piitLi^6 April, 1949 A Carrier System for 8000-Cycle Program Transmission— i?. A. Leconle, D. B. Penick, C. W. Schramm and A.J. Wier 165 Delay Equalization of Eight-KiloOycle Carrier Program C^ircuits— C. H. Dagnall and P. W . Rounds 181 Band Pass Filter, Band Elimination Filter and Phase Simulating Net- work for Carrier Program Systems— F. S. Farkas, F. J. Hallenbeck and F. E. Slehlik 1^6 MAY 1 7 1950 iv BELL SYSTEM TECHNICAL JOURNAL A Precise Direct Reading Phase and Transmission Measuring System for Video Frequencies — D. A. Alsberg and D. Leed 221 Physical Principles Involved in Transistor Action — J. Bardeen and W. H. Bratlain 239 Lightning Current Observations in Buried Cable — H. M. Trueblood and E. D. Sunde 278 The Electrostatic Field in Vacuum Tubes with Arbitrarily Spaced Elements — W. R. Bennett and L. C. Peterson 303 Transconductance as a Criterion of Electron Tube Performance — T. Slonczewski 315 July, 1949 Editorial Note regarding Semiconductors 335 Hole Injection in Germanium — Quantitative Studies and Filamentary Transistors — W. Shockley, G. L. Pearson and J. R. Haynes 344 Some Circuit Aspects of the Transistor — R. M. Ryder and R. J. Kircher 367 Theory of Transient Phenomena in the Transport of Holes in an Excess Semiconductor^ — Conyers Herring .- . 401 On the Theory of the A-C. Impedance of a Contact Rectifier — /. Bardeen 428 The Theory of p-n Junctions in Semiconductors and p-n Junction Transitors — IF. Shockley 435 Band Width and Transmission Performance — C. B. Feldman and W. R. Bennett 490 October, 1949 Reactance Tube Modulation of Phase Shift Oscillators — F. R. Dennis and E. P. Fetch 601 A Broad-Band Microwave Noise Source — IF. IF. Mumford 608 Electronic Admittances of Parallel-Plane Electron Tubes at 4000 Mega- cycles— Sloan D. Robertson 619 Passive Four-Pole Admittances of Microwave Triodes — Sloan D. Robertson 647 Communication Theory of Secrecy Systems — C. E. Shannon 656 The Design of Reactive Equalizers — A. P. Brogle, Jr 716 Index to Volume XXVIII Admiltances, Klcctronic, of Parallel-Plane Electron Tubes al 4000 Megacycles, Sloan D. Robertson, page 619. Admittances, I^assive Four-Pole, of Microwave Triodes, Sloan D. Robertson, page 647. Alberslieim, W. J., Propagation of TEoi Waves in Curved Wave Guides, page 1. Alsher^, D. A. and D. Leed, A Precise Direct Reading Phase and Transmission Measuring System for Video Frequencies, j)age 221. Amplification b)' Interaction between Two Electron Streams, Experimental Observation of, .4. ]'. Hollenberg, page 52. Amplifier, High-Frequenc\-, A New Type of, J . R. Pierce and W. B. Hebenstreil, page ii. B Band Width and Transmission Performance, C. B. Feldman and W . R. Bennett, page 490. Bardeen, J., On the Theory of the A-C. Impedance of a Contact Rectifier, page 428. Bardeen, J. and IV. H. Brattain, Physical Principles Involved in Transistor Action, page 239. Bennett, \V. R. and C. B. Feldman, Band Width and Transmission Performance, page 490. Bennett, W. R. and L. C. Peterson, The Electrostatic Field in Vacuum Tubes with Arbi- trarily Spaced Elements, page 303. Brattain, W . H. and J. Bardeen, Physical Principles Involved in Transistor Action, page 239. Broad-Band Microwave Noise Source, A, W. W. Muniford, page 608. Brogle, A. P. Jr., The Design of Reactive Equalizers, page 716. Cable, Buried, Lightning Current Observations in, H. M. Trueblood and E. D. Sunde, page 278. Carrier Program Circuits, Eight-Kilocycle, Delay Equalization of, C. H. Dagnall and P. W . Rounds, page 181. Carrier Program Systems; Band Pass Filter, Band Elimination Filter and Phase Simulating Network for, F. S. Farkas, F. J. Hallenbeck, and F. E. Slehlik, page 196. Carrier System for 80C0-Cycle Program Transmission, A, R. A. Leconte, D. B. Penick, C. W . Schramm, and A. J. Wier, page 165. Communication Theorj- of Secrecy Sj'stems, C. E. Shannon, page 656. Dagnall, C. H. and P. W. Rounds, Delay Equalization of Eight-Kilocycle Carrier Program Circuits, page 181. Dennis, F. R. and E. P. Felch, Reactance Tube Modulation of Phase Shift Oscillators, page 601. Diflerential Ecjuations, A Set of Second-Order, Associated with Reflections in Rectangular Wave Guides — Application to Guide Connected to Horn, 5. O. Rice, page 136. £ Editorial Note regarding Semiconductors, page 335. Electron Streams, Two, Exjjerimental Observation of Amplification bj- Interaction Be- tween, A. V. Hollenberg, page 52. Electron Tube Performance, Transconductance as a Criterion of, T. Slonczeicski, page 315. Electron Tubes, Parallel-Plane, at 4000 Megacycles, Electronic Admittances of, Sloan D. Robertson, page 619. vi BELL SYSTEM TECHNICAL JOURNAL Electrostatic Field, The, in Vacuum Tubes with Arbitrarily Spaced Elements, W. R. Bennett and L. C. Peterson, i)age 303. Equalizers, Reactive, The Design of, A. P. Brogle, Jr., page 716. Equalization, Delay, of Eight-Kilocycle Carrier Program Circuits, C. H. Dagnall and P. W. Rounds, page 181. F Parkas, F. S., P. J. Hallenbeck, and P. E. Stehlik, Band Pass Filter, Band Elimination Filter and Phase Simulating Network for Carrier Program Systems, page 196. Felch, E. P. and F. R. Dennis, Reactance Tube Modulation of Phase Shift Oscillators, page 601. Peldman, C. B. and W. R. Bennett, Band Width and Transmission Performance, page 490. Filters: Band Pass Filter, Band Elimination Filter and Phase Simulating Netwcrk for Carrier Program Systems, P. S. Parkas, P. J. Hallenbeck, and P. E. Stehlik, page 196. G Germanium, Hole Injection in — Quantitative Studies and Filamentary Transistors, W. Shockley, G. L. Pearson, and J. R. Haynes, page 344. H Hallenbeck, P. J., P. E. Stehlik, and P. S. Parkas, Band Pass Filter, Band Elimination Filter and Phase Simulating Network for Carrier Program Systems, page 196. Haynes, J. R., W. Shockley, and G. L. Pearson, Hole Injection in Germanium — Quantitative Studies and Filamentarj- Transistors, page 344. Hebenslreit, W. B. and J. R. Pierce, A New Type of High-Frequency Amplifier, page 33. Herring, Conyers, Theory of Transient Phenomena in the Transport of Holes in an Excess Semiconductor, page 401. Hole Injection in Germanium — Quantitative Studies and Filamentary Transistors, W . Shockley, G. L. Pearson, and J . R. Haynes, page 344. Holes in an Excess Semiconductor, Theory of Transient Phenomena in the Transport of, Conyers Herring, page 401. Hollenberg, A. V., Experimental Observation of Amplification by Interaction between Two Electron Streams, page 52. I Impedance, A-C, of a Contact Rectifier, On the Theory of the, /. Bardeen, page 428. K Kircher, R. J. and R. M. Ryder, Some Circuit Aspects of the Transistor, page 367. L Leconte, R. A., D. B. Penick, C. W. Schramm, and A. J. Wier, A Carrier System for 8000- Cycle Program Transmission, page 165. Leed, D. and D. A. Alsberg, A Precise Direct Reading Phase and Transmission Measuring System for Video Frequencies, page 221. Lightning Current Observations in Buried Cable, H. M. Trueblood and E. D. Sunde, page 278. M Microwave Frequencies, A Method of Measuring Phase at, Sloan D. Robertson, page 99. Microwave Noise Source, A Broad-Band, W. W. Mumford, page 608. Microwave Triodes, Passive Four-Pole Admittances of, Sloan D. Robertson, page 647. Microwaves: Reflection from Corners in Rectangular Wave Guides— Conformal Trans- formation, S. O. Rice, page 104. A Set of Second-Order Differential Equations .\sso- ciated with Reflections in Rectangular Wave Guides— Application to Guide Con- nected to Horn, .S'. O. Rice, page 136. Modulation, Reactance Tube, of Phase Shift Oscillators, P. R. Dennis and E. P. Pelch, page 601. Mumford, W. W., A Broad-Band Microwave Noise Source, page 608. INDEX vii N Noise Source, A Broad-Hand Microwave, W . W. Mumford, page 608. O Oscillators, Phase Shift, Reactance Tube Modulation of, F. R. Dennis and E. F. Felcli, page 601. Pearson, G. L., J. K. Haynes, and W. Shockley, Hole Injection in Germanium — Quantita- tive Studies and Filamentary Transistors, page 344. Fenick, D. B., C. W. Schramm, A. J. Wier and R. A. Leconte, A Carrier System for 8000- Cycle Program Transmission, page 165. Feterson, L. C. and W. R. Bennett, The Electrostatic Field in Vacuum Tubes with Arbi- trarily Spaced Elements, page 303. Phase at \Iicrowave Frequencies, A Method of Measuring, Sloan D. Robertson, page 99. Phase, A Precise Direct Reading; and Transmission Measuring System for Video Fre- quencies, D. A. Alsberg and D. Leed, page 221. Phase Shift Oscillators, Reactance Tube Modulation of, F. R. Dennis and E. P. Fetch, page 601. Pierce, J. R. and W. B. Hebenstreit, A New Tj-pe of High-Frequency Amplifier, page 33. Program Circuits, Eight-Kilocycle Carrier, Delay Equalization of, C. H. Dagnall and P. W. Rounds, pa.ge 181. Program Systems, Carrier; Band Pass Filter, Band Elimination Filter and Phase Simu- lating Network for, F. S. Farkas, F. J. Hallenbeck, and F. E. Stehlik, page 196. Program Transmission, 8000-Cycle, A Carrier System for, R. A. Leconte, D. B. Penick, C. W . Schramm, atid A. J. Wier, page 165. Propagation of TEoi Waves in Curved Wave Guides, W. J. Albersheim, page 1. R Reactive Equalizers, The Design of, A. P. Brogle, Jr., page 716. Rectifier, Contact, On the Theory of the A-C. Impedance of a, /. Bardeen, page 428. Reflection from Corners in Rectangular Wave Guides — Conformal Transformation, S. 0. Ri^:e, page 104. Rice, S. 0., A Set of Second-Order Differential Equations Associated with Reflections in Rectangular Wave Guides — Application to Guide Connected to Horn, page 136. Reflection from Corners in Rectangular Wave Guides — Conformal Transformation, page 104. Robertson, Sloan D., A Method of Measuring Phase at Microwave Frequencies, page 99. Electronic Admittances of Parallel-Plane Electron Tubes at 4000 Megacycles, page 619. Passive Four-Pole Admittances of Microwave Triodes, page 647. Rounds, P. W. and C. II. Dagnall, Delay Equalization of Eight-Kilocycle Carrier Program Circuits, page 181. Ryder, R. M. and R. J. Kircher, Some Circuit Aspects of the Transistor, page 367. Schramm, C. W., A. J. Wier, R. A. Leconte, and D. B. Penick, A Carrier System for 8000- Cycle Program Transmission, page 165. Secrecy Systems, Communication Theory of, C. E. Shannon, page 656. Semiconductor, Excess, Theory of Transient Phenomena in the Transport of Holes in an, Conyers Herring, page 401 . Semiconductors, Editorial Note regarding, page 335. Semiconductors and p-n Junction Transistors, The Theory of {)-n Junctions in, W. Shockley, page 435. Semiconductors: On the Theory of the AC. Impedance of a Contact Rectitier, J. Bardeen, page 428. Shannon, C. E., Communication Theory of Secrec)- Systems, page 656. The Synthesis of Two-Terminal Switching Circuits, page 59. Shockley, W., The Theory of p-n Junctions in Semiconductors and p-n Junction Transis- tors, page 435. viii BELL SYSTEM TECHNICAL JOURNAL Shockley, W ., G. L. Pearson, and J . R. Haynes, Hole Injection in Germanium — Quantita- live Studies and Filament ;iry Transistors, page 344. Slonczewski, T., Transconductance as a Criterion of Electron Tube Performance, page 315. Slelilik, F. E., F. S. Farkas, and F. J. Ilallenbeck, Band Pass Filter, Band F^limination Filter and Phase Simulating Network for Carrier Program Systems, page 196. Siinde, E. D. and II. M. Trueblood, Lightning Current Observations in Buried Cable, page 278. Switching Circuits, Two-Terminal, The Synthesis of, Claude E. Shannon, page 59. Transconductance as a Criterion of Electron Tube Performance, T. Slonczewski, page 315. Transformation, Conformal — Reflection from Corners in Rectangular Wave Guides, S. 0- Rice, page 104. Transient Phenomena in the Transport of Holes in an Excess Semiconductor, Theory of, Conyers Herring, page 401. Transistor, Some Circuit Aspects of the, R. M. Ryder and R. J. Kircher, page 367. Transistor Action, Physical Principles Involved in, J. Bardeen and W. II. Braltain, page 239. Transistors: Editorial Note regarding Semiconductors, page 335. Theory of Transient Phenomena in the Transport of Holes in an Excess Semiconductor, Conyers Herring, page 401. Transistors, Filamentary, and Quantitative Studies — Hole Injection in Germanium, W. Shockley, G. L. Pearson, and J. R. Haynes, page 344. Transistors, p-n Junction, The Theory of p-n Junctions in Semiconductors and, W. Shock- ley, page 435. Transmission, 8000-Cycle Program, A Carrier System for, R. A. Leconte, D. B. Penick, C. W. Schramm, and A. J. Wier, page 165. Transmission Measuring System for Video Frequencies, A Precise Direct Reading .Phase and, D. A. Alsberg and D. Leed, page 221. Transmission Performance, Band Width and, C. B. Feldman and W . R. Bennett, page 490. Triodes, Microwave, Passive Four-Pole Admittances of, Sloan D. Robertson, page 647. Trueblood, H. M. and E. D. 5««de, Lightning Current Observations in Buried Cable, page 278. Tube Performance, Electron, Transconductance as a Criterion of, T. Slonczewski, page 315. Tubes, Parallel-Plane Electron, at 4000 Megacycles, Electronic Admittances of, Sloan D. Robertson, page 619. Tubes, Vacuum, with Arbitrarily Spaced Elements, The Electrostatic Field in, W. R. Bennett and L. C. Peterson, page 303. Vacuum Tubes with Arbitrarily Spaced Elements, The Electrostatic Field in, W. R. Bennett and L. C. Peterson, page 303. Video Frequencies, A Precise Direct Reading Phase and Transmission Measuring System for, D. A. Alsberg and D. Leed, page 221. W Wave Guides, Curved, Propagation of TEoi Waves in, IV. J. Albersheim page 1. Wave Guides, Rectangular, Reflection from Corners in — Conformal Transformation, S. 0. Rice, page 104. Wave Guides, Rectangular, A Set of Second-Order Differential Equations Associated with Reflections in — Application to Guide Connected to Horn, S. 0. Rice, page 136. Wier, A. J., R. A. Leconte, D .B. Penick, and C. W. Schramm, \ Carrier System for 8000- Cycle Program Transmission, page 165. VOLUME XXVIII JANUARY, 1949 no. i THE BELL SYSTEM *^^5* TECHNICAL JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION Propagation of TEoi Waves in Curved Wave Guides W. J. Albersheim 1 A New Type of High -Frequency Amplifier J. R. Pierce and W. B. Hebenstreit 33 Experimental Observation of Amplification by Interaction Between Two Electron Streams. . . .A. V. Hollenberg 52 The Synthesis of Two-Terminal Switching Circuits Claude E. Shannon 59 A Method of Measuring Phase at Microwave Frequen- cies Sloan D. Robertson 99 Reflection from Comers in Rectangular Wave Guides — Conformal Transformation. S. O. Rice 104 A Set of Second-Order Differential Equations Associated with Reflections in Rectangular Wave Guides — Ap- plication to Guide Connected to Horn . . . . S. O. Rice 136 Abstracts of Technical Articles by Bell System Authors ... 157 Contributors to this Issue 162 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 50^ per copy $1.50 per Year \K THE BELL SYSTEM TECHNICAL JOURNAL Published quarterly by the American Telephone and Telegraph Company 195 Broadway, New York, N. Y. »»ii««»t« ■"■ ' EDITORS R. W. King J. O. Perrine EDITORIAL BOARD C. F. Craig O. E. Buckley O. B. BlackweU M. J. KeUy H. S. Osborne A. B. Clark J. J. PiUiod F. J. Feely ■ii«ii»ii« ■ i«ii» ■ It ■ ■ SUBSCRIPTIONS Subscriptions are accepted at $1.50 per year. Single copies are 50 cents each. The foreign postage is 35 cents per year or 9 cents per copy. Copyright, 1949 American Telephone and Telegraph Company PRINTED IN U.S. A. The Bell System Technical Journal Vol. XXVIII Janitarv, 1949 No. 1 Propagation of TEqi Waves in Curved Wave Guides By W. J. ALBERSHEIM TFloi waves transmitted through curve wave guides lose power by conversion to other modes, especially to TMn. This power transfer to coupled modes is explained by the theor\- of coupled transmission lines. It is shown that the power interchange between coupled lines and their propagation constants can be derived from a single couiJJing dis- criminant. Earlier calculations of TEoi conversion loss in circular wave guide bends are confirmed and extended to S-shaped bends. Tolerance limits for random deflections from an average straight course arc given. THE TEoi mode of propagation in circular wave guides has great potential value for the transmission of wide-band signals because its attenuation decreases with frequency. In order to take full advantage of this property one must use sufficiently large wave guides to operate well above the cutoff of the lowest transmitted frequency. The difficulty of this transmission method lies in the fact that TEoi is not the dominant mode and that energy may be lost by transfer to the many other modes capable of transmission in the wave guide. In an ideal wave guide, which is perfectly straight, perfectly circular and perfectly conducting, the propagation is undisturbed; but slight imperfections and especially a slight curvature of the wave guide axis may produce serious disturbances. The character of these disturbances has been investigated in several publications by Prof. M. Jouguet^ and in unpublished work by Mr. S. O. Rice of the Bell Telephone Laboratories. Both Jouguet and Rice use the method of perturbations, which is a form of calculus invented by astronomers to compute the deviations from the exact elliptical orbits of the planets which are caused by the disturbing influences of their fellow planets. Although the above-mentioned authors obtained valuable results, the interpretation of their solutions is difficult due to this rather abstract mathematical formulation. To most engineers the understanding of a physical problem is greatly helped if it is possible to use a method of analysis which is elementary in character and easily interpreted in familiar physical terms. The familiar concept on which the present treatment will be based is that of coupled circuits. ' See References 2 and 3, listed on page 7. 1 2 BELL SYSTEM TECHNICAL JOURNAL It has been stated by the earlier authors that the curvature of the wave guide produces a coupling between modes. Before going into a detailed analysis one may estimate by inspection the nature of this coupling and the kind of modes that are most strongly coupled to each other. Figure la shows the cross section and the longitudinal section of a straight cylindrical wave guide. The location of every point inside the wave guide is determined by three coordinates: the radial distance r from the cylinder axis; the azimuth angle (p from an arbitrary 0 line and the axial distance z from the a- CYLINDRICAL CO-ORDINATES IN STRAIGHT WAVEGUIDE TOROIDAL CO-ORDINATES IN CURVED WAVEGUIDE Fig. 1 origin. If the wave guide is bent as shown on Fig. lb, but a wave front at right angles to the cylinder axis is to be maintained, the waves must be shortened at the inside of the bend and lengthened at the outside of the bend. Regarding compression as a positive and expansion as a negative deformation, one sees that the distortion of the wave shape is proportional to the curvature of the wave guide multiplied by the cosine of the azimuth angle. It is natural to assume that the coupling between modes is propor- tional to this distortion. Now it is known that all modes of propagation in a circular wave guide CURVED WAVE GUIDES 3 can be derived from functions Jnix^) cos }up. In these functions, n is called the aximuthal index because it indicates the type of symmetry around the circumference of the wave guide. When these characteristic functions are multiplied by the distortion factor cosine (p, the resulting expressions are proportional to the sum of cosine (w -{- I) (p and cosine (n — 1) tp. This means that the bending of the wave guide couples mainly those modes which dififer by ±1 in azimuth index. Since the TEoi mode has the azimuthai index 0, it is coupled to all modes of the type TEi™ and TMi^ . In the above qualitative discussion we have claimed that coupling exists without defining the physical coupling parameters and their effects. We must now supply this definition and show that the TEoi mode is particularly susceptible to coupling losses. ^2 a-COUPLED TRANSMISSION LINES Fig- 2 b-COUPLED RESONATORS Our investigation is guided by S. A. ScheUcunoff's statement^ that a wave guide mode has the same equation of propagation as a high-pass transmission line. Schelkunoff further points out^ that the high-pass character of circular wave guide modes can be interpreted as the effect of interfering plane waves whose directions of propagation deviate from the wave guide axis by a constant slanting angle. We therefore approach the problem of coupled wave guide modes by studying the behavior of two coupled transmission lines such as shown on Fig. 2a. Each transmission line is schematically shown as an array of small ladder sections. The series impedances per unit length of the lines are Zi and 22 ; their shunt admittances per unit length, yi and y2 . The two lines are loosely coupled by small mutual series impedances per unit length (Zm) and by small mutual shunt admittances per unit length (>»„,)• A network of coupled ladder sections is more tractable than a wave guide structure, but still somewhat complicated. Let us therefore carry the analogy one step further. Figure 2b shows two resonant circuits, each 2 Ref. 4, pp. 378 and 381 of the book. « Ref. 4, p. 410 of the book. 4 BELL SYSTEM TECHNICAL JOURNAL consisting of single capacity C, and an impedance Z which includes an inductance L and a damping resistance R. The resonators are coupled by a small mutual inductance Z„, and by a small mutual capacity Ym ■ The behavior of coupled resonators is very well known to radio engineers. They occur as tuned transformers in amplifier circuits, as band-pass filters and as "tank circuits" in radio transmitters. Even before the advent of radio, their acoustical equivalents were studied in the form of resonant tuning forks. The mathematical aspects of this problem were already clearly set forth in a paper by \Men written in 1897^ He showed that the interaction between the free vibrations of two tuned circuits depends on the coupling coefficient and on the ratio of their complex resonance fre- quencies. The closer the two frequencies are to each other, the less coupling is needed to transfer energy between the two circuits. The reason is that the individual free vibrations of two nearly synchronous circuits remain in step long enough to accumulate the small energy transfer impulses of many vibrations. Now consider the two transmission lines of Fig. 2a and assume that a constant frequency signal is impressed upon the input of one or both of them. The signals are carried along the two lines as traveling waves. Again it is true that loosely coupled signals affect each other strongly if they remain in step. With traveling waves "remaining in step" means that they must travel with approximately equal phase velocities. We conclude that the phase velocities or phase constants of coupled transmission lines play a similar role as the resonant frequencies of coupled tuned circuits. This intuitive reasoning is confirmed by analysis (see Section 1 of the analytical part of this paper). We thus find that we must expect trouble for TEoi wave guide trans- mission if a mode with an azimuth index 1 has a propagation constant close to that of the TEoi . It so happens that there exists one mode, the TMu , which in an ideal wave guide has exactly the same propagation constant as the TEoi . This then should be the principal source of trouble — and from previous work it is known that such is the case. Our discussion of coupled transmission lines has shown that the interaction effects are functions of their relative uncoupled propagation constants and of the coupling coeflicient. The propagation constants of the TEoi and TMu wave guide modes are known but their coupling coefficient remains to be found. Since the energy of the transmission modes is located in the dielectric inside the wave guide, we consider first the coupling between the plane "slant wave" groups from which the modes are built up. ^ Reference 5. CURVED WAVE CI IDES 5 As shown in the analytical part, the couphng coefficient of these slant waves may be defined as the energy interchanged between the modes per unit length of line divided by the geometric mean of the energies per unit length stored up in each of the modes. b'rom the coupling coefficient of the slant waves the coupling coefficient of the wave guide modes is derived. On the basis of the above physical interpretation the analysis is carried out and the properties of TEoi propagation through curved wave guides of various shapes are derived in the analytical part of this paper which is subdivided into the following nine sections: Section 1 develops an approximate theory of loosely coupled, weakly damped circuits. The theory is first derived for coupled resonators which are familiar to communication engineers, and then applied in similar form to coupled transmission lines. It is shown that the important interaction properties of coupled lines are functions of a single coupling discriminant. The relative energy content of the two lines in each of the two possible coupled modes is plotted as a function of the coupling discriminant. Section 2 contains the field equations of a straight circular wave guide and their modification by a toroidal bend. Section 3 gives the solutions of the field equations for the uncoupled TEoi and TjMu modes in wave guides with infinite, and with small but finite conductivity. Section 4 applies the coupling theory to the TEoi and TMh modes in circular wave guide bends. The coupling coefficient, coupling discriminant and energy division between the two modes are deri\'ed as functions of the wave guide diameter bending radius and conductivity and of the signal frequency. Section 5 derives the critical bending radius and the attenuation of TEoi waves in long wave guides of constant curvature. Two numerical examples are given. Section 6 shows that in a curved section of wave guide which follows a long straight section or other source of pure TEoi the energy fluctuates back and forth between a condition of pure TEoi and of predominant TMu . 'llie length and magnitude of the fluctuations are derived. Section 7 computes the increase in average attenuation caused by serpen- tine bends of regular shapes. Numerical examples are tabulated. Section 8 shows that the results of Section 7 can be applied to helical bends and to small two-dimensional random deviations from a straight course. Section 9 shows that for any given statistical distribution of random angular deviations the average attenuation is minimized by an optimum 6 BELL SYSTEM TECHNICAL JOURNAL wave guide radius for each signal wave length and by an optimum signal wave length for each wave guide radius. Numerical examples are given for sinusoidal bends. Summary of Results 1. The energy loss of TEoi waves in curved wave guides by conversion into the TMu rnode is interpreted as a case of coupling between resonant transmission lines. 2. In a pair of coupled lines the energy cannot be confined entirely to a single Une but travels through both in one or both of two possible combination modes. 3. All important properties of coupled circuits, including wave guide modes, are functions of a single discriminant. 4. When the discriminant is much smaller than one, most of the energy can be carried in one line or component mode. 5. When the discriminant is much larger than one, the energy flow is nearly equally divided between the two lines or component modes. 6. In wave guides of typical dimensions the coupling discriminant becomes one for a "critical" bending radius greater than a mile. For all sharper bends, that is for most practical installations, the discriminant is greater than one. 7. In a long wave guide section with more than critical curvature the average attenuation constant is the arithmetic mean between those of the TEoi and the TMu modes. 8. If a wave guide region carrying pure TEoi is followed by a curved region, the energy in the curved region fluctuates back and forth between pure TEoi and predominant TMu . The location of TEoi minima and maxima is a function of the signal frequency, the wave guide diameter and the total bending angle. 9. For highly supercritical curvatures the bending angles at which minima and maxima occur are nearly independent of the curvature and approach the limiting values previously computed by Jouguet and Rice. The minima approach zero. When the bending radius approaches or exceeds the critical value, the maxima and minima become shallower and their spacing is increased by a function of the coupling discriminant. 10. For regular serpentine bends or random angular deviations from an average straight course which are much smaller than the first extinc- tion angle, the percentage increase in average attenuation is propor- tional to the square of the maximum deviation and to the fourth power of wave guide diameter and signal frequency. 11. Wave guide installations of practical dimensions for frequencies now CURVED WAVE GUIDES 7 attainable are tolerant to random angular deviations of the order of 1 degree. 12. For any expected distribution of random angular deviations there exists an optimum wave guide radius for each signal wave length and an optimum signal wave length for each wa\'e guide radius, which minimize the average attenuation. References 1. Jahnke & Emde, Tables of Functions, Dover Publications, New York, 1943. 2. M. Jouguet, Propagation dans les tujaux courbes, Comptes Rendus — Academie des Sciences, Paris, Feb. 18, 1946, March 4, 1946 and Jan. 6, 1947. 3. M. Jouguet, Effets de la courbure dans un guide a section circulaire, Cables & Trans- mission, 1 No. 2, July 1947, pp. 133-153. 4. S. A. Schelkunoff, Electromagnetic Waves, D. Van Nostrand Company, Inc., New York, 1943. 5. M. Wien, Ueber die Rueckwirkung eines resonierenden Systems, Ann. d. Physik, 1897, Vol. 61, pp. 151-189. ANALYSIS 1. Interaction of Coupled Circuits 1.1 Free Oscillations of Coupled Resonators {Fig. IB) The circuits are coupled according to the following four equations: ci = —Z\i\ -\- Zmii 1.1-1 n = YxCx + F„g2 1.1-2 ei = —Ziii -\r Zrnii 1.1-3 /o = y^eo + F„ei 1.1-4 where index i refers to circuit 1, index o to circuit 2 and index ^ to the mutual coupling impedance and admittance. The coupled oscillations have the solution: ei = Eue'"'' 4- Elbe"'' 1.1-5 f2 = E.ae""' + Eobe"" 1.1-6 In the limiting case of zero coupling (I'm = 0,Zm = 0) the obvious solution shows independent oscillations in the two separate circuits: eio = A'l/io = Eioe''^ 1.1-7 ^20 — K^iio = £20 f 1.1-8 The wave impedance A'l of the primary circuit is found by dividing equation 1.1-1 by 1.1-2 8 BELL SYSTEM TECHNICAL JOURNAL Similarly, --!=4/^ By multiplying equation 1.1-1 by 1.1-2 one finds -Zil'i = 1 1.1-9 from which one can compute the exponent px. In the specific circuits shown in Fig. lb Zi = Lxpi + Ri and 1.1-10 Yx - Cxpx 1.1-11 From 1.1-y, 10 and 11 ?' = - ^' + ^- - - 2X. + -'■ /SFl' '■'-'' and by analogy p..^-,, + ;.. = - ^1 + ; ^^^ - g 1.1-13 In equations 1.1-7 and 1.1-8, £io and £20 are ampUtude constants determined by boundary conditions. In equations 1.1-12 and 1.1-13, bx and 62 are the decay or damping constants, coi and C02 the radian frequencies. f^ With finite but loose coupling and small damping the circuits can oscillate with either or both of the two frequencies. p^ = p±±i^j^p±^-^r+^^ = P: + 0.5 p,(\ - vrr^^) 1-1-14 p, ^ ti^LJt - ^^vnw^ = p-^ + 0.5 px{\ - vrw) 1-1-15 In the last two equations, the symbol k, defined by '^ = \/ r ^r\, '^' "^^^^^ be called the coupling discriminant. The first term of the product on the right side of this expression is the reciprocal of the fractional difference between the uncoupled frequencies; the second term k is the "coupling coefficient." When there is only one coupling impedance, the coupling coefficient is usually defined as the mutual circuit impedance divided by the geometric mean of the separate circuit impedances. A broader definition which applies to all combinations of mutual impedances and admittances is k = -%- = -^ 1-1-16 CURVED WAVE GUIDES 9 In this equation 1\ is the energy stored in circuit 1, F-, the energy stored in circuit 2 and Pn is the energy- transferred from one circuit to the other. One finds F.^ :i+'^ = ^ = ilK, 1.1-17 1.1-18 2 .2r. 2 €i tiK ei 2K "^2 Ki = iiKi el .2 ,. euc-: . . „ -jr^ = 112 12^2 = A 2 Ai ii A'l Pl2 = -~ = ?I2/2A'2 = ; , - + /21/lAi 1.1-19 A2 Ai Equations 1.1-5 and 1.1-6 contain four ampHtude constants. Two of these, for instance Eu and £26 , can be adjusted to satisfy boundary conditions. The other two are fixed by the equation -fi2« A'l _ pa — Pi _ pb — p2 _ EibKi ElaK2 pa - P2 pb - pi Elf,Ki The square root of this expression, 2a /Ki _ Eib /Ki _ \a y K,- E,b V Ki~ '^ may be called the normalized amplitude ratio. It is a vector quantity denoting the amplitude ratio and phase relation of each oscillation frequency in the two circuits, assuming that they have been normalized to equal resistances by an ideal transformer. The absolute value ■EL A'l _ ,., _ 1 7-2 ,- —Ha— ,., iila A 2 *' b is the ratio of the energies stored in the two circuits oscillating at frequencies pa and Pb respectively. From 1.1-14, 15, 16 and 18 IVa Vl + k' - 1 Vl + k' + 1 4„ = Vl + K-- - K~ = If When the indexes are left off, II' < 1 and | .-1 [ < 1 by definition. One sees that energy, amplitude and phase relations between the coupled circuits at each oscillating frequency are governed by the coupling dis- criminant. This also applies to the damping coefficients and frequencies 10 BELL SYSTEM TECHNICAL JOURNAL of the coupled oscillations. It can be shown by combining and transforming equations 1.1-14, 15, 19, that the coupled damping coefficients are 6i + 62 ir , \\\a . , IV^a Oa — —. ; — ^7^ — Ol?77 r (>2 1-i-W Wtot.X IFtotal ^ 8iW -i- 82 ^ 81 Wib 82 1^26 ' 1 -\-W IFtotal W^total The damping constants of two coupled resonances are found by combining the uncoupled damping constants in the same proportion as the energies oscillating in the two resonators. The coupled frequencies are 03a = 0)b = 1 - W (J^2 — Wul and 1 - W 1.2 Forced traveling waves in coupled transmission lines (Fig. lA). The two lines are coupled according to the four equations Tci = Ziii + Zmi2 Til = y\e\ + jmei Ve' total "2 + ayW ]V\b . TF26 1 T Q "'' = 1 I w ^ "' w — + "' w — 1 "T n Ik total It total The attenuation constants of the coupled waves are found by combining the uncoupled attenuation constants in the same proportion as the energies traveling in the two lines. The coupled phase constants are ft = ^i^' 1.2-% From equations 1.2-5 to 1.2-8 one sees that the coupled propagation constants are conveniently described in terms of the power ratio W. JT itself is a known function of the complex coupling determinant k which is shown on the attached Fig. 4 for the following three special cases: Case 1. The two lines have equal phase constants and diferenl attenuation con- stants:/32 = /Si 0L2 ^ «i K is an imaginary number. W changes its character abruptly at the critical coupling. I K critical [ = 1 For I K I < 1 W < 1; OCb ^ OLa \ l^b = ^a For I /c 1 ^ 1 IF =1; cvb = a„ ; ^b ^ Ha Case 2. The lines have dif event phase constants and equal attenuation consia.n\.s. K is a real number IF changes asymptotically from PFo = 0 to TFi = 0.172 and to IFoc = 1 CURVED WAVE GUIDES 13 Case 3. The phase and allciiiuilioii constants dijler by equal amounts. As shown below, in section 4, this case apphes to the coupHng between the TEoi and TMii modes in curved circular wave guides with finite conductivity. ^'- is an imaginary number. \V clianges asymptotically from IFo = 0 to Wi = 0.217 and W^ = 1 For K » 1 all three cases approach the limit ,5. - ^'-^ ^' (l+l) 1.2-10 (■^0 (■*?) 2. Deriv.ation of Field Equations Consider a straight circular cylinder with an inside radius such as shown on Fig. 1 A. Let the radial coordinate equal r, the azimuthal coordinate equal (p and the longitudinal coordinate equal 2. Let the dielectric losses inside the cylinder be negligible. The field equations inside the cylinder are* rd(p dz dEr BEz . ----- = -iwnrH^ dz dr d(rE^) dEr ■ „ — = —joifxr Hi and dr dip dH^ dz =^'''^' dih . J, dHr . p dip '> See Ref. 4, pg. 94 of the book. 14 BELL SYSTEM TECHNICAL JOURNAL The natural transmission modes which satisfy these equations have the form E = /„(r) e^'^'^+^o^ . e^'""-^' 2-0 Each of these modes conforms to the same equations as a wave traveling in a transmission line with an impedance and phase velocity dependent upon the mode. In a straight cylinder with perfectly conducting walls, there exists no coupling between the different modes so that any and all can exist without interacting. If the conductivity of the walls in a straight circular cylinder is finite, it produces a resistive coupling between modes of equal azimuthal index {n in equation 2-0). In copper tubing and at the frequencies now obtainable (co < 10'-) this coupling effect is negligible. A stronger coupling may be caused by deviations of the wave guide from the shape of a straight circular cylinder. The deformation considered in the present analysis consists in a circular bend of the axis, as shown schematically on Fig. lb. In such a circular bend the longitudinal coordinate is replaced by the product of the bending radius R by the bending angle d: z = Re This transforms the first two component equations of curl E into diREe) dE, Rrdip Rdd dEr d(REe) = — icon H^ Rdd Rdr The variable R can be eliminated by the relation R = Ro — r cos (f where Ro is the bending radius of the cylinder axis. The coordinate 6 can be replaced by a longitudinal coordinate s, measured along the cylinder axis. Hence, 5 = BRo The progressive modes which we investigate have the approximate form Hence — = Rn — = — Ra V dd ds — r= jn for all field components. CURVED WAVE GUIDES 15 — may be expressed by a prime : dr Thus the equations with curl E become jnEs Es sin (p RoTE^ „ r Kq — r cos

r One may therefore approximate Ro . 1 I '' p ^ 1 + P cos (^ K(j — r cos ^ ico It is convenient to introduce the symbol a which is proportional to the couphng coefficient. All powers of c greater than the first will be neglected. One can now write the approximate field equations in the curved cylinder : iflE C T — ' + TE^ -\- - E, sin (f -\- cV - E^ cos

