-axis, which is also the direction of magnetization and the propagation direction of a circularlj^ polarized wave incident on the boundary z = 0 (see Fig. 4). In terms of the parameters p and a of Section 2.1, Part I, the effective permeability' for a circularly polarized wave is fJL ± K =^l = /Xo(l± P 1 ± (7 the upper sign referring to right circular polarization. The correspond- ing propagation constants in the slab are then /3± = coVeMoy 1 ± Y^a Fig. 4 — Normally niagnotizod foirite slab. 1154 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 Avhere e is the dielectric constant of the ferrite. If d is the thickness of the slab, the electrical thickness is e± = I3±d = -- where 9o = co-\//xo€ d is the electrical thickness of the umnagnetized sample. Let us now confine the discussion to the right circular wave. If the incident electric field is taken to be e~^^°', where /3o is the free space propagation constant, co \//io€o , the incident magnetic field ^^•ill be WMo and if the reflected electric field is pe^^"', the reflected magnetic field will be CO Ho since |8o reverses sign. Inside the slab, the electric field consists of forward and backward travelUng parts Tie~"'^+^ and 726^^+^, and the corresponding magnetic fields are and 0}IJ.+ Finally the transmitted electric and magnetic fields wiR be denoted by and — — T^e respectively. In general p, as well as the t's, wiil be complex. To obtain T3 , we write down the equations of continuity of all tangential fields across the boundaries 2 = 0 and z = d. Since the fields are confined to a plane normal to the 2-direction, these equations are: 1 + P = Tl + 72 , — d — p) = — (ri — T2) GUIDED WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA. Ill 1155 and T3 = Tie + + Tie -iP+d MO M+ / - Ti = {Tie coMo WMo T-ye j3+d^ These four equations in four unknowns are easily solved for t-.^ . Writing and noting that /3+ /^o Vr M+/ Mo V «o a where «=/-: x+ say, one finds the solution to be T-i where and dox -\- ^^ I — + —] sin ^0^;+ cos — J*H = I ^3 l+e tan$,=l(-^+^ 2 \x+ a tan ^o-'c+ ^3 1+ = 2 ^ , 1 / a a;+ cos ^o.T+ + - I — H 4 \a:+ a sui 6^0.1:+ -1/2 (18) (19) Similar results apply to a left circular A\ave ; it is necessary only to reverse the signs of o-, y in the expression for .t+ . Equations (18) and (19) show that equal right and left circular incident waves emerge with tUfferent amplitudes and phases; hence an incident plane polarized wave emerges elliptically polarized with its major axis inclined to the polarization direc- tion at incidence. The inclination and the ratio of minor to major axis \ (20) 1156 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 are determined as follows: The right and left circular fields, upon emer- gence, may be written in terms of rectangular components (with the polarization of the total incident field along the x-axis) E/ - jEy' = r^e'-\ E- + jEy- = T_e^"', from which the resultant field in the x-direction is seen to be E,T = E,-^ + E- = \t+\ cos M -'!>+) + I T_ I cos {o^t - *_)] and in the y direction EyT = Ey^ + Ey~ = | T- | si u {wt " $_) " | T+ | siu {uit - 4>+).J The amplitude at time t, {Ej-t + Eyr')^'^, is thus given by ExT^ -{- EyT^ = \ T+\ + I T_ I + 2| T+ I • I r_ I COS [2cof - ($_ + $+)] (21) The major axis of the ellipse is the maximum of {Ext" + Eyr^Y' ^Wth respect to wt. It equals | 7+ | + | t_ | and is attained at ^t = >^($_ + 4>+). Similarly the minor axis is the mhiimum and equals | t+ | — | t_ |. The ratio of minor to major axis is therefore I r+ I - i r_ I r+ I + I T_ The angle between the x'-axis (the incident polarization direction) and the major axis is found by substituting oit = 3^^(4>_ + ^+) in (20). This gives E^T = (I r+ I + I T_ I) cos Eyr = (I T+ I + I r_ I) sin which shows that the angle is 2 I T I and $ are plotted versus a; in Fig. 5. 7+ , $+ and t_ , $_ at given GUIDED WAVK l'H()rA(;ATI()\riIK(»r(ilI (JYHOMAGNKTIC MKDIA. Ill 1 I ")7 o .."^ '-2 X 2 08 ^*H ^ I NEAR X2 = ^ AND |7^ = /-. THE FLUCTUATIONS IN 1 SUBSIDE, SINCE THE SLAB _*" / X V \ \ IS THEN MATCHED. BEYOND X^ - j- i THE FLUCTUATIONS RISE AGAIN ° X o ^ ^ \ \ \ a I 0 o V M A l^ ^ »o = "- IT- X "• o "- 0 6 \ -1.2 T 1 2.0J 1.6J 1.2J O.aj 0.4J 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 X Fig. 5a 1.6 A 1.2 u-- "^ ] (\ A /\ 1 1 AAi \f lA /> u Vv v v\ / \ 11^ \J Vi i* 9S-0OX 1 I.OU 0.8L 0.6L 0.4L 0.2L 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 X Fig. 5b Fig. 5 — Phase and amplitude of transmitted circularly polarized wave as a / p 6 function of j = A/ 1 + : , with — = 10. (a), top, for ^0 = tt and Stt; (b), r 1 + o- €0 bottom, $ for 0o = Ttt; (c), | ts | for do = w and Stt; and (d), | T3 | for 00 = 7-ir. I o- I, \ p \ are found by choosing positive and negative signs for a and p in ■^ = / + ri^- Note that .r can be imaginaiy corresponding to cut-off in the range -1 < a < -1 - p (p < 0). 1158 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 ,# l\j 1^ \3/- / \i 1 w r «o = ^-''' Je^^^n 2.0J 1.6J 1.2J 0.8J 0.4J 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 X Fig. oc • « • • / » 1.0J O.ej 0.6J 0.4J 0.2J 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 X Fig. 5c1 Fig. 5(c), top and (d), bottom — See Fig. 5. Near re = 1 (corresponding either to small^ or to sufficiently large c), $ differs from its value $o for the isotropic case by a small amount 5$. The rotation, to first order, is then just one half the difference between the 5$ for positive and negative p, a. Writing .T± = 1 + 5.r± = 1 ± i ^ i 2 1 ± 0- GUIDED WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA. Ill 1150 and expanding equation (18), we obtain where ^ is defined by cosh '^ sec^ do — ° sinh ^ ^0 1 + cosh2 ^ tan2 ^o a =/«/- = e* Co [This result holds even near do = (n -\- 3^)x where tan do = ^o , as can be seen by expanding the reciprocal of equation (18).] The quantity }4do{8x+ — 8xJ) is the rotation corresponding to a single trip through the sample. The actual rotation is M(5$+ — d^-). Hence we may define a rotation gain as the ratio ^9o{8x+ — 8x^) , ^ 2 ^ tan 00 . , T ^^^^ cosh ^ sec 00 — — - — smh ^ 1 + cosh2 ^ tan2 Bo In many cases, 0o » 1, that is, the thickness of the sample is much greater than a reduced wavelength in the specimen. Then the second term in the numerator of gr is always negligible compared with the first, g then simplifies to cosh ^ gi = cos^ do + cosh^ ^ sin^ do This expression is plotted in Fig. 6 as a function of d for various a = e*. For given ^ it has minima equal to — -. — at do = [ n-\- - I tt, and max- cosh ■^ \ 2/ ima equal to cosh ^ at 0o = nirin = 0, 1,2, • • •). When a » 1 (a '^ 3 for many ferrites), cosh ■^ is replaced by }^e = }/2^, and then 1 _ 2 ylmin — a It is to be noted that when d = nir, the condition for maximum gi , the unperturbed reflected amplitude is zero, and the elhpticity vam"shes to first order in 8x. 1160 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 195-1: 91(^0) .^.^^^ 2 8 (n-i);?- (n-;^)77 r\TT (n + -j)77 (n+i);7 Fig. 6 ■ — Rotation gain, gi{Oa), versus electrical thickness, Bq , for small mag- netization. Perturbation theory enables us to solve approximately, for weak mag- netization, the problem just considered for the case in which a right circular cylinder of ferrite snugly fits a cylindrical waveguide. We will show that, "\\ith a suitable reinterpretation of the constants, equation (22) for the rotation gain of the extended slab continues to apply here. Before magnetization, a particular right circularly polarized mode, say the rri , is present in the sample. The small magnetization distorts the pattern of this mode and changes the propagation constant slightly. The distorted pattern can be expanded in the series of normal modes of the unperturbed material. In these expansions only the coefficient of the Gi ii)i:u ^^■AVE propagation thuoigii gyuomagxktic media, hi 11(11 originally present //?* ' mode Avill be large; all others will at most be of the order of the perturbation. Denoting perturbed quantities by the supeifix + , we have for the distorted fields Elm = e '" ' ^"=1 PmnEtn , J^ tm — C Zw 1=1 9mn" tn , where pmm and gmm are large compared AAith the other coefficients, ^ti and the p's and g's are determined from equations (7a) and (7b). a„ in these equations is identified with e""'*^'" ' p^r , hn ^^■ith e"-'^"' ' q^n ■ Since all perturbation integrals involving Et and H^ , E^ vanish in the present case, we obtain PnQmn PmPmn "T fit, - Ho)H'[mHtn dS -j f KH^*Htn clS and In the first of these equations, Htn , the perturbed magnetic field, con- sists of QmrnHtm , plus an admixture of other modes with coefficients them- selves of order m — Mo , «• Therefore, to first order, it suffices to write in the integrand, with the result: PnQmn PmPmn ■» mnQmrn j Pmn ^ Qmn > where IL = -^ [ ifi - H,)H„nHtn - JkHLHu] dS. An Elimination of p,„„ gives (/3^„ - ^m^)qmn = ^nltnqmn • The case n = m determines ^m': ^t' = /3^(l - li,n,^J. (23) All other cases give, to first order, B 7"^ Ki-* "in 1162 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 which, again to first order, equals mn ^l - &i Pmm . (24) One of the quantities pmm , qmm , is still arbitrary. If it is required that the perturbed field is, to zero order in 7^„ , normalized to the same value as the unperturbed field, it is readily found that one can take _ 1 _ i3m Pm We are now in a position to consider the problem of a right circular mode, say the r*^, incident upon the end plane, 2 = 0, of the ferrite cylin- der extending to z = d. One simplifying feature of this problem is that the unperturbed modes inside, and the modes outside, the sample have the same dependence on radius since the sample fills the whole guide- cross-section. However the modes inside and outside may have different numerical coefficients. Thus, if we distinguish quantities outside the sample by primes, the TE^r mode can be represented outside and inside the sample by / Pr Sgn iS'r ^ , coMo /3r Sgn I3r ^ Here sgn /S means: sign of the propagation direction, +1 and —1 for forward and reverse respectively. The function \{/sr is given by where

0) and X2(<0) satisfying the L-equation can then immediately be read off, but while they will give ^ = X1X2 , they do not necessarily, for given o-, p, satisfy the Polder re- lation, equation (35). To ensure this, the quadrant, X < 0, 0, and the L-contours therein are transformed on to X > 0, cr > 0 by the Polder relation for fixed p : X3 = '^ + P - ^' = rao. 1 — crXi The surfaces L\ = L(Xi , a) and Li = L(T(\i), a) will intersect in a number of curves, along whose base curves in the X-o- plane both the Polder equation and the L equation are satisfied, and along which /3^ = — X1X2 is known as a function of a, p. The zero and infinity curves of L(X, a) are denoted by 0, / when X > 0, and by 0', /' when X < 0, The transforms of 0', /' onto X > 0 are denoted by (0') r , (I')t . The suflfix n denotes the infinity (zero) curves corresponding to 1 — X^ / l\^ ^ /1 — X" ^\ r^7x = ("+2J^' ir:^X = "'5^j n integral. The lines X = 0, +1, —1, are all zero curves denoted b}^ 0^ , 0^ , Ob', respectively. The line Xo- = 1 is a conditional infinity curve, called Ic . It is an / curve when viewed from crX > 1 for X < 1, and when ^'ie^^•ed from o-X < 1 for X > 1. Otherwise (for aX < 1, X < 1 and o-X > 1, X > 1) it is a limit curve of all possible curves L = const., where the constant takes on any value indefinitely many times. (See Part I, Section 4, where the curve o-X = 1 is a conditional zero curve, Oc .) Fig. 8, drawn for a '^ 1, p '^ 0.5, shows the part of the first ({uadrant allowed b}' the Polder relation divided into regions of like and unlike sign of L(X, a) and L(T(\i , a)) by the various 0, / and (0')?- , (I')t curves. The un- GllDKD WAVE PR(1P.\0.\TIO\ 'rilliorcil (i V KO.M A(;\ KTK ' MKDIA. Ill 1 1 7!^ shaded regions are regions of like sign, and nil those carry solution curves (dotted hues), by the same reasoning as was employed in Part I, Section 4.11. Two branches of the TE]\I limit mode exist; in the area bounded by (0,4) r , 0^ , (Ob')t and in the area Ob, /i , (Ob')t ■ The branch in the first region begins at o- = 0 Anth Xi,X2 given by XiVl - Xi^ tan aVl - Xr = XaVl - Xo^ tan a\/l - Xj^, Xi + X2 = p and ends at the intersection of (0,i)t , 0^ with a = 1 — p and /3^ = 0 (since X2 = 0 on (0^)t). The branch in the region Ob , h , {Ob')t begins with /32 = 00 , 0- = 1 and proceeds towards a = ^ ^^"^ = 1 . The region bounded by 7i , {Oa)t , h contains an infinity of solution curves, the incipient modes. The n ^ of these begins at o- = 1, Xi = 1, the intersection of /„ and Ic (the line Ic is also the transform of X2 = — «= , Oav I.^. A ObV^(0'b)t (Oa)t (Ob)t 0 •A2 A,' Fig. (S - Zero ;iii(l infinity contour.s of L(Xi , I, the TEM mode has no lower branch. Behavior near resonance (a = 1): (37) TEM mode: /3' = :r-^p, n ' incipient mode: j8 = \ 2a2 /l - a (n = 1,2, •••) Cut-off of the n incipient mode (parametric representation) : Xi = e ; a = e I 1 ^"2 I + -^-2 « V a — Xi; = 0 7o2 2 2/ 2 2 rf/3 mir f m T - 1 coth e 2 2 m T (38) (39) , . _e , 2 tan a -e e 1 e + — z — e — e (The reader may note the similarities of these formulas to those for the cylindrical waveguide). Spot-point for the n incipient mode _2 1 - a r (40) p = (1 L- a)(/3' - 1\ GUIDED WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA. Ill 1175 Approximate foniuiltu; for all antisymmetric modes at small p, a ^ 1: TM modes: /3' = fi^j ^^^ , (41) 1 — 0-^ TE modes: /S' = /3„,>' M - 7-^. ) ' (42) where 2 2 ^2 -, n TT iSn.i = 1 - -^ n = 1. 2. and iSn/ = 1 - ^ zf^~ n = 0,1, 2, TEMmode: = 1 - 1 - a ^^ = l-r^. (43) TEM mode for large a: = 1 + ^ + 0(^1. (44) .2 . . P , ^/l , Hr = {H+ -f //_) cos $, Hg = (H_ - H+) sin $, (46) ^_(r) = (h 2xi«/i(Xi^o) 1 Xi J2{xir) Ml- 2xiJi{xin) 1 + H-{r) = -- 1 — Xi Xi 2xi/i(xiro) J2(xir) (47) 1 + 1 — Xi ^o(r) = 2xi^i(xiro) Uxir) ^ /o(xir) - ^i(xiro) Hoir) = -y^^^^'^ Xi JKxi^o) The terms in square brackets are in each case the same as the cor- responding unbracketed terms, X2 and X2 replacing Xi and xi • The ciuan- tities £"+ and E^ are the amplitudes of the left-handed and right-handed components of circular polarization into which the transverse £'-field may be resolved at each point. i/+ and i/_ have a similar significance Gl'IDKD WAVK I'H()I>A(;.\T1()N" TlIHOrCll C.Y KOM VCXKTir MKDIA. Ill 1177 for the transverse //-field. Hiis may be readily verified hy examining the vectors (E^, + jEy)e~'"' and (H^ + jHy)e~'"', which represent the transverse field vectors in the laboratory system. The transverse fields are elliptically polai'ized at any point and the ratio of minor to major axes of the ellipses are I I ^+ I + I ^- I 1 l\H^\ + \H_\\ • The ficld.s so far are normalized only hy the choice of a .sinii)lc foiin foi- the function, Eo(r); all components may, of course, be multiplied by the same arbitrary constant. There is some virtue to a normalization })ased upon power flow. The power flow is given in unreduced units by f {EX //). dS. •'guide Using the scaled units of part I with r 'actual ^^ / and // replaced by a/ ^ H to gi\'e it the same dimensions as E (e is the dielectric constant of the ferrite), the power flow l)ecomes in the present variables a/- -^ [ (E^H^ + ^_//-) r dr, We shall normalize the E and H fields by dividing the values given by equations (47) by /^'^ This makes the power per (isotropic wavelength in the ferrite)" a constant. In Fig. 9 we show the normalized fields for the TEii-limit and TMu-limit modes as a function of r for the case To = 5.75, 1 p I = 0.6 and several values of a. The beha\ior of the modes as a function of a and p for this rachus is shown in Fig. 14 of Pail T. AVe also show the amplitudes for the isotropic cases, o- = p = 0. It may be recalled from the discussion of cut-off points in Part I that, for the T^I mode at this radius, when cut-off is reached at a = —0.4, the am- plitude of the H^ field is overwhelminglj^ greater than that of the others. Even when normalized to the same power flow, the field amplitudes for a given a are undetermined to a factor of ±1. This factor has been 0.2 i ° ^^ TE„ <. P = -0.6 *^^ ^ ^ ^^ E-^ \. H° ^ ^ , — ^— — -— — " E° "^ k -^ *=^ z:^:^ H" r.ss^"' rr:: b^^:::;;^ E" ■ ■ h->r .-- _^-'" ^.^-^ ^^ -.-'- i 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5^ 5.5 ' W) RADIUS O -0.3 TE„ "^ •* n ^ .S A n A r =, .0 ■= .5 1 6 Fig. 9(c) — See Fig. 9. 1180 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 195J: 0.5 0.3 0.2 TE„ — *^ P = -0.6 / s \ / / / \ / \ i — N V / / 1 \ Y x \ \ / / \ \ / / \ \h° / ' \ ^^' \ H" / / \ ,/ \ \ "•n / / / / \ \ \ -1 - ,,_ v-' \ \ < -^^ / / ,^-^ """■1 N^. V \ V V \ ^ •"-— ^ .H + A ■1 7 \ N \ \ :y ^ ^\ r^-- ■>'^ \ \ \ \ / V A \ / E>- y / \ \ V y ^ /^ \ N >< y / 1 \ \ J^ / \ \ ^ \ s 1 0.5 I.O 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 I 6.0 RADIUS ^° Fig. 9(d) — See Fig. 9. GUIDED WAVK PROPAGATION TIlHorOIl GYKOM AGNKTIC M101)l\. Ill I ISl ,-'' ^^^, TE„ ,/ N p = -0.6 1/ <^ ^^ ^r '"'■'-.. / .^^ .^-' ^^' ho\ V ^> ^N^ <-"'A .-.^__ ^—.^ ^^^ \ \ < .^-- .-'"- "^--J ^ _El \ k> ^^ ^ ^s "*-^ V \ s. \ V "^'^ -^^; _/ -^ \ s_ N V fy >'-.. \ / \ ^ N ,^ y >< — - ^ T-t-r RADIUS 0.1 -0.3 f TE„ ^ ^ p = 0 ^ ^ E+'" ^ — — — "ho" ' 55 .11^ — • — H" ~~^ ■ E° E^^ :^ .tl'- ^ • ■' .^** ^^ — — -"* ^^ -***"^ 1 - - n 1 «! =^ n s s 1 fi RADIUS Fig. 9(e), top, and (1), bottom — See Fig. 0. 1182 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 TEn p = 0.6 ^^ "^-^v^ < i— ^^ ^ ^^ ,--' ^'' -^^ -^ ..--•" ^•. ^--- -.— --"""■" 1 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ' 6.0 RADIUS ^ TE„ 0.3 ^^ 0 — u.o p = 0.6 "^ 0.2 ««^,^^^ v^ ^ < S 0.1 -HO- \ ^ ^' ,^^ 1- _l a 1 0 .-— *s^ =:=^ =s=- =: -^^ -E°- t :s ^' E" 'y' -0.1 ^.-' ^ H> ,-'^ -a2 ^-^ .»'' ^^' '^'' -0.3 — — ' — 0 04) 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 I 6.0 RADIUS *() Fig. 9(g), top, and (h), bottom — See Fig. 9. GUIDFD WAVP: PUOPAGATIOXH TIIROUG GYHOMAGXKTIC media. Til 11 S3 O.t UJ o 0. 2 < -0.2 -0.4 TE,, **^ ^^ 0—1,0 p = 0.6 TYPE 2 CUTOFF ^ ^ ^ \, ^ H° K Ss. t 1 — >-^ .»•'— .^— — "H '""*""«< 1 LfO ^^ ^^o -«d .*^ s^^ t 1 : V^to E- ^ 5?-*h ,.-'^ h:> ,.-' .^-" ,4* 4 g .^-- ^-^ 1 o — ^-•^ 1 -0.3 — — ' 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 I 60 RADIUS fo 0.2 ,^'' ^^. TE„ /' N. 0 — \.d P = 0.6 H0> \ \ ~' \ '^^ ^ ,-"'' '"h'" — — •-.^ y / / / :>< \ x; »^^ / / f — 1 - .--1 '^^. :>r-i _^^ — - N V"^ ^ ^^ — — s 'Tr-- V ^^ X ^ N| Sv N "'*'• >^ \ k \ s^ \ y *** s^ :^^ V N i:^^ ^ ^' / r ^-N ^^ y / X ^" y -T+T Fig. 9(i), top, and (j), bottom — See Fig. 9. 1184 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 a2 TE„ P = 0.6 H° ,^-' ^«"'"' H~^ '*'^>« !^5-< — — ■^^^ C^ --— .r ^«>" -'r- + ■ ^-..1 "*-» y t^^ K, r^-i — " ^ K^^ ""•^^ < v^ ^ \ \^ '--^ ^-^^ ^ ^^ "\ •»^,,,_^^^ U^ ^'^^' E° U: Z^ ^ • D 0 6 1 0 1. 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 S 1 fi Q 0 1-0.1 — — TM„ ■^^ ^V., P = -0.6 X '^X E*, ^ K ^ "^^^ k^ N V __^^. >^- "Tr ;-'' .**•**" /' V. ---•" ,-'■■ \ ""I^ H°^^ —J u/, N y ^ -^ N ^:- k..^ y / ^ ^"^ iia, / ^^ N *-^ L^^ / ^•^ Y^ --«= ^^ V^ / X /- / r y / ^^^ X 1 0 0 5 1 0 1 5 2 .0 2 .5 3 .0 3 5 4 0 4 .5 5 0 5 S 1 6 Fig. 9(k), top, and (1), bottom — See Fig. 9. GUIDED WAVE PEOPAGATION THROUGH GYROMAGNETIC MEDIA. Ill 1185 0.3 0.2 O.t 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 TM„ P = -0.6 >» y "-X J /' "e^ "n A H°>L ^ \ K /- "/ /___ ?^~" \ _,^' \ -•^v "7 / ^^- ,^^ N J / ,^ V f" \ ■— ,c ^ LU \ / '^-^ 5 ^ "^v 'k "--.^ 0.2 / / \ ■*^^, -tl^ / / > \ *""*"" -~ 0.1 -^ k \ \ / V \ > ^— ^ L / / ^^ LlV-^- \ K-»^ ^ H" -^ui" K ^^^r">^ \^ > <^ ^l^ ^^^^^sO\ / \ ^~ ^y X' ■"«( :^ L_ ^ r ^.3> ^ n ? ^ r -.,.H0__ 1 < 1 n S 1 n 1 s ? 0 ? .?. 3 0 3 S 4 0 4 5 S .0 5 s~ ft RADIUS 0.3 Q 0.2 TM„ ''*"»^ , p = -0.6 "^^ "-. "v. \.l \ .--^ '' ""•"v 'v "->. H* > \ ^^^ *«»^ ,/ t ^^- \ ^H~ ^v^ "^-^ — < -ir-H --'" < ^'*'*' bJ^ ^ h^ \ kH° i^ r y \ rV- '''•^ ^V^d^ s -->^ \^-f^ r ^^ ""^ k. , f=J -^^ ^^ — 1 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 I 6.0 RADIUS ^O Y\^. 9(n), toj), and (o), bottom — See Fig. 9. GlIDKI) AVAVK n{( )!' AO ATK 1\ TMHOUGM (iYHC )M ACX K'I'K ' MKDIA. U\ 1IS7 as 1 0 -a4 -as TM„ """"■^ «» p = 0 > N. \ \ k \ .1"-^ > \ .^-- ,--■ '"-•> ^x V «» - ,-' \ k _,> -'" \ A \^ ^^- ^^^ > y H° ' / \ V / \ "^ / \ \H+ / N V ?^ x^. N ^. . / \ > r ^*-«> "---^ _E~_ -A -^ \ s. > / / / ** \ / 5/ .y ^^ y / . .^^ s ^ y 1 . .. 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 I 6.0 D A Pi I I IC ' n Fig. 9(p) —See Fig. 9. 1188 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 0.4 0.3 0.2 O.t -0.. -0.3 -0.5 \ V / ^E^'f^ / \ y / / -N i-" \ /- ^— -• ' H" / \ r^ — — 7 zll. E" '^ r r ^ 1 \? ■^ V /^ V /ho >\ R \ f \, / eV y > •'' \ \ ^.' /^. .^-^ • "^ V s. / / J y H-"" \ \ \ / ,y^ \ ^ ^^/ / \ c y / / / r / "^ 1 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ' 6.0 RADIUS fb Fig. 9(q) — See Fig. 9. GUIDED WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA. Ill 1180 0.4 =— ^ tu Q 1- 3 -0. CL < -0.3 -0.5 -0.6 TM„ P=0.6 ^^^ """*•> "n.^ / \ \ / \ y 1^ / ^^ —-7 ^ A v X / H, / / \ y \ \ . ^ / \ / y \ / H°/ / .•^^i < \ ^^ J:, V " __ — • •"■""" 1 y y \ K v\ i f \ /e° — y\ \ \\ / N X / / / \ \ \ \ / > H"-^ y \ \ \ s. / y / / \ 1 \. ^y t y / / \ \ / / 1 \ \ / / / / \ \ / / / \ / / \ / / / \ / / / \ \ ,/ ^ / "w^ ,y y r 1 3 0 5 I 0 1 5 2 .0 2 .5 3 .0 3 .5 A .0 4 .5 5 .0 5 .f, 1 6 Fig. 9(r) — See Fig. 9. 1190 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 TM„ 0.4 0 1,(3 P = 0.6 TYPE 2 CUTOFF t — -^ 1 0.3 "n. \, !>■ / / 8 0.2 \ \ > 7 1-s X p ^ / N /^ s. 1t 0.1 x< / / \, >r y \ 1 1 X^ \ ,r-^ ^ \ — --1_ ik- -\ J K > r \ 5>y r r )s.' -0.1 \\ s. / N ^^„/ // hO / \ \ / /- ^- -It-- .,'" -0.2 \ \ ^ / y / / / \ \ "7 / / -a3 \ / 1 \ \ \ / 1 -0.4 \ \ / 1 1 \ \ J i 1 -0.5 X J 1 J -0.6 / \ k. / ^A ^ ^N. _-»'' -0.7 ,0^ i ( ) 0 5 1 0 1 S 2 0 2 5 3 .0 3 .S 4 a 4 5 ^ o s 5 ' fi. Fig. 9(s) — See Fig. 9. 0.5 TM„ C.4 p = 0.6 -N 0.3 ^N. '^^J \, > \, O.I _,•»•" N s^H- ,..-"! ^\. H" ^^ 0 r^ . " "■ \ LlO >^ y "^^^^ ^ ^^ > r^ P" 1 -o.t \ s. Al K \ \ y E" ^>.^ rsrrrT r^-' ^'' -0.2 Ej> ^-^ -0.3 — ""^ -0.4 ( 3 0 5 1 0 1 5 2 .0 2 5 3 .0 3 .5 4 .0 4 .5 5 .0 5 .5 1 6 -0.2 ^^^ TM„ ■"-V "n,^ «/ - Kb P =0.6 "^, -s ^, ^v \ N. **> < H" 1>, — 'S -rr. r—— — '~^-^^ «cr^ ---^ "^-^^ a^;:* V 1 *l ^^.^ .^ ^ ":>> ^ , ' — — _ E~_ i>-e ^ '"v^,^ ^^^^ .- ^ >— ^y ■4 ^ ^ ^ "^ '^ — 1— 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 RADIUS ° Fig. 9(t), top, and (u), bottom — See Fig. 9. 1191 1192 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 chosen for each pattern according to the fol]o\\-ing considerations. For large | 1, the field at most points in the guide was locally rotating in the opposite direction to that of the whole pattern. The points of similarity to the present case are clear and it is also evident that the TE mode more nearly approaches the large guide situation because r^ = 5.75 is much further above the cut-off radius for this mode than it is beyond the cut-off for the TM-mode. ACKNOWLEDGMENTS We are indebted to M. T. Weiss for frequent consultations and for permission to use some of the data computed for him. We are also obliged to J. H. Rowen and to S. P. Morgan, Jr., for advice in some matters relating to Part III, and to R. Kompfner for useful discussions concerning the "non-reciprocal helix." R. W. Hamming arranged the integration of the Ricatti equation of Part II. Special thanks are due to Mrs. A. Rebarber for the many computa- tions relating to Parts I and III, to Mrs. C. A. Lambert for her calcu- lations on Part II, and to Miss M. J. Brannon for a number of results computed for Part III. REFERENCES 1. A. A. Th. M. Van Trier, Applied Sci. Res., Sec. B, 3, p. 305, B. Lax, Private communication. 2. R. B. Adler, Res. Lab. of Electronics, M. I. T., Teclmical Report No. 102, May, 1949. 3. J. H. Rowen, B. S. T. J., 32, pp. 1333-1369, Nov., 1953. Bell System Technical Papers Not Published in this Journal Allison, H. W., see Moore, G. E. Bangert, J. 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R.^ Spin Resonance of Donors in Silicon, Letter to the Editor, Physical Review, 94, No. 5, pp. 1392-1393, June 1, 1954. 1 Bell Telephone Laboratories, Inc. 2 American Telephone and Telegraph Company. TECHNICAL PAPERS 1197 Fuller, C. S.,^ Struthers, J. D./ Ditzenberger, J. A.,' and Wolf- STIRN, K. B} Diffusivity and Solubility of Copper in Germanium, Pliys. Rev., 93, PI). llS2~llSi), Mar. 1."), 1954. Fuller, C. S., see Ciiapin, D. M. Geballe, T. H.,1 and Hull, G. W.^ Seebeck Effect in Germanium, Phys. Rev., 94, No. 5, pp. 1134-1140, June, 1, 1954. Green, E. I.^ The Decilog — A Unit for Logarithmic Measurements, Elec. Eng., 73, No. 7, pp. 597-599, July, 1954. Gross, A. J., see Tanenbaum, M. Heffner, H.^ Analysis of the Backward Wave TraveUng Wave Tube, I.R.E. Proc, 42, No. 6, pp. 930-937, June, 1954. HoiiN, F. E.i The Conference on Training in Applied Mathematics, Am. JNIatli. Monthly, 61, No. 4, pp. 242-245, April, 1954. Holden, a. N., see Fletcher. R. C. Hull, G. W., see Geballe, T. H. Jaffe, Hans, see Mason, W. P. Ladd, F. E.i 50 Mc TVI — Its Causes and Cures, Part I, QST, 38, No. 6, p. 21, June, 1954. Part II, QST, 38, No. 7, p. 32, July, 1954. Mason, W. P., see Shockley, W. 1 Bell Telephone Laboratories, Inc. 1198 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 Mason, W. P.,^ and Jaffe, Hans^ Methods for Measuring Piezoelectric, Elastic and Dielectric Co- efficients of Crystals and Ceramics, I.R.E. Proc, 42, No. 6, pp. 921- 930, June, 1954. Matthias, B. T.,^ Corenzwit, E.,' and Miller, C. E.^ Superconducting Compounds, Letter to the Editor, Phys. Rev., 93, p. 1415, Mar. 15, 1954. Matthais, B. T.^ and Corenzwit, E.^ Superconducting Alloys, Letter to the Editor, Phys. Rev., 94, No. 4, p. 1069, May 15, 1954. May, J. E., Jr.^ Characteristics of Ultrasonic Delay Lines Using Quartz and Barium Titanate, J. Acoust. Soc. Am., 26, No. 3, pp. 347-355, May, 1954. McKAY, K. G} Avalanche Breakdown in Silicon, Phys. Rev., 94, No. 4, pp. 877-884, May 15, 1954. Mendel, J. T.,i Quate, C. F.,^ and Yocum, W. H.^ Electron Beam Focusing With Periodic Permanent Magnetic Fields. Proc. LR.E., 42, pp. 800-810, May, 1954. Merritt, F. R., see Fletcher, R. C. Miller, C. E., see Matthias, B. T. Moore, G. E.,^ Allison, H. W.,^ and Wolfstirn, K. B.^ Reduction of SrO by Methane, J. Chem. Phys, 22, No. 4, p. 726, April, 1954. Morin, F. J.i Electrical Properties of NiO, Phys. Rev., 93, pp. 1199-1204, Mar. 15, 1954. ' Bell Telephone Laboratories, Inc. ^ Brush Laboratories Company. TKCIIMCAL rAl'KHS 1190 MORIN, F. J.' Electrical Properties of of-Fe.O;; , Phys. Rev., 93, i)p. 1195-1199, .Mar. 15, 1954. Pkarson, G. L., see Ciiapix, D. ^F. Pkarson, G. L., see Fletcher, R. C. Pfaxx, W. G., see Tanenbaum, M. Prince, M. B.^ Drift Mobilities in Semiconductors: II — Silicon, Phys. llcw, 93, pp. 1204-1200, Mar. 15, 1954. QuATE, C. F., see Mendel, J. T. Pvead, W. T., see Fletcher, R. C. Shockey, W.,^ and Mason, W. P.^ Dissected Amplifiers Using Negative Resistance, Letter to the Editor, J. Appl. Phys., 25, No. 5, p. 077, i\Iay, 1954. Slepl\n, David^ Estimation of Signal Parameters in the Presence of Noise, I.R.E. Trans. P.G.I .T.-3, pp. 08-89, Alar. 1954. Stansel, F. R} An Improved Method of Measuring the Current Amplification Factor of Junction Transistors, I.R.E. Trans. PGl-3, pp. 41-49, April, 1954. Struthers, J. D., see Fuller, C. S. Tanenbaum, ]\I.,^ Gross, A. J.,^ and Pfann, W. G.^ Purification of Antimony and Tin by a New Method of Zone Re- fining, J. Metals, 6, No. 6, pp. 762-703, June, 1954. Thomas, D. E.^ 1 Bell Telephone Laboratories, Inc. 1200 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 A Point Contact Transistor VHF FM Transmitter, I.R.E. Trans. ED-1, No. 1, pp. 43-52, April, 1954. Turner, E. C.^ Telephone Growth Forecasting. Part I, Telephony, 146, No. 24, pp. 17-19, 48, June, 12, 1954. Part II, Telephony, 146, No. 25, pp. 23-25, June 19, 1954. Vance, R. L.^ and Maggs, C.^ Magnetron Heater Design to Avoid Undesirable Magnetic Effects, (Abstract), I.R.E. Trans. ED-1, No. 1, p. 64, Feb. 1954. Varney, R. N.^ Liberation of Electrons by Positive-ion Impact on the Cathode of a Pulsed Townsend Discharge Tube, Phys. Rev., 93, pp. 1156-1160, Mar. 15, 1954. Walker, L. R.^ Stored Energy and Power Flow in Electron Beams, Letter to the Editor, J. Appl. Phys., 25, No. 5, pp. 615-618, May, 1954. Wick, R. J.^ Solution of the Field Problem of the Germanium Gyrator, J. Appl, Phys., 25, No. 6, pp. 741-750, June, 1954. Windeler, a. S.^ Polyethylene-Insulated Telephone Cable, A.I.E.E. Commun. and Electronics, No. 12, pp. 106-111, May, 1954. Wolfstirn, K. B., see Moore, G. E. Wolfstirn, K. B., see Fuller, C. S. Yager, W. A., see Fletcher, R. C. YocuM, W. H., see Mendel, J. T. 1 Bell Telephone Laboratories, Inc. 6 New York Telephone Company. Recent Monographs of Bell System Technical Papers Not Pnblished in This Journal* Benxett, W. Telephone-System Applications of Recorded Machine Announce- ments, Monograph 2213. BOGKKT, B. P. On the Band Width of Vowel Formants, ^Monograph 2167. Browx, W. L. N-Type Surface Conductivity on P-Type Germanium, Monograph 2173. Burton, J. A., Hull, G. W., Morin, F. J., and Severiens, J. C. Effects of Nickel and Copper Impurities on the Recombination of Holes and Electrons in Germanium, Monograph 2193. Burton, J. A., Prim, R. C, Slighter, W. P., Kolb, E. D., and Strutiiers, J. D. Distribution of Solute in Crystals Grown from the Melt, Monograph 2231. Coy, J. A. Heat Dissipation from Toll Transmission Equipment, Monograph 2214. Conavell, E. M., see Debye, P. P. * Copies of these monographs may be obtained on request to the Publication Department, Bell Telephone Laboratories, Inc., 463 West Street, New York 14, N. Y. The numbers of the monographs should be given in all requests. 1201 1202 the bell system technical journal, september 1954 Debye, p. p. Electrical Properties of n-type Germanium, Monograph 2220. Ellis, W. C, and Pageant, Jacqueline. Orientation Relationships in Cast Germanium, Monograph 2219 Pageant, Jacqueline, see Ellis, W. C. Felch, E. p., see Potter, J. L. Preliminary Development of a Magnettor Current Standard, Mono- graph 2198. Felker, J. H. Arithmetic Processes for Digital Computers, Monograph 2208. Goertz, M., see Williams, H. J. Grisdale, R. O. The Properties of Carbon Contacts, Monograph 2206. Hagstrum, H. D. Electron Ejection from Ta by He"^, He"^"*", and Het, Monograph 2148. Hopkins, I. L. The Ferry Reduction and the Activation Energy for Viscous Flow Monograph 2203. Hull, G. W., see Burton, J. A. Karnaugh, M. The Map Method for Synthesis of Combinational Logic Circuits, Monograph 2199. Kolb, E. D., see Burton, J. A. RECENT MON'oaUAIMlS 1203 Lander, J. J. Auger Peaks in the Energy Spectra of Secondary Electrons from Various Materials, Monograph 2215. Lewis, \\'. D Electronic Computers and Telephone Switching, Aloiiograiih 2187. LiNVILL, J. G. A New RC Filter Employing Active Elements, Monograph 2221. May, J. E. Characteristics of Ultrasonic Delay Lines Using Quartz snd Barium Titanate Ceramic Transducers, Monograph 2223. McSkimin, H. J. Measurement of Elastic Constants at Low Temperatures by Means of Ultrasonic Waves Data for Silicon and Germanium Single Crystals and for Fused Silica, Monograph 2171. MoRiN, F. J., see Burton, J. A. Pennell, E. S. A Temperature-Controlled Ultrasonic SoUd Delay Line, Monograph 2222. Pfann, W. G. Change in Ingot Shape During Zone Melting, Monograph 2218. Pierce, J. R. Spatially Alternating Magnetic Fields for Focusing Low-Voltage Electron Beams, Monograph 2169. Potter, J. L., see Felch, E. P. Prim, R. C., see Burton, J. A. 1204 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 Prince, M. P. Experimental Confirmation and Relation Between Pulse Drift Mobil- ity and Charge Carrier Drift Mobility in Germanium, Monograph 2168. Read, W. T., Jr. Dislocations and Plastic Deformations, Monograph 2216. Reiss, H. Chemical Effects due to the Ionization of Impurities in Semiconduc- tors, Monography 2172. Ryder, E. J. Mobility of Holes and Electrons in High Electric Fields, Monograph 2159. Schnettler, F. J., see Williams, J. H. Severiens, J. C, see Burton, J. A. Sherwood, R. C, see Williams, J. H. Shockley, W. •Transistor Physics, Monograph 2217. Slighter, W. P., see Burton, J. A. Snoke, L. R. Soil-Block Bioassay of a Creosote Containing Pentachlorophenol Monograph 2212. Struthers, J. D., see Burton, J. A. Struthers, J. D., see Thurmond, C. D. Thurmond, C. D. Equilibrium Thermochemistry of Solid and Liquid Alloys of Ger- manium and of Silicon, Monograph 2189. KIX'KXT MOXOCHAIMI.S 1205 TiEX, P. K. Traveling-Wave Tube Helix Impedance, Moiio^rapli 220',). W'lLLAUD, ( 1. W . Ultrasonically Induced Cavitation in Water - A Step-by-Step Process, M()ii(»i>;rai)li 2170. Williams, TI. J., Shkrwood, R. C, Goertz, M., and Sciinettler, F. J. Stressed Ferrites having Rectangular Hysteresis Loops, jMonof^ruph 2200. Contributors to this Issue Orson L. Anderson, B.S., M.S. and Ph.D., University of Utah, 1948, 1949 and 1951; Institute of Rate Processes, University of Utah, 1949-1952; Bell Telephone Laboratories, 1952-. Dr. Anderson has been engaged in the investigation of mechanical and electrical properties of solids, with emphasis on glasses, and in studies of the mechanism of plastic flow of amorphous bodies. A member of the mechanics division of the Mathematics Department, he is now studying the strength and flow properties of glasses under high pressure. Member of American Physical Society, American Ceramic Society and Society of Glass Tech- nology. J. K. Galt, A.B., Reed College, 1941; Ph.D., M.I.T., 1947; O.S.R.D., M.I.T. and Harvard University, 1943-1945; National Research Council Fellow, Bristol, England, 1947-1948; Bell Telephone Laboratories, 1948-. Dr. Gait has been engaged in research on the properties of solids, especially of ferrites, with emphasis on their magnetic properties. Fellow of the American Physical Society and member of Phi Beta Kappa. W. P. Mason, B.S. in E.E., University of Kansas, 1921; M.A., Ph.D., Columbia, 1928. Bell Telephone Laboratories, 1921-. Dr. Mason has been engaged principally in investigating the properties and applications of piezoelectric crystals, in the study of ultrasonics, and in mechanics. Fellow of the American Physical Society, Acoustical Society of America and Institute of Radio Engineers and member of Sigmi Xi and Tau Beta Pi. J. L. Merrill, Jr., B.S. and M.S., Pennsylvania State University, 1928 and 1930; ElUot Research Fellow, 1928-1930; American Telephone and Telegraph Company, 1930-1934; Bell Telephone Laboratories, 1934-. Mr. Merrill spent his first years with the Laboratories on trans- mission features of such projects as the time and weather announcement systems and operator training programs. During World War II, he en- gaged in planning system operation of air raid warnings as well as work on tactical wire and radio networks for the armed forces. Since the war he has been concerned with the design and application of negative 1206 CONTRIBUTORS TO THIS ISSUE 1207 iinpedaiice repeaters for the improvemont of (exchange transmission. He holds several patents and is the author of imnu'rous Icchnical ai'ticlcs. Member of Theta Alpha I'hi. Irad S. Rafuse, B.S. in E.E., Cooper Union, 1927; Columbia TTni- versity; Western Electric Company, 1920-1925; Bell Telephone Labo- ratories, 1925-. He worked on the development of high quality vertical disc recording for a number of years before luining to measurements and testing of switching apparatus. During World War II he engaged in th(> development of sonar equipment in cooperation with the N.D.R.C. aiul the Bureaus of Ships and Ordnance from which he received a com- mendation for his work. He was later in charge of a group engaged in the development of a new wire-spring, multi-contact relay and now is in charge of a group developing glass sealed switches. Arthur F. Rose, B.S. in E.E., Colorado College, 1914; American Telephone and Telegraph Companj^, 1914-. Mr. Rose immediately joined the General Engineering Department of the A.T.&T. Co. Upon completing the student training course for new employees, he was assigned to the development work then under way on the New York- San Francisco route which culminated in the first transcontinental tele- l)hone service in 1915. As a result of this initial acquaintance with tele- phone repeaters, he continued in transmission work dealing particularly w ith these devices. In 1919, when the General Engineering Department was divided and the Operating and Engineering Department formed, Mr. Rose was assigned to the group that was concerned primarily Avith the application of repeaters and carrier systems in toll engineering. In 1939 he was transferred to the Plant Extension Section and in 1953 re- turned to the Transmission Section as Exchange Transmission Engineer. J. 0. Smethurst, B.S. in Communications, Tufts College, 1929; Bell Laboratories, 1929-. For many years he was concerned with overseas telephony, concentrating especiall.y on control terminals for radio tele- phone circuits. During World War II he was associated with various government projects and after the war he worked on NIKE. Since 1953 he has concentrated on E2 and E3 repeaters. Harry Suhl, B.Sc, University of Wales, 1943; Ph.D., Oriel College, University of Oxford, 1948. Admiralty Signal Establishment, 1943-46; Bell Telephone Laboratories, 1948-. Dr. Suhl conducted research on the properties of germanium until 1950 when he became concerned with 1208 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 electron d3aiamics and solid state physics research. His current work is in the applied physics of solids. Member of the American Institute of Phj^sics and Fellow of the American Physical Society. Laurence R. Walker, B.Sc. and Ph.D., McGill University, 1935 and 1939; University of California, 1939-41. Radiation Laboratory, INIassachusetts Institute of Technology, 1941-1945; Bell Telephone Laboratories, 1945-. Dr. AValker has been primarily engaged in research on microwave oscillators and amplifiers. At present he is a member of the physical research group concerned with the applied physics of solids. Fellow of the American Physical Society. HE BELL SYSTEM Jeck / meat journa 5 VOTED TO THE SCIENTIFIC ^^^ AND ENGINEERING iPECTS OF ELECTRICAL COMMUNICATION NOVEMBER 1954 *' ^^*^* NUMBER >LUME XXXIII i"j ■ ». '■■.•\r DEC 2 1954 Waveguide as a Communication Medium 8. b. miller 1209 A Governor for Telephone Dials — Principles of Design W. PFERD 1267 In -Band Single-Frequency Signaling A. WEAVER AND N. A. NEWELL 1S09 Centralized Automatic Message Accounting System g. v. kino 1S31 The Wave Picture of Microwave Tubes j. r. pierce 1343 Theory of Open-Contact Performance of Twin Contacts M. M. ATALLA AND MISS R. E. COX 1373 I Bell System Technical Papers Not Published in this Journal 1387 Recent Bell System Monographs 1392 Contributors to this Issue 1398 COPTBIGHT 1954 AMEHICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD S. BRACKEN, Chairman of the Board, Western Electric Company F. R. KAPPEL, President, Western Electric Company M. J. KELLY, President, Bell Telephone Laboratoriet E. J. M c N E E L Y, Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE W. H. DOHERTY, Chairman F. R. LACK A. J. B U S C H W. H. N U N N G. D. EDWARDS H. I. R O M N E 6 J. B. F I S K H. V. S C H M I D T E. I. GREEN G. N. THAYER R. K. HONAMAN J. R. WILSON EDITORIAL STAFF J. D. T E B O, Editor M. E. 8 T R I E B Y, Managing Editor R. L. SHEPHERD, Production Editor THE BELL SYSTEM TECHNICAL JOURNAL is published six times a year by the American Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. Qeo F. Craig, President; S. Whitney Landon, Secretary; John J. Scanlon, Treasurer. Subscriptions are accepted at $3i)0 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed in U. S. A. THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXXIII NOVEMBER 1954 numberG Copyright, 19BJ,, American Telephone and Telegraph Company Waveguide as a Communication Medium By S. E. MILLER (Manuscript received March 23, 1954) The circular electric wave in round metallic tuhing has an attenuation coefficient which decreases as the freciuency of operation is increased. A corol- lary to this behavior is the fact that any preselected attenuation coefficient can in theory be obtained in any predetermined diameter of pipe through the choice of a suitably high carrier frequency. The attenuation which is charac- teristic of microwave radio repeater links, about 2 db/mile, is in theory attainable in a copper pipe of about 2" diameter using a carrier frequency near 50,000 mc. Scale-model transmission experiments, conducted at 9,000 mc, showed average transmission losses about 50 per cent above the theoretical value. These extra losses were due to (1) roughness of the copper surface and (2) transfer of power from the low-loss mode to other modes which can also propagate in the pipe. The latter effect may have serious consequences on signal fidelity because power will transfer (at successive waveguide imperfections) from the signal mode to unused modes and, after a time delay, back to the signal mode. This effect has been studied experimentally and theoretieally , and it is con- cluded that (1) either mode fillers must be inserted periodically to absorb the power in the unused modes of propagation, or (2) the medium itself 7nust be modified to continuously provide large attenuations for the unused modes of propagation. The latter approach is attractive in that it also provides a solu- tion to the problem of bending this form of low-loss guide. 