1 5-3 . OTE + TM z a «& > r -^-* 22 BELL SYSTEM TECHNICAL JOURNAL In a long wave guide the "i" mode will die down due to its greater attenua- tion, no matter how much of it was initially present, so that one need only consider the "a" mode. This mode has a phase velocity slightly smaller than that of the uncoupled TEoi wave and an attenuation nearer to that of the TEoi than the TMn wave. The magnitude of the critical radius is illustrated by the two examples of Table I. Table I Characteristic Values Parameter Symbol a Equation Example 1 Example 2 Wave guide radius .05 m .05 m Free space wave length Cutoff ratio Xo V 3-11 .03 m .366 .01 m .122 Attenuation constant «TK(cu) 3-15,4-17 2.04 X 10-" 3.58 X 10-s Attenuation constant «TM(cu) 3-19 neper/ m 1.53 X 10-^ neper/m 2.41 X 10-3 Critical Radius Rcrit 4-10 neper/m 2.12 km. neper/m 3.44 km. Table II Relative Attenuation Versus Radius of Curvature General formulae Example 1 Example 2 K Ro/RcT w a/aa Rokm oo a/ao Rokm a/ao 0 00 0 1 1.00 00 1.00 0.1 10 0.0025 1 -1- 0.0025 (v^ - 1) 19.7 1.02 34.15 1.17 0.2 5 0.01 1-fO.Ol 9.85 1.06 17.08 1.66 0.5 2 0.06 1-f 0.057 3.84 1.38 6.83 4.45 1 1 0.22 l-fO.18 1.97 2.16 3.42 12.94 2 0.5 0.48 14-0.32 0.98 3.11 1.71 22.2 5 0.2 0.75 1 4- 0.43 0.38 3.83 0.68 26.6 10 0.1 0.87 1+0.46 0.20 4.05 0.34 27.9 00 0 1.00 1 4- 0.50 0 4.30 0 34.1 The increase of attenuation in long wave guides with uniform curvature is shown on Table II, with numerical values for the same examples as in Table I. 6. Propagation in a Uniformly Curved Section of Wave Guide Following a Long Straight Section. (Fig. 4) No matter what mixture of modes may prevail at the beginning of the wave guide, all modes except the TEoi die down in the long straight section due to their higher attenuation, so that the wave form at the beginning of the curved section is pure TEoi . Since it has been shown in Sections 1 and 4 that each of the two possible CURVED WAVE GUIDES 2i modes of propagation in a curved wave guide consists of both TEoi and TMu waves, it follows that both modes must be superimposed in such a manner that at the transition point the TM comi)onents cancel each other by interference. Let the relative ampUtudes of the two TE components equal a and b; then the corresponding amplitudes of the TM modes are aAa and bAt, 20 Fig. 4 respectively, where Aa and A^ are the normalized voltage ratios per 1.2-5. .\i the beginning of the curved section a -\- b = \ (TE amplitude) aAa + bAb = aAa - b/Aa = 0 (TM amplitude) Hence b = 1 + Ai Al = W The two waves have different phase velocities and therefore interfere with each other. According to 1.2-9, the difference between the phase constants is I -4- M^ /35 - ^„ - (/32 - /3i) — ^^. 1 — W By means of equations 4-9, this can be transformed into /36 - /3a = 032 - iSiXp + q) 6-1 24 BELL SYSTEM TECHNICAL JOURNAL The length of one complete interference cycle is '"' ~ (pi- gm - ^i) If both components had equal attenuation, the beats superimposed on the decaying envelope of the TEoi wave would correspond to an amplitude ratio eo max 1 + W ^ , ^ ^ 1 nr ^ P '^ ^1 eo mm 1 — Ir However, during the progress of the mixed wave through the curved section, the intensity of the fluctuations is reduced by the greater attenuation constant of the faster and weaker "6" component. In one complete inter- ference cycle, the dififerential attenuation reduces the weaker component to A^w -(a.,-ai)so^ -(2wlp + q) Approximation for Weak Coupling For I K I « 1 /36 - /3a = (/32 - /3i)(l + 0.5 I . \') From 3-15 and 3-19 at] 1.2 X 10' at] /(I v%u (J + 0.5 i /c 1^) radian/m V ^og a For intermediate coupling, 6-1 may be transformed into ^6 - |Sa = k^lf{K) .,. , ±±± _ Vl + Vf+W^ + V- 1+ Vl+l^' with /(«) = /T, I I — ■ — i — i — Approximation for Strong Coupling For 1 K I » 1 /u, = 1 + 0.125 1 K \-' ^b - i^a = k^iU + 0.125 1 K \-') Substituting the value of k from 4-13 and transforming, /?6 - i8a = ,-p /(«. = ^^ (1 + 0-12^^ \k\ ) 6-2 The phase difference between the two components is 2.32a5 , 2.32ad r ,^„ /: ^ AoAq Ao CURVED WAVE GUIDES 25 where d is the beiuHnj; angle of the wave guide. The power carried by the TEoi wave is Fte = cos - . 0-4 Minima of TEoi occur when this phase difference is an odd multiple of tt. Hence, the bending angles producing minima of TEm amplitudes are: 1.36Xo . (2n + 1)2.22. a{{ + 0.12.-> \k\ •*) /(«) The initial fluctuation ratio approaches Cl min which is a large value tending to intinity. The relative attenuation of the slightly weaker component during one beat cycle is A2t^ _ -2T/p+g _ -v/St/M ^ 4 _ ■i^'i Ao \k\ which is a small reduction tending to zero. Hence, the fluctuations persist through a large number of beats. The power is transformed back and forth between the TEoi and the TMu modes. In Section 5, it was shown that in a long, uniformly curved wave guide the attenuation is intermediate between that of the TEoi and TMn modes. But from equations 1.2-7 and 8 it follows that the two modes contribute to the attenuation in proportion to their relative power flow. Since at the beginning of the bend the power of the TMn component is zero, it is to be expected that the initial rate of attenuation equals that of the TEoi wave alone. This is proved by differentiating with regard to s. One finds for all values of k that — ae + oe : = — ai as \ l«=o Discussion of Results Equation 6-2 corresponds directly to an equation derived by S. O. Rice and, after allowing for the different choice of variables, to M. Jouguet's equation (75)'^. It differs from the results of these earlier calculations by the factor /(k) = 1 + 0.125 | k j "* which is a reminder that the simplihed form of the equations given by the earlier authors is an e.xtrapolation to infinite conductivity or infinite curvature of the wave guide. * Reference 3, pg. 150 of Cables and Transmission, July 1947. 26 BELL SYSTEM TECHNICAL JOURNAL From equations 6-3 and 6-4 it is seen that the TEoi wave is recovered by bends which are an even multiple of dm\n ■ But such bends are efficient transmitters of TEoi waves only over a narrow frequency range since dmin varies with frequency. If the circular bend is followed by a long straight section, the TEoi and TMu components existing at the end of the bend are carried over into the straight section, but the TMu component dies down due to its greater attenuation and constitutes a total loss. Numerical examples for first extinction angle. Using the same dimensions as in Table I of Section 5, one finds from eq. 6-5 for: Example 1: dmin — 0.816 Radians — 46.8° Example 2: dmin — 0.272 Radians — 15.6° 7. Serpentine Bends Sections 5 and 6 dealt with bends continued with uniform curvature over large angles. The present section considers the small random devia- tions from a straight course which are unavoidable in field installations. Actual deviations are expected to be random both with regard to maximum deflection angle and to curvature ; they are likely to approximate a sinusoidal shape. For purposes of computation, the following analysis assumes as a first case circular S-bends which consist of alternate regions of equal lengths and equal but opposite curvatures. An exaggerated schematic of such S-bends is shown on Fig. 5A. Each circular bend tends to produce a single mode with an attenuation per equation 1.2-7. However, the discontinuous reversals of curvature at the inflexion points produce mixed modes, and the initial part of each region reduces the amplitude of the TM components produced in the previous region. Each region may be treated as a discrete 4-terminal section of a trans- mission network. Regardless of the wave composition at the input terminal, differential attenuation will establish in a long serpentine wave guide a steady state condition. In this steady state each region produces equal attenuation. This attenuation per region and the resulting average attenuation constant will now be derived. The TEoi and TMu waves each consist of "a" and "b" components with separate amplitude ratios and propagation constants, as derived in Sections 1 and 4. In the first region (between points 1 and 2 of Fig. 5A) Ro is taken as positive, and A - Vl + K-' - i^-' 1-2-6 CURVED WA VE GUIDES 27 In the second region, between points 2 and 3, the polarity of Ro , and consequently of k. and the ratio of TM to TE ampUtudes, are reversed. a -CIRCULAR S BEND5 b- SINUSOIDAL BENDS Fig. 5 Except for this change of polarity the amplitude ratios at points 1 and 2 are equal. Introducing the symbols — TsTrt S = e gm = h(ga + gb) gd = Ugh- go) one can tabulate point 1 2 a + b ^TM aA A '' + bg'i ) g"^ ( ^- - j Sd Calling = y, one finds a I I 2 i2 , 2 1 + ygd ^ -A + yga 1 + y A^ - y ^average 7-1 7-2 28 BELL SYSTEM TECHNICAL JOURNAL 1 1 + >' r._.e = r^ + - lo.' i-^-y^i with y = J >c-\i + gf) + .4 ^gr- + \^ (1 + ^r)^ This formal solution is hard to evaluate. It can be greatly simplified for the subcritical and supercritical cases. 1. Subcritical Curvature | k ] « 1 o ^ ~ 2 -TS J. - V f ^ ^ '^' ^ ~ •iA ^ r = r \ ^ 1 + grf/ For very low curvatures, the average attenuation approaches that of the "fl" mode, and this in turn approaches that of the TE(t\ wave. 2. Supercritical Curvature. 1^1^ 1 The differential attenuation constant is small compared to the differential phase constant. Substituting these values into 7-1 and 2, one finds -0.5j9m y = e Expressed as a function of 6: ye = cos i/' + j sin \}/ with ^|y ^ M{d - 0.5 d„.) 7-3 M has the value per eq. 6-18. The power ratio of the combined TMn and TEoi waves is We = tan^ ^/2 In view of equation 1.2-7 the instantaneous rate of energy loss is ae = ai cos" - + a-jsm" - = ai + (as — aj sm - /-4 2 2 ■^ "averuge = " / a. d, = — ag dO Sm Jo "vi •'0 , . . /l sin MdA In view of .^-20 Vv^^ + 1 v""^ - 1 sin M0„n a^verage '-''l Mdr CURVED WAVE GUIDES 29 For small deflection angles, Mdm « 2 Q^avernge OL\ -2 1+^ - -M'el'^ 6 If a p% increase in attenuation is the tolerance limit Qm ^ ,^,> A/ - . . Substituting the value of M from 6-3, ^ _ 0.105 V? j/Xo Urn — , . The maximum deflection equals Afl — 0.5 dm. Hence, in view of 3-11 . ^ 0.032V?X^ ,. 1.84X^ ^ /""^ ^ Afl ^ -„ — , — -^— radians = — ^— A / - — ^ — ^ degrees = J^ n ~h /72 'y 1 _ „2 ^"o a Vl — »'" 1.84X^ / /> a^ y 1 - y' 3. Sinusoidal Bends with Predominantly Supercritical Curvature. Sinusoidal bends cannot be supercritically curved over their entire length because at the inflection points the curvature drops to zero. For sufficiently short bends, however, no great error is caused by treating the entire length as supercritical. In that case, equations 7-3 and 7-4 remain valid. 6 takes the new value a Gm , 0m . 7r(5 — Sm) 6 — — + — sm — Zi Zr ^^ffl Hence Mdm . t(s — Sm) \p = ——- sm 2 2s„ . (X2 — ai f .2 "average = CCi + / SHl Sm Jo For small deflection angles, Mdm « 2 r2/,2 Mdm . Tr(s — Sm)! , a-z — ai M dm f . 2 7r(i- — Sm) , ttuverage = "i + — - / Sm dS Sm 4 Jo 2Sm = ai + [ai - aO -^ - aj 1 + M'dm . 0.026\Wp >. 1.49X5 /~7~, Aj = -5 — ■ - radians = — r— A/ -^ — ^degrees o S/l - v^ a2 |/ 1 - ;/2 « The tolerance limit for sinusoidal deflections is 20% smaller than for circular S bends. 30 BELL SYSTEM TECHNICAL JOURNAL The effect of supercritical but shallow circular and sinusoidal S bends is illustrated by the following numerical examples. Table III Increase or Attenuation in S Bends Maximum deflection AO (in degrees) Attenuation Increase ^% Example 1 (i- = .366) Example 2 (r = .122) Circular Sinusoidal Circular Sinusoidal 10 20 30 40 50 2.25 3.18 3.89 4.50 5.03 1.82 2.58 3.15 3.64 4.07 0.23 0.33 0.41 0.47 0.52 0.19 0.27 0.33 0.38 0.42 8. Helical Bends and Random Two-Dimensional Devlations A helical bend may be treated as a bend which has a constant absolute magnitude, but a changing direction of curvature. As indicated in eq. 3-17, the TMii wave can be polarized in all directions. At any differential element of wave guide length, the TMn component polarized in the local bending plane is coupled to the TEoi wave; the TMn component polarized at right angles is not coupled and persists unchanged. By requiring that the absolute magnitude of the TMu/TEu amplitude ratio remain constant, a steady state solution can be found. Shallow helical bends of small curvatures may be treated as the super- position of two sinusoidal bends offset by 90° in the longitudinal direction and in the bending plane. The increases in attenuation due to these two sinusoidal bends are computed from eq. 7-5 and added. It is believed that random deviations from a straight course approach sinusoidal shape more closely than circular shape, hence equation 7-5 may be used to establish a tolerance limit for such random deviations. For quantitative results the statistical distribution of the squared deviation maxima must be taken into consideration. 9. Optimla of Wave Guide Radius, Signal Wave Length .and Attenuation as a Function of Angul.ar Deviation In a straight wave guide the attenuation decreases with wave guide radius and signal frequency. However, the deterioration due to wave guide curvature increases with wave guide radius and frequency. Hence, for a given tolerance limit to angular deviation from the straight course there exists an optimum radius for each wave length and an optimum wave length for each radius. This will be shown for the case of uniform sinusoidal bends, under the simplifying assumption that the cutoff radio »» « 1. CURVED WA VE GUIDES 31 Solving 7-5 for p one obtains p = ()A5^\^'\-' 9-1 where p si the percentage increase in attenuation, and A the deviation angle in degrees. Hence the average attenuation a^ = a(l + O-Ol/*) 9-2 From 3-15 and 3-11 . V Ri . 3 2-8 ^ - a = = 10 A, A a 9-3 ai] Introducing the Ri value from 4-8 a = 4.5 10-5piXi-6a-» 9-4 where p is the high-frequency resistance of the wave guide relative to copper. From 9-1, 2 and 4 «A = 4.5'lO~Vxi-5a-3(l + qk-^a^) 9-5 with q = 4.5 10-»A2 The attenuation reaches a minimum when fix, a) = A^-^a~* + qX~--^a = minimum C a^e /. X is given 8f/8a = -3X'-^a-' + ^X-^-^ = 0 aopt = 1.32X?-"" = 5.2XA-"-s From 9-5 a^„p, = 4a = 1.29 10~V-'-'Ai-"^ Case 2. a w given bf/SX = 1.5X'^-5a-^ - 2.5qX-'-'a = 0 X„^, = l.Uaq'-' = 0.294aA»-5 From 9-5 -1.5. 0.75 ctAopt ~ 1-6q: — 1.15 10 pa * A Numerical Example Let A = 0.42° a = 0.05 w X = 0.01 m 32 BELL SYSTEM TECHNICAL JOURNAL From Table III ttA = 1.50 a = 5.4 10~*p neper/w Case 1 : X fixed at 0.01 m Oopt = 0.08 m Uopt = 2.76 10~^p neper/meter Case 2 : a fixed at 0.05 vi \opt = 0.0097 w "opt = 5.36 10~V neper/w Assuming sinusoidal bends with a 0.42° maximum deviation, the attenua- tion of centimeter waves can be reduced to one half by increasing the wave guide radius from 5 to 8 cm. For a 5 cm wave guide radius, 1 centimeter wavelength is close to the optimum. A New Type of High-Frequency Amplifier By J. R. PIERCE and W. B. HEBENSTREIT This paper describes a new amplifier in which use is made of an electron flow consisting of two streams of electrons having different average velocities. When the currents or charge densities of the two streams are sutTicient, the streams interact to give an increasing wave. Conditions for an increasing wave and the gain of the increasing wave are evaluated for a particular geometry of flow. 1. Introduction IN CENTIMETER range amplifiers involving electromagnetic resonators or transmission circuits as, in klystrons and conventional traveling-wave tubes, it is desirable to have the electron flow very close to the metal circuit elements, where the radio-frequency field of the circuit is strong, in order to obtain satisfactory amplification. It is, however, difficult to confine the electron flow close to metal circuit elements without an interception of elec- trons, which entails both loss of efl&ciency and heating of the circuit elements. This latter may be extremely objectionable at very short wavelengths for which circuit elements are small and fragile. In this paper the writers describe a new type of amplifier. In this ampli- fier the gain is not obtained through the interaction of electrons with the field of electromagnetic resonators, helices or other circuits. Instead, an electron flow consisting of two streams of electrons having different average velocities is used. When the currents or charge densities of the two streams are sufficient, the streams interact so as to give an increasing wave. Electro- magnetic circuits may be used to impress a signal on the electron flow, or to produce an electromagnetic output by means of the amplified signal present in the electron flow. The amplification, however, takes place in the electron flow itself, and is the result of what may be termed an electromechanical interaction. '" While small magnetic fields are necessarily present because of the motions of the electrons, these do not play an important part in the amplification. ■ Some electro-mechanical waves with a similar amplifying effect are described in "Possible Fluctuations in Electron Streams Due to Ions," J. R. Pierce, Jour. A pp. Plivs., Vol. 19, pp. 231-236, March 1948. -While this paix;r was in preparation a classified report by .\ndrew V. Haeff entitled "The Electron Wave Tube — A Novel Method of Generation and Amplification of Micro- wave Energy" was received from the Naval Research Laboratory. Dr. Haeff 's report (now declassified) contains a similar analysis of interaction of electron streams and in addition gives experimental data on the performance of amplifying tubes built in ac- cordance with the new principle. We understand that similar work has been done at the RCA Laboratories. 33 34 BELL SYSTEM TECHNICAL JOURNAL The important factors in the interaction are the electric tield, which stores energy and acts on the electrons, and the electrons themselves. The charge of the electrons produces the electric field; the mass of the electrons, and their kinetic energy, serve much as do inductance and stored magnetic energy in electromagnetic propagation. By this sort of interaction, a traveling wave which increases as it travels, i.e., a traveling wave of negative attenuation, may be produced. To start such a wave, the electron flow may be made to pass through a resonator or a short length of helix excited by the input signal. Once initiated, the wave grows exponentially in amplitude until the electron flow is terminated or until non-linearities limit the amplitude. An amplified output can be ob- tained by allowing the electron flow to act on a resonator, helix or other output circuit at a point far enough removed from the input circuit to give the desired gain. There are several advantages of such an amplifier. Because the electrons interact with one another, the electron flow need not pass extremely close to complicated circuit elements. This is particularly advantageous at very short wavelengths. Further, if we make the distance of electron flow between the input and output circuits long enough, amplification can be obtained even though the input and output circuits have very low imped- ance or poor coupling to the electron flow. Even though the region of amplification is long, there is no need to maintain a close synchronism between an electron velocity and a circuit wave velocity, as there is in the usual traveling-wave tube. A companion paper by Dr. A. V. Hollenberg of these laboratories describes an experimental "double stream" amplifier tube consisting of two cathodes which produce concentric electron streams of somewhat different average velocity, and short helices serving as input and output circuits. No further physical description of double stream amplifiers will be given in this paper. Rather, a theoretical treatment of such devices will be presented. 2. Simple Theory For simplicity we will assume that the flow consists of coincident streams of electrons of d-c. velocities Ui and «2 in the .v direction. It will be assumed that there is no electron motion normal to the x direction. The treatment will be a small-signal or perturbation theory, in which products of a-c. quantities are neglected. M.K.S. units will be used. All quantities will be assumed to vary with time and distance expj{ul — )3.v). The wavelength in the stream, Xg , is then related to /3 by /3 = 2t/X. (1) /l ,/2 Ih U2 Poi , P02 Pi, P2 Vl, V2 Vi ,V2 V IIIGII-IKEQUENCY AMPLIFIER 35 Tlic following additional nomenclature will be used: eo dielectric constant of vacuum €0 = 8.85 X 10-'- farad/meter 7} charge-to-mass ratio of the electron ■q = 1.76 X lO'i coulomb/kilogram d-c. current densities d-c. velocities d-c. charge densities Poi = —Ji/ui , po2 = —Jilut a-c. charge densities a-c. velocities d-c. voltages with respect to the cathode a-c. potential ;Sl = wlU]_ , ^2 = C0/U2 Although the small-signal equations relating charge density to voltage V have been derived many times, it seems well to present them for the sake of completeness. For one stream of electrons the first-order force equation is di'i dvi , dii dV ■-^ — + «l = 7? dt dt dx dx (co — I3ui)vi = —r]^V _ -r?/3K ^'1 = -7^ ^. (2) Wi(|Si - ^) From the conservation of charge we obtain to the first order dpi ^ r \ \ ^ = — — IPor^'i + piih) dt dx wpi = PoiiSl'l + Uilipi From (2) and (3) we obtain We would find similarly ^ - U\{^2 - ^f ^'^ It will be convenient to call the fractional velocity separation b, so that 2(wi - U2) , . 36 BELL SYSTEM TECHNICAL JOURNAL It will also be convenient to define a sort of mean velocity Wo uo = (7) th + ^2 We may also let Vo be the potential drop specifying a velocity mq , so that tio = V2vVo (8) It is further convenient to define a phase constant based on uq CO /3o = - We see from (6), (7) and (9) that /3i = ^o(l - b/2) /32 = /3o(l + b/2) We shall treat only a special case, that in which Ji Ji Jo 3 Ui 3 — 3 W2 Wo (9) (10) (11) (12) Here Jo is a sort of mean current which, together with Uo , specifies the ratios Ji/ui and /2/W2 , which appear in (4) and (5). In terms of these new quantities, the expression for the total a-c. charge density p is, from (4) and (5) and (8) p = Pi + p2 /or 2uo Vo .W'-l)-^I^W'+0-^L (13) Equation (13) is a ballistical equation telling what charge density p is produced when the flow is bunched by a voltage V. To solve our problem, that is, to solve for the phase constant /3, we must associate (13) with a circuit equation which tells us what voltage V the charge density produces. We assume that the electron flow takes place in a tube too narrow to propa- gate a wave of the frequency considered. Further, we assume that the wave velocity is much smaller than the velocity of light. Under these circumstances the circuit problem is essentially an electrostatic problem. The a-c. voltage will be of the same sign as, and in phase with, the a-c. charge density p. In other words, the "circuit efi'ect" is purely capacitive. Let us assume at first that the electron stream is very narrow compared with the tube through which it flows, so that V may be assumed to be con- stant over its cross section. We can easily obtain the relation between HIGH-FREQ UENC Y A MPLIFIER 37 V and p in two extreme cases. If the wavelength in the stream, X, , is very short (fi large), so that transverse a-c. fields are negligible, then from Poisson's equation we have P = coiS" F (14) If, on the other hand, the wavelength is long compared with the tube radius (jS small) so that the fields are chiefly transverse, the lines of force running from the beam outward to the surrounding tube, we may write p = CV (15) ^^«— Fig. 1 — A "circuit" curve for a narrow electron stream in a tube. The ratio of the a-c. charge density p to the a-c. voltage V produced bj' the charge density is plotted vs. a parameter /3/|3o , which is inversely proportional to the wavelength X, in the flow. Curve 1 holds for very large values of /3//3o ; curve 2 holds for very small values of i3//3o , and curve 3 over-all shows approximately how p/V varies for intermediate values of /3//3o . Here C is a constant expressmg the capacitance per unit length between the region occupied by the electron flow and the tube wall. We see from (14) and (15) that if at some particular frequency we plot p/V vs. /3//3o for real values of /3, p/V will be constant for small values of jS and will rise as /S^ for large values of /3, approximately as shown in Fig. 1. For another frequency, /3o would be different and, as p/7 is a function of /?, the horizontal scale of the curve would be different. Now, we have assumed that the charge is produced by the action of the voltage, according to the ballistical equation (10). This relation is plotted in Fig. 2, for a relatively large value of Jq/uqVq (curve 1) and for a smaller value of Jq/uqVq (curve 2). There are poles at /3//3o 1 ± - , and a mini- mum between the poles. The height of the minimum increases as Jo/uqVo is increased. A circuit curve similar to that of Fig. 1 is also plotted on Fig. 2. We see 38 BELL SYSTEM TECHNICAL JOURNAL that for the small-current case (curve 2) there are four intersections, giving four real values of /3 and hence four unattenuated waves. However, for the larger current (curve 1) there are only two intersections and hence two unattenuated waves. The two additional values of /3 satisfying both the circuit equation and the ballistical equation are complex conjugates, and represent waves traveling at the same speed, but with equal positive and negative attenuations. Fig. 2 — Curve 3 is a circuit curve similar to that of Fig. 1. Curves 1 and 2 are Ijased on a ballistical equation telling how much charge density p is produced when the voltage V acts to bunch a flow consisting of electrons of two velocities. The abscissa, /3//3o , is proportional to phase constant. Intersections of the circuit curve with a ballistical curve represent waves. Curve 2 is for a relatively small current. In this case inter- sections occur for four real values of (3, so the four waves are unattenuated. For a larger current (curve 1) there are two intersections (two unattenuated waves). For the other two waves 13 is complex. There are an increasing and a decreasing wa\'e. Thus we deduce that, as the current densities in the electron streams are raised, a wave with negative attenuation appears for current densities above a certain critical value. We can learn a little more about these waves by assuming an appro.ximate expression for the circuit curve of Fig. 1. Let us merely assume that over the range of interest (near /3//3o = 1) we can use p = a-eoiS-F (16) Here a- is a factor greater than unity, which merely expresses the fact that the charge density corresponding to a given voltage is somewhat greater HIGH IREQUENCY AMP/JEIER 39 than if there were held in the x direction only and equation (11) were vah'd. Combining (16) with (13) we obtain w tie re 1 1 .o^t)-^)" (^(^ + 'y:^"'''" '''' I- = ;r"2 ^2 77 (^8) Za eoPn Mo I o In solving (17) it is most convenient to represent /3 in terms of (So and a new variable 5 IS = ^o(l + 5) (19) Thus, (14) becomes 1 1 1 + ,„.v'(,,.v-.= Solving for d, we obtain The positive sign inside of the brackets always gives a real value of d and hence unattenuated waves. The negative sign inside the brackets gives unattenuated waves for small values of U/b. However, when j) > I (^^) there are two waves with a phase constant 0o and with equal and opposite attenuation constants. Suppose we let Um be the minimum value of U for which there is gain. From (22), t/A/2 = b'/8 (23) From (21) we have for the increasing wave The gain in db/wavelength is db/wavelength - 20(2x) logio e\8\ (25) = 54.6 I 5 I i = jj lUj[A/i + s[r^) - 1 ) - 1 40 BELL SYSTEM TECHNICAL JOURNAL We see that by means of (24) and (25) we can plot db/wavelength per unit h vs. (U/UmY- This is plotted in Fig. 3. Because U"^ is proportional to current, the variable {U/U mY is the ratio of the actual current to the current which will just give an increasing wave. If we know this ratio, we can obtain the gain in db/wavelength by multiplying the corresponding ordinate from Fig. 3 by b. JU 25 -— -— — 20 / y / / / / / 10 5 0 1 1 1 j._ _^ _j_ 1 ..L_ _L j_ 40 60 80 100 400 600 1000 (W/Wm)^= (U/Um)' Fig. 3— The parameter {W fW mY = ((tVt\\f)-isproportional to current. As the current is increased above a critical value for which {W/Wm)"^ = 1, there is an increasing wave of increasing gain. In this curve the gain per wavelength per unit b, called F{W/WMy, is plotted vs. {IV /Wm)^- For large values of (W/Wm)-, /^(H'/ll'-v/)' approaches 27.3 and the gain per wavelength approaches 27.3 b. We see that, as the current is increased, the gain per wavelength at first rises rapidly and then rises more slowly, approaching a value of 27.36 db/wavelength for very large values of {U/U m)'- We now have some idea of the variation of gain per wavelength with veloc- ity separation b and with current {U/U mY- A more complete theory would require the evaluation of the lower limiting current for gain (or of U m) in terms of physical dimensions and an investigation of the boundary condi- tions to show how strong an increasing wave is set up by a given input signal. The latter problem will not be considered in this paper; the former is dealt with in the third section and in the appendix. HIGH-FREQUENCY AMPLIFIER 41 3. Design Curves It is proposed to present in this section material for actually evaluating the gain of the increasing wave for a particular geometry of electron How. In this section there is some repetition from earlier sections, so that the material presented can be used without referring unduly to section 2. In order to avoid confusion, much of the mathematical work on which the section is based has been put in the appendix. The flow considered is one in which electrons of two velocities, «i and u-i , corresponding to accelerating voltages Vy and Vo , are intermingled, the corresponding current densities /i and J2 being constant over the flow. The flow occupies a cylindrical space of radius a. It is assumed that the surrounding cylindrical conducting tube is so remote as to have negligible effect on the a-c. fields. It will be assumed, according to (12), that the current densities and the voltages Vi and Vo are specified in terms of a "mean" current Jo and a "mean" voltage Vo corresponding to a velocity Uo , by _Ji_ 2l. 1± r.r. -s yzn - yzn - yzn (12aj The gain will depend on the beam radius, the free-space wavelength X, and on Jo and Fo , and on the fractional velocity separation b = 2(^^L^J^) (6) wi + W2 The wavelength in the beam, Xs , which is associated with the voltage Fo is given by X. = X^ = X^2^ c c X, = 1.98 X KT'xVVo (26) Here c is the velocity of light. A dimensionless parameter IF is defined to be w = -; = -^ (27) TI'^ = 8.52 X 10^ 7—^ (28) Here We is the electron plasma frequency associated with the average space charge density Jo/uo , and co is the radian frequency corresponding to the wavelength X. In (28), the constant is adjusted so that Jo is expressed in 42 BELL SVSTIiM TECIIXICAL JOLRXAL amperes per square centimeter rather tlian in amperes per square meter, while / is expressed in megacycles. Below a minimum value of \V, which will be called Il'.v/ , there is no gain. W i, is a function of the velocity separation b and of the ratio of the beam radius a to the beam wavelength, X, . A plot of (Il\w/7>)- as a function of (a/Xs) is shown in Fig. 4. The variation of gain in the interval, 11 '^ < IT < x , is shown in Fig. 3 where "Decibels gain/ wavelength/unit b" is plotted as a function of (W/Wm)'- This is the same curve which was derived in section 2. The 8 6 4 2 1.0 0.8 0.6 - \ \ - - s. \ v - \ - \ v~ — 0.4 - — ^ — 0.2 \^ ^^"-v. ^ 0.1 1 ..L —l- _^ 1 ,....L ., __L. , 0.2 0.4 0.6 0.8 1.0 ^ Fig. 4 — As the ratio of beam radius a to wavelength in the beam, X, , is increased, the critical value of 11', W \f , decreases and less current is needed in order to obtain gain. Here (ll/w/^)', which is called H{a/X,), is plotted vs. (a/X,). ratio (U'/Wm)- is the same as the parameter {U/U mY used there, although V and W are not the same. The curve in Fig. 3 is useful in that it reduces the interdependence of a large number of parameters to a single curve. However, there are cases as, for example, when one is computing the bandwidth of an amplilier, in which it would be more convenient to have the curve in Fig. 3 broken up into a family of curves. We can do this by the following means: We can write the gain in db/'wavelength in the form db/wavelength = bF{W/WM)- (29) HIGH-FREQUENCY A MPUI'IER 43 Here F{W/Wm)- is the function plotted in Fig. 3. If (. is the total length of the tlow, the total gain in db, G, will thus be As (30) We will now express (W/IVm)' in such a form as to indicate its dependence on wavelength in the beam, X^ . We can write from (27) 11' = X? Here K is a "plasma wavelength," defined by the relation Wo X« = (coe/27r) We further have Wl = PH(a/\s) Here H(a/\s) is the function of (a/Xs) which is plotted in Fig. 5. (31) (32) 25 ^ 20 ^ 10 1 ^ ^^- ^ ^ :::;-> ^, Vb/le^" .^' s ^^ S^ R ^ ^ N ^o v \\ ^ y \ A V / \ V 1 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 bcy/oje Fig. 5 — In these curves the total gain in db, G, divided by the ratio of the length / to the plasma wavelength \e , is plotted vs. bco/ue, which is proportional to frequency, for several values of the parameter {a/bXe)^- Changing b, the velocity seperation, changes both the parameter and the frequency scale. Now, from (26), (27), and (29) we can write G -(-)( — I ( ;■ ) /' kJ \ a bXeJ Hia/\s)_ (34) For a given tube the parameters {(/\e) and (a/b\e) do not vary with fre- quency, while (a/Xs) is proportional to frequency. Hence, we can construct universal frequency curves by plotting G/{C/Xe) vs. (a/Xs) for various values of the parameter (a/bXe). It is more convenient, however, to use as an abscissa bXg/Xs — bu/wg , and this has been done in^Fig. 5. 44 BELL SYSTEM TECHNICAL JOURNAL In order to use these curves it is necessary to express the parameters bo}/(i)e , \e and (a/bXeY in terms of convenient physical quantities. We obtain 6co/coe = .545 X 10-'' bV"'u,/Jl'^ \e = 2.04 X \0-WT/jT (35) {a/bKY = 767 h/bW^''' Here /o is current in amperes and Jo is in amperes / cm. The broadness of the frequency response curves of Fig. 5 is comparable to that of curves for helix-type traveling-wave tubes. It is interesting to note that the maximum value of G/{(/\e) varies little for a considerable range of the parameter a/b\e , approaching a constant for large values of the parameter. This means that, with a beam of given length, velocity and charge density, one can obtain almost the same opti- mum gain over a wide range of frequencies simply by adjusting the velocity- separation parameter b. 4. Concluding Remarks There is a great deal of room for extension of the theory of double-stream amplifiers. This paper has not dealt with the setting up of the increasing wave, nor with other geometries than that of a cylindrical beam in a very remote tube, nor with the effect of physical separation of the electron streams of two velocities nor with streams of many velocities or streams with con- tinuous velocity distributions. This last is an interesting subject in that it may provide a means for deal- ing with problems of noise in multivelocity electron streams. Indeed, it was while attempting such a treatment that the writers were distracted by the idea of double-stream amplification. APPENDIX Derivation of Results Used in Section 3 Consider a double-stream electron beam whose axis coincides with the z-axis of a system of cylindrical coordinates (r, (p, z) and which is subject to an infinite, longitudinal, d-c. magnetic field. The radius of the beam is a and each of the streams is characterized by d-c. velocities, Ui and W2 , which are vectors in the positive s direction, and d-c. space charge densities, po2 and po2 . All d-c. quantities are assumed to be independent of the coor- dinates and time, except, of course, for the discontinuities at the surface of the beam. Small a-c. disturbances are superimposed upon these d-c. quanti- ties and they are small enough so that their cross products can be neglected compared with the products of d-c. quantities and a-c. quantities. It is IIIGII-FREQUENCY AMPLIFIER 45 further assumed that only those a-c. quantities are allowed which have no azimuthal variation, that is, — = 0. Fig. 6 shows the electron beam. Outside the beam the appropriate Maxwell's equations are r dr rjo ^ ^ irH,) = j-E, (A-1) (A-2) (A-3) where dz = - 170 dE, dr dEr dz = jkr]cH^ c 70= 1 /mo Co = 377 ohms (A-4) (A-5) Inside the beam, equations (A-2) and (A-3) remain the same, but instead of equation (A-1) we have - I- (rH^) ^ j~E.+ q, + q, - (A-6) r dr 7/n where ^i and q^ are the first order a-c. convection current densities of the two streams. These quantities can be calculated from the force equation and the equation for the conservation of charge. Assuming that all a-c. quanti- ties vary as expj(cot — /3z), the force equation is (for stream number one, say) juvi — jlSuiVi = — (e/ni)Ez (A-7) and the equation for the conservation of charge is jl3poiVi + jlSuipi = +iwpi (A-8) Equations (A-7) and (A-8) can be solved for vi and pi : — {e/m)E2^ (A-7a) (A-8a) where Vl "M-D /3poi Pi "-(■-1) ^1 CO Ml Vl 46 BELL SYSTEM TECHNICAL JOURNAL Combining equations (A-7a) and (A-8a) one has _ j^poi{e/m)E^ '""('- IT ^^'^ The first order a-c. convection current density is given by qi = Poi^'i + piui (A-10) which, by combining with (A-7a) and (A-8b), becomes j{k/r]oKpoi/meo)E^ 9i = CO- 1 Similarly {-S [f we now define and let where coe , the plasma-electron angular frequency given by nteo Equations (11) and (12) become (Z - B,) Thus equation (A-6) becomes (A-11) . k e J - POl J^z 92 = ^ ^2 (A-12) ;8o = K/3i + /32) (A-13) B. = f; 5. = f (A-14) po Po Z = ^ (A-15) W\ = ^ ; W, = ^ (A-16) CO CO co!. = -^\etc. (A-17) men IREQi-EXCV AMl'IJIIER 47 where If we assume tliat the tube whicli surrounds the beam be taken as infinitely remote, the appropriate solutions outside the beam are //^o = A,K,{yr) (A-21) and inside the beam j"^^ A,K,M (A-22) ^vi = Aihi^r) (A-23) E,i - -i-'^^.li/oaO (A-24) \/L where 7- = (S^ - k'~ ^ (3'^ (A-25) The /'s and i^T's in equations (A-21)"(A-24) are modified Bessel functions.' At the surface of the beam (r = a), one has the following two independent boundary conditions B^i = H^o (A-26) ^zi = -£ro (A-25a) which, using equations (A-21)-(A-24), yield VLh(^a) K,{ya) ^'''" ^ From equations (A-13), (A44), (A-15) and (A-24) one has ^a = Z/3oa\/Z (A-28) ya = ZjSoa (A-29) If we now define a beam wavelength, X^ , by the relations /3o - f (A-30) and assume for the purpose of simplifying the calculation that in the ex- pression for L in (A-20) wIbI = wIbI = W^ (A-31) ^ See A Treatise on the Theory of Bessel Functions, G. N. Watson, Chapter 3. 48 BELL SYSTEM TECHNICAL JOURNAL We easily see that where We = - Jo/(oUo (A-32) (A-33) We obtain from (A-20), (A-28), (A-29) and (A-30) raZ (2TraZ\ ( ry2irt ,, (l-waZX r { rr 2TaZ\ = L (A-34) = 1 - W + W^ .{Z-B^y {Z-B^)\ Equation (A-31) is equivalent to Equation (12) of the text or to (A-35) 2a _i_ ■1 II ^ N Fig. 6 — The diameter of the electron flow considered is 2a, and the length is /. Letting F = — Z and making use of the following well known relations between the Bessel functions U{jx) = Jo{x) hijx) = jJxix) Equation (A-34) becomes w (A-36) (A-37) W (Z - B,Y (Z - ^2)- Let the right-hand side of equation (A-37) be denoted by Fi{Z) and the middle of Ft{Z). In order to find the real roots of equation (A-37) one can plot F\ and F^ as functions of Z on the same chart. The abscissae of the intersections of the two curves will then be the real roots. In Fig. 7, F\ is plotted as a function of Z for B]. = 0.9 and B^ = 1.1. HIGH-FREQUENCY AMPLIFIER 49 In view of the definitions in equations (A-13) and (A-14), both Bi and B2 are uniquely defined by a single parameter, namely, the fractional velocity separation, b. That is and b = 2(wi - U2)/{ux + Wo) = 2(/32 - ^i)/(/32 + iSi) = B2- Bi B,= 1- ib/2) ^2 = 1 + {b/2) (A-38) (A-39) 500 ^ 0.75 0.80 1.00 z Fig. 7 — A curve illustrating conditions giving rise to various types of roots. A complete plot of F2 , for any value of the parameters W and (a/Xj), would show that equation (A-37) has an infinite number of real solutions. A real solution of equation (A-37) means an unattenuated wave. Thus there are an infinite number of unattenuated waves possible. The waves which will actually be excited in any given case, however, depend upon the boundary conditions at the input and output of the tube. Ordinarily only those waves will be excited which do not have a reversal in phase of the longitudinal E vector, say, as r varies from 0 to a. Attention, therefore, 50 BELL SYSTEM TECHNICAL JOURNAL will be given only to those waves for which Ez does not change sign over a cross-section of the beam. By inspection of equation (A-23), it is evident that this requirement is automatically satisfied if L > 0. On the other hand, if L is negative, one has Ezi ^Jo(Vv ^-?- z) ( A-23a) Thus attention will be limited to those roots which satisfy ^■'^l^Z < 2.405 (A-40) As where 2.405 is the first zero of the Bessel function in equation (A-21a). Returning to Fig. 7, portions of three different F2 curves are plotted: one for W^ = 0.01, one for IP = 0.0152 and one for W- = 0.02. All three curves are for (a/A,,) = 0.16. The intersections which represent roots which satisfy the inequality (A-40) are marked with arrows. Evidently there are either four real roots of this type or there are two real roots and a complex conjugate pair, the distinction being determined by the value of W. Thus there is a critical value of IF- (in this case it is 0.0152) for which two of the real roots are identical. This identical pair is indicated by two arrows near the minimum of the Fi curve at Z = 1. A pair of conjugate complex roots means that there are an increasing wave and a decreasing wave. Thus for each value of b and (a/Xs) there is a least value of IP below which the tube will have no gain. It can be shown that the critical tangency of the Fi and F2 curves occurs at a value of Z which is less than b'^ away from unity. Very little error will be incurred, then, by assuming that this critical point occurs at Z = 1 if b is small. Letting Z = 1 in equation (A-37), and using equations (A-39) one has 8(IIV^)^ - 1 = f^^(2WXj/o(V8aiV^F^2WxJ\ (A.41) \A'o(27ra/X.)/i (\/8(IlV^)' - 1 2ira/K)J whe:e Wm is the critical value of H'. Equation (A-41) determines (IFV^)^ as af unction of (a/X,). This relationship is plotted in Fig. 4. We will find that there will be an increasing wave in the range Wm < IF < oc . The calculation of the gain in this interv-al would be very laborious since Bessel functions of complex argument would be involved. However, a good approximation can be made when b is small. The real part of Z will always be near unity and the imaginary part will be found to be less than b/2. Therefore one can let Z = 1 in equation (A-37) where it multiplies the factor (lira/Xa) in the argument of the Bessel functions and let Z — 1 = U in the right-hand side of Equation (A-37). With these HIGH-FREQUENCY AMPLIFIER 51 assumptions I' can be determined as a function of (a/Xs) and U can be deter- mined as a function of 1'. We have from Equation (A-37) ' + ,„^^. = l+Jf (A-37a) {U + 6/2)- iU - h/2Y W^ When t/ = 0, U'" = W\, = W\, , so that 1 + 1' = 8{\VWbT- (A-42) and equation (A-37a) becomes (FTV2)' + (ir^f = (8/s')(n-.»/ir)' (a-37« the solution of which, for the increasing wave, is u = i(V2)[(i/2)(ir/ir.„)-(A/i + s{Wm/w)' - i) - i]^ (a-43) and the gain will be given by Gain/6 = 27j[(l/2)iW/\VMr-(VfT^W7/W " l) " l]^ (A-44) db/wavelength/unit b "Decibels gain/wavelength/unit b" is plotted against (W/Wm)~ in Fig. 3. As (TF/ir.v)- becomes very large, the gain per wavelength approaches 27.3 b db. Experimental Observation of Amplification by Interaction Between Two Electron Streams By A. V. HOLLENBERG The construction and performance of an amplifier employing the interaction between two streams of electrons having different average velocities are de- scribed. Gain of 33 db at a center frequency of 255 Mc has been observed with bandwidth of 110 Mc between 3 db points. 1. Introduction ANEW type of amplifier in which the gain is obtained by an interaction between streams of electrons of two or more average velocities is proposed in a companion paper by Pierce and Hebenstreit.^ This amplifier contains input and output portions in which signals are impressed on and extracted from the electron flow by electromagnetic circuits and a central portion in which gain occurs purely by interaction between streams of elec- trons without any circuits being involved. A small signal theory for coinci- dent electron streams of two d-c. velocities is presented in Pierce and Heben- streit's paper. In this paper a description of the construction and operation of an amplifier of this kind will be presented. Departures of the actual conditions in the amplifier from the assumptions of the theory limit the expectations of quanti- tative agreement. It is believed, however, that the evidence for gain arising from the interaction between two streams of electrons is clear, and that the broad frequency response predicted by the theory has been confirmed. 2. Description or Amplifier The frequency range near 200 Mc was chosen for the first experimental test of the proposed method of amplification for reasons of convenience. The theory indicates that current density requirements increase with fre- quency, but that these requirements become severe only at the higher micro- wave frequencies. Availability of circuit parts and test equipment, rather than anticipated difficulties at higher frequencies, led to the choice that was made. The essential features of one of the double-stream amplifier tubes which has been constructed and operated are represented in Fig. 1. The out- put helix was identical with the input helix in construction and connection 1 A New Type of High Frequency Amplifier, J. R. Pierce and W. B. Hebenstreit, this issue of the B. S. T. J. 52 INTERACTION BETWEEN ELECTRON STREAMS 53 54 BELL SYSTEM TECHNICAL JOURNAL to the coaxial line. The two identical probes pi , and p2 , extending from coaxial lines into the two electron streams at the beginning and end of the central portion of the tube between the two helices were inserted for com- parison of the signal amplitudes at the be;.^innini? and end of the region in which no circuit is present. A similar tube containing an output gap in place of the output helix sec- tion is represented in Fig. 2. In both cases concentric tubular electron streams originate at the ring- shaped emitting surfaces of the two cathodes at potentials Vi and V2 , pass through their respective control grids and through a common accelerating grid. An axial magnetic field of approximately 700 gauss is applied in order to maintain the definition of the beams. The outer and inner tubular beams .\^^^^^ kwvvvC- ( k ^ k t k k-rt Ik k k k k k k I.WVrr-'.W l|l|l|l l|l|l|l| C, C,- INNER AND OUTER CATHODES S S _ II I. II ELECTRON STREAMS T -HELIX TERMINATION H -INPUT HELIX G -OUTPUT GAP Fig. 2 — Representation of double-stream amplifier with gap output. have mean radii of 0.215" and 0.170" respectively and a wall thickness of 0.030" in each case. The short sections of helix which are used for input and output are wound of 0.013" diameter molybdenum wire, 44 t.p.i., and mean diameter of 0.500". The axial velocity of signal propagation along this helix is equal to that of 54- volt electrons. The helix sections are each 2" long. Ceramic supporting rods on each helix section are sprayed with aquadag, over Ij" of their length on the end nearest the center of the tube, for terminating purposes. The thickness of the spray coating increases toward the center of the tube. The distance between helices is 8.7". The gain produced by the electronic interaction depends upon a difference in velocity between the two electron streams. The signal is impressed upon one of the streams by the helix when its velocity is that at which travel- ing wave amplifier interaction between the stream and the helix occurs. It IXTERACTIOX BETWERX ELECT ROX STREAMS ,55 is required, therefore, for this helix that one of the streams travel at a velocity corresponding to a potential in the neighborhood of 54 volts. Useful interaction occurs from 50 to 60 volts. The inner stream is adjusted for helix interaction in this amplifier, and the outer stream travels at a lower velocity to bring about the interaction between the two streams. Operation about a mean voltage of about 50 volts was planned in designing the ampli- fier, and in estimating its expected performance. The amplifier is 16 wave- lengths long in terms of the wavelength associated with a mean voltage of 50 volts and a frequency of 200 Mc. Eleven of these wavelengths are in the center portion between the helices. The conditions in the amplifier tube differ from those assumed in the derivation of the theory of the double-stream interaction in the following significant ways: 1. The beams are separated in space and not completely intermingled. Calculations on the effect of this separation have been made. Numeri- cal examples of the calculated magnitude of the effect on gain will be given below. 