1209 1210 THE BELL SYSTEM TECHNICAL JOURXAL, NOVEMBER 1954 The general outlook, based on 'present knowledge, is that a waveguide sys- tem micjht transmit hasehand widths as large as 100 to 500 mc using a rugged modulation method such as PCM. Some form of regeneration is likely to be required at each of the repeaters, which may be spaced on the order of 25 miles. A total rf bandwidth of about 40,000 mc may be available in a single guide. Table of Contents Introduction 1210 ( )idinary Versus Circular Electric Waves 1211 Theoretical Characteristics of the Circular-Electric Wave 1213 Some Results of Transmission Experiments 1219 Mode Conversion and Reconversion as a Signal-Loss Effect 1229 Mode Conversion and Reconversion as an Interference Effect 1230 Analysis for Continuous Mode Conversion 1239 Direct Evaluation of Mode Conversion Magnitudes 1243 The Bend Problem 1247 Improved Forms of Circular Electric Waveguide 1250 Surface Roughness 1252 Circular Electric Wave and Millimeter Wave Techniques 1253 Modulation Methods 1256 Conclusion 1259 Appendix — Theoretical Analysis for Continuous Mode Conversion 1261 INTRODUCTION The circular electric wave in round metallic tubing possesses a prop- erty so unique that some early research workers doubted the reality of the wave. This unique property is an attenuation coefficient which, in a given pipe, decreases without limit as the frequency of operation is in- creased. In parallel wire, coaxial, or ordinary waveguide lines the "skin effect" at the surface of the conductor causes the loss to increase as the frequency increases indefinitely, so the predicted circular-electric-wave loss characteristic aroused considerable interest as soon as it was dis- covered by S. A. Schelkunoff and G. C. Southworth in the early 1930's. Since that time considerable work has been done at Holmdel to explore the reaUty of the circular electric wave and to evaluate its usefulness to the Bell Sj^stem. It is the purpose of this paper to report on the status of this work and to give a description of some of the basic characteristics of circular electric wave propagation. The Bell System is interested in knowing whether waveguide can be used as a long distance communication medium in the manner in which coaxial cable or the radio relay system is now employed. Our interest in long distance waveguides is due in part to the fact that radio-wave propagation through the atmosphere becomes progressively more se- verely' handicapped by rain, water vapor and oxygen absorptions at WAVEGUIDE AS A t'OMMUXICATlON MEDIUM 1211 iVeMluencios above 12,000 mv. V^c of the sj)e('tium a])()ve tlie 10,000- 20,000 me region secMiis to reiniire a slielhMvd transniission medium. Circular electric waxc transmission may also liiul application in slioi't coimecting links, such as hetweiMi subscribers re(iuii'ing \-ery broad band circuits, between two central oflic(\s as a multi-channel carrier link, or between a radio relay antenna site and a somewhat remote transmitter- receiver location chosen for accessibility. In each of these cases, the broad bands a\ailal)le in the microwave l)orti()n of the spectrum, the complete shielding afforded by waveguides generally, combined with the low-loss properties of the circular ele(;tric wa\'e would seem to pro\'ide an ideal transmission medium. We there- fore seek knowledge of the precision required in the w^aveguide and some indication of general system complexity to facilitate a judgment as to whether the cost will be competitive. ORDINARY VERSUS CIRCULAR ELECTRIC WAVES Let US approach a discussion of circular electric waves by considering their relation to the waveguides which are now used in oui" I'adio relay systems and which found widespread use in the radars of World War II. The vast majority of waveguides in commercial use now are rectangular in cross section and have dimensions large enough so that one and only one wave-tj^pe, usually called the ''dominant mode", can i)ropagate. To simplify this discussion, such waveguides will be called ordinary wave- guides. Ordinary ^vaveguides are analogous to coaxial or parallel-wire lines in manj^ respects. Because only one mode can propagate, departures, from an absolutely straight tube of constant cross section show up as reactance effects only. A dent in the side wall of the guide or of the co- axial, an abrupt change in cross section, or a twist or bend of the line all appear as non-dissipative reflection effects which may be cancelled at one frequency (or in one band of frequencies) by the addition of another compensating reactance at a point suitably located. A great many of the components used in ordinary waveguides, including the frequency selective filters, depend on such reactance cancellation effects in order to achieve satisfactory operation. Since the techniques for employing ordinary waveguides have been thoroughly explored, it is natural to inquire as to whether we can use them for communication purposes. We do use ordinary waveguides in lengths of the order of 100 feet and more to connect the antennas and repeaters in the 4,000 mc (TD-2) radio relay syst(^m. The attemiation is excessive, however, for long-distance applications. The particular type 1212 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 of brass rectangular waveguide used for TD-2 transmission lines has attenuation in excess of 50 db per mile, and use of the very best copper would only reduce the theoretical loss to about 40 db per mile. In order to reduce the loss in ordinary waveguides, just as in coaxial or parallel wire lines, one must go to lower frequencies. In particular, the theoreti- cal loss at a carrier frequency near 1000 mc is about 2 db per mile, which is about the same as the transmitter-to-receiver attenuation in our radio relay systems. The waveguide in the 500-1000 mc region would have cross-sectional dimensions on the order of one foot, would be cumber- some to handle and would involve rather large material cost. In addi- tion, it turns out that such a waveguide would be useful in signal band- widths only a few mc wide as a result of delay distortion, which will be discussed further in the ensuing discussion. Thus, we have concluded that ordinary waveguide is not very attractive as a transmission medium over distances on the order of a mile or more. It is true that the attenuation in any hollow metallic waveguide can be reduced to any desired extent at a given frequency by making the cross sectional area larger by a suitable factor. The penalty is that the transmission medium becomes capable of propagating energy in several characteristic ways, known as modes. The striking feature of a multi- mode transmission medium is that energy in one mode is entirely in- dependent and unaltered by the presence or absence of energy in one of the other modes. This situation is sketched diagrammatically in Fig. 1. Energy can theoretically propagate between 1 and 1', between 2 and 2', and between 3 and 3' at the same time and in the same frequency hand without mutual interference. The separate modes represent independent transmission lines which occupy the same space. The distinguishing fea- tures of the various modes in a multi-mode waveguide are: (1) ^>locity of propagation or phase constant, (2) Attenuation coefficient, and (3) Configuration of electric and magnetic field lines within the waveguide. 000 ©00 Fig. 1 — Diagram of multi-mode waveguide transmission. WAVEGUIDE AS A COMMUXIPATIOX MIIDIUM 1213 The fact that it is necessary to use a waveguide wliose dimensions are large enough to permit the existence of a numhcr of mcxles has far- reaching influence on the research being discussed lici'c. Practically, the iii(le])(Mulenc(> hetwecMi the \-arious modes of ])ro])agation is limited by tolerances of various kinds. In the multimode \va\-eguide, changes in cross section or bends or twists I'ciiuii'e design attention with regai'd to mode purity as well as with regartl to impedance match, and it is not jxnmissible to insert arbitrarily shaped probes or irises for impedance matching purposes as is the common practice in ordinary waveguides. This means that a complete new technifjue is required for tlu^ old com- ponents, such as frctiuency-selective filters, hybrids, and attenuators, as well as for a new series of components such as pure mode generators and mode filters. TIIEOHETICAL CHAKACTEHISTirS OF THE CIHCULAU ELECTKIC WAVE Sinc(> it has been found necessary to use a waveguide in the multimode i-egion in order to get the tlesired losses in a reasonable size waveguide, we may inciuire as to which of the modes is best suited to our problem. At a gi\'en fre([uency the loss for any one of the modes may be reduced as much as is desired by making \hv cross sectional area of the guide large enough, but there is a mode for which the loss decreases with in- creasing guide size much more rapidly than for any other mode. This is the circular electric (TEoi) mode in straight round pipe. It turns out that no current flows in the direction of propagation in the metallic walls of a straight round pipe carrying the circular electric mode. It is the ab- sence of current in the direction of propagation which p(»rmits the circu- lar-electric-wave attenuation to decrease indefinite^ as the frequency increases, and this difference between ordinary transmission lines and the circular-electric wave is further illustrated in Fig. 2. In the familiar parallel-wire line the electric field extends directly from one conductor to the other, resulting in charge accumulations at half-wave intervals along the axis of propagation and associated conduction cuirents in the copper wires. These conduction currents in the direction of propagation do not diminish as the frecjuency of operation increases, since they are associated with the energy transmitted to the end of the transmission line. With the circular electric wave the electi'ic field lines close upon themselves, are always tangential to the conducting wall, and do not n^sult in a charge accumulation on the walls due to the main energy flow. The wall currents which do flct ric wave family. Under certain conditions of mode coupling, which will he de- scril)ed at a later point, it is undesirahle for the medium to be able to propagate modes with attenuation coefficients comparable to that of the mode which is used for communication purposes. SOME RESULTS OF TRANSMISSION EXPERIMENTS Transmission experiments have been conducted on the 500-ft wa\'e- guide line shown in Fig. 9.* Supports for the line were set in concrete 40 1.0 0.9 0.8 0.7 0.6 \\ \\ 1 \\ y. Vt = 03 TEo\ . \ T \teoA E,3\ \ ^ \ w \ s^ V \ \ \ \ \ \ N ^ ^ y \ ^ ^ \ \ \ \ \ \ \ \ \ \ s \ \ V \ V \ s. \ 10 XIO^ 20 30 40 50 60 80 100 FREQUENCY, f, IN MEGACYCLES PER SECOND ^'° Fig. 7 — Attenuation versus frequency for a 2" dianieter round waveguide. * This is the same line used for the work r('i)ortod in Reference 1. Some of the e.xperiments described in Reference 1 arc ;d.so doscriliod hero in order to furni.sh background for the new material. 1220 THE BI:LL system TECHXICAL journal, NOVEMBER 1954 1 1 ^5 700 E5 500 y?: o y O 200 / / J f / / / / TE„,/ / / ^ / / / J / / / i 02 / / / FREQUENCY IN MEGACYCLES PER SECOND Fig. 8 — Base band width per channel versus frequency for a 2" diameter pipe (one-mile waveguide length). and optically aligned so as to provide a waveguide straight within about \i," over its entire length. The philosophy behind this installation was the familiar one of providing for experimental purposes, as close to the ideal line as possible so that deviations could be created in a controlled manner. The inside diameter is about 4.73", chosen to obtain the de- sired theoretical loss of about 2 db per mile at 9,000 mc, where measuring equipment was readily available. The difference between the major and minor inside diameters of the pipe was in the range 0.005" to 0.008" at the ends of the sections which averaged 20 ft in length. At the time this work was initiated, in 1946, generators of higher frequencies which would permit the use of smaller waveguides had not yet become available for use in this research. Tests were conducted on this line using a technique due to A. C. Beck,^ and involving the layout of equipment shown in Fig. 10. Short bursts of RF energy approximately ^.fo microsecond in duration, were injected into the line at intervals of about three hundred microseconds. Except for two small holes through which to couple to the transmitter and WAVKni'IDK AS A COMMI^XICATIOX MKDIUM 1221 receiver, the wavegui(l(> line was short-circuited at both ends. The in- jected • 10 microsecond pulse occupied at any instant a space interval of 100 feet and lIuM-eforc this pulse, wiiile travellinj»; from one end to tiie other between the siiort circuits, iJioduccd :il the receiver coujjlinji; hole spurts of energy corresponding to the time when the pulse ])asscd the sending end. Each such pulse passing through the receiver (•()ui)liiig hole was am])lified, detected, and placed as a \-crtical deflection on the oscil- loscope. The horizontal d(>flcction on the ()S('illosco])e was a linear time Fig. 9 — Experimon(;il wavoKuido installutioii, 5" diainclor lloliiulcl Hue 1222 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 base having a duration of a few microseconds. In order to observe the received pulse after a selected number of trips back and forth down the waveguide, a variable delay was placed between the trigger and the oscilloscope horizontal deflection. The discussion which follows will refer to photographs of the oscilloscope under different conditions. By way of preparation, it may be stated that the pulse transmitted through the small hole in the end plate of the waveguide will excite a large num- ber of modes. There are, at 9,000 mc, approximately 40 modes which can propagate in this waveguide. Also, the coupling through the holes is so weak and the energy lost due to dissipation in the shorting plates is so small as to represent an attenuation which is small compared with the theoretical wall loss in the 500-foot long line. Therefore, as the pulse shuttles back and forth in the line, it will decay as though it had trav- elled on a straight long section of waveguide made up of 500-foot long segments identical to the single 500-foot section actually constructed. Fig. 11 shows a photograph of the oscilloscope displaying the time interval immediately following the transmitted pulse. The pulse at the extreme left represents the transmitted pulse which passes directly from the transmitter hole to the receiver hole on the end plate of the wave- guide. The blank time interval immediately following the transmitted pulse is about one microsecond long and represents the time of travel of energy down to the far end of the 500-foot line and back to the sending end. During this interval no pulses were received because the joints in the line produce little reflection. The first pulse after the transmitted pulse represents energy travelling in the mode which has the highest [* - 500 FT --^ Fig. 10 — Diagram of equipment used for pulse tests of waveguide transmission. WAVEGUIDE AS A COMMUNICATION MEDIUM 1223 Fig. 11 — Photograph of cathode-ray tube presentation during the time inter- val immediately following the transmitted pulse. group velocity. Pulses immediately following this first received pulse represent energy travelling in other modes whose velocities are lower and which therefore recjuire more time for the one round trip of travel. At the time 2Af, we begin to observe pulses which have made two round trips in the Une. If the transmitted pulse width were short enough, we could theoretically identify the mode in which the energy travelled by observing the time of arrival, since the velocities of propagation and the distance are known parameters. The ^o microsecond pulse used in these experiments is not short enough to allow this kind of resolution on an individual mode basis. Something on the order of five or six modes have velocities so nearly the same that they cannot be resolved as sep- arate pulses with the Ho microsecond pulse after a single round trip in the 500-foot line. Fig. 12 represents the same condition as Fig. 1 1 , except that the horizontal time base has been changed to display the interval 0 to 14Af instead of the interval 0 to 2At. Fig. 12 shows fewer pulses in the time interval 6At to 12Ai than in the time interval 0 to 6A^. This is because energy travelling in some modes is attenuated more rapidly than that in other modes. For time delays greater than 10 At the received pulses 1224 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 SG WAVEGUIDE AS A COMMUNICATION MKOIUM 1225 appear at r('«j;ulai' iiit('r\-als and with sinoollilx- (Iccaxiiit; atnplil udc 'i'liis l)('lia\i()r iiulicatcs thai the major ])<)rli(>ii of Ihc ciici'^y in I lie line was t laA'clliiiji; ill a siiijilc mode, and we deduced that this mode was 'ri']|ii as I'oHows: We ohsei'Ncd that the Nclocily of propatial ion was near thai for the 'ri*]|ii mode hy measiirinji; tlie al)soiule lime Itelwcen pulses, a\'er- aged over many round trijjs. Tliis (^xehided all hut about (» modes wiiose {'ut-off t'reciuencies are near tluit of TEm . Measurement of t I'ansmission loss was made liy observin<2; the rate of (l(>cay of the received pulses a\-eraged over 10 or more round trips. 'Die measured loss was found to he approximately 3 db per mile eompared to a theoretical \'alue of 1.9 db ])er mile for TEoi pr()))agation. Il follows that ))i-opai!;atioii must have Fig. 13 — Record of pulses after 40 miles of repeated traversal over the 50()-foot been taking place in the TEoi mode, for all other modes near TEoi in velocity have theoretical losses well in excess of the observed value. To summarize the effects shown in Fig. 12, a great many modes in- cluding TEoi w-ere launched by exciting the waveguide through a small aperture in the end plate. All these modes propagated back and forth in the line for a while, but due to the fact that TEoi has appreciably less loss than the other modes, the energ}^ remaining in the line after a suit- able time delay Avas substantially all in the TEoi mode. This permitted measuring TEoi loss over a distance of many miles by allowing the energy to traverse the 500-foot line many times. Fig. 13 records three successive trips of a pulse which had tra\-elled up and down the 500-foot waveguide for a total distance of 40 miles. The pulse shape was still essentially the same as that of the transmitted pulse, although background noise had become clearlj' visible. We cer- 1226 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 tainly can conclude from this that circular electric wave transmission over great distances is possible. The long waveguide line also provided a very convenient way of dem- onstrating additional multi-mode transmission effects. For example, in a multimode medium we maj^ use mode filters. One such filter may have a very low loss for the circular electric waves but very high loss to other waves. Such mode filters have been built, and Fig. l-l shows the trans- mission changes which result when introducing one. The upper half of Fig. 14 shows the photograph of the oscilloscope in the time interval 0 to llAi with no mode filter in the line. (This is the same as Fig. 12.) On introducing the mode filter into the waveguide, the received signal is altered as shown in the lower half of Fig. 14. We observed that energy propagating in the undesired modes has largely disappeared; it was absorbed by the mode filter. There are still a few small pulses in the lower half of Fig. 14 which cannot be in the TEoi mode because of their time position. Starting at time l.loA^, there is a series of regularly spaced pulses in Fig. 14 labeled TEo2 , and a single small pulse labeled TEos . The geometric placement of resistive material in the mode filter leads us to anticipate low filter losses for the entire circular electric (TEon) family of modes, and there- fore the extra pulses were suspected of being in higher-order circular electric modes. Only two such modes, TE02 and TE03 , were above cut- off. The TE03 pulse was tentatively identified by noting that its group velocity was 55 to 60 per cent of that of the TEoi pulses. High attenua- tion in the TE03 mode prevented additional TE03 pulses from being observed. In the case of the TE02 series of pulses, it was possible to get a fairly accurate measure of relative group velocity, confirming the identifica- tion as TE02 . Note that the seventh TEoi pulse coincides with the sixth TE02 pulse, and that the pulse at 7 At shows on the TEoi train as being too large in amplitude. The smooth decay of the TEoi train in the lower half of Fig. 14 in the interval At to 6A^ is in marked contrast to the corresponding pulse train in the upper half of the figure, and is graphic illustration of the im- portant effects that mode filtering can produce. Another very important transmission observation appeared during Mr. Beck's experiments with the 5" diameter line. He observed that the attenuation, as measured by the amplitude decay of the shuttling pulse, was a function of the position of the piston at the far end of the line. Translations of the far end piston on the order of 10 to 40 centimeters changed the overall transmission from a condition in which the original WAVEGUIDE AS A COMMUNICATION MEDIUM 1227 pulse sliape was preserved for as manj' as 40 miles of travel (Fig. 13), to a condition wherein the shape of the pulse was l)adly distoi-ted after only 3 or 1 miles of tra\'el. This general beluiN'ior is illustiatcd liy (he ))hoto- gra])hs shown in Fig. 15. We will concentrate for the moment on tiic top two rows of photographs which record tiie pulse transmission in the l)are waveguide as a function of distance of pulse travel for both favorable and unfavorable piston settings. However, all of the rows of the })hotographs were taken under such conditions as to permit direct comparison. The photographs at the extreme left end represent the outgoing pulse and the first echo pulses which travelled one round trip in the line, ap- proximately 340 yards. All the other photographs show two principal pulses which record two successive trips of the pulse as it passed the transmitting end (Fig. 10). The second photograph from the l(^ft re])re- sents the pulse as it passed the sending end after 10 and 1 1 round trips. The third picture from the left records the 20th and 21st trip, the fourth picture the 30th and 31st trip, etc. The numbers placed directly beneath the individual photographs represent the relative sensitix'ity of the re- ceiver for that particular photograph. Reference receiver sensitivity was taken as the condition under which the 10th and 11th trip in the bare waveguide were recorded with a favorable piston setting (0 db beneath the photograph), and the designation — 6 db under the adjacent photo- graph indicates that 6 db more receiver sensitivity was used in the lattei' case. The relation between display amplitude and actual pulse ampli- tude was approximately square law. The distance of pulse travel asso- ciated with each of the pictures is given at the bottom of the figure. Comparing the top two rows representing favorable and unfavorable piston positions, we note that the attenuation was appreciably differ- ent ■ — the values being 2.6 db per mile and 3.1 db per mile respectively. Serious distortion of the transmitted pulse also occurred for the un- favorable piston setting. Since the receiver in the experiment was sensi- tive to very many modes, one might suspect that the spurious i)ulses which appeared at more than 7,000 yards with the unfa^'orabl(> piston setting might represent energy present in some of the other modes. Actually, all of the pulses and wiggles shown in the photographs at ranges greater than 3,500 yards were in the circular electric mode. This was deduced by first noting that every two successive pulses were not appreciably different from each other (see Fig. 15). If some of the dis- tortion effects shown by the received i)ul.se were due to energy l)eing received in modes other than the signal mode, then successive pul.ses would be different in shape because the \arious modes have different phase constants. The very gradual change in pulse shape which did s > o < 1228 WAVEGUIDE AS A COMMUXICATION MKDIUM 1229 occur as the pulse travelled n\) ami down the line liad llic ^ciicral I'oiin of an amplitude component which led oi' hiii^cd Ihc sif>;iial ])ulsc l)\- a constant interval but which ^rachuUlx' inci'cascd in am))lilndc with in- creasing distance that the signal jiulsc had Irax'ellcd. Note, loi- example, the growth of spurious ]:)eaks b(>f()i'e and after the signal i)ulse in low 2 of Fig. 15 at 3,500, 7,000, 10,500 and 1 1,000 yards of travel. The explanation of this behavior involves transfer of energy from the circular electric mode to one or more of the unused modes of propaga- tion and reconversion of the same energy l)ack to the circular electric mode. This process is one of the characteristic features of multimode waveguide systems and is discussed at greater length in the following sections. MODE CONVERSION AND RECONVERSION AS A SIGNAL-LOSS EFFECT In beginning discussion of the mode conversion-reconversion phe- nomena, we take an idealized case of a dissipationless waveguide con- taining two deformities. Fig. 16. We assume a c-w signal entirely in the circular electric mode entering this waveguide. After passing the first deformity there will be energy present in some other mode, and this is designated as TXi . When the combination of TEoi + TXi strikes the second deformity, another conversion takes place and the outi)ut will be a large TEoi component, two smaller components in thv muised mode, TXi and TX2 , and a still smaller circular electric wave component, TEo/ , which is due to reconversion of energy- from TXi to the circular electric wave in traversing the second deformity. It can be shown' that for the proper distance between two identical symmetrical deformities, the wa\'e emerging from the line may be purely circular electric; the two compo- nents TXi and TX2 cancel each other under this condition. Another separation between the deformities results in a maximum energy trans- fer from circular electric to other modes. Therefore, it follows that any mechanism which varies the effective spacing between conversion points will produce conversion loss variations. This accounts for the change in the attenuation of the circular electric wave pulses in Fig. 15 as a function of the far end piston setting. FIRST SECOND DEFORMITY DEFORMITY TEoi+TX,+TX2+TE;, Fig. 16 — A distorted waveguide and the associated mode-coiivension signal- loss effects. 1230 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 MODE CONVERSION AND RECONVERSION AS AN INTERFERENCE EFFECT The mode eoiiverision-reconversion phenomena can also produce a signal distortion or interference effect. Fig. 17 shows another idealized waveguide containing two deformities, but in this case we assume a short pulse in the circular electric wave as the input signal. The ampli- tude-time plots below the waveguide show the energy present in the circular electric and unused waves respectively at various physical points along the line. The key item in this diagram is the time displacement between the converted energy TX and the circular electric wave energy at the input to the second deformity. This time difference appears as a result of propagation over an identical line length at two different group velocities which, in general, the circular electric and unused modes will possess. Since the signal and unused mode pulses strike the second de- formity at different times the second conversion process results in energy appearing back in the circular electric wave at a time separated from the signal pulse itself. When the distance between deformities is too short for the pulses to be resolved at the second deformation, the result will be a distortion of the signal pulse rather than the appearance of a separate pulse. The above very much simplified picture of the mode conversion and reconversion effects allows one to visualize several general properties of this phenomenon: 1. In general, there will be a large number of unused modes which will be coupled to the signal mode through the various imperfections in the transmission line. Since these unused modes have unequal phase con- stants, and since the imperfections will be randomly spaced along the line, the reconverted signal pulses in a time-division system w^ill be spread WAVEGUIDE DEFORMATIONS PURE TF° Fig. 17 — Signal interference effects due to mode conversion and reconversion. WAVEGUIDE AS A COMMUNICATION' MKDIUM 1231 out on the lime scale I'atluT than ai)])(':ir as a sinj^lc wcll-ddiiicd distor- tion pulse. 2. Because some of the unused modes of propajiation haxc <2;r()Ui) velocities greater than the grou]) \-elocitA' of the circulai- electric wave, the recouverted euerg}' pulses may reach the receiving end of the trans- mission system before the signal pulse itself ai)i)ears. 3. The reconverted energy will, in general, be out of phase with the signal from which it was derived. When the differential time of travel in the unused mode is short, the principal effect of the conversion-recon- version process will be to distort the signal wave. When the differential time of traN'el in the umised mode becomes as large as the r(M'i])rocal of the modulation frec]uencies iiu'olved, the reconverted energy will appear more like an echo. In a time-division system such echo pulses would interfere with the pulses representing other signal components. Because of the large number of conversions contributing at random time delays, this "echo-interference" may be unintelligible. In this sense the inter- ference may be thought of as a noise effect, just as multi-channel cross- talk due to amplitude non-linearities in a single sideband AM system may be thought of as noise. 4. The general case of a signal pulse, both preceded and succeeded by a series of reconverted energy pulses, is sketched in Fig. 18. It is quite apparent that if the reconverted signal pulses are allowed to become of the same order as the signal pulse itself, even a pulse code modulation system will be rendered inoperative. Other types of modulation will experience difficulty at appreciably smaller magnitudes of reconverted energy. 5. The level of the reconverted energy relative to the signal is deter- mined by the transmission medium. It is not possible to avoid this inter- ference by using more power at the sending end of the transmission link, for the interference rises with the signal. The need for low-noise receivers is just as acute as in other transmission systems, because better noise figure means that correspondingly less power is required from the trans- mitter. 6. If the loss to the mode TX is very large in the region between suc- cessive waveguide imperfections, the TX pulse can be attenuated to a negligibly small value before reaching the second deformation, thus pre- venting any significant reconversion back to the signal mode. 7. Limits can be placed on the time delay between the signal energy and the reconverted energy returned to the signal mode. The lower limit on this time delay is obviously zero, corresponding to a series of imi)er- fections very close together. The upper limit can be taken as the differ 1232 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 ence between the times it takes the signal mode and the unused mode to travel a distance in which there is a large difference between the ohmic losses in the unused-mode and signal mode. This concept may be ex- panded as follows: Energy is transferred at an imperfection located at coordinate Zi on the axis of propagation, producing an amplitude in unused mode x. At Zo on the axis of propagation the amplitude in mode X will be attenuated relative to the signal-mode amplitude at the same point by the factor — (aj;— ai)(02— zi) where a^ and on are the normal heat loss coefficients in mode x and the signal mode respectively. When the exponential factor is small enough (i.e., 02 large enough), reconversion will no longer be important com- pared to reconversion near Zi . For order of magnitude we might assume that 10 db more attenuation for the re -mode amplitude than for the sig- nal-mode amplitude would render further reconversion unimportant. i^ Fig. 18 — Schematic of signal distortion due to conversion and reconversion effects in a line with randomly placed conversion points. WAVEGUIDE AS A COMMUNICATION MEDIUM 1233 TluMi we know the distaiico (s.j — ^i) from 1.15 (--.. - zi) = The correspoiuling "upper limil" on lime dohiy hclwccn tlic signal and tlie reconverted energy in the fsignal mode i.s t= fe-^x)(---) (1) where Vx and Vs are the groiij) velocities in the mode x and tlie signal mode. It is well known that the unused modes of a circular electric waveguide have attenuation coefficients which are appreciably larger than attenua- tion coefficients for the circular electric wave itself. In the light of this attenuation to the unused modes plus the fact that the reconverted energy has undergone two mode conversion losses before it reaches the signal mode again, one might wonder whether the mode conversion- reconversion phenomenon wn)uld really be an important effect. The first indication that the magnitudes of the reconversion amplitudes are sig- nificant came during experimental work on the 5'' diameter 9,000-mc line, described above in connection with Fig. 15. A more quantitative theoretical discussion which follows shows that the reconversion phe- nomena will continue to be important even when mode filters are intro- duced into the line. The effects of the mode conversion-reconversion process are very simi- lar to the effects of multipath transmission through the atmosphere. In microwave radio there are under unusual fading conditions 2 or 3 sub- sidiary signals, and these are representative of propagation over different path lengths in space but at the same velocity of propagation. In the waveguide, there will in general be a large number of subsidiary trans- mission paths, each of which corresponds to the identical distance of propagation but at velocities of propagation which are different for the \arious modes. The radio multipath phenomenon exists only occa- sionally, whereas the waveguide multipath phenomenon is a steady characteristic present 100 per cent of the time. Long-distance radio trans- mission in the 6-20 mc region by way of the ionosphere encounters mul- ti-path effects more like those expected in the waveguide, with the excep- tion that wa\-eguide multi-path effects are expected to have far greater short-time stability. Quantitative relations describing the conversion-reconversion process may be derived by considering an infinitely long waveguide composed of 1234 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 a series of identical sections each containing a single mode-conversion discontinuity at the midpoint. (The actual 500-foot line contains many conversion points, as will be discussed later, and is effectively repeated over and over as the pulse traverses the identical line many times.) A very short signal-pulse of unit amplitude is assumed as an input to the idealized line containing identical conversion points. After the first con- version point the amplitude in the unused mode is k and the amplitude in the signal mode is (1 — t^)^'', where fc is a measure of the size of the conversion irregularity. There is no reconverted wave at this point since there is no input to the first conversion point in the unused mode. After the second conversion point, however, there is a reconverted-wave am- plitude /c^e*-^, where 6^ is defined below. After the n"' conversion point, it may be shown that the amplitude in the signal mode is ^(„-l)«, (^ _ j^2^nl2 (2) the amplitude in the unused mode is («-i) A;(l _ /,2^^-,("-l)<'x (n-1) -f etc. to + A;(l - k') ^ ,(«-i)»i and the reconverted wave amplitude is (n - 1)/^(1 - A:^)^,<'x+(n-2)«x + (n - 2)k\l - A;2)-^V''^+(-«)«i (4) + (n - 3)k\l - ^-2)-^,39.+(„-4)fl, + etc.tofc'd - A;^)'-^e^"-i)«x in which z = distance between adjacent conversion points ai and ax are the heat loss coefficients (applying to wave amplitudes) for the signal and unused modes respectively. j8i and /3x are the phase constants for the signal and unused modes respectively. ^1 = - (ai + j^i)z WAVEGUIDE AS A COMMUNICATIOX MKDIUM 1235 ^x = - («x + MZ k = amplitude of the converted wave for a single conversion ]ioint and for unit wave incident on the conversion ])<)int. In deriving the series represented by (2), (3) and (4), it is assumed that all of the converted power travels in the forward direction, and that re- flection effects are negligible. These conditions are tyi)ical of imperfec- tions in practical multi-mode waveguides. When the input pulse for the idealized line is sufficiently short, the various terms of (3) and (4) (representing successive conversions and reconversions) are non-overlapping pulses. It is instructive to write down the ratio of the signal-wave amplitude to the reconverted wave ;uni)litu(l(' which is separated from the amplitude of the signal wave at the same point by the time difference z{l/vx — 1/vi), in which v^ and Vi are group velocities. This ratio is [ratio of (2) to the first term of (4)] (1 - 1^ ,_. (o) (n - l)/cV^"'~"^^^ It is clear from this ratio that the signal wave may be smaller than the reconverted wave if n is sufficiently large. Physically, what happens is that the reconverted amplitude created at each successive conversion point adds in phase with the reconverted wave amplitude present at that point due to previous conversions and reconversions. This liappcMis of course because the line contains identicall}^ spaced conversion points. ^\'ith random location of conversion points, a less severe build-up of reconverted wave energy would certainly occur. The first and second reconverted pulses [i.e., the first and second terms in (4)] are separated by the time difference 2(1 /t'x — l/vi) and the ratio of the amphtude of the second to the first reconverted pulse is (n - 1) The largest amplitude existing in the unused mode at a point immedi- ately following the n'^ conversion is the component converted at the n"' conversion point and the ratio of this component to the amplitude of the signal pulse after the n^'' conversion is: ^ (7) (1 - k-y- Xote that this ratio is independent of n. The unu.sed-mode anijjlitude converted at the first conversion point is, after n trips and relative to 1236 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 the signal pulse at the same point, 1^ —{n—l){ax—ci{)z /q\ (1 - l^^l- ' ^^^ These expressions show that the largest unused mode amplitude at any point in the line will be the one most recently converted from the signal wave and will be nearest to the signal pulse in time position. The sketch shown in Fig. 19 represents schematically the signal pulse, the unused mode pulse and the reconverted pulse amplitudes after n trips past the conversion point. It is interesting to note that the most recent (the n"*) conversion to the unused mode appears in a time position close to the signal pulse whereas the most recent reconversion appears at a time far removed from the signal pulse. Let us investigate the ratios (5) through (8) under conditions repre- sentative of those in the 5" diameter waveguide line. Row 2 of Figure 15 shows that after 40 trips down and back on the 500-foot line, i.e., after 80 trips past the center of this line (where there is assumed a single con- version point), the amplitude of the reconverted pulses which appear just before and just after the signal pulse are about equal to the signal z ^A there are several modes for which the delay factor 2(1/^1 — 1 t'l) does not result in a separation hetween the reconverted pulse and the signal pulse until z is on the order of 5,000 feet. Thus, we might (^xpect the conversion-recouN-eision i)henomenon to broaden the signal pulse. This does indeed take i)lacc ev(Mi for the fa\-orahle piston setting of the bare waveguide, as shown by tlic toj) line of pulses in Fig. 15. The pulses at the distance 3,500 yards are sharper than those pulses at the distance 27,700 yards. However, addition of the mode filter (which introduces negligible signal attenuation) does a])i)reciab]y shar])en the pulse at the 27,700 yard distance (row 3). Thus, on the basis of pulse transmission observations on the 5" line and a simple theoretical analysis, we conclude that the conversion- reconversion phenomenon w'ill be important in a waveguide system, and that it is important to have as much dissipation as possible present in the unused modes of propagation. ANALYSIS FOR CONTINUOUS MODE CONVERSION The traveling pulse type of theoretical analysis utilized in the pre- ceding section can be extended to describe a more realistic spatial dis- tribution of conversion points and to include a series of unused modes instead of only one. An extension of this type is reciuired in order to cal- culate directly the behavior which might be expected in a waveguide composed of randomly disposed irregularities. A much simpler mathematical treatment, originally suggested to the writer by J. R. Pierce, is to assume uniform mode conversion along the direction of propagation and to represent this condition by a differential equation. An analysis of this type is attached as an appendix. The work includes the assumption of quadrature addition of conversion and recon- version components, and the total magnitude of such components given by the analysis may be thought of as the rms average of the conversion magnitudes in waveguide lines containing randomly located imperfec- tions. Any single line might show somewhat more or less conversion effects, a factor of ±10 db probably being adeciuate to cover most lines containing randomly located imperfections. If practical lines show ap- preciable correlation between the spacings of the conversion ] joints, the reconverted-wav(> magnitude would become greater. The analysis has the advantage of being simple and understandable and should give overall trends accurately. This analysis shows that the waveguide performance with regard to conversion-reconversion effects is completely specified witii knowledge 1240 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 of (1) the ratio of the conversion coefficient (ai^) to the true dissipation coefficient* in the signal mode (ai/,), (2) the ratio of the heat-loss coeffi- cient of the unnsed-mode to that for the signal-mode (a^h/aik) , and (3) the length of the transmission line specified in terms of decibels of heat loss to the signal wave. For a given ratio of conversion-loss to heat-loss, the same ratio of signal-power to reconverted-wave power will be present for a long low-loss waveguide as for a short high-loss waveguide. This makes it important to determine the ratio of conversion-loss to heat-loss for waveguides of several nominal attenuation coefficients and to predict these effects theoretically insofar as it is possible. Another result of this analysis is plotted in Fig. 20, which shows the ratio of the signal power to the power in the unused mode at the end of the line, with transmission-line heat loss as the abscissa. These curves have been plotted for a fixed magnitude of conversion loss coefficient (au) equal to 50 per cent of the true heat loss coefficient (aih) and for ratios {axh/ciih), heat loss in the unused mode to heat loss in the signal mode, between 2 and 100. These values are typical of solid round wave- guide without mode filters. It is interesting to note in Fig. 20 that the magnitude of the unused mode power relative to the signal mode power reaches very nearly a constant value in a transmission line length of only ^2 to 1 db, except for extremely low ratios axh/am . Physically what is happening is that the unused mode power becomes dissipated through heat loss about as rapidly as it is created by mode conversion, after an initial short transmission line length. Fig. 21 shows the ratio of signal power to reconverted wave power as a function of transmission line length for the same conditions described in connection with Fig. 20. A heat loss ratio on the order of 2 to 10 is typical of important modes in solid round waveguide without the addi- tion of mode filters,! and Fig. 21 shows that the ratio of signal-to-recon- verted wave power for such a medium becomes poorer than 20 db for transmission line lengths longer than 1.5 to 2 db. Although there is some uncertainty as to the precise interpretation which may be placed on the signal power to reconverted wave power calculated in this manner, since the time relations in connection with a definite modulation method are not included, it seems evident that a solid copper tube without mode * There is a v(M-y sigiiifieaiit difiereiice between the effects of signal power loss to other modes through conversion and signal power loss due to dissipation in the waveguide walls. However, it does not matter here whether the latter be due to surface roughness, chemical impurity or just the theoretical minimum heat loss for ideal copper. Therefore, all of the heat loss effects are combined into the single coefficient, au • t See the appendix for further discussion. WAVEGUIDK AS A COMMrXlC ATIOX MKOITM -I X q-Iq. 100 _ .