2. Hollow tubular beams are used, instead of "solid" beams of uniform current density over their cross-sectional area. The theory indicates that, for the beam dimensions and currents used here, the parameters which depend upon beam radius and total current in the beam are nearly the same whether the current is concentrated in an infinitely thin cylindrical shell or uniformly distributed over the cross-section of a cylinder of the same radius. 3. The metal wall surrounding the beams is not infinitely remote. Its diameter was chosen as a compromise between the requirements of preventing serious d-c. space charge depression of potential in the beam and of being far enough removed from the beam to prevent a large effect on the interaction due to its presence. Its proximity tends to increase the minimum current required for producing gain, and there- fore to reduce the ratio of actual to critical current on which the gain depends. 4. The beams are not perfectly confined to hollow cylinders of the dimen- sions given. There is evidence that some spreading outside of these dimensions occurs. The currents reaching the collector can be meas- ured and these are used as "beam currents" in the discussion to follow and in comparisons between theory and experiment. Somewhat larger currents than these were initially launched, and the lost fraction may have contributed to the interaction before striking the walls. Although the assumptions of the theory are not fulfilled in the actual amplilier, estimates of its performance were first made without correction for the discrepancies. With voltages of 40 and 60 volts on the outer and 56 BELL SYSTEM TECHNICAL JOURNAL inner streams and currents of 0.5 and 0.8 milliampere, a gain of 40 db at 200 Mc due to the double-stream amplification was predicted, with bandwidth to 3 db points of 90 Mc, centered about 200 Mc. A later estimate, in- cluding the effect of the separation of the hollow beams in space, reduced the 40 db figure to 23 db. Prediction of the performance of the device as a whole also requires an evaluation of the couphng of the helix sections to the electron stream. The length of the active portions of the helix sections was chosen to give gain of order unity as estimated from single-stream traveling wave amplifier con- siderations for the proposed operating current. o ~^^ -20 -30 -40 ^^ / X Fig. 3 — Power output versus power input for double-stream amplifier at 200 Mc with beam potentials of 54 volts and ii volts and 1.1 ma current in each beam. 3. Experimental Observations The two beam potentials and the two beam currents were varied until good operating conditions were reached. In the amplifier tube with helix output, good operation with 29 db gain at 200 Mc was observed at low input signal levels with inner beam potential 54 volts, outer beam potential 2>i volts, and 1.1 milliampere current in each beam. Measurements of output and input signal power are shown in Fig. 3, in which output power is plotted as a function of input power. The gain is seen to be constant at 29 db from low levels up to an output power of 0.03 milliwatts, at which point compression sets in. Maximum power output is 0.3 milliwatts. INTERACTION BETWEEN ELECTRON STREAMS 57 The variation of gain with frequency under very nearly the same condi- tions as above is shown in Fig. 4. The input signal was in the linear region of Fig. ^. The figure shows a bandwidth of 110 Mc between points 3 db down from maximum. The center of the band is at approximately 255 Mc. The currents required to realize the above results are some tenths of a milliampere higher than those used in making estimates of performance of the amplifier, and the voltages for best performance are lower. The use of the actual operating values leads to a prediction of maximum gain at 240 Mc for the double-stream interaction in better agreement with the 40 30 \ / ^^> /' K 150 ZOO 250 300 350 400 FREQUENCY - MEGACYCLES Fig. 4 — Gain versus frequenc>- for double-stream amplifier, operating in linear portion of Fig. 3. observed maximum at 255 Mc than the value of 200 Mc originally estimated from the lower currents and higher voltages. Estimated double-stream gain at 240 Mc from theory for the separated hollow beams is 40 db. Evidence that the gain of the amplifier resulted from the double-stream interaction was obtained in the following ways: by comparison of the trans- mission through the amplifier with both streams with that for each stream alone, and by comparison of signals obtained from the two identical probes in the electron streams at the ends of the central portion of the tube shown in Fig 1. In a typical comparison of the first type, 1 db gain for the device was observed with the inner stream alone. This increased to 29 db when the outer stream was turned on. With the outer beam alone the loss in the 58 BELL SYSTEM TECHNICAL JOURNAL device was very large, for tlie velocity of the outer beam was far from that at which interaction with the helices occurs. A signal from the probe at the end of the central portion of the tube 23 db greater than that from the probe at the beginning of this section was observed in a comparison of the second type. This can probably be taken as a measure of the increase in signal in this portion of the tube due to the double-stream interaction alone, although the probe arrangement may also be subject to some remaining complicating effects. Overall gain for the device in this measurement was 32 db. Further interaction of the same kind occurs in the portions of the tube outside of the space between the probes. Measurements of the gain of an amplifier with helix output as a function of velocity separation between the streams have been made. For fixed mean voltage and current, theory predicts an increase in gain from zero db at zero separation to a maximum and then a decrease to zero as the velocity separation is further increased. A maximum gain was observed experi- mentally as velocity separation was varied, and in the neighborhood of the predicted optimum value of velocity separation for the current used. In the amplifier tube with gap output it was possible to evaluate the a-c. component of current in the electron stream produced by the amplified signal since the impedance across the gap was known. The power output from this tube at saturation was 0.1 mw, a little less than the maximum shown in Fig. 3. For 75 ohm output impedance this power corresponds to 1.15 milliampere r.m.s., or about one third of the total d-c. current to the collector in both streams. The output power, although relatively low, is thus of the right order of magnitude for the currents used. Acknowledgment The writer wishes to acknowledge his indebtedness to J. R. Pierce for valuable suggestions and discussion, and for supplying unpublished calcu- lations concerning the relation between hollow and solid beams, the efifect of the proximity of the conducting wall, and the effect of the separation of the beams in space. Thanks are also due to A. R. Strnad for assistance in mechanical design and to R. E. Azud for construction of the amplifier tubes. The Synthesis of Two-Terminal Switching Circuits By CLAUDE. E. SHANNON PART I: GENERAL THEORY 1. Introduction THE theory of switching circuits may be divided into two major divi- sions, analysis and synthesis. The problem of analysis, determining the manner of operation of a given switching circuit, is comparatively simple. The inverse problem of finding a circuit satisfying certain given operating conditions, and in particular the best circuit is, in general, more difficult and more important from the practical standpoint. A basic part of the general synthesis problem is the design of a two-terminal network with given operating characteristics, and we shall consider some aspects of this problem. Switching circuits can be studied by means of Boolean Algebra.'- This is a branch of mathematics that was first investigated by George Boole in connection with the study of logic, and has since been applied in various other fields, such as an axiomatic formulation of Biology,^ the study of neural networks in the nervous system,^ the analysis of insurance policies,^ prob- ability and set theory, etc. Perhaps the simplest interpretation of Boolean Algebra and the one closest to the application to switching circuits is in terms of propositions. A letter A^, say, in the algebra corresponds to a logical proposition. The sum of two letters A' + Y represents the proposition ".Y or I'" and the product XY represents the proposition "A" and F". The symbol A'' is used to represent the negation of proposition X, i.e. the proposition "not A"". The constants 1 and 0 represent truth and falsity respectively. Thus A -\- Y = 1 means .Y or Y is true, while -Y + YZ' = 0 means A' or (!' and the contradiction of Z) is false. The interpretationof Boolean Algebra in terms of switching circuits*^-^''-'"' is very similar. The symbol A' in the algebra is interpreted to mean a make; (front) contact on a relay or switch. The negation of A', written .Y', represents a break (back) contact on the relay or switch. The constants 0 and 1 represent closed and open circuits respectively and the combining operations of addition and multiplication correspond to series and parallel connections of the switching elements involved. These conventions are shown in Fig. L With this identification it is possible to write an algebraic 59 60 BELL SYSTEM TECHNICAL JOURNAL NETWORK » 0^0 • a 0^0 • HINDRANCE FUNCTION 0 (PERMANENTLY CLOSED CIRCUIT) 1 (PERMANENTLY OPEN CIRCUIT) X (MAKE CONTACT ON RELAY X) X' (BREAK CONTACT ON RELAY X) X + Y (SERIES CONNECTION) XY (PARALLEL CONNECTION) -• w [x + y(z+ X')] Fig. 1 — Hindrance functions for simple circuits. expression corresponding to a two-terminal network. This expression will involve the various relays whose contacts appear in the network and will be called the hindrance or hindrance function of the network. The last net- work in Fig. 1 is a simple example. Boolean expressions can be manipulated in a manner very similar to ordinary algebraic expressions. Terms can be rearranged, multiplied out, factored and combined according to all the standard rules of numerical algebra. We have, for example, in Boolean Algebra the following identities^ 0 + Z = X OX = 0 \-X = X X -\-Y = Y + X XY = YX X + (F + Z) = (X -f F) + Z X(YZ) = iXY)Z X{Y -h Z) = XY -\- XZ The interpretation of some of these in terms of switching circuits is shown in Fig. 2. There are a number of further rules in Boolean Algebra which allow SINTIIESIS OF SWITCHING CIRCUITS 61 « o"o- -o o- X Y + X X (Y2) (XY) 2 -0 o- -o^o- XY2 hC ^> X (Y+Z) = XY *• XZ Fig. 2— Interpretation of some algebraic identities. simplifications of expressions that are not possible in ordinary algebra. The more important of these are: X = X+X = X^- X + X = etc. X= X-X = X-X-X = etc. X + 1 = 1 X+YZ= {X+ Y){X+ Z) X + X' = 1 X-X' = 0 (X+ Y)' = X'Y' {XY)' = X' + r The circuit interpretation of some of these is shown in Fig. 3. These rules make the manipulation of Boolean expressions considerably simpler than ordinary algebra. There is no need, for example, for numerical coefficients or for exponents, since nX = X" = X. By means of Boolean Algebra it is possible to find many circuits equivalent in operating characteristics to a given circuit. The hindrance of the given circuit is written down and manipulated according to the rules. Each different resulting expression represents a new circuit equivalent to the given one. In particular, expressions may be manipulated to eliminate elements which are unnecessary, resulting in simple circuits. Any expression involving a number of variables Xi , X2 , • • • , X„ is 62 BELL SYSTEM TECHNICAL JOURNAL called a. Junction of these variables and written in ordinary function notation, f{X, , X2 , • • • , Xn). Thus we might have /(X, F, Z) = X + I"Z + XZ'. In Boolean Algebra there are a number of important general theorems which hold for any function. It is possible to expand a function about one or more of its arguments as follows: /(Xi , X2 , • • • , X„) = X:/(l, X2 , • • • , X„) + X7(0, X2 , • • • , Xn) This is an expansion about Xi . The term /(I, X2 , • • • , X„) is the function -• = •- X + X + X (X + Y) (X + Z) -c;:> = XX' = 0 Fig. 3 — Interpretation of some special Boolean identities. /(Xi , X2 , • ■ • , X„) with 1 substituted for A^, and 0 for X', and conversely for the term/(0, X2 , • • • , X„). An expansion about X'': and A'2 is: /(Xi,X2,--- ,X„) = X:X2/(l,l,X3,---,X„) + XiX2/(l,0,X3,---,X„) + X(X2/(0, 1, X3 , • • • , X„) + X(X2/(1, 1, X3 , • • • , X„) This may be continued to give expansions about any number of variables. When carried out for all n variables, / is written as a sum of 2" products each with a coefficient which does not depend on any of the variables. Each coefficient is therefore a constant, either 0 or 1. There is a similar expansion whereby/ is expanded as a product: /(Xi,X2,---,X2) = [Xi + /(O, X2 , • • • , X„)] [X( + /(I, X2 , • • • , X„)] = [Xi+ X2+/(0,0, • • • ,X,0] [Xi + X2+/(0, 1, • • • ,X„)] [x; + X2 + /(1, 0, • • • , X,.)] [x; + Xo + /(1, 1, • • • . Xn)\ = etc. svxTiiEsis or swnnnxc; circuits 63 The following are some further identities for general functions: A'+/(.v, y,z,---) = A-+/(0, V,Z, •••) A"+/(.v, i',z, ■••) = x'+/(i, r,z, •••) Xf(X, Y, /,-••) = A7(l, I',Z, •••) X'fiX, Y, Z, ■••) = A7(0, Y,Z,---) X + f(X,Y, Z,W) X + f(0,Y, Z,W) 'o— o o— o o— f (X,Y, Z,W) l— o o— o o— o Xf(l,Y,2,W) + X'f(0,Y,Z,W) X 1 °''° 1 X' X' 0 0 Y' o' n Z. W' 0^0 z w 1— o O— 0 0 — 1 = XYf(l, 1,Y, Z) + X Y'f (1,0,Y, Z) + X'Y f (0, 1,Y,Z) + X' Y' f (0, 0, Y,Z) Fig. 4 — Examples of some functional identities. The network interpretations of some of these identities are shown in Fig. 4. A little thought will show that they are true, in general, for switching circuits. The hindrance function associated with a two-terminal network describes the network completely from the external point of view. We can determine •from it whether the circuit will be open or closed for any particular position of the relays. This is done by giving the variables corresponding to operated relays the value 0 (since the make contacts of these are then closed and the break contacts open) and unoperated relays the value 1. For example, with the function / = W[X + Y{Z + A^')l suppose relays A' and Y operated and Z and W not operated. Then/ = 1[0 + 0(1 + 1)] = 0 and in this condition the circuit is closed. 64 BELL SYSTEM TECHNICAL JOURNAL A hindrance function corresponds explicitly to a series-parallel type of circuit, i.e. a circuit containing only series and parallel connections. This is because the expression is made up of sum and product operations. There is however, a hindrance function representing the operating characteristics (conditions for open or closed circuits between the two terminals) for any network, series-parallel or not. The hindrance for non-series-parallel net- works can be found by several methods of which one is indicated in Fig. 5 for a simple bridge circuit. The hindrance is written as the product of a set of factors. Each factor is the series hindrance of a possible path between the two terminals. Further details concerning the Boolean method for switching circuits may be found in the references cited above. This paper is concerned with the problem of synthesizing a two-terminal circuit which represents a given hindrance function /(A^i , • • • , X„). Since any given function / can be realized in an unlimited number of different f = (w+X)fZ+S)(W + Y + S)(Z+Y + X) Fig. 5 — Hindrance of a bridge circuit. ways, the particular design chosen must depend upon other considerations. The most common of these determining criteria is that of economy of ele- ments, which may be of several types, for example: (1) We may wish to realize our function with the least total number of switching elements, regardless of which variables they represent. (2) We may wish to find the circuit using the least total number of relay springs. This requirement sometimes leads to a solution different from (1), since contiguous make and break elements may be combined into transfer elements so that circuits which tend to group make and break contacts on the same relay into pairs will be advantageous for (2) but not necessarily for (1). (3) We may wish to distribute the spring loading on all the relays or on some subset of the relays as evenly as possible. Thus, we might try to find the circuit in which the most heavily loaded relay was as lightly loaded as possible. More generally, we might desire a circuit in which the loading on the relays is of some specified sort, or as near as possible to this given distribution. For example, if the relay Xi SVNTHKSIS OF SWITCH fNC, CIRCUITS 65 must operate very quickly, wliile .V2 and .V3 have no essential time limitations but are ordinary U-type relays, and .V4 is a muiticontact relay on which many contacts are available, we would probably try to design a circuit for/(.Vi , .Y2 , A'3 , .Y4) in such a way as, hrst of all, to minimize the loading on .Yi , next to equalize the loading on Y2 and A'3 keeping it at the same time as low as possible, and finally not to load A'l any more than necessary. Problems of this sort may be called problems in spring-load dislribulioii. Although all equivalent circuits representing a given function / which contain only series and j)arallcl connections can be found with the aid of lioolean Algebra, the most economical circuit in any of the above senses will often not be of this type. The problem of synthesizing non-series-parallel circuits is exceedingly ditlicult. It is even more dilBcult to show that a circuit found in some way is the most economical one to realize a given function. The difficulty springs from the large number of essentially different networks available and more particularly from the lack of a simple mathematical idiom for representing these circuits. We will describe a new design method whereby any function /(-Yi , A'2, • • • , X„) may be realized, and frequently with a considerable saving of elements over other methods, particularly when the number of variables n is large. The circuits obtained by this method will not, in general, be of the series- parallel type, and, in fact, they will usually not even be planar. This method is of interest theoretically as well as for practical design purposes, for it allows us to set new upper limits for certain numerical functions asso- ciated with relay circuits. Let us make the following definitions: \(n) is defined as the least number such that any function of n variables can be realized with not more than \{n) elements.* Thus, any function of n variables can be realized with X(//) elements and at least one function with no less. • IJL(n) is defined as the least number such that given any function/ of n variables, there is a two-terminal network having the hindrance / and using not more than n(n) elements on the most heavily loaded relay. The first part of this paper deals with the general design method and the behaviour of X(;/). The second part is concerned with the possibility of various types of spring load distribution, and in the third part we will study certain classes of functions that are especially easy to synthesize, and give some miscellaneous theorems on switching networks and functions. 2. Fundamental Design Theorem The method of design referred to above is based on a simple theorem deal- ing with the interconnection of two switching networks. We shall first * An element means a make or break contact on one relay. .\ transfer element means a make-and-break with a common spring, and contains two elements. 66 BELL SYSTEM TECHNICAL JOURNAL State and prove this theorem. Suppose that M and iV (Fig. 6) are two (w + 1) terminal networks, M having the hindrance functions Uk (k = 1, 2, • • • w) between terminals a and k, and N having the functions Vk between b and k. Further, let M be such that Ujk = l(j') ^ = 1. 2, • • • , n). We will say, in this case, that M is a disjunctive network. Under these con- ditions we shall prove the following: Theorem 1: If the corresponding terminals I, 2, • • • , ti of M and N are connected together, then Uab=^Il{Uk-\- Vk) (1) where Uab is the hindrance from terminal a to terminal b. — — • 1 1 • 92 2« ——•3 3« M 1 N — Fig. 6 — Network for general design theorem. Proof: It is known that the hindrance Uab may be found by taking the product of the hindrances of all possible paths from atob along the elements of the network.* We may divide these paths into those which cross the Hne L once, those which cross it three times, those which cross it five times, etc. Let the product of the hindrances in the first class be TFi , in the second class Ws , etc. Thus Now clearly Uab= Wi-WrW, Wi = n (U + Vk) (2) and also Ws = TFb 1 since each term in any of these must contain a summand of the type Ujk which we have assumed to be 1. Substituting in (2) we have the desired result. V, The method of using this theorem to synthesize networks may be roughly SYNTHESIS OF SWITCH I XG CIRCUITS 67 described as follows: The function to be realized is written in the form of a product of the type (1) in such a way that the functions Uk are the same for a large class of functions, the Vk determining the particular one under consider- ation. A basic disjunctive network M is constructed having the functions Uk between terminals a and k. A network N for obtaining the functions Vk is then found by inspection or according to certain general rules. We will now consider just how this can be done in various cases. 3. Design of Networks for General Functions — Beh.a.vior of X(w)- a. Functions of One, Two and Three Variables: Functions of one or two variables may be dismissed easily since the number of such functions is so small. Thus, with one variable X, the possible functions are only: 0, 1, X, X' and obviously X(l) = 1, m(1) = 1- With two variables X and F there are 16 possible functions: 0 X F XY XV X'Y X'Y' XY' + X'Y \X' Y' X + F X+Y' X' +Y X' + F' XF + X'Y' so that X(2) = 4, m(2) = 2. We will next show that any function of three variables /(X, F, Z) can be realized with not more than eight elements and with not more than four from any one relay. Any function of three variables can be expanded in a product as follows: /(X, F, Z) = [X + F + /(O, 0, Z)][X + F' + /(O, 1, Z)\ [X' + F + /(I, 0, Z)] [X' + F' + /(I, 1, Z)]. In the terminology of Theorem 1 we let f/i = X + F Ki = /(O, 0, Z) U,= X+Y' V,^ /(O, 1, Z) f/3 = X' + F F3 = /(I, 0, Z) f/4 = X' + Y' V, = /(I, 1, Z) so that 4 f'a6 =/(X, F,Z) = ]l{Vk+ Vk) k-X The above Uk functions are realized with the network M of Fig. 7 and it is 68 BELL SYSTEM TECHNICAL JOURNAL easily seen that U jk — \ {j, k = 1,2, 3, 4). The problem now is to construct a second network iV having the Vh functions Vi , V2 , V3 , Vi . Each of these is a function of the one variable Z and must, therefore, be one of the four possible functions of one variable : 0, 1, Z, Z'. Consider the network N of Fig. 8. If any of the F's are equal to 0, connect the corresponding terminals of M to the terminal of N marked 0; if any are equal to Z, connect these terminals of M to the terminal of N marked Z, etc. Those which are 1 are, of course, not connected to anything. It is clear from Theorem 1 that the network thus obtained will realize the function /"(X, F, Z). In many cases some of the elements will be superfluous, e.g., if one of the F,- is equal to 1, the element of M connected to terminal i can Fig. 7 — Disjunctive tree with two bays. be eliminated. At worst M contains six elements and N contains two. The variable X appears twice, F four times and Z twice. Of course, it is completely arbitrary which variables we call X, F, and Z. We have thus proved somewhat more than we stated above, namely. Theorem 2: Any function of three variables may be realized using not more than 2, 2, and 4 elements from the three variables in any desired order. Thus X(3) < 8, m(3) < 4. Further, since make and break elements appear in adjacent pairs we can obtain the distribution 1, 1, 2, in terms of transfer ele- ments. The theorem gives only upper limits for X(3) and n{i). The question immediately arises as to whether by some other design method these limits could be lowered, i.e., can the < signs be replaced by < signs. It can be shown by a study of special cases that X(3) = 8, the function X @ Y ® Z^ X{YZ + Y'Z') + X' {YZ' + Y'Z) requiring eight elements in its most economical realization. m(3), however, is actually 3. SYNTIJE^IS OF SWITCIJING CIRCUITS 69 It seems probable that, in general, the function Ai e A'2 e ■ • • e a'„ requires 4(;/ — 1) elements, but no proof has been found. Proving that a certain function cannot be realized with a small number of elements is somewhat like proving a number transcendental; we will show later that almost all* functions require a large number of elements, but it is difBcult to show that a particular one does. -•b 0 • Fig. 8 — Network giving all functions of one variable. Y Fig. 9 — Disjunctive tree with three bays, b. Functions of Four \'ariables: In synthesizing functions of four variables by the same method, two courses are open. First, we may expand the function as follows: f{\\\ X, Y, Z) = [W + X + 1' + V^{Z)\-{W + A' + I" + V.{Z)]. [W + X' -V Y + Vz{Z)\-[W + X' + I" + V,{Z)\. [W + X^ Y + V,{Z)\-{W' + X + y + ^5(Z)]. [w + X' + r + v,{z)\-[w' + r + y' + v,{z)]. By this expansion we would let Ui = W + A^ + Y, U2 = IF + A' + I ', • • • , f/g = II" 4- A"' + I" and construct the M network in Fig. 9. X would * We use the expression "almost all" in the arithmetic sense: e.g., a property is true of almost all functions of n variables if the fraction of all functions of n variables for which it is not true — > 0 as « — > « . 70 BELL SYSTEM TECHNICAL JOURNAL again be as in Fig. 8, and by the same type of reasoning it can be seen that X(4) < 16. Using a shghtly more complicated method, however, it is possible to reduce this limit. Let the function be expanded in the following way: f{W, X, Y, Z) = [W-\-X+ Fi(F, Z)HW -\-X' + ^2(7, Z)] [W + X + V^{Y, Z)]-[W' + X' + V,{Y, Z)\. We may use a network of the type of Fig. 7 for M. The V functions are now functions of two variables Y and Z and may be any of the 16 functions: 0 Y Y' Z c< YZ Y'Z YZ' D. Y + Z Y -\- Z' Y'-\-Z Y'Z + YZ' e\ YZ + Y'Z' Z' Y'Z' Y' + Z' B{ We have divided the functions into five groups, A, B,C, D and E for later reference. We are going to show that any function of four variables can Fig. 10 — Simplifying network. be realized with not more than 14 elements. This means that we must construct a network N using not more than eight elements (since there are six in the M network) for any selection of four functions from those listed above. To prove this, a number of special cases must be considered and dealt with separately: (1) If all four functions are from the groups, A, B, C, and D, X will certainly not contain more than eight elements, since eight letters at most can appear in the four functions. (2) We assume now that just one of the functions is from group E; without loss of generality we may take it to be YZ' + Y'Z, for it is the other, replacing Y by Y' transforms it into this. If one or more of the remaining functions are from groups A ov B the situation is satisfactory, for this func- tion need require no elements. Obviously 0 and 1 require no elements and F, Y', Z or Z' may be "tapped oflf" from the circuit for YZ' + Y'Z by writing it as (F + Z)(Y' + Z'). For example, Y' may be obtained with the circuit of Fig. 10. This leaves four elements, certainly a sufficient number for any two functions from A, B,C, or D. SYNTHESIS OF SWITCHING CIRCUITS 71 (3) Now, Still assuming we have one function, VZ' -\- Y'Z, from E, suppose at least two of the remaining are from D. Using a similar "tapping off" process we can save an element on each of these. For instance, if the functions are Y + Z and I'' + Z' the circuit would be as shown in Fig. 11. (4) Under the same assumption, then, our worst case is when two of the functions are from C and one from D, or all three from C. This latter case is satisfactory smce, then, at least one of the three must be a term of YZ' -\- Y'Z and can be "tapped off." The former case is bad only when the two functions from C are YZ and Y'Z'. It may be seen that the only » o o- Z' .b Y' Fig. 11 — Simplifying network. Fig. 12 — Simplifying network. essentially different choices for the function from D are Y -{- Z and Y' + Z. That the four types of functions/ resulting may be realized with 14 elements can be shown by writing out typical functions and reducing by Boolean Algebra. (5) We now consider the cases where two of the functions are from E. Using the circuit of Fig. 12, we can tap off functions or parts of functions from A, B or D, and it will be seen that the only difficult cases are the fol- lowing: (a) Two functions from C. In this case either the function/ is symmetric in F and Z or else both of the two functions may be obtained from the circuits for the E functions of Fig. 12. The symmetric case is handled in a later section, (b) One is from C, the other from D. There is only one unsymmetric case. We assume the four functions are Y © Z, Y @ Z', YZ and Y + Z'. This gives rise to four types of functions /, which can all be reduced by algebraic methods. This completes the proof. 72 BELL SYSTEM TECHNICAL JOURNAL Theorem 3: Any function of four variables can be realized with not more than 14 elements. c. Functions of More Than Four Variables: Any function of five variables may be written f(Xx, ••• , X,) = [Xs + fiiXr, ■■■ , X,)]-[Xi + /2(Xi, ••• , X,)] and since, as we have just shown, the two functions of four variables can be realized with 14 elements each, f{Xi , • • ■ .Y5) can be realized with 30 x'»- -00- - X' yV X' Y»- X'+Y'»- X'Y'»- X +Y'»- X' + Y»- x' xyV X Y'+ X'Y H::>k:;n- X Y + X' Y'l <^:} Fig. 13 — Network giving all functions of two variables. Now consider a function /(Xi , X2, ■■■ , Xn) of n variables. For 5 < « < 13 we get the best limit by expanding about all but two variables. /(Xi , X2 , • • • , Xn) = [Xi + X2 + • • • + X„_2 + Fi(X„_l , X„)] [X( + X; + • • • + xLo + F.(X„_x , \\)] (4) The F's are all functions of the variables .Y„_i , A'„ and may be obtained from the general X network of Fig. 13, in which every function of two variables appears. This network contains 20 elements which are grouped into five transfer elements for one variable and five for the other.* The M network for (4), shown in Fig. 14, requires in general 2" ' - 2 elements. Thus we have: * Several other networks with ihe same property as Fig. 13 have been found, but they all require 20 elements. SYNTHESIS OF SWITCH I XG CIRCUITS 73 Theorem 4. \(n) < 2"-' + 18 d. Upper Limits for X(;0 witli Large //. Of course, it is not often necessary to synthesize a function of more than say 10 variables, but it is of considerable theoretical interest to determine as closely as possible the behavior of X(;0 for large n. n-2 Fig. 14 — Disjunctive tree with {n — 2) bays. Fig. 15— Network giving all functions of {in + 1) variables constructed from one giving all functions of m variables. We will tirst prove a theorem placing limits on the number of elements required in a network analogous to Fig. 13 but generalized for m variables. Theorem 5. An N network realizing all 2'"' functions of m variables can be constructed using not more than 2-f'" elements, i.e., not more than tico ele- ments per function. Any network ivilh this property uses at least il — e) elements per function for any e > 0 with n sufficiently large. The first part will be proved by induction. We have seen it to be true for m = 1,2. Suppose it is true for some m with the network .V of Fig. 15. Any function of m + 1 variables can be written g= lX„,+i+/J[.YWi+/6l 74 BF.LL SYSTF.M TECHNICAL JOURNAL where /a and/b involve only m variables. By connecting from g to the cor- responding fa and fb terminals of the smaller network, as shown typically for ^3 , we see from Theorem 1 that all the g functions can be obtained. Among these will be the 2'"' f functions and these can be obtained simply by connecting across to the / functions in question without any additional elements. Thus the entire network uses less than (2^ - l-"")! + 2-t elements, since the N network by assumption uses less than 2 • 2" and the first term in this expression is the number of added elements. The second statement of Theorem 7 can be proved as follows. Suppose we have a network, Fig. 16, with the required property. The terminals can be divided into three classes, those that have one or less elements di- ►fa .f,2"l Fig. 16 — NetWork giving all functions of m variables. rectly conriected, those with two, and those with three or more, The first set consists of the functions 0 and 1 and functions of the type (X+/) = X+/x=o where X is some variable or primed variable. The number of such functions is not greater than Im-l^"" for there are 2m ways of selecting an "X" and then 2"'" different functions /x=o of the remaining w — 1 variables. Hence the terminals in this class as a fraction of the total -^ 0 as w -^ <» . Functions of the second class have the form g= (X+/0(F+/2) In case X 9^ V this may be written XY + XY'gx^i.y^o + X'Ygx=o.y=i + X'Y'g^^o.v^o and there are not more than (2m)(2m — 2)[2"'" f such functions, again a vanishingly small fraction. In case X = Y' we have the situation shown in Fig. 17 and the XX' connection can never carry ground to another terminal since it is always open as a series combination. The inner ends of these elements can therefore be removed and connected to terminals SYNTHESIS OF SWITCHING CIRCUITS 75 corresponding to functions of less than m variables according to the equation g= (X-\- U){X' + U) = {X+ /lX=0)(X' + /2x=.l) if they are not already so connected. This means that all terminals of the second class are then connected to a vanishingly small fraction of the total terminals. We can then attribute two elements each to these terminals and at least one and one-half each to the terminals of the third group. As these two groups exhaust the terminals except for a fraction which -^ 0 as « ^ 20 , the theorem follows. If, in synthesizing a function of n variables, we break off the tree at the (« — m)\\\ bay, the tree will contain 2"'""^^ — 2 elements, and we can find an N network with not more than 2"'"- 2 elements exhibiting every function of the remaining m variables. Hence \{n) < T -2+22' < 2"^'"'^^ + 2 2^" Fig. 17 — Possible situation in Fig. 16. for every integer m. We wish to find the integer M = M{n) minimizing this upper bound. Considering m as a continuous variable and n fixed, the function /(m) = 2"-'"+'+ 2'"" -2 clearly has just one minimum. This minimum must therefore lie between m\ and Wi + 1, where /(wi) = /(wi + 1) I.e., or ^n ^ 2'"i+i(2''"'"'' _ 2''"') Now wi cannot be an integer since the right-hand side is a power of two and the second term is less than half the first. It follows that to find the inleger M making /(M) a minimum we must take for M the least integer satisfying 2" < 2^^'2''''" 76 BELL SYSTEM TECHNICAL JOURNAL Thus M satisfies: This gives: M + 1 + 2"^' >n>M+2' n < 11 11 < n < 20 20 < n < 37 37 < « < 70 70 < n < 135 etc. M = 2 iW = 3 M = 4 M = 5 M = 6 (5) X ^ N /■ S / \ / ■\ / \ ^ , / \ — '^ \yl v/ 1/ V v V y 3 4 5 6 7 8 9 10 11 LOGgn Fig. 18 — Behaviour of ^(m). Our upper bound for X(;0 behaves something Hke " — with a superimposed saw-tooth oscillation as n varies between powers of two, due to the fact that m must be an integer. If we define giji) by 2.-M+1 _^ f^^ gin) n M being determined to minimize the function (i.e., M satisfying (5)), then g{n) varies somewhat as shown in Fig. 18 when plotted against log2 n. The maxima occur just beyond powers of two, and closer and closer to them as w ^ oc . Also, the saw-tooth shape becomes more and more exact. The sudden drops occur just after we change from one value of M to the next. These facts lead to the following: Theorem 6. (a) For all n \{n) < ^^—, n {b) For almost all n \{n) < SYNTIIESrS OF SWITCHING CIRCUITS 77 (r) There is an injhiile sequence of nifor which \(n,) < — (1 + e) e > 0. n These results can be proved rigorously without much difTiculty. e. A Lower Limit for X(;0 with Large //. Up to now most of our work has been toward the determination of upper limits for X{n). We have seen that for all n X{n) < B- . n 2" We now ask whether this function 5 — is anywhere near the true value n of X(»), or may \{n) be perhaps dominated by a smaller order of infinity, e.g., n^. It was thought for a time, in fact, that \{n) might be limited by n~ for all //, arguing from the first few values: 1, 4, 8, 14. We will show that 2» this is far from the truth, for actually — is the correct order of magni- tude of \{n): A-< X(n) < B- n n for all n. A closely associated question to which a partial answer will be given is the following: Suppose we define the "complexity" of a given func- tion / of )i variables as the ratio of the number of elements in the most economical realization of/ to X(77). Then any function has a complexity lying between 0 and L Are most functions simple or complex? Theorem 7: For all sufficiently large n, all functions of n variables excepting a fraction 8 require at least (1 — e) — elements, where e and 8 are arbitrarily small positive numbers. Hence for large n \{n) > (1 - e) - n and almost all functions have a complexity > |(1 — e). For a certain sequence Hi almost all functions have a complexity > ^(1 — e). The proof of this theorem is rather interesting, for it is a pure existence proof. We do not show that any particular function or set of functions 2" requires (1 — e) — elements, but rather that it is impossible for all functions 78 BELL SYSTEM TECHNICAL JOURNAL to require less. This will be done by showing that there are not enough 2" networks with less than (1 — e) — branches to go around, i.e., to represent n all the 2^" functions of n variables, taking account, of course, of the different assignments of the variables to the branches of each network. This is only- possible due to the extremely rapid increase of the function 2" . We require the following: Lemma: The number of two-terminal networks with K or less branches is less than (6K) . Any two-terminal network with A^ or less branches can be constructed as follows: First line up the K branches as below with the two terminals 0 and b. a. 1—1' 2—2' 3—3' 4—4' b. K—K' We first connect the terminals a,b,l,2, ■ ■ ■ ,K together in the desired way. The number of diferent ways we can do this is certainly limited by the num- ber of partitions oiK-\- 2 which, in turn, is less than for this is the number of ways we can put one or more division marks between the symbols a,\, • ■ ■ , K,b. Now, assuming a, \,2, • ■ ■ , K, b, intercon- nected in the desired manner, we can connect 1' either to one of these ter- minals or to an additional junction point, i.e., 1' has a choice of at most A+ 3 terminals, 2' has a choice of at most A -f 4, etc. Hence the number of networks is certainly less than 2^+i(iC -i- 3) (A + 4) (A + 5) • • • {2K -f 3) <{6KY K>2> and the theorem is readily verified for iv = 1,2. We now return to the proof of Theorem 7. The number of functions of n variables that can be realized with elements is certainly less than n the number of networks we can construct with this many branches multi- syxruEsis or switciiinc circiits 79 plied by the number of assignments of the variables to the branches, i.e., it is less than (l-.)(2«/n) H = (InY'-''''""'' h{\ - e) ^ Hence log2 // = (1 - e) - log In + (1 - e) - log (1 - e) - • 6 // n 11 = (1 — e) 2" + terms dominated by this term for large n. By choosing n so large that - 2" dominates the other terms of log H we arrive at the inequality log2 H <{\- ei) 2" But there are 5 = 2 functions of n variables and 22" — » 0 as n — » 06 . Hence ahnost all functions require more than (1 — €1)2" eleriientS. Now, since for all n> N there is at least one function requiring mofe than 1 2" (say) - — elements and since \{n) > 0 for « > 0, we can say that for all n, 2 n 2* X(w) > A~ n for some constant A > 0, for we need only choose A to be the minimum number in the finite set: 1 Ml) X(2) X(3) XC^) 2 ' 2* ' 2^ ' 2' ' " " ' 2^ I 2 3 iV 2" Thus X(w) IS of the order of magnitude of — . The other parts of Theorem 8 follow easily from what we have already shown. The writer is of the opinion that almost all functions have a complexity nearly 1, i.e., > 1 — e. This could be shown at least for an infinite sequence Hi if the Lemma could be improved to show that the number of networks is less than {6K)^'^ for large K. Although several methods have been used in counting the networks with K branches they all give the result (6K)'^. 80 BELL SYSTEM TECHNICAL JOURNAL It may be of interest to show that for large K the number of networks is greater than This may be done by an inversion of the above argument. Let/(/\') be the number of networks with A' branches. Now, since there are 2" functions of 2"+- n variables and each can be realized with (1 + e) elements {n sufficiently // large), ./Yd + e) Y V2;o"^'>'^''"^'"^ >r for n large. But assuming /(A') < (6A)^ reverses the inequality, as is readily verified. Also, for an infinite sequence of A, /(A) > (6A)^''^ Since there is no obvious reason why /(A) should be connected with powers of 2 it seems likely that this is true for all large A'. We may summarize what we have proved concerning the behavior of 2n+l \{n) for large n as follows. \{n) varies somewhat as — ; if we let n ryn + l \{n) = An — n then, for large n, An lies between 5 — e and (2 + e), while, for an infinite sequence ofw, 5— e<^„ 00 and also to get a good evaluation of m(«) for small n. The problem is rather interesting mathematically, for it involves additive number theory, a subject with few if any previous applications. Let us first consider a few simple cases. Suppose we are realizing a function with the tree of 'Fig. 9. The three variables appear as follows: W, X, Y appear 2, 4, 8 times, respectively or, in terms of transfer elements* 1,2,4. Now, W, X, and Y may be interchanged in any way without altering the operation of the tree. Also we can interchange X and Y in the lower branch of the tree only without altering its operation. This would give the dis- tribution (Fig. 19) 1,3,3 * In this section we shall always speak in terms of transfer elements. 82 BELL SYSTEM TECHNICAL JOURNAL A tree with four bays can be constructed with any of the following dis- tributions w X Y Z 2, 4, 8 = 1,2,4,4-1,2,4 2, 5, 7 = 1,2,4 -M, 3, 3 2, 6, 6 = 1, 2, 4 -f 1, 4, 2 3, 3, 8 = 1, 2, 4 -f- 2, 1, 4 3, 4, 7 = 1,3,3 +2, 1,4 3. 5, 6 = 1,4, 2 +2,1,4 4, 4, 6 = 1,3, 3 + 3, 1, 3 4, 5, 5 = 1,4, 2 +3, 1,3 and the variables may be interchanged in any manner. The "sums" on the right show how these distributions are obtained. The first set of numbers represents the upper half of the tree and the second set the lower half. They are all reduced to the sum of sets 1, 2, 4 or 1, 3, 3 in some order, and these sets are obtainable for trees with 3 bays as we already noted. In general it is clear that if we can obtain the distributions fli , flj ) fl3 > • • • > fln - . bi ibi f bi ^ ' • • , bn for a tree with n bays then we can obtain the distfibution 1, fll + ^1 , ^2 + ^2 , • ' ' > a„ + bn for a tree with n + 1 bays. Now note that all the distributions shown have the following property: any one may be obtained from the first, 1, 2, 4, 8, by moving one or more units from a larger number to a smaller number, or by a succession of such operations, without moving any units to the number 1. Thus 1, 3, 3, 8 is obtained by moving a unit from 4 to 2. The set 1, 4, 5, 5 is obtained by first moving two units from the 8 to the 2, then one unit to the 4. Further- more, every set that may be obtained from the set 1, 2, 4, 8 by this process appears as a possible distribution. This operation is somewhat analogous to heat flow — heat can only flow from a hotter body to a cooler one just as units can only be transferred from higher numbers to lower ones in the above. These considerations suggest that a disjunctive tree with u bays can be constructed with any load distribution obtained by such a flow from the initial distribution 1,2,4,8, ••• ,2-> We will now show that this is actually the case. SYNTHESIS OF SWITCHING CIRCUITS 83 First let us make the following definition: The symbol (ci , oa , • • • , On) represents any set of numbers bi , bi , ■ ■ • , bn that may be obtained from the set oi , a2 , • • • , «« by the following operations: 1. Interchange of letters. 2. A flow from a larger number to a smaller one, no flow, however, being allowed to the number 1. Thus we would write 1, 2, 4, 8 = (1, 2, 4, 8) 4, 4, 1, 6 = (1, 2, 4, 8) 1, 3, 10, 3, 10 = (1, 2, 4, 8, 12) but 2, 2 5^ (1, 3). It is possible to put the conditions that bi , bt , ■ ■ ■ , bn = {ai , a2 , • ■ ■ , an) (6) into a more mathematical form. Let the a, and the bi be arranged as non- decreasing sequences. Then a necessary and sufficient condition for the relation (6) is that » t (1) llbi>T.ai s = 1,2, ••• ,n, i-l i n n (2) ^ bi ^ J2 di , and (3) There are the same number of I's among the Oi as among the bi . The necessity of (2) and (3) is obvious. (1) follows from the fact that if a,- is non-decreasing, flow can only occur toward the left in the sequence ai , a2 , as , • • ■ , Gn a and the sum ^ a,- can only increase. Also it is easy to see the suflSciency of 1 the condition, for if 6i , 62 , • • • , bn satisfies (1), (2), and (3) we can get the bi by first bringing Ci up to bi by a flow from the c,- as close as possible to ci (keeping the "entropy" low by a flow between elements of nearly the same value), then bringing 02 up to 62 (if necessary) etc. The details are fairly obvious. Additive number theory, or the problem of decomposing a number into the sum of numbers satisfying certain conditions, (in our case this definition is generalized to "sets of numbers") enters through the following Lemma: Lemma: If ci , 02 , • • • , fln = (2, 4, 8, • • • , 2") then we can decompose the Oi into the sum of two sets Ci = bi + Ci such that 6i , 6a , • • • , 6n = (1, 2, 4, • • • , 2"-0 84 BELL SYSTEM TECHNICAL JOURNAL and ci , C2 , • • • , c„ = (1, 2, 4, ■ • • , 2''-i) We may assume the a, arranged in a non-decreasing sequence, Oi < C2 < di < ■ ■ ■ < Cn- In case ai = 2 the proof is easy. We have 1, 2, 4, • • • , 2-1 B 1, 2, 4, • • • , 2-1 C 2,4, 8, •••,2" A and a flow has occurred in the set 4, 8, 16, • ■ • , 2" to give a2 , az , ■ ■ ■ , an ■ Now any permissible flow in C corresponds to a permissible flow in either A or B since if Cj = aj + bj > Ci — ai -\- bi then either a, > ai or bj > bi Thus at each flow in the sum we can make a corresponding flow in one or the other of the summands to keep the addition true. Now suppose Qi > 2. Since the a, are non-decreasing (« - 1) 02 < (2"+i - 2) - ai < 2«+i - 2 - 3 Hence a,-\< i -^ - 1 < 2"-^ n — 1 the last inequality being obvious for n > 5 and readily verified for n < 5. This shows that (oi — 1) and (c2 — 1) lie between some powers of two in the set 1,2,4, ••• ,2"-i Suppose 2-^-1 < (ci - 1) < 2« 2P-1 < (02 - 1) < G" q< P<{n- !)• Allow a flow between 2' and 2«-i until one of them reaches (ai — 1), the other (say) R\ similarly for (02 — 1) the other reaching S. As the start toward our decomposition, then, we have the sets (after interchanges) L {a, - 1) 1 1 02-1 2, 4 • • • 2""^ R 2'^^ • • • 2""^ 2" 2"^^ • ■ • 2"~^ 2, 4 • • • 2'"" 2""^ 2' ■■■ 2''~^^S 2"^^ ■ ■ ■ 2""^ fli 4, 8 ■ • • 2'"' • • • 2"^^ • • • 2" L SYNTFIESIS OF SWITCHING CIRCUITS 85 We must now adjust the values to the right of L — L to the values 03 , 04 , • • • , o„ . Let us denote the sequence 4, 8, • • • , 2«-S (2-^' + R), 3-2«, 3- 2"+', • • • (2" + S), 2"+\ • ■ • , 2" by n\ , jji-i , ■ • • , yin-2 ■ Now since all the rows in the above addition are non-decreasing to the right oi L — L, and no I's appear, we will have proved the lemma if we can show that S Mi < S '^. i = 1, 2, • • ■ , (n - 2) 1 = 1 i=3 since we have shown this to be a sufficient condition that 03 , 04 , • • • , On = (mi , Mn , • • • , Mn-2) and the decomposition proof we used for the first part will work. For i< q— 2, i.e., before the term (23-i -f R) E M. - 4(2'' - 1) and smce Hence i+3 3 q< P i i+3 m "i < XI o» i < q — 2 3 Next, for {q - \) < i < {p - 3), i.e., before the term {2^ + S) Z M. = -^(2'"' - \) + R + 3-2^(2' "^^ - 1) < 3-2'^' - 4 < 3-2*^' smce also again i? < 2« i+3 Z ai > 12"-' 3 SO that in this interval we also have the desired inequality. Finally for the last interval. 86 BELL SYSTEM TECHNICAL JOURNAL i 2_] Ml = 2' — C] — 02 ^ 2' — fli — fl2 — 2 and since <+3 3 1 22 ^t = 2_/ <^i — Ol — «2 > 2'"^^ — 01 — 02 ai , 02 , • • • , On = (2, 4, 8, • • • , 2") This proves the lemma . 5. The Disjunctive Tree It is now easy to prove the following: Theorem 8: A disjunctive tree of n bays can be constructed with any dis- tribution ai , 02 , • • • , On = (1, 2, 4, • • • , 2"-^). We may prove this jby induction. We have seen it to be true for n = 2, 3, 4. Assuming it for n, it must be true for » + 1 since the Lemma shows that any fli , ^2 , • • • , fln = (2, 4, 8, • • • , 2") can be decomposed into a sum which, by assumption, can be realized for the two branches of the tree. It is clear that among the possible distributions (1, 2, 4, • ■ • , 2"-0 for the tree, an "almost uniform" one can be found for all the variables but one. That is, we can distribute the load on (n — 1) of them uniformly except at worst for one element. We get, in fact, for w = 1 1 n= 2 1, 2 n = 3 1, 3, 3 n = 4 1, 4, 5, 5, n = 5 1, 7, 7, 8, 8, n = 6 1, 12, 12, 12, 13, 13 n = 7 1, 21, 21, 21, 21, 21, 21 etc. as nearly uniform distributions. 6. Other Distribution Problems Now let us consider the problem of load distribution in series-parallel circuits. We shall prove the following: syMnE:iis or s\\ itci/i.xg cimclits 87 Theorem 9: Any fwiclion f{Xi , X2 , • ■ ■ , Xn) may be realized with a series- parallel circuit with the following distribution: (1,2,4, ••■ ,2-2), 2-2 ill terms of transfer elements. This we prove by induction. It is true for ;/ = 3, since any function of three variables can be realized as follows: f{X, Y, Z) = [X + /i (F, Z)][X' + /2 (F, Z)] and/i(F, Z) and/2(F, Z) can each be realized with one transfer on V and one on Z. Thus /(A', I', Z) can be reahzed with the distribution 1, 2, 2. Now assuming the theorem true for {n — 1) we have /(Xi ,X2,--- ,Xn) = [Xn + /i (Xi , X2 , • • • , Xn-l)] [X„ -{- f2{Xi , X2 , • ' - , Xn-l)] and 2, 4, 8, • • • , 2-3 2, 4, 8, • • • , 2-3 4, 8, 16, • • • , 2-2 A simple appHcation of the Lemma thus gives the desired result, Many distributions beside those given by Theorem 9 are possible but no simple criterion has yet been found for describing them. We cannot say any distribution (1, 2, 4, 8, • • • , 2-2, 2-2) (at least from our analysis) since for example 3, 6, 6, 7 = (2, 4, 8, 8) cannot be decomposed into two sets ai , 02 , 03 , 04 = (1, 2, 4, 4) and bi,b2,b,,b,= (1,2,4,4) It appears, however, that the almost uniform case is admissible. As a final example in load distribution we will consider the case of a net- work in which a number of trees in the same variables are to be realized. A large number of such cases will be found later. The following is fairly obvious from what we have already proved. 88 BELL SYSTEM TECHNICAL JOURNAL Theorem 10: It is possible to construct m dijferent trees in the same n variables with the folknving distribution: ai , a2 , ■ ■ ■ , an = (ni, 2m, 4m, ■■ ■ , 2''~'^m) It is interesting to note that under these conditions the bothersome 1 disap- pears for m > 1. We can equalize the load on all n of the variables, not just n — 1 of them, to within, at worst, one transfer element. 7. The Function n{n) We are now in a position to study the behavior of the function m('0- This will be done in conjunction with a treatment of the load distributions possible for the general function of n variables. We have already shown that any function of three variables can be realized with the distribution 1,1,2 in terms of transfer elements, and, consequently ix{i) < 4. Any function of four variables can be realized with the distribution 1, 1, (2, 4) Hence m(4) < 6. For five variables we can get the distribution 1, 1, (2, 4, 8) or alternatively 1, 5, 5, (2, 4) so that /i(5) < 10. With six variables we can get 1, 5, 5, (2, 4, 8) and m(6) < 10 for seven, 1, 5, 5, (2, 4, 8, 16) and m(7) < 16 etc. Also, since we can distribute uniformly on all the variables in a tree except one, it is possible to give a theorem analogous to Theorem 7 for the function m('0- Theorem 11: For all n m(") < ^^ n+3 For almost all n 2' m(") < - SYNTHESIS or SWITCHING CIRCUITS 89 I'or an inlinilc number of nt , fi(n) < (1 + e) V n n+l 2~ The proof is direct and will be omitted. PART III: SPECIAL FUNCTIONS 8. Functional Relations We have seen that almost all functions require the order of 2n+i elements per relay for their realization. Yet a little experience with the circuits encountered in practice shows that this figure is much too large. In a sender, for example, where many functions are realized, some of them involving a large number of variables, the relays carry an average of perhaps 7 or 8 contacts. In fact, almost all relays encountered in practice have less than 20 elements. What is the reason for this paradox? The answer, of course, is that the functions encountered in practice are far from being a random selection. Again we have an analogue with transcendental numbers ^although almost all numbers are transcendental, the chance of first en- countering a transcendental number on opening a mathematics book at random is certainly much less than 1. The functions actually encountered are simpler than the general run of Boolean functions for at least two major reasons: (1) A circuit designer has considerable freedom in the choice of functions to be realized in a given design problem, and can often choose fairly simple ones. For example, in designing translation circuits for telephone work it is common to use additive codes and also codes in which the same number of relays are operated for each possible digit. The fundamental logical simplic- ity of these codes reflects in a simplicity of the circuits necessary to handle them. (2) Most of the things required of relay circuits are of a logically simple nature. The most important aspect of this simplicity is that most circuits can be broken down into a large number of small circuits. In place of realizing a function of a large number of variables, we realize many functions, each of a small number of variables, and then perhaps some function of these functions. To get an idea of the effectiveness of this consider the following example: Suppose we are to realize a function f{Xi , X2 , • ■ ■ , Xzn) 90 BELL SYSTEM TECHNICAL JOURNAL of 2n variables. The best limit we can put on the total number of elements necessary is about . However, if we know that / is a function of two In functions /i and/2 , each involving only n of the variables, i.e. if / = g{h,h) f\ = f\ {Xl , X2 , • • • , Xn) fi = j2\Xn-^\ , -X^n+2 , ' " ' , -^2n) then we can realize / with about 4 • — n elements, a much lower order of infinity than — — . If g is one of the simpler functions of two variables ; for example if g(/i , /2) = /i + /2 , or in any case at the cost of two additional relays, we can do still better and realize /with about 2 elements. In general, the more we can decompose a synthesis n problem into a combination of simple problems, the simpler the final circuits. The significant point here is that, due to the fact that / satisfies a certain functional relation /=g(A,/2), we can find a simple circuit for it compared to the average function of the same number of variables. This type of functional relation may be called functional separability. It is often easily detected in the circuit requirements and can always be used to reduce the limits on the number of elements required. We will now show that most functions are not functionally separable. Theorem 12: The fraction of all functions of n variables that can he written in the form f = g(h(X, ■ ■ • X.), X.+ 1 ,-■■ ,Xn) where 1 < s < n — 1 approaches zero as n approaches 00 . We can select the 5 variables to appear in // in ( j ways; the function h then has 2^' possibilities and g has 2"" ' possibilities, since it has n— s-\- \ arguments. The total number of functionally separable functions is there- fore dominated by SYNTHESIS OF SWITCHING CIRCUITS n-2 91 i;(^)2^'2^'*""' < {n - 3) — 2 2 and the ratio of this to 2" — > 0 as n — > <» . Fig. 20 — Use of separability to reduce number of elements. Fig. 21 — Use of separability of two sets of variables. In case such a functional separabiUty occurs, the general design method described above can be used to advantage in many cases. This is typified by the circuit of Fig. 20. If the separability is more extensive, e.g. / = gOhiXx ■ ■ ■ Xs), h2(Xs+y ■ ■ ■ X,), Z.+i , • • • , X„) the circuit of Fig. 21 can be used, using for '7/2" either hi or ht , whichever requires the least number of elements for realization together with its negative. We will now consider a second type of functional relation which often occurs in practice and aids in economical realization. This type of relation may be called group invariance and a special case of it, functions symmetric '>2 BELL :iYSTEM TECUNICAL JOURNAL in all variables, has been considered in (6). A function /(Xi , ••• , Xn) will be said to be symmetric in .Vi , .Y2 if it satisfies the relation f{\\ ,X,,---, A'„) = f{X, . A'l , • ■ • , Xn). It is symmetric in A'l and A''2 if it satisfies the equation fix, ,X,,---, Xn) = f{X', ,X[,X,,--- ,Xn) These also are special cases of the type of functional relationships we will consider. Let us denote by Xoo ••■0^1 the operation of leaving the variables in a function as they are, Xioo • • • o the operation of negating the first variable (i.e. the one occupy- ing the first position), A^io • • • o that of negating the second variable, Xuo ■ ■ • o that of negating the first two, etc. So that NioifiX, V, Z) = f(X'YZ') etc. The symbols Ni form an abelian group, with the important property that each element is its own inverse; NiNi = / The product of two elements may be easily found — if A^- Nj = Nk , k is the number found by adding i and j as though they were numbers in the base two but wilhotd carrying. Note that there are 2" elements to this "negating" group. Now let ■5'i,2,3,...,w = I = the operation of leaving the variables of a function in the same order S2,i,i....n = be that of interchanging the first two variables 'S'3,2,i,4,....w = that of inverting the order of the first three, etc. Thus SnoJiX, Y, Z) = fiZ, X, Y) SsufiZ, A, ]') = Sl,f(X, Y, Z) = f{Y, Z, X) etc. The Si also form a group, the famous "substitution" or "symmetric" group. It is of order n !. It does not, however, have the simple properties of the negating group — it is not abelean (w > 2) nor does it have the self inverse property.* The negating group is not cyclic if n > 2, the symmetric group is not if n > 3. The outer product of these two groups forms a group G whose general element is of the form A\5> and since i may assume 2" values andj, n I values, the order of G is 2"«1 It is easily seen that SjNi = NkSj, where k may be obtained by per- * This is redundant; the self inverse property impUes commutativity for if A'A' = / thenXF = (XF)-' = F-^X"' = YX. SyNTIIKSLS O/' SWITCHING CIKCmi'S 93 forming on i, considered as an ordered sequence of zero's and one's, the permutation Sj . Thus liy tliis rule an\' product such as N' i Sj N k N i Sm ^^ n S p can be reduced to the form A^•iVy ••• N„SpS, ■■■ Sr and this can then be reduced to the standard form A^iSj . A function/ will be said to have a non-trivial group invariance if there are elements XiSj of G other than / such that identically in all variables N,S, f = f. It is evident that the set of all such elements, NiSj , for a given function, forms a subgroup Gi of G, since the product of two such elements is an ele- ment, the inverse of such an element is an element, and all functions are invariant under /. A group operator leaving a function / invariant implies certain equalities among the terms appearing in the expanded form of /. To show this, consider a fixed I^^iSj , which changes in some way the variables (say) Xi , X2 , • • • , Xr . Let the function f{Xi , • • • , X„) be expanded about Xi , ■ ■ • , Xr : / = [Xi + X2 + • • • + A% + /i(Z,+i , • • • , X„)] [X[ + X2+ ■■■ + Xr + MXr+l ,■•■ , Xn)] [X[ + X: + • • • + Xl + MiXr+l ,■■• , Xn)] If/ satisfies XiSjf — f we will show that there are at least j2'' equalities between the functions fi,fo, ■ • • , f-ir. Thus the number of functions satisfying this relation is since each independent /, can be any of just 2~ functions, and there are at most f 2'' independent ones. Suppose A^^*; changes Xi , X2 , • • ' , Xr A into Xai , Xaj , • • • , Xar B where the *'s may be either primes or non primes, but no Xa, = A', . Give 94 BELL SYSTEM TECHNICAL JOURNAL Xi the value 0. This fixes some element in B namely, Xai where ai = 1. There are two cases: (1) If this element is the first term, ai = 1, then we have 0X2,--- ,X, 1 Xa^ , ' • ■ , Xa^ Letting X2 , • • • , Xr range through their 2^~ possible sets of values gives 2"^"^ equalities between different functions of the set fi since these are really f{X\ , X2 , • • • , Xr , Xr+1 , • ■ ■ , Xn) with Xi , X2 , • • • , Xr fixed at a definite set of values. (2) If the element in question is another term, say Xaj , we then give X2 in line A the opposite value, X2 = {X^^ = {X2 )'. Now proceeding as before with the remaining r — 2 variables we establish 2"^^ equalities between the fi . Now there are not more* than 2"w! relations NiSjf = f of the group invariant type that a function could satisfy, so that the number of functions satisfying any non-trivial relation < 2"w!2*'". Since 2"w! 2^^72^" -^0 sisn-^ 00 we have: Theorem 13: Almost all functions have no non-trivial group invariance. It appears from Theorems 12 and 13 and from other results that almost all functions are of an extremely chaotic nature, exhibiting no symmetries or functional relations of any kind. This result might be anticipated from the fact that such relations generally lead to a considerable reduction in the number of elements required, and we have seen that ahnost all functions are fairly high in "complexity". If we are synthesizing a function by the disjunctive tree method and the function has a group invariance involving the variables -^1 , X2 , • • • , Xr at least T ^ of the terminals in the corresponding tree can be connected to * Ourfactorisreally less than this because, first, we must exclude iV, 5, = /; and second, except for self inverse elements, one relation of this type implies others, viz. the powers {NiSM = f. SYNTHESIS OF SWirCIIING CIRCUITS 95 other ones, since at least this many equahties exist between the functions to be joined to these terminals. This will, in general, produce a considerable reduction in the contact requirements on the remaining variables. Also an economy can usually be achieved in the M network. In order to apply this INDEPENDENT OF X.Y Fig. 22 — Networks for group invariance in two variables. Fig. 23 — Networks for group invariance in three variables. method of design, however, it is essential that we have a method of deter- mining which, if any, of the iV, Sj leave a function unchanged. The following theorem, although not all that might be hoped for, shows that we don't need to evaluate NiSJior all NiSj but only the N if and Sjf. Theorem 14: A necessary and sufficient condition that NiSjf = f is that N' symmetric functions. The corresponding entry Z'Y'X in the group table refers us to circuit 9 of Fig. 23. The asterisk shows that the circuit may be used directly; if there is no asterisk an interchange of variables is required. We expand f about X, Y, Z and only two different functions will appear in the factors. These two functions are realized with two trees extending from the terminals of the network 9. Any such function/ can be realized with (using just one variable in the A^ network) 9 + 2(2"-' - 2) -t- 2 + 7 elements, svxTfiEsis or swirciiixc. cikci its 97 a much hotter limit than the corresponding 2"-' + 18 for the general function. Table I CiRori' Invarianck Involving Two Variables (Superscripts Refer to Fig. 22) -V„„ {x y) (v xy* Nox {x y'T'* (y x')^* Xio (x'y)^ iy'x)^* Xn {x'y'r (y'x'y Table II Group Invariance Involving Three Variables (Superscripts Refer to Fig. 23) .S"l-23 SU2 S,u .V231 .S-3:'. Sin A'ooo xvz XZY 1 YXZ •* I'ZX 2* ZA'F2* ZYX- X.„i XI'Z'3* XZY' '* FA'Z' ' FZA" » ZXY' ^ ZYX' * Xou XY'Z^ XZ'Y'* ]A"Z * FZ'A' ^ ZX'Y^ ZY'X - X„n A'F'Z'5 XZ'Y'- YX'Z' 8 YZ'X' 2 ZX'Y'2 ZY'X' 8 Xu. X'VZ^ X'ZY-* 1"A'Z ' F'ZA' 9 Z'XY ^ Z'YX* i^^lOl X'YZ''^ X'ZY' »* Y'XZ' 8* I"ZA" 2 Z'XY'^ Z'YX' 1 A^iio X'Y'Z^* X'Z'Y^* ]"A''Z 1 Y'Z'X 2 Z'X'Y^ Z'Y'X^* xViii X'Y'Z'^ X'Z'Y'- Y'X'Z' - I-'Z'A" 9 Z'A"F'9 Z'Y'X'-' 9. Partially Symmetric Functions We will say that a function is "partially symmetric" or "symmetric in a certain set of variables" if these variables may be interchanged at will without altering the function. Thus ATzir + (AT' + A'Dir + irz' is symmetric in A' and Y. Partial symmetry is evidently a special case of the general group invariance we have been considering. It is known that any function symmetric in all variables can be realized with not more than n~ elements, where // is the number of variables.'' In this section we will improve and generalize this result. Theorem 15: Any function f(Xi , X-2 , ■ ■ ■ , X„, , I'l , I'o , • • • I'„) sy)n- metric in A'l , A'o , • • • , X„, can be written /(A 1 , A^2 , ■ ■ ■ , X,n , I 1 , I 2 , • • • = [5„(Xi,A%, ••■ ,A„,)+/,.(I'i, Y,, {s,{x,,x,, ■■■ ,x„.) +/i(ri, Y,, I'l , I': ■I , , r n) ,r n)]. , Y .)] [S^X, ,X,,--- , A„,) + /„,(I'i , 1'2 , ■ ■ • , Y,,)] (6) .'here A-(Fi , r^ , • • • , F„) = /(o, 0, ••• ,0, 1, 1, ■•• , 1, r, , r.,--- , r„) y^O's (m - k) Vs 98 BELL SYSTEM TECHNICAL JOURNAL and Sk{Xi , Z2 , ■ • ■ , A^„) is the symmetric function of Xy , -Y2 , • • • , X,^ ivith k for its only a-numhcr. This theorem follows from the fact that since / is symmetric in -Yi , A'2 , • • • , A"„, the value of / depends only on the number of JY's that are zero and the values of the I"s. If exactly K of the X's are zero the value of / is therefore /a- , but the right-hand side of (6) reduces to Jk in this case, since then Sj{Xi , A^2 , • • • , X„) = l,j^K, and Sk = 0. The expansion (6) is of a form suitable for our design method. We can realize the disjunctive functions Sk{Xi , X2 , • • • , A^,,) with the symmetric function lattice and continue with the general tree network as in Fig. 24, one tree from each level of the symmetric function network. Stopping the trees at F«_i , it is clear that the entire network is disjunctive and a second application of Theorem 1 allows us to complete the function/ with two ele- ments from Yn ■ Thus we have Theorem 16. A ny function of m -\- n variables symmetric in m of them can be realized ivith not more than the smaller of im + 1)(X(//) + m) or {m + 1)(2" + m - 2) + 2 elements. In particidar a function of n variables symmetric in n — 2 or more of them can be realized with not more than n- - n-\- 2 elements. If the function is symmetric in Xi , X2 , • • • , X„, , and also in Fi , F2 , • • • , Yr , and not in Zi , Z2 , • • • , Z„ it may be realized by the same method, using symmetric function networks in place of trees for the F variables. It should be expanded first about the A''s (assuming m < r) then about the F's and finally the Z's. The Z part will be a set of (w + l)(r + 1) trees. References 1. G. Birklioff and S. MacLane, "A Surve}- of Modern Algebra," Macmillan, 1941. 2. L. Couturat, "The Algebra of Logic," Open Court, 1914. 3. J. H. Woodger, "The Axiomatic Method in Biolog}'," Cambridge, 1937. 4. W. S. McCulloch and VV. Pitts, "A Logical Calculus of the Ideas Immanent in Nervous Activity," BuU. Matti. Bioptiysics, V. 5, p. 115, 1943. 5. E. C. Berkelev, "Boolean Algebra and Applications to Insurance," Record {American Institute of Actuaries), V. 26, p. 373, 1947. 6. C. E. Shannon, "A Symbolic Analj'sis of Relay and Switching Circuits," Trans. A. 1. E. E., V. 57, p. 7"13, 1938. 7. J. Riordan and C. E. Shannon, "The Number of Two-Terminal Series Parallel Net- works," Journal of Matliematics and Pliysics, V. 21, No. 2, p. 83, 1942. 8. A. Nakashima, Various papers in Nippon Electrical Communication Engineering, A^n\, Sept., Nov., Dec, 1938. 9. H. Piesch, Papers in Archiv. from Etcctroleclinik XXXIII, p. 692 and p. 733, 1939. 10. G. A. Montgomerie, "Sketch for an Algebra of Relav and Contactor Circuits," Jour. /. ^/£. £., V. 95, Part III, No. 36, July 1948, p. 303. 11. G. P(Mva, "Sur Les Types des Propositions Composees," Journal of Svmbolic Logic, V. 5," No. 3, p. 98, 1940. A Method of Measuring Phase at Microwave Frequencies By SLOAN D. ROBERTSON A method of measuring microwave phase differences is described in which it is unnecessary to compensate for amphtude inequalities between the signals whose phases are being compared. The apjmratus descril^ed is also suited for the measurement of the magnitude of a transfer impedance as well as the phase. WITH the increasing interest in wide-band amplifiers and circuits for microwave communication systems the measurement of the transfer phases of such components has become a necessary procedure. A commonly used technique for measuring phase at microwave frequencies is to sample the signal at the input and output of the device to be measured and to obtain a null balance between the two signals by varying the phase of one signal by a known amount. If the two samples are not of nearly equal amplitudes, it is necessary to attenuate the larger one with an attenuator of known phase shift. The latter operation presents difficulties. A method of phase measurement has been developed which overcomes these difficulties by permitting measurements to be made with samples of unequal amplitudes. The method uses the homodyne detection principle and operates in the following manner: The output energy of a signal oscil- lator is divided into two portions. One portion is applied to a balanced modulator where it is modulated by an audio-frequency signal. The sup- pressed-carrier, double-sideband signal from the modulator is applied to the device to be measured. As before, means are available for sampling the signal at both the input and output of the device. The other portion of the oscillator power is fed through a calibrated phase shifter and is applied to a crystal detector in the manner of a local oscillator in a double-detection receiver. The signal samples are then alternately applied to the crystal detector where they are demodulated by the action of the homodyne carrier. In each case the phase shifter is adjusted so that the audio signal is a mini- mum in the detector output. This occurs when the phase of the homodyne carrier is in quadrature with the signal sidebands. The difference in [)hase between the two adjustments of the phase shifter is equal to the phase dif- ference between the two samples. Figure 1 shows the apparatus used for measuring phase in this manner. Radio frequency power from a suitable oscillator is applied to the ll-plane branch of an hybrid junction' where it divides and emerges in equal portions 1 W. A. Tvrrell, "Hybrid Circuits for Microwaves," Proc. I. R. E., Vol. 35, Xo. 11. pp. 1294-1306; Noveml)er'l947. 99 100 BELL SVSTlUr TECHNICAL JOURNAL from the two lateral branches. The })ortion applied to the calibrated variable phase shifter at the top of the hgure becomes the homodyne carrier. The remaining portion is applied to a balanced crystal modulator- through a second variable phase shifter which need not be calibrated. The latter was introduced in order that the phase of any modulated power reflected due to an imperfect balance in the modulator could be shifted so that it would be in quadrature with the homodyne carrier and would, therefore, not produce an audible signal in the detector. The portion of the power which enters the modulator is modulated by a signal derived from an audio-frequency oscillator. The suppressed-carrier, double-sideband signal which leaves the modulator is applied, after a certain amount of attenuation, to the input of the device to be measured. Probes are provided at the input and output of the latter for sampling the signal. Provision is made for connecting either probe to a crystal detector of the type used for detecting an amplitude-modulated signal. The homodyne carrier emerging from the calibrated phase shifter is attenuated to a level of about one milliwatt and is applied to the crystal detector. The output of the detector is connected to an audio-frequency amplifier terminated by a pair of headphones or an output meter. An attenuator may be placed between the amplifier and the detector as an aid in measuring the magnitude of a transfer impedance. The procedure for adjusting the apparatus and measuring phase is as follows: With both sampling probes disconnected from the detector the variable phase shifter between the oscillator and modulator is adjusted until the output of the detector is zero. This balances out the effect of any signal reflected by the modulator. The input probe is then connected to the detector and the calibrated phase shifter is adjusted until the signal disap- pears in the audio output. When this occurs the homodyne carrier is in quadrature with the signal sidebands, and the resultant signal applied to the detector is equivalent to a phase-modulated wave having a low modulation index, and consequently is not demodulated by a detector of the type used here. The input probe is then disconnected from the detector and the output probe connected. The phase shifter is again adjusted for a null in the audio output. The difference in phase between the two adjustments of the phase shifter is equal to the phase shift between the input and output of the device. If the probes are not located exactly at the input and output termi- nals of the unknown it may be necessary to make a correction in the meas- - (', F. Edwards, "Micruwave Conveners," Proc. /. A'. E., Vol. 35, Xo. 11, pp. USl- 11<)1; November 1947. MEASURING MICROWAVE FREQUENCIES 101 ured phase by allowing for the known phase shift in the line between the probes and the actual terminals of the unknown. So much for the general method. Certain precautions are necessary in order to avoid errors in measurement. In practice the carrier is not com- pletely suppressed in the output of the balanced modulator. It may be at a CRYSTAL DETECTOR Fig. 1 — Schematic circuit for microwave phase measurement. SUPPRESSED CARRIER Ec UPPER \. . LOWER SIDEBAND ^ \ / SIDEBAND RESULTANT CARRIER A4>__ ____-- - \ REFERENCE CARRIER ALTERNATE CONDITION OF BALANCE Fig. 2— Vector diagram of balanced condition with the resultant carrier in quadrature with the signal sideband. level of the order of 10 to 20 decibels below the sidebands. Since the residual carrier will be added to the homodyne carrier in the detector, and since the null adjustment will be reached when the resultant carrier is in quadrature with the sidebands, it is desirable that the residual carrier be low in level compared with the homodyne carrier. The error in phase A0 introduced by the residual carrier is shown in the vector diagram of Fig. 2. A difference in level of about 40 decibels between the homodyne and residual carriers will give an error of not more than half a degree in phase. The 102 BELL SYSTEM TECHNICAL JOURNAL homodyne method of detection has all the conversion efficiency of the usual double-detection arrangements and, in addition, has the advantage in this particular application of having a very low noise level due to the relatively narrow band required for the audio signals. The 40-decibel level difference mentioned above is accordingly not a serious handicap. Other precautions must be observed. The homodyne carrier can be brought in quadrature with the signal for two different phases 180° apart. This is illustrated in Fig. 2. In many applications, where only the variation Fig. 3 — Variable phase shifter using a polystyrene vane. in phase difference is of importance, this uncertainty of 180° can be ignored. The correct setting of the homodyne carrier phase can, however, be deter- mined very easily. Assume that the input probe is connected to the receiver and that the phase has been adjusted for a balance. Then disconnect the audio frequency drive from one of the crystals in the balanced modulator. The residual carrier will now no longer be suppressed and the error angle A(/> of Fig. 2 will become larger. Whether the homodyne carrier is lagging or leading the signal carrier can be determined by observing whether more or less phase shift, respectively, must be introduced to restore balance. A similar test performed with the output probe will indicate whether or not it is necessary to add 180° to the measured phase difference. If either probe MEASURING MICROWA VE FREQUENCIES 103 test indicates a lead, whereas the other probe indicates a lag, then the addi- tion of 180° is indicated. In microwave circuits it frequently happens that the transfer phase varies quite rapidly with the frequency, particularly if some part of the circuit is at or near resonance. In measuring the phase characteristics of a circuit of this type over a band of frequencies it is necessary, therefore, to take the points of measurement close enough together to avoid phase errors corres- ponding to multiples of 360°. When a balance has been established so that the signal is minimized in the detector output, one may observe the presence of the second harmonic of the audio tone. This harmonic is a distortion term generated in the detector. If it is objectionable, it can be eliminated either by a low-pass filter in the audio output, or by using a balanced detector. In measuring transfer impedances it is desirable to know the ratio of the magnitudes of an output voltage and an input voltage as well as the phase difference. The equipment described here can be used for measuring ampli- tudes by adjusting the phase shifter for a maximum signal in the audio output. Maximum signal levels can then be compared with the aid of an audio-frequency attenuator and output meter connected as shown in Fig. 1 . The apparatus was assembled with standard 4000-megacycle waveguide components. A satisfactory phase shifter was made of an ordinary vane- type variable attenuator by replacing the resistance strip with a vane of quarter-inch thick polystyrene six inches in length. This phase shifter gave a total shift of about 100°. Constructional details of this phase shifter are shown in Fig. 3. Other phase shifters could have been used with equally satisfactory results. It is desirable, however, that the phase shifter be impedance matched to the line in which it is located in order that reaction back on the oscillator shall be a minimum. In the shifter of Fig. 3 the ends of the polystyrene vane have been tapered two inches at each end to accom- plish this result. The phase shifter can be readily calibrated by using a standing wave detector fitted with a sliding probe as a standard of phase. The standing wave detector is terminated on one end and connected to the modulated signal source on the other. The signal picked up by the sliding probe is applied to the crystal detector. Knowing the guide wavelength in the standing wave detector, known phase shifts can be introduced by sliding the probe along the guide. By adjusting the phase shifter in the homodyne carrier path for balance, calibration points can be established. The measuring procedure described above has been tested experimentally at 4000 megacycles with very satisfactory results. With ordinary care it was possible to measure phase differences with an accuracy of better than half a degree. Reflection from Corners in Rectangular Wave Guides — Conformal Transformation* By S. O. RICE A conformal transformation method is used to obtain approximate expressions for the reflection coefficients of sharp corners in rectangular wave guides. The transformation carries the bent guide over into a straight guide filled with a non- uniform medium. The reflection coefficient of the transformed system can be expressed in terms of the solution of an integral equation which may be solved approximately by successive substitutions. When the corner angle is small and the corner is "not truncated the required integrations may be performed and an exphcit expression obtained for the reflection coefficient. Although appUed here only to corners, the method has an additional interest in that it is applicable to other types of irregularities in rectangular wave guides. Introduction THE propagation of electromagnetic waves around a rectangular corner has been studied in two recent papers, one by Poritsky and Blewett^ and the other by Miles-. Poritsky and Blewett make use of Schwarz' "alternating procedure" in which a sequence of approximations is obtained by going back and forth between two overlapping regions. Miles derives an equivalent circuit by using solutions of the wave equation in rectangular coordinates. Several papers giving experimental results have been pub- lished. Of these, we mention one due to Elson^ who gives values of reflection coeflScients for various types of corners. Here we shall deal with the more general type of corner shown in Fig. 1 by transforming, conformally, the bent guide (in which the propagation "constant" of the dielectric is constant) into a straight guide in which the propagation "constant" is a function of position— its greatest deviation from the original value being in the vicinity of points corresponding to the corner. This type of corner has been chosen for our example because it possesses a number of features common to problems which may be treated by the transformation method. The essentials of the procedure used are due to Routh* who studied the vibration of a membrane of irregular shape by transforming it into a rectangle. After the transformation the density (analogous to the propaga- tion constant in the guide) was no longer constant but this disadvantage was more than offset by the simplification in shape. Until this paper was presented at the Symposium I was unaware of any * Presented at the Second Symposium on Applied Mathematics, Cambridge, Mass. , July 29, 1948. 1 See list of references at end of paper. 104 CONFORM A /, 7'A'. 1 XSl-ORMA TION 105 other wave guide work based on conformal transformations (as described above) except that of Krasnooshkin'. At the meeting I learned that the transformation method had also been discovered (but not yet published) by Levine and by Piloty independently of each other. Levine has studied the same corner, see Fig. 1, as is done here. However, his method of approach is quite different in that he obtains expressions for the elements in the equivalent pi network representing the corner, whereas here the reflection coefficient is considered directly. This is discussed in more detail at the beginning of Section 6. Piloty's work is closely related to the material presented in a companion paper and is discussed in its introduction. In this paper the partial differential equation resulting from the trans- fomiation, together with the boundary conditions, is converted into a rather complicated integral equation. Numerical work indicates that satisfactor>' values of the reflection coefficient, in which we are primarily interested, may be obtained by solving this integral equation by the method of succes- sive substitutions. However, the question of convergence is not investigated. Although they are here applied only to corners, the equations of Sections 3, 4 and 5 are quite general. In order to test their generality they were used to check the expression^ for the reflection coefficient of a gentle circular bend in a rectangular wave guide, E being in the plane of the bend. The work has been omitted because of its length. It was found that the essential parts of the transformation may be obtained by regarding the inner and outer walls of the guide system as the two plates of a condenser, solving the corresponding electrostatic problem (using series of the Fourier type), and utilizing the relation between two-dimensional potentials and the theory of conformal mapping. When the angle of the corner is small we may obtain the series (7-5) and (7-11) for the reflection coefficients corresponding to simple (i.e. not truncated) E and H corners, respectively (a corner having the electric intensity E in the plane of the bend will be called an E corner or an electric corner. H corners are defined in a similar manner). When the angle of the general E corner shown in Fig. 1 is small we may use the series (7-18). The series (7-5) and (7-11) giving the reflection from small angle corners are related to the series giving the reflection coefficients for gentle circular bends. In fact, if the radii of curvature of the latter be held constant while the angle of bend is made small, the series for the circular bends reduce to those for the corners. As for the limitations of the method, note first that it can be used onl}- for wave guide systems in which the dimension normal to the plane of transformation is constant throughout. Moreover, the integral equations of the present paper, except for the work of Appendix III, are derived on the assumption that the dimensions of the guide approach constant 106 BELL SYSTEM TECHNICAL JOURNAL values at minus infinity and the same values at plus infinity. WTien this assumption is not met, a conformal transformation may still be used to carry the system into a straight guide. However, there appears to be some doubt as to the best way of dealing with the resulting partial differential equation. One method, discussed in the companion paper , leads to an infinite set of ordinar>' linear differential equations of the second order. Again, possibly the Green's functions appearing in Sections 3 and 5 may be replaced by suitable approximations. /. Representation oj Field for Corner or Bend in Rectangular Guide Quite often waves in rectangular wave guides are classed as "transverse electric" or "transverse magnetic". However, for our purposes it is more convenient to class them as "electrically oriented" or "magnetically oriented" waves.