^^ "^^"^^^ __30 —'2. •- 1 ^«, ~~- * S"""'' ^ "" 1 1 1 1 HEAT LOSS RATIO ^^=2^ 111 1 1 \ 1 ^ 0.1 0.2 0.4 Q6 0.8 1.0 2 4 6 8 10 TEoi HEAT LOSS IN DECIBELS P'ig. 20 — Ratio of TEoi signal power (Pi) to X-mode power (P^) versus line lougtli for conversion coefficients (ai^ and Oxi) equal to one-half the licat loss co- efficient (fli;,). filters is very unlikely to be satisfactory for long distances as a communi- cation medium. However, the addition of mode filters will rai.se the heat loss to the inidesired waves, and the latter improves the signal to re- con^'erted-power ratio directly as the ratio of heat loss in the unused wave to heat loss in the signal wave. Thus, as shown on Figure 21, a heat loss ratio on the order of 500 would produce a ratio of signal ])ow(>r to reconverted wave power on the order of 20 db at a transmission line length of 60 db. It may be shown that the magnitude of the signal to reconverted wave power varies as the square of the conversion to heat loss coefficient ratio aix/fli/j . The sharp break downward on the right hand end of the curves for CLjh/a^h — 2 in Figs. 20 and 21 represents the condition wherein the power in the reconverted wave becomes comparaljle to the power re- maining in the signal wave. We next consider the improvement in signal fidelity which results from the introduction of ideal mode filters. We shall assume the ideal mode filters have a matched impedance for all modes, very high trans- mission loss to the unused modes, and no transmission loss for the signal mode. The impro\em(Mit in tlu^ ratio of signal power to reconverted wave power due to the addition of such filters is shown in Fig. 22. This plot has been calculated for a total line length of 20 db heat loss, but the conclusions are valid for any line length wherein the signal wave power remains appreciably larger than the reconverted waxc ))()\vcr. We ob- served that mode filters improve the signal-to-recon\-ertt'd-\va\'e powers very slowly when placed far apart, typical improvements ranging be- 1242 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 tween 2 and 8 db for 1 db reparation between mode filters. The larger improvement is obtained when the heat loss in the undesired mode is more nearly comparable to the heat loss in the signal mode before the addition of the mode filter. For very small spacing between the mode filters, addition of the ideal mode filters improves the signal-to-recon- verted wave power by very large factors. A somewhat different form of effect due to the addition of mode filters is observed if the line before the introduction of the mode filter has a reconverted wave power larger than the signal power. Under this con- dition, relatively large mode-filter spacings bring about a large improve- ment in signal-to-reconverted wave power, but this is due to the very poor condition present before filtering. It is doubtful whether the trans- mission line would become very useful without rather strong mode filtering of the type represented in Figure 22. We may conclude that the mode filters should be placed very close together, preferably at spacings of less than .1 db heat loss to the signal wave. We may also conclude that the transmission of signals over dis- tances corresponding to the order of (50 db heat loss will require either oTltf ^ "^ N ^ ^ >^ ^ ^*>, ^ 5 ^ •«^^1000 - ^ s^ ^ ^ \ \ V \, ^ ''"^ *^ HEAT L .OSS RA TIC axh 2V^ \, 1 1 1 .1. 1 ^ 0.4 0.6 0.8 1.0 2 4 6 8 10 TEo, HEAT LOSS IN DECIBELS 40 60 100 Fig. 21 — Ratio of TEoi signal power (Pi) to reconverted TEoi power (P„) versus line length for aix = 0.5 au . WAVi:(;rinK as \ coMMrxic \ ri(»\ midk m 1213 25 t '0 V \ HEAT LOSS <3xh RATIO -^ - \2 '■' \, N \ \ \jo \ s N \ ^ ^ Xn,,.30 1 ^ \ ^ 1 1 1 1 0.1 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 20 40 60 100 LINE LENGTH BETWEEN MODE FILTERS -TEqi HEAT LOSS IN DECIBELS Fig. 22 — Improvement in Pi/P„ due to addition of mode filters. Total line length equals 20 db TEoi heat loss. (Plotted for an = 0.5 Ou). the spacing of mode filters at less than 0.1 db to the signal wave or use of a continuous form of transmission line having a ratio of heat loss in the unused mode to the heat loss in the used mode on the order of 500. DIRECT EVALUATION OF MODE CONVERSION MAGNITUDES The important influence that mode conversion effects are expected to exert on signal fidelity lead us to make direct evaluations of the con- version coefficients. The direct evaluation consisted of transmitting (actually or in imagmation) a pure circular-electric wave into one end of a w^aveguide section and, by measurement or by calculation on the basis of known geometry, determining the relative magnitude of the power converted to the unused modes. The simplest experimental technicjue for analyzing mode impurities consists of a short radial probe at the guide wall. The radial probe re- sponds to energy in any mode of propagation except the circular electric family, and serves as a versatile instrument for measuring the order of magnitude of mode conversion effects. The limitations of the technicjue stem from (1) the fact that the probe responds to the vector sum of the amplitude of the radial electric field compon(Mits of about 35 modes (in the 5" line case), and this sum varies with circumferential and longitudi- nal position of the probe even though the power present in the modes is constant; and (2) the fact that the sensitivity of the probe response to a given magnitude of power in the guide is variable from mode to mode, 1244 THE BELL SYSTEM TECHXICAL JOURXAL, NOVEMBER 1954 being (in an extreme case) 25 db greater for TE91 than for TE13 in the 5'' pipe. For the majority of modes, howe\'er, the latter variation is ±3 db, and the maximum probe response as a function of curcumferential and (to a hmited extent) longitudinal position can be determined with- out excessive labor. The probe technique of mode conversion evaluation Avas first applied by M. Aronoff to the individual sections of the h" diameter experimental line. He found that the average indication of conversion for the (ap- proximately) 20 ft. lengths of pipe was 29.5 db below the signal wave power; since the power loss due to dissipation in the walls for a 20 ft. section is about 27 db below the signal wave power, the individual pipe measurements gave an order-of -magnitude estimate of 0.55 for the ratio of conversion loss to heat loss (aixA^ift)- Four mechanically distorted sections of line, previously considered satisfactory, were identified and discarded on the basis of this approach. A. C. Beck and M. Aronoff next applied the probe technique to the 5" diameter line assembled into lengths of 145 feet, 270 feet, and 500 feet. The indications of converted power were —17 db, —10.5 db, and — 13 db respectively (at wavelengths near 3.2 cm) which is compatible with the hj^pothesis of random addition of a number of conversion com- ponents. Since the heat-loss power for the 500-foot line is about — 13 db compared to the incident signal power, the 500-foot line radial probe measurement gave an order of magnitude estimate of 1.0 for Oix/ai/, , in fair agreement with the value of 0.55 from single pipe measurements. The probe indication of conversion as a function of frequencj^ for the 500-foot hne is plotted in Fig. 23, which shows that quite a number of 3.04 3.08 3.12 3.16 3.20 3.24 3.28 3.32 3.36 3.40 3.44 3.48 3.52 3.55 WAVELENGTH, Aqj '^ CENTIMETERS Fig. 23 — Probe recording of converted power in the 500-foot line. WAVEGUIDE AS A COMMUNICATION MEDIUM 1245 conversion comjx^u'nts were ])r('s(>nt ; similar data were ohscrvcd on llic sliorter h^ijiths of lino. Azinmthal (listril)uti()ns of pi'oUc rcs))()iisc showed coiu-lusixrly lliat v(My little conversion was ])i-cseiit to the \'A '\'\'],,„ and T.M,,,,, modes having an index "/?" of four or moic A much more precise though more eiahoiale met hod olC\ alual ing the eon\'ersion coefficients invoh'es use of coupled waxc transducers. ' Such devices resj-jond to only one mod(^ and ha\-e known s(Misiti\-ities, and therefore permit truly (iuantitati\-e m(>asur(Miients. (.\t t!i(> "pi'es(Mit time this ap]n-oach requires a separate transducei' foi' each mode 1o he e\ahi- ated, a somewhat cumbersome pi'ocedure, hut in principal the mode transducers can l)e made "tunable" for a series of modes.) A. ('. Heck and ]\[. Aronoff applied this more accurate method of conx-ersion coeffi- cient determination to several modes of the oOO-foot hue, and 'liable 1 shows the vakies averaged over the frecpiency band. Table I — Average Ratio of Conversion to TEm Heat Loss WITH TEoi Excitation Mode flii/aiA TEu 0.21 TiMu 0.05 TE^i 0.14 TEs. 0.05 TMoi <0.001 Total 0.47 The estimated absokite accuracy is between 10 per cent and 20 per cent for these ratios. The variation of the conversion coefficients as a function of frequency is shown in Fig. 24 for two of the important modes and for the total of the modes given in Table I. Th(> total of the modes measured is consistent with the probe indications, though the acou'acy of the latter is low enough that this should not be interpreted as proxing there are no other important conversion contributors. The above direct evaUiation of mode conversion in the .")()0-fooi line yielded magnitudes that are sufficient to explain the conx'ersion-recon- version process as already outlined. We wished to extend our experience with this particular fine to higher-fre(|ueiicy lines which might be built and to absolute tolerances that might l)e ])laced on the construction of a new hue. Toward this end, theoretical relations wei'c derived by S. P. Morgan, Jr., for the mode conversions to be exp(>cted due to waveguide ellipticity, and due to the tilt and offset which may be exi)ected to occur at the junction of two sections. Experimental work was done by M. 1246 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 3.24 3.28 3.32 3.36 3.40 3.44 WAVELENGTH, \o» 'N CENTIMETERS 3.48 3.52 3.56 Fig. 24 - TE21 , TM, Observed conversion from TEni to TEu , TE21 , and the sum of TEa , , TMoi , and TE31 for the 500-foot line. Aronoff at 9000 mc on accurately created imperfections of the above types, and he found excellent agreement. We proceed to use the theory to compare the present line to a hypothetical 50,000 mc line. The re- sults are given in Table II. These computations represent a single oval section or a single tilted or offset joint. The converted power varies as the scjuare of the tilt angle, as the sciuare of the offset distance, and as the square of the difference between the major and minor diameters. The total ratio of converted power to heat loss power depends on the number of conversions per unit length. For the accuracy of constructing a 2" diameter line assumed in Table II, the amount of mode conversion to be expected is not appre- ciably different from what appears attributable to known mechanical imperfections in the 500-foot line already discussed. Another waveguide property of interest is the way the mode conver- sion magnitudes vary across the frecjuency band in a fixed pipe, and Table III shows this for the 2" diameter line. "WAVECIJIDE .\S .\ rOMMrXlC ATIOX MKDUM 1247 Table II — Mode C'onversio.n Compahison ok 'JOOO-mc 1.732" Diameter Line with a Hvi'otiietu-al o(),()(X)-mi" 2" Diameter IjIne Imperfection I'ipr l)i:imfUT I''ifi|uciuy Magnitude of Converted Power Type Magnitude Percentage Mode inches Mc. Ovalitv Ovality 16* mils 4* 4.732 2.0 50,000 0.53%t 0.18%t Tlvu TKn "' Tilt Tilt 1° 1° 4.732 2.0 9000 50,000 0.34% 2.0% TK,, TK„ Offset Offset lot mils 2.5t 4.732 2.0 0000 50, ooo 0.008%, 0.003% TK,2 TE,, * Difference between major and minor diameters. t Separation of guide axes. X These are upper-limit values, based on the length of the oval section of pipe which would produce maximum mode conversion, and based on a cross-sectional shape (trifoili which would jjroduce the maximum of mode conversion. It is interesting that two of the three ('on\'ersion effects are essentially independent of frequency. Tilt at a waveguide junction introduces a phase-front error and would be expected to cause greater conversion effects at increasing fre(iuencies. We shall see that bends ])roduce a similar mode conversion, also due to a phase front error, that increases with increasing frequency. the bend problem The problem of transmitting the circular electric wave around bends was recognized as being important at an earlj^ date, and contributions to its solution were made by M. Jouguet,*'^ W. J. Albersheim,^ S. 0. Rice/ and the writer.^ The essence of the problem is as follows: .V bend in a Table III — Frequency Variation of Mode Conversions in 2" Diameter Guide Imperfection Mode Magnitude of Converted Power Type Magnitude / = 24,000 mc / = 50,000 mc / = 75,000 mc Ovality Tilt Offset 4 mils 1° 2.5 mils TE31 TE12 TE12 0.22%* 0.41% 0.003% 0.18%* 2.0% 0.003% 0.20%* 4.8% 0.003%, These are upper-limit values, based on the length of the oval section of pipe which would produce maxinuun mode conversion, and based on a cross-sectional shape (trifoilj which would produce the maximum of mode conversion. 1248 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 round guide causes coupling l)etween the circular electric wave (TEoi) and the TMu wave, and in round solid pipe the TEoi and TMu wa^'es are degenerate, i.e., they have identical phase constants. This degen- eracy has the effect of bringing in jihase all components transferred to TMu 110 matter how gradually the bend might take place. Theory neglecting dissipation ' shows that in a round pipe bent with any radius of curvature, the power will flow from th(^ TEoi mode to the TMu mode and back as a function of the total bend angle 6 according to the relations TEoi amplitude = cos f ^ - j (9) TMii" amplitude = sin (^^ j\ (10) where TT ^o 2^0" Z.6Z a Xo = free-space wavelength a = waveguide raduis A solid round pipe is unsatisfactory for transmission of the circular electric wave around bends in the broad bands we seek to use. One method of making the guide satisfactory in bends is to break the degeneracy between the TEoi and TMu waves. Use of an elliptical pipe has been shown theoretically to be one way of doing this. For a 2" di- ameter guide at 50,000 mc an eccentricity of about 0.3 permits a bending radius of 700 feet with theoretical bend losses in the range 0 to 0.17 db for any total bend angle; the heat loss coefficient for such an elliptic guide is about 35 per cent higher than for a perfectly round guide.^ Another method of avoiding bend losses is to introduce dissipation to the TMii wave without adding loss to the TEoi w^ave. It has been shown that a large difference between the attenuation coefficients of two coupled waves reduces the power transferred from the low-loss wave to the high-loss wave. Applied to the bend problem, this means that a struc- ture with increased TMu loss may be bent with less signal (TEqi) loss even though the phase constants might be degenerate. The reader is referred to the earlier paper for a more complete discussion. There exist several alternate forms of circular electric waveguide (to be discussed) which have an attenuation coefficient for TMn more than 5,000 times the attenuation coefficient for TEoi • The calculated extra loss in the bend region for such structures and for solid round pipe has been plotted WAVEGUIDE AS A COMMUNICATION MEDIUM 1249 1.0 0.8 0.2 r N \ \ * \\ V - V " DIA 2"DI^ » 1" DIA \ (solid, ROUND pipe) \ 2" DIA \(SOLID^ ROUND PIPE) - \ \ \ \ \ \ % / \ / \ \ \ X V *TM„/«TEo, = 1200 \ \ \ \ \ \ \ \ 4800\ > ,12 \ - \ \ \ \ - * \ \ \ - \ \ \ \ » \ \ \ » \ - \ \ \ \ \ \ \ \ \ \ t 1 \ \ V 1 \ 1 ^- \ 1 \ BEND RADIUS IN FEET Fig. 25 — Computed ch;uige in the 50,000 mc TKoi attenuation coefficient due to bends in 1" and 2" diameter solid pipes and in modified guides having TMn attenuation coefficients larger (than in solid pijic) by a factor of 100. ub and a, are the bend-region and straight-line attenuation coefhcients, respectively. as a function of bending radius in Fig. 25, assuming the TjMh-TEoi coupling due to the bend is the same in the altered guide as in solid round pipe. Whereas a bending radius of 17,500 feet causes a 100 per cent increase in TEoi heat loss for 50,000 mc waves in 2" diameter solid pipe, the modified structure with a TMn attenuation coefficient that is larger by a factor of 100 should tolerate a bending radius of 1,750 feet for the same heat loss increase. (The 50,000 mc attenuation coefficients for ideal 1" and 2" copper pipes are 14.8 and 1.79 db^ mile respectively.) For estimating purposes, the ratio of the extra loss per unit fine length in the bend region to the straight line loss ma.y be calculated for solid round pipes from the relation as — ccs 2.5 X 10'"a' (11) and the ab.solute increase in attcnuntion due to a IkmuI is* , 9.7 X lOV ru/^f ■ 1 ^ (as - as) = — ^ „,,p, (db/meter for copper giude) (12) * For small bend radii and bend angles less than 0c , this relation gives a greater loss than the correct value. See Figs. 22 and 23 of Reference 8 and also see Refer- ence 2. 1250 THE BELL SYSTExM TECHNICAL JOURNAL, NOVEMBER 1954 where as = straight hne attenuation poefficient ub = bend region attenuation coefficient a = guide radius R = bending radius Xo = free space wavelength The approximations used in deriving (11) and (12) are good when the operating frecjuenc}^ is at least 50 per cent greater than cut off for TEoi . For guides modified to have higher TMn attenuation both (11) and (12) may be divided by the factor m 0 m 0 (13) where a^ and a° denote attenuation coefficients for the modified guide and solid round guide respectively. On the assumption that the mode coupling is the same in the modified guide as in the solid round guide, use of (13) with (11) or (12) provides an estimate of bend losses in modified circular-electric waveguides. For a fixed ratio of bend-region attenuation to straighthne attenua- tion, the allowable bending radius varies inversely as the scjuare root of the ratio of TMn heat loss to TEoi heat loss, varies inversely as Xo'^ , and varies directly as the third power of guide radius. For fixed bending radius, the absolute bend loss varies inversely as Xo^^; since the straight line TEoi loss varies directly as Xo'^, bend losses tend to equalize the overall heat loss versus frequency characteristic of the waveguide. IMPROVED FORMS OF CIRCULAR ELECTRIC WAVEGUIDE In the preceding discussion it has been indicated that added dissipa- tion for the unused modes of propagation has the effect of decreasing signal losses and of reducing the interference effects associated with mode conversion. Dissipation can be introduced to the unused modes of propagation through the addition of mode filters at intervals along the line, but it appears very desirable to introduce the dissipation to the unused modes on a continuous basis. Several ways of making the line lossy to the non-circular electric modes have been found, and one is illus- trated in Figure 26. The copper rings lie in planes transverse to the direction of propagation and provide the conductivity required as a boundary for the circular electric wave family. Successive rings are insulated from each other, howe-\'er, and the guide provides very poor conductivity in the longitudinal direction. All modes other than the WAVKC.UIDK AS A fOMMUXICATK )\ Mi;i)irM 1251 circular olectrie wave family have wall ciirrcnts in the loiigitiidiiuil (iirco- tion and oxperienec con-sidcrahly increased loss in the spaced-rinii slruc- t are compared to a solid-walled wavi'ji;uide. A. (".. i'ox lirst ohserxcd lliat the spaced rinp; structure could \)o used to transmit circular electric waves around bends, and since that time additional work has been car- ried out by A. P. King and M. Aronoff . The observed loss for the spaced- rins structure under ])ro])er conditions was observed to be about ()() per cent more than the theoretical loss for an ideal copper tube, whereas the observed loss for the unused modes of propagation was on the order of 1,000 to 5,000 times the circular electric wave value. The spaced-ring structure therefore has the electrical properties we seek. The higher-order circular waves exist with losses comparable to their values in a solid copper pipe, but fortunately the magnitudes of conversion between the waves of the circular-electric family have been found to be small. The spaced-ring structure does present some difficult problems with regard to fabrication. An analogous structure composed of a continuous helical conductor supported within a lossy housing has electrical properties which approxi- mate tho.se of the spaced-ring .structure, and the helix should be con- siderably easier to manufacture in long lengths. The helix might be expected to support a wave-type approximating the circular electric wave both from the standpoint of field distribution and loss when one observes that a helix of very small pitch presents almost circumferential conductivity as required by the circular-electric wave, and the very small longitudinal component necessary due to the finite wire size tends toward zero as the helix pitch tends toward zero. James A. Young of these laboratories has constructed helices in the 2 db/mile waveguide size (4.73" diameter at 9,000 mc) and found a heat-loss coefficient on the order of 1.75 times the theoretical value for ideal copper pipe. These large ex- B-B A-A Fig. 26 ■ — Spaced-ring circuhir electric wtivoguide. 1252 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 perimental helices were known to be impei-fect, and in a smaller diameter of helix, losses within 15 per cent of the theoretical values for perfect copper pipe were achieved. These results lead us to regard the helical line as a verj- promising medium for circular electric waves. SURFACE ROUGHNESS Since frequencies on the order of 50,000 mc are desirable for low-loss waveguide use, it was recognized that roughness of the surface at the waveguide walls might appreciabh^ increase the heat loss in a practical waveguide. The first approach to this problem was made by VV. A. Tyrrell using sections of 5 diameter copper pipe from the experimental line. Tyrrell measured the heat-loss coefficients of the pipe when used as a resonant cavity at 9,000 mc in lengths on the order of 4 to 8 feet. Care- fully selected resonant conditions were employed to avoid bringing the unused modes of propagation into resonance at the same time that the circular electric wave was resonated. Whereas Tyrrell observed that the heat loss coefficient in the pipe as originallj^ drawn was about 21 per cent higher than the value computed using the measured dc conductivity, he found that rotary* grinding and polishing the inner surface of the guide reduced the excess loss to about 12 per cent. Tyrrell also observed that commercially drawn brass and 2S aluminum tubing of approximately the same dimensions showed measured losses 11 per cent and 20 per cent respecti^'ely greater than the ^'alue predicted from the measured dc conductivity'. Therefore, the indication from Tj'rrell's work was that surface roughness did indeed account for increased losses even at 9,000 mc, and that the excess losses could be reduced either by polishing the surface or through the use of lower conductivities (which have the effect of increasing the skin depth). A parallel approach to the measurement of surface-roughness effects was made bj' A. C. Beck and R. W. Dawson, also at 9,000 mc.^ Beck and Dawson used small wire samples as the center conductor of a coaxial cavityf and found that commercially drawn copper, aluminum and sil- ver wires showed loss values 10 per cent and 15 per cent higher than those expected from the measured dc conducti^'ity. By mechanically polishing * Because the wall currents for the circular electric wave are circumferential, the longitudinal surface scratches produced by drawing are in the worst possible orientation. Polishing was carried out in a rotary manner so that the current would not cross the scratches so induced. t In a coaxial, the currents are longitudinal, as are the scratches from drawing, so the measurements in the coaxial would be expected to show somewhat less excess loss due to surface roughness than the measurements made in circular-elec- tric waveguide cavities. WAVEGUIDE AS A COMMUNICATION MEDIUM 1253 these same wires the losses were reduced to 5 to 8 jxt cciil nhox-c llie de values and, by electropolishiug, ('()j)])er wires were l)i()u for backward-wave 1254 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 oscillators which has already resulted in 5 to 6 mm oscillators and which appears suitable for use at shorter wavelengths.* Recently, E. D. Reed produced a 5 to 6 mm reflex klystron.* R. S. Ohl has continued his pioneering work on point-contact crystal converters and has made units for use at 6 mm having conversion losses of less than 8 db and output noise ratios on the order of three times basic thermal noise. Ohl also made point contact silicon units for use as har- monic generators to permit the conversion of 24,000 mc power to 48,000 mc power. His harmonic generators have proven invaluable as a source of millimeter wave power — essentially all of the radio research work done to date has been carried out using them. In order to evaluate crystals and millimeter wave oscillators, it is essential to have an absolute power reference in the millimeter region, and work* has been done by W. M. Sharpless to establish such a reference. Up to the present time all of the amplifiers, oscillators and other circuit elements have employed dominant-mode rectangular waveguides in order to simplify the circuit design. Therefore, it is of importance to know how to transform a signal from a dominant-mode rectangular guide to the circular electric wave in round pipe. The first circular-electric- wave transducer made in these Laboratories was designed by A. P. King and had the form sketched in Fig. 27. This transducer is of the general type in which the metallic boundary of the waveguide is shaped to force the field lines in the cross section of the guide into the pattern character- istic of the desired output wave. In Fig. 27 the dominant-mode rectan- gular guide at the left end is gradually tapered to the sector of a circle; the size of this sector is small enough so that only one wave type can exist at this point, and the electric field lines are arcs of a circle. Next, the angle of the sector is gradually increased along the axis of propagation until at one point a cross section of the guide has the shape of a half circle. The size of the sectoral angle is continually increased, however, until finally the metallic sector of the circle disappears as a radial vane. When the taper is done gradually (an overall length of approximately 10 to 15 wavelengths) the electric field lines remain ares of a circle as Fig. 27 — Circular-olectric wave transducer (due to A. P. King). * A portion of this work was carried out under Office of Naval Research Con- tract Nonr 687(00). WAVEGUIDE AS A COMMTTXirATIOX MEDTT-M 1255 Fig. 28 — ^Nlode filters which pass only circuhir electric waves. they were in the sector at the left-hand end of Fig. 27, and the circular- electric wave emerges in the round pipe. This type of transducer has been shown to have transfer losses from dominant-mode rectangular guide to circular-electric wave in round pipe of approximately 0.3 dh at 24,000 mc. Similar models have been made by A. G. Fox for use at 48,000 mc. Another important component is the mode filter previously referred to and which attenuates all wave types other than the circular electric wave family. One type of mode filter to perform this function is the spaced ring structure of Fig. 26 and another type, due to A. P. King, consists of resistive sheets along radial planes as shown in the photo- graph of Fig. 28. The circular electric wave family has no electric field in a radial direction or in a longitudinal direction. All other wave types, however, have radial-electric or longitudinal-electric field components and experience attenuation due to the presence of the resistive sheets. The coupled-wave type of transducer sketched in Fig. 29^ is useful in connecting from dominant rectangular guides to the circular electric or other modes in round guide. This type of transducer makes use of the fact that the various modes in the multimode guide have unequal j^hase constants. The transfer of power from rectangular guide to the round guide takes place only to the particular round-guide mode whose phase constant is equal to that of the wave in the rectangular guide. This type 1256 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 of transducer has the same geometric appearance (aside from exact dimensions) for any mode in the round guide and is attractive in that it presents a matched impedance to all the modes of propagation. This property may be used to combine a series of signals onto different modes in a single transmission line. The coupled wave transducer may also be employed to multiplex a series of frequency bands into one pipe. We may wish to employ the long-distance waveguide over the frequency band from perhaps 35,000 mc to 75,000 mc and will require a series of transducers to go from dominant mode guides in various portions of this band to the circular-electric wave in the round guide. Frecjuency-selec- tive coupled-wave transducers may be employed in the manner sketched in Fig. 30 to multiplex these frequency bands into the pipe for the long distance transmission. A. G. Fox'* has shown that dielectric waveguides are attractive as a flexible connecting link for terminal equipment in the millimeter wave region and may also be employed in circuits such as hybrids. On all of these items of millimeter wave technique and multimode waveguide technique, individual publications will appear as soon as the work has reached the point where this becomes possible. MODULATION METHODS The modulation method to be used for the transmission of intelligence on a waveguide system will probably be dominated by the conversion- reconversion phenomenon already discussed. In order to evaluate the ,-z COUPLING APERTURE Fig. 29 ■ — Coupled-wave transducer for generating circular-electric or other .veeruide modes. waveguide modes. WAVEGUIDE AS A COMMUNICATION Mi;i)HM 1257 iiitci'lVrinn cITccts of the coiixci'sioii jjroccss, it is iiu])()il;iiit to lake iiilo acfowiil the time relations liclwccii the siiiiial coinpoiiciils aii1 Appendix tiiix)i;ktu'al analysis for coxTixuors modk coxversion Whereas the travelling-pulse type ol' thcorclicnl analysis utilized in the body of this paper can be extended to a realistic spacial distribution of conversion points and to a series of modes instead of only one, J. I^ Pierce suggested that the assumption of iniiform motie conversion along the axis of propagation would lead to a solution in closed form and would probably show the general properties being sought. 'J'his suggestion was adopted and Pi is designated as the signal jjower, 1\ as the power in the unused mode, and P,, as the power which has transferred from mode-a; l)ack to the signal mode, mode 1. We assume (luadrature addition of conversion components, and write a series of dilt'crential e(juations ex- ])ressing the power flow between the modes along the axis of propagation, including the heat loss effects: ~ = -auPi - ai.Pi (9) dz dP V = -a^hPx - a.iPx + OixPi + auPn (10) dz dPn -J^ = -aihPn — CluPn + dxlPx (11) dz in which the symbols have the following definitions: ttih = the heat loss coefficient - mode 1 flix = the mode conversion coefficient from mode 1 to mode x Qxh — the heat loss coefficient — mode x 0x1 = the mode conversion coefficient from mode x to mode 1 z = distance along the axis of propagation Note that the above heat loss coefficients are those associated with power rather than attenuation coefficients associated with amplitudes, (2ai = ciih , 2ax = ttjch)- The above equations also imply mode con\-ersion in the forward direction only. In a phenomenological way, these eciuations represent the decay of power in the signal mode and the build up of power in both the unused mode X and reconverted energy P„ in mode 1. The general plan is to solve these equations for P. and P„ in terms of the input wave power. P„ is maintained separate mathematically from Pi , even though both of them are in the same mode, so that we can clearly identify tlie energy which has been at one time in the unused mode x. 12G2 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 For mathematical solution, (9), (10) and (11) may be put in the fol- lowing form: ^1 + aPi = 0 (12) dz ^ + /3Px - auP^ - auPn = 0 (13) dz dP ^ + aP„ - a.iP. = 0 (14) dz in which a = aih -\- dix 13 = a^h + ctxt The general solution for (12), (13) and (14) is given by the following Pi = ki e~"' (15) p, = fc2 8^^' + ks e'" (16) P„ = -k,e-"' + ^ (ri + ^)i'' + - (r2 + ^y (17) where n _ -(« + ^) ± V(« - iS)- + 4ai,a,i ^2 (18) The positive sign is to be associated with ri and the negative sign with r2 . For the boundary conditions, at 2 = 0, Pi = Po P. = 0, Pn = 0, that is to say, the input to the transmission medium being zero in both the X mode and reconverted energy mode, then the solution takes the following form: Pi = Poe""' (19) p _ dlxPo 7- 12 (llxPo rzz /()f)\ (r-i - ra) (n - r^ wavp:guide as a commuxicatiox mkdium 1203 rn - — i oe + — ^ e — r- e (21) (^1 - ro) (n - r.) It is informative to note that (ri — ra) is always positive and is e(|ual to Via -/3)2 + 4aua,i We are usually interested in the ratio of the power in the x-modctothe signal power Pi and the ratio of the reconverted energy P„ to the signal ]ioAV(n- Pi . These ratios are given by the following exi)ressions P„ _ (n + /3) ^r,+a)z _ (/•2 + /3) (n+aU _ . /orjN* Pi " (ri - r,) ' W^=^) " ^^^^ [1 - e-'-'-'''-] (23) Px (ri - r.) Thus, we have explicit solutions for the iniiform transmission medium containing mode conversions. In order to make the most general study of these relations, we shall express the mode conversion coefficients in terms of the heat loss coeffi- cient in the signal mode — i.e., as the ratio au/dih . This is natural enough phj'sically, for Ave are interested in the relative magnitudes of the heat loss and mode conversion effects. It is found that knowledge of the ratios a\x!a\h , 0x1 Qu and axh/cbih enables us to completely determine Pi/Pi and Pn Pi in terms of the distance parameter e""""'. The latter is the heat loss in the signal mode, another familiar physical characteristic. characteristic conditions in bare round waveguide In order to use the theoretical relations derived iii the preceding sec- tion, we need to know typical values of the parameters. In particular, we need to know the magnitude of typical conversion coc^fficicnts au and values of the heat loss coefficients a-iu and Oxh for the modes of interest. One set of heat loss coefficients wliicli is of immediate interest may be made up fiom the calculated values for the 5-inch diameter round wave- guide used in waveguide experiments at Holmdel. Table V shows the ratio of attenuation coefficients for several modes in this fine. The circular electric wave TEoi has the lowest attenuation coefficient (absolute value * When using these relations, it is helpful to note that (ri + Tt) = -(a -f- 0) at all times. Hence (rj -|- a) = -(rj -}- /3) and (r^ + a) = -(r, -|- /3). 1264 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Table V — Ratio of Attenuation Coefficients for Several Modes in 4.732" Diameter. Waveguide at Xo = 3.0 Cm Modes Attenuation Coefl5cient Modes Attenuation Coefficient Ratio ax/a\ or axh/aVi Ratio axlai or Oxh/aih TE12/TE01 2.45 TE31/TE01 12.5 TE02/TE01 3.82 TE41/TE01 17. TEii/TEoi 4.56 TEsi/TEoi 21. TE22/TE01 4.63 TEei/TEoi 27. TEis/TEoi 6.61 TEn/TEoi 34. TE21/TE01 8.51 TEsi/TEoi 44. TMn/TEoi 10.78 TEsi/TEoi 61. TE03/TE01 11.3 TEio,i/TEoi 100. equal to 1.6 db/mile), and the ttxh/ciih ratios for the other modes range from 2.5 to 100 times the TEoi value. Table V represents a selection of the various modes which can exist in the 5-inch diameter line, but the number tabulated is not an indication of the density of the ratio of at- tenuation coefficients near a given value. Actually, there are eight modes having ttxh/cLih ratios in the range 2.5 to 10, 19 modes in the range 10 to 20, and 15 modes greater than 20. Experimental work reported elsewhere shows that significant con- version takes place between TEoi and the TEn , TE21 , TE31 and TMu modes. There is some likelihood that conversion to TE12 takes place, but the magnitude has not been measured. The experience gained by measurement, therefore, shows that most typical conversions occur be- tween TEoi and modes having attenuation ratios axh/am in the range 2.5 to 12. The absolute magnitudes of the conversion coefficients aix have in some cases been measured directly, and may also be inferred from meas- urements of total signal attenuation on the 500 ft. experimental line and separate knowledge of the heat-loss values; the inference is that a^ must fall in the range 0.1 to 1.0 au for the particular line studied. References 1. S. E. Miller and A. C. Beck, Low-Loss Waveguide Transmission, Proc. I. R E., 41, pp. 348-358, March, 1953. 