*'' Thus, the electric and magnetic intensities are obtained )y multiplying \ d^A _dB iiae dxSr dy H,= dA 1 d^B dy ioin dxd^ " iwt dydi dx Hy = dA 1 d'B dx i(j)yL dydt; (1-1) 1 a'^ „ - „ , 1 d^B £{. = - ico^xA + — rrr: H^ ^ - iweB + . ' toeaf' 7WMar- by e'"' and taking the real part. Here w, p, and e are the radian frequency, the permeabiUty of the medium filling the guide (m = 1.257 X 10~* henries per meter for air), and the dielectric constant of the same (e = 8.854 X 10""'- farads per meter for air), respectively, .t, y, and <: constitute a right-handed set of rectangular coordinates in which the f axis is normal to the plane of the bend. Equations (1-1) may be verified by substituting them in Maxwell's equations. The potentials A and B satisfy the wave equation d'^A , d-A , d'A 2 , + + TT — (^ A dx^ ^ df ar (1.2) a = tcoViue = i2ir/Xo where Xo is the wave length in free space corresponding to the radian frequency w. WTien the electric vector lies in the plane of the bend, as shown in Fig. 1, and the incident wave contains only the dominant mode we set ^ = 0, B = Qsin (ir^/a) (1-3) CONFORM A L TRA NSFORM A TION 1 07 where a is the wide dimension of the rectangular cross-section, the guide walls normal to the f axis are at f = 0 and f = a, and (J is a function of x and y such that r,o = /27rx;r'(i - Xna~V4)."' The guide walls are assumed to be perfect conductors and hence the tan- gential component of E must vanish at the walls. This requires the normal derivative of Q to vanish at those walls which are perpendicular to the plane of the bend : ^ = 0. (1-5) dn When the magnetic vector lies in the plane of the bend and the incident wave consists of the dominant mode, we set ,4 = P, B = 0 (1-6) where P is a function of .\ and v such that ^ + ^{ - TloP = 0, Too = t2x/Xo (1-7) dx^ oy^ and P = 0 (1-8) at the walls perpendicular to the plane of the bend. In this case the guide walls parallel to the plane of the bend are at f = 0 and ^ = b. 2. Electric Vector in Plane of Bend Figure 1 shows a section of the bend taken parallel to the electric vector. b is the narrow dimension of the guide. Let the frequency and the wide dimension a of the guide (measured normal to the plane of Fig. 1) be such that only the dominant mode is freely propagated. The position of any point in this section is specified by the complex number 2 = .v + iy where the origin and the orientation of the axes have been chosen somewhat arbitrarily. The constant k and related propagation constants which appear in the formulas dealing with Q and electric bends are given by k = (26/Ao) [1 - (Xo/2a)'f = -il\ob/Tr yl - m' - k^; m = 0, 1, 2, • • • ; To = ik (2-1) Xo = free space wavelength lOH BELL SYSTFAf TECHNICAL JOURNAL Since, by assumption, only the dominant mode is freely propagated, k and 7m for w > 0 are real and positive. We imagine an incident wave of unit amplitude coming down from Z5 in the upper left portion of Fig. 1. WTiat arc the amplitudes of the reflected wave traveling back toward Zs and the transmitted wave traveling outward to the- right towards Z3? Our task is to find a Q{x, y), satisfying the wave equation (1-4) and the boundary condition (1-5), which represents a disturbance of the assumed type. Z5 = oo e L{7r-2a) Zn = 00 v = -t 1Z4 v = t Fig. 2 "T" 77- ■e-TT V *■ CD 61 = 0 The first step is to find the conformal transformation z = x+ iy = f{v + id) = f{w) (2-2) which carries the bent guide (shown in Fig. 1) in the (;v, y) plane over into the straight guide (shown in Fig. 2) in the (v, 6) plane. This may be done by the Schwarz-Christoffel method discussed in Appendix I. This trans- formation carries the wave equation (1-4) and the boundary condition (1-5) into 9r dd- ^ = 0 at ^ = 0 and 0 = tt (2-3) (2-4) CO.\FOR.\rA L TR. I XSFOR.UA TIO.Y 109 where the upper and lower guide walls are carried into ^ = 0 and d = ir, respectively, and g(v, 6) is given by 1 + g(^ e) = \J'{v + id) |- iT/b' (2-5) „/., n) ^ [7^ V + cos or _ ^^'' ^ [ch{v - t) - cos 0Hc^(t' + /) - cos 8]- ^ ^ ^ Here ch denotes the hN'perbolic cosine, /'(r + id) denotes the first deriva- tive of /(k')) and from Appendix I, lira is the total angle of the bend. / is a parameter which depends upon a and the ratio d/do where d = \ Zi — zq \ and do = | Z4 — ze ] in Fig. 1. A table giving values of / for a 90° bend (a = 1/4) appears in Appendix I. That the propagation constant is no longer uniform in the transformed guide shows up through the fact that the coefficient of k'-Q in (2-3) is now a function of the coordinates (i', 9). g(v, 9) measures the deviation of the propagation constant from its value ?Lt v — — x . For example, if we consider a wave front coming down from z-, we expect it to get past S4 before it reaches Zo . In Fig. 2 the same wave front is tilted forward corresponding to a high phase-velocity (or small propagation constant) at Zi where v = 0 and 9 = TT. This is in line with the fact that the coefficient of k-Q in (2-3) vanishes at Z4 by virtue of (2-6). Similar considerations hold at Zi and zo • What is our reflection problem in terms of the transformed guide? In addition to satisfying the two equations (2-3) and (2-4) Q must behave properly at infinity. For large negative values of v, Q must represent an incident wave plus a reflected w^ave. The incident wave is of unit amplitude and the reflected wave is of the, as yet, unknown value Re- For large positive values oi v Q must represent an outgoing wave. Thus Q must also satisfy the two equations Q ^ g-ikv ^ R^e'''\ ^ ^ - 00 (2-7) Q = Tec-''^' , ^ ^ 00 (2-8) where the subscript E appears on the "reflection coefficient" Re and the "transmission coefficient" Te to indicate that here we are dealing with an electric corner. Our problem is now to take the four equations (2-3, 4, 7, 8) and somehow or other obtain the value of Re ■ We are not so much interested in Te because it does not have the practical importance of the reflection coefficient. There are at least two different ways we may proceed from here. One is to transform the differential equation plus the boundary conditions into an integral equation which may be solved approximately by iteration. Another way is to assume () to be a Fourier cosine series in 9 whose co- efficients are functions of v. Substitution of the assumed series in the no BELL SYSTEM TECHNICAL JOURNAL differential equation (2-3) gives rise to a set of ordinary differential equations having v as the independent variable and the coefficients as the dependent variables. The integral equation method is used in this paper. The second method is discussed in the companion paper. 3. Conversion of Differential Equation into an Integral Equation The differential equation (2-3) may be converted into an integral equation by using the appropriate Green's function in the conventional manner. The only modifications necessary are essentially those given by Poritsky and Blewett^ in a similar procedure. The conversion is based upon Green's theorem in the form where the integration on the right extends over the rectangular region Vi• — oo . Equation (3-5) is a general equation which may be applied to a number of wave guide problems by choosing a suitable function g{v, 6). For the corner of Fig. 1 g{v, 6) is given by (2-6). If g{v, 6) approaches zero when | v \ becomes large, as it does for the corner, expressions for the reflection coefficient Rg and the amplitude Te of the transmitted wave may be obtained by letting I'o -^ ±20 in (3-5). For ver>' large values of | r© | the contributions of all the terms in the summa- tion e.xcept the first (w = 0) vanish. Comparison of the resulting expression for Q(vo , do) with the limiting forms (2-7) and (2-8) defining Re and Tb gives Re= -^ r dv f dd g(v, e) Q(i, d)e-''' (3-6) Zir J-x, Jo T,= \-^ r dvf dd g(v, d) Q(i, 0)6'" (3-7) Zir J-to Jo Since the integrands involve the as yet unknown Q(v, 6) these expressions are not immediately applicable. In fact, if we knew Q{v, 6) it would not be necessary to use these integrals for Re and Te — we could simply let i; — > ± oc and use (2-7) and (2-8). Nevertheless, (3-6) and (3-7) are useful in obtain- ing approximations to Rb and Te when approximations to Q are known. In Appendix IV it is shown that Rb is the stationary' value, with respect to variations of the function Q, of an expression made up of integrals containing Q in their integrands. From the integral equation it follows that when k-^Q, i.e., when the frequency decreases toward the cut-off frequency of the dominant mode, Q becomes approximately exp {—ikv). Furthermore, Rg approaches zero. This is in contrast to the apparent behavior of Rh which, according to the discussion given in Section 5, may possibly approach — 1 under the same circumstances. Thus reflections from the two types of corners, or more generally, irregularities in the E plane and in the H plane, appear to behave quite differently as the cut-off frequency is approached. Rb and Tb are not independent. Since the energy in the incident wave is equal to the sum of the energies in the reflected and transmitted waves we expect ReR*e + TeT*e - 1, (3-8) where the asterisk denotes the conjugate complex quantity. In addition, there is a relation between Rb and Tb which for a symmetrical irregularity, i.e. for g{v, 6) an even function of v, states that the phase of Re differs from that Te by ±7r/2. In this special case Tb is determined to within a plus or 112 BELL SYSTEM TECHNICAL JOURNAL minus sign when Re is given. These relations may be proved by substituting various solutions of equation (2-3) for Q and Q in the equation ^ dv ^ dv Qf-Qf] (3-9) where I'l and V2 are large enough (t'l negative and v^ positive) to ensure that Q and Q have reduced to exponential functions of v. Equation (3-9) follows from Green's theorem. When Q is taken to be the solution for which (2-7) and (2-8) holds and Q its conjugate complex Q*, equation (3-8) is obtained. Keeping the same solution for Q but now letting Q denote the solution corresponding to an incident wave of unit amplitude coming in from the right : Qi = Tie'''" , v-^ —CO gives T = Ti where we have dropped the subscript E and have assumed that g{v, 6) may be unsymmetrical. Taking Q to be Qi gives RTt + RtT = 0 which is the relation sought. In the symmetrical case R — Ri, R/T + R*/T* is zero and hence R/T is purely imaginary as was mentioned above. The same relations hold for R„ and Th ■ These results are special cases of a more general result which states that the "scattering matrix" is symmetrical and unitary for a lossless junction.^" 4. Approximate Solution of Integral Equation A first approximation to the solution of the integral equation (3-5) is obtained when we assume that the non-uniformity of the propagation constant has no efifect on Q. Thus we put Q^'\v, 9) = e-"-" (4-1) in the integral on the right and obtain an expression for the second approxi- mation Q'-'^v, d), and so on. Here we shall not go beyond Q^-^(v, 6). It is convenient to expand g(v, 6) in a Fourier cosine series 00 g(^', ^) = S «n(t') cos nd (4-2) an(v) = -" / g('', 0) cos nd dd, eo = 1; e„ = 2, n > 0. TT Jo CONFORM A L TRANSFORMATION 113 The second approximation, obtained by substituting (4-1) in (3-5), may then be written as 00 Q^-\v,, do) = e^''"" + k'2-' X y^' cos mdo (4-3) L The ;/th approximation i?i"^ to the reflection coefficient (when the electric vector lies in the plane of the bend) is defined in terms of Q^"^ by Limit Q'^'iv, d) = e-'" + R'.-'e"" (4-4) V— » — 00 Re"^ is also equal to the integral obtained by replacing Q in (3-6) by (2^"~". We have 7?y> = 0, Rf = -ik2^' r ao(T)e-''''' dv, J— cc rT =R';-' -ik'j:(^yme^r' (4-5) • / dvoam(io) I dvam(v)( J— 00 *J — oo -ik(v+VQ)—lv—vo\ym where jm is given by (2-1). The results of this section have the same generality as the integral equation (3-5) in that they are not restricted to corners. 5. Truncated Corner — Magnetic Vector in Plane of Bend When the magnetic vector lies in the plane of the bend the reflection may be calculated by a similar procedure. The wide dmaension a of the wave guide now replaces the narrow dimension h in Fig. 1. We shall call the result of making this change the "modified Fig. 1". We again assume the frequency to be such that only the dominant mode is propagated without attenuation. In place of equations (1-3, 4, 5) involving Q we have those of (1-6, 7, 8) involving P. The conformal transformation which carries the modified Fig. 1 into Fig. 2 leads to ^ + ^ + [1 + ^0', ^)]'<^P = 0 9r- dd- (:>-l) P = 0 at ^ = 0 and 6 ^ ir 114 BELL SYSTEM TECHNICAL JOURNAL where K = 2a/Xo = -iTooa/ir, c = (k'' - l)"^ = ak/b dl = ni" - I - c"" = m^ - k\ 5i = ic (5-2) Xo = free space wave length, w = 1, 2, 3 • • • and \fL^{v + id) l^xVa^ = 1 + g{v, 6). (5-3) Here /mod(w) pertains to the modified Fig. 1. Since the expression for f'{w) given in Appendix I is proportional to b and since the modified trans- formation contains a in place of b, it follows that g{v, 9) for the magnetic corner is exactly the same function, given by (2-6), as for the electric corner. It is again assumed that the incident wave coming down from the left in the modified Fig. 1 is of unit amplitude and of the dominant mode. At large distances from the corner P = [,-"' -f i?He'"] sin 0, v-^-cc (o-4) P = TbC "" sin d, V ^ -\-x which serve to define the coefiicients of reflection and transmission. The subscript H on the reflection and transmission coefficients indicate that here we are dealing with a magnetic corner. The conversion of the differential equation into the integral equation now employs the Green's function G = 2 Z 5-' sin me, sin w^^-'^-'""-" (5-5) which corresponds to V = 0 at 0 = 0 and 0 = TT The integral equation for P is found to be P(vo , ^o) = e"""" sin 9o + ^ [ dv I dd g(v,e)Piv,e) J^28Z' sinmdosmmde'^" ""'*" 2ir J— 00 •'0 "'=-1 (5-6) where the parameters are given by (5-2). This is a general equation. For the corner of the modified Fig. 1 g{v, 6) is given by (2-6). CON FORMAL TRANSFORMATION 115 Hy letting ro — ^ — °° we ()l)taiii the exact expression R„=-— dv dd f;(v,e)P{v,d)e~'" sine (5-7) TTC J— 00 •'0 When dealing with the electric corner we saw that Re —^ 0 a.s k —^ 0. The presence of c in the denominator of (5-7) suggests the possibility that /?// ^ — 1 as c — ^ 0. For Rh must remain finite and this may perhaps come about through P{v, 6) —^ 0 in the region, say around v = 0, where g(v, 6) is appreciably different from zero. This and the fact that P{v, d) must contain a unit incident wave suggest that for 2; < 0 the dominant portion of P{v, 6) is 2i sin cv which gives Ru = — 1. Incidentally, it is apparent that the approximations for P{v, 6) given below in (5-8) and (5-10) (and therefore also the approximations (5-11) for Rh) fail when c becomes small. The first approximation to the solution of the integral equation (5-6) is P^^\v, 6) = e-^'" sin 0 (5-8) When we introduce the coefficients 2 r bn{i) = - / g(v, e) sin 6 sin nd dd (5-9) sin dg{v, 6) = ^ bn{v) sin nd bi(v) = aoiv) — a2{v)/2, bn(v) = [on-i(u) — an+i{v)]/2, n > 1 we find that the second approximation is P^^\vo , do) = e"'"" sin do + k^2~' Y. C sin mdo (5-10) The successive approximations to the reflection coefficient are R\!^ = 0, R\r = -i f\vbMe~''- 2c J-oc RT = R'k^ - u E (4c5„)~' dvob,,{vo) (5-11) m=l J— 00 J. A /■- \ „—ic(vi-Vn)—\ii — VQ[i„ dvbm{vo)e " '" . ■X f no BELL SYSTEM TECHNICAL JOURNAL 6. Series for 7?'^' When Corner Has No Truncation The integrals which appear in the approximations for the reflection coefficients are difficult to evaluate in general. This section serves to put on record several expressions which have been obtained for R^'-^ when the corner is not truncated. Corresponding evaluations of R'^^' would be welcome since the work of Section 7 for small angle corners indicates that j^s) _ 7^(2) is of the same order as R^-K However, I have been unable to go much beyond the results shown here. As mentioned in the introduction, H. Levine has studied the effect of a corner in a wave guide by representing it as an equivalent pi network having an inductance for the series element and two equal condensers for the shunt elements. Early in 1947 he derived the following expressions (in our notation) for the elements corresponding to a simpe E corner :* Ba/Yo = k {^•) ^r-^n-M-^ Bt,/Yo = {kiv)-' cot (/3x/2) where Fo is the characteristic admittance of the straight guide, iB^ the admittance of one of the two equal shunt condensers, — iBb the admittance of the series inductance, ^(:v) the logarithmic derivative of r(.v + 1), and /37r is the total angle of the simple corner (for no truncation we set /3 = 2a). When the reflection coefficient for the corner is computed from the equivalent network for the case /3 -^ 0 it is found to lie between the approxi- mate value Re^ given by (7-3) and the considerably more accurate value R^E^ given by (7-5). All three approximations are of the form Aff- + 0{^^) where A differs from approximation to approximation but is independent of /3, and 0(/3*) denotes correction terms of order |S^ Since Re^ gives the exact value of .4, it may be regarded as the standard when the three approxima- tions are compared. If this comparison be taken as a guide, it suggests that the rather cumbersome expressions (6-2) and (6-5) for R^e'^ given below are not as accurate as the simpler expressions resulting from Levine's work. Dr. Levine has also obtained corresponding results for the general £-corner of Fig. 1. It is hoped that his work will be published soon. When the corner is not truncated it is convenient, as mentioned above, to replace 2q: by ^ so that fiir is the total angle of the bend. For no trunca- tion / = 0 and (2-6) becomes gii\ 0) chv -+- cos - 1. (6-1) chv — cos 0_ * I am indebted to Dr. Levine for communicating these expressions to me. cox FORM A L TRA NSFORMA TION From (4-2) and (4-5), or from (3-6), Tin - ik)T(n + ik) ^ (^2^)^n. (/3) n- 117 (6-2) = -ik z n'.iii - 1)!2 ro(2«.)"(" - ^0! where we have expanded g(v, 6) as given by (6-1) in powers of cos d/ch V and integrated termwise. The notation is (o;)o = 1, («)„ = a{a + 1) • • • (a + « — 1). For a right angle corner /3 = 1/2, and a more rapidly convergent series may be obtained by subtracting the sum of the series corresponding to yfe = 0, namely log 2 = E (l/2)„ „=i n\2n (6-3) Thus for /3 = 1/2 R'i' = -ik 'log.2-Z^"(l-^4„)], „=i n\ln J Ai = TT^/sinh -wk, An = Ai IT (1 + ^""^ ^). « > 1 I = 1 (6-4) The rate of convergence of the more general series (6-2) may be increased in a somewhat similar way. It is found that R'l' J - 2^{\ - AO -^(2 + /3')(1 -Ad 2^ 135 (23 -f- 20/3' + 2/3') ( 1 - ^3) - J = K -\- L {^)n (6-5) K = Z ^^f" = j^ - ^(1 - 0) - .5772 „=i w!« 1 — /3 ^ = Z (-2/3),„ z (^)n-. ^1 (2m)! „=m (» — M)\n y (1/2 - /3)^ i'i(l/2)„m(w-/8) where .5772 • • • is Euler's constant, '^(.v) is the logarithmic derivative of II(.t) = r(.v + 1), and An is given by (6-4). 118 BELL SYSTEM TECHNICAL JOUkNAL The results corresponding to Rj, are quite similar. WTien the corner is not truncated r,(2) . 2 v^ T(n - ic)T(n + id v^ (- 2/3) 2m (i3),i_m ^w = —''K 2^ , i — T-ST/^ TVTTT 2^ ;f='i in + !)!(« - l)!2c^o (2w)!(« - w) 2c • 2 y - ^'(1 - iO - ^3 (2 + ^')(l - i2) (6-6) - 2^(23 + 20/3^ + 2/3')(l - ig) - i - 1 - ^(1 - /S) - .5772 (1/2 - ^),. - ^(1 - /3) E ^i(l/2)„m(m - ,8)(m - ^+ 1) in which ^„ is obtained by replacing c by ^ in the expression (6-4) for A^ . The evaluation of the integrals for i?^ and R^ for general values of / appears to be difficult although it is possible to obtain approximate expres- sions for the case when t is large. 7. Reflection from Small Angle Corners The expressions for i?^^^ and R^^^ may be evaluated approximately when the angle of the corner is small. It turns out that, for / = 0, they are of the same order of magnitude and both of them must be considered. Moreover i?<"' for n> 3 differs from R^^^ by terms of the same order as those neglected in our approximations so that there is no point in going to the higher values of n. We first obtain the approximation for Re for a corner with no truncation having the total angle t^. Since ,8 is very small (6-1) may be written as g{v, &) - exp \^ 0. \\'e consider only positive values of v since g{v, d) and the a„(t>)'s are even functions of v. Thus (4-5) yields R'^' = -ik2^' Z «"'('^' + kY' + 0(i3') (7-3) n-1,3.5.- • This is an approximation to the exact value given by the double series in (6-2). Comparison of (6-5) and (7-3) when ^ and k approach zero gives, incidentally, ±m-'t(n- l/ir =ltfn-'. m=-l n"»l ^ m—1 From (4-2), (7-1) and the expansions (A2-2) of log (chv ± cos 6) it follows that fl„(iO = 4/3m-ie-"l''l + 0(/32), m = 1, 3, 5, • • • (7-4) aM = 0(/32) , m - 0, 2, 4, 6 • • • Equations (4-5), (A2-4), the relation 7„ = m^ — k-, and (A2-8) give us the answer we seek: ^^3) ^ ^(^2) _ .j^z^^ Yl y-' in-' J{m, m, k, y„ , 0, 0) + 0(^') m=1.3.5.--- = -ik2^ Z 7»'^"' + 0(^') (7-5) m-l.3.5,--- It is not necessary to go to i?^ ' because it differs from Rg^ by only 0(/3^). When // lies in the plane of the bend the reflection from a small angle corner with no truncation may be obtained by much the same procedure. For brevity we shall not write down the order of magnitude of the remainder terms. From (5-9), (A2-1), and (A2-3) b,{v) = ao(r) - a2(r)/2 (7-6) = ^[h - h - (h - h)/2] 120 BELL SYSTEM TECHNICAL JOURNAL where we have written /„ for Imiy, v) and assumed v > 0. Then, using (5-11), n=3,5.... -2 z «o/-ir^(n^ + cV] n=»2.4.-" When we put br.{v) = [an-x{v) - a„+iW]/2, n > 1 (7-8) bn{v) = 2/3[(;j - l)-ig-("-i)|^'l - (n -\- iyh-^-+'^\% n = 2, 4, 6, • • • bn{v) = 0(i32), w = 1, 3, 5, • • • in (5-11) and use the results of Appendix II we obtain R'^' = i?y^ - iA-y E 5;^[0^ - l)~V(;z - 1, w - 1,5„, 0,0) 71=2,4,6- •• + (w + ly'Jin -f 1, w + 1, c, 5„ , 0, 0) . (7-9) -2(n - ly' J{n - 1, w + 1, c, 5„, 0, 0)] The values of the first two J's, obtained by setting m = n ± 1 in (A2-7), may be simplified by using c2 + (n ± 1 4- 5)2 = 2(ii ± l)in + 5) where we have dropped the subcript n from 5„ . In order to eUminate 5 from the denominator we multiply both numerator and denominator by n — 8 and use (n - 8) {8+ In ± 2) = (». ± 1)^ -f c^ - 8{n ± 2) «2 _ 52 = 1 + ^2 = ^2 Setting in the value, given by (A2-9), of the last / and separating the terms (into those which contain the first power of 8 and those which do not) enable us to write the term within the square brackets in (7-9) as \n^ _ 8 J (» - 1) - 1 (m + 1) + 1 K\n^ - 1)2 k2 [(^ _ 1)2{,2 + (w - 1)2} "^ (/Z + 1)2{C2 + (W + 1)M , 2n{2(>z^ + c^) -/cV-l)}] .710) "^ k2(«2 - 1)2(«2 + c2) J It is found that when (7-10) is put in (7-9), the contribution of the first two terms within the square bracket of (7-10) exactly cancels the summation CON FORMAL TRANSFORMATION 121 which is taken over 3, 5, 7 • • ■ in the expression (7-7) for /?// \ Moreover, if we make use of E Mn - 1)"' = 1 n=2.4.6,--- we see that the contribution of the last term within the square brackets of (7-10) cancels the remaining terms in Rli\ Only the contribution of the first term in (7-10) remains and it gives Ri,^> = -U^'c-' E n('^ - ly'C + 0(/3') (7-11) 71=2.4.6.- • ■ The relative simplicity of this result indicates that there may be another method of derivation which avoids the lengthy algebra of our method. Recently approximate expressions for the reflection coefficient of gentle circular bends have been published^. In our present notation these may be written as Rg ~ — ib" pi sm " i L y^ cos ti — e 24 m=i.3.5.-- ir^m'^ym [-uijc 2 n sm M r. '^^ cos u — e n Sir-C- n-2.4.6.--- T*C8„ {ll' — 1)'J where /Stt is the angle of the bend, pi is the radius of curvature of the center line of the guide and u is 2-k times the length of the center line in the bend divided by the wavelength in the guide: u = ^Trkp\/b = ^TTcpi/a The first expression for u is to be used in Re and the second in Rh . If we now let /3 -^ 0, keeping pi fixed, then ii -^ 0. The trigonometric and expo- nential terms may be approximated by the first few terms in their power series expansions, and part of the series which make their appearance may be replaced by their sums given, for example, by equations (4.1-7) and (4. 1-8) of reference^. After some cancellation, the above expression for Re and Rh , which hold for gentle circular bends, reduce to (7-5) and (7-11), respectively, which hold for the sharp corners. In other words, the reflection coefficients for both the sharp and the circular bends approach zero as /? -^ 0, and furthermore their ratio approaches unity. We shall merely outline the derivation of the approximation i?^ for a truncated corner. Instead of (7-1) we have from (2-6), g{v, e) = e.xp M - 1 = a

0, w > 0, a^iy) = 2a{v - t^v) + 2-'a^^h{v, v) + hiv - t, v - l) + Ii(v -\- t,v-\-t) - -ihiv - t,v) - 4/2(7) -}- t,v) + 2 h{v - l,v-^ t)], amiv) = 2am-'[-2(-)"'e-^\''\ + g-^l'^'l + g-^l^+'l] (7-13) where xpiv) = 1 when 0 < v < t and i^(d) = 0 when v > t. Substitution of the values (A2-3) for /i and h gives ao(i') = [2a{v - 0 + 2a2(^ - 0']^(i') (7-14) n=l -f g-2nlt^-(| ^ 2g-''l''-«l-"l"+'l — 4(— )"e-"I'^'l^''] The second approximation to the reflection coefficient is R'i^ = iak-' sh-i'kt - ia^k-n-'{2kt - sin 2kt) - ika'^T. n-'in' + k')'' {2-(-)"2e-»' (7-15) + [1 - 2(-)"e-"' + e-2"']cos 2kt + wyfe-'[e-2»« _ (-)"2e-«']sin 2kt] The typical term in the summation (4-5) ior Rg^ is - .— f ^" ^^0 a.(.o) r ^^' ^^(tO^"''''''^"'^"'''"'"''''" (M6) When w = 0, eo = 1, To = ik, and 00(1*) is 2a(i) - 0 + 0(«") for 0 < i' < / and is 0(a^) for f > /. The integral may then be approximated by replacing the upper limit 00 in (A2-14) by /. The value of (7-16) for m = 0 is found to be, to within 0(a^), 2-ia¥(e-2'fc' - 1) - (3/4)to2^-*(sin 2kt - 2kt) (7-17) When m > 0, €„ = 2,^1, = m^ — k^, and the substitution of the value (7-13) for amiv) enables us to express (7-16) as the sum of six /'s where J is defined by (A2-4). The /'s may be evaluated with the help of (A2-7) and (A2-8). Substitution of this value of (7-16) and the value (7-17) for m = 0, together with R^e^ given by (7-15), in the expression (4-5) for R^ gives our final result R^P = iak-h\n^ kt + a2/22-Kg-2"' - 1) (7-18) + ia^[^-'k-\2kt - sin 2kt) - Bsm 2kt + ^Z >7--7^'^n] CONFORM A L TRANSFORMATION 123 where 00 B - J2 n-'ie-'"' - 2(-)"e-"'] n=l ^„ = COS 2kl - [2cos kt - (-)"r''."']^ Equation (7-18) is an approximation, to within terms of order a-, for the reflection coefficient of a truncated corner which turns through a small angle lira. The electric vector lies in the plane of the bend. When t = 0, (7-18) reduces to (7-5) by virtue of 2a — /3. APPENDIX I CoNFORM.'VL Transformation of Truncated Corner We shall use a Schwarz-Christofifel transformation* to carry the guide of Fig. 1 into the straight guide of Fig. 2. The first step is to transform the interior of Fig. 1 into the upper half of an auxiliary complex plane which we shall denote by f . Let the points 21,22,23,24, Z5 in Fig. 1 correspond to the points — h, h, 1, 1/2 respectively. Now that the guide of Fig. 1 has been transformed into the upper half of the f plane, the next step is to transform this upper half into the straight guide of Fig. 2. We want f = — 1, i.e. Ss , to go into v = —x and f = 1, i.e. Zs , to go into i' = -\- ^ . Again using the Schwarz-Christofifel formula with w = r -\- id (the exterior angles at r == ±=o are equal to ir) w = A + £i f (r + 1) \t - 1)"-' dr (Al-3) We take the point So in Fig. 1 to correspond to r = 0, ^ = 0 in Fig. 2. Since this corresponds to i' = 0, Di must be zero. Also dw/d^ is real because w traverses the walls of the guide of Fig. 2 as f moves along the real axis in the .t plane. Hence £i is real. As f goes from 1 — e to 1 + e around a small circular indentation above .t = I, w changes from x to x + /tt. Thus iir = Ex2-'{-iir) or £1 = -2 (Al-4) When (Al-3) is integrated, (Al-4) inserted, and the result solved for f we obtain i" = tanh w/2 (Al-5) COMORMA L /"AM XSFORMA I'lON I2.S Tlu' function we require is obtained by differentiating (Al-1) and (Al-.V): = (1 - hYif - h^-b/n bT ch'w/2 T IT \_sh^(w — /) sh^{w + /)J TT \_{e^-' - l)(e«'+' - 1)J (Al-6) where h = tanh t/2 (Al-7) For a 90 degree corner a = 1/4 and i = 2''^(1 - d/do) (Al-8) where, in Fig. 1, d — | S4 — 2o | and do = \ Zi — Ze \. In order to obtain the relation between /, defined by (Al-7), and d/do various values of h- were picked and the corresponding values of / and d/do (using (A 1-2) and (Al-8)) computed. Representative values are given in the following table. d/do / d/do / .000 0 .5796 1.2302 .9041 .0633 .5385 1.4910 .8565 .1417 .5000 1.7594 .8292 .2007 .4615 2.0634 .7745 .3500 .3727 2.8872 .7196 .5421 .2804 4.0096 .6919 .6549 .1708 5.987 .6273 .9624 .0959 8.294 APPENDIX II Integrals Associated with Corners of Small .\xgle The derivation of the integrals encountered in Sections 7 and 8 will be outlined here. The first ones are 1 r /i(«, v) = / log(c/; u — cos 6) log {ch v — cos 6) dd (A2-2) 126 BELL SYSTEM TECHNICAL JOURNAL 1 r Ii{u, v) = j \o^(ch u — cos 6) lo,:; (ch v + cos 6) dO IT Jo (A2-1) 2 r Iz{u, i) = I COS 26 log {ch u — cos 6) log {ch v — cos 6) dd ■w Jo Ii{u, v) = COS 26 log {ch u — cos 6) log {ch v + cos 9) dd TV Jo Assuming m and v to be positive and using the expansions 00 log(c^ u — COS 6) = log(e"/2) — 22^ w~'g~""cos n6 n=l 00 log(cA u + cos <9) = log(e"/2) - 2^ (-)"«-'e-""cos w0 n = l leads to oo /i(m, v) = log(e"/2) log(eV2) + 2X) ^-^e-""-"" n=l 00 /2(w, v) = log(e«/2) log(gV2) + 2X) (-)";;- V"-"" n=l /3(m, t)) = -e-2"log(eV2) - e-2''log(6V2) + 2^-"-" 00 + 22 '^~K» + 2)-ie-"»-"''(e-'-" + g-^") h{u, v) = -g-2"log(eV2) - g^-'log (e"/2) - 26-"^" 00 + 2^ {-Yn-\n + 2)-ie-"»-"''(e-2" + g-^") n=-l When 2f or v are negative they are to be replaced by their absolute values in the expressions (A2-2, 3). Now we consider the double integral (A2-3) /+00 - +00 di'o I d-j ■ 00 J — 00 (A2-4) •exp [ — /i I To — r I —m \v — s\ —ic{v + 7'o) —8\v — Tq | ] in which n, m, c, 5 are real and positive and r and s are real. The double integral may be reduced to a single integral by substituting —i\v—v e o' = ^ T" (6== + :t')-V"<"-''»^ dx, (A2-5) TT J— 00 CON FORM A L TRA NSFORMA TION 127 interchanging the order of integration, and integrating with respect to v and j'o . Assuming 5 — /- > 0, the integral is then evaluated by closing the [lath of integration by an infinite semicircle in the upper half plane and rakulating the residues of the integrand at the poles ib, c + im, —c + i^l•. /« 1^ ii(»— r)— tc(r-(-«) « 7r(52 + x-')W Ab^inie^ dx = AdjjLin + l' + ix+ cy][m' -\- {x - cy] -(6+ic)s+(«-ic)r LMm- + (c + i8y][m^ + {c - idY] — m»+(m— 2 »c)r (A2-6) m[8^ + (f + imYln^ + (2c + iniY] jir—(.ft+2ic)s + m[52 + (c - i^iYlm'' + (2c - !>)']_ Substituting special values for the parameters gives the results required in the text. Thus, J(fn, m, k,y;t,t) = g-2«>< /(^^ ,„^ ^, 7; 0, 0) J(m, m, k,y;—t, t) = e^'*' J{m, m, y^, 7; 0, 2/) /(w, w, ^, 7; -/, 0) - e-'*' J(m, ni, yfe, 7; 0, /) (A2-7) w s n n\ 2w(6 + 2m) J{m, m, c, 8; 0, 0) - — — — ,..,,. ' ,.., (c2 + m2)[c2 + (w + 5)2] which hold irrespective of any relations between the parameters. The derivation of the last result is simplified by setting a = c + im, a = c — im and factoring the denominators in (A2-6) so as to obtain terms of the form a ± id, a ± id. WTien 7' = w^ — k- considerable simplification is possible and we obtain 1 m ~\ J{m, m, ^, 7;0, 0) = Jim, m, ^; 7, 0, /) = ^2 1^^ ^2 _|_ ^2^ ye k"- e '' _ e "" (w cos kt — k sin kl) _ 7 vi^ -\r k^ (A2-8) If we put M = w — 1, w = w + 1, and set 6- == n- — 1 where k- = 1 + c^, (A2-6) yields, after some reduction, J{n - 1, w + l,c,5;0,0) = + n — \ + (« - 1)5 (en + i5)2 2(« + 1)(1 - icy{n - ic) {n +J)5_ 2(w - 1)(1 - ic)2(7r+ ic) (A2-9) kXw2 - 1) + nb[2hi + c') - k^(h'' - 1)] K^n" - l)(w2 + c2) 128 BELL SYSTEM TECHNICAL JOURNAL The form of the final expression has been chosen so as to be suited to the use we shall make of it. Another double integral which appears in our work is /+C0 dvQaivo) ■ CO (A2-10) • / dv a(v) exp [ — ik(v + vo) — y \v — vo |] J_oo where a(v) is an even function of v and is such that all of the integrals encountered converge. We begin our transformation by dividing the interval of integration (— =o , oo) for z'o into (— co, 0) and (0, x). Making the change of variable Vo = —vo,v= —v' in the first interval, dropping the primes and using a{—v) = a{v) leads to j(k^y)^2f dvoa(vo) f dv aii^e^"^'-'"^ cos k{v -\- Vo) (A2-11) Jo J-oo We now split the interval of integration of v in (A2-11) into the intervals (— oOj 0), (0, I'o), {vo , °o). In (— Gc, 0) we change the variable from v to — v', drop the prime, and use a(—v) — a{v). By paying attention to the sign of V — ro we may remove the absolute value sign. By changing the order of integration in the double integral arising from the third interval (in which 0 < z'o < "^ , vq < v < oo ) we may show that it is equal to the double integral arising from the second interval. Thus I{k^ y) = 2 f dvo a(vo) [ dv a(v)e""~""'° cos Hvo - z') + 4 f dvo a(vo) f " dv a(v)e-"">^'''' cos Hvo + v) Jo Jo When a{v), y and k are real we may write (A2-12) as I{k,y) = 2! r dv aiv)e-'"-''''\ I Jo I + 4 Real C dvoa{vo)e-" "''''' f" dv a(v) e'"^'"' Jo Jo and when 7 = ik we have I{k,ik) — 2 dvoa{vo) / dv a{v}e'^''"'' Jo Jo + 2 f dvoaivo) f " dv a(r)[/''^" + e-^"""]. Jo Jo (A2-12) (A2-13) (A2-14) CON FORM A L TRA NSFORMA TION 1 20 APPENDIX III Integral Equation When Guides Entering and Leaving Irregularity Are of Different Sizes Here we shall indicate how the integral equation method may be extended to cover the case mentioned in the above title. It is supposed that only the dominant mode is propagated freely in both guides. E in Plane of Irregularity Let the notation for the guide carrying the incident wave be the same as for the ^-corner, b denotes the narrow dimension of the guide and the quantities k and 7^ are given by (2-1). Both guides have the same wide INCIDENT WAVE 0 Fig. 3 dimension a. The narrow dimension of the guide shown on the right of Fig. 3 is ii . We introduce the new quantity ki = [(26i/Xo)- - (b,/ayf" (A3-1) to correspond to k. Since, by assumption, only the dominant mode is freely propagated in both guides both k and ki are real positive quantities less than unity. Let z = f{w) carry the system of Fig. 3 into a straight guide of width tt in the w = v -\- id plane (see Fig. 2), and let g{i', 6) be defined by ^ + giv,e) = |/'(ziO|-. The behavior of g{v, 6) at infinity is shown by the table V dz/dw g{v, &) — b/ir 0 + 00 bi/r kikT — 1 where bi/b = ki/k has been used. It is convenient to introduce the ap- proximation g{v) to g{v, 6). g(v) may be chosen at our convenience subject only to the conditions that it be differentiable, g{— 00) = 0, and |(°o) = klk-'- - 1. When we define G by equation (3-3) so that, as before, it is the Green's 130 BELL SYSTEM TECHNICAL JOURNAL function corresponding to a guide of width b, we may use equation (3-4) to derive the new integral equation de Q{v, , do) = e-""-'^ + ^ f dv f • [g{v, e)Q(v, 6) - i(v)Tse-''^'}G(vo , do ; v, d) + T,F{v,) in which N-(vo) = 2-'k{k, - kV f'g'{i)e''"'''''"dv iV+(ro) = 2-'k{k, + k)-' f g'{i)e-'''^-'''' dv (A3-2) (A3-3) Here ^'{v) denotes d^(v)/dv. Equation (A3-2) and Limit Q(v, d) = T^e'"''' (A3A^ D— ♦OO are to be solved for the unknown function Q{v, d) and the unknown quantity T g . The method of successive approximations may be used in somewhat the same fashion as in the simpler case but we shall not give a general discussion. The first approximations are found to be ry^ = lAV-(°o), R'-^^ = -N+i-ccyN-i'^) (A3-5) where the A^'s may be obtained by setting I'o = ± ^o in equations (A3-3) . One of the simplest choices for g{v) is to let it be zero for negative values of V and to have the value ^(°o ) = kik~^ — 1 for positive values of v. Then Ti'^ = 2k{ki + k)-\ R'^^ = {k- ki) (k + yfei)-i (A3-6) These are quite similar to tlie corresponding expressions for a transmission line which have been used extensively in wave guide work. In working with these formulas, when k is small, it is sometimes convenient to use the result r dv r ddg(v,d) = ir'b-' r dv \ dd\f'{u)f - ds - z-Ot (A3-7) Jvi Jo Jfi Jo where the evaluation of the double integral on the right is made easier by the fact that it represents the area in the original guide (in the (x, y) plane) enclosed by the lines corresponding to z; = di and v — v^ . r? and vi are CONFORM. I L TRA NSFOR.UA TION 1 3 1 chosen to be moderately large positive and negative numbers, respectively. It turns out that, when ki and k are very small, this is related to the "excess capacity" localized at the irregularity whose effect must be added to that of the mismatch, indicated by (A3-6). When the entering and leaving guides are of the same size it is still possible to use the formulas of this appendix. N~(vn) may be replaced by an expres- sion which now has for its limiting value ,V-(^) = 1 + i(k/2) f i(v) dv (A3-8) // •/;/ Plane of Irregularity Let the figure corresponding to the irregularity be Fig. 3 with b and bi replaced by a and ai , respectively. In addition to the quantities c and k defined by equations (5-2) we define Ki = 2ai/Xo , c, = {k\ - 1)''' (A3-9) where we assume k and ki to lie between 1 and 2. At u = — =0 P{v^ 0) still consists of the unit incident wave plus the reflected wave given by the first of equations (5-4) and g{v, 6) is still zero. However, now, a.t v = x , P{v, 6) - Tac'^''" sin e |(.x>) - kik"' - 1 = K~\c\ - c") (A3-10) The integral equation for P{v, 6) and Th is P{v, , do) - e-'"" sin ^0 + ;^ dv Zw •'—00 ^0 ■dd{g(T, e)P{v, e) - g{i)T„er''''' sin e\G{v, , e, ; V, d) + Til sin doFni^'o) in which Fh(vo) = e-''^"'g(To)/g(^) - e-'-^ATivo) - e''"> M"- (z'o) J— 00 (A3-11) dv (A3- 12) -{-c)v M^ivo) = ^(2cr\c + c:)-' [ i'{i)e-''''^"' dv J Da First approximations are ri" = l/M-(oo), R*-^^ = -M+(-oo)/M-(oo) (A3-13) 132 BELL SYSTEM TECHNICAL JOURNAL which, when we choose g{v) to be zero for w < 0 and kik~ — 1 for i) > 0, become n'^ = 2c{c, + c)-\ R^J^ = {c- ci)(ci + c)-i (A3-14) which again agrees with results obtained from transmission Hne considera- tions. WTien the entering and leaving guides are the same size we may use M"(oo) = 1 + u\2c)~' ( g(i) dv (A3-15) J— 00 It seems difBcult to give any general rules for the choice of g{v). Since for Rh and Th , the factor sin d reduces the effect of the singularities on the walls of the transformed guide, the choice g(v) = g(y, 7r/2) suggests itself. The factor sin d is not present in the formulas for Re and Te and regions near the walls are more important. In this case the selection i(v) = T-' [ giv, e) dd may be useful, especially since it allows us to use the result (A3-7) when k and ki become small. APPENDIX IV Variational Expressions for Reflection Coefficients The reflection coefficients are proportional to the stationary values of certain forms associated with the integral equations. In order to obtain these forms we proceed as follows. It is readily seen that the values of Xi and Xi which satisfy the symmetrical set of equations (A4-.1) cinXi -\- a22X2 = ^2 are the ones which make / = aiiX"i -j- 2auXiX2 + 022^:2 — 2biXi — 262.T2 (A4-2) stationary when xi and X2 are given small arbitrary increments. This stationary value of J is Js = —biXi — b2X2 If we take the integral equation to be the analogue of the set of linear equations, the reflection coefficient turns out to be proportional to /, . In order to set down the actual expressions it is convenient to write r for CONFORM A L TRANSFOR\fATION 133 (v, d) and dS for the element of area dvdd so that the integral equation (3-5) for Q{v, 6) may be written as Q(ro) = e-''"'" + k'dT)-' I g(r)Q(r)G(ro , r) dS (A4-3) where the integration extends over the interior of the guide and G(ro , r) denotes the dreen's function (3-3). If the number of equations in the set (A4-1) were increased from two to a large number .V, the set of .v's would correspond, say, to the values of Q{r) or of g(r)Q(r), and the 6's would correspond to the values of exp{—ikvo). In any event, we take the analogue of / to be Je= f g(r)Q{r)[Q(r) - 2e-'''] dS (A4-4) ■- k-K2irr' ff g{r)Q{r)g(ro)Q(ro)Giro , r) dSo dS where the subscript E indicates tiiat we are dealing with an electric corner. It may be verified,* by giving Q{r) a small variation 8Q(r), that the function Q{r) which makes Je stationary is the one which satisfies the integral equation (A4-3). Furthermore, when we assume Q(r) to satisfy the integral equation, the expression for Je reduces to an integral which is proportional to the integral (3-6) for the reflection coefficient Re . More precisely, Re ^^ given by ik Re ^ ■;r- [Stationary value of Je] (A4-5) 2ir It follows that if, by some means, we have obtained a fairly good approxi- mation to Q, we may obtain a better approximation to Re by computing Je and using the formula Re = ik{2ir)-'JE When we use the first approximation exp(— i^ti) for Q to compute Je it turns out that the above formula gives the third approximation, R^e\ to the reflection coefficient. The magnetic corner may be treated in much the same way. The integral equation (5-6) for P{v, d) becomes, in the notation of this appendix, P{h) = g-'^'" sin ^0 + Kilir)-' j g{r)P{r)G{ro , r) dS (A4-6) in which the v in dS = dvdd is integrated from — qo to -j- oo and d from 0 to tt, * See Courant and Hilbert, Methoden der Mathematischen Physik, Julius Springer, Berlin (1931), page 176, where a similar problem is treated. 134 BELL SYSTEM TECHNICAL JOURNAL as before, and G{ro , r) now denotes the Green's function (5-5). We define Juhy Jh= f g{r)P{r)[F(r) - le'''" sin d\dS (A4-7) -kK2-k)-' II g(r)P(r)g(r,)P(ro)G(r, , r) dS, dS. J H is stationary with respect to small variations in P{r) when P{r) satisfies the integral equation (A4-6). Furthermore, from the integral (5-7) for Rh , Rh = iK-(Tc)~'^ [Stationary value of Jh] (A4-8) which may be used in the same way as equation (A4-5) for Re ■ J. Schwinger has used variational methods with considerable success to deal with obstacles in wave guides.* However, his variational equations differ somewhat from those given here. Some light on the relation between Schwinger's equations and the present one may be obtained by returning to the simple algebraic equations (A4-1) and (A4-2). A rough analogue of the expression required to be stationary in Schwinger's theory is (aii.vi + 2ai2.vi-Vo -|- a22.V2)/(^i-Vi + ^2X2)' (A4-9) The essential point here is that the stationary value of the expression corresponding to (A4-9) gives the value of an impedance or combination of impedances appearing in some equivalent circuit. Expression (A4-9) may be obtained by expressing /, defined by (A4-2), as a function of .vi and y = X2/.V1 . / is still to be made stationary but now it is a function of Xi and y. Solving dj/dxi = 0 for Xi and setting this value of .Vi in / gives the following function of y -(bi + b^yT- (an + 2aviy + ^22/)-', which is the stationary value of / with respect to variations in xi when y is held constant. This function is still required to be stationary with respect to y. The same is true of its reciprocal which becomes (A4-9) when both numerator and denominator are multiplied by .vi and the definition of y used. When (A4-1) is replaced by a larger number of equations similar considerations lead to a generalized form of (A4-9). The expression required to be stationary by Schwinger is obtained when the sums in the general- ized form are replaced by integrals. * An account of the method together with applications is given in "Notes on Ix-ctures by Julian Schwinger: Discontinuities in Waveguides" by David S. Saxon. An account is also given by John VV. Miles." COXFORM. 1 /. TK. 1 .V.SV'OA' \/, I 770.V 1 .IS Refkrkncks 1. Poritskv ami Hlcwctl, A McllioJ of Solution of Field Prol)lcms 1)V Means of Over- lapping Regions, Qiuirt. Jl. Appl. Math., 3, 336-347 (1946). 2. J. W. .Miles, The K(|uivalent Circuit of a Corner Bend in a Rectangular Wave (iuide, Proc. I. R. E.. 35, 1313-1317 (1947). 3. N. Klson, Rectangular Wave Guide Systems, Hends, 'i'wisls and Junctions, Wireless Eiif;iiicer, 24, 44-54 (1947). 4. 1'',. J. Routh, .Vdvanccd Rigid Dynamics, 6th edition, jip. 461-467, London (1905). 5. W Rrasnooshkin, .\coustic and Electromagnetic W^ave Guides of Complicated Shape, Jl. of Physics {Acad. Sci. i'.S.S.R.), 10, 434-445 (1946). 6. S. O. Rice, A Set of Second-Order Differential E(|uations .Associated with Reflections in Rectangular Wave Guides — .Vpplicalion to Cjui Ic Connected to Horn, this issue of the B. S. T. J. 7. S. O. Rice, Rellections from Circular Bends in Rectangular Wave Gui les —Matrix Theory, B. S. T. J., 27, 305-349 (194S). In addition to the earlier work of R. H. Marshak referred to there, the interesting, hut as _\et unpublished, work of H. Levine should be mentioned. He uses variational methods to obtain the values of the impedances ajipearing in the ec|uivalcnl network for the bend. 8. S. A. Schelkunoff, On Waves in Bent Pipes, Quart., Jl. Appl. Math., 2, 171-172 (1944). 9. H. Buchholz, Der Eintluss der Krummung von rechteckigen Hohlleitern auf das Phasenmass ultrakurzer Wellen, E.N.T., 16, 73-85 (1939). 10. Montgomery, Dicke. and Purcell, Principles of Microwave Circuits, McGraw-Hill (1948), pige 149. 11. J. W. Miles, The Equivalent Circuit for a Plane Discontinuity in a Cylindrical Wave Gui ie, Proc. I.R.E., 34, ll^-Ul (1946). A Set of Second-Order Differential Equations Associated with Reflections in Rectangular Wave Guides — Application to Guide Connected to Horn* By S. O. RICE In dealing with corners and similar irregularities in rectangular wave guides it is sometimes helpful to transform the system, conformally, into a straight guide. Propagation in the straight guide may then be studied by an integral equation method, as is done in a companion paper, or by a more general method based upon a certain set of ordinary differential equations. Here the second method is developed and applied to determine the reflection produced at the junc- tion of a straight guide and a sectoral horn — a problem the first method is unable to handle. The WKB approximation for a single second-order differential equation is extended to a set of equations and approximate expressions for the reflection coefhcient are derived. IN A companion paper^ the disturbance produced by a corner in a rec- tangular wave guide is examined by transforming the system, con- formally, into a straight guide. Although the medium in the straight guide is no longer uniform, an integral equation may be set up and approximate solutions obtained. In that paper the wave guide is assumed to have the same cross-section at -f CO as at — oc . WTien this is not so, a conformal transformation may still be used to transform the system into a straight guide provided one dimension of the original cross-section is constant. However, now some advantage appears to be gained by replacing the integral equation by a set of differential equations. Since two cases appear, corresponding to E and // corners, there are two sets of equations to be considered. These two sets of equations are studied in the present paper. After their derivation in Sections 1 and 2 several remarks are made in Section 3 con- cerning their solution, special emphasis being laid on the problem of deter- mining the reflection coefficient. In the remainder of the paper the general theory is applied to a system formed by joining a rectangular w^ave guide to a horn (with plane sides) flared in one direction. The reflection coeffi- cients for sectoral horns flared in the planes of the electric and magnetic intensity, respectively, are given approximately by equations (6-1) and (7-1). These approximations assume the angle of flare to be small so that, as it turns out, only the first equations of the respective sets need be considered. As was mentioned in the companion paper, Robert Piloty has recently made use of conformal transformations in wave guide problems. In his * Presented at the Second Symposium on AppUed Mathematics, Cambridge, Mass., July 29, 1948. 'See list of references at end of paper. 136 RECTANGULAR WAVE GUIDES 137 method the propagation function g(v, 6) is derived graphically from the geometry of the wave guide irregularities and the result used in one or the other of two sets of differential equations which are equivalent to those derived below. Piloty's work is scheduled to appear soon in the Zeihclirifl fiir angrii'audle Physik under the title "Ausbreitung el.-magn. W'ellen in inhomogcnen Rechteckrohren." 1. Differential Equations when Electric Vector is in {x, y) Plane The partial differential equation to be solved is, from equation (2-3) of the companion j)aper', g + 0 + u + sO, «)i*"e = 0 (1-1) where — = 0 at 0 = 0 and 0 = tt dd 00 1 + g{v,e) = 1 + E ancosne = \f'{-v + id) fir'/b" (1-2) k — [(26/Xo)" — {b/aYY , Xo = free space wavelength In (1-2), z = X -]- iy ^ f{v + id) is the transformation which carries the wave guide system in the (x, y) plane into the straight guide of width 6 = v in the (v, 6) plane. For the sake of simplicity we shall always assume that far to the left the system becomes a straight wave guide of dimensions a, b {b < a) such that only the dominant mode is propagated without attenuation. This insures that the a„'s (which are functions of v) will approach zero a.sv—^ — oo . The dimension (of our system) normal to the {x, y) plane is a throughout. Since the normal derivative of Q vanishes on the walls at 0 = 0 and 6 = t we assume Q= Fq + FiCosO -\- FiC05 2d+ ••' , (1-3) where Fi , F^ , ■ ■ ■ are functions of v, and substitute it together with the Fourier series (1-2) for 1 + g(v, 6) in (1-1). The equations obtained by setting the coefficients of the resulting cosine series to zero are Fo + (1 + ao)k'Fo + ^ Z <7„/'„ - 0 (1-4) 2 n=l F'rl + [(1 + ao + a2m/2)k' - m']F,n + a^k'Fo (1-5) ,2 00 + ^ Z^' {a\n-m\ + an+m)Fn = 0 Z n=l 138 BELL SYSTEM TECHNICAL JOURNAL where m = 1, 2, 3, • • • , /%„ = cPF,n/dv", and the prime on ^ indicates that the term ;/ — m is to be omitted. In groui)ing the terms we have assumed that Fo is the major part of Q. The principal problem is to solve equations (1-4) and (1-5) when the fundamental mode Fq is of the form (1-6) Fo = Te{v), 1- ^ +0C in which Re is a constant and Te(v) represents a wave traveling towards I! = 00 . At z' = ±20 Fi , F2 , ■ ■ ■ have the form of waves traveling (or being attenuated) away from the region around v = 0. As before, we shall be mainly interested in determining the reflection coeflicient R. It is assumed that only the dominant mode is propagated without attenu- ation in the straight wave guide far to the left and hence Fi , Fo , ■ ■ ■ all become zero as i' — > — =0 . 2. Differential Equations when Magnetic Vector is in (x, y) Plane The partial differential equation is now given by equation (5-1) of the companion paper'^ ^+^+ n + dv,e)U'P = ^) (2-1) dv^ dd- where the dimension of the system normal to the (.v, y) plane is now b, a is the dimension (in the (x, y) plane) of the straight guide at the far left and p = 0 at e = 0 and 6 ^ ir 1 -f g{v, ^) = 1 + E a« cos ne (2-2) n = l K = 2c/Ao , Xo = free space wavelength C = (k2_ 1)1/2 Since P = 0 at ^ = 0 and 6 = ir we assume P = J^Fn sin nd (2-3) n = l where the F's are functions of v to be determined by the equations 2 « Fi + [k(\ + flo - 02/2) - l]Fi + ^ E ('/«-i - «n+.)/^„ = 0 (2-4) F'J, + [k(\ +Co - a2m/2) — m'\Fm + 2 ('^'"-1 ~ am+i)Fi 2 « -f •;r E' (a\m-n\ " flm+„)Fn = 0 -i n=2 (2-5) RECTANGULAR WAVE GUIDES U'> in which w = 2, 3, 4, • • • and tlic primes on /•',„ and /I have the same significance as in (1-4) and (1-5). The principal problem here is to solve equations (2-4) and (2-5) simul- taneously subject to (2-6) Fi = Tf,(T), v-^ -\-x which again corresponds to a unit wave in the dominant mode incident from the left. Tn(v) and the remaining F's correspond to outward traveling waves as before. F-i , F3 , ■ ■ ■ aW approach zero as z; — > — =0 . 3. Remarks tvi Solving the Equalioiis of Sections I and 1 for the Reflection Coefficient Suppose that we have a system in which the wave propagation is governed by the single differential equation —^ - h'y = 0 (3-1) av~ where // = h(v) is a positive imaginary function of v, twice differentiable and such that h -^ ic, c being a constant; as r ^ — x . We desire the solution of (3-1) which, together with its first derivative, is continuous everywhere and at ± X satisfies the conditions y = g-icv _^ J^gicv^ ^, ^ _ oo (3-2) y' + (// + h'/{2h))y -^ 0, r -> ^ {i-i) The constant R (the reflection coefficient) is to be determined. Condition {i-i), in which the primes denote differentiation with respect to v, is sug- gested by the fact that we want y to represent a wave traveling in the positive V direction (the factor exp (/oj/) is suppressed). In writing ii-i) we have assumed that // is such that for large values of v the two solutions of (3-1) are asymptotically proportional to* y = h-'e"-", (3-4) ^ ^ ^(r) = icv + [ {h - ic) dv. (3-5) Physical considerations suggest that solutions satisfying (3-2) and {^-i) exist in most cases of practical importance. However, if the function h is picked arbitrarily the corresj)onding solutions may be incapable of satisfying * S. A. SchelkunotT- mentions that this approximation, sometimes designated by "WKB", goes back to Liouville. The ideas we shall use are quite similar to those in SchelkunotT's paper. 140 BELL SYSTEM TECHNICAL JOURNAL the conditions. For example, if h — ic/{\ -\- exp v) then h —^ ic exp {—v) as z) ^ 00 , and the solutions of (3-1) behave like Bessel functions of order zero and argument c exp {-v). It may be verified that these solutions do not satisfy {i-2>). Again, condition {2>-i) may be satisfied without y having much resemblance to an outgoing wave at i) = oo . Thus if /? — > ia/v as z; — )■ oo , y inc eases like v" whe e )i' — )i — a — 0. When 0 < a < 1/2 both values of n He between 0 and 1, and both solutions satisfy {S-3). Despite these sho-tcomings it still seems best to etain (3-3) to specify the behavio" oi y Bit V — 00 . It should be mentioned that P. S. Epstein^ has obtained the reflected wave by transforming the hypergeometric differential equation into the form (3-1). This method has been extended by K. Rawer^ who gives a number of references in which the approximation (3-4) is used to study propagation in a medium having a variable dielectric "constant". An interesting paper on the general subject of reflection in non-uniform trans- mission lines has been written by L. R. Walker and N. Wax*. 1. When most of the reflection occurs in a short interval, say near v = 0' R may be obtained by numerical integration of (3-1). One method is to start at z) = 0 with the initial conditions y = 1, y' = 0 and work outVvards in both directions. Let Ya(v) denote this solution and Vb{v) the solution obtained by starting with y = 0, y' = 1. The general solution is y = C^Yaiv) + C^Ybiv). (3-6) Ci and C2 are to be determined by the conditions y — (constant) lr^'-e~^ , v > V2 (3-7) y = (/c//?)'/-[e-f + Re^ , V < vi ' (3-8) where I'l and V2 are large negative and positive values, respectively, of v. These conditions lead to equations for Ci ,€2 , R- [y'+ ^y],.„, = 0 [y' - e-y + 2(khy' e-~^l=,, = 0 (3-9) [y' + ry - 2(ichy'Re^],^,, = 0 in which ^ is given by (3-5) and e^ = hzL h'/(2h). ■ (3-10) The required value of R is obtained by letting di -^ — 00 , ^jg ^ °° in the expressions, which follow from (3-9), RECTANGULAR WAVE GUIDES 141 y = C2/C, = -[(r: + d+ya)/{yl + e+Y,)U,, r = [y'/yUv, = [(rl + yY',)/{ya + Tn)].-«. (3-11) ie = [(^ + r)/(r - r)]..„, exp -ikv, - 2 j^ {h - ic) dv\ where the arguments of Ya{v) and Yb{v) have been omitted for brevity. If // should change from a positive imaginary quantity to a positive real quantity in {vi , v^) and remain greater than some fixed positive number for z) > z;2 it may be shown that | i? | = 1 (7 and T are real and Im ^ = Im d~, Real ^ = —Real ^ at d = vi). This complete reflection is to be expected .from physical consideration. 2. An exact expression for the reflection coefficient which holds when h satisfies the conditions following (3-1) (in particular it must not pass through zero anywhere in — co < v < qo ) is R = i(ic)-^ r e-^y{v) ^, h-^ dv (3-12) J- M dv- where ^ is given by (3-5). Before this integral for R may be evaluated y{v), and hence R itself, must be known. Nevertheless, when R is small a useful approximation may be obtained by using the WKB approximation y{v) = (ic/hyh-^ (3-13) Thus R^- r e-'^h"' ^h-Uv 2 J- 00 dv- = 2,L' U^ ( 5 ,,-5/2 (dK\ 1 -,-3/2 d K Til 1 _ t (3-14) dv / 4 dv dv in which K = — li\ The expression (3-12) for R is obtained by letting vo^ — ^ in the integral equation y{ro) = {ic/hfe-^' - f_^ Ga(vo,v)yiv)h^~^Jt-Uv, , ^ {e^-^\ V < ro ^ ^ Ga(ro,r)=-P^^/rM ^ ^ (3-l:>) ^0 - ^ = / lidv, ho = //(z'o), ?o = ^(I'o). Go(^o , "v) is the approximate Green's function suggested by (3-13). The 142 BELL SYSTEM TECHNICAL JOURNAL integral equation may be obtained from the differential equation (3-1) and the boundary conditions (3-2) and ii-?>) by the one-dimensional analogue of the method used in Section 3 of the companion paper^ If we multiply both sides of ^ - Jh = s(v) (3-16) (where s(v) has been added for generality) by Ga(i'u , ^0, integrate twice by parts over the intervals (vi , Vq — e), (vo + e, Vo) with 6 > 0 and Vi < I'o < Vo , and finally let e ^ 0 we obtain Kfo) = f Ga{vo,v) s{v) - y{v)Jf ^, ir + Ga(To,n)ly' - e-yU,; - Ga(T,,vd{y' + d-'yUr,. dv (3-17) Equation (3-15) follows when we put s(v) = 0 and let vi -^ — x , vo -^ ^ . It will be recognized that (3-17) and (3-15) are closely related to integral equations occurring in the work of R. E. Langer^ and E. C. Titchmarsh^. When // has, for example, one or more simple zeros in — x < v < x the integral in (3-15) contains a factor which becomes infinite and the integral equation fails. However, we shall not concern ourselves with this case beyond remarking that it involves results obtained by H. Jeffreys'", Langer^, Furr>''' and others. 3. So far we have been considering the solution of only one equation whereas we really require the solution of a set of equations. If it is apparent that most of the disturbance is given by the first equation of the set it may be possible to proceed by successive approximations, each of the remaining equations being of the form (3-16) with s{v) determined by the solution of the first equation. Another method of dealing with a system of -V equations is that of numeri- cal integration. As a contribution towards obtaining the boundary condi- tions at large positive and negative values of v we shall state a generalized form of the WKB solution. Although this solution is related to the general results obtained by Birkhoff'-, Langer^ and XewelF^ concerning the asymp- totic forms assumed by the solutions of a system of ordinary linear differen- tial equations of the first order, it is worth mentioning explicitly. Let the wth equation of the set be A' A'm = S Amnyn , W = 1 , 2 , • • • , iV (3-18) n=l where the .lm«'s are relatively slowly varying functions of v (see equations RECTANGULAR W AVE GUI DOS 143 (vS-22) for a more precise statement of the assumptions) and the dots denote (lifTerentiation with respect to v. We shall reserve primes to denote trans- l)osition of matrices. It is supposed that Amn = A^m (equations(2-4) plus (2-5) satisfy this condition and (1-4) plus (1-5) may be made to do so by setting Fo = 2'/-7''o). The solution of (3-18) is approximately .V y,n = Z S,n,[e'un + e-^^/|] (3-19) (=\ where the d( are the 2N constants of integration and N n=l ^tjls\(=\ (3-20) n=l ^( = ipc dv serve to determine ^( , ^t , and S,n( (the last to within a plus or minus sign). We assume the .V roots (^J , (fo , • • • <^^ of the determinantal equation arising from the lirst of equations (3-20) to be unequal, and denote by (pc that square root of <^/ which has a positive real part or, if the real part be zero, which has a positive imaginary part, v^f is any convenient constant. The approximation (3-19) may be obtained by setting the assumed form y,n = gm e^\ ^ = I

p + gni^ and sufficient condition that (3-24) be consistent^*, namely, ^ hm{2gmO

- oo (3-30) y = Se~"g+, z) -^ oo (3-31) where the column matrix /+ (corresponding to the incident wave) is given and/~ (corresponding to reflected wave) and g+ (corresponding to the trans- mitted wave) are to be determined. The elements of /+, /~, and g+ are constants. It is further supposed that S satisfies the conditions S'S - S'S = 0, z)^±oo which are certainly met if the elements of A approach constants at ± «> . In the wave guide problem we assume Vo ■ and integrating by parts twice. It is seen that each column of Ga(vo , v) is an approximate solution of y = Ay, in which the columnar constants of integration are the columns of -So , and represents a wave traveling away from Vo in both directions. Gaii'o , ^') is continuous at v = Vo and ^ Ga(Vo , V) dv -~ Ga(Vo , V) dv = So^oSo = I Thus the Hth column of Ga(vo , v) gives the approximate values of yi(v), y-ii"!-'), ■ ■ • , yn(v), subject to the conditions that all these and all of their first derivatives are continuous at t; = Vq except y„{v) which has the jump y„(i'o+0) - y„ivo-0) = 1. The presence of 2*^' + $5' = *5' - ^S'SS~' = ^{S'S - S'S)S~' in (3-29) and (3-32) makes the X variable case somewhat different from the case xV = 1. vS. When Z,„,i and I^„„ are slowly varying functions of v the approximate solution of the transmission line equations dv dJr, = -T z J .V = - Z Vmn Vn (3-32) RECTANGULAR WAVE GUIDES 147 where Z,„„ = Z„,„ and I',,,,, = I',,,,, is, as in (-^-l^), .V V,n = T.S,„< \e^' di: + e^' d"[\ X (=\ Here ^c is the integral of ip( as given by (3-20), and ip( is determined by setting the determinant of the matrix v?-/ — ZV to zero. When<|0/ is known, S,n( and T„,f are determined (to within a plus or minus sign which may be absorbed by the constants d( of integration) by the relations n=l X (PfTmt = — Z^ ymnSmt (3-34) 71=1 .V 2-j Sm( Tml — 1 n=l The last condition, which arises from the condition that the equations for the second-order terms be consistent, may be regarded as a generalization of Slater's^^ result for the case .Y = 1. 4. Transjormalionfor Wave Guide Plus Horn The system to which we shall apply some of the preceding equations con- sists of a straight wave guide starting at x = — oc and running to .v = 0 where it is connected to a sectoral horn. The horn is flared in the (.v, y) plane only. The dimension of the system normal to the (.v, y) plane is constant and equal to a or i according to whether the electric or magnetic vector is in the plane of the horn. One might expect that the field in this system may also (in addition to our method) be determined by an alternating procedure of the type described by Poritsky and Blewett"' using the equations obtained by Barrow and Chu" for transmission in the horn. However, we shall not investigate this j)ossibility as we are primarily interested in using the system as an example to which we may apply the foregoing equations. If the total angle of the horn is lair, and if the sides of the straight guide are at y = 0 and y = b, (assuming the electric vector to be in the plane of the horn), the equation of the lower side, i.e., the continuation of the side y = 0, of the horn is y = — .v tan air and that of the upper side is y = ^ + .v tan ar. li z = X -\- iy and w = v -\- id then the Schwarz-Christofifel transformation 148 BELL SYSTEM TECHNICAL JOURNAL z = f(w) which carries the guide plus horn in the z plane into the straight guide with walls at 9 = 0, 6 = r in the w plane may be obtained from ^ = (1 -e'Tb/T (4-1) aw This gives, upon setting = |/'(i + id) p = [1 - le'" cos 26 + e'Tb'/T dz dw the relation (4-2) 1 -\- g(v^ ^) = [1 _ 2g2- cos 2d + ey from which the a„'s may be obtained in accordance with (1-2). 5. Expressions for the an s for Horn The Fourier coefficients of 1 + g{v, 6) appearing in (1-2) and (2-2) are the same. It may be shown from (4-2) that U"''F{-a,-a;\;e''') , v >0 1 + flo = jr(l -f 2a)/P(l -fa) , I- = 0 • (5-1) [F{-a,-a-\;e') , z;<0 and {^2e'"''-''\-a)rF{-a, r - a; r -^ V, e^''')/r\ , v>0 a^r = \ 2(-a).(l + ao),.=o/(l + a). , ^ = 0 (5-2) [2e'"{-a)rF{-a, r - a; r + 1; e'")/r[ , v <0 where the F's denote hypergeometric functions, r = 1, 2, • • • and we have used the notation (/3)o = 1, {0)r = i3(/^ + 1) • • • (/3 + r - 1) {S-?>) When n is odd, a,i = 0 because of symmetry about d = 7r/2. The expres- sions for i- > 0 in (5-1) and (5-2) may be verilied by expanding the two factors in 1 + g(iS d) = e {\ - e ) {I - e ) by the binominal theorem and picking out the terms containing e""^ When ^ < 0 we use the relation 1 + g{v, B) = e^^^l + g{-v, 6)], and when v = 0 we may sum the hypergeometric series. Differentiation of (5-1) and (5-2) leads to r4ae''""F(-«, 1 - a;l;e-'") , i' > 0 ^ (1 + a„) = 2a(l + ao).=o , Z) = 0 (5-4) ^' \^a-e''F{\ - cc,\ - cc\'^\e\ i'<0 RECTANGULAR WAVE GUIDES 149 / I6a'e'""(l - e-'y^'Ha, a; 1; e"'^), v > 0 . ., (1 + oo) = „ , , ., , , (5-5) ^^''- []6a'e"'(\ - e'T F(a, a; 1; e'"), r < 0 where in obtaining (5-5) use was made of Euler's transformation F(a, b;c; x) = (1 - xy-''-''F(c - a, c - b;c;x) It is seen that 0 F{-a, -a; l;e'-) + ae'''F{-a, 1 - a; 2; e*''), V < 0, (5-8) 6. Approximation to Reflection Coefficient of Horn, FJcctric Vector in (.v, y) Plane When the flare angle 2a:7r of the horn is very small the reflection coefficient may be shown to be R, la 2k + 0(a2) (6-1; 130 BELL SYSTEM TECHNICAL JOURNAL where 0(a-) denotes correction terms of the order a-. This result is based upon the fact that when terms of order a' are neglected the set of dif- ferential equations (1-4) and (1-5) reduce to the single equation Fo + (1 + a,)k''Fo = 0 where, from (5-1, 4, 5), 1 + a. V > 0 V 0 k = K-(4a e^" - 2ae--'') (7-4) A = K.-{\()a-e^'- + Aae--') K = /c-(l + ae-') - 1 = c- + K-ae-' , z' < 0 A - laK-e-" (7-5) A = iaK-e-" where the dots denote differentiation with respect to v and c- = k- — 1. We have retained the a'- in A as given by (7-4) because at this stage we do not know whether it may be neglected or not. When V < 0, the dehnition (3-5) of ^ and (7-5) yield ^ = icv -\- i i (K' — c) dv (7-6) = icv + ic I [(1 + K c"' ae'")' — IJ dv = icv + 0(a) J— 00 152 BELL SYSTEM TECHNICAL JOURNAL and we have J— 00 •/— 00 (7-7) O(a^) which may be neglected. The other integral suggested by (3-14) is -2J r— 3/2 ;•- , / -2icv -3 , 2 2r , e K K av — I e c 4a:K e dv ■00 *'— 00 = 2aK c~ /(I — ic) (7-8) When V > 0, (7-9) ^ = e.=o + i f [Kie'"' -I- ae--'") - 1]' dv Jo = i f {K-e'"" - \f dv + {){(x), Jo — -— [x — tan~ X — c -\- tan" c] + 0(a), za .-v = (k e "" — 1)% 2q: -3 2, , (7-11) -f- e c K ia dv Jo ^_,(,_tan l.-o+tan-lc]/a_^._,^^. _^ 2aK'c~'/il + ic) = Oia) + 2aK-'rV(l + ir). That the integral having x as the variable of integration is 0(pr) may be shown as in (7-10). When we combine our results in accordance with (3-14) we obtain 2l J-oo dv _OCK C ^ r 1 1 4i \_1 — ic 1 + ic ia/(2c') + Q(a) + O(a^) (7-12) which is (7-1). If, instead of discarding (7-10) because it is 0(a-), we retain it and the corresponding integral in (7-11) (in the hope that they represent most of the difference between the approximate value (7-1) for Rh and the true value) we obtain the approximation i?/? = 2~3 "" T / ^ (:>x + .T ) dx (/-13) in which the integral may be evaluated by numerical integration. The approximations (6-1) and (7-1) for the reflection coefficients may also be obtained from an equation given by N. H. Frank. ''^ However, care must be taken to suitably define the wave guide characteristic impedance which appears in his expression. 8. Speculation on the Reflection Obtained from Horn blared in Both Directions All the work from Section 4 onward applies only to a horn tlared in one plane. Nevertheless, it is interesting to speculate on how close an estimate of the reflection from a three-dimensional horn may be obtained by super- posing the two reflection coefficients (6-1) and (7-1). It must be kept in mind that the flare angles (the a's) may be different in the two directions, 154 BELL SYSTEM TECHNICAL JOURNAL that k is given by (1-2) and c by (2-2), and finally the difference (not the sum) of Re and Rh must be taken. In (6-1) Re is the retlection coetBcient of the component of the magnetic vector normal to the (.v, y) plane (which is proportional to Q), while in (7-1) Rh is the reflection coefficient of the transverse electric vector (which is proportional to P) and there is a difference in sign just as in the case of voltage and current reflection coefficients. If a > b and Xo is the wavelength in free space, the superposition gives the following expression for the reflection coefficient of the electric vector: R = Rh — Re (8-1) - '- [(2a/\of - \r(ocH/[{2a/\,f - 1] - aa^/&) where liran and Itvue are the total horn angles in the planes of H and E, respectively. Of course this approximation can be expected to hold only when uh and a^ are small. 9. Numerical Calculations — Rh for 60° Horn The value of Rh , the reflection coefficient when the magnetic vector lies in the plane of the flare, was computed on the assumption that only the dominant mode need be considered.* Thus, instead of the system of equations (2-4) and (2-5), only their simplified version, namely the single second order differential equation (7-2), was used. This equation may be written as ^^ + TiTFi = 0 (9-1) where, according to (5-6), K - -//2 = ,-2(1 j^ a^_ a.,/2) - \ (9-2) 1 + ao - 02/2 = (e-" + l)-«F(-a, 1/2; 2; sech- i')- The problem was to obtain the Rh appearing in that solution Fi of (9-1) which satisfies the boundary conditions (2-6). No computations for Re were made. In the first method of calculation the integrals in the approximation (3-14), namely R„= 1 f\-^A-^'^f,A'-''^J. (9-3) 2/ J-oo dv- 2^ = 2icv + 2i / {K^ - c) dv, * I am indebted to Miss M. Darvillc for carrying out the computations of this section. RECTANGULAR W AVE Gi IDES \55 were evaluated by Simpson's rule. The second derivative of A'"""* was computed from the even order central (Hflferences of A' . For a = 1/6, corresponding^ to an anj^le of tt 3 Ijetween tlie two sides of the horn, calcula- tions at two representative wave lengths led to the table (9-4) Xo c K- = l + C- R„ 1.549a .8173 1.6680 -.0420 + /.0724 1.61()a .7376 1.5441 -.0551 + /.0878 An idea of the variation of A may be ol^tained from its values at — ^, — .6, 0, .6, 1.8, 3.6 which are approximately .67, .76, .98, 1.62, 4.56, 17.4, respectively. The range of integration was — 3 < r < 4.4. The second method of computation used the formulas (3-11) with Fi playing the role of y. The differential equation (9-1) was integrated by the Kutte-Runge method, the interval between successive values of z- being 0.2. For c = .8173 the values obtained were vx V2 y T Rh (9-5) -.6 .6 -.202 - /.981 -.142 - /.794 -.0167 + /.0658 -1.2 1.2 -.218 - /1.004 -.049 - /.696 -.0525 + /.0754 -1.8 1.8 -.225 - /.989 +.086 - /.716 -.0512 + /.0753 -2.4 2.4 -.220 - /1. 000 .136 - /.842 -.0424 + /.0722 In order to gain an idea of the meaning of these values of v it should be recalled that ic — v + id and the walls of the guide are at ^ = 0 and 9=7:. An interval of length tt = 3.14 ■• • in the v direction therefore corresponds roughly to a distance equal to the width of the guide. The above table indicates that, loosely speaking, most of the reflection occurs close to the junction of the horn and wave guide. The last value of Rh in (9-5) agrees quite well with the value —.0420 + /.0724 obtained from the approximate expression (9-3). It appears that the method leading to (9-5) is superior to the one based on (9-3) since, in theory, it may be made as accurate (insofar as the single equation (9-1) may replace the set of equations (2-4, 5)) as desired. Moreover, less actual work seems to be required. The approximation (7-1) yields, for c = .8173, ^"~ 2? ~ 2X8173? " '•^''^ which is considerably in error, as we might expect, since a = 1 6 is not small. However, if we use the a{)i)roximation (7-13) and evaluate the integral by Simpson's rule we obtain 156 BELL SYSTEM TECHNICAL JOURNAL R„ = /.153 - (.061 + 1.077) = -.061 + /.076 which is in better agreement with the earlier values of Rh . No similar computations have been made to test the corresponding approximation for Re obtained when the correction term (6-5) is added to the leading term in (6-1). However, it appears that for a = 1/6 and the representative value k = .38, (6-5) is only about one sixth as large as ia/(2k) and hence is relatively unimportant. References 1. S. 0. Rice, Reflection from Corners in Rectangular Wave Guides — Conformal Trans- formation, this issue of the B. S. T. J. 2. S. A. Schelkunoff, Solution of Linear and Slightly Non-linear Differential Equations, Quart. Jl. Appl. Math., 3, 348-360 (1946). 3. P. S. Epstein, Reflection of Waves in an Inhomogeneous Absorbing Medium, Proc. Nat'l. Acad. Set., 16, 627-637 (1930). 4. K. Rawer, Electrische Wellen in einem geschichteten Medium, Ann. der Physik, 35, 385-416 (1939). 5. L. R. WaUicr and N. Wax, Non-Uniform Transmission Lines and Reflection Coeffi- cients, //. Appl. Phys., 17, 1043-1045 (1946). 6. R. E. Langer, The Asymptotic Solutions of Ordinary Linear Differential Equations of the Second Order, with Special Reference to the Stokes Phenomenon, Bull. Amer. Math. Soc, -10, 545-582 (1934). Many references to the earlier work are given. 7. R. E. Langer, On the Connection Formulas and the Solutions of the Wave Equation, Phys. Rev., 51, 669-676 (1937). 8. R. E. Langer, The Boundarv Problem of an Ordinary Linear Differential Sj'stem in the Complex Domain, Trans. Am. Math. Soc, 46, 151-190 (1939). 9. E. C. Titchmarsh, Eigen-function Transformations, Oxford (1946). 10. H. Jeffreys, On Certain Approximate Solutions of Linear Differential Equations of the Second Order, Proc. London Math. Soc, (2), 23, 428-436 (1924). 11. W. H. Furry, Two Notes on Phase Integral Methods, Phys. Rev., 71, 360-371 (1947). 12. G. D. Birkhoff and R. E. Langer, The Boundary Problems and Developments Asso- ciated with a System of Ordinary Linear Differential Equations of the First Order, Proc Amer. Acad. Arts and Sci.,'58, 49-128 (1923). 13. H. E. Newell, The Asymptotic Forms of the Solutions of an Ordinary Linear Matric Differential Equation in the Complex Domain, Duke Math. JL, 9, 245-258 (1942) and 10, 705-709 (1943). 14. See, for example, Frank and von Mises, Differential und Integralgleichungen, Vol. 1, page 56, Braunschweig (1930). 15. J. C. Slater, Microwave Transmission, page 74, McGraw-Hill (1942). 16. Poritsky and Blewett, A Method of Solution of Field Problems bv Means of Over- lapping Regions, Quart. Jl. Appl. Math. 3, 336-347 (1946). 17. Barrow and Chu, Theory of the Electromagnetic Horn, Proc. I.R.E., 27, 51-64 (1939). 18. N. H. Frank, Reflections from Sections of Tapered Transmission Lines and Wave Guides, Radiation Lab. Rep. 43-17, Jan. 6, 1943. 19. Whittaker and Robinson, Calculus of Observations, page 64, Blackie (1926). Abstracts of Technical Articles by Bell System Authors Pulse Echo Measurements on Telephone and Television Facilities} L. (i. Abraham, A. W. Lebert, J. B. Maggio, and J. T. Schott. Pulse echo measurements have been used on telephone and television facilities since 194() to locate impedance irregularities and control quality in manufacture and installation. These sets send a pulse into a line and observe on an oscilloscope the echoes returned from irregularities. The shape and width of the pulse, the rate at which it is repeated and the pulse magnitude are important in determining the accuracy of the results and the requirements of the measuring apparatus. The "coaxial pulse echo set" is used for factory and field testing of coaxial cables. The "Lookator" was developed for use on much narrower band systems such as spiral-four field cable and open wire lines. Television Neticork Facilities.- L. G. Abraham and H. I. Romxes. This paper describes television network facilities which are needed to connect studios and other pickup points to transmitters in the same and in distant cities, and discusses their transmission characteristics. Short-haul tele- vision circuits may be by microwave radio or over wire circuits. Long- haul television connections may be by radio relay or over coaxial systems of the type originally developed for carrier telephone circuits. Transmission requirements include adequate frequency band, accurate gain and phase equalization, and freedom from interference resulting from excessive noise, crosstalk, or modulation. Radio and wire systems are under development to provide extensive high-quality television networks. A Carrier Telephone System for Rural Service.^ J. M. Barstow. The Ml carrier telephone system was designed for the purpose of extending tele- phone service into areas served by rural power lines, but not served by co- existing rural telephone lines. To the local office operator and to a carrier subscriber the service provided is the same, so far as procedures involved in establishing a connection are concerned, as a voice-frequency line. At the office end of the system a telephone wire line extends from the office to a point near the power line. Here is located a converter (called common terminal) which converts the voice-frequency signal to be trans- mitted to the subscriber to an amplitude-modulated double-sideband carrier signal. This signal is coupled to the power line through a coupling unit 1 Trans., A. I.E. E., vol. 66, 1947 (pp. 541-548). "^ Transactions, A. I. E. E., vol. 66, 1947 (pp. 459-464). ' Trans. A. I.E. E., vol. 66, 1947 (pp. 501-507). 157 158 BELL SYSTEM TECHNICAL JOURNAL and high-voltage capacitor. At the subscriber's location the signal is taken off by similar means and led by separate wires to the subscriber premises, where it is reconverted to voice frequency by means of a subscriber terminal. A signal transmitted from the subscriber to the central office goes through similar conversions. The usual number of parties per two-way channel may be assigned ac- cording to local custom, and divided-code or full-code ringing is provided. Equipment is available for live two-way channels over a single-power line employing frequencies in the range 150 to 425 kilocycles. A sixth channel has been discontinued because of radio interference. A description is given of the manner in which the power line should be treated in order to reduce reflection effects. The power line treatment does not affect its capabilities in regard to power transmission. Application of Rural Carrier Telephone System} E. H. B. Bartelink, L. E. Cook, F. A. Cowan,* and G. R. Messmer. This paper deals with the application of a carrier system developed primarily for providing rural telephone service over power distribution circuits in areas where this means of extending telephone service may be more attractive than other avail- able methods. The modifications required in the power circuits to permit carrier frequency transmission are described, including the effect of these modifications on the operation of the power system. Construction fea- tures also are discussed. The use of the rural carrier telephone system over open wire telephone pairs is discussed briefly. An Improved Cable Carrier System} H. S. Black, F. A. Brooks, A. J. WiER and I. G. Wilson. A new 12-channel cable carrier system is described which is suitable for transcontinental communications. Important fea- tures are negative feedback amplifiers of improved design, new arrangements for accurate equalization of the cable loss, and automatic thermistor regu- lators which continuously control the transmission of each system. Joint Use of Pole Lines for Rural Power and Telephone Services.^ J. W. Campbell,* L. W. Hill, L. M. Moore, and H. J. Scholz. The use of poles to carry both power and communication circuits is not new, having been employed before 1890. There are today more than 6,000,000 poles used jointly by power and telephone organizations in the United States. The great bulk of these poles are located in urban areas where the voltages of the power circuits concerned are generally less than 5,000 volts and the span lengths between poles generally do not exceed about 150 feet. As power and telephone lines were extended into rural territory, new ' Trans. A. /. E. E., vol. 66, 1947 (pi). 511-517). '' Trans. A.L E. E., vol. 66, 1947 (pp. 741-746). 0 Trans. A. /. E. E., vol. 66, 1947 (pp. 519-524). * Of Bell Tel. Labs. ABSTRACTS OF TECHMCAL ARTICLES 159 j)r()l)lems were encountered in the application of joint construction because of the use of longer spans and higher voltages for the power circuits and the increased noise induction in the necessarily longer exposures. However, progress in the art through cooperation of the telei)hone industry with the Edison Electric Institute and the Rural Electrification Administration has brought about the develoi)ments reviewed in this paper which now make long span higher voltage rural joint use feasible where conditions are favor- able. Atomic Energy.' Karl K. Darrow. {The 1947 Xorniaii Wail Harris Lectures at Xorthiicstern L'niversity.) This little book, which reproduces four lectures substantially as they were given, is at once a very readable and a very accurate account of enough of the facts of nuclear physics to convey a good understanding of the atomic bomb and the possibilities of atomic power. The scientitic accuracy of the presentation is instanced by the author's apologies for his title; he emphasizes that in reality his subject is unclear energy, but that on the day of Hiroshima somebody wrote of an atomic bomb and the misusage spread like a chain reaction. The role of electrons, protons and neutrons in atom building is told in a simple and entertaining style (but with a degree of ornamentation that may disturb some readers), and the discussion of rest mass and the Einstein rela- tion between mass and energy is pointed up by well-chosen numerical il- lustrations beginning with the lightest composite nuclei. The role of fast- particle bombardment in increasing and decreasing the size of nuclei is also explained. The reader thus acquires a clear understanding of the basic phenomena for which nuclear fission is famed. The text is augmented by well-chosen cloud chamber photograj)hs. Though the author's treatment is accurate, his style and marshalling of facts are very readable. This is well illustrated by the closing paragraph of the third lecture, which follows immediately upon the author's develop- ment of the idea of the chain reaction: Here is the climax of my lectures, and here is where you should be frightened; and' if I had an orchestral accompaniment, here is where the orchestra would have mounted to a tumultuous fortissimo, with the drums rolling and the trumpets blaring and the tuba groaning and the strings in a frenzy, and whatever else a Richard Wagner could contrive to cause a sense of Gotterdammerung; for, let there be no doul)l of it, this is something that could bring on the twilight of civilization. Hut at this crucial junc- ture I have only words to serve me, and all the words are spoiled. We speak of an awful headache, a dreadful cold, a frightful bore, and an api)alling storm; and now when something comes along that is really awful and dreadful and frightful and ajjpalling, all these words have been devaluated and have no terror in them. I have to fall back on the saying, of unknown origin and dul)ious value, that the strongest emphasis is understatement. Let then this picture, with its circles and its symbols and its numbers, be considered an emphatic understatement of the most terrific thing yet known to man. ' Publishedbv John Wiley & Sons,Inc., New York, and Chapman & Hall, Ltd., London. 80 pages. $2.00. 