2. S. E. Miller, Coupled-Wave Theory and Waveguide Applications, B. S. T. J., 33, pp. 661-720, May, 1954. 3. M. Aronoff, Radial Probe Measurements of Mode Conversion in Large Round Waveguide with TEoi Mode Excitation, presented orally at the March, 1951, I.R.E. National Convention. AVAVEGUIDE AS A COMMUNICATION MEDIUM 1265 4. M. Jouf^uct, I'ltTccts of the (Jurvaturo on the Propagation of Electromagiiolic Waves in (iuidcs of Circular (^ross Section (^al)l(>s and Transmission (Paris), 1,, No. 2, pp. i;» 153, July, 1947. 5. M. JoU{2;uet , Wave Propagation in N(>aily Circular Waveguides: Transniission- Over-liends Devices foi' 11 u Waves, CaMes and Trans (Paris) 2, No. 4, p]). 257-2S4, Oct., 1948. 6. W. J. Alherslieini, Propagation of TlCoi Waves in Curved Waveguides, 15. S. T. J., 28, pp. 1 :]2. Jan., 1949. 7. 8. (). Pice, unpublished work. 8. S. E. Miller, Notes on Methods of Transmitting the Circular Electric Wave ^. ^\. Around Bends, Proc. I.R.E., 40, pp. 1104-1113, Sept., 1952. . 9. A. C. Beck and R. W. Dawson, C'onductivit}' Measurements at Microwave t> ' Frequencies, Proc. I.R.E., 38, i)p. 1181-1189, Oct., 1950. 10. J. B. 1 jf tie, Ami)lification at 6-mm Wavelength, Bell Labs. Record, Jan., 1951. 11. S. Milhnan, .\ Spatial Harmonic Travelling-Wave Amplifier for Six-Milli- meters Wavelength, Proc. I.R.E., 39, pj). 1035-1043, Sept., 1951. 12. R. Konipfner, Backward-Wave Oscillator, Bell Labs. Record, Aug., 1953. R. Kompfner and N. T. Williams, Backward Wave Tube, Proc. LR.E., 41, pp. 1602, Nov., 1953. 13. A. Karp, Paper presented orally at the June, 1951, Conference on Electron Tube Research. 14. A. G. Fox, New Guided-Wave Techniques for the Millimeter-wave Range, presented orally at the March, 1952, LR.E. National Convention. 15. J. R. Pierce and W. G. Shepherd, Reflex Oscillators, B. S. T. J., 26, pp. 460- 681, July, 1947. 16. G. D. Sims, The Lifluence of Bends and Ellipticity on the Attenuation and Propagation Characteristics of the Hn Circular Waveguide Mode, Proc. Institution of Electrical Engineers (London), 100, Part IV, No. 5, pp. 25- 34 Oct., 1953 17. S. P. Morgan, Jr., Mode Conversion Losses in Transmission of Circular Elec- tric Waves through Slightly Non-Cylindrical Guides, J. Appl. Phys., 21, pp. 329-339, April, 1950. A Governor for Telephone Dials Principles of Design By W. PFERD (iManuseript received July 29, 1954) This is a report on the development of a new type of governor for regulating the speed of rotary dials. The paper includes derivation of the equations of motion which determine the theoretical speed of the governor during dial run-down and an analysis of the operating characteristics of the governor as influenced by vanjing friction and input torque. Experimental verification of the relations is presented. A theoretical analysis which explains "governor chatter^' or positional instability for friction-centrifugal governors is also given. INTRODUCTION Machine switching telephone systems depend on the telephone dial for originating information used in completing a call. During run-down, the dial originates current pulses which operate step-by-step switching etiuipment or are registered for use in common-control panel or crossbar systems. For reliable functioning of dial pulse controlled switching ecjuip- ment, the pulses must be closely controlled in frequency and form. Since the pulses are produced during run-down of the dial after release by the customer, the run-down speed most be constant. Friction-centrifugal governors are commonly used to provide this required control of speed. If the pulses reaching the central office were exactly like those gen- erated by the dial, the designers of dials and central office switching ap- paratus and circuits would find themselves far less restricted. Unfor- tunately the dial pulses are distorted by the electrical characteristics of the customer's loop. To compensate for this distortion and insure ac- curate registration of the pulses at the central office, the dial and central office efiuipment must be designed to operate to close limits of perform- ance. The designs must also be such that there is negligible change of 1267 1268 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 adjustment resulting from use, time or weather conditions, which might affect the abihty to accurately send or receive dial pulses. To achieve the accuracy of timing reciuired of dial governors, it was recognized that a general theoretical analysis which defined speed of governors would be beneficial. The late C. R. Moore investigated this problem in the thirties, and derived from theoretical considerations, general ec[uations of motion relating to governors. The relationships derived by the Moore analysis are extremely useful in that they can be used to indicate the influence of various design factors on the performance of a governor. This theory was applied in developing the new go^'ernor used in the 7-type dial of the 500-type telephone set* and will be pre- sented in this paper. To better demonstrate the operating characteristics of the new governor, it will be compared with a previous governor which was used in an older type dial. Photographs of the new governor as it is assembled in a dial are shown in Figs. lA and IB. Fig. 1A — Front view of 7-type dial. * Inglis, A. H., and Tuffnell, W. L., An Improved Telephone Set, B. S.T.J. , 30, pp. 239-270, April, 1951. A GOVERXOK FOR TELEPHONE DIALS 1269 DIAL AND GOVERNOR OPERATION In dialing, the tingerwheel is rotated through an angle proportional to the number being dialed and then released. Energy stored in the motor spring, Fig. 2, causes the mechanism to return to the start position. For each 30° rotation of the fingerwheel during run-down, the intermediate gear rotates one-half revolution and the cam pinion and pulsing cam ro- tate one full revolution. (3nce the pulsing pawl is in position, each revolu- tion of the pulsing cam in the run-down direction results in a pulse being placed on the telephone loop. The intermediate gear also meshes with a governor pinion which is coupled to the governor shaft and governor through a spring clutch.* This clutch decouples the governor from the fingerwheel on windup to reduce the windup torque. On run-down the go\'ernor rotates two full revolutions for each 30° rotation of the finger- wheel. Fig. IB — Rear view of 7-tj'pe dial. Left, speed governor, center, off-normal contact and right, pulsing mechanism. Wiebusch, C. F., The Spring Clutch, J. Appl. Alech., Sept., 1939. 1270 THE BELL SYSTEM TECHNICAI JOURNAL, NOVEMBER 1954 CONTACT SHUNTS (GOVERNOR RECEIVER DURING PULSING CONTACTS Fig. 2 left. Simplified diagram of 7-type dial mechanism; governor appears top As shown in Fig. 3, the weights of the new governor are free to pivot at the ends of the fly-bar which, in turn, is allowed to rotate with respect to the shaft. Rotation is imparted to the system by the drive-bar which presses against each weight at a specific point. As the mechanism begins to rotate during dial run-down, the two weights are caused by centrifugal force to move outward against the tension of a spring. Movement of the A GOVERNOR FOR TELEPHOXE DIALS 1271 weights about their pivots continues until the friction studs touch the case. At this instant governing begins, and controls the dial speed until the end of run-down. The speed attained by the governor will be de- pendent on the friction between the studs and case, the magnitude of the driving torque, and the tension to which the spring of the governor is adjusted. In the schematic of the new governor, Fig. 4, the driving force is desig- nated as F. By applying this dri^'ing force between the weight pi\-ot and the center of the go\'ernor, the mechanism behaves during operation as a true friction-centrifugal governor and also as a brake. This configura- tion results in a gain in the ability of the governor to resist the increase in speed which normally results from an increase in the applied rotational force. The drive-bar force, f, and the torque produced by the stud-to-case force about the pivot assists the centrifugal force in pressing the friction studs against the case. WEIGHT FRICTION DRUM SPRING DRIVE BAR GOVERNOR CASE Fig. 3 — The 7-type dial drive-bar governor. 1272 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 The comparative design shown in Fig. 5 is a conventional fly-bar type governor consisting of two weights each pivoted at the end of a fly -bar which is fixed to a shaft. As the shaft accelerates during run-down, the weights move outward under the influence of centrifugal force and are restrained in their motion by the tension of the go\Trnor spring. At a certain speed the friction studs contact the inner surface of the governor case. The governor gradually decelerates until the input torc[ue to the governor is balanced by the stud-to-case frictional loss and the governor shaft and dial theoretically rotate at constant speed. It will be noted in this configuration that only the torque produced by the stud-to-case GOVERNOR ,^ CASE RUN-DO\WN Fig. 4 — Schematic of drive-bar governor. F — Force applied by torque on governor weights Fn — Normal force of case acting on studs F s — Force exerted by spring when studs touch case Fm — Centrifugal force acting at center of gravity of weight /x — ■ Coefficient of friction /o — Moment of inertia of the governor about center shaft CO — Angular velocity of governor wo — Critical angular velocity at which studs just touch the case in — Mass of each weight ?'o — Radius to the center of gravitj^ of each weight r — Radius of governor case a — Stud angle Neg. Rotation — Run down of governor A GOVERNOR FOR TELEPHONE DIALS , —WEIGHT 1273 Fig. 5 — Fly-bar type governor. friction about the pivots aids the centrifugal force in pressing the studs against the case. EQUATIONS OF MOTION In deriving the general equations of motion for a governor, two as- sumptions are made concerning the action. During the interval of time that the go^'ernor is approaching the critical velocity when contact of the friction surfaces first occurs, the motion is assumed to be that of a simple fly-wheel, constantly accelerating. The angular velocity of the governor during the time from rest to the critical velocity, wo , is then given by 9!l I (1) where G = applied torque t' = time from start of motion / = moment of inertia of the governor assembly about the shaft 1274 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 After stud-to-case contact occurs, it is assumed that there is no further pivoting of the weights or fiy-bar, and the assembly rotates as a rigid body. During this time the general equation for angular velocity is CO = g tanh 0^^ + In \/a) (2) where q ^ regulated or final angular \-elocity h = design constant A = design and adjustment constant t = time measured from the moment of initial braking The deriv^ation of this general speed equation as it applies to the new 7-type dial governor is given in Appendix I. For this drive-bar type governor the following terms apply: j^rjG/e h - G(d - yc) + Myu)o~ Io{d — yc) 9 = My. hid - yc) A = _ g + coo g — coo (4) (o) ^ aA}= a /^(d — yc) -f Mycop- — yrjG/e /^x y g y My The derivation of the theoretical equation for speeds in excess of the critical speed for the comparative fly-bar design results in the following relationships: CO = g tanh ( - + ^n ^/A j where , G{d - yc) -\- Myoia .^. /o(a — yc) Io{d — yc) = /■ ^ _ . /<^(f^ - MC) + il/MCOo- (9) g y My The form of the general speed equation is the same for all types of A GOVERNOR FOR TELEPHONE DIALS 1275 friction-centrifugal go\'crnors but each particular governor will have different terms in the values for ^, h and q. The theoretical e(iuation of motion can be used to calculate the speed of the dial at any time, t, after the critical velocity is reached or the time required to reach any given speed once the governor studs touch the case. The equation shows that for large values of t, u approaches q, so that steady state speed is given by the value of q for each type governor, and is in terms of the operating ^'alues and design constants of the mechanism. For the drive-bar governor the steady state speed equation is 0} = q = i /^(^ ~ ^g) + ^Mcoo' - nrjG/e ^^^ Mn THEORETICAL SPEED-TIME CURVES Drive-Bar Governor The design constants and physical data given in Table I apply to the drive-bar governor, and were used to calculate the theoretical speed Table I — Refer to Figs. 3 and 4 d = 0.390 cm /o == 7.40 gm cm* (experimental) c = 0.361 cm G = 7,500 dyne-cm (steady-state)* r- = 1 . 180 cm 13,500 dyne-cm (initial) ro = 0.635 cm M = 0.25 (assumed) b = 0.920 cm m = 3.9 gms e = 0.498 cm K = To/r = 0.538 ;■ = 0.236 cm M = 2mr^bk = 5.38 * Appendix II — Governor Input Torque. versus time curve shown in Fig. 6. The dial governor is initially adjusted so that signaling is at the rate of 10.0 pulses per second, requiring a steady state governor shaft velocity of 125.6 radians per second. This steady state velocity was used to determine the critical velocity coo by substitut- ing the values in Table I in equation (10) : coo = 121.8 radians /second and from equations (3), (4), (5), and (6) g = 0.606 h = 9,520 h/q = 75.9 A = 72.7. 1276 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 S -J O _1 < Z < P OVER ETIC MEN O S a: CC UJ Q. < I X £D J— UJ lij ! > ! q: I a 1 1 a: < UJ CD t UJ ^:- v; ^^ 3 "^^ ^^ (E O z is o z 1 1 1 JOVER ORET ERIME 1 FLY-BAR ( THE EXP 1 < Ul tr (Q 4 UJ < 2 A < UJ a. ffl -i UJ < i < UJ (T CD J < i i 1 v ^ ^~^ ^ CO O) 6 II 3 X X ^^ ^ c\J Z d o qil G c3 P. QNOoas a3d SNviava ni a33ds aoNasAoo A GOVERNOK FOR TELEPHONE DIALS 1277 Table II / 7.S.9< + 2.142 w = 12S.6 tanh (75.9t + 2.142) 0.000 2.142 121.8 0.004 2.446 123.7 0.008 2.750 124.6 0.012 3.054 125.1 0.016 3.358 125.3 0.020 3.662 125.5 Substituting these values in the general speed eciuation, CO = g tanh { - -\- In \/Z ) gives the velocity of the dri\'e bar governor at any time t measured from the moment the governor reaches the critical velocity, i.e., when the friction studs first touch the inside of the governor case. (Table II.) For governor speeds from start of rotation up to the critical velocity, it is assumed that the system rotates as a simple fly-wheel, therefore from ecjuation (1) t' = ~g7 121.8(7.4) 0.0668 seconds 13,500 This time of 0.0668 seconds determines the slope of the straight line portion of the theoretical speed-time curve shown on Fig. 6. Fhj-Bar Governor The data given in Table III applies to the fly-bar governor shown schematically in Fig. 7. Substituting the values given in this table in the steady state speed eciuation for the fly-bar governor, (ji — q = -4 G(d - nc) + Mmcoq^ coo = 118.5 radians/sec. >nd from equations (7), (8) and (9): g = 0.609 h = 9,560 h/q = 76.4 A = 36.35 Substituting these values in the general speed equation gives the velocity of the fly-bar governor at any time (t) measured from the moment braking begins, Table IV. For this particular fly -bar design, the 1278 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 time required to reach the critical velocity is, from efjuation (1), / coo/o 118.5(7.36) nnrir J ' = G7 = 13,500 = °°''* ''"""^' experimental speed-time curves — normal torque Experimental velocity versus time curves were obtained for the fly-bar and drive-bar governors constructed to the specifications listed in Tables I and III. Data used in determining the true speed versus time picture for the experimental governors was taken from photographic traces ex- posed on a recording oscillograph. A thin disc having 36 radial slots spaced every 10° was fastened to the end of the governor shaft. Light detected through the slots of the moving disc by the element of a photo tube was used to deflect one of the strings of the oscillograph. The trace Table III — Refer to Figs. 5 and 7 d = 0.390 cm G = 7,500 (steady state) c = 0.361 cm 13,500 (initial) r = 1 . IS cm fi = 0.25 (assumed) To = 0.63.5 cm m = 3.9 gm b = 0.920 cm A' = ro/r = 0.5.38 7o = 7.36 gm cm2 M = 2mr%K = 5.38 of this string appeared on the photographic paper as a distorted sine wave. The distance between two successive wave peaks represented 10° of rotation of the governor. By noting on the trace the time between peaks, it was possible to determine the average velocity of the governor at 10° intervals after release of the finger- wheel, or start of rotation of the governor mechanism. The experimental speed curves for the drive-bar and fly-bar governors are plotted on Fig. 6 along with the theoretical speed curves. It was assumed in the theoretical analysis that while accelerating up to the critical velocity, the governor assembly rotates as a simple fly- wheel. This requires that the velocity increase linearly. The theoretical and experimental velocity curves for both type governors during the initial accelerating period show the fly-wheel assumption to be justified. The slope of the velocity curve, or rate of acceleration is generally con- stant. For that portion of the theoretical and experimental curves which show speeds from the critical velocity to 98 per cent of rated speed, agreement is not too clearly defined. The theoretical curve is naturally smooth in shape. An oscillating type characteristic appears in the experimental A GOVERXOR FOR TET.EPHOXE DIALS 1279 speed curves of both types of governors. This probably results from the shock and grabbing when the friction studs first touch the case and con- tinues initil the forces tending to move the weights outward increase to a value sufficient to hold them against the case for governing. That part of the experimental curve, during which governing actually occiirs at rated speed is relatively smooth through full run-down. Both the fly-bar and new drive-bar governors exhibit excellent speed regula- tion. The waves present on the trace do not necessarily indicate hunting or ^'ibration since the variations in speed which appear are actually smaller in magnitude than the degree of accuracy present in measiu'ing the experimental photographic trace. RUN-DOWN Fig. 7 — Schematic of fly-bar governor. Fn — Normal force of case acting on studs Fs — Force exerted by spring when studs touch case Fm — Centrifugal force acting at center of gravity of each weight IX — Coefficient of friction Zo — Moment of enertia of the governor about center shaft CO — Angular velocity of governor wo — Critical angular velocity at which studs just touch the case m — Mass of each weight Co — Radius to the center of gravity of each weight r — Radius of governor case a — Stud angle Neg. Rotation — Run down of governor 1280 THE BELL SYSTEM TECHNICAL JOURXAL, NOVEMBER 1954 THEORETICAL OPERATING CHARACTERISTICS The sokition of the e(iuatioii of motion for any governor is based on specific design constants and certain assumed and estimated values. Such dimensions as the governor case inside diameter r, and distance from the shaft center to the weight pivot h, are two examples of design constants. These constants establish the arrangement of the various component parts of the mechanisms and are, therefore, subject to practical manu- facturing and space considerations as well as considerations from a speed regulation standpoint. The design constants are in effect static considera- tions. They are not subject to appreciable variation once established, and on any particular governor do not vary significantly over the life of the governor. Table IV / 76.4/ + 1.798 0)= 125.6 tanh (76.« + 1.798) 0.000 1.798 118.5 0.004 2.103 121.9 0.008 2.408 123.6 0.012 2.714 124.5 0.016 3.010 125.0 0.020 3.325 125.3 0.024 3.631 125.4 During actual operation there are two factors which directly affect the degree of speed control afforded by a governor; i.e., the coefficient of friction which exists between the governor case and friction studs, and the value of input torque to the governor shaft. Design control over these factors is present, but to a lesser degree than for the design constants mentioned previously. These factors are considered fixed in arriving at a given design but actually vary from governor to governor and over the life of a governor. It is therefore necessary to consider carefullj^ the effect of variations in friction and torque if close regulation of speed is re- c^uired. The input torc^ue to the governor shaft vnW vary because of the dimen- sional variations of the motor springs, the tolerance permitted for the driving torque at full windup of the dial, dial friction and the variation in pulsing spring forces. These variables appear at the governor as a range of input torques during iiin-down of the mechanism. For the motor spring used in the 7-type dial, input torque referred to the governor shaft decreases during run-down on the average from 20,000 dyne-cm to 13,000 dyne-cm. Torque required to overcome bearing and gear fric- A GOVERNOR FOR TKLIOPHONE DIALS 1281 tion and the loads imposed by the i)ulsing mechanism result in an average torciue oi 7,500 dyne-cm at the governor. Over the life of a dial this input torcjue at the governor will vary as the dial efficiency varies. Initially the dial mechanism is lubricated and the bearings and gears turn freely. With time and continued operation, the accumulation of dirt and wear products affect the dial so that more torc^ue is needed to dn\-e the mo\-ing parts. This causes a decrease in the remainder toi-(iue going to the gON'ernor. Another aspect of torciue riMiuiring consideration is that resulting from forcing of the finger-wheel during run-down. This action can produce torque values at the go\'ernor of the order of 1 10,000 dyne-cm or a tortiue of approximately fifteen times that which appears at the governor during normal operation. The second factor, which can vary during dial life, is the value of the stud-to-case coefficient of friction, fx. Both the drive-bar and fly-bar governors have studs of Ebonite compounded with 40 per cent by weight of hard rubber dust and cases of ASTM B16 brass. Actual service tests show the satisfactory wearing ability of these materials. Each governor is initially adjusted for speed by changing the tension of the governor spring. At the time this adjustment is made, a particular friction condition exists between the governor studs and case. With time or continued operation there is always the possibility of a change oc- curring in this friction value. Such factors as very high humidity, lubri- cation products traveling to the stud operating region, or the accumula- tion of foreign-material or wear particles may produce different values of friction and hence result in variation in governor and dial speed from the initially adjusted value. The range of coefficient of friction values (jl expected for rubber on brass is from 0.05 to 0.35. These are the extreme conditions produced by oil in the governor case for the 0.05 value and very low unit pressure on a scored brass surface for the 0.35 value. For this study a representative figure for the average stud-to-case friction value was taken to be 0.25. The problem of variation in steady state governor speed with changes in the coefficient of friction and mput torque can be analyzed by con- sidering the derivatives of speed with respect to these values. This is done by operating on the equations for terminal speed. Speed with Respect to Coefficient of Friction For the drive-bar governor, the partial derivative of speed, with respect to coefficient of friction, is as follows: 1282 THE BELL SYSTEM TECHNICAL JOURXAL, XOVEMBER 1954 Taking the derivative of w with respect to /x gives ^ = __Gd_ dn 2wMm2 where M = 'Imrrob. For the fly-bar governor, dco/dij., is as follows from (11) CO = q = /j/' G{d - nc) + MfjLOJo'^ Mfx (12) Sco ^ __Gd_ dn 2coikf/i2 Since the equations are identical for both governors the same considera- tions exist in holding to a minimum the change in speed caused by a change in the coefficient of friction. For optimum speed regulation, the partial derivative, du/d^, should be a minimum. Small values of du/dn can be obtained by operating on the design constants, controlling the value of n, or ha\'ing high governor speeds. Specifically, the design constants m, ro, r and h should all be large. Inspection of the drive-bar governor schematic. Fig. 4, shows that there are physical limitations to the arbitrary enlargement of these values. Space, manufacturability, and cost of materials must be consid- ered in establishing these terms. In selecting materials for fixing the coefficient of friction value the wearing quality of the materials to be used must be of first consideration. A high governor speed is advantage- ous but must be weighed against the primary disadvantage of high inertia loads for the entire mechanism. The temis G and d in the numerator of equations (11) and (12) indicate that low input torque and a small value for d would be desirable For G, one must consider the anticipated change in dial efficiency and variation in torque produced by the pulsing mechanism plus the torque necessary for the governor to maintain adequate speed control. As shown in the drive-bar governor schematic. Fig. 4, d is the dis- tance from the weight pivot to the noraial of the point of contact be- tween the rubber stud and governor case. Its magnitude is controlled by the stud angle a, and the distance between the stud and case when the weights are in the closed position. As indicated, d should be as small as possible for best regulation with friction change. This requirement imposes a difficult design problem because as d decreases, the stud and bearing hole in the weight approach A GOVERXOR FOR TELEPHONE DIALS 1283 each other. A minimum d is therefore fixed by interference of the parts themselves. Inspection of the governor mechanism also shows that even if the interference problem were not Hmiting, the weight turning angle imposes a further restriction on the (/ value. The life of a governor is considered to end when the friction material is worn to the point of allowing the weight to touch the inside of the case. Because of this initially large weight motion, little material is provided for wearing and hence the possible life of the mechanism is reduced. In the design of the drive-bar governor, the stud angle, a, was made as small as manufactur- ing techniques would permit. The stud angle is 22° from the weight piA'ot with a corresponding d = 0.390 cm. Since both governors under study were designed to operate in the case of the same dial the terms shown in Tables I and III which apply in the friction equations (11 and 12) are identical. This identity of terms indicates that both type governors should exhibit identical frictional characteristics. Fig. 8 represents a plot of the dw/dii for ^^arious governor input torque values. One set of curv^es applies. The range of torciue values covered by the curves is from zero to the forcing condition at fifteen times normal motor spring torciue. Curves for coefficient of friction values, 0.05, 0.10, 0.20, 0.35, are shown as encompassing the extremes in operating range. Speed with Respect to Input Torque For the drive-bar governor, the partial derivative of speed with respect to torcjue is determined as follows from the steady state speed equation. G{d - ijlc) + Mmcoo^ - firjG/e CO — Q — ' ' Mil Taking the derivative of w with respect to G gives: ^ = 1 dG 2 G{d — iic) + ilZ/xwo" — nrjG/e and doi 1 {d — fjLC — fjLTJ/e) {d — ijlc — fJirjG/e) (13) dG 2 MfjLu where M = 2 ynrrob For the fly-bar go\ernor the deri\-ative of co with respect to G can be 1284 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 developed similarly to give do3 dG lid 2 fxc) MfjiO} (14) Here also the design constants, 7n, ro , r and h, the coefficient of fric- tion n, and the governor speed co, should all be high values. Minimum change in dial speed would then occur for a given change in the input torque. Inspection of the equations for do)/dG, indicates that it is possible to have perfect torque regulation for at least one value of n. For this limit- ing condition, doi/dG, would equal zero and equation (13), for the drive- bar governor, equates to d — fxc — ixrj/e = 0 (15) and equation (14) for the fly-bar type governor d — fxc = 0 (16) If these equations were satisfied, there would be zero change in speed for a given change in input torc^ue to the governor. As stated previously n is predetermined and has in this design a maximum known value of 0.35. A margin of safety is considered by taking n = 0.425 for the limit- ing case and equation (15) becomes for the new drive-bar governor d - 0.425 (c -f rj/e) = 0 and equation (16) for the fly-bar governor d - 0.425 c = 0 (17) (18) dco dG 12 10 8 6 4 2 n 1 1 1 1 do; Gd ^ 1 1 1 =1 2a»M;u2 ^^ ^ ^^ IP i! ^ Itr lo 2i 1 ^ ^^ IZ Ix 1 1 1 ^ 0.10 1 1 ^ 0.20 0.36 1 1 ■H H 20 30 40 50 60 70 80 90 100 GOVERNOR INPUT TORQUE IN DYNE CENTIMETERS 120 XlO^ Fig. 8 — Derivative of governor speed with respect to coefficient of friction versus input torque to the governor. A GOVERNOR FOR TELEPHONE DIALS 1285 1.2 !2o.8 FLY- BAR TYPE GOVERNOR d AND C VALUES DETERMINED BY GOVERNOR CASE AND FLY-BAR / 1 /Ot = 120» \ ^=51 / L ^^ / ^^ ^ ~- ^ <1 a = 90° bP .osV ix \ a = 2 V \ \ ^ ^ / H-0 Z5C^ \ — J % -r 06 = 160^.^ V ^ ^ ^^^ ^^ ^ /- ^ d-0.05C = 0 \ L. 0.8 1.0 1 1.2 1.4 C IN CENTIMETERS 2.2 ^- FLY-BAR ARM LENGT (0.920 CM) r = 1.18CM Fig. 9 — Design diagram for fly -bar type governor. Comparison of these two equations shows a very important differ- ence in the term multiplied by [x = 0.425. For the drive-bar governor, there are four variables which can be operated on to satisfy the eciuation; i.e., c, r, i and e. For the fly-bar governor only c is available. The im- portance of these additional terms can be realized when one considers that the value for c results from our choice of d in making dw/dyi a mini- mum. For both type governors c is equal to 0.361 cm. Substitution of the d and c values in the limiting eciuation, (18), for the fly-bar governor does not lead to a solution when p. = 0.425. Solving for this limiting ju in the fly-bar governor gives (0.390) - (0.361) M = 0 /x = 1.08 This value of ju is far beyond that encountered in actual governor operation and, in effect, represents useless margin. This is graphically shown in Fig. 9 where all d and c values which conform to the geometry of the fly -bar governor mechanism are plotted as a design diagram. The three straight lines radiating from the origin represent plots of the limiting equation for ju = 0.425, 0.25 and 0.05. The intersection of these lines ^\'ith the d and c semicircle, noted at points 1, 2 and 3, give the particular angle at which the stud should be located for optimum regula- 1286 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 DRIVE-BAR TYPE GOVERNOR d AND C VALUES DETERMINED BY /GOVERNOR CASE AND FLY-BAR , / / /^ ■^ \ 1 VC= 50° / >< \ ^ \^-oJ^ ^ a.= i2oV . ^ a=22 La/] X^-f^ ^j' ^ « =160°- T 1 ^^ \ ^ r 4- ^A ^--^ " d- 0.05C = 0.028 1 0.2 [ 0.4 0.6 0.8 1.0 1 1.2 1.4 1.6 1.8 2.0 2.2 C IN CENTIMETERS L. U- _FLY-BAR ARM LENGT (0.920 CM) -J Fig. 10 — Design diagram for drive-bar type governor. tion for these particular friction values. The point A on the d and c circle denotes the d and c values which result from the stud being at a. = 22°. The off-set appears because stud-to-case contact is not made on the center line of the stud. The points 1, 2 and 3 are all below the point, A, established by the minimum pennissible angle, a = 22°. This indicates that the fly-bar governor has its dw/dG equal to zero at some coefficient of friction value higher than /x = 0.425. As previously deter- mined in this value is ju = 1.08. Fig. 10 represents the design diagram for the new drive-bar governor. Here again, the large semi-circle is a plot of all d and c values which con- form to the geometry of this governor mechanism. Point A is determined by the stud angle a = 22°. The straight lines represent the limiting equa- tions for jLi = 0.425, 0.25 and 0.05 and are shown intersecting the d and c circle at 1, 2 and 3 respectively. For this particular governor, the line representing ju = 0.425, d - 0.425 (c 4- rjfe) = 0 intersects the d and c circle at the point A . This is possible by making the term (nrj/e) equal to 0.238. The intersection of this curve at A indicates that there will be zero change in speed for a given change in input torque to the governor when the stud-to-case coefficient of friction value is 0.425. A GOVERNOR FOR TELEPHONE DIALS 1287 The additional terms, r, j and e in the limiting equation make it pos- sible to design the governor for optimum regulation for any particular value of n desired. Since the term r, the case inside radius, is controlled by space reciuirements, the terms j and e assume added importance. They are determined by the point at which the drive-bar arms act against the weights and can be made anj^ value required to meet the design objec- tive. I Figure 11 is a plot of doo/dG at various values of /x for the drive-bar and fly-bar governors specified in Tables I and III. It indicates that for any coefficient of friction the new drive-bar governor should exhibit less change for a change in input torcjue than is possible with the fly-bar governor. EXPERIMENTAL DATA To substantiate the theoretical conclusions drawn from the analysis of the steady state speed eciuations, experimental data were compiled from a nvimber of models of each type of governor. Drive-bar and fly-bar governors, made to the specifications listed in Tables I and III were investigated to determine their response to changes in input torque and changes in the stud-to-case coefficient of friction value, m- The tests were conducted on 7-type dials manufactured by the Western Electric Com- pany as standard product. Dial Speed Versus Coefficient of Friction The theoretical analysis of the equation doi Gd indicates that the two types of governors should exhibit identical fric- tional characteristics. Fig. 12 represents the theoretical plot of dial speed in pulses per second versus coefficient of friction. The single curve satisfies both the fly-bar and drive-bar governors as specified in Tables I and III. This cun-e shows the change in speed of a dial initially adjusted to 10.0 pulses per second, operating at normal torc^ue, as the coefficient of fric- tion varies. It indicates that if there is a decrease in the value of n from that which existed at the time of initial adjustment, there will be a cor- responding increase in the dial speed. The curve is drawn with m = 0.25 as representing the normal stud-to-case condition and a normal governor input torque of 7,500 dyne-cm. The experimental data were compiled for the following operating con- 1288 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 5.0 3.0 2.0 1 DRIVE- -BAR TYPE: r\/p)] ec 20jUfjL FLY- BAR TYPE: do; (d->uc) I w V FLY -BAR ^'GOVERNOR \ .^' DRIVE-BAR GOVERNOR <^ ^ \ \ 0.5 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 COEFFICIENT OF FRICTION, /y Fig. 11 — Derivative of governor speed with respect to governor input torque versus coefficient of friction. ditions: as received, governor case cleaned with acetone, damp atmos- phere, and SAE 10 petroleum base oil in the case. Each of the governors were initiallj' adjusted to 10 pulses per second with the governor cases in the "as received" condition. The gov^ernors were then removed and the cases and governor studs were cleaned with acetone. The governors were then reassembled and the new speed recorded. During this procedure extreme care was exercised so as not to disturb the governor spring ad- justment. Speed was next recorded for the damp atmosphere condition, and finally for the condition with one drop of SAE 10 oil in the governor case. For these last two conditions the governors were not removed from the dials. The average speeds recorded for the four conditions are plotted on the theoretical speed curve of Fig. 12. The points were arbitrarily placed on the theoretical curve. Xo attempt was made to determine the exact co- efficient of friction values corresponding to the four conditions. In this A GOVERNOR FOR TELEPHONE DIALS 1289 respect, the speed attained with oil in the case shows a value of /x = 0.06 which is very close to the ^l = 0.05 taken as the lower limit. To produce a decrease in governor speed one must increase the \aliie of /x. This is difficult to do on a controlled basis, since it is brought about by the progressive action of wear particles and foreign material scoring the surface of the brass case during the life of the go\'ernor. The the- oretical analysis indicates that when the coefficient of friction increases to ju = 0.35 the speed of the governor w^ill decrease from 10.00 to 9.65 pulses per second. This assumes that there would be no speed change due to wear in any part of the governor. In practice of course the gover- nor studs Avear as the case is scored and, therefore, are made progres- sively shorter. For this condition the increased outward motion of the weight required for stud-to-case contact produces an increase in the spring force, Fs , acting on the weights. The speed change which results from increasing the spring force is in the direction to compensate for the loss in speed due to increase in coefficient of friction. Therefore, con- sidering only stud friction and wear as effective in causing change in speed, generally the governor and dial speed should increase from its initially adjusted value during life. It can be concluded from the experimental and theoretical evidence that there is the possibility of a speed change due to varying coefficient z o ,"r! 13 D- 12 DRIVE- BAR FLY -BAR GOVERNOR GOVERNOR o EXPERIMENTAL ' SAE 10 I ^^IN CASE \ DAMP ATMOSPHERE K CASE ^..CLEANED . NORMAL SPEED ' ^ 1 '\ AS j RECEIVED 0.10 0.15 0.20 0.25 030 03b COEFFICIENT OF FRICTION, ju 0.40 0.45 Fig. 12 — • Dial speed versus coefficient of friction. 