160 BELL SYSTEM TECHNICAL JOURNAL The book will be welcomed particularly by those who at one time or another have had a general acquaintance with radio-activity, cosmic rays and the results of cloud chamber research, but whose vocational activities have forced such special knowledge well into the nebulous regions of their memories. Xehcork Theory Comes of Age. ^ R. L. Dietzold. The third decade in the growth of modern network theory, the decade of maturity, is considered in this review of the advances in network theory evolved over the past ten years. New types of networks developed during the war are included. Thermistors as Components Open Product Design Horizons^ K. P. DowELL. These thermally sensitive resistors with high negative tempera- ture coefficients have come a long way since they were laboratory curiosities and are now available in a wide range of types with diverse and stable char- acteristics. You may be able to transfer to your own problems some of the unusual design ideas described here. Gas Pressure for Telephone Cables}^ R. C. Giese. Communication cables consist of a number of electric conductors insulated from one another and encased in a metal sheath. This encasement is subjected to numerous hazards, such as those caused by electrolysis, crystallization, various kinds of mechanical damage, and lightning burns. Any damage to the sheath which will permit water to enter the cable will decrease the effectiveness of the insulation material and thus cause an impairment or an interruption to the service. The entrance of moisture through small openings in the sheath can be materially retarded when the space inside the cable, not occupied by the conductors or insulation, is filled with a gas maintained under controlled conditions. Nitrogen is the gas usually used for this purpose because it is inert and does not combine chemically with the conductors or insulation. In addition the use of the gas provides a method of locating openings in the sheath by means of a pressure gradient, which is a material aid in cable main- tenance. Rural Radiotelephone Experiment at Cheyenne Wells, Colo}'^ J. Harold Moore, Paul K. Seyler and S. B. Wright. The first rural party-line telephone service by radio installations operating on the subscribers' prem- ises was inaugurated August 20, 1946. This paper describes the equipment used, how it operates, and the results obtained during the preliminary testing and the initial period of regular operation. Radio is one of several new methods which the Bell System is exploring in its program for extension of telephone service in rural areas. It is expected that experience gained in 8 Electrical Engineering, September 1948 (pp. 895-899). ' Elec'l. Mfg., August 1948 (pp. 84-91). 10 Transactions, A. I. E. E., vol. 66, 1947 (pp. 471-478). '1 Trans. A. I.E. E., vol. 66, 1947 (pp. 525-528). ABSTRACTS OF TECHNICAL ARTICLES 161 this experiment will aid in developing a standard rural radiotelephone system. Efect of Passive Modes in Traveling-Wave Tubes}- J. R. I'ikrce. As the beam current in a traveling-wave tube is increased, the local fields due to the bunched beam become appreciable compared with the llelds propa- gating along the circuit. The eflfect is to reduce gain, to increase the electron speed for optimum gain, to introduce a lower limit to the range of electron speeds for which gain is obtained, and to change the initial loss. New Test Equipment and Testing Methods for Cable Carrier Systems^ W. H. TiDD, S. Rosen and H. A. Wexk. Three portable test sets developed for the improved cable carrier telephone system are described : A high sen- sitivity selective transmission measuring set covering 10 to 150 kc, a decade oscillator for frequencies from 2 to 79 kc, and a tube test set. .4 New Microwave Television System}* J. F. Wentz and K. D. Smith. A microwave point-to-point radio system is described which is designed for the transmission of television programs. This system is intended to sup- plement wire facilities for local distribution of television signals from pickup points to studios or from studios to broadcast transmitters and to long dis- tance network terminals. The circuits and equipment are described in detail. Performance obtained in tests during 1946 is given. 12 Proc. I. R. E., August 1948 (pp. 993-997). ■3 Trans. A.I. E. E., vol. 66, 1947 (pp. 726-730). '* Transactions, A. I.E. E., vol. 66, 1947 (pp. 465-470). Contributors to this Issue W. J. Albersheim, Technical Colleges, Aachen and Munich, Germany; B.Sc, Aachen, 1920; E.E., Aachen, 1922; Doctor of Engineering, Aachen, 1924; Professional Engineer, University of State of New York, 1937. Elec- trical Research Products, Inc., 1929-41; Bell Telephone Laboratories, 1941-. Dr. Albersheim was concerned with radar and jamming work during the war; he is now engaged in broadband FM transmission problems. W. B. Hebenstreit, A. A., University of Chicago, 1935; B.S., Cahfornia Institute of Technology, 1941. Bell Telephone Laboratories, 1941-47. While at the Bell Telephone Laboratories Mr. Hebenstreit worked on var- ious microwave vacuum tubes including magnetrons, traveling wave tubes and the new double stream amplifier. He is now with the Hughes Aircraft Company. A. V. HoLLENBERG, A.B., Willamette University, 1931 ; M.S., 1933, Ph.D., 1938, New York University; Instructor in Physics, Queens College, 1938-42. Columbia Radiation Laboratory, 1942-45; Bell Telephone Laboratories, 1946-. Dr. Hollenberg was engaged in research and development work on microwave magnetrons at the Columbia Radiation Laboratory. At Bell Telephone Laboratories he has been concerned with traveling wave ampli- fiers. J. R. Pierce, B.S. in Electrical Engineering, California Institute of Tech- nology, 1933; Ph.D., 1936. Bell Telephone Laboratories, 1936-. Engaged in study of vacuum tubes. S. (). Rice, B.S. in Electrical Engineering, Oregon State College, 1929; C^difornia Institute of Technology, 1929-30, 1934-35. Bell Telephone Laboratories, 1930-. Mr. Rice has been concerned with various theoretical investigations relating to telephone transmission theory. Slo.^x D. Robertson, B.E.E., University of Dayton, 1936; M.Sc, Ohio State University, 1938, Ph.D., 1941; Instructor of Electrical Engineering, University of Dayton, 1940. Bell Telephone Laboratories, 1940-. Dr. Robertson was engaged in microwave radar work in the Radio Research Department during tiie war. He is now engaged in fundamental micro- wave radio research. 162 COM'KIBLWKS TO THIS ISSI'E 16.i Ci.AUDK E. Shannon, R.S. in Electrical Eiif^inccrin^, Iniversity of Michi- gan, 1936;S.M. in Electrical Engineering and Ph.D. in Mathematics, M.ET., 194(). National Research Eellow, 194(). Bell Telephone Laboratories, 1941-. Dr. Shannon has been engaged in mathematical research princi- pally in the use of Boolean Algebra in switching, the theory of communi- cation, and cryptography. VOLUME XXVIII APRIL, 1949 NO. 2 THE BELL SYSTEM TECHNICAL JOURNAL Ol 'It DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION A Carrier System for 8000-Cycle Program Transmission R. A. Leconte, D. B. Penick, C. W. Schramm, A. J. Wier 165 Delay Equalization of Eight-Kilocycle Carrier Program Circuits C. H. Dagnall and P. W. Rounds 181 Band Pass Filter, Band Elimination Filter and Phase Simu- lating Network for Carrier Program Systems F. S. Farkas, F. J. Hallenbeck, F. E. Stehlik 196 A Precise Direct Reading Phase and Transmission Measur- ing System for Video Frequencies D. A. Alsberg and D. Leed 221 Physical Principles Involved in Transistor Action J. Bardeen and W. H. Brattain 239 Lightning Current Observations in Buried Cable H. M. Trueblood and E. D. Sunde 278 The Electrostatic Field in Vacuum Tubes with Arbitrarily Spaced Elements W.R. Bennett and L. C. Peterson 303 Transconductance as a Criterion of Electron Tube Performance T. Slonczewski 315 Abstracts of Technical Articles by Bell System Authors .... 329 Contributors to this Issue 332 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 504 per copy S^-^O P^r ^^^^ V THE BELL SYSTEM TECHNICAL JOURNAL Published quarterly by the American Telephone and Telegraph Company 195 Broadway, New York, N. Y. EDITORS R. W. King J. O. Perrine EDITORLA.L BOARD C. F. Craig O. E. Buckley O. B. BlackweU M. J. KeUy H. S. Osborne A. B. Clark J. J. PiUiod F. J. Feely »■»«»«»««■» SUBSCRIPTIONS Subscriptions are accepted at $1.50 per year. Single copies are 50 cents each. The foreign postage is 35 cents per year or 9 cents per copy. ■■■■■■■■II Copyright, 1949 American Telephone and Telegraph Company PRINTED IN UiS.Ai The Bell System Technical Journal Vol.XXVIIl April, 1949 No, 2 A Carrier System for 8000-Cycle Program Transmission By R. A. LECONTE, D. B. PENICK. C. W. SCHRAMM, A.J. WIER With the raj)id expansion of hroad-band carrier lelc|)hone s_vslems throughout the country, the use of these facilities for program transmission has become desirable. This paper describes a carrier program system capable of trans- mitting a band up to about 8000 cycles wide. Introductioni T7R0M the beginning of radio the Bell System has supplied the broad- -^ casting industry the needed interconnecting links between broadcasting stations, studios, and other program originating points. For many years these facilities have been provided at audio frequency over loaded cable pairs, or over open-wire lines. Because present growth of message facil- ities over main traffic routes is predominantly in broad-band carrier tele- phone circuits, it has become desirable to adapt these new carrier facilities for the transmission of high-quality program material. The carrier program system to be described operates in conjunction with message circuits and can be used to provide a band width of either 5000 or 8000 cycles. It can be applied to type K multipair cable,^ type L coaxial cable, and type J open-wire carrier systems.^^ Use of the 8000-cycle band of course requires more complete equalization than the 5000-cycle band, and requires the frequency space normally occupied by three message channels. It is e.xpected that the 5000-cycle band can be accommodated by displacing two message channels. The carrier program system was developed by 1942 but, owing to the war, its first commercial application was not made until early in 19-46 on the transcontinental type K route west of Omaha. It is now in use in all sections of the country, particularly the west and south, on type L as well as type K s}'stcms and has been successfully tested on type J. In general, a band width of 5000 cycles is used in these applications. Objectives Existing audio-frequency program circuits may be as long as 7000 miles, may have 100 or more dropj^ing or bridging i)oints, any one of which may occasionally transmit to all of the others, and may be arranged for auto- matic reversal of the direction of transmission by means of a control signal. 165 166 BELL SYSTEM TECHNICA L JOURNA L In order to coordinate with these existing circuits and studio loops, a carrier program system must be capable of duj)licating this flexibility while maintaining the desired standards of quality of transmission. In setting an objective for the standards of transmission quality of this new system the trend towards wider band widths has been recognized. Most of the major networks now use a 100 to 5000-cycle band width. A large part of the present audio-frequency cable facilities, however, can be arranged to transmit a band from 50 to 8000 cycles. It was decided to match this grade of transmission in the design of the new carrier system. For the cases where still higher quality is desired, a 15-kilocycle carrier program system has been developed and is now available. Design Features The 12-channel bank of message circuits forms the basic building block of the broad-band carrier telephone systems. In the channel bank, each of the 12 voice-frequency channels modulates one of 12 carriers spaced 4 kilocycles apart from 64 to 108 kilocycles. The lower sideband resulting from each modulation is selected by a band filter and combined with the other 11 lower sidebands to give a channel group occupying the frequency space from 60 kilocycles to 108 kilocycles. This channel group is then further modulated as a unit to its appropriate place on a broad-band spec- trum for transmission over the line. In order to arrange a channel bank for program transmission, message channels, 6, 7, and 8 are disabled, clearing a space from 76 kilocycles to 88 kilocycles in the group-frequency spectrum. In a program terminal sepa- rate from the channel bank, an audio frequency program modulates an 88-kilocycle carrier derived from the message channel carrier supply. Its lower sideband is selected by a band filter and, combined with the lower sidebands of message channels 1 to 5 and 9 to 12, gives a group-frequency spectrum shown diagrammatically in Fig. 1. This figure also shows the same spectrum after it has been modulated with a 120-kilocycle group carrier for transmission Over a type K line. Other line-frequency spectra are similarly produced in type J and type L group modulators. The reversing and control signal in an audio-frequency program circuit is a d-c. signal superimposed on the program pair. It may be applied at the studio which originates the program, and conditions all of the amphfiers along the line to transmit away from the originating studio. As long as the signal is applied, the direction of transmission is locked so that no other control station can inadvertently break the network. When the transmis- sion from this studio ends, the signal is removed, and the next originating point applies it. This efifects such reversals as are required for trans- mission and again locks all amplifiers. By this means it is possible to use CA RRIER SYSTE.\f FOR PROGRA M TRA .\S\f/SSIOX 167 a single pair of wires and one set of amplifiers for transmission in either direction as required. The carrier system over which the carrier program channel is transmitted is constantly in operation in both directions simul- taneously, and therefore requires no reversal. The program terminal itself, however, must be switched between transmitting and receiving lines if equipment is not to be needlessly duplicated for transmitting and re- ceiving. In any case, a control signal must be carried through the carrier circuit and delivered to connecting audio-frequency circuits at the receiving end as a d-c. signal. This is accomplished by means of a 78-kilocycle con- trol signal (42 kilocycles at K line frequencies) which is transmitted along PROGRAM CHANNEL BAND FILTER 79.6KC 88KC CHANNEL FREQUENCIES INPUT TO GROUP MODULATOR t^\^c°'" I lONc PILOTS 64KC 76KC 92 KC CHANNELS 12 3 4 5 PROGRAMI 9 10 11 12 ' """^^ FREQUENCIES I I r '^°" CABLE CARRIER OUTPUT TONE 11'-"^ GROUP MODULATOR PILOTS 12 KC 28 KC 42 KC 56KC60KC| BLOCKING FILTER AT BRANCHING POINTS |32KC 10 20 30 40 50 60 70 80 90 100 HO FREQUENCY IN KILOCYCLES PER SECOND Fig. 1 — Frequency allocation for one program channel and nine message channels in cable carrier systems. with the program channel outside of its frequency band. This signal is generated in the transmitting program terminal whenever the d-c. signal is impressed from the transmitting audio-frequency circuit. At the re- ceiving program terminal, the tone is converted into a d-c. signal which is impressed on the receiving voice-frequency facility. When there is no transmitted d-c. signal, there is no high-frequency signal and no received d-c. signal. Each program terminal, then, is ready either to receive d-c. from the voice circuit and send out 78 kilocycles to the carrier circuit or to receive 78 kilocycles from the carrier circuit and send out d-c. to the voice circuit. The program transmission path is maintained in the last estab- lished direction, regardless of the presence or absence of control, until a reversing signal is received. The arrangement of the circuit elements in a carrier program terminal is shown in the block schematic of Fig. 2. The transmission circuit wiring is 168 HKLI. SYSTEM TECHNICAL JOURNAL UJ < t— Otr cc - H LL CL O tt Ouj< O '-^^^ LL y UJ ct O - u < Lu q: uj o ff^2 1"A X^ -I LU < < <7 5 z£ H II << croLu OOtt mz ^ tr o c Q. < I ^^ I o=> <^ 5 fc ">/^ aanid qnvb 13nnvh3- asvssavN do svinvs oi 1 70 BELL SYSTEM TECH NIC A L JOURNA L shown in heavy lines. The reversing and control circuits, indicated in light lines, are permanently connected to the external audio-frequency circuit and to the transmitting and receiving carrier line circuits regardless of the condition of the switching relays. Figure 3 shows a carrier program system including two terminals and a branching point as it is connected to a type K system. The program equipment is identified by double-line blocks. The carrier program terminals are connected into the networks in the same way as the audio-frequency facilities, through equalizers, ampli- fiers, bridges, and reversing circuits. Connected as one leg of a reversible bridge, a carrier program circuit may feed or be fed by any of the other legs, which may include cable, open-wire, studio loop, or other carrier circuits. Terminal Circuit As Fig. 2 indicates, a carrier program terminal consists of three elements: a modulator-demodulator or modem, a demodulator amplifier, and a re- versing and control circuit. The heart of the terminal is the modem, which translates the program material from its original audio band to its desired position in the carrier-frequency spectrum or vice versa. It consists essen- tially of the non-linear varistor to which the carrier and program material are applied, and the band filter which selects the desired sideband from the modulation products. The varistor is connected in the double-balanced bridge arrangement in which the signal, carrier, and sideband circuits are each balanced against the other two. It is composed of copper-oxide ele- ments and, in order to meet the conflicting requirements for high carrier-to- signal ratio and low transmitted carrier leak, a high degree of balance be- tween the varistor bridge arms must be maintained. This is accomplished by building up each bridge arm of 16 copper-oxide elements connected in series-parallel. This modulator as compared to one using single-element bridge arms, has the same impedance, 12 decibels better carrier balance, 12 decibels greater carrier power capacity, and with the higher carrier power, 12 decibels lower non-linear distortion products. An amplifier provides the required power and a narrow-band filter gives additional suppression to carrier frequencies of other channels which are fed from the same carrier supply. The band filter, which represents a major development in itself and is described in another paper, introduces a considerable amount of delay distortion. This is corrected by delay equalizers incorporated in the modem circuit as shown in Fig, 2. Most of the delay correction is done in the audio- frequency branch of the circuit by a 31-section network which also includes equalization for the small residual attenuation distortion of the filter in its pass band. At the lower end of the audio-frequency band, however, attain- ment of the required phase characteristic with audio-frequency elements is CARRIER SYSTEM FOR PROGRAM TRAXSM ISSfO.X 171 more difficult. Consequently, the delay correction for that portion of the band below 1000 cycles is actually done at sideband frequency, using quartz crystal elements. The design of these delay equalizers is described in another paper. ^* Transmission through the resulting modem unit is essen- tially constant in both attenuation and delay over the usable frequency range. The demodulator amplifier is a conventional two-stage resistance-coupled amplitier. It is stabilized by 25 decibels of feedback to a nominal gain of 38 decibels, variable over a 12-decibel range by a potentiometer in the feed- back circuit. The transmission characteristic is flat within 0.3 decibel over the 35 to 15,000-cycle frequency range. The output impedance is stabilized by the use of an output bridge for obtaining the feedback voltage. This amplifier feeds a — 10 vu point in the circuit and can deliver up to +18 decibels above one milliwatt of output. Noise is kept to a minimum by operating the input stage vacuum tube at reduced voltages, mounting it and the magnetically shielded input transformer on a vibration-reducing sus- pension, and providing heavy iiltering for the A and B batter>' circuits. The limiting source of noise in any communication system is usually the transmission medium. In the carrier program system, the transmission medium is a carrier system which introduces noise energy equally dis- tributed over the program band. The program energy being transmitted, however, is heavily concentrated at the lower frequencies. In order to increase the signal-to-noise ratio without an increase in total transmitted power, a predistorting network is introduced ahead of the modem, which attenuates the lower frequencies relative to the higher. The total dis- crimination is about 18 decibels, distributed symmetrically on a logarith- mic frequency scale above and below 1500 cycles. A restoring network having an inverse characteristic is inserted in the receiving program path to return the program energy distribution to normal. The noise improvement thus obtained is about 7 decibels. The reversing circuit consists of a set of five relays and a 78-kilocycle amplifier-oscillator. Two of these relays, as shown in Fig. 4, set up the transmission circuits for transmitting or receiving. The transmitting relay connects the predistorter in the audio-frequency circuit and connects the modem output to the transmitting high-frequency line. The receiving relay connects the modem to the receiving high-frequency line and inserts the restorer and demodulator amplifier in place of the predistorter. These relays are interlocked so that only one at a time can be operated. Their operation is supervised by two other relays, one transmitting and one receiving, which respond to the transmitting and receiving control signals respectively. The supervisory relays are similarly interlocked so that the control signal from only one direction at a time can be eflfective. They are 172 BEI.L SYSTEM TECHNICAL JOV RN A L SO connected to the transmission relays that, when no control signal is applied, both supervisory relays are released and the transmission relays maintain the circuit condition established at the last reversal. A two-stage, tuned, feedback-stabilized amplifier is used to raise the level of the 78-kilocycle receiving control signal selected from the receiving high-frequency line by a narrow-band crystal filter. A copper-oxide recti- fier converts the amplified signal to d-c. to operate a sensitive relay connected TO AUDIO FREQUENCY PROGRAM CIRCUIT ■t— Hi PROGRAM ^_ TRANS- L_-~| MITTING FROM J, K OR L RECEIVING GROUP CIRCUIT — *■ TRANSMITTING — •- RECEIVING TO J, K OR L TRANSMITTING GROUP CIRCUIT Fig. 4 — Schematic of reversing and control circuit. to the receiving supervisory relay. The supervisory relay, besides control- ling the transmission circuits, also sends on a control signal as a d-c. simplex on the audio-frequency pair leaving the program terminal. The same 78-kilocycle amplifier used for receiving the control signal is also used as an oscillator to generate the high-frequency control signal in the transmitting direction. The transmitting supervisory relay, under the control of a d-c. control signal coming in on the audio-frequency pair, dis- connects the receiving control signal rectifier and connects instead a varis- LAKKIEK SYSTEM l()l< I'ROGKAM I k'.WS \f/SSf()\ 17.^ tor limiter across the out{)ul and a 78-kilocyfli' crystal from the oulj)Ut to the input, yihasod for positive feedback. TvPK K Bkanchinc; C'lkrurr liecause of the operatin = — ^„ + ;/a)„ perpendicular to the plane of the paper and that a unit positive charge is placed at an arbitrary point, CO, along the real frequency axis. The component of the force normal to the o) axis exerted on the unit charge may be written in the form 1 (3) Jn J , (w — Oin) When distances in equation (3) are identified with frequencies in equation (2), the two expressions are identical. A similar argument applies to the other zero, and also to the two poles provided that the filaments passing through the poles have charges of the opposite polarity. Thus we may say that the network of Fig. 2 will have a delay proportional to that component of the electric field intensity which is normal to the co axis, when a positive filament passes through each zero and a negative filament through each pole. Fig. 4 indicates the character of the delay as a function of frequency. Parenthetically we may note that the component of the field intensity parallel to the w axis is proportional to the derivative of the loss. Since this component is zero, the loss will be constant at all frequencies. In the case of the reactance networks with which we are dealing here, the loss is zero. Although the usefulness of the electrostatic analogy lies principally in its application to more complex networks, several conclusions may be drawn from Fig. 4. The right-hand zero and pole, because of their sym- metrical spacing and opposite charges, make equal contributions to the total delay. The same statement holds true for the left-hand zero and pole combination. As the zeros and poles approach the real-frequency axis, the delay peaks become sharper and higher because of the increased local field intensity. The figure also shows that the slope of the delay curve is zero at zero frequency and that, unless lOn is large compared to kn , the delay at zero frequency is of appreciable magnitude. These isolated facts will be exploited later in considering more complex networks. Assume, now, a tandem series of sections of the type shown in Fig. 2, in which the zeros and poles are so selected that they are evenly spaced at intervals, a, along straight lines parallel to the real-frequency axis as shown in Fig. 5. It was pointed out by H. W. Bode^ that the resulting field in- tensity may be approximated by distributing the total of the discrete charges on the plates of an equivalent condenser passing through the zeros and poles and extending a distance of a/2 beyond the extreme zeros and DEL.l y K - PLANE OOn -t*■a-^a-' yri -e e & © — — o o- 1T -ft H M X X- + P Fig. 5— Zeros and poles of a complex delay network liased on the condenser-plate design. resulting from the condenser charge is proportional to the angle subtended by the plates at that frecjuency. It is also proportional to the charge per unit length of plate or, in other words, to the density, 1/c, of the filament spacings. The geometry is illustrated in Fig. 6, where 2(r + d) is the 186 BELL SYSTEM TECHNICAL JOURNAL angle subtended by the plates at the frequency co. From this figure it may be seen that the field intensity in the region between the plates will have a fairly uniform value which falls off sharply as the edges are reached and becomes vanishingly small at frequencies remote from the plates. Along with this simple determination of the average delay characteristic, D. F. Tuttle in an unpublished memorandum has derived expressions for the magnitude of the delay ripple. As shown in the appendix, the field W - PLANE DELAY + P Fig. 6 — Delay-frequency characteristic of the network of Fig. 5. intensity or delay for an infinitely long set of charged filaments may be expressed in the form 27r lirkn 1 = — tanh a a cos Iwo} cos liro) cosh 2irkn + cosh 2irkn (4) For reasonably large values of 2irkn/a, this relation may be replaced by the approximate expression r = ro[l - 5 cos Toco] (5) where Tq = l-w/a is the average delay and 5 = 2^"^" " is the percentage ripple about the average value. The ratio ^„/a may thus be determined from the percentage delay ripple in accordance with the formula a Zir 0 (6) DELAY EQUALIZATION OF CARRIER CIRCUITS 187 To equalize the low-frequency and high-frequency filter delay shown in Fig. 1, a condenser plate of the form shown in Fig. 6 might be visualized. Although the high-frequency delay approximates that desired, the low-fre- quency delay shows insufficient shaping to be complementary to the filter characteristic because of the contribution of the negative-frequency plates. By bringing the plates closer to the frequency axis, that is by decreasing the ratio y^n/oji , a sharper-breaking low-frequency characteristic could be obtained. However, to achieve a sufficiently small delay ripple, the spacing, a, as determined from equation (6) would then have to be decreased with the result that the number of sections would be correspondingly increased. In attempting to reduce the total number of sections required, it was observed that a carrier-frequency delay equalizer would not be subject to AUDIO FREQUENCY - p CO- PLANE CARRIER FREQUENCY / AUDIO / FREQUENCY / ^ ' ^' ^BM^-^^^ CO + CO + p Fig. 7 — Condenser-plate design for a combined carrier and audio-frequency delay equalizer. the same low-frequency limitation, since the negative-frequency plates would be removed from the single-sideband signal by approximately twice the carrier frequency of 88 kc. However, since high-frequency delay sections are more expensive to construct than those operating at audio frequencies, a compromise is made in which the first few sections are built to operate at carrier frequencies and the remaining sections at audio frequencies. The equivalent condenser plates, referred to the audio- frequency signal, are shown in Fig. 7. A condenser-plate design has thus been achieved which allows the low- frequency and high-frequency delay to be equalized at least approximately. Further modifications must be made in the design, particularly in the middle of the band, to shape the characteristic so that a more accurate complement of the filter delay may be obtained. The delay in a condenser- 188 BELL SYSTEM TECHNICAL JOURNAL l)late design is directly j)roportional to the charge density along the plate. Up to now, this density has been assumed to be uniform. When the charge is located on discrete filaments, the restriction of uniform density is no longer necessary and it is possible to modify the delay characteristic as desired by changing the spacing of the filaments in inverse proportion to the desired change in delay. The assumption of a flat plate is also useful in simplifying the analysis; in the actual design the equivalent plate is bowed out over the major por- tion of the frequency range to reduce the delay ripple. The final zeros and poles obtained are shown in Fig. 8, in which the carrier-frequency zeros and poles are plotted on an equivalent audio-frequency basis. A total of 29 OOO OOOOOOO O O O O OOO OOOOoOOo< - p CO - PLANE o ooooooOOO°°° OOOOOOOOO OOOooC + p Fig. 8 — Plot of the zeros and poles of the delay equalizer for 8-k.c program terminals. delay sections are required, of which three are assigned to the carrier-fre- quency equalizer and 26 to the audio-frequency equalizer. Audio-Frequency Equ.alizer To complete the design of the audio-frequency equalizer some means must be found for absorbing the effects of dissipation in the coils and con- densers so that the final dissipative network will exhibit the theoretical non-dissipative performance plus a loss which is constant with frequency. It can be shown that a non-dissipative all-pass section plus a flat-loss pad can be replaced with a dissipative all-pass section (of modified constants) in tandem with a minimum-phase loss equalizer as in Fig. 9. It would be uneconomical to associate a loss equalizer with every phase section; and it is in fact unnecessary, since any minimum-phase device accomplishing the same result will exhibit the same performance.' The problem is then re- duced to equalizing the loss of the network composed of dissipative dela}- sections. DEI.. 1 1 ■ l-\)l A I.I /.A I'lOX or ( \ I KRIEN ( 7 A'( 7 ' / 7-.V 189 The dissij)ative loss of these sections may he determined from the approxi- mate relation Dissipative Loss in nc[)ers = 36 + 9 (7) wiiere R 1 '' G C resistanre-inchutance ratio of coils in ohms per henry = conductance-capacitance ratio of condensers in micromhos per microfarad T = delay of network in seconds FLAT -LOSS PAD NON-DISSIPATIVE DELAY SECTION DISSIPATIVE DELAY SECTION MINIMUM - PHASE LOSS EQUALIZER Fig. 9 — Four-terminal equivalence showing the method of absorliing the etTecls of dissipation in the audio-frequency equalizer sections. This expression indicates that, when the quantity {R/L + G/C) is nearly constant with frequency, the shape of the loss characteristic will be gen- erally similar to that of the delay characteristic. The ripples in the delay characteristic have been made sufficiently small so that the corresponding loss ripples may be ignored and only the general trend considered. A schematic of the resulting equalizer is shown in Fig. 10. The attenuation equalizer sections, in tandem with the delay sections, produce a loss char- acteristic complementary to that of the band filter over the cSOt)()-cycle program range. Resistors have been added to the crossarms of each lattice delay section to allow the dissii)ative losses to be adjusted to the nominal values assumed in the design. For manufacturing convenience, the sec- tions are assembled in seven separate containers which are mounted on an 8f inch by 19 inch relay-rack panel as illustrated in Fig. 11. 190 BELL SYSTEM TECHNICAL JOURNAL ■MSu — I •ATTENUATION SECTIONS -DELAY SECTIOf.'S- Fig. 10 — Schematic of audio-frequency equalizer. Fig. 11 — Photograph of audio-frequency equalizer. Carrier-Frequency Equalizer The critical frequencies of the carrier-frequency equaUzer are located at 318, 610 and 890 cycles on an audio basis. Since the carrier is at 88 kc DKLA Y EQL'A LIZA TION OF CARRIER CIRCUITS 191 and the lower sideband is transmitted, the corresponding carrier frequencies are 87682, 87390 and 87110 cycles, respectively. The required change of phase per cycle is the same as at audio frequencies, but the percentage rate of change is eleven times that of the audio-frequency sections operating at 8(X)0 cycles. This requires that the arms of the sec- tions have proportionately stiffer reactances, higher Q's, and greater tem- Ll n r^lHP — IF- 1-2 C2 '-rrn^ \^ , ,C3 L4 o — ^wr^ L6 c (a) (b) (c) Fig. 12 — Schematic of the lattice equivalent of three tandem sections of the type shown in Fig. 2. (0) (b) Rb-Ra (^) zl (C) Rb Rb-Ra — VW — \Rb-R'stals Yl to Y4, inclusive, and the repeating coils Tl and T2. The fixed condensers and resistances are mounted above the panel. Results Curves A and B in l*'ig. 16 show the delay-frequency characteristics of the audio-frequency equalizer and the carrier-frequency equalizer, respec- tively. Curve C shows the equalized delay of one terminal, which is the sum of the delays of the equalizers added to the unequalized delay of Fig. 1 . 194 BELL SYSTEM TECHNICAL JOURNAL Listening tests over ten carrier links in tandem indicate that the design objectives are sound and that a satisfactory reduction in delay distortion has been achieved. b 5 w W^^ c ^^ ^ 4 7 O o UJ A ^^-^ ■ "^ _I /^ ^^^^^.^ \ 3 1 \ s / \ z B / \ V r^ / < I \ / 2 UJ / ' o 1 J I V V 8 FREQUENCY IN KC Fig. 16 — Delay of audio and carrier-frequency equalizers and delay of equalized pro- gram terminal. APPENDIX For an infinitely long set of charged filaments of the type shown in Fig. 3 and located at co = c/2, 3a/2, Sa/2, etc., the insertion loss and phase may be expressed by the infinite-product expansion of equation (1), [p^- K- j(n - l)a][p + K+ j{n - ^)a] A+i '-U n-1 [p 1 - = n kn - j{n - \)a\[p - K-\- Jin i2(p + kn)Tr/a (In - \)jr ^)a] 1 - i2{p — kn)Tr/a (8) (2n - l)7r Expression (8) is a standard form of product expansion and may be written A+jB ^ cos j(p + k„)ir/a cos j{p — kn)ir/a (9) or A-\-jB = logcosi(/> + ^Jr/a — logcosy(/> — *„)7r/a (10) DELA Y EQUALIZA TION OF CARRIER CIRCUITS 195 Substituting ju for p and differentiating with respect to co, we obtain dA , .dB . 2ir sinh 27r^„/a ,..^ -4- f =: 1 V 1 1 / do3 d(j) a cosh lirkn/a -\- cos 2iroi/a from which dA/dia is zero. Equation (11) may be written dB 27r ,.,„,, V / 1 \ = — ■ (tanh l-wkn/a) doi a " I 1 I ^®s 2Tro}/a I (12) \ cosh 27r^„/c/ which, when expanded, gives equation (4). References 1. "Wave Transmission Network," H. W. Bode, United States patent 2,342, 638. 2. "Network Analysis and Feedback Amplitier Design" (l)ook) Chapter 11, H. W. Bode, D. Van Nostrand Co., Inc., New York, N. Y., 1945. 3. Reference 2, Chapter 14. 4. "Network Theory Comes of Age," R. L. Dietzold, Electrical Engineering, Volume 67, Number 9, September 1948, page 898. 5. "A Carrier System for 8000-cycle Program Transmission," R. A. Leconte, D. B. Penick, C. W. Schramm, A. J. Wier. A companion paper. This issue of BSTJ. 6. "Band Pass Filter, Band Elimination Filter and Phase Simulating Network for Carrier Program Systems," F. S. Farkas, F. J. Hallenbeck, F. E. Stehlik. A companion paper. This issue of BSTJ. Band Pass Filter, Band Elimination Filter and Phase Simulating Network for Carrier Program Systems By F. S. FARKAS, F. J. HALLENBECK, F. E. STEHLIK* A paper In- Leconte, Penick, Schramm and VVier' discusses the system aspects of 8-kc program circuits over carrier facihties and outMnes the functions of several filters and networks. This paper describes in detail two of the filters and one network. These are: 1. The channel selecting crystal band pass filter used at program terminals of all broad-band carrier systems, 2. The band elimination filter which blocks the program at branching points on type K carrier systems, 3. The network used at branching points on type K carrier systems to simu- late the phase shift of the band elimination filter. Channel Selecting Crystal Band Pass Filter AN IMPORTANT component of the modulator-demodulator circuit at " the carrier program terminal is the band pass filter which selects the lower side band resulting from modulation of the audio frequency program material with the 88-kc carrier. This step of modulation locates the pro- gram frequencies in their allotted position in the carrier frequency spectrum of the standard broad-band terminal. System flexibility requires that long program circuits be established by tandem connections of carrier links. A link consists of a transmitting and a receiving carrier program terminal connected by the appropriate trans- mission medium. The original objectives were based on a ten-link carrier circuit. This means that each terminal must introduce no more than five per cent of the total allowable system distortion. Assuming the band filter introduces the major part of the terminal distortion it is seen that the re- quirements placed on each band filter are extremely severe. One of the transmission objectives of the system is to transmit audio frequencies as low as 50 cps. Hence the band filter must transmit the wanted carrier frequency sideband to within 50 cps of the carrier and must suppress the unwanted sideband beginning at 50 cps above the carrier. This sharp cut-ofif and the need for low distortion in the pass band requires the use of filter elements with so little dissipation that the only possibility of realizing the desired performance is by the use of quartz crystal elements. In addition to suppressing the unwanted sideband above the carrier the filter must also provide suflkient discrimination above and below the pass * Phase simulating network by F. S. Farkas. Band elimination filter by F. J. Hallcnbeck. Band pass filter by F. K. Stehlik. 196 FILTERS FOR CA RRIER SYSTEMS 197 band to prevent crosstalk of the adjacent message channels into the pro- gram channel. The necessity of using quart/, crystal elements limits the maximum band width of filter which can be realized. This limitation is the result of the comparatively poor electromechanical coupling of quartz.- The resulting filter band width is 8.5 kc with the upper cut-off located near the 88-kc carrier. This is slightly greater than the S-kc nominal band width of the system. The crystal band pass filter designed for the single sideband j^rogram channel weighs approximately 30 lbs. and occupies 7 inches of mounting space on a standard 19 inch relay rack. A total of 44 lilter components are required for its construction, half of which are balanced quartz crystal plates. The remaining components consist of eight adjustable air capaci- tors, three fixed mica capacitors, seven balanced retardation coils, three of which are adjustable, and four resistors. A schematic which shows the relative placement of these parts in the filter is given in Fig. 1. The measured insertion loss characteristic of the filter between resistive terminations is shown in Fig. 2. The pass band and the vicinity of the upper cut-off are given in greater detail in the enlarged characteristics of Figs. 3 and 4. The extreme sharpness of the upper cut-off is evident in the latter figure. At 40 cps above the 88-kc carrier the discrimination has reached 20 db while the slope of the insertion loss versus frequency curve through this point is about 1 db per cps. Since at least two filters are con- nected in tandem in any program circuit a minimum of 40 db discrimination is provided to all frequencies in the unwanted sideband. The loss realized at frequencies outside the band also is shown in Fig. 2. The delay distortion in the pass band of the filter, computed from the slope of its measured insertion phase characteristic, is given in Fig. 5. For short program systems, where no more than six filters are used in tandem, the delay distortion would not exceed the limits set for a high quality system. For longer systems it is necessar>' to equalize this delay distortion. The design and performance of the delay equalizers for this purpose are given in a separate paper.^ These equalizers also include some attenuation equalization to correct for the systematic distortion of the filter. Figure 6 shows an exterior view of the filter. On both sides of the mount- ing panel are metal containers which are provided with covers that can be soldered on to make a hermetic-sealed enclosure. In a corner of one can is a terminal box which contains the input and output terminals. These terminals are of the metal glass seal type which are vacuum tight. Mounted on brackets in each of the containers is a brass panel supporting the tilter elements. One side of one of these panels is visible in Fig. 6, the other side is shown in Fig. 7. 198 BELL SYSTEM TECHNICAL JOURNAL ^W^ i -^M^ lliiii I — 1 1 — 1 1 — 1 1 — II — II — I TTTTTT ^ = ■^ nrmiTTTTi I — 1 1 — II — 1 1 — II — I it::=t I — 1 1 — 1 1 — II — 1 1 — I TTTjTTTTLTTT ^ llliii I — 1 1 — 1 1 — 1 1 — 1 1 — 1 1 — I TTTTTT TTTTll I — 1 1 — 1 1 — 1 1 — II — II — I TTTTTT ■^ % rTTmrTTm I 1 I II II 1 I 1 rj:: :i= I 1 (=1 1=1 [=1 (^^ ttT,Tttttt,ttj ■^Ir ilTTlT I — 1 1 — 1 1 — II — 1 1 — 1 1 — I TTTTTT ■vGM^ 1^ ■vM^ Fig. 1 — Schematic of the channel selecting crystal band pass filter as constructed. FILTERS FOR CARR/FR SVSTF.\fS 199 In Fig. 7 the two large cylindrical containers parallel to the panel house eleven of the balanced quartz crystal elements. The smaller cylindrical cans contain adjustable retardation coils while the rectangular cans house fixed coils. Adjustable air capacitors can be seen mounted on the hard rubber plate between the two crystal units. The adjustment side of the brass panel is e.xposed in Fig. 6. Screwdriver adjustment of the retardation coils is possible through the circular holes at -12 -to -6 -2 0 2 4 (CARRIER) FREQUENCY IN KILOCYCLES PER SECOND (FROM 88-KC CARRIER) Fig. 2 — The insertion loss-freciuency characteristic of the filter. OO 3 1 V -^ ?