1290 THE BEIL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 of friction to values between 9.65 and 12.0 piils(;s per second during life for dials initially adjusted to 10.00 pulses per second. An increase in speed may result from a decrease in stud friction due to the existence of lubri- cant or high humidity in the stud operating region or, the speed may decrease due to high friction. The above changes represent the extremes in dial speed determined solely by reaction of the governor to change in stud friction. In practice it is anticipated that dials adjusted to 10 pulses per second initially can vary from 9 to 11 pulses per second during normal usage. A reduction in speed will occur as more torque is required to compensate for the in- crease in bearing friction caused by the accumulation of dirt and wear products during ordinary life. This additional bearing drag will cause a decrease in the torc^ue available for governoring and therefore the dial will be regulated at reduced speed. For extreme cases of wear and con- tamination, it is of course possible that the dial will stop altogether dur- ing run-down. Such cases are not controllable by the governor. They result from the expected attrition during extended life or unfavorable environment. To guard against excessive increase in dial speed from the value at time of initial adjustment, precaution is taken during manufacture. As stated previously, lubricant traveling into the governor case after initial adjustment will cause a marked increase in dial speed. To avoid this sort of contamination, the governor case is washed after machining and swabbed with clean chamois prior to insertion of the governor onto the shaft. Care is also taken to see that no lubricant is placed in the governor case during lubrication of the shaft bearings. These practices assure that initially the friction surfaces are relatively free from contamination. The increase in dial speed up to the 11 pulses per second possible during dial life will result primarily as a result of stud wear, increase in efficiency of the mechanism, and operation during periods of high humidity. Dial Speed Versus Governor Input Torque To substantiate the theoretical conclusion that the drive-bar governor should exhibit better regulation due to changes in input torque, experi- mental data were compiled on the dials equipped with the two type governors. The results of this test are plotted on Fig. 13, along with theoretical forcing curves for both governors. A coefficient of friction value of 0.25 was assumed in arriving at the theoretical curves. The dials were initially adjusted to 10.00 pulses per second by bending the governor spring to have proper tension. Loads of 1, 3 and 5 lb were A GOVERNOR FOR TELEPHONE DIALS 1291 r ^,^ rr tU 1 ^-«- |UJ 13 0^ Oi t- 1 FLY- BAR GOVERNOR H ^^ „ - -1 0^1 ^ ^ DRIVE -BAR GOVERNOR < Oi ^ -^ |Z 1 X JL^ ^ "^^ ^ NOR MAL SF EED HEORETICAL XPERIMENTAL 0,A E 20 30 40 50 60 70 80 90 100 GOVERNOR INPUT TORQUE IN DYNE CENTIMETERS 120 XIO-' Fig. 13 — Dial speed versus governor input torque. suspended from the fingerwheel at %" radius and released. Average experimental speeds were recorded for the three forcing conditions and are noted in Fig. 13 for both the fly-bar and the new drive-bar governors. Good agreement, between the theoretical and experimental values, is evident. For a forcing condition of fifteen times normal motor spring torque, an average speed of 15.6 pulses per second is shown for the fly-liar governor Avhile an average speed of 13.4 pulses per second is noted for the drive-bar type. Theoretically the speeds should be 15.3, and 13.2, respectively. This type agreement is also present for the 1 and 3 lb forcing conditions, and therefore, it may be concluded that for any input torque resulting from forcing the fingerwheel a dial equipped with a drive-bar governor will exhibit less speed increase than one having the flj^-bar governor. The theoretical analysis indicates that for the torque available during normal rundown, drive-bar governors as specified in Table I will de- crease in speed from 10.00 to 9.80 pulses per second and fly -bar governors as specified in Table III will decrease in speed to 9.70 pulses per second. These theoretical speed changes were checked experimentally hy record- ing on a rapid record oscillograph a trace of the make and break times of the pulsing contacts during rundown of the dial from digit zero. This information was used to determine the average dial speed in pulses per second for each seciuence of make and break times. Actual loss in speed from the first to the ninth pulse for dials equipped with drive-bar gover- 1292 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 nors averaged 0.24 pulses per second while dials with fly-bar governors decreased 0.33 pulses per second. The slight additional loss in speed noted experimentally was probably due to friction in the gear mechanism which is not considered by the theoi'etical analysis. However, since the differences recorded are quite small, one can conclude that the theoretical and experimental results are in good agreement even when concerned with small changes in torque experienced during normal rundown of a dial. CHATTER IN GOVERNORS It is not uncommon for governors with fine regulating ability to produce an objectionable chattering noise when operated near or at the vertical position. This chattering, while in most cases not particularly adverse from a regulation or wear point of view creates in the mind of the lis- tener grave doubts as to the correctness of the design. In severe cases a sharp noise is heard during every half revolution of the governor shaft as each weight alternately leaves the case and strikes against the end of the other governor weight. During every revolution of the governor shaft, each weight is alternately supported as show^n in the schematic. Fig. 4. At this instant, the gravity moment about 5 is a maximum, and along with the spring moment, opposes the centrifugal moment. If the gravity component, or effective mass of the weight, is sufficiently large, a new system is produced which has a critical velocity in excess of the regulated speed. Since the governor speed is continually regulated by the bottom weight at a speed lower than the new critical velocity, the top weight falls from contact with the case. The magnitude of the gravity component is a function of the angle at which the governor operates and is a maximum when the dial is in the vertical position. As the operating plane of the dial decreases to the horizontal, the gravity effect decreases to zero. Chatter will not occur when the operating angle produces a gravity component smaller than the difference between the centrifugal force and the spring force. Since the chattering effect is the result of a balance of forces on the governor, it is apparent that a relationship can be derived which ^\^ll express the effect in terms of governor constants. This derivation is given in Appendix III and shows the chatter equation for a conventional fly-bar governor to be .m S fW-^ (19) 2rn(, sui /? A GOVERNOR FOR TELEPHOXE DIALS 1293 This expression must be satisfied if the governor is to operate free of chatter. By substituting the constants for the fly-bar governor, in Table III, we have for this governor operating in the vertical plane ^QrQfin^ < 7,500(0.390 - (0.25) (0.361)) ^•^^^^^^ - 2(1 18) (0.25) (1.092) or 3,820 dyne-cm g 2,790 dyne-cm. Since the equation is not satisfied instability should be present and governors of this fl\'-bar de.sign do chatter loudly when operating in the vertical plane. The chatter equation for the fly-bar governor indicates that by adjust- ing the design constants, one can eliminate the instability effect. This is true. A set of values could be used which would result in a fly-bar gover- nor which operates free of chatter. Unfortunately such a governor would also have reduced ability to govern. The relationship between chatter and governing is explained as follows.^! The equations which define changes in governor speed with respect to changes in friction and torque for the fly -bar governor ^ = ^^ dy. 2uMfx^ and dco _ 1 (d — fic) dG ~ 2 Mmco and the chatter equation, (19), show that operation without chatter and good speed regulation are totally incompatible. Those terms in the equations which should be small for good speed regulation; i.e., torque G and stud location d and c, should be large to avoid chattering of the governor. Those temis which should be large for good speed regulation; i.e., case radius r, friction n, and the distance from the pivot to the center of gravity I, must be small for no chatter. As the theoretical analysis indicates there is no term in the fly-bar governor chatter equa- tion which can be operated on to eliminate chatter without impairing regulation of speed. A similar analysis, given in Appendix IV, shows the chatter equation for the new drive-bar governor to be < G(d - MC - yrj/e) (20) 2rti€smfi 1294 THE BELL SYSTEM TECHXICAL JOURXAL, XOVEMBER 1954 This expression must be satisfied if there is to be no chattering during operation. A comparison of this ecjuation with that determined for the fly-bar governor, equation (19), shows an additional term (nrj/e). Sub- stitution of the design constants for the drive-bar governor given in Table I leads to the following. , . < 7,500(0.390 - (0.25)(0.361) - (0.25) (1.18) (0.236)7(0.498)) rf.y^y»Uj = 2(1.18)(0.25)(1.092) or 3,820 dyne-cm ^ 1,862 dyne-cm. This indicates that instability should be present in this governor, and that the additional term (nrj/e) causes a greater difference between the mass term and the torque tenn, than for the conventional fly-bar gover- nor. This is to be expected, since for the drive-bar governor also, adequate speed regulation, and a design w'hich has no chatter, are totally incom- patible. The sensitivity of speed to torque change and changes in co- efficient of friction, dd) _ 1 (d — fxc — ijLVJ/e) dG ~ 2 McoM ~ and ao) Gd dfx 2Mo}fi were made as small as possible for best regulation and this results in a small value to oppose chatter. This chatter analysis indicates that a new approach in design is needed in order to provide a governor which will operate without excessive noise and still regulate speed as required for use in the 7-type dial. This is found in a design which prohibits rapid movement of the governor weights during the unstable period. Referring to the assembly drawing of the drive-bar governor, Fig. 3, which shows the governor in the rest position, one can see that each governor weigh rests on the end of one of the arms of the drive-bar. By supporting the weights in this manner the following two beneficial effects are achieved. One, during operation in new assemblies, the weights mov^e outward to touch the case only a nominal distance of 0.007". This small allowable motion in the drive-bar governor results in a low velocity of the weight at closure and hence, less impact noise. Two, because the drive-bar presses to rotate the governor weights, impact occurs as the A GOVERNOR FOR TELEPHONE DIALS 1295 unstable weight skids against its arm of the drive-bar. This introduces fri(!tion damping to still further reduce the noise on closure of the weight. Some additional damping is provided by making the drive-bar and weight of powdered metal rather than wrought material. These features, which result from the particular physical arrangements of the component parts make possible the relatively quiet operation of the drive-bar governor. Experience with dials eciuipped with drive-bar governors indicates that they are effective since the noise due to chattering has been satis- factorily reduced so as to not be objectionable. SUMMARY This study has carried forward the work of C. R. Moore by presenting the derivation of theoretical equations which define speed for the drive- bar type governor. Design considerations necessary for optimum speed regulation indicated by the theory were applied in establishing the shape and working relationship of various components in the drive-bar gover- nor. Governors constructed to these dimensions have operated as forecast by the theory. The excellent agreement between theory and practice indicates that it is both desirable and practicable to apply the Moore theory in the design of governors. The initial requirements imposed on the design of a governor for the 7-type dial were two-fold. The new governor had to provide speed regu- lation at least equal to the conventional fly-bar type and no objectionable noise could be created by the governor during operation. To better under- stand the reasons for noise in governors, suitable theory was developed for investigating this phenomenon. Application of this theory to any type governor results in an equation which defines chatter in terms of the constants of the governor. This equation and the ecjuations deter- mined by the Moore theory for speed regulation indicate the existence of an interrelationship between speed regulation and noise in governors. The theory indicates that noise free operation and good regulation are totally at variance. The fly-bar governor supports this conclusion since this governor having fine control of speed also produces a chattering noise during operation. To satisfy both requirements, good regulation and quiet operation, it is first necessary to design a governor which will regulate properly and secondlj^, if the constants selected indicate that chattering will occur, prohibit excessive noise by providing means for restraining the system during the unstable period. This method of attack was taken in the design of the new drive-bar governor. By applying the Moore theory, a governor was developed for 1296 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 use in telephone dials which is an improvement over the conventional fly-bar type governor. Fine regulation is provided by the drive-bar governor for a given change in the coefficient of friction between the studs and case. This is achieved by locating the studs as close to the weight pivot as manufacturing techniciues will permit. Improved speed regula- tion is provided for varying input torque in the new governor as com- pared with the fly-bar governor. The experimental data shows the new de- sign able to control speed approximately twice as well. It is an effective non-forcing governor, prohibiting excessive increase in dial speed as a re- sult of forcing the fingerwheel during rundown. This nonforcing feature is achieved by applying the driving torcjue to the weights at a point to develop a moment about the weight pivot. This drive-bar moment assists the centrifugal force in maintaining pressure of the friction stud against the case for friction governoring. Having established a design which provided the degree of dial speed regulation considered necessary, it was then possible to investigate the second requirement of noise free operation. Application of the chatter theory to the drive-bar governor indicated the design to be unstable. This situation was controlled by using the ends of the drive-bar to limit the fall of the governor weights. This configuration of parts allows only small movement of the weights during the unstable period and provides damping as the weights close on the arms. The small motion and friction damping in the assembly results in a governor which is relatively free from noise during operation. Experience with dials ecjuipped with drive- bar governors indicate that the chattering effect has been controlled. ACKNOWLEDGMENT The unpublished work of C. R. Moore covering the theory of fly-ball, fly-bar and band type governors has served as a foundation for this paper and the experimental work of R. E. Prescott aided considerably in determining the final drive-bar governor design. The writer also wishes to express appreciation to Mr. Prescott and H. F. Hopkins for their helpful comments and suggestions during preparation of this paper. Appendix I DERIVATION OF THE DRIVE-BAR TYPE GOVERNOR SPEED EQUATION Referring to Fig. 4, as the governor mechanism rotates in a clockwise direction each weight tends to move outward vuider the influence of centrifugal force and the torque force, F. The centrifugal force, Fm , A GOVERNOR FOR TKLEl'HOXE DIALS 1297 acts through the center of gravity of the weights radially from the turn- ing center of the governor shaft. The torciue force, F, is applied on the weights by the drive-bar arms. These forces are opposed by the spring force, Fs . A stud-to-case force, F„ , and a frictional component of this force, fiFn , act on the weights when the friction studs are in contact with the case. In deriving the eciuation of motion for speeds in excess of the critical velocity the following symbols will be used as noted on Fig. 4. F — Force applied by tonjue on governor weights F„ ■ — ■ Normal force of case acting on studs Fs — Force exerted by spring when studs touch case Fm — Centrifugal force acting at center of gravity of each weight ju ■ — ■ Coefficient of friction 7o — Moment of inertia of the governor about center shaft u — Angular velocity of governor m ■ — Critical angular velocity at which studs just touch the case m — Mass of each weight ro ■ — Radius to the center of gravity of each weight r — Radius of governor case a — Stud angle Neg. Rotation — Rundowai of governor From the schematic. Fig. 4, taking moments about B we have FJ) - F,h + nF„c - FJ-^ F = 0 (1) collecting terms b{F^ - Fs) - Fnid - Mc) + Fj = 0 b{F^ - Fs) + Fj (2) F = (d — ixc) The driving torque on the governor is G^ = 2Fe; the retarding torciue, 2nF„r. The difference between the driving torque and the retarding torcjue is as follows: ho: = G - 2nFnr (3) where h = Moment of inertia of the governor about the shaft center CO = Angular acceleration about shaft center equating F^ = ^^_^ (4) 2nr 1298 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Combining equations (2) and (4) and solving for the acceleration, d), 6(F^ - F.) + Fj ^ G- I<^ (d — nc) 2nr 2Fmbnr 2iFsb - Fj)nr ., ^ . (d — nc) (d — fxc) or G 2FJ)txr , 2{Fsh - Fj)nr CO + /„ /(i(f/ — nc) Iu((l — nc) Substituting the following values for F,„ , F^ , F F„, = mcoVo Fs = mwlro then F = ^ 2e G 2mw~rtthixr . 27ncoo'robur jurG oi = T — 1-n ^ + /o /o((i — ixc) Io{d — nc) Io{d — nc)e Substituting for the design constants K = To/r M = 2mr%K G Myoii , il/ucoo arjG 0} = -7- — -i-n ^ + /o Io{d — lie) U{d — /ic) U{d — fic)e or . Mil 2 G , MiJL 2 nrjG .,. '^ + TTTt \ <^ = 7- + 7-7:1 v ''^o — 7-— ^ (5) io{d — lie) U Io{d — IXC) hid — iic)e This is of the form doi 2 7 ^ + SCO = A or dw h — g(xP- A GOVERNOR FOR TELEPHONE DIALS 1299 Separating variables and integrating* t = -V Ln ^-- — + c where o" = - (0) 2/1 q — CO g Applying the initial conditions to solve for the constant c CO = co(i at f = 0 q ^ g + coo c = - -^ Ln 2/1 g — ^too Substituting in etiuation (G) / - '^ T ,. g+ ^ 9 T .. g + ^o f — —- Ijh — ~Y i^n 2/i q — (ji 2h g — coo Letting Then q — m t = ^ Ln 1 ^^+^^ (7) 2/1 .4 (q - co) ^^ and = q tanh f — + Ln y/A ) (8) Eciuation (8) applies as the equation of motion for the drive-bar governor for speeds in excess of the critical speed when /o(rf — IXC) j^ ^ Gjd - (xc) + Mnwo^ - nrjG/e , . Io{d — nc) h ^ ^ /Gid - nc) + Mho' - nrjG/e ^^^ g V MfjL Appendix II GOVERNOR INPUT TORQUE In order to apply the theoretical speed eciuations and the chatter equations de\'eloped for governors one must determine suitable values * Short Table of Integrals, Pierce, B. O., pp. 8, No. 50. 1300 THE BELL SYSTEM TECHXICAL JOURNAL, NOVEMBER 1954 for the stud-to-case coefficient of friction, ju, and governor input torciue, G. Experimental evidence indicates a m of 0.25 exists during normal operating conditions for hard rubber on brass. The values for governor input tor(iue given in Tables I and III and used in the theoretical analy- sis for the governors were determined as follows. The initial torcjue applied to the governor for the period up to the critical velocity was calculated from oscillograph string traces. These traces were obtained by mounting on the end of the governor shaft a thin disc having 36 radial slots spaced uniformly about the circumference. Light, detected through the slots of the rotating disc on the element of a photo tube, appeared as a distorted sine wave on the photographic paper. The distance between two successive wave peaks represented 10° of rotation of the governor. By noting on the trace the time between peaks, it was possible to determine the average velocity of the governor at 10° intervals after release of the fingerwheel, or start of rotation of the gover- nor mechanism. The complete plot of these velocities appear as the experimental speed curve on Fig. 6. Inspection of the experimental curve for the drive-bar governor shows constant acceleration immediately after release. This appears as a straight line in the velocity time cun^e. Using the slope of this line and the moment of inertia, Jo = 7.4 gm cm , the initial governor torque was calculated as follows: ^ J, h 100(7.4) .oA^f^A Gi = Ico = -J = — — — = 13,480 dyne-cm. t 55 The governor torque value during normal rundown was found by first determining the governor stud-to-case force, F„ , and using this value in the equation G = 2nFnr The fly-bar governor mechanism was used to determine the F„ force since the moment equation for this type governor contains measurable values. Referring to Fig. 7, the schematic of the fly-bar governor, the moment equation about B is as follows: F„,b - F,h - Fnd + nFnC - 0 To solve this equation for F„ one must determine the centrifugal force, Fm , and the spring force, Fg . Using electronic flash equipment with an exposure time of Ho,ooo of a second, it was possible to take distortion free photographs of the governor mechanism at the middle of the run- down. These photographs were taken with the governor in the horizontal A GOVERNOR FOR TELEPHONE DIALS 1301 Table V — Type Dla.l Governor Input Torque at Pulse No. 5 Dial No. P.P.S. Fm F, Fn G 1 9.89 37,820 32,820 15,320 9,050 10.05 39,050 35,950 12,580 7,420 9.92 38,100 34,300 11,650 6,880 2 10.02 38,920 35,230 11,340 6,700 10.01 38,600 34,880 11,400 6,730 9.96 38,400 35,000 10,420 6,160 Ni 3 10.03 38,950 34,100 14,850 8,770 10.22 40,600 35,420 15,880 9,360 10.08 39,400 35,000 13,522 7,980 Average 7,500 dyne-cm, approximately. position, thus eliminating the gravity effect. The deflection of the governor spring measured on the photograph was used to determine the Fg force, and the governor speed, necessary to determine Fm , was taken from the oscillograph string trace. Three dials were tested, each having the same maximum motor spring torque, 490,000 dyne-cm, and clean governor cases and studs to produce an assumed coefficient of friction value, u = 0.25. For three 7-type dials with fly-bar governors, the ex- perimental data given in Table V applies. As determined, this 7,500 dyne-cm torque at the governor exists at the middle of rundown of the dial. In order to compare it with the 13,500 dyne-cm initial torque previously determined, it is first necessary to consider the effect of the motor spring. As stated previously, the torque provided by the motor spring during dial rundown decreases approxi- mately 35 per cent from the initial value of 490,000 dyne-cm. Logically, the torque at the governor would decrease by the same percentage. Applying this factor to the torque value for the middle of rundown gives a value of 10,500 dyne-cm which can be compared with the 13,500 dyne-cm torque. A difference of 3,000 dyne-cm exists for the value of initial torque at the governor as determined by the two test methods. This remaining difference can be explained by considering the frictional losses in the dial mechanism during rundown. This analysis follows: During rundown of the dial mechanism, a pair of pulsing springs ten- sioned against components on the pulsing shaft are alternately raised and lowered. This action allows contacts on the springs to open and close for pulsing in the telephone line. A portion of the input torc^ue provided by the motor spring is required for performing this function. On Fig. 14 are plotted the torque curves for these pulsing springs as the pulsing 1302 THE BELL SYSTEM TECHNICAL JOURXAL, XOVEMBER 1954 aaaav aagaosav - sa3l3^MllN3^ bnaq ni anoaoi A GOVERNOR FOR TELEPHONE DIALS 1303 shaft rotates during riiiulowii. Fig. 15 is a schematic of the pulsing springs as they appear when the contacts are closed and when open. For a short period during each revolution of the pulsing shaft, the puls- ing springs aid the motor spring in driving the gear system. This occurs when the springs are being lowered by the cam just prior to opening of the contacts. For the remaining portion of each revolution the motor spring must pro^•ide energy to overcome frictional losses and lift the springs. These changes in energy required for moving the springs have been combined to give the total instantaneous torque curve also shown on Fig. 14. As indicated by the pulsing spring torque analysis, the pulsing mech- anism absorbs an average of only 200 dyne-cm during the period when accelerating up to the critical velocity. For rotation after the critical velocity, the average torque needed to drive the pulsing mechanism is approximately 3,000 dyne-cm. The difference between these average torque values appear at the governor shaft as a 1,500 dyne-cm torque difference. That is, 1,500 dyne-cm more torque is available for driving the governor prior to the time the critical velocity is reached as com- pared to that available after this time. This accounts for one half of the 3,000 dyne-cm difference in initial governor torques as calculated by the BIFURCATED SPRING ~- CAM FOLLOWER ---'SPRING =— CONTACTS PAWL CONTACTS OPEN CONTACTS CLOSED Fig. 15 — Pulsing springs of 7-type dial. 1304 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Table VI Average torque at governor — Test No. 1, up to the critical velocity 13,500 Average torque at governor — Test No. 2, after reaching the critical velocity 10,500 Torcjue at pulsing mechanism 1,500 Torque required to overcome friction in the bearings 1,500 Total 13 500 dyne-cm 13,500 dyne-cm two test methods. The remaining 1,500 dyne-cm difference can be accounted for by considering the friction in the mechanism before and after the critical velocity. Initially the system is accelerating as a simple fly wheel under the influence of the motor spring. At the critical velocity the governor studs engage the case and the dial rotates at virtually constant speed for the rest of the rundown period. This implies that the average speed of the moving parts in the mechanism will be twice as great for the period after the critical velocity as before. By considering the friction which exists in the dial bearings,* one can see the effect of this change in speed on the torque required to drive the mechanism. In sliding bearings with film lubrication the coefficient of friction is a function of speed. Specificalh', as the speed of rotation of the journals increases, the coefficient of fric- tion in the bearings will increase. Since the regulated dial speed is greater than the average speed while accelerating, friction in the system will also be greater. One can, therefore, justify the remaining 1,500 dyne-cm torque difference w^hich exists before and after the critical ^'elocity by considering it to result from the increased friction at the higher speed. Therefore, by considering the effect of the pulsing mechanism and friction in the system, it has been possible to account for the difference in torciue determined by the two test methods. Table VI shows the dis- position of the torque. Appendix III DERIVATION OF CHATTER EQUATION FOR FLY-BAR TYPE GOVERNOR Consider the governor rotating in the vertical plane at constant speed CO. Referring to the schematic Fig. 7 and taking moments about B FJ) - F,h - Fnd + y.F^c - ml sin /3 = 0 * Design of Machine Members, Valence and Doughtie, p. 255. A GOVERXOK FOR TELEPHONE DIALS 1305 where m = mass of weight I = distance from C.Cl. of the weight to the pivot (B) 8 = operating angle of governor weights with respect to a hori- zontal phme For chattering to occur we know that the weight must lea^'e the go\'ernor case. Theiefore, at some angle 13 the gra\'ity component will equal the forces tending to move the weight outward. For this condition pressure of the friction stud against the case will be equal to zero and F„d = 0 fxFnC = 0 and the moment ecjuation becomes for this e(iuilil)rium condition FJ) = F,h + ml sin 8 (1) Centrifugal ]Moment = Spring Moment + Gravity Moment By applying the steady-state speed eciuation and the equation for cen- trifugal force we can transpose equation (1) to contain only design con- stants of the governor. From the steady-state speed equation G{d - nc) + Mmcoo- Mil Substituting Fs = nioio' Tq and M = 'Itnrrob 2 G ., >, , mrrobfiFs mrrobnu = — {d — nc) -{- or 2 mro 2 G (d - iic) Fs = mno} — 2 rhu also, the centrifugal force is Fm = mwra Substituting in equation (1) the values for {Fs) and (F«) 2 7 2, G{d — ixc) . T • a mwroh = mruu h — + m/ sm 0 Irn 130G THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 195-t or G{d - nc) ml sin j8 = 2rn where the gravity moment (vil sin (3) must be just ecjual to or smaller than [G(d — pic)]/2riJi to have no chatter occur in the governor. Expres- sing this in terms of the mass of the weight we have the chatter equation for the fly-bar governor as -^1!^ Appendix IV DERIVATION OF CHATTER EQUATION FOR DRIVE-BAR TYPE GOVERNOR Referring to the schematic, Fig. 4, consider the governor rotating at constant speed o) in the vertical plane. Taking moments about B F„J) - Fsb + F, + fxFnC - Fnd - ml sin 0 = 0 where m = mass of weight / = distance from C.G. of weight to pivot B (S = operating angle of governor weights Assume that at some angle d the gravity component will be large enough to make the F„ force equal to zero and a condition of equilibrium exists. For this condition FJ = 0 nFnC = 0 and therefore, Fj , the torque component on the weight, must also equal zero. The moment equation becomes FJ) = Fsh + ml sin 3 (1) Using the equation for centrifugal force Fm = mJ^n A GOVERNOR FOR TELEPHONE DIALS 13()'( and steady-state speed for the drive-har governor / G{d - nc) + M/iojo^ - iirjG/e where Fs = ??/coo"ni and M = 'lynn-nb Solving for (F.,) C Fs = mr oo)^ — ^r-^ {d - nc — nrj/e) Substituting in equation (1), the values for Fm and F., C mcoVo6 = mrocJ^b — - — (d — ixc — firj/e) + ml sin 8 2r/i or ml sin /3 = - — {d — nc — iirj/e) 2ryi Expressing the equation in terms of mass of the weight we have the chatter equation for the drive-bar governor. m ^ g(l- M^ - ^rj/e) ^2) 2rixl sin /3 In-Band Single-Frequency Signaling By A. WEAVER and N. A. NEWELL (]Manuscri])t recoivcd June 7, 195-1) Single-frequency signaling liberates dial systems from the restrictions of dc signaling methods. This freedom^ as might be expected, is most important in the long distance telephone plant where trunks are frequently too long or have no conductors for dc signaling. The general plan of signal frequency (SF) sigrialing is based upon contiyiuous signaling because of it's speed and reliability. In this respect it is like the usual dc trunk signaling schemes. SF uses steady current in the trunk signaling path for the normal idle trunk condition and no current in the signaling path for the other and alternate busy (talking) trunk condition. This choice of signal conditions is essential for SF signaling in-band systems, which as the name implies operate within the standard voice channel, to avoid conflict between signal and voice trans- mission. The same conditions are also used in SF out-of-band and separate line systems. The in-band SF system can be used with any type or length of line facility that meets normal voice transmission requirements and is therefore the pre- ferred 7nethod used by the Bell System to meet requirements for toll dialing on a national basis, with other signaling arrangements limited to the shorter trunks. The requirements, design considerations, main features, and method of operation for the in-band system are outlined in this paper. INTRODUCTIOX The signaling requirements for dial telephone operation are naturally more exacting than those for manual switching methods. This means a high order of signaling system is needed to satisfy the recjuirements for the toll telephone plant and for automatic toll switching systems de- scribed in recent papers in this Journal/-^ Indeed the advantages in speed and economy of dial telephone systems depend to a large extent upon the type of signaling provided for them. The signaling arrange- ments for intertoll telephone trunks which are the links between tele- phone s^^'itching systems, therefore, become most important. Dial operation in the past has been based upon dc signaling which is 1309 1310 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 limited to relatively short distances and to line facilities having dc paths available to them. In planning for natiomnde dialing of long-distance calls the need for an ac signaling system for dial telephone trunks be- came apparent. The length of intertoll trunks and the extensive growth in carrier line facilities which do not have associated dc paths made it necessary to develop ac dial signaling systems. The single-frequency sig- naling plan was developed for this purpose and is the first of its kind to satisfy the conditions associated with long distance intertoll dialing in the Bell System. There are now several trunk signaling means that may be grouped as using the SF signaling plan. These are (1) the adaptation of VF carrier telegraph requiring an additional line channel independent of the voice transmission line facilities, (2) Nl and 01 carrier signaling furnished as part of these carrier terminals using 3,700 cycles outside but adjacent to the voice paths, and (3) 1600-cycle and 2400-cycle signaling systems, the in-band systems that use the voice paths. Both in-band and out-of-band signaling have advantages and dis- advantages. The in-band single-frequency signaling system uses ac in the voice frequency range to pass full supervision and dial pulsing sig- nals over the same paths that are furnished for voice transmission in telephone trunks. This is accomplished without any loss in band width, change in line facility or addition of intermediate signaling eciuipment. On most calls voice and signal transmission are not required at the same time. On the few calls going to intercept operators voice trans- mission is impaired slightly by the effect of signal tone being on in one direction. On calls encountering busies, it is desirable to return both flashing supervision and interrupted audible tones. This can be done with out-of-band signaling but in-band signaling can return either but not both. The signaling system allows remote build-up and breakdown and provides for supervision of the temporary connections ordinarily used. Control and supervision of distant ends of trunks is required con- tinuously whereas dial pulsing is required only at the start of calls and speech transmission is required only when connections are established. TRUNK SIGNALS Before going into the details of the signaling system itself it seems appropriate to review the trunk signals it is called upon to transmit. Most intertoll trunks are arranged for two-way operation, which means that a connection can originate at either end. To permit this operation, the signaling in each direction must be symmetrical and the trunk must allow the direction in which the connection is established to determine IX-BAXD SIXGLE-FREQUENCY SIGNALING 1311 Table I — Calling to Called Direction Trunk Condition Idle (disconnect) Connect Dial pulsing* Ring forward Disconnect (idle) Signaling Frequency On Off On, then off, on ])ulses corresponding with dial break intervals On, then off, one pulse On * Multifrequency pulsing,^ a separate a-c signaling sj'stem, is a faster means of transmitting number information often used instead of dial pulsing. Its use eliminates only the dial pulsing signals. Table [I — Called to Calling Direction Trunk. Condition Signaling Frequency Idle (on-hook) Stop pulsing* Start pulsing* Flashingt Off-hook (answer) Ring back On-hook (idle) On Off On Off, then on, off" pulses corresponding with off hook supervision Off On, then off, on for duration of ring On * Stop- and start-pulsing control signals are required only in connection with common control switching equipment. t Flashing supervision signals are required only for operators; the on intervals light cord circuit lamps to inform operators of the status of calls independently of the position of cord circuit talking keys. the signaling interpretation. The latter is conveniently identified by different names for the trunk signals in the two directions. Only two signal conditions, that is, tone on or tone off, in each direction are re- quired for all dial trunk signals. Continuous dependence upon these two conditions assures a high degree of reliability because of signal redun- dancy. Tables I and II show the required dial intertoll trunk signals, together with the action taken in regard to the signaling frequency. The signaling system must be able to handle minimum and maximum length signals. The minimum times occur in dial-pulsing where the shortest signal element may be as low as 30 milliseconds. All other types of signals have longer durations. The maximum peiTnissible transmission time for signals between trunk terminals is determined by the allowable imguarded intervals on two- way trunks, during which double connections may occur, and also by the stop-pulsing signal recognition interval. This time is limited to about 175 milliseconds. The distortion permitted in the transmission of signals is proportional 1312 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 to tho time duration of each signal. In general, variations in signal time should be within ±5 per cent. All effects of the trunk signal medium should be confined within the trunk terminals or be of such character as to have no adverse reaction in connected circuits. This is necessary for proper operation of switched connections. BASIC PLAN The in-band single-fret juency signaling system, although fairly com- plex in detail, is very simple in principle. Normally, i.e., when the circuit is idle, steady tone is transmitted over the line and holds relays operated at the receiving end. Signaling is accomplished by removing and re- applying this tone, which in turn releases and reoperates the distant relays. Independent operation is obtained in each direction with one signal frecjuency on four-wire lines, which have separate one-way trans- mission pathes from terminal to terminal, and with two signal fre- quencies, one for each direction of transmission, on two-wire lines. The signaling system is provided as a separate entity. It is connected in series with the transmitting and recei\dng branches of the line circuit at each end of a trunk and to the terminal relay circuit (trunk circuit) by two one-way signaling leads. A typical arrangement for a four-wire line terminating in a two-wire switching office at the West terminal and a four- wire switching office at the East terminal is shown in Fig. 1 . -^ TRUNK CIRCUIT -13 DB LEVEL BALANCING NETWORK +4 DB LEVEL +4 DB LEVEL SF SIGNAL CIRCUIT 4-WIRE TOLL LINE X R '*" SF SIGNAL CIRCUIT TRUNK CIRCUIT -13 DB LEVEL > ^ o CONTROL LEADS Fig. 1 — Application of single-frequencj^ signaling to trunks with four-wire lines. IX-BAXD SINGL.E-FKEQUENCY SIGNALING 1313 In the case of two-wire lines, the signaling e(}nipment is applied to the four-wire transmission paths of terminal repeaters, using a different freciuency for signaling in opposite directions. Band elimination networks are provided ahead of each receiving circuit to block the transmitting frequenc}', which would otherwise come into the receiver via echo paths and interfere with its operation. GENERAL DESIGN CONSIDERATIONS The successful use of the voice path for signaling, especially for con- tinuous as contrasted to "spurt" signaling, is feasible only l)y a compro- mise among a num]:)er of conflicting factors. These factors or design con- siderations are (a) choice of signal freciuency, (b) signaling power and receiver sensitivity, (c) imitation of signal by speech or tones, (d) inter- ference to signal by other tones and noises, and (e) audibility of signaling tone to operators and subscribers. (a) Choice of Signal Frequency The choice of signal freciuency is determined mainly by considerations of signal imitation by speech. As will be shown later on, signal imitation decreases rapidly as the signaling frequency is raised with the result that the highest frequency that can be reliably handled by the transmission path is used. In the case of some four-wire type lines, such as EB carrier, the highest frequency that should be used is 1,600 cycles. However, the use of 1,600 cycles results in an expensive signaling system and it is de- sirable to have another system using a higher frequency (2,600 cycles) for application to lines that can handle this frequency. These systems are basically the same in principle and both are described in the present article. For application to two-wire lines the second freciuency used is 2,000 cycles in the case of the older 1 ,600-cycle system and 2,400 cycles in the new 2,600-cycle system. (6) Signaling Power and Receiver Sensitivity To limit cross talk into adjacent voice channels and to avoid adding much signal power to the repeaters it is desirable to use the lowest prac- ticable signal power consistent with a usable signal-to-noise ratio. A value of —20 dbm referred to zero transmission level for the steady idle tone is satisfactory for this purpose. In order to obtain an overall margin of 8 db the sensitivity of the receiver is set at —28 dbm. A higher power 1314 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 is used for a short time at the beginning of each application of signaling tone to help overcome line noise and attenuation variation. This power increase is 14 db for the 1600-cycle system and 12 db for the 2,600-cycle system. (c) Imitation of Signal by Speech or Plant Tones An in-band signaling system requires that the receiver respond to signal- ing tone and at the same time be non responsive to speech formed cur- rents. The principal design factors employed to achieve this feature are (1) the use of "guard action," (2) the employment of as narrow a band- width as practicable for the signal selective network, (3) the use of volume limiting, (4) the use of the longest operate time practicable, consistent with trunk signaling reciuirements, and (5) the use of the highest frequency that can be handled by the voice path. "Guard action" is the principal means used in protecting the receiver against operation on speech. It consists in the use of nearly all fre- ciuencies in the voice band other than those in a narrow band centered on the signaling frequency to generate a voltage which is used to oppose that resulting from the signal frequency. The sum of these two voltages, plotted against frequency, for a typical receiver is shown in Fig. 2. A term used to specify the magnitude of the guard action is "guard-signal ratio" {G/S in Fig. 2) or just "guard ratio." The amount of guard which can be used is limited by signal to noise ratio because noise, like speech, tends to oppose operation of the receiver. A guard ratio in the range of 6 to 10 db has been found to be practicable. Protection against signal imitation is also provided by narrowing the signal frecjuency band as much as practicable, since this reduces the effective operating power of voice and noise frequencies. However, the extent of this narrowing is limited since the operating bandwidth must be sufficient to allow for frequency variation in the signal supply, for carrier shift in the transmission path, for variation in the elements of the tuned circuit in the receiver and to allow for the transmission of the needed side bands of the signaling pulses. A bandwidth of 60 to 75 cycles at the 3 db points at dialing power (about — 6 dbm at zero level) has been adopted as about the minimum that is practicable. Because of limiting and guard action the effective bandwidth is a function of input power and in the particular designs adopted approaches about 150 cycles at the just operate point. Volume limiting is another means used to help prevent false operation on high levels of speech. The explanation of this action is illustrated in IN-BAND SIN(iLE-FREQUEXCY SIGNALING ]31J 15 -20 -30 -35 f\ *- - 0 DBM ' *■- -16 A S *- 1 -25 \ \ ^ \ A G I / t ?, 7 \ \ -i. 1 1 1 i "^ y / / \ -16 \ y 1 I 1 1 r 0 dbm\ INPUT \ V y ^-^ 1 f ( i 1 1 / \ \ ^ -x/ i / \ \ -/ 1 V 1 1 1, 0.2 0.3 0.4 0.5 0.6 0.8 1.0 2 3 4 5 6 FREQUENCY IN KILOCYCLES PER SECOND 8 10 Fig. 2 — Signal-guard characteristics of 2,600-cycle receiver. Fig. 3. The dotted line shows a characteristic for a receiver with no limiting, while the solid lines are for one with limiting. As shown, a given large input would produce an output of Ei , the difference between the signal voltage and guard voltage components, for the former case and an output of E^ , which is about half as much, for the latter case. This will be less likely to operate the receiver when applied for a short interval of time, although either will produce an operation if applied long enough, because either exceeds the just operate value Ei . Having established the basic design parameters of sensitivity, band- width, guard to signal ratio, limiting characteristics and speed of re- sponse it is important to know the relationship between signal imitation and the frequency used for signaling. To obtain information on this subject a series of tests were made using a number of guard channel receivers as nearly alike as possible except for the frequencies of signal response, which were 800, 1,350, 1,800, 2,400 1316 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 ^ GUARD VOLTAGES Fig. 3 — Limiter characteristics. o 0 1000 1500 2000 2500 FREQUENCY IN CYCLES PER SECOND Fig. 4 — Signal-guard characteristics used in signal imitation tests. Table III Receiver sensitivity (0 level) Receiver signal bandwidth (3 db points), at just operate level '■" approx Guard to signal ratio Start of limiting above just operate, approx Minimum duration signal for operate -28 dbm 150 cycles 6db 5db 50 ms IN-BAND SINGLE-FKEQUEXCY SIGNALING 1317 and 2,000 cycles. The signal-guard characteristics of all of these re- ceivers are shown in Fig. 4. Other parameters used are given in Table III. The results of these tests are shown in Fig. 5, where frequency is plotted against signal imitations per 100 calls. The receivers were located at New York and were coiniectcd at ditterent times in trunks to Boston, Toronto, Buffalo, Pittsburgh, Washington and Miami, so as to get some geographical speech distribution. There was' no' detectable geographic effect. The tj^pe of speech sound causing the signal imitation is also of in- terest even though we have not as yet been able to put this information 0.8 0.6 0.08 0.06 0.02 - N > 800 ON CYCLE DATA BASED ABOUT 9000 CALLS, \ \, ALL OTHER BASED ON ABOUT 20,000 CALLS - X v - \ - > o \ - \ \1 > IMITATIONS 0F\ RING FORWARD \ SIGNAL > 50 MS \ - V - S ) \ - N s. \ V - N \ < \ I IMITATIONS Of\ DISCONNECT > SIGNAL > 170 MS \ \ - \l > - \ y - \ V - \ ^ \ \ \ \ \ \ 800 1200 1600 2000 2400 FREQUENCY IN CYCLES PER SECOND Fig. 5 — Results of signal imitation tests. 1318 THE BELL SYSTEM TECHNICAL JOURXAL, NOVEMBER 195-t Table IV Circuit "singing" or momentary circuit transient 10 Adult whistle 2 Uncertain 8 Tone spurt 1 Speech imitations 48 Total 69 to use in the design of single frequency guard type receivers. In observa- tions on many thousands of calls it was noted that vowel type sounds were the predominant cause of signal imitations, with all except a few being formed by female speech. At the highest frecjuency tested (2,600 cycles) over 90 per cent were caused by the long e vowel sound (as in jeet). At the intermediate frequencies 1,350 and 1,800 cycle) most ^'owel sounds were noted, while at 800 cycles signal imitations were caused principally by two sounds, namely 0 (as in hole') which accounted for about 50 per cent of the total and aw (as in awX) and similar sounds like ah as in father. Signal imitations from vowel sounds are to be expected because of their relatively large energy and sustained nature. For instance it is well known that a sustained long e sound can have a large component in the high frequency range with very little energy in the range from 500 to 2,000 cycles where the guard action is effective. Likewise a sustained long 0 sound can have a large peak in the 500-cycle to 1,000-cycle range with little energy in the 1,000-cycle to 3,000-cycle range where the guard action is effective for the 800-cycle receiver. Speech formed currents are not the only source of signal imitation. In one series of observations using 2,600-cycle receivers in\'ol\'ing circuits from New York to a number of other cities including Toronto, Boston, Baltimore, Washington and Miami a total of 69 signal imitations were observed. In each case an attempt was made to determine the sound that caused the false operation, with the results given in Table IV. NOISE CONSIDERATIONS Noise affects the signaling system in a \^ariety of ways depending upon the nature of the noise and upon the particular signaling function being performed. When tone is first applied it is of course desired that the re- ceiver operate. However at this time the "guard" circuit is functioning because it is also desired that the receiver be non responsi\-e to speech. Noise, which acts on the guard circuit like speech, will therefore tend to prevent operation of the receiver. If the noise is steady and large enough IN-BAND SINGLE-FREQUENCY SIGNALING 1319 it would of course pcM-manciitly i)re\(Mit operation, while if it is of short duration and occurred at the beginning of a signal inter\'al it would only delay operation. E\'en a short delay would be harmfid to ring forwai'd or dialing signals ])ut could l)e tolei'ated in disconnect or flashing signals. An example of how a short duration high le\'el noise affects receiver operation is illustrated in Fig. G. As can be seen this particular noise transient, which ha})pened to be caused by a relay in the trunk circuit, occurred just prior to and overlapping the tone interval. It would se- riously damage a ring forward signal. The solution to this particular problem is to absorb the noise at its source, or prevent it from reaching the ^'oice path. After the recei\^er is operated for a short time (0.2 sec or so) the guard action is removed. Later on when the tone is removed it is desired that the receiver release, but noise at this time will tend to prevent release. The solution to this problem is a compromise in receiver sensitivity, i.e., it must be sensitive enough to hold up on the weakest signaling tone and yet release on the maximum noise that can be tolerated from a speech point of view. Fortunately such a compromise is achieved with a sensitivity that will cause the receiver to hold with the tone about 8 db 5.0 2.5 -2.5- -5.0 L BEGINNING OF TRANSMITTING TONE TRANSIENT VOLTAGE AT TRANSMITTING END (-16 TL) BEGINNING OF RECEIVED TONE 20 30 40 TIME IN MILLISECONDS TRANSIENT VOLTAGE AT RECEIVING END (+7TL) Fig. 6 — Example of transient voltages generated by relay operation. 1320 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 IN-BAXD SIXGLE-FREQUEXCY SIGNALING 1321 Iwlow normal and yet permit release in the presence of as much as 50 dba of thermal noise at zero transmission level. SPECIFIC DESIGNS The two signaling systems 1,600 and 2, GOO cycles, are basically similar in principle. However, as can be seen by reference to Fig. 5, a simple guard type receiver having a frequency of 1,600 cycles would have too many signal imitations. To ON'ercome this the guard ratio during the talking condition was increased from 6 to 10 db, the minimum signal interval to just cause a response was increased to 100 milliseconds during the talking condition and the sensitivity was decreased to —16 dbm. As a result fairly complicated timing and switching circuits are needed to assure that both transmitting and receiving circuits have the right condition at the right time. Table V gi^'es a sununary of the principal design parameters of the two systems. DESCRIPTION OF 1,600-CYCLE DESIGN A front view of the 1,600-cycle main unit is shown in Fig. 7. This panel is 8 inches high by 23 inches Ande and weighs about 20 lb. The essential elements of the circuit are shown in Fig. 8. It connects to the trunk relays over two leads e and m and into the line circuit via leads labeled t, r, Ti and Ri . The transmitter, shown in the upper portion of the figure, uses dc biased germanium varistors (diodes) to control the application of signal current to the line, and for control functions, uses four relays designated m, co, hl and rr. The functioning of the first three except for tone control are described under the heading "Descrip- tion of 2,600 Cycle Design" later in this article. The rr relay (not shown) in conjunction with the m relay lengthens the sent pulse for the ring for- Table V Dialing Condition Talking Condition 1,600 2,600 1,600 2,600 Sensitivit3% dbm Bandwidth, cycles Low level High level Guard ratio, db Minimum signal for just operate, ms Start of limit above just operate, db -28 150 60 0 35 5 -28 150 75 -6 35 5 -16 150 60 10 100 5 -28 150 75 6 50 5 1322 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 iinoaiD xNnai oi Fig. 8 — Simplified diagram of 1,600-cjele signaling circuit. IX-BAND SINGLE-FREQUENCY SIGNALING 1323 ward signal because the far end receiver at this time has a long operate time. The signal receiver is connected in series with the receiving hraiich of the voice transmission path and is provided with a voice amplifier lo provide a l)lo(*king function so noises originating in the switciiing eciuip- ment or beyond will not interfere with operation of the sigiuding receiver, and to compensate for the signaling l)ri(lging loss. The receiving portion of the circuit is shown in the central and lower portions of Fig. 8. The idle condition of the trunk is shown, tone is being received, the r, rg and rf relays are operated and the band elimination filter is inserted in the receiving l)ranch to prevent signaling tone from entering a connected circuit and interfering with signaling there. The signal cm-rents coming in from the line are passed through the signal amplifier, limiter and low pass filter and applied to the signal- guard netW'Ork from which signal voltage is applied to the dc amplifier tube to operate the above mentioned relays and open the e lead, which extends into the trunk circuit. Typical wave forms at several points in the circuit are shown in Fig. 9. The extra operate time provided during the talking condition is obtained from slow relays (not shown) W'hich at this time are in the path from the r to the rg relay. These relays also change the sensiti\-ity, and guard ratio. INCOMING SIGNAL RECTIFIED SIGNAL VOLTAGE e,, FIG. 8 r r RELAY OPERATES \ RECTIFIED GUARD VOLTAGE Qz^ FIG. 8 NET VOLTAGE AT DC AMPLIFIER OUTPUT SIGNAL (NO CORRECTION) Fig. 'J — Tj-pical wave forms in 1,600-cycle receiver. 1324 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Fig. 10 — Front view of the 2,600-cycle signaling panel. The R delay, because of guard action and the fact that its secondary winding is closed through a varistor and resistance, is relatively slow to operate and fast to release. For this reason some foiTn of pulse correction is necessary to get good dial operation. This is obtained with the rg (regenerate) relay and its associated cr timing network, and there re- sults an output pulse within the needed limits, even though the signal on the R relay is shortened considerably. DESCRIPTION OF 2,600-CYCLE DESIGN The 2,600-cycle unit, shown in Fig. 10, is just half the size of the 1,600- cycle unit, costs less than half as much, and is of the "plug in" type so it can be readily replaced for maintenance action. A simplified diagram of the new signaling circuit is shown in Fig. 11, with the transmitting portion in the upper part of the figure and the receiver in the lower part. The transmitting portion employs three re- lays designated m, hl, and co which are interconnected to perform the following functions: relay m is used to key the signaling tone; relay hl (high level) adds 12 decibels to the tone power at the beginning of each signal tone application to improve signal reliability in the presence of line noise and variations in attenuation; and relay co (cut off) cuts the line momentarily to prevent noises originating in the switching equip- ment from interfering with signaling. IN-BAND SlXCiLE-FREQUENCY SIGNALING 1325 The operation of the receiver will be explained by describing its action (except for pulsing which will be described later) when signal frequency is received. This ac tone is amplified (or limited if it is too large) and then passed on to the signal and guard networks where a relatively liigh voltage results in the signaling channel and a lower voltage in the j -40 V "=" Fig. 11 — Simplified diagram of 2,600-cycle signaling circuit. 1326 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 guard channel. These voltages are then applied to rectifying circuits where positive and negative dc voltages are developed and passed on to the dc amplifier tubes a and b, respectively. The rf relay operates first and cuts in the band elimination filter thereby preventing signaling tones from entering a connected toll line and interfering with the signal- ing there. However, a short spurt of tone will get through because of the finite time required to operate this relay, and the r relay must therefore be made slow enough so that it will not operate on this tone. This action is obtained with the resistor-capacitor network (ot and Cs) in the grid circuit of the associated dc amplifier. A short time (about 200 ms) after the r relay operates, a relay (not shown) releases, which short circuits the guard network and inserts enough resistance in series ^\'ith the signal network to substantially re- move its tuning. The purpose of removing the guard action during the idle condition is to prevent release of the receiver which would otherwise be caused by occasional bursts of line noise. The signal network is made broad at this time for the following reason. In connections to an intercepting operator, "off hook" supervision is not returned to the originating end to avoid charging for the call. This means that tone remains on the line to continue to hold up the receiver. At the same time the intercepting operator's speech must of course be trans- mitted over the line so that both speech and tone enter the receiver. Speech can be of a relatively high power as compared to the tone with the result that the action of the limiter tends to suppress the tone and could falsely release the receiver if the signal tuning were present. How- ever, with broad tuning either speech or tone will hold up the receiver and no trouble is encountered. The blocking amplifier seen in Fig. 11 has the same function as in the 1,600-cycle design described previously. Among the new features in this unit, one of the most significant is the pulse-correcting circuit. This feature is a very important element in the entire long distance connection since it serves to keep the length of the dial pulses within specified time limits. The dial pulses on many calls may have to go through a number of central offices and all their associated equipment, and in each stage of the transmission path the ideal 60-millisecond dial pulse may be distorted so that it becomes too long or too short. The pulse-correction is accomplished by generating appropriate tran- sient voltages, whose duration is determined by capacitor-resistor net- works. These voltages are then applied to the grid of dc amplifier b in Fig. 11 to perform the elongation or shortening of the pulses as required. IN-BA\D SlXCiLE-FUKQllONCY SR!\AL1.\ y (b) 10 PPS -T^ -^ m 6 LINKS .-' [^ ^ - .--■' J LINK ^ y (C) 12 PPS ^^ 6 LINKS^ [i. 1 LINK ■ ■^ ^^"^^ ** """ *^^" 35 40 45 50 55 60 65 70 75 80 65 90 PER CENT BREAK IN Fig. 14 — Per cent break input versus output churacteristics for (a) 9 pulses per second, (b) 10 pulses per second, and (c) 12 pulses per second. 1330 THE BELL SYSTEM TECHXICAL JOURNAL, NOVEMBER 1954 This "something" consists of using the large negative transient voltage generated at the plate of tube a resulting from the application of a posi- tive pulse to its grid. This transient, shown at (c) drives the grid of b rapidly and heavily negative, thereby forcing the k relay to release. The remainder of the pulse-correcting action consists in using this same transient to delay the reoperation of the r relay. This is accomplished by storing some of the energy in the Ri Ci network. This slows the building up of the voltage as shown at (e), and the relay r reoperates at (f). The resultant repeated signal is shown below, where a 75-millisecond signal has been pulse-corrected to 65 milliseconds, further corrections being effected in the subsequent sf units. Dialing performance of typical signaling units is shown by graphs in Figs. 14(a), (b), and (c). These curves show per cent break input plotted against per cent break output for 9, 1 0 and 1 2 pulses per second for one and 6-link operation. If the system were linear the input-output charac- teristic w^ould be a 45 degree straight line. When the slope is less than 45 degrees there is pulse correction, and if the slope were zero with an output at 60 per cent, pulse correction would be perfect. It is noted that the pulse correction action improves as the speed increases and at 12 pulses per second the output is nearly independent of input. ACKNOWLEDGEMENTS The success of this project is the result of contributions by many people, and all cannot be named specifically. However special mention should be made of F. A. Hubbard, who designed the equipment arrange- ments of the 2,600-cycle system and W. W. Fritschi, C. W. Lucek, R. 0. Soffel and A. K. Schenck who made important contributions to the cir- cuit design. REFERENCES 1. J. J. Pilliod, Fundamental Plans for Toll Telephone Plant, B. S.T.J. , 31, pp. 832-850, Sept., 1952. 2. F. F. Shipley, Automatic Toll Switching Systems, B.S.T.J., 31, pp. 860-882, Sept., 1952. 3. C. A. Dahlbom, A. W. Horton, Jr., and D. L. Moody, Application of Multi- frequency Pulsingin Switching, A.I.E.E. Transactions, 68, pp. 392-396, 1949. 4. H. Fletcher, Speech and Hearing, Van Nostrand. Centralized Automatic Message Accounting System By G. V. KING (Miinuscript received May 6, 1954) A centralized automatic message accounting si/stem. (CAM A) has been (Jrveloped so that the hilling data can be recorded at a centralized crossbar tandum office for message unit and toll calls originated by telephone customers served by a large number of local dial central offices. It is an essential part of facilities for economical nationwide customer dialing through central offices with older types of switching equipment and through other central offices which could not otherwise economically give this service. The new sys- tem records the billing data on paper tapes in the same form now used by local automatic message accounting systems. Tapes for both local and cen- tralized automatic message accounting systems are processed in the same accounting center. INTRODUCTION One broad objective of the Bell System is to extend the customer's dialing range so that ultimately he will be able to dial his own calls to any telephone in the country in much the same way as he now dials local calls. Several steps toward this goal have already been taken. A revised fundamental plan for automatic toll switching has been adopted which involves among other things the use of a nationmde numbering plan covering the United States and Canada. In accordance with this plan each customer will be given a distinctive 10-digit designation which will consist of a 3-digit regional or area code, a 3-digit central office code, and a 4-digit customer's number. In many parts of the country automatic toll switching systems are now in use by operators who complete more than 40 per cent of all toll calls by dialing directly to the called telephone in distant cities. In order that customers may use these switching systems to dial their own toll calls, some automatic means for recording the necessary billing information on such calls must be provided. 1331 1332 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 PRESENT RECORDING AND CHARGING METHODS Several types of automatic recording equipment are now in servdce in the Bell System. Multi-unit registration (zone registration) has been in use for many years in a number of panel and No. 1 crossbar central offices whose rate structure permits bulk Ijilling. This method can be used for calls which cost 6 or less message units for the initial period. Although zone registra- tion is economical, it does not pro\dde a detailed record of each call but merely scores the number of message units on a register associated with the customer's line. Remote control zone registration has been serving customers in panel central offices since 1941. It is similar to multi-unit registration, but the timing and register control equipment is located in a tandem office in- stead of in each originating panel central office. Automatic ticketing,^ which was developed some years ago for use in step-by-step central offices, does make a record of the details of each customer dialed call. A simple ticket printer is permanently associated ^\dth each outgoing trunk to produce an individual typewritten ticket for each call. Common relay equipment is used to furnish the called num- ber, calling number, etc. to the printer. The information printed on the ticket is in detailed form and is similar to that prepared by the operator in manual operation. It can be used for billing the customer manually either on a detailed or a message unit basis. A greatly improved form of recording, the Automatic Message Ac- counting^ (AMA) system, was introduced into the Bell Sj^stem in 1948. In central offices having this equipment, all of the data required for billing of customer dialed calls are automatically perforated in code on paper tapes. These tapes are taken to an accounting center where they are processed by suitable machines to produce customers' bills. The re- cording machines are associated with the transmission circuits onlj^ when required to make a record, one recorder ser^ang up to 100 such circuits. Recorders, together wdth their associated equipment, are installed in each central office arranged for local AMA recording. The information for each call is recorded on the tape in three stages, or entries. The initial entry is recorded after the customer has finished dialing. One time entry is recorded when conversation starts and another when conversation ends. For short-haul calls that are to be billed on a message unit basis, the initial entry contains only the calling office code and telephone num- ber, and the charging rate. This information, together with the duration of the call, is sufficient for determining the charges. On toll calls which are to be billed in detail, the called office and telephone number are also CENTRALIZED AUTOMATIC MESSAGE ACCOUNTING SYSTEM 1333 required. This system permits individual and two party customers in No. 1 and No. 5 crossbar offices to dial calls to telephones in their home area. In addition, customers in some No. 5 crossbar central offices may now dial directly to other areas. These facilities have been installed in No. 5 crossbar offices in Englewood, N. J., and in several other locations whose customers now may dial directly to about 13 million telephones in 13 metropolitan areas. Relatively expensive recording eciuipment is reciuired in each central office in the local AMA system. For new central offices this recording equipment is economical only if the toll and message unit calling rates are relatively high. The addition of local AMA recording ecjuipment to existing offices is, in most cases, uneconomical. CENTRALIZED AUTOMATIC MESSAGE ACCOUNTING The Centralized Automatic Message Accounting system (CAMA) pro- vides an economical means of recording billing data for customer dialed calls from many central offices that cannot justify local AMA. This sj^stem is economical because one group of recording equipment, located at a crossbar tandem office, can serve as many as 200 local central offices without requiring major changes in, or additions to, those offices. The first crossbar tandem equipment arranged for CAMA was placed in service in Washington, D. C, in November, 1953. This equipment serves the customers in 85 central offices in Washington and in suburban Virginia and Maryland. They are able to dial each other directly and to dial their own calls to Baltimore and to other nearby toll points. Even- tually, they will be able to dial their own calls to most points in the United States and Canada. Similar crossbar tandem CAMA equipments have been installed in Detroit, New York, San Francisco and Phila- delphia. The CAMA installation at Detroit enables the customers served by approximately 800,000 telephones in 99 Detroit panel and No. 1 crossbar local central offices to dial station-to-station multi-unit interzone and toll calls to 63 communities in Michigan and in nearby Canada. The map of Fig. 1 shows this dialing area. THE CROSSBAR TANDEM SYSTEM The crossbar tandem system into which CAIMA has been introduced is used today in panel-crossbar and step-by-step areas. It receives calls from local dial central offices and completes them to other local central offices and to the toll network. It is also arranged to receive calls over 1334 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Fig. 1 — Points reached via Detroit CAMA at cutover in Dec, 1953. intertoU trunks and complete theraito other intertoll trunks or to local central offices. In many situations, it provides more efficient trunking facilities between central offices than do direct trunks and connects to- gether offices with different signaling systems. Since the present crossbar tandem offices have in themselves no means for recording billing data, their use has been restricted to operator dialing, to customer dialing of CENTRALIZED AUTOMATIC MESSAGE ACCOUXTIXG SYSTEM 1335 flat rate calls from all types of central offices, and to customer dialing of message unit and toll calls from central offices using one of the present methods of charging. FIELD OF USE FOR THE CAMA SYSTEM The CAMA system, as now developed, is suitable for use in panel- crossbar local areas. It is arranged to serve 7-digit calls only since fa- cilities for 10-digit dialing are not available for panel and No. 1 crossbar central offices. Thus, in general, it can complete calls only to its own numbering area. However, provision is made for completing calls to one adjacent area, this area being selected by dialing "one-one" ahead of the listed 7-digit number. BRIEF DESCRIPTION OF THE CAMA SY'STEM If a customer makes a call that requires CAMA treatment, the call will be routed by the local central office to a crossbar tandem office arranged for CAMA recording. Until automatic means for identifying the calling customer's number for billing purposes is developed, an oper- ator will be bridged on the connection at the tandem office to obtain the calling number and register it in the CAMA equipment by keying. The CALLING TELEPHONE ORIGINATING CENTRAL OFFICE CROSSBAR TANDEM OFFICE SWITCHING EQUIPMENT OPERATOR'S POSITION TO NATIONWIDE TOLL NETWORK Fig. 2 — Simplified switching diagram. 1336 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 information necessary for correctly charging for the call will then be re- corded on paper tape by the automatic message accounting equipment located in the tandem office. This method of operation is sho\\n in simplified form on Fig. 2. A more detailed block diagram of the principal eciuipment units and their interconnections for a crossbar tandem CAMA office in a panel- crossbar area is shown in Fig. 3. Here the switching equipment consists of the conventional trunks, sender link frames, senders, markers and trunk link and office link frames for switching calls through the office. Such new iniits as the position link frames, positions, transverters, billing indexers, recorders, call identity indexers, and master timer constitute the major AM A equipments needed for recording the billing information for each call. Most of these have functions similar to corresponding local AMA equipments. AMA features have also been added to the trunks, sender links and senders. TRUNK LINK FRAME OFFICE LINK FRAME FROM PANEL OR CROSSBAR CENTRAL OFFICE INCOMING TRUNK SENDE_R_UNK CONTROL CIRCUIT PCI SENDER CALLING NO. REGISTER POSITION LINK CONTROL CIRCUIT OUTGOING TRUNK l^-' THRU - MARKER CONNECTOR \THRU TRANSVERTER CONNECTOR TRANSVERTER BILLING T I INDEXER 1:^ THRU CONNECTORS RECORDER MASTER TIMER CALL IDENTITY INDEXER Fig. 3 — Block diagram of CAMA system in panel -crossbar areas. CENTRALIZED AUTOMATIC MESSAGE ACCOUNTING SYSTEM 1337 FUNCTIONS OF THE CAMA SYSTEM The functions of the system in establishing a connection and in re- cording the billing data may be divided into seven major groups as follows : 1. Operation of the sender link and control circuit in selecting an idle sender and connecting the selected sender to the incoming trunk. 2. ReceiAdng and registering the called office code and number in the sender. The sender does not pulse the entire number forward until the Aj\IA functions are completed. 3. Operation of the marker in establishing the connection through the switches and furnishing the sender with directions for completing the call. 4. Operation of the position link and control circuit in selecting an idle occupied position and connecting it to the calling customer. 5. Obtaining the number of the calling telephone verbally from the customer and keying it into the sender. 6. Connection of the sender to a transverter and billing indexer and the derivation of the billing data from the called and calling office codes and the rate class of the calling customer, and recording the charging information on the AM A tape. 7. Operation of the sender in transmitting information of the proper type to the terminating office, or if a toll call, to the next toll office in the chain. Calls from Panel and Crossbar Customers The first three functions in a crossbar tandem CAMA-equipped office in a panel-crossbar area are the same as in a non-CAMA office. Since published information on these features is available, they will not be described in detail. The fourth major function is handled by the position link and control circuit which is sho^Nii in block diagram form in Fig. 4. This circuit con- sists of primary and secondary crossbar s^^atch links and control circuits in duplicate. Each group functions independently to serve calls to the same 40 senders and can connect to two different groups of 50 positions. In case of failure of one link group, the other link group ■u'ill continue to serve calls to the 40 senders. The control circuits are arranged in such a manner that all senders and all positions receive essentially equal treat- ment. WTien the position link connects the sender to an idle CAMA position, the operator obtains the calling number verbally from the customer and 1338 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 records that number in the sender by the operation of numerical keys located at her position. The CAMA switchboard shown in Fig. 5 is a cordless board of modern sheet metal design. The sixth group of functions ■ — that of converting the data into the desired form and perforating it on the paper tape — is performed jointly l)y the transv^erter, the billing indexer, the recorder and the call identity indexer. The trans\'erter is very similar to that used in local A]\IA. It registers the calling and called nimiber received from the sender, registers the billing data received from the billing indexer and controls the re- corder in the perforation of the initial entry. The billing indexer is strictly a translating circuit. It receives the calling and called office codes and the customer rate class from the trans- verter and converts this information into a form which the transvertei and recorder can use. It provides a 1-digit billing index, which denotes the charging plan to be used, and provides a type of initial entry indica- A LINK SENDER GROUP 3 -- SENDER GROUP 0 _""_ POSITION _^_ GROUP 9 POSITION _0_ GROUP 5 POSITION _M_ GROUP 4 POSITION GROUP 0 CONTROL B Fig. 4 — Position link and control circuit. CENTRALIZED AUTOMATIC MESSAGE ACCOUNTING SYSTEM 1339 Fig. 5 — Switchboard. tion which tells the transverter whether to perforate a two- or a four-line initial entry. The two-line initial entry used for calls billed on a message unit basis contains only the calling office code and telephone number, the billing index and the trunk identity whose function is discussed later. This information, together with the duration of the call, is sufficient for billing. Four-line entries contain, in addition, the called office code and telephone number. They are used for detail billed toll calls and for those bulk billed calls on which all details of the call are required for record purposes. Fig. 6 shows a simplified schematic of the billing indexer. The calling office code combined with the customer rate class, chooses a par- ticular rate treatment relay. This rate treatment is common to all cus- H31H3ASNVai Oi zx>- - UJ< ±zui £D ~ cr 6 bi3ia3ASNVfcli /^oad 1340 CENTRALIZED AUTOMATIC MESSAGE ACCOUNTING SYSTEM 1341 tomers in the area who are charged alike for their calls through the tandem office. To actually determine the billing index or charge treat- ment for each call, the rate treatment is modified by the called office code by means of the rate treatment relays. Since calls with the same })illing index may recjuire a 2-line entry if originated in some offices and 4-line entry if originated in other offices, the billing index and calling office code information are translated jointly by means of the entry combina- tion relays to produce either the 2-line or 4-line indication. The recorder, call identity indexer and master timer circuits are of the same type as used in local AIMA and perform the same functions. The recorder perforates initial entries as directed by the transverter. It also perforates a timing entry at the beginning of conversation and an- other at the end of conversation. The call identity indexer, one of which is associated with each recorder, identifies the trunk used on a call as a particular one of the maximum 100 served by a recorder. This enables the recorder to perforate that identity on initial and timing entries. The identity is used by the accounting center to gather together the three entries involved on each call. The master timer keeps the recorders continually informed as to the correct time. When the initial entry is completely recorded on the AMA tape, the sender completes its task of pulsing the called number forward and then releases. The transmission path is now completed through the tandem office and the only CAMA functions remaining are the perforations of the timing entries mentioned above. ACCOUNTING CENTER PROCESS The accounting center process for CAMA is the same as for local AMA. It automatically assembles the three bits of information pertain- ing to each call, computes the conversation time on all calls, sorts by the type of call, prices each call either in terms of message units for bulk- billed calls or in ternis of dollars for detail billed toll calls and brings together the records of all calls made by each customer. MAINTENANCE FEATURES To properly maintain the AMA recording facilities, test circuits are provided for testing the major features of the CAMA equipment. An automatic incoming trunk test circuit tests the CAMA trunks. An auto- matic sender test circuit tests the CAMA senders in much the same way that the present sender test circuit tests non-AMA senders. Facilities are also provided for making operating tests of the position links, posi- 1342 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 tions, transverters and billing indexers. As in the local AMA system, testing of the recorder features is done by the test unit on the master timer frame. The AMA circuits are provided A\-ith many self-checking features which detect trouble while a call is being handled. When a trouble is detected, the AMA circuits connect momentarily to a recording circuit called a "trouble indicator" which lights lamps to indicate how far the call has progressed and which of the common control circuits were used on the call. This information aids the maintenance force in locating the trouble. FUTURE DEVELOPMENTS As stated earlier, use of Centralized Automatic Message Accounting by panel and crossbar customers will be restricted initially to calls to the home area and one foreign area. The centralized recording will be done initially at crossbar tandem offices with operators identifying the calling telephones. Ultimately, customers served by all types of dial local central offices will be able to dial their own calls — local or nation- wide. Operator identification of individual and two-party lines will be replaced in many cases by automatic identification. The centralized re- cording equipment A\dll be located in various types of tandem and toll offices as determined by the economics of each case. CONCLUSION The development of Centralized Automatic Message Accounting arrangements is another major step toward nationwide customer dialing from central offices which cannot be economically equipped with local AMA recording equipment. REFERENCES 1. Pilliod, J. J., Fundamental Plans for Toll Telephone Plant, B.ST.J., 31, pp. 832-850, 1952. 2. Clark, A. B., and Osborne, H. S., Automatic Switching for Nationwide Tele- phone Service, B.S.T.J., 31, pp. 823-831, 1952. 3. Nunn, W. H., Nationwide Numbering Plan, B.ST.J., 31, pp. 851-859, 1952. 4. Shipley, F. F., Automatic Toll Switching Systems, B.S.T.J., 31, pp. 860-882, 1952'. 5. Friend, O. A., Automatic Ticketing of Telephone Calls, A.I.E.E. Trans., 63, pp. 81-88, 1944. 6. Meszar, J., Fundamentals of the AMA System, A.I.E.E. Trans., 67, Part I, pp. 255-269, 1950. 7. Collis, R. E., Crossbar Tandem System, A.I.E.E. Trans., 69, Part II, pp. 997- 1004, 1950. 8. Cahill, H. D., Recording on AMA Tape in Central Offices, Bell Labs. Record, 29, p. 565, Dec, 1951. 9. Jordan, W. C, The AMA Timer, Bell Labs. Record, 30, p. 122, March, 1952. The Wave Picture of Microwave Tubes By J. R. PIERCE (Manuscript received March 12, 1954) Many microwave tubes make use of a long electron beam. The radio fre- quency excitation on such a beam can be expressed in terms of two space- charge waves, one of which has negative energy and negative power flow. The electron beam may pass through resonators, through lossy surroundings, through slow-wave circuits. In this paper the low-level operation of klystrons, resistive-wall amplifiers, easitrons, space-charge-wave amplifiers, traveling- wave tubes and double-stream amplifiers is explained in terms of waves on electron beams and on circuits. Noise is discussed in terms of such waves. INTRODUCTION There are many different ways in which one can make a valid analysis of the low-level or small-signal behavior of the many types of microwave tubes which use long electron beams. Which way one should choose de- pends partly on one's purpose in making the analysis, and partly on the particular problem to be solved. All of these analyses lead at some point to waves or modes of propaga- tion: waves which travel along an electron stream, along a circuit, or along the two together; waves which are unattenuated or which increase or decrease with distance. Sometimes, the analysis starts out with elec- tron current, electron velocity and circuit dimensions as the fundamental physical ciuantities, just as network analysis can start out with induc- tance, capacitance and resistance. However, an analysis can start out instead with waves, their propagation constants and their characteristic impedances as the fundamental physical bases of the analysis. We might argue that as w^e are to end with waves, we may well start with waves. As it turns out, the picture of the operation of various tubes in terms of waves is simple and pleasing. It is the purpose of this paper to present a picture of the operation of microwave tubes in terms of waves. This may be of some interest to those outside of the tube field, in that it gives an account of many recent devices. For experts in the field it can serve as an introduction to a method of analysis which is fairly recent and which may be unfamiliar. 