-2 - -9 -8 -7 -6 -5 -4 -3 -2 -1 0 FREQUENCY IN KILOCYCLES PER SECOND (CARRIER) (FROM 88-KC CARRIER) Fig. 3 — Enlarged insertion loss-frequency characteristic of the filter pass band. the top left and right of the panel. The rotors of three of the four air capacitors are visible inside the square cut-out in the panel. The panel in the lower half of the filter contains the remaining elements mounted and wired in a similar manner. The schematic which was found to be most useful during the design of the filter is shown in I-'ig. 8. Thus the electrical circuit consists essentially of two comple.x lattice sections sei)arated by one constant-k ladder section 200 BELL SYSTEM TECHNICAL JOURNAL 80 70 !3 ^ 60 u LU Q / 1 / V _^ / Z 50 n / I o / ^40 / 1 / / 1 / 1- / ] / i;^ 30 2 / / / V. / 20 ^ / 10 ^^.^ 0 20 30 40 50 60 70 80 100 200 300 400 FREQUENCY IN CYCLES PER SECOND (ABOVE 68 KC) Fig. 4 — The sharpness of the upper cut-off of the filter is shown in this enlarged loss characteristic. tfl 5 o H 2 AT 86 KILOCYCLES, DELAY IS 0.56 MILLISECOND 1 / V . ^ —- ■\ -- J Fif -7 -6 -5 -4 -3 -2 -1 0 FREQUENCY IN KILOCYCLES PER SECOND (CARRIER) (FROM 88-KC CARRIER) -Dela_\' distortion in the pass band of the filter. and terminated at each end by half-sections of the constant-k ladder type. The performance of the filter results almost entirely from the lattice sections since they control the flatness of the pass band, the sharpness of the cut-of¥ I- 1 ITERS lOR CAKRIKK SYSTEMS 201 and give practically all the discrimiiiution recjuired. It will be noted that the filter uses the equivalent of l.^U electrical elements consisting of 63 in- ductors. 63 capacitors and 4 resistors. The use of complex filter sections permits the realization of filter char- acteristics which have low distortion in the pass band and high discrimina- tion outside the pass band with a more efiicient utilization of elements than is possible with a larger number of simpler sections. Improved mathe- matical methods of network analysis developed in the past several years Fig. 6 — Exterior view of the tiller with one cover removed. After adjustments arc completed the cover is soldered on to seal the assembly. have made the design of such complex tilter sections possible while recent developments of precise and stable filter elements and improved measuring circuits have made it possible to manufacture such filters. It has been mentioned before that the use of quartz crystal elements re- stricts the filter band width which can be realized. In the frequency loca- tion selected for this filter (lower sideband of an 88-kc carrier frequency) a complex lattice section of the type shown in Fig. 8, when used alone, will permit the use of physical crystal elements for bands not over 7.^()0 cps 202 BELL SYSTEM TECHNICAL JOURNAL FILTERS FOR CARRIER SYSTEMS 203 KmL^ Fig. 8 — The schematic used during the design of the filter contains 130 electrical ele- ments. 204 BKLL S YSTKM TECH NIC A L JOURNA L wide. A wider band was obtained in this case by combining the complex lattice sections with hidder sections as shown in the figure. For this filter a combination of sections was designed which gave physically realizable crystal elements for a band width of 8.5 kc. This was the maximum band width possible without increasing the distortion in the band. Summarizing, the filter design process consists of: 1. Design of wide band lattice sections which have quartz elements which cannot be realized in practice. 2. Design of electrical ladder sections of still wider band which introduce little distortion at the pass band frequencies of the lattice section. At this point the schematic is as shown in Fig. 8. 3. Combination of like elements, electrical transformations, and replace- ment of groups of elements consisting of an inductor and capacitor in series shunted by a second capacitor by their equivalent crystal ele- ments. This gives the final schematic shown in Fig. 1, in which the crystal elements are physically realizable. The general steps in the design of lattice filters--'* are as follows: 1. Choice of filter cut-oflfs. 2. Determination of number and location of impedance controlling fre- quencies to give a good match of image impedance to the termination. 3. Location of peaks of infinite attenuation to give the necessary transfer loss at frequencies removed from the pass band. 4. Determination of impedance level which gives the most reasonable element values. Theoretically a filter could be designed which contains only one lattice section. The decision to split the filter into two sections was based on a desire to simplify the design to ease the manufacturing problems. The attenuation burdens of each section were reduced sufficiently to allow wider tolerances to be placed on the filter components. The last design steps are to determine the schematic of each section and to compute the theoretical element values in accordance with previously dezcribed methods.-* Although the filter elements computed were physically realizable they represented such extreme values as to introduce difficult problems. This was true especially of the crystal elements where the equivalent inductances of the eleven crystal elements in one section varied from 16 to 465 henries, a range of 1 :29. A similar situation existed in the other section. Crystal elements of the -|-5 degree X-cut type vibrating in their funda- mental longitudinal mode are used in this filter. The equivalent inductance of such crystal elements varies directly with the thickness and inversely with the width of the plate. Therefore the high inductance plates are thick and narrow and the low inductance plates are thin and wide. In one sec- tion of the filter the dimensions of the plates required varied in width from I-ILTERS lOR (• 1 KKIEK SYSTEMS 205 0.67 to 0.17 inch, in thickness from 0.119 to 0.012 inch and in lenj^lh frf)m 1.40 to 1.23 inches. The small variation in length is due to the fact that the length is determined primarilx' !)}• the frequency of resonance of the plate and this change is small across the tilter band. Tlie temperature coeflicient of the -{-S degree X-cut quartz crystal element used in this filter is su[)erior to the —18 degree X-cut longitudinal type which has been used in many other crystal filters but otherwise they are similar in use and in manufacture. The filter attenuation distortion in the vicinity of the cut-ofTs is depen- dent on the dissipation in the elements which resonate there. In order to minimize this distortion, it has been found necessary to impose minimum Q requirements of 80,000 on the high-impedance crystal elements which re- sonate near the cut-ofifs. This high Q is realized by suspending the quartz crystal plates from fme wires^ and operating them inside of evacuated con- tainers. The low-impedance crystal elements which resonate at frequencies removed from the cut-offs require a minimum Q of 15,000. This compara- tively low Q is realized by quartz crystal elements vibrating in air at atmos- pheric pressure. In the equivalent electrical circuit of a quartz crystal element the large ratio of the shunt capacitance to the internal capacitance is a measure of the poor electromechanical coupling of quartz. For the +5 degree X-cut quartz crystal element this ratio of capacitances is about 140 for a plated blank before fabrication. It is obvious that fabrication, wiring and para- sitic capacitances which may be in parallel with the quartz plate will make this ratio still higher and thus will reduce further the filter band width obtainable. For this reason it is important to keep to a minimum any capacitances which appear across any arms of the crj^stal lattices. One method used to minimize these capacitances was to assemble the eleven crystal elements required for each section in two containers instead of eleven separate ones. The five high-impedance elements requiring mini- mum ^'s of 80,000 are assembled in one evacuated metal container and the si.x low-inductance elements having the lower Q's are assembled in another hermetic sealed container filled with dry air. A photograph showing the method of assembly used in given in Fig. 9. A method was found to reduce the ratio of capacitances of the crystal elements. This method consists of dividing the plating on the surface of the quartz so that the driving voltage is removed from the end portions of the quartz plates. This plating division increases the equivalent induc- tance of the quartz plate but also decreases the direct capacitance between the plated surfaces. It has been found that the decrease in shunt capaci- tance with removal of plating is more rapid than the increase in equivalent inductance up to a certain point. If the plating is removed up to this opti- mum point it has been found possible to reduce the shunt capacitance about 206 BELL SYSTEM TECHNICAL JOURNAL s riLTERS FOR CA RK/ER .'iVSTEMS 207 17% below what it would be with a fully plated crystal element having the same inductance. This method of capacitance reduction was used on the six low-inductance crystal elements in each section. Another step in minimizing the unwanted capacitances was to design the retardation coils which connect to the terminal ends of each lattice to have as little capacitance as possible. Finally precautions were taken to keep the wiring capacitances to a minimum and the air condensers used inside the lattice for adjustment purposes are of special design having a minimum capacitance of only 0.5 mmf. The resonant frequencies of each of the twenty-two crystal elements must be adjusted to the desired nominal frequencies within very close toler- ances. On the ten high-impedance crystal elements the tolerance is ±2 cps while on the 12 low-impedance crystal elements the tolerance is ±5 cps. This precise frequency adjustment is accomphshed by careful grinding of the length of the quartz plate. The equivalent inductance of each of the 22 quartz crystal elements is required to be within two per cent of its nominal value. This specification is met primarily by close dimensional tolerances in the manufacture of the quartz plate. Any small adjustments which are necessary to meet this requirement are done by the aforementioned method of isolation of a small amount of plating from near the end of the quartz plate. The four fixed retardation coils are adjusted to be within two per cent of their nominal inductance values. The variations from nominal are par- tially absorbed in the filter adjustment procedure where the coils are tuned with their associated variable capacitors to give the desired resonance fre- quency. The fixed mica capacitors are manufactured to be within 0.5 per cent of the desired nominal value. The three adjustable retardation coils are constructed to permit an inductance variation of five per cent on either side of their nominal values. This is done by moving a permalloy core in the field of the coil. Adjustment of these coils in the filter is accom- plished by tuning them with their associated precision capacitor to give the desired resonance frequency within ±25 cps. This type of adjustment procedure gives the correct LC product. The correct L/C quotient is obtained also since C is accurate to ±0.5 per cent. The two resistors at each end of the filter compensate for the dissipation in the end retardation coils and thus restore the terminating impedance to the value required for optimum filter performance. Each lattice of crystal elements and capacitors is a four-terminal bridge which is adjusted for maximum bridge balance at a particular frequency by means of the variable air capacitors in two of the arms. The precision of inductance adjustment of the crystal elements insures that the other peaks of attenuation will be sufficiently close to their nominal locations. 208 BELL SYSTEM TECIINICA L JOURNAL To obtain maximum loss at the filter peaks it is necessary to secure a conductance balance in each lattice section as well as a susceptance balance. This can be done if care is exercised in the choice of materials used in fabri- cating the crystal elements and capacitors which appear inside the lattice. In this case the crystal element insulators and dielectrics consist of glass, mica, quartz and clean dry air or vacuum while the air capacitors use glass, ceramic and air for their insulators and dielectrics. If these materials are clean and dry they have very low conductance and do not influence the bridge balance. A complete discussion of the effects of impedance un- balances on crystal lattice performance has been given in a recent paper by E. S. Willis." To further insure that dirt and moisture will not influence its performance the filter is adjusted, tested and hermetically sealed in an air conditioned room where the relative humidity does not exceed 40 per cent. Since manufacture started about the beginning of 1946 several hundred of these filters have been made and are functioning satisfactorily in the telephone plant. Band Elimination Filter at Branching Points When broad-band carrier systems are equipped for the transmission of a carrier program channel, it is frequently necessary to provide between carrier terminals intermediate or branching points at which the program may also be received. If only receiving facilities are involved, rather simple bridging arrangements can be provided. However, program network needs often require a more flexible arrangement at the branching point so that a line may be cleared of the program originating at one terminal and a new program introduced for transmittal toward the next terminal. To do this without affecting the message channels also being trans- mitted on the line, a filter has been developed to block the program channel already on the line while freely transmitting the message channels. With this filter in the circuit the high-frequency line between the branch point and the following terminal is free of program frequencies and the program originating at the branch point may be sent toward that terminal. Since the program channel occupies frequency space near the center of the 12-channel message group, the remaining message channels appear above and below the program frequencies. Therefore the blocking filter at the branching points must be of the band elimination type. The circuit em- ploying this filter may be designed to block either at line frequencies or at basic group frequencies. The latter method, of course, requires that a demodulation process be provided to translate line frequencies to basic group frequencies before the blocking filter is inserted in the circuit. IIIJ ERS lOK CAKRIEK SYSTEMS 209 The band elimination filter described herein was developed for the type A' carrier system (Carrier-on-Cable) for which the hrst option mentioned above was chosen. This filter operating at the line frequencies of the type A' system is required to transmit frequencies from 12 to 31.6 kc and 44.2 to 60 kc while blocking those from 32 to 43.2 kc. Actually the filter will transmit frequencies below 12 kc and above 60 kc but these do not appear on the type A' line and therefore there are no requirements in these ranges. The filler which i)erforms these functions is shown schematically in Fig. 10. Several factors made its design difficult. A high level of dis- crimination of the order of 75 db is required over a wide frequency range of about 12 kc. Also the allowable waste interval between wanted and un- wanted frequencies is very small. The filter must transmit with a maxi- mum distortion of 0.2 db to within 97.5% of the first unwanted frequencies at which a discrimination level of 75 db is required. Because of the severe requirements the familiar image parameter design method was not employed. In this, as is well known, the composite filter first presented b\- Zobel" is made up of sections with matched image im- pedances but different transfer constants depending upon the attenuation requirements. Instead, it was felt that a design method proposed by Dar- lington^ offered a better possibility of meeting the requirements with a reasonably sized filter. This procedure known as the inserlion loss method is based upon the determination of a four-terminal transducer of reactances which, when inserted between definite resistance terminations, will produce a specified loss characteristic. A filter so designed has an advantage over image parameter filters in that the attenuation obtainable is greater for the same effective cut-off and an equal number of elements. Elective cut-of as used here means the last frequency of interest in the transmitted band. It is possible, therefore, with an insertion loss filter to use fewer elements for a given attenuation, or to obtain a wider transmission band with the same number of elements. The advantage inherent in the newer design method is not derived from a difference in structure. In configuration there is no way to distinguish such a filter from one of conventional image design. The difference lies solely in the element values. A simple way to visualize how the insertion loss design varies from image design is to consider that the newer method removes an arbitrary restriction placed upon the image theory to simplify the mechanics of design. The restriction is that the nondissipative image attenuation must be identically zero over continuous frequency ranges including the transmitted bands and other than zero everywhere else. This leads to the familiar ladder image filter composed of matched sections, or the lattice filter with coincident critical frequencies. o n^W 0 Fig. 10 — Schematic of the band elimination filter used at branching points of the type A' system. 210 FILTERS FOR CARRIER SYSTEMS 211 Analysis of an insertion loss ladder tilter shows Ihat it may be considered a composite of image sections which are not matched in image impedance. As a composite filter this means that the effective pass band has been split into a number of pass bands each separated by a small attenuation region. Darlington has formulated the process by which these bands can be so arranged that advantage can be taken of the fact that the image attenua- tions in these bands for small mismatch are comparable to the terminal effects and that reflection gains up to 6 db are possible in the same regions. The combination of these effects, which can be controlled up to and in- cluding the cut-off, gives the insertion loss -filter its improved performance since the efective and theoretical cut-offs can be made identical, with no FREQUENCY, f — ^ Fig. 11 — Non-dissipative filter characteristic obtained by use of Tchebychefif param- eters in pass bands and attenuation band. frequency space needed for the rounding due to the terminal effects in image filters. In general the mathematical steps required to design a filter by this method are as follows: An insertion loss frequency function is chosen which will satisfy the filter requirements and will lead to a structure economical of elements. From this are found the open and short circuit impedances of the proposed network which is normally of the standard lattice or ladder forms. Finally from these expressions the element values are determined. The particular form of insertion loss design employed for the filter de- scribed here is a special case of the general theory. The filter requirements lent themselves to the use of Tchebycheff parameters simultaneously in the pass bands and attenuation band. The application of these param- eters was first described by Cauer.^ The typical non-dissipative charac- teristic resulting from their use is shown on Fig. 11. It is seen that the 212 BELL SYSTEM TECHNICAL JOURNAL pass band characteristic is of the ripple type with equal maxima and equal minima. In the attenuation region the valleys of loss are of equal value. The general form for the insertion power ratio to obtain the desired char- acteristic is: [1 + (e'"" - 1) cosh' Bi] In this equation R\ and R2 are the resistive terminations and ap is the maximum ripple in the pass band as shown in Fig. 11. di represents a function of frequency so chosen that cosh di is an odd or even rational function of frequency. Also di must be a pure imaginary throughout the passing band and must be of the form («/ + nWi) in the attenuation region. The term ai is real at all attenuation frequencies becoming infinite at those required by the specification of minimum a„ in Fig. 11. Darlington further showed that di closely conforms with the image transfer constant of an image parameter filter if the effective pass band of the insertion loss filter coincides with the theoretical pass band of the image filter. Based on this conclusion a design method was formulated which permits a reference filter derived from image parameters to be used as the basis of the insertion Joss filter. There is, of course, no correspondence between the elements of the reference image filter and the insertion filter. This reference filter is not a requisite to the development of the insertion theory but it does offer a convenient and well known transfer constant which is the right functional form for use in the insertion power ratio stated above. Referring again to Fig. 11, the approximate minimum loss, aa , deter- mines the number of peak sections required in the reference filter from the relationship : aa = 20 log (e'"" - 1) - 10(2w + 1) log 9 - 18 where "w" is the number of peaks required and ap is the band ripple func- tion as before. The new term introduced here is "9" which is directly tied up with the selectivity demanded of the filter, i.e., the amount of fre- quency space avilable between the last useful frequency or ejfective cut-off and the first frequency at which attenuation equal to aa is needed. The relationships are as follows: 1 Tl - a/F^""* q = iz\ — — ^.-- I + 16 Ll + where K' = Vl - K"^ and K _h-h /4-/1 I- 1 ITERS lOR C'.l RRIRR SYSTEMS 213 The filter described here actually consists of two filters connected in tandem, each derived from a different power ratio. This step was taken because of the relatively low dissipation factor realizable with coils of reason- able size. By dividing the total attenuation between two power ratios, lower overall distortion due to dissipation was achieved. The distortion represented by the non-dissipative ripj^le "ap" was minimized by so assign- ing the frequencies of infinite attenuation to the two functions that phasing in of the ripples was avoided as far as possible. The two power ratios selected are: /«i = 1 + (e2«P _ Dcosh'e,^ For these the peak frequencies were assigned on an alternate basis as follows : To 0/, : nil , W3 , W6 and im To di2 : Wi , '"i; , ffh and m^ with the value of "m" decreasing from Wi to W7 and ttii = 1. The param- eter "m" has the same meaning as in image filter theory. The next step in the process is the finding of the roots of the two power ratios. These may be obtained from the following expansions: For e"^ representing a reference filter of 3^ sections: (W3 -f xfims + x)\m, + .v)-(l + :v) + I - — r— i ) (ws - .v)"(w6 - xfinii - xfil - x) = 0 which is expressed in the form Ki[x + Ci.v + 02 X + Qsx -f fl4-^" + ObX -f ae-v + ^7 = 0] For e'"' representing a reference filter of 4 sections: (;«2 -f .v)(w, -f .x:)(w6 -f a;)(l -f x) /.Up 1 J c^p -j- 1 which is expressed by Ki[x^ + a^x^ -f c^x^ -f (710.V + an = 0] ^ = 4A+^ In the above expressions x — \/ \ +75 where p = iw and a^ for the filter 214 BELL SYSTEM TECHNICAL JOURNAL discussed here is 0.1 db. From the roots obtained from the above equations of 7th degree and 4th degree complexity, the open and short-circuit im- pedances are determined which in turn lead to the element values. The complete development of the process resulted in the filter portion of the network shown on Fig. 10. The remainder of the schematic shows the equalizer which corrects the rounding of the filter characterisic near the cut-ofifs due to dissipation. The equalizer is of conventional bridged "7" design with constant "i?" impedance in tandem with a simple series section. 60 o !2 30 f\ 1 \J \k t- V. 25 30 35 40 45 FREQUENCY IN KILOCYCLES PER SECOND Fig. 12 — Insertion loss-frequency characteristic of the band elimination filter. Repeating coils are required as shown because the filter was designed at a 600-ohm level to give commercial elements whereas it is required to oper- ate between 135 ohm resistances. In the schematic a resistance will be noted in series with one termination. This is^eeded because the "insertion" design with inverse impedance terminations as shown here requires unequal terminations to produce the specified loss characteristic. Usually this would be taken care of by proper design of the repeating coil but, in this case, economic reasons dictated the use of the same repeating coil at both ends of the structure. The termination was therefore built out with a FILTERS FOR CA RRIER SYSTEMS 215 physical resistance. This of course introduces a flat loss but in this case enough gain was available in the circuit to permit it. On Fig. 12 is shown a typical transmission characteristic when the filter is operating between 135-ohm resistances. A variety of component parts are required to give this performance. The Alter portion employs mica condensers throughout and a mixture of molybdenum permalloy and air- core retard coils. As many permalloy coils are used as possible in order Fig. 13 — Exterior view of the band , elimination filter. to obtain high "Q". The air-core coils, of an adjustable type, are used in those arms which control the peak frequencies near the pass band. These arms must be adjusted very accurately for resonance in order to maintain the steep slope of loss in the cut-off region. The equalizer sections employ duolaterally wound air core coils also adjustable in order to set the pass band losses accurately. Mica and paper condensers are used in the equalizer, the latter being used where capacity values make mica condensers extremely expensive. 216 BELL SYSTEM TECHNICAL JOURNAL A completed filter is shown in the photograph of Fig. 13, while the internal arrangements of one portion of the assembly are shown on Fig. 14. Fig. 14 — Interior view of one portion of the assembly of the band ehmination filter. Phase Simulating Network When program rearrangements at a branching point are required, the band elimination filter must be switched into or out of the through trans- mission path. This transfer is accomplished without opening the through path. Thus, for a brief time during the switching interval, message chan- nels are transmitted simultaneously through the filter and the non-blocked circuit. A large phase difference between the two parallel paths is intro- duced by the filter which, in the absence of phase correction in the through circuit, could cause errors in the transmission of voice frequency telegraph signals. Therefore a network having phase shift similar to that of the filter over most of the message range is provided in the through circuit. 218 BELL SYSTEM TECHNICAL JOURNAL The phase simulating network is shown in schematic form in Fig. 15. The network is a balanced structure and consists of the following pieces of apparatus connected in tandem: 1. An input repeating coil to improve the longitudinal balance at the sending end, 2. A half-section high-frequency cut-off low-pass filter to mop up the phase shift introduced by two repeating coils of the band elimination filter, U- --M — > < -- P - > ■« M >j < X > 1000 900 ^ -^ r^ >^ 800 RANGE: M - MESSAGE ^ P = PROGRAM / X 1 ^ 1 T = TELEGRAPH 1 / 1 1 700 / 1 / / 1 1080° 1 ADDED 600 500 400 300 / f / 1 1 / 1/ / / ALL-PASS NETWORK /' f BLOCKING FILTER 200 100 ^,^^'- <'-' ^^^ ir:-' 0 24 28 32 36 40 44 48 FREQUENCY IN KILOCYCLES PER SEC0N3 Fig. 16— Phase shift-frequency characteristic of the phase simulating network. 3. A resistance pad to equalize the over-all loss level of the all pass network to within dzO.l db of the pass band loss of the band elimina- tion filter, and 4. Three delay sections, self equalized for loss, for simulating the phase shift of the band elimination filter. The network simulates the phase shift of the band elimination filter over the frequency ranges covered by message channels 1 to 4 and 10 to 12 to within 20 electrical degrees as shown in Fig. 16. As phase simulation is FILTERS FOR CARRIER SYSTEMS IV) incomplete in the frequency ranges occupied by channels 5 and 9 due to the steep phase shift slope of the band elimination fdter near its cut-off points, no telegraph channel are assigned to these channels of tyi)e "A'" carrier circuits equipped with branching points. The phase shift of the band elimination filter is discontinuous between its cut-off frequencies and has a positive slope with frequency in its pass bands. As the phase shift of a delay section increases continuously with frequency, it is impossible to provide the exact counterpart of the filter in a delay network. However, the addition of any multiple of 211 radians does not change the transmission characteristic. Hence 611 radians (3 CD 90 O 32 33 34 35 36 37 38 39 FREQUENCY IN KILOCYCLES PER SECOND Fig. 17 — Delay of the phase simulating network at program^frequencies. CHANNEL NUMBERS 9.2 IS) 111 1 2 3 4 5 6 7 8 9 10 11 12 ■^ ^ ^ ~~^ \ 09.I UJ Q " ^-^ Z 90 24 28 32 36 40 44 48 FREQUENCY IN KILOCYCLES PER SECOND Fig. 18 — Insertion loss-frequency characteristic of the phase simulating network. revolutions) are added to the phase shift of the elimination filter above the upper cut-off to simulate its phase characteristic in the 10 to 12 message channel range, as well as to provide an almost linear phase slope in the 32 to 40 kc program channel range resulting in minimum delay distortion. The delay distortion of the network over the program channel is approxi- mately 16 microseconds as shown in Fig. 17. The loss distortion over the program channel is approximately 0.05 db and over any one message channel it is less than 0.05 db as shown in Fig. 18. The self loss equalizing feature of a delay section is evaluated at zero frequency in the form of a resistance pad by making the pad loss approxi- mate the insertion loss at the critical frequency of the delay section. The 220 BELL S YSTEM TECH NIC A L JOURNA L resistance Rx located in the series branch may be evaluated from the ex- pression t in which t is the transfer loss in nepers at the critical frequency, R is the 135 ohm resistance termination and Rdc is the DC resistance of the induc- tance coil. The resistance Ry located in parallel with the series resonant R} branch may be evaluated from the expression Ry = -^ . Rx By changing the loss, e , of the derived resistance pad at zero frequency slightly from the measured loss at the critical frequency of the delay section, a suitable loss compensation may be realized to produce an optimum loss equalization over the message and program channel ranges. It is satis- factory to follow this technique when the condenser Q factor is much greater than the coil Q factor. When this condition exists, the insertion loss about the critical frequency becomes geometrically dissymmetrical, that is, the loss falls off more rapidly for frequencies above the critical frequency be- cause of the controlling condenser Q factor. References 1. "A Carrier System for 8000-Cycle Program Transmission," R. A. Leconte, D. B. Penick, C. VV. Schramm, A. J. Wier., a companion paper. This issue of BSTJ. 2. "Electromechanical Transducers and Wave Filters" (book), W. P. Mason, D. Van Nostrand Co., New York, N. Y., 1942. 3. "Delay Equalization of 8-kc Carrier Program Circuits," C. H. Dagnall and P. W. Rounds, a companion paper. This issue of BSTJ. 4. "Communication Networks," Vol. II (book), E. A. Guillemin, John Wiley & Sons, New York, N. Y., 1935. 5. "The Mounting and Fabrication of Plated Quartz Crystal Units," R. M. C. Greenidge, Bell Sys. Tech. Jour., Vol. 23, July 1944, page 234. 6. "A New Crystal Channel Filter for Broad Band Carrier Systems," E. S. Willis, Elec. Eng., Vol. 65, March 1946, Page 134. 7. "Theory and Design of Uniform and Composite Electric Wave Filters," O. J. Zobel, Bell Sys. Tech. Jour., Vol. 2, 1923, Pages 1-46. 8. "Synthesis of Reactance 4-Poles which Produce Prescribed Insertion Loss Charac- teristics," S. Darlington, Journal of Mathematics and Physics, Vol. 18, No. 4, Sept. 1939. 9. "Ein Interpolationsproblem mit Funktionen mit Positiven Realteil," W. Cauer, Mathematische Zeitschrift, 38, 1-44, 1933. 10. "Distortion Correction in Electrical Circuits with Constant Resistance Recurrent Networks," O. J. Zobel, Bell Sys. Tech. Jour., Vol. 7, July 1928, Pages 438-534. A Precise Direct Reading Phase and Transmission Measuring System for Video Frequencies By D. A. ALSBERG and D. LEED THE evolution of transmission networks for communications systems progresses through three fairly well-defined phases — design, syn- thesis and final adjustment. The design phase ordinarily involves no problem of measurement. In the synthesis stage, during which the physical model is constructed from the paper design, precise equipment is often needed for measuring the magnitude of the various components comprising the network. The adjustment stage, in which the network is actually tested as an element in a transmission circuit, generally requires the most complex instrumentation. In the latter category we may include insertion loss, gain, and phase measurement systems. Television and broad-band carrier facilities, such as the New York- Midwest video cable link, employ vast numbers of transmission networks. These include, for example, filters, equalizers, and repeaters. The fmal adjustment of these networks requires a large number of precise insertion phase and transmission measurements during both development and manu- facturing stages. Consequently, the measurement equipment must com- bine laboratory accuracy with speed of measurement suitable for use in production testing. The quantities measured are defined in Fig. 1. Conforming with current usage, the term Transmission is used herein to designate insertion loss and gain. The performance of the system with respect to frequency range, measure- ment range and accuracy is as follows: Frequency Range: 50-3600 kilocycles Generator and Network Termination Impedance: 75 Q Transmission Range: -|-40 db to — 40 db; Accuracy ±0.05 db — 40 db to — 60 db; reduced accuracy. Insertion Phase Shift Range: 0-360°; Accuracy ±0.25 degree (+40 db to -40 db) The measuring circuit is based on the heterodyne principle whereby the phase and transmission of the unknoicn are translated from the variable frequency to a constant intermediate frequency at which the phase and transmission standards operate. Accurate phase-shifters and variable attenuators with negligible phase shifts are constructed readily for fixed 221 222 BELL SYSTEM TECHNICAL JOURNAL frequency operation. This advantage more than offsets resulting problems of modulator design and automatic frequency control. Conforming with the definitions of insertion phase and transmission, the measurement system compares, with respect to phase and amplitude, the outputs of two transmission channels energized from the measurement frequency source, one of which serves as a standard or reference channel, while the other contains the apparatus under test. This is illustrated by the block drawing in Fig. 2. For loss measurements the range attenuator I (Fig. 2) is set at 0 db. Measurement frequency F from the master oscillator is appHed to both standard "5" and unknown "X" channels through spUtting pad /. The voltages at "S" and "X" modulator inputs, points A and B respectively Z. % 1 >/\A/' * • — T INSERTION LOSS OR INSERTION GAIN = 20 LOG INSERTION PHASESHIFT = Z-E^-Z-E, Fig. 1 — Definition of quantities measured. in Fig. 2, differ with respect to phase and amplitude because of the trans- mission differential introduced between the two channels by the apparatus under test. By frequency conversion in the "S" and "X" modulators these amplitude and phase differences at frequency F are translated at points C and Z> to a constant intermediate frequency, 31 kc. The second input to the "5" and "X" modulators, of frequency F + 31 kc, is supplied by the slave oscillator which automatically tracks at constant 31 kc dif- ference with respect to the master oscillator. By selective filtering, only the difference frequency appears at the modulator outputs C and D. 31 kc has been chosen as the intermediate frequency, primarily on the basis of filtering requirements in the modulators. The detector (Fig. 2) compares the voltages of the "X" and "S" channels at K and L as to magnitude ATTENUATOR )0 db .STEPS 31 kc "X" 3lkc DETECTOR PHASE DIFFERENCE DETECTOR AMPLITUDE DIFFERENCE DETECTOR d-c -^ ^ K 270° r\ 180° [^ 1 31 kc 3lkc HASE SHIFTER )US 0°-360° d-c TRANSMISSION INDICATOR r~ -1 - 0 + APPARATUS UNDER TEST X MODULATOR ->-» «-► OW^ MASTER OSCILLATOR SLAVE OSCILLATOR RANGE ATTENUATOR I 50 TO 3600 kc 4 SYNCH 05C f+31kt F.3lkc ■>— * ♦-►■ MEASURING ATTENUATOR 0-60 db 0.1 db. STEPS PHASE INDICATOR DETECTOR ^- ^ y RANGE ATTENUATOR JL 0^|3 H t> TRANSMISSION INDICATOR S MODULATOR Fig. 2— Uluck schcmalic of the phase and tra MEASURING PHASE SHIFTER CONTINUOUS 0°-360° MEASURIMG SYSTEM FOR VIDEO 226 and phase, and indicates their difference on the direct reacHng scales of the indicator meters. If the measuring attenuator is set at 60 db loss, and the range attenuator // at 40 db loss, ".S"" and "X" channels are in balance when the apparatus under lest is replaced by a zero loss strap. The phase-shifter has, by design, 20 db loss; so that under these conditions "5'" and "X" channels are nomi- nally in balance, except for small residual phase and transmission dif- ferentials which may be zeroed-out by initial adjustment of the phase-shifter and of the relative gain between "5" and "X" channel amplifiers within the detector. Null readings on the phase and transmission difference indicating meters tell when exact phase and transmission balance between the two channels has been established. The phase-shifter and attenuator dials are arranged to read zero after this initial balance has been made. To measure apparatus transmission and phase, the strap is replaced by the apparatus under test and the balance restored by adjustment of the phase- shifter and the measuring attenuator. The insertion phase and trans- mission of the apparatus under test are then read directly from the cali- brated dials of the phase shifter and attenuator. When measuring loss, attenuation in the measuring attenuator is reduced by the amount of attenuation introduced in the high-frequency portion of "X" channel by the "apparatus under test." In measuring gain, the attenuation through the measuring attenuator must be increased by the amount of apparatus gain. To insure that "5" and "X" channel modulators are not overloaded by excessive input, range attenuator I is set to 40 db loss during gain measurements. This attenuator is common to both chan- nels and therefore introduces no phase differential. Simultaneously and automatically, the range attenuator // immediately following the "6"" modulator, is operated, removing 40 db loss from the 31 kc standard channel. The measuring attenuator is self-computing and indicates directly in illuminated figures the gain or loss of the apparatus under test, A simple switching arrangement automatically controls the dial-hghting circuit of the measuring attenuator. When measuring gain the dial indications increase in one direction, and when measuring loss the indications increase in the opposite direction (Fig. 3). In addition to the null-balance method, a deflection method of measure- ment using direct reading scales of the phase and transmission difference indicating meters is also possible. An automatic volume control circuit assures invariance of the indicator scale factors with either the modulator frequency-transmission characteristic, or input voltage variation at the "5" modulator caused by reflections from apparatus under test. The auto- matic volume control circuit regulates the output voltage of the slave oscillator to maintain the amplitude of the "5" channel input to the dif- MEASURING SYSTEM FUR VIDEO 223 and phase, and indicates their difference on the direct reading scales of the iiKhcator meters. If the measuring attenuator is set at 60 db loss, and the range attenuator // at 40 db loss, "6"' and "X" channels are in balance when the apparatus under test is replaced by a zero loss strap. The phase-shifter has, by design, 20 db loss; so that under these conditions ".S" and "X" channels are nomi- nally in balance, except for small residual phase and transmission dif- ferentials which may be zeroed-out by initial adjustment of the phase-shifter and of the relative gain between "5" and "X" channel amplifiers within the detector. Null readings on the phase and transmission difference indicating meters tell when exact phase and transmission balance between the two channels has been established. The phase-shifter and attenuator dials are arranged to read zero after this initial balance has been made. To measure apparatus transmission and phase, the strap is replaced by the apparatus under test and the balance restored by adjustment of the phase- shifter and the measuring attenuator. The insertion phase and trans- mission of the apparatus under test are then read directly from the cali- brated dials of the phase shifter and attenuator. When measuring loss, attenuation in the measuring attenuator is reduced by the amount of attenuation introduced in the high-frequency portion of "X" channel by the "apparatus under test." In measuring gain, the attenuation through the measuring attenuator must be increased by the amount of apparatus gain. To insure that "5" and "X" channel modulators are not overloaded by excessive input, range attenuator I is set to 40 db loss during gain measurements. This attenuator is common to both chan- nels and therefore introduces no phase differential. Simultaneously and automatically, the range attenuator II immediately following the "5" modulator, is operated, removing 40 db loss from the 31 kc standard channel. The measuring attenuator is self-computing and indicates directly in illuminated figures the gain or loss of the apparatus under test. A simple switching arrangement automatically controls the dial-lighting circuit of the measuring attenuator. When measuring gain the dial indications increase in one direction, and when measuring loss the indications increase in the opposite direction (Fig. 3). In addition to the null-balance method, a deflection method of measure- ment using direct reading scales of the phase and transmission difference indicating meters is also possible. An automatic volume control circuit assures invariance of the indicator scale factors with either the modulator frequency-transmission characteristic, or input voltage variation at the "5'" modulator caused by reflections from apparatus under test. The auto- matic volume control circuit regulates the output voltage of the slave oscillator to maintain the amplitude of the "5" channel input to the dif- 224 BELL SYSTEM TECHNICAL JOURNAL MEASrRIXG SYSTEM lOR VIDEO 225 ferential detector constant. As the control action simultaneously affects both "^"' and "A'" modulators uniformly, the system zero is undisturbed. Careful attention has been given to the problem of obtaining an electrical match between '\