1343 1344 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 111 this analysis, certain simplifications are made. One underlying simplification is that of linearity; it is assumed that at low signal levels the behavior of the electron stream, which is inherently non-linear, can be represented by that of a truly linear system. As this paper purports to give an accurate and useful picture of the low-level operation of microwave beam devices rather than an exhaustive discussion, some details have been omitted or passed over lightly because they seemed to be of secondary importance. Material which may be un- familiar to workers in other fields but which is important as background is presented in appendices A to C. Various points can be pursued further in the literature. References to publications and to those responsible for various advances are not given in the body of the paper or in the ap- pendices; they are given for each topic in Appendix D. SPACE-CHARGE WAVES Many microwave tubes embody a long, narrow electron beam sur- rounded by a conducting tube and focused or confined by a longitudinal magnetic field. At low levels of operation, the radio-frequency disturb- ances on such an electron stream can be expressed in terms of space- charge waves. In these waves, two forms of energy are of primary importance: electrostatic energy associated with the bunching together of electrons, and kinetic energy, associated mth differences in the velocities of the electrons. Thus, the waves may be called electromechanical; the electric energy which we associate with waves in transmission lines and wave- guides is present, but the magnetic energy is replaced by kinetic energy. In circuit terms, we have an electrical capacitive element, but the in- ductive element is inertial, not magnetic in nature. When the electron and wave velocities are slow compared with the velocity of light, the magnetic fields produced by the electron convection current are negli- gible. There may be many space-charge modes or waves in an electron stream, some with complex radial and angidar variations of amplitude over the electron stream. Two waves predominate in the operation of tubes, however, and one simplification we mil make is to deal mth these only, and to disregard other modes of propagation on the electron stream. Appendix A discusses such a pair of waves in a simplified physical system. We can associate with these two waves an ac electron convection cur- rent i and an ac electron velocity v. Either we can assume that the elec- tron beam is narrow and disregard the fact that these quantities vary WAVE PICTURE OF MICROWAVE TUBES 1345 across the beam, or we can deal with peak or effective vakies much as in the case of voltages and currents in waveguides. These ac quantities are assumed to contain a factor That is, they vary sinusoidally with time and with distance, and (as- suming /3 to be positive) propagate in the -\-z direction. The phase con- stants |S of the two waves \\all be called /3i and /So . For beams of mod- erate charge they are very nearly CO . Wn iSi = - + -^ Bo = — - ^ Here iio is the electron speed, w is the operating radian frec^uency and w^ is the effective plasma radian frequency. The plasma frequency of the electron beam cop is given by e -po 2 m Up = e Here e/m is the charge-to-mass ratio of the electron, po is the charge density and e is the dielectric constant of vacuum. In terms of Wp , Wg may be expressed COg = R(j)p Here R is a, factor somewhat less than unity which depends on the geometry of the electron beam, on co and cop , and on the velocity distribu- tion of the electrons (see Appendices A and C). Let us consider the simple case in which R is unity and the effective plasma frequency is equal to the plasma frequency. The phase velocities Vi and V2 of the two waves, which are co divided by /3, are Vi V2 = Uo 1 + "^ CO Uo 1 0}p CO Thus, the first wave has a phase velocity less than that of the electrons; it is a slow wave, and the second wave is a fast wave. 1346 THE BELL SYSTEM TECHXICAL JOURXAL, XOVEMBER 1954 Suppose we make up a radio-freciuency pulse out of various fre- quency components of one wave. The pulse envelope generally travels with a different velocity from that of the rf sinusoids under the envelope. The velocity of the en\'elope is called the group velocity. The group velocity is the velocit}^ with which a signal is transmitted. The direction of the group velocity is the direction in which causality acts (for some waves the phase \'elocit.y and the group velocity have opposite direc- tions). The group \'elocity tells in which direction energy flows, and the power flow P is the stored energy per unit length, W, times the group velocity, Vg . P = WVg The group velocity is given by We see that for our assumption Wq is ec^ual to cop , the group velocity for each wave or mode is «o , the velocity of the electrons in the beam Vg = «0 Thus, of the two waves, the first has a phase velocity slower than that of the electrons, the second has a phase velocity faster than that of the electrons, and each has a group velocity equal to that of the electrons. A simple discussion of power flow is given in Appendix B. In describ- ing the excitation of the electron stream we can use the convection current i together with a c^uantity U which is analogous to voltage. In terms of the ac electron velocity v, U = — — V III m The real power flow P is given by p = mw* + i*u) This relation is justified in Appendix B. For each of the two waves the voltage U bears a constant ratio to the current i; this ratio is the characteristic impedance K of the wave. We find that K = — = - 9^^ Z? ii CO /o u CO 7( WAVE PICTURE OF MICROWAVE TUBES 1347 Here Vo is the accelerating voltage specifying the electron velocity Uo and /o is the total beam current. We see that the characteristic impedance 7vi of the .slow wave is negative. This means that the power flow in the +2 direction is negative. We could also say that positive power flows in the —z direction, but this may carry an unfortunate implication as to the direction in which causality acts. An example may be helpful. Fig. 1 shows an electron beam acted on by the fields of two devices A and B. The fields in A are such as to set up the slow wave only. This travels between A and B. The fields of B are such as to just remove the slow wave entirely, so that the electron beam leaves B with no ac dis- turbance on it. The electron velocity Wo , phase velocity v, group velocity Vg and negative power flow —P are all directed in the -\-z direction, that is, to the right. We must remove a power P from A to set up the slow w^ave. A power — P flows from A to B. We must add a power P to Bto remove the slow wave from the electron beam. Causality acts from A to 5. To change the ampUtude of the slow wave betw^een A and B we must change the fields in A, not the fields in B. The power flow is the group velocity times the stored energy per unit length. As the group velocity for the slow wave is positive and the power flow is negative, we see that the stored energy must be negative. If we moved with the electrons and observed the weaves, we would find that the average kinetic energy associated with the ac electron velocity was equal to the average potential energy of the electric field, and that both were positive ; this is characteristic of waves in a stationary medium. The kinetic energy of the electrons relative to a fixed observer is proportional to the square of their total velocity, that is, the ac velocity plus the average velocity. The average velocity is larger than the ac velocity, so that energy terms involving the product of the average UNMODULATED BEAM ELECTRON VELOCITY Uo- PHASE VELOCITY GROUP VELOCITY POWER FLOW P I TOUT UNMODULATED BEAM P lIlN Fig. 1 — Device A sets up the slow space-charge wave only, and device B removes it. uo , v, Vg and —P are respectively the electron velocity, the phase velocity, the group velocity and the power flow between A and B. 1348 THE BELL SYSTEM TECHXICAL JOURNAL, NOVEMBER 1954 velocity and the ac velocity are larger than terms involving the square of the ac velocity. The product terms may be negative or positive. We can understand the negative energy of the slow wave qualitatively through a simple argument of a somewhat different sort. In the slow wave, the charge density is greatest in regions of less-than-average velocity and least in regions of more-than-average velocity, so that the electron beam has less total kinetic energy in the presence of the slow wave than it does in the absence of the slow wave. How does this come to be? Suppose that we move with the wave; we then see electrons moving in an electric field which is constant wdth time, and hence, as electrons move through the field their velocities vary as the square root of the potential. Relative to the Avave, the electrons move slowest in the low-potential regions, and correspondingly, they are bunched together in regions of low potential. Now, for the slow wave the total electron velocity is the arithmetic sum of the wave velocity and the electron velocity relative to the wave, so if the electrons are bunched in regions of lowest velocity relative to the wave they are necessarily bunched in the regions of least total electron velocity, and the kinetic energy of the slow wave is thus negative. In the case of the fast wave, the electrons travel backward relative to the wave. The total electron velocity is the arithmetic difference between the wave velocity and the electron velocity relative to the wave. Hence, the total electron velocity is greatest at the bunches, where the velocity relative to the wave is least, and the kinetic energy of the fast wave is positive. THE KLYSTRON We can explain the operation of a number of types of vacuum tubes in terms of space-charge waves. Consider the klystron, illustrated in Fig. 2. The voltage produced across the input resonator by the input signal sets up on the electron beam both the slow and the fast space-charge waves in equal magnitudes and so phased that the velocities v, or the voltages U add, while the currents cancel. Thus, just beyond the input resonator, the beam has an ac velocity; it is velocity modulated, but it has no ac convection current. Because the two space-charge waves, one mth negative power flow and one with positive power flow, are set up with equal magnitudes, the ac power flow in the beam between the input and the output resonators is zero. The input resonator neither adds power to nor subtracts power from the beam. Because the two wa\'es have different phase velocities, their relative phase changes as they travel along the beam. If we go along the beam a WAVE PICTURE OF MICROWAVE TUBES 1349 distance L such that 2 ^' L = r Uo we will find that the ac velocities of the two wa^'es cancel and their cur- rents add. If at this point Ave put an output resonator, the current will produce a Voltage across the resonator which will act on the electron beam to set up new components of the slow and the fast waves. If the resonator is on tune, so that it acts as a resistive impedance, the phase of the voltage is such with respect to the space-charge wave producing it that the new component of the fast space-charge wave . r INPUT RESONATOR 2-— L = 77 Uo OUTPUT RESONATOR Fig. 2 — In a klystron the input resonator sets up slow and fast space-charge waves so phased that the velocities add and the currents cancel. At the output resonator the currents add and the velocities cancel. The voltage across the output resonator increases the amplitude of the slow, negative-power wave and decreases the amplitude of the fast, positive-power wave. subtracts from the old component, while the new component of the slow space-charge wave adds to the old component. Thus, while to the left of the output resonator the two space-charge waves have equal magni- tudes, so that the net power flow is zero, to the right of the output resonator the slow space-charge wave has a greater magnitude than the fast space-charge wave, so that the poAver flow in the beam is negative. The missmg power appears as the output from the output resonator. Of course, klystrons are frequently used in the nonlinear range of operation, and the distance L between resonators may be chosen differ- ently from other considerations. THE RESISTIVE-WALL AMPLIFIER Consider a tube much like a klystron, but in which the electron beam is surrounded by a glass tube coated Avith lossy material, such as graphite, as shoAvn in Fig. 3. 1350 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 As in the klystron, the mput resonator produces both the slow and the fast waves with equal magnitudes. As each wave travels, it induces currents in the resistive wall surrounding it and dissipates power in the wall. Thus, the power in each wave must decrease as the wave travels. The fast wave has a positive power, and so for its power to decrease the amplitude must decrease. Thus, in the resistive-wall region the amplitude of the fast space-charge wave decreases exponentially ^^ith distance. Because the slow space-charge wave has a negative power, its power can decrease only if the amplitude of the wave increases, so that the power flow becomes less (more negative). Thus, in the resistive-wall region the amplitude of the slow space-charge wave increases exponen- tially ^\ith distance; the wave has a negative attenuation; it is amplified as it travels. If we put the output resonator far from the input resonator, the ampli- tude of the fast space-charge wave will be very small there, but the amplitude of the slow space-charge wave may be very large. Its current will produce a large voltage across the output resonator. As in the case of the klystron, this voltage ^nll increase the amplitude of the slow space- charge wave, thus decreasing the power flow in the electron stream. The resistive wall amplifier has a feature which the klystron lacks ; the process of amplification involves an actual growing wa^^e along the elec- tron stream. THE EASITRON; increasing WAVE IN A LOSSLESS SYSTEM Consider a tube somewhat similar to the resistive wall amplifier, but in which the beam is surrounded, not by a lossy tube, but by a series of pill-box resonators, as shown in Fig. 4. Imagine that the resonators are so tuned that at the operating frequency they present a lossless negative susceptance to the electron beam. TUBE COATED WITH RESISTIVE MATERIAL :^ Fig. 3 — In a resistive-wall amplifier the currents excited in the lossy wall bj- the slow, negative-power wave decrease the power in the wave, so that the ampli- tude of the wave must increase. WAVE PICTURE OF MICROWAVE TUBES 1351 The impedance an electron beam sees in traveling through free space or in a concentric lossless tube is capacitive. In section 1 the space-charge waves were described as in\'olving the stored energy of the electric field, capacitive in nature, and the kinetic energy of the electrons, which has an inductive effect. We might liken the beam and its capacitive circuit to the ladder network of Fig. 5. We know that such a network supports waves. When the charge of the beam sees a negative susceptance, the be- havior is much as if the capacitances in the ladder network of Fig. 5 were negative.* In this case the waves characteristic of the circuit are not traveling waves, but are a pair of waves, one of which decays with dis- tance and one of which increases with distance. Neither has any net stored energy. We can express the propagation constants of the waves much as in the section on, ''Space-Charge Waves," but the effective plasma frequency cog is now imagmary; we will call it jbiq . The phase constants j8i and ^2 ARRAY OF RESONATORS ^ Mm Fig. 4 — In the easitron, resonators surrounding the beam change the suscep- tance the electrons see from positive to negative. The system, no longer supports two traveling waves, but rather, a growing and a decaying wave. .^-rw^ ■^Wo rmp' Fig. 5 — If the capacitances in this ladder network were negative it would sup- port growing and decaying waves rather than traveling waves. * Some care must be used in arriving at proper equivalent circuits. For instance, neither of the electric waves on a ladder network has negative energy if the net- work is set in motion, but we have seen that one of the longitudinal space-charge waves does have negative energj'. If both the capacitances and the inductances of a ladder network are negative, the waves on the network will have negative energies. 1352 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 become /3i -\- J Mo Mo P-2 CO , . CJq = — + J — llo Mo The characteristic impedances become CO io CO io The fact that the characteristic impedances of the two waves are imaginary means that neither of the waves alone has any power flow. Neither of the waves can very well carry power. The amplitudes change with distance; hence for each wave Ui* and iU* mcrease or decrease with distance. But, the circuit and the electron beam are lossless, and the power cannot change with distance. As the ^^'aves do have a group velocity, neither has any stored energy. Does this mean that the beam cannot carry any power? The beam can carry power, just as a filter in its stop band can carry some power from a source at one end to a resistive load at the other end. The power flow is still given properly in terms of the total current i and the total voltage U by the same expression used in section 1. Suppose that the two waves have currents ii and 12 . Then the total power flow is P = V2[(ii + ^2){K^*h* + Iu%*) + (zi* + t2*)(Iuii + /v2^2)] P = 3^[(i^i + i^l*)(^■l^l*) + (Ko + K2*)i2i2* + iii2*(Ki + A%*) + iiii2*(K, + K2*))*] First consider the case in which cog is real and for which the characteris- tis impedances are real and 7^1 = -lu In this case P = K\iii* + /C2i2i2* This is the familiar case of unattenuated waves. The total power is the power of each wave calculated individually. Let us now consider the case in which the effective plasma frequency WAVE PICTURE OF MICROWAVE TUBES 1353 is imaginary. In this case we can write ivi = -jKo lU = +jKo where Kn is real. We have P = [-jni2*Ko + (-ifif2*/vo)*] Either wave alone carries no power; there is power flow only when the two waves are present simultaneously. The two waves vary with distance as — j(w/«o)z (a>g'/«o)2. -jiuluQh) —{uq'luQlz SO the iiii* is constant \Aith distance. If this were not so the power would change with distance, but as the resonators have been assumed to be lossless, neither taking power from the beam nor adding power to the beam, this is impossible. Thus, in a lossless system an increasing wave is always one of a pair, and the other member decreases with distance in such a way as to keep the product of the amplitudes of the two waves constant with distance. Neither the increasing wave alone nor the de- creasmg wa\'e alone carries any power, but the two together can carry power. We will note that in the easitron the direction of the group velocity, that is, the direction of causality, is the direction of electron flow. Thus, the waves are both set up at the input resonator; it is there that boundary conditions on both current and voltage must be satisfied. COUPLING OF MODES OF PROPAGATION We know that waves which increase and decrease exponentially with distance are characteristic of a ladder network in which the susceptances of the shunt and series arms have the same signs. They occur in other networks as well. Consider a smooth transmission line loaded periodically with shunt capacitances, as shown in Fig. 6. Each capacitance reflects Fig. 6 — Capacitances connected across a smooth transmission line periodically couple the forward and Ijackward waves and produce stop bands characterized by growing and decaj'ing waves. 1354 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 part of a wave approaching it. In other terms, each capacitance acts to couple one mode or wave (say the forward wave) to another (say, the backward wave). When the distance between the capacitances is such that the couplings reinforce, that is, near a half wavelength in this case, the system is a filter in its stop band ; it does not transmit traveling waves, but supports rather a wave which increases exponentially with distance and a wave which decays exponentially with distance. Neither of these waves alone carries any power. The space-charge waves of an electron stream can be coupled to one another, to a space-charge wave of another stream, or to an electromag- netic A\ave. In any of these cases we can have increasing waves. THE SPACE-CHARGE-WAVE AMPLIFIER Consider an electron beam surrounded by a series of metallic tubes A, B, A, B ■ • • , alternately at different potentials with respect to the cathode from which the electrons come, as shown in Fig. 7. The impe- dances of the space-charge waves will be different in tuloes ^4 from what they are in tubes B. The behavior of this system is much like that for the transmission line system shown in Fig. 8, in which we have alternate line sections of different characteristic impedances Ka and Kb. We know that such a series of line sections forms a filter ^vith stop bands. L {]{ ?U^U (; 1 i _ llllllltllllllll TO llllllllllllllll Fig. 7 • — The impedances of waves in an electron beam passing through elec- trodes at alternate!}' higher and lower potentials differ in regions of different po- tentials. This can result in stop bands characterized by growing and decaying waves. Such a device is a space-charge-wave amplifier. Fig. 8 — A transmission line with alternating sections of impedances K^ and Kb is somewhat analogous to the space-charge-wave amplifier. WAVE PICTURE OF MICROWAVE TUBES 1355 In the case of the space-charge-wave structure of Fig. 7, the stop band occurs for conditions near that in which for both sections A and B the section lengths La and Lb are such that 2 ^ L,, = T Uo 2'^Ls = T Here Wy.i and w^s are the effective plasma frequencies for sections A and B. A structure such as that of Fig. 7 can be interposed between input and output circuits, such as resonant cavities, to give a space-charge-wave amplifier dependent for its action on the growing wave of the pair. THE TRAVELING-WAVE TUBE In the space-charge-wave tube, the two waves which are coupled to- gether ha\'e different \'elocities, just as the forward and backward waves on an electron stream have different velocities. Hence, they can be coupled strongly only through the use of some periodic structure in which the period is related to the difference in phase constants of the two waves. In a traveling-wave tube we can hstve coupling between a space- charge wave and a wave traveling on a circuit, and both of the waves can have velocities which are nearly or exactly the same. Here we must consider two different cases. If both of the coupled waves carry power in the same direction (that is, if the power is positive for both, or negative for both), coupling cannot result in a stop band, but only in transfer of po^^'er between one wave and the other. In order to ha\'e a stop band, power which we try to send in on one wave must come back to us on the other. Hence, to produce a stop band and gaining waves, the two coupled waves must carry powers \\ith opposite signs. A traveling-wave tube can consist of a helix of ^\•ire, which can sup- port a slow electromagnetic wave, surrounding an electron beam, as sho^^^l in Fig. 9. I TOUT Fig. 9 — The vital elements of a traveling-wave amplifier are an electron stream and a slow-wave circuit which may be a helix surrounding the electron stream. 1356 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Traveling-wave tubes really involve at least four waves: two space- charge waves and two circuit waves. Usually, the backward circuit wave is so far out of synchronism with the space-charge waves that we can neglect its coupling with them. Further, if the space-charge waves are well separated in velocity, that is, when cog is large enough, then when one is coupled to the circuit wave the other isn't, and so we can get some idea of traveling-wave tube operation by considering waves in pairs. The simple mathematics of such coupling is given in Appendix D. In Fig. 10, the behavior of various phase constants, plotted as a func- tion of oj/wo , is shoAvn qualitatively. Here co is radian frequency and Uo is electron velocity. We may consider that co/uq is varied by changing the electron velocity Vo and keeping the frequency w constant. The horizontal line (3c is the phase constant of the forward circuit wave in the absence of electrons, or when the coupling to the electrons is zero. jSc does not change with electron velocity. /8« and ^f are the phase con- stants of the slow and fast space-charge waves, respectively, with zero coupling to the circuit wave. For the slow space-charge wave, the power flow is negative, while for the circuit wave and the fast space-charge wave the power flow is positive. Thus, for coupling between the slow 0.5 / / ^y P>z Af / / / 1.0 1.5 2.5 Uo Fig. 10 — Suppose that at a constant radian frequency w we change the electron velocity wo in a traveling-wave tube. If the waves of the electron stream were not coupled to the waves of the helix, the phase constants, /3<- of the forward circuit wave, (3s of the slow wave, and /?/ of the fast wave, would vary approximately as shown. AVAVE PICTURE OF MICROWAVE TUBES 1357 Fig. 11 — Because of coupling of the space-charge waves to the forward circuit wave, gain is produced near /3c = j3, , while the curves sheer off from one another near jic = 13/ . space-charge wave and the circuit wave we can have a stop band, while for coupHng between the circuit wave and the fast space-charge wave we cannot. The consequences of the couplings between the circuit wave and the space-charge waves near the intersections of /3c with jSs and (3/ are il- lustrated in Fig. 11. We see from Fig. 11 that near synchronism between the circuit wave and the fast space-charge wave ((3c = /3/ for no coupling) these waves combine so that for any given value of u/uo there are always two dis- tinct real values of /3. This is typical for coupling between modes with power flows of the same sign. At /3c = /3/ each of the two mixed waves has equal energies in the circuit and in the electron stream. Near synchronism between the circuit wave and the slow space-charge wave (/3c = /3s in absence of couplmg) these two waves combine so that over a range of oj/uq near /3c = /3s , /3 has two complex values with the same real part and with equal and opposite imaginary parts. We can write this as /3 = /3i ±ja; -jfi = -j/3i + a This corresponds to an attenuated and a groAAing wave with the same phase velocity. In Fig. 11, i3i and a are plotted as dashed lines. 1358 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Over the range of w/wo for which the waves are attenuated {a j^ 0) the net power flow in each of the modes is zero. The power flow in the electron stream is equal and opposite to that in the circuit. Such behavior is characteristic when two modes with power flows of opposite signs are coupled. It is characteristic of the stop band of an electric wave filter. The curves of Fig. 11 exhibit the same behavior that has been found by other means, although similar curves are sometimes plotted somewhat differently. When an input signal is applied to the helix of a traveling-wave tube, all three forward waves are set up. The increasing wave grows until it predominates, and it forms the amplified output of the tube. The total ac power of the increasing wave is zero. How can we obtain power from it? In the increasmg wave we have a positive electromagnetic power flow in the circuit and an ec^ual negative power flow in the elec- tron stream. If we terminate the helix we can draw off the electromag- netic power; the electron stream is left with less power than it had on entering the helix. DOUBLE-STREAM AMPLIFIERS A double-stream amplifier makes use of two streams of electrons which have different velocities, as shown in Fig. 12. The behavior of a double- stream amplifier is very similar to that of a tra\'eling-wave tube. In such a device each electron stream supports a slow, negative-energy wave and a fast, positive-energy wave. At a constant frequency w let the velocity U]_ of one stream be kept constant and let the velocity Ui of the other stream be varied. The behavior of the phase constants /3 of the waves is shown quahtatively in Fig. 13. /3«i and jS/i are the phase con- stants of the slow and fast waves of the constant-velocity stream, and /3g2 and /3/2 are the phase constants of the slow and fast waves of the stream whose velocity is changed. There are two ranges of velocity Ui for which gam is obtained; for u-i a little larger or a httle smaller than Ux . INPUT I I I Fig. 12 — Two nearby electron streams of different velocities u\ and u-i consti- tute a double-stream amplifier. WAVE PICTURE OF MICROWAVE TUBES 1359 NOISE WAVES ON ELECTRON STREAMS Consider the electrons of the beam as they leave the cathode. If the velocity distril)ution is MaxweJlian, and if the electrons leave inde- pendently, there will be a mean-square fluctuation in convection current, i", given by I = 2eIoB and an uncorrelated mean square fluctuation in ac velocity, v , given by ^-'>(fj(~> Here /o is beam current, e and m are electron charge and electron mass k is Boltzmann's constant, To is cathode temperature and B is bandwidth' Usually, space-charge-limited flow is used. In this case the beam cur- rent is only a part of the emitted current ; the rest is turned back at the potential minimum. In this case we may use the above relations, counting Jo as the beam current, as some sort of approximation for the current passing the potential minimum. The wave picture we have been discussing may be seriously inaccurate near the cathode where the relative spread in electron velocities is large. Suppose that we hope for the best and apply it. We find that in the most general case our electron stream will have on it a noise standing-wave pattern. If imin and Vax are the minimum and the maximum noise currents, I ^'min I I ?max I 1 / CO \ f kT, I'm ax 2ehB 2 \o}J \eVo Here a is a constant near to unity. The noise pattern is made up of two uncorrelated noise standing-wave patterns, one from i at the cathode and the other from v at the cathode; these patterns have amplitudes ?"i and 12 at their maxima; the minima are of course zero. We have I imin I I tmax I = U'l I h'2 | slu ^ Here ^ is the relative phase angle of the standing- wave patterns associ- ated with t'l and Z2 . That is, if the maximum of the 12 pattern is at that of t'l ,■*• = 0, Avhile if the maximum of the 12 pattern is midway between maxima of ii , then ^ = 7r/2. The first of these theorems says something about the noise current at the maximum and that at the minimum, but it does not directly say how 1360 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 2.0 2.5 Fig. 13 — In a double-stream amplifier, gain is obtained when the phase con- stants of the slow wave of the faster stream and the fast wave of the slower stream are nearly equal. large the maximum is. For an ordinary two-potential electron gun, ^max is very large compared ^\dth ?'min . NOISE DEAMPLIFICATION Early traveling-wave tubes made use of a two-potential electron gun spaced a critical distance from the circuit, as shown in Fig. 14. More recently it has been found possible to reduce the noise figure considerably by the use of space-charge-wave amplification, as discussed in the sec- tion on "The Space-Charge-Wave Amplifier." The structure used is indicated in Fig. 15. The gun has a low-potential anode followed by a kGUNj-L^ir. Fig. 14 — When a simple, two-potential electron gun is used, the noise figure of a traveling-wave tube can be optimized by adjusting the drift-space between the gun anode and the helix. WAVE PICTURE OF MICROWAVE TUBES 1361 drift tube. At the point where the noise current is a minimum the voltage is "jumped" to the hehx voltage. A second drift tube follows, so that there is a critical distance between the jump and the helLx. The effect of this "voltage jump" gun is to deamplify the component of the space-charge wa^'es which is associated with the noise current at the current maximum. In space-charge-wave amplifier terms, this com- ponent sets up the decreasing wave only. Thus, in the second drift tube the ratio | ?max/?min I is smaller than in the first. By usmg a single velocity jump, traveling-wave tubes with noise figures around 8 db have been made. The use of more velocity jumps has been proposed. It can be shown, however, that as z'max is deamplified, imia must be amplified. This sets a theoretical limit of around 6 db to the noise figure attainable by means of space-charge-wave deamplification alone. iNO.I-H"" NO. 2- DRIFT TUBES Fig. 15 — When a two-potential or "velocity jump" gun is used, the noise figure can be reduced by space-charge-wave deamplification of the noise on the electron stream. NOISE CANCELLATION It would be highly desirable to build a travelmg-wave tube such that the electromagnetic input would excite an increasing wave, but the noise in the electron stream would excite only some combination of the decreasmg and the unattenuated waves. If we succeeded in this, the noise introduced by the tube could be made as small relative to the signal as desired, merely by making the tube long enough. Can we ac- complish this by means of some special structure near the input end of the tube? We can represent the noise on the electron stream at some reference point by means of a velocity fluctuation v and a current fluctuation i; we have seen that neither can be zero. Because the system is linear, super- position applies, and the amplitudes of the growing, attenuated and unattenuated waves which are excited are the sums of the amplitudes excited by i and v independently. Suppose that v = 0. Then the beam carries no power. Thus, i cannot 1362 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 excite the unattenuated wave, for that wa^-e carries power. Let us assume that it excites the decreasing wa^■e alone, which, when present alone, carries no power. So far there is no contradiction, and we can believe that it is possible to arrange matters so that the current alone does not excite the increasing wave. Suppose we have so arranged matters that i does excite the decreasing wave only. Consider what happens when / = 0. Can v alone excite the decreasing wave only if i alone excites the decreasing wave onl}^? If it can, then v and i together must excite the decreasing wa^'e onl3^ But suppose V and i are of the same frequency and in phase. Then the beam carries power. But, the decreasing wave alone cannot carry power, and hence what we have assumed is impossible. If the i excites the decreasing wave only, then v must excite at least a component of the growing wave. Hence, we cannot cancel out the noise from the beam completely. FINAL COMMENTS We have seen that the properties of space-charge waves and the behavior which must follow when space-charge waves are coupled to other space-charge waves or to circuit waves can be used to explain the operation of seemingly diverse types of microwave tubes. The wave picture gives a clear and quantitative picture of energy relations and power flow. It enables us to understand simply the effect of thermal velocities on the operation of tubes through their effect on the phase constants of the space-charge waves. It is useful in detailed considera- tions of noise, and in one case it has enabled us to draw a general con- clusion without resorting to formal mathematical manipulation. It may well be that the wave picture can be of further use both in calculating detailed behavior of tubes and in understanding their general properties. Appendix A SPACE-CHARGE WAVES Consider a narrow electron stream in which we may assume that elec- tron velocity and charge density do not vary across the stream, and in which the electrons are free to move in the ^-direction only. An axially symmetrical electron focusing system immersed, cathode and all, in a strong magnetic field approximates this. Let all ac quantities contain the factor —jSz jut e e and let the total charge densitj^, current and electron velocity be made WAVE PICTURE OF MICROWAVE TUBES 1363 UJ3 of dc and ac parts as follows:* charge density: — po + p con\'ection current densit}^: — /o + t velocit}^ : Vo + v Here po , h and ?/o are positive dc quantities. The quantities on the right are the ac components. AVe have from the definition of con\'ection current (-/o + i) = (-P0 + p)(uo + v) (Al) In the case of very low level operation, we neglect products of ac quantities in comparison with products of ac and dc quantities. Doing this, we obtain from (Al) the dc and ac convection currents /o = poUo (A2) or i = —pov + vop (A3) i + po?' Uo (A4) We can apply the continuity equation, or, the equation of conserva- tion of charge, to the ac convection current (A5) ^i di dz dp dt ^i = (jop tlo/ {jwi - pojcov) — copoV (A6) 0) — /3mo Thus, if we have a wave with a given phase constant /3, and if we know Po and cj, (A6) gives the convection current in terms of ac electron velocity. How can we find what j8 will be? To find this we must consider the effect of the electric field on the electrons. Consider an ac electric field E^ in the z direction, which also varies with time and distance as * It will be convenient elsewhere to use — /o and i as currents rather than current densities and — po and p as charge per unit length. 1364 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 do the other ac quantities. We can write i'=--E. (A7) at in Here e/m, the charge-to-mass ratio of the electron, is taken as a posi- tive quantity. In (A7), dv/dt is the rate of change of v with respect to t for a single electron; that is dv/dt observed riding along with the electron. If we ride along with the electron for a time dt we move along distance dz dz = {uq -f- v) dt For small signals we neglect v in this expression and write dz = Ua dt Hence, the total change dv in the velocity of the electron in the time dt is dv = — dt -{- — Uodt dt dz Hence, we find that dv ,^ = i(a) - l3uo)v (A8) dt Using (A7) and (A8), we see that m (co — /3uo) (A9) We can combme (A9) with (A6) and write for the convection current density . ^ ~^'£"^"^- (AlO) Let us now consider a special, hypothetical case in which the electric field is in the z direction only, so that there are no transverse electric WAVE PICTURE OF MICROWAVE TUBES 1365 fields and no transverse displacement current. Then the total ac current density it is the sum of the convection current density and the displace- ment current density, or, ii = i -\- ju€E^ It = (All) Let us use a quantity Wp , which was long ago named the plasma fre- quency (radian frequency) e m^' (A12) Using cop , (AlO) can be written as According to Maxwell's equations the divergence of the total current is zero. Both components of it vary mth z. If, as we have assumed, there is no current normal to the z direction, then it must be zero. If this is to be so, we must have (co - /3wo)' = ^ = -±-. (AM) ■Uo Wo In actual electron beams there is transverse electric field away from the beam, and hence it is not zero. It is found, however, that when ojp is small compared with co, we can write quite accurately ^^-d.''-^ (A15) Here Wg , which is knowai as the effective plasma frequency, is smaller than cop . As CO is raised, so that the wavelength of the space-charge Avaves becomes smaller compared with the diameter of the electron beam, the electric field tends to become largely longitudinal and w, approaches Up . 1366 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 The upper sign in (A15) gives the phase constant of the slow wave, a wave with a phase velocity less than that of the electrons. The lower sign gives that of the fast wave, a wave with a phase velocity faster than that of the electrons. From (A15) and (A6) we note that i= ±- pov (A16) The upper sign holds for the slow wave; the lower sign for the fast wave. It has been convenient to use — /o and i as current densities and — po and p as charge densities. In subsequent work and in the text, — /o and i will be used as beam current and — po and p as charge per unit length. All the relations of this appendix except (A11)-(A13) will hold if the quantities are so interpreted. Appendix B power flow in space charge waves The purpose of this appendix is to justify the expression for power flow in the beam. Consider that the electron beam is acted on over a short distance by an ac voltage. Imagine, for instance, that the beam passes through two very closely spaced grids which form a part of a resonator, and that a voltage AV appears between the grids. What does the voltage do to the beam? The voltage AV changes the velocity of the electrons but it does not change the convection current. To find out how much the velocity is changed we need only consider the case in which the beam has no ac velocity on reaching the grids, smce in a linear system the change will be the same in all other cases. The total velocity ^lo + t^ is given in terms of the total accelerating voltage F + AF by uo+ V = J 2 £ (Fo + AF) (Bl) We assume A F to be small, so that AF - AF ^ ^ = -^ (B2) 2 ^ Fo ^° / m WAVE PICTURE OF MICROWAVE TUBES 1367 The change AU in the "voltage" U is ATT '^'<^^ AT/ m The con^'ection current i flows against the voltage Al^, so that a power AP is transferred from the beam to the resonator which is attached to the grids. AP = -ReAVi* (B4) Thus, the change in the power in the beam in passing through the grids must be — AP -AP = Re {- AVi*) = ReAUi* (B5) AP = -ReAUi* According to the expression we ha\'e used in calculating beam power, if the "\-oltage" of the beam on reaching the grids is U, and the convec- tion current is i, then the beam power Pi on reaching the grids is Pi = ReUi* (B6) After passing through the grids, U is increased by an amount AU while the current is unchanged, so that the power P2 of the beam leaving the grids is P2 = Pe(f/ + AU)i* (B7) The loss of power in the beam, AP, is AP - Pi - P2 = -RcAUi* (B8) This agrees with (B5), in which AP was calculated as the poAver lost from the beam to the resonator. Appendix C the effect of the velocity distribution in the electron beam on the effective plasma frequency Consider an electron beam in which electron motion is confined to the ^-direction, and in which the electrons have a velocity spread with a 1368 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 mean square deviation {u ) about the mean value Wo . If cor/o is the effective plasma frequency for {u") = 0, then taking the velocity spread into account, the effective plasma frequency co, is given approximately by OiQ — 03,0 I i -r O TT W50/ / If the velocity distribution is the same as for electrons accelerated individually by a voltage Vo , that is, if electron interactions do not affect the velocity distribution appreciably (as they probably do not) {u') _ 1 /A-r.V _ 1 ul 4:\eVo/ 4\ll,600Fo Here k is Boltzman's constant and 7\ is cathode temperature. Thus, from this assumption Following our wave picture, we can take into account the thermal velocity spread by using this corrected value for the effective plasma frequency in all our formulae. For all practical purposes, the change in effective plasma frequency due to thermal velocities is negligible. In a paper which wiW appear in the Journal of Applied Physics, D. A. Watkins has used a somewhat different approach in treating the effect of thermal velocities on the operation of traveling- wave tubes. Appendix D phase and attenuation curves for coupled modes When two unattenuated modes of propagation are coupled together periodically in a lossless manner, they combine to form two new modes. For each of these new modes the amplitude is changed in one period of the coupling structure by a factor where M is a root of M' - 2VTWT^ cos (^^ - ^+0y- ^3) ^1 = 0 (D2) Here A- is a coupling coefficient which is zero for zero coupling. The upper sign applies if the power flow in the two modes have the same WAVE PICTURE OF MICROWAVE TUBES 1369 signs while the lower sign applies if the power flows have opposite signs. dq and dp are phase lags per coupling period associated with the two orig- inal modes and ^i and Oo are phase angles associated with the coupling device. We can treat the case of continuous coupling by letting the period of coupling L be very short, the angles dq , dp , 6i , 6^ be A-ery small, and the coupling per period, k, be very small. In this case the cosine can be represented by the first terms of a power series and we find that the phase constants ^ of the modes are given by ^-n^-{'^1V^4^)' (.3) Here ^a and ^b are the phase constants for K ^ 0 (zero coupling) ^. = '-^ (D4) (D5) L and K is the coupling per unit length /^ = I (D6) As before, the upper sign in the radical applies when the power flows have the same signs and the lower sign when the power flows have opposite signs. In applying (D3) to the case of traveling- wave tubes and backward- wave oscillators, the effect of all but two modes was of course neglected when the two phase constants would have had nearly the same value in the absence of coupling; the curves for such regions were then joined smoothly to give the overall plots of Figs. 11 and 13. In Fig. 11 the parameters chosen arbitrarily were: /3o = 1 /3s = u/uo + ^i /?/ = 0}/Uo — }i K = 0.1 The complex portion of the phase constant, or, the real portion of the propagation constant, in a stop band caused by the coupling of two 1370 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 modes Avith power flows of opposite signs is designated by a and is plotted as the dashed ellipses about the horizontal axis. Appendix E This appendix comments briefly on various sections and cites refer- ences, which are listed at the end of the appendix. The list of references is not exhaustive, but it should enable the interested reader to follow work back to its source. 1. Space-Charge Waves Space-charge waves of the general sort considered are related to the plasma oscillations of Tonks and Langmuir. Waves in long beams were first discussed by Hahn^ and Ramo.^ The efl"ects of a velocity distribution are discussed by Pierce and by Bohm and Gross. ^ The negative energy of the slow space-charge wave has been reported b}^ Chu^ and by Walker.^ Chu gave the effective "voltage" U and the characteristic impedance K for the waves. 2. TheKkjstron Beck gives an adequate description of and references to klystrons. 3. The Resistive Wall Amplifier The effect has been discussed by Pierce, and a tube using it has been described by Birdsall, Brewer and Haeff.^*^ 4- The Easitron; Increasing Wave in a Lossless System The original easitron was a tube built by L. R. Walker at Bell Tele- phone Laboratories ; it was a 3-cm tube using half -wave wires as resonant elements. It has not been described in the literature. Pierce has discussed the operation of this sort of multi-resonator klystron on page 195 of Traveling Wave Tubes ^ and elsewhere.^ 5. Coupling of Modes of Propagation The operation of traveling-wave tubes was first explained in terms of coupling between an electromagnetic wave and a space-charge waA'e by C. C. Cutler in unpublished work. JMathews has made an analysis in WAVE PICTURE OF MICROWAVE TUBES 1371 these terms." Such coupHiig has been considered in general terms by- Pierce. 6. The Space-Charge-Wave Amplifier This tube was invented by Tien, Field and Watkins^* and is described in more detail by Tien and Field. 7. The Traveling Wave Tube Adequate descriptions and references are available in work by Kompf- ner/*^ Pierce," and Beck. 8. Double-Stream Amplifiers Descriptions and references are given by Pierce" and by Beck. 9. Noise Waves in Electron Streams Cutler and Quate have published experimental results. The theorems quoted are given by Pierce. 10. Noise Deamplification This was suggested by Tien, Field and Watkins^^ and is described in detail by Watkins^^ and Peter.' 11. Noise Cancellation Noise cancellation was first proposed by C. F. Quate. REFERENCES 1. Lewi Tonks and Irving Langmuir, Pliys. Rev., 33, pp. 195-210 and p. 990, 1929. 2. W. C. Hahn, Gen. Elect. Rev., 42, pp. 258-270, 1939. 3. Simon Ramo, Phys. Rev., 56, pp. 276-283, 1939. 4. J. R. Pierce, Jour. App. Phys., 19, pp. 231-236, 1948. 5. D. Bohm and E. P. Gross, Phys. Rev., 75, pp. 1851-1876, 1949, 79, pp. 992-1001, 1950. 6. L. J. Chu, paper presented at the Institute of Radio Engineers Electron Devices Conference, University of New Hampshire, June, 1951. 7. L. R. Walker, J. App. Phys., 25, pp. 615-618, May, 1954. 8. Thermionic Valves, A. H. W. Beck, Cambridge University Press, 1953. 9. J. R. Pierce, B. S.T.J. , 30, pp. 626-651, 1951. 10. Charles K. Birdsall, George R. Brewer and Andrew V. Haeff, Proc. I. R. E., pp. 865-871, 1953. 11. Traveling Wave Tubes, J. R. Pierce, Van Nostrand (1950). 12. W. E. Mathews, J. App. Phys., 22, pp. 310-316, 1951. 1372 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 13. J. R. Pierce, J. App. Phys., 25, pp. 179-183, Feb. 1954. 14. Ping King Tien, Lester M. Field and D. A. Watkins, Proc. I.R.E., 39, p. 194, 1951. 15. Ping King Tien and Lester M. Field, Proc. I.R.E., 40, pp. 688-695, 1952. 16. R. Kompfner, Rep. Progress. Phys., 15, pp. 275-327, 1952. 17. C. C. Cutler and C. F. Quate, Phys. Rev., 80, pp; 875-878, 1950. 18. J. R. Pierce, J. App. Phys., 8, pp. 93-933, 1954. 19. D. A. Watkins, Proc. I.R.E., 40, pp. 65-70, 1952. 20. R. W. Peter, R.C.A. Review, 13, pp. 344-368, 1952. 21. C. F. Quate, paper presented at the Institute of Radio Engineers Electron Devices Conference, University of New Hampshire, 1952. Theory of Open- Contact Performance of Twin Contacts By M. M. ATALLA and MISS R. E. COX (Manuscript Received June 17, 1954) The first 'part is a presentation of an analytical study of the open-contact performance of twin contacts. It provides means for predicting their per- formance from single contact data. It is shown that the prohability of failure of twin contacts is generally apprcciahly greater than the square of the prohahility of failure of single contacts. This is supplemented with the results of an experimental study which determines the effects of a few design parameters on the performance of single contacts. These are the parameters that determine the magnitude of improvement in performance obtained hy replacing single contacts hy twin contacts. INTRODUCTION The present s^\itching apparatus normally operates in atmospheres that may be contaminated with dust particles and foreign matter. Some apparatus components, particularly the contacts, are relatively sensitive to such contaminations which may interrupt the proper functioning of a pair of contacts. Normally, a single switching operation in a central office requires the operation of as many as a thousand relays or 10,000 contacts. To secure the high level of performance desired, it is evident that superla- tive performance and high degree of reliability of the contacts are es- sential. Many attempts have been and are being made to reduce the so-called "open" contact troubles due to foreign matter. Examples of environ- mental precautions are filtering the air supply to the central office, en- closing apparatus in cabinets, limiting personnel activities in the office, etc. An additional precaution incorporated in the apparatus design is the use of twin contacts. Such a scheme, when properly used, should result in substantial improvement in performance since an open can only take place when both members become open simultaneously. It may occur to one that the probability of a twin-contact open is the square of the prob- 1373 1374 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 ability of a single-contact open. Such a performance, however, has never been observed in practice where an improvement of only 10 : 1 is usually more tj^pical. A study of the mechanisms involved has revealed that only a very small number of opens are obtained due to the simultaneous oc- currence of opens on both members of a twin contact. The majority of opens, however, occur by having an open in one member of a twin con- tact which persists long enough to allow the occurrence of an open on the other member. By expressing this physical process in mathematical terms it was possible to develop a theory of performance of tmn contacts in terms of the characteristics of single contacts. NOTATION d Diameter of dust particle / Fractions of opens in single contacts cleared after A^ operations /^ The asymptotic value of / corresponding to A^ = oo n Average number of operations required to clear an open on a single contact r Distance of particle from center of circular open contact zone To Radius of "open zone" s Fraction of the twin contacts that are half open at any time, S = S„ + Sn s^ Fraction of twin contacts that are permanently half open Sn Fraction of t\\'in contacts that are temporaril}' half open w Alechanical wipe X Average displacement of a dust particle per contact operation F Contact force N Number of contact operations Ps Probability of occurrence of a single-contact open in opens/con- tact operation Pt Probability of occurrence of a twin-contact open in opens/contact operation X Total displacement distance to clear an open a = (2 - P.)(l - /J ^ = nf^Ps(2 - P.) 6 Angle of displacement if Slope of contact surface irregularity PRESENTATION OF THEORY Outline and Assumptions Consider a large group of t\\Tin contacts each constituting a pair of identical and entirely independent single contacts. After a period of OPEN-CONTACT PERFORMANCE OF TWIN CONTACTS 1375 operation a certain number of the twin contacts will become half -open.* These contacts will behave as if they were single-contacts until either: (1) the open half clears itself by operation, or (2) the other half becomes open leading to a twin-contact open. It is assumed that a failing twin- contact is cleared by an operator and then put back into service. In developing the theory it was necessary to represent by analytic expressions the rate of occurrence of opens on single contacts and the rate of their clearing by operation. These are approximations of a fairly large amount of experimental data consistently obtained from a number of tests on a variety of actual telephone relay contacts. (1) Rate of opens of single contacts "Ps": For a large number of single contacts at one set of operating conditions, the rate of opens is usually constant. This constant depends primarily on the quality and concentration of the offending foreign matter involved, the design of the contacts and their mechanism of actuation. Fig. 1 shows the results of three tests on single contacts of different design at different test condi- tions. The}'' all substantiate the assumption that the rate of opens of 1.2 y / ^REL/ \Y TYPE 1 A Y Ps = 1 V\ ) " I2,500y ^^ r / y ^ ^^^L I ^^HIGH DUST ^CONCENTRATION RELAY T ^^'low dust concentration^ rPE 2 c / 1 c ]^^^^ 27,000^ y/i Y ^ U^ r""! u= '^^ 40,000 6 8 10 12 14 THOUSANDS OF OPERATIONS Fig. 1 — Rate of opens of single relay contacts. * A half -open is defined as one where only one member, of a pair in a twin-con- tact, is open. In practice, a half -open is not normally detected. Only a simultane- ous open on both members of a twin contact will cause a circuit failure. 1376 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 single contacts is a constant: Ps = constant (1) where Ps is defined as the number of opens per "contact operation." In some cases, a certain transient period may precede the equilibrium char- acteristic "Ps = constant characteristic." This transient period is usually relatively short and is neglected in this analysis. (2) Clearing of opens on single-contacts by operation: If an open single contact is allowed to operate mechanically, it is possible that it will clear itself after a number of operations. As discussed in a later sec- tion, the opens obtained are never identical in nature. They, instead, have a certain statistical distribution which usually accounts for a wide spread in their clearing rate. If, however, the operating conditions are under control, the clearing characteristic of a set of contacts is found to follow a well defined and reproducible statistical distribution. Fig. 2 shows an accumulative distribution curve for clearing opens produced by cotton lint fibres.* The ordinate represents the fraction of the open single contacts that clear after N operations as given by the abscissa. In general, these relations have the following typical characteristics. The first operation following the occurrence of the open is the most efficientf single operation in clearing opens. It is usually responsible for clearing 10 to 30 per cent of the total number of opens. The subsequent opera- tions are progressively less efficient and in general a certain fraction foo Cj 3 rr— — V ■■ ^ o / u y f^ ^ 1 ^ ( ^ 1 1 1 5 6 8 10 20 NUMBER OF OPERATIONS, N 30 40 50 60 80 100 Fig. 2 — Distribution of clearing opens caused by lint. * This is one of the major causes of open contacts in central offices. These fibres are usually in ribbon form of various configurations. t This apparent efficiency is only due to the presence of opens that are more easy to clear than others. These will readily clear after one or a few operations. OPEN-CONTACT PERFORMANCE OF TWIN CONTACTS 1377 (1 — /„) of the opens will persist for a relatively large number of opera- tions. A study of a variety of these clearing characteristics generally indicates a rapid rise to the asymptotic value /„ in less than 100 opera- tions, and to / = 0.5 in less than 10 operations. As will be shown the fractional persistency (1 — /«,) is of major importance in determining twin-contact performance. In general, ho^^•ever, opens on twin-contacts are due to both the persistent half-opens and the temporary half-opens that might develop into twin-opens before clearing takes place. DEVELOPMENT OF THE THEORY Consider a large number of contacts operating at steady conditions. After N operations, let s^ be the fraction of the contacts that is per- manently half -open and Sn be the fraction that is temporarily half -open. As discussed, the number of operations necessary to clear a half-open is not constant and, for the majority of the contacts, is of the order of a few operations. To simplify the treatment, it is assumed that each temporary half-open \n\\ clear in an average of n operations from the time it first occurred. The fraction sn must, therefore, have been produced during the n operations directly preceding the time t. Since n is usually relatively small, the universe can be assumed to have had a negligible change during the operations n. Hence, Sn = (rate of formation of temporary half -opens) X n = 77[2(1 - s)Ps - (1 - s)P:']f^ where s = total fraction of half-opens = s^i + s«, • Substituting /3 = fif^Ps{2 - Ps) one gets s-n = 5-^ (1 - sj (2) Also, after A'' operations, the incremental change ds^ due to dN opera- tion is: ds^ = [2(1 - s)P. - (1 - s)P/](l - U dN - s^Ps dN where the second term is the reduction in s^ due to occurrence of twdn- opens. Substituting a = (2 — Ps)(l — /«) and combining with equation 2 to eliminate s-^ give: ds^ = 1 + « + ^ a PsdN A' (3) (4) 1 +/3 1 1 7^- -(l+a+|3)/(l+/3)P '" 1 + a + /3 1378 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Avhere K is an integration constant. Let at .V = 0, s^ = So , which is a description for the initial conditions of the contacts. The sokition be- comes : s^ = 1 + " + ^ „ \ -(l+a+^)/{l+^)P3.V 1 - 1 - — '-^ So ]e (5) 1 + « +/3 L The rate, or probabiUty, of twin-contact failure is determined from: P, = sP« + (1 - s)P/ (6) Substituting from 2 and 5 into 6 and reducing gives: P, = P/ 4- P.(l - Ps) Y^ l+a+iS\\ OL 7 /_ where, one may repeat for convenience: a = (2 - P.)(l - fj and /3 = 7y^P,(2 - P.) For all practical cases, Pg « 1, a = 2(1 — /«,) and /3 = 2nf^Ps < 1. Sub- stituting in 7 gives P, = P/ -f 2P, 1 fx, I -, I 1 3 ~ 2/^ ^ ^ _(3_2/ )p^A^ «/„P.+ _^^^l-^l___^^,,. (70 This is a general expression, relating the expected performance of twin- contacts to that of single contacts. It is evident that the idealistic per- formance of Pt = Ps", i.e., the probability of a twin-contact failure is the square of that for smgle contacts, can only be achieved if: (a) f„ = 1.0, i.e., persistent half-opens never occur, (b) il = 0, i.e., each temporary half-open occurring during one operation will clear during the subse- quent operation, and (c) So = 0, i.e., there is no initial contamination. These conditions are never obtained in practice and generally Pt is much greater than P/. Equation 7' also indicates that at the beginning of operation, when the exponent is much less than 1.0, Pt is given by: (Pt)o = P/(l + 2nU + PsSo (8) Numerically if il = 50, /„ = 1.0 and So = 0, Pj = 101 P/ which is 101 times worse than the idealistic performance of Pt = P^. The initial rate OPEN-CONTACT PERFORMANCE OF TWIN CONTACTS 1379 of failure of twin contacts is also quite sensitive to initial contact con- tamination. If, for example, So = 10~ , i.e., Hooo of the twin contacts are permanently half-open to start with, and for the same numbers used above and Ps = Ko^ equation 8 gives Pt = 1.1 X 10~ . This corresponds to an 11 fold increase in twin-contact failures just due to an initial con- tamination So of 0.1 per cent. This performance is also 1100 times worse than the idealistic performance of Pt = P/ = 10~^\ By operation, the performance of twin contacts will exponentially deteriorate according to equation 7. It will asymptotically approach a constant rate of failure given by: (Pt)^ = Ps\l + 2/I/J + 2Ps 1 -/o. (9) This is independent of the initial contamination So and is practically reached in a number of operations: (nth = 3/(P.(3 - 2/J) (10) The worst performance of twin contacts is obtained when f„ = 0, i.e., h d DUST PARTICLE _- CAUSING OPEN CONTACT ---^DUST PARTICLE JUST CLEARED Fig. 3 — Particle in open zone. 1380 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 1.0 / 0.8 -y / / Q LLI / % 0.6 LU -I / / z o 1- 0.4 < y / ,/* y / 0.2 y 0.1 n — 0.1 0.15 0.2 0.3 0.4 0.5 06 0.8 1.0 1.5 2 NX9J d Fig. 4 — iNx

1 ' o A A A^ A ( 1 " t > < 1 . . A A A A -• 1 • 1 " ( 0 ) k A A • • • • < ( 1 , I ) |) i i • • 0-°°' D i 1 O 0 0 F=10GM W O 0.006 CM • 0.010 CM A 0.018 CM A 0.025 CM » c o ) 1 I 1 1 1 1 2 4 6 8, 10 10 NUMBER OF OPERATIONS, N Fig. 5 — Effect of wipe. For a certain surface roughness and particle size d. 1382 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 tion function / should be unique if plotted against NvfF^ where the a and h are constants to be determined experimentally. Experiment Cotton lint fibres and clean contact surfaces were used exclusively in this study. The fibres were essentially in the form of ribbons a few microns thick and of variable width and length. By using a dust sepa- rator,* lint fibres A\ith a well controlled size distribution were collected on a glass plate. The setting used gave a rather uniform monolaj^er of fibres 80 per cent of which had a A\idth between 10 and 20 microns. The contacts tested were flat and made of palladium, f They were cleaned with methyl alcohol and distilled water, then dried. The col- lected lint fibres were transferred to the surface of one contact by a special adapter which allows the pressing of the contact on the glass LU 0.60 a. < LU do.50 z o 0.1 o9- ■ 1 1 • t ■ • [I '' • 1 c I < -t^ i A \ ' 1 . ' 1 1 c o J) > ) *■ A '^ A A < 1 ( 1 ( ) ( ) , o A!^ • • < b ♦ • • o oO»° ■ • • 0 0 W= 0.0025" F n 5 GMS O 10 GMS • 20 GMS A 30 GMS A 40 GMS ■ 80 GMS T A 1 , o A i 1 1 1 1 1 10 10"^ NUMBER OF OPERATIONS, N Fig. 6 — Effect of force. * Based on controlling sedimentation by adjusting air speed in a two-stage separator. t Contact surface roughness was controlled by frequently polishing the contact surfaces by a fixed process. OPEN-CONTACT PERFORMANCE OF TWIN CONTACTS 1383 plate with the fibres. The pressing force was the same as that used in the subsequent operation of the contacts. The contacts were then operated at four operations per second in a sealed compartment. After each closure a checking circuit using 48 volts, and a maximum current of 0.50 amp., checked the continuity in the contacts. When the open was cleared the unit automatically stopped and the corresponding number of operations was obtained from a counter. The maximum number of opera- tions allowed for each run was 2,000. For one set of operating conditions, it was necessary to repeat the above for at least 150 times before a repre- sentative clearing distribution was obtained. Results Effect of Mdpe: Fig. 5 shows the results obtained for a range of wipes between 0.006 and 0.025 cm at a constant force of 10 grams. As expected, the clearing rate was higher for larger wipes. Effect of force: Fig. 6 was obtained at a constant Avipe of 0.006 cm and a set of forces between 5 and 80 grams. Large forces gave higher clearing rates. The effect of changing the force, however, is not as significant as that of changing the wipe. As outlined in the preceding introduction, the above data was replotted as fraction clearing / versus Nw"F'^. The results are shown in Fig. 7 with a = 3 and 6 = 1.0. As indicated, the pomts converged to a single average line with comparatively small spread.* This shows that, at least for the range covered, the change in clearing rate obtained by changing the wipe say by a factor of two can also be obtained by changing the force by a factor of 8. To determine the persistency (1 — /„), one may choose an arbitrary number of operations for defining it. If 2,000 operations is chosen, one may determine from the above data the fraction, (1 — /2,ooo), that will persist to beyond 2,000 operations. This was done and the results are plotted in Fig. 8 as (1 — /2,ooo) versus F^'^w. This suggests the follow- ing relation: — /2,ooo) = e (11) where F is in grams and w in cms. This expression allows the determina- tion of the effects of force and wipe on the performance of twin contacts by substituting in Equations 7' through 10. Similarly the average number of operations n, used in the above equations, may be obtained. This may * This same convergence was obtained, but not presented here, by plotting / versus NF at constant iv and / versus Nw^ at constant F. 138-1 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 • - • - < < - V - \ ^i \ <1 - :\ - ■ oooooooo >OfM^— oooo oqooqqoq dddddddd 11 OOOOOOOiD 0»<1- - 1 XI X •■Vj 1^ - • \ c ■• V o < - ^ ■ X ■ < <1 - V n - X <1 - \ o IX <] ■ D \ - t a \ - '^^ ^\ <] - 1 ■o \ ^ o X o o ' \° - C o \^ > ■ \i o V X - f - %N \ - c K - N \\ ^ J- aadvano NOiiovad OPEN-CONTACT PERFORMANCE OF TWIN CONTACTS 1385 < 0.6 0.06 0.05 0.04 0.03 0.02 O \^ \-o — 0 0.005 0.010 0.015 0.020 0.025 0.030 F'/3w Fig. 8 — Persistency of opens at 2,000 operations. ▲ m ■ ^ i - k. ▲ ■ / • t k ■ ■ y ^ • ,y x^ • • CURVE FROM FIG.7 FINE POLISHING ROUGH POLISHING MASS =4 GRAMS • ■ ▲ j^ -2 2 Nw^F Fig. 9 — Effect of other parameters. 1386 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 be arbitrarily defined as the number of operations at which 50 per cent of the opens will have cleared. At this or any other value of / one obtains from Fig. 7 fiFw^ = constant or il is inversely proportional to Fw . OTHER EFFECTS Fig. 9 shows the results obtained by varying other parameters. The solid line, obtained from Fig. 7 is shown for comparison. Indicated are the effects of fine polishing and rough polishing of the contact surfaces and of increasing the mass of the moving contact from 0.5 to 4 grams. Bell System Technical Papers Not Published in this Journal Ahearn, a. J., see Hannay, N. B. Beach, A. L., see Guldner, W. G. BiDDULPH, R.i Short Term Autocorrelation Analysis and Correlatogram of Spoken Digits, J. Acous. Soc. Ain., 26, pp. 539-541, July, 1954. Brattain, W. H., see Garrett, C. G. B. BULLINGTON, K.^ Reflection Coefficients of Irregular Terrain, Proc. I.R.E., 42, pp. 1258-1262, Aug., 1954. DeWald, J. F.,^ and Lepoutre, Gerard^ The Thermoelectric Properties of Metal — Ammonia Sodium and Potassium at —33°, J. Am. Chem. Soc, 76, pp. 3369-3373, July, 1954. Fine, M. E.,^ and Kenney, Nancy T.^ Moduli and Internal Friction of Magnetite as Affected by the Low- Temperature Transformation, Phys. Rev., 94, pp. 1573-1576, June 15, 1954. ^ Bell Telephone Laboratories, Inc. 2 Faculty Libre des Sciences, Lille, Nord, France. 1387 1388 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Fletcher, K. C.,^ Yager, W. A.,^ Pearson, G. L.,^ and Merritt, F. R.^ Hyperfine Splitting in Spin Resonance of Group V Donors in Silicon, Letter to the Editor, Phys. Rev., 95, pp. 844-5, Aug. 1, 1954. Gambrell, J. B., Jr.i What is a Printed Publication Within the Meaning of the Patent Act, J. Patent Office Society, 36, pp. 391-405, June, 1954. Garrett, C. G. B.,^ and Brattain, W. H.^ Self Powered Semiconductor Amplifier, Letter to the Editor, Phys. Rev., 95, pp. 1091-1092, Aug. 15, 1954. Green, E. I.^ Creative Thinking in Scientific Work, Elec. Eng., 73, pp. 489-494, June, 1954. Gremling, R. C., see Wright, Marie G. Guldner, W. G.,^ and Beach, A. L.^ Gasometric Method for Determination of Hydrogen in Carbon, Analytical Chemistry, 26, pp. 1199-1202, July, 1954. Hannay, N. B.^ A Mass Spectrograph for the Analysis of Solids, Rev. Scient. Instr., 25, pp. G44-G48, July, 1954. Hannay, N. B.,^ and Ahearn, A. J.^ Mass Spectrographic Analysis of Solids, Analytical Chemistry, 26, pp. 1056-1058, June, 1954. Herring, Conyers^ Pole of Low Energy Phonons in Thermal Conduction, Phys. Rev., 95, pp. 954-965, Aug. 15, 1954. ' Bell Telephone Laboratories, Inc. TECHNICAL PAPERS 1389 HoGAN, C. L., see Van Uitert, L. G. Karlin, J. E., see Munson, W. A. Kelly, H. P.i Differential Phase and Gain Measurements in Color Television Sys- tems, Elec. Eng., 73, pp. 799-802, Sept., 1954. Kelly, H. P.^ Color Video Tester Checks Distortion, Electronics, 27, pp. 128-131, Sept., 1954. Kenney, Nancy T., see Fine, M. E. King, R. A.,^ and Morgan, S. P.^ Transmission Formulas and Charts for Laminated Coaxial Cables, Proc. I.R.E., 42, pp. 1250-1258, Aug., 1954. KisLiuK, Paul^ Arcing at Electrical Contacts on Closure — Part V. The Cathode Mechanisms of Extremely Short Arcs, J. Appl. Phys., 25, pp. 897- 900, July, 1954. Legg, V. E.,1 and Owens, C. D.^ Magnetic Ferrites: New Materials for Modern Applications, Elec. Eng., 73, pp. 726-729, August, 1954. Lepoutre, Gerard, see DeWald, J. F. Lovell, L. C., see Vogel, F. L. Maita, J. P., see Morin, F. J. Mason, D. R} Considerations on Chemical Engineering Design Problems (in French); Chimie et Industrie, 71, pp. 477-481, March, 1954. 1 Bell Telephone Laboratories, Inc. 1390 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 MCHUGH, K.3 Speculation on the Failure of Telephony, Telephony, 147, pp. 17-19, 42-43, Aug. 14, 1954. Merritt, F. R., see Fletcher, R. C. Merz, W. J.i Domain Formation and Domain Wall Motions in Ferroelectric BaTiOs Single Crystals, Phys. Rev., 95, pp. G90-698, Aug. 1, 1954. Morgan, S. P., see Kjng, R. A. MoRiN, F. J.,1 and Maita, J. P.^ Conductivity and Hall Effect in the Intrinsic Range of Germanium, Phys. Rev., 94, pp. 1525-1529, June 15, 1954. MuNSON, W. A.,^ and Karlin, J, E.^ The Measurement of Human Channel Transmission Characteristics, J. Acous. Soc. Am., 26, pp. 542-553, July, 1954. Owens, C. D., see Legg, V. E. Pearson, G. L., see Fletcher, R. C. Read, W. T., see Vogel, F. L. Schafer, J. P., see Van Uitert, L. G. SCHAWLOW, A. L.^ Nuclear Quadrupole Resonances in Solid Bromine in Iodine Com- pounds, Chem. Phys., 22, pp. 1211-1214, July, 1954. Shockley, W., see van Roosbroeck, W. 1 Bell Telephone Laboratories, Inc. 3 New York Telephone Company. technical papers 1391 Tien, Ping Kjng^ Bifilar Helix for Backward Wave Oscillators, Pioc. I.K.E., 42, pp. 1137-1143, July, 1954. Van Roosbroeck, W.,^ and Shockley, W.^ Photo-Radiative Recombination of Electrons and Holes in Ger- manium, Phys. Rev., 94, pp. 1558-1560, June 15, 1954. Van Uitert, L. G.,^ Schafer, J. P.,^ and Hogan, C. L.^ Low Loss Ferrites for Applications at 4,000 Millicycles per Second, Letter to the Editor, J. Appl. Phys., 25, p. 925, July, 1954. Vogel, F. L.,1 Read, W. T.,i and Lovell, L. C.^ Recombination of Holes and Electrons at Lineage Boundaries in Germanium, Letter to the Editor, Phys. Rev., 94, pp. 1791-1792, June 15, 1954. Walker, A. C.^ Hydrothermal Growth of Quartz Crystals, Ind. and Eng. Chem., 48, pp. 1670-1676, Aug., 1954. Wolff, P. A.^ Theory of Secondary Electron Cascade in Metals, Phys. Rev., 95, pp. 56-66, July 1, 1954. Wright, Marie G.,^ and Gremling, R. C.^ Xerographic Short Cut, Special Libraries, 45, pp. 250-251, July- August, 1954. Yager, W. A., see Fletcher, R. C. ^ Bell Telephone Laboratories, Inc. Recent Monographs of Bell System Technical Papers Not Puhlished in This Journal* Anderson, P. W., Merritt, F. R., Remeika, J. P., and Yager, W. A. Magnetic Resonance in a FCiOs , Monograph 2229. Barstow, J. M., and Christopher, H. N. Measurement of Random Monochrome Video Interference, Mono- graph 2259. Bond, W. L. Notes on Solution of Problems in Odd Job Vapor Coating, Monograph 2302. Christopher, H. N., see Barstow, J. M. Clark, M. A. An Acoustic Lens as a Directional Microphone, Monograph 2291. Clogston, a. M., and Heffner, H. Focusing of an Electron Beam by Periodic Fields, Monograph 2267, Cory, S. I. A New Portable Telegraph Transmission Measuring Set, Monograph 2248. Darrow, K. K. Solid state Electronics, Monograph 2253. * Copies of these monographs may be obtained on request to the Publication Department, Bell Telephone Laboratories, Inc., 463 West Street, New York 14, N. Y. The numbers of the monographs should be given in all requests. 1392 RECENT MONOGRAPHS 1393 DiTZENBERGER, J. A., See FuLLER, C. S. Fine, M. E., and Kenney, Nancy T. Moduli and Internal Friction of Magnetite as Affected by the Low- Temperature Transformation, Monograph 2251. Fracassi, R. D., and Kahl, H, Type-ON Carrier Telephone, Monograph 2296. Fuller, C. S., Struthers, J. D., Ditzenberger, J. A., and Wolf- STIRN, K. B. Diffusivity and Solubility of Copper in Germanium, Monograph 2270. Galt, J. K., Yager, W. A., and Merritt, F. R. Temperature Dependence of Ferromagnetic Resonance Line Width in a Nickel Iron Ferrite: A New Loss Mechanism, Monograph 2245. Gilbert, E. N. Lattice Theoretic Properties of Frontal Switching Functions, Mono- graph 2246. Green, E. I. The Decilog: A Unit for Logarithmic Measurement, Monograph 2254. Hagstrum, H. D, Instrumentation and Experimental Procedure for Studies of Electron Ejection by Ions and Ionization by Electron Impact, Monograph 2256. Heffner, H. Analysis of the Backward-Wave Traveling-Wave Tube, Monograph 2285. Heffner, H., see Clogston, A. M. Jaffe, H., see Mason, W. P. 1394 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Johnson, J. B., and McKay, K. G. Secondary Electron Emission From Germanium, Monograph 2279. Kaiil, H., see Fracassi, R. D. Kenney, Nancy, T., see Fine, M. E. KoMPFNER, R., and Williams, N. T. Backward-Wave Tubes, Monograph 2295. Kretzmer, E. R. An Amplitude Stabilized Transistor Oscillator, Monograph 2239. Lewis, H. W. Search for the Hall Effect in a Superconductor I. — Experiment, Monograph 2255. LiNVILL, J. G. RC Active Filters, Monograph 2260. LiNVILL, J. G. Transistor Negative-Impedance Converters, Monograph 2294. Maita, J. P., see Morin, F. J. Mason, W. P., and Jaffe, H. Methods for Measuring Piezoelectric, Elastic, and Dielectric Co- efficients of Crystals and Ceramics, Monograph 2241. McKay, K. G., see Johnson, J. B. Mendel, J. T., Quate, C. F., and Yocum, W. H. Electron Beam Focusing With Periodic Permanent Magnet Fields, Monograph 2240 Merritt, F. R., see Anderson, P. W. RECENT MONOGRAPHS 1395 Merritt, F. R., see Galt, J. K. MoRiN, F. J., and Maita, J. P. Conductivity and Hall Effect in the Intrinsic Range of Germanium, Monograph 2300. MoRiN, F. J., see Pearson, G. L. Pearson, G. L., Read, W. T., Jr., and Morin, F. J. Dislocations in Plastically Deformed Germanium, Monograph 2230. Pfann, W. G. Redistribution of Solutes by Formation and Solidification of a Molten Zone, Monograph 2290. Pierce, J. R. Coupling of Modes of Propagation, Monograph 2252. Prince, M. B. Drift Mobilities in Semiconductors I, Germanium II, and Silicon, Monograph 2271. QuATE, C. F., see Mandel, J. T. Read, W. T., Jr., see Pearson, G. L. Remeika, J. P. Method for Growing Barium Titanate Single Crystals, Monograph 2247. Remeika, J. P., see Anderson, P. W. Ryder, R. M., and Sittner, W. R. Transistor Reliability Studies, JNIonograph 2263. Shive, J. N., see Slocum, A. 1396 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Shockley, W., see Van Roosbroeck, W. SiTTNER, W. R., see Ryder, R. M. Slocum, a., and Shive, J. N. Shot Dependence of P-N Junction Phototransistor Noise, Mono- graph 2265. Smith, C. S. Piezoresistance Effect in Germanium and Silicon, Monograph 2233. Snoke, L. R. Observations on a Possible Method of Predicting Soil-Block Bio- assay Thresholds by Distillation Characteristics of the Weathered Creosotes, Monograph 2238. Thomas, D. E. A Point-Contact Transistor VHF FM Transmitter, Monograph 2234. Valdes, L. B. Resistivity Measurements on Germanium for Transistors, Mono- graph 2261. Van Roosbroeck, W., and Shockley, W. Photon-Radiative Recombination of Electrons and Holes in Ger- manium, Monograph 2306. Varney, R. N. Liberation of Electrons by Positive-Ion Impact on the Cathode of a Pulsed Townsend Discharge Tube, Monograph 2232. Walker, L. R. Stored Energy and Power Flow in Electron Beams, Monograph 2264. RECENT MONOGRAPHS 1397 Wick, R. F. Solution of the Field Problem of the Germanium Gyrator, Alono- graph 2301. Williams, N. T., see Kompfner, R. Wolff, P. A. Theory of Plasma Waves in Metals, Alonogiaph 2262. WoLFSTRiN, K. B., see Fuller, C. S. Yager, W. A., see Anderson, P. W., and Galt, J. K. YocoM, W. H., see Mendel, J. T. Contributors to this Issue M. M. Atalla, B.S., Cairo University, 1945; M.S., Purdue Univer- sity, 1947; Ph.D., Purdue University, 1949; Studies at Purdue under- taken as the result of a scholarship from Cairo University for four years of graduate work. Bell Telephone Laboratories, 1950-. For the past three years he has been a member of the Switching Apparatus Develop- ment Department, in which he is supervising a group doing fundamental research work on contact physics and engineering. Current projects in- clude fundamental studies of gas discharge phenomena between con- tacts, their mechanisms, and their physical effects on contact behavior; also fundamental studies of contact opens and resistance. In 1950, an article by him was awarded first prize in the junior member category of the A.S.M.E. He is a member of Sigma Xi, Sigma Pi Sigma, Pi Tau Sigma, the American Physical Society, and an associate member of the A.S.M.E. Rosemary E. Cox, B.A., Ladycliff College, 1949; M.S. Fordham University, 1950; Bell Telephone Laboratories, 1951-. Miss Cox, who taught high school mathematics for a year before coming to the Labora- tories, has been engaged in fundamental studies of contact physics. She won a New York State University Scholarship and scholarships from Ladycliff and Fordham. Gerald V. King, B.S., Carnegie Institute of Technology, 1920; Western Electric Company, 1921-24; Bell Telephone Laboratories, 1925-. Mr. King analyzed customer orders for step-by-step and manual systems for three years before turning to the design and checking of manual and dial PBX and community dial offices. From 1932 to 1939 he engaged in fundamental studies and development of new local and toll crossbar systems. He was involved in military work from 1939 to 1944 and since then he has been concerned with design and development of AMA accounting centers, central offices and crossbar tandem sys- tems. He was appointed Switching Systems Development Engineer in 1952. Stewart E. Miller, University of Wisconsin, 1936-39; B.S. and M.S., Massachusetts Institute of Technology, 1941. Bell Telephone 1398 CONTRIBUTORS TO THIS ISSUE 1399 Laboratories, 1941-. Since June 1954, ]\Ir. Miller has been Assistant Director of Radio Research at Holmdcl and has been in charge of re- search on guided systems and associated millimeter and microwave techniques. During World War II, he workcnl on airborne radar systems. He also worked on coaxial carrier transmissions systems. Mr. Miller holds patents in connection with automatic frequency control, an os- cillator control scheme and the D-C amplifier. Member of the I.R.E., Eta Kappa Nu, Tau Beta Pi and Sigma Xi. Norman A. Newell, E. E., Lehigh University, 1920. Mr. Newell joined Bell Telephone Lalwratories in 1934 after fourteen years with the American Telephone and Telegraph Company. He has formed reciuire- ments for the development of circuit and equipment arrangements for intertoll trunk signaling systems, local and toll manual switchboards, toll switchboard trunking systems and automatic toll systems. He helped coordinate these projects and prepared descriptions of the systems for the operating companies. Mr. Newell holds patents in the fields of straightforward trunking, toll-call timing and single-frequency signaling. He is a member of the A.I.E.E., Tau Beta Pi and Pi Delta Epsilon. William Pferd, B.S. in M.E. Rutgers University, 1947; M.S. in M.E., Newark College of Engineering 1951. Mr. Pferd has recently been con- cerned with coin collector development. Previously, he worked on design and de\^elopment of the station ringer for the 500-type telephone set and the dial mechanism for the same set. During World War II he served as a photographic intelligence officer in Italy with the 98th Bomb Group. John R. Pierce, B.S., M.S., Ph.D., California Institute of Tech- nology, 1933, 1934 and 1936; Bell Telephone Laboratories, 1936-; Ap- pointed Director of Electronics Research, 1952. Dr. Pierce has specialized in the development of electron tubes and microwave research since join- ing the Laboratories. During the war he concentrated on the develop- ment of electronic devices for the armed forces. Since the war he has done research leading to the development of the beam traveling- wave tube for which he was awarded the 1947 Morris Liebmann Memorial Prize of the Institute of Radio Engineers. Dr. Pierce is the author of two books: Theory and Design of Electron Beams, published in second edition this year, and Traveling Wave Tubes (1950). He was voted the "Out- standing Young Electrical Engineer of 1942" by Eta Kappa Nu. Mem- ber of the A.I.E.E., Tau Beta Pi and Eta Kappa Nu. Fellow of the American Physical Society and the I.R.E. 1400 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 Allan Weaver, B.S. in E. E., University of Nebraska, 1921. Since 1945 Mr. Weaver has been in charge of a group concentrated on toll sig- naling with particular attention to the development of single-frequency signaling for use in connection with nationwide dialing systems. He joined Bell Telephone Laboratories in 1934 after thirteen years with American Telephone and Telegraph Company and was first concerned with the development of telegraph, telephotograph and teletypewriter systems. During World War II he was assigned to radar development. He holds thirty-seven patents in the fields of telegraphy, teletypography and sig- naling. Mr. Weaver is a member of the A.I.E.E., I.R.E. and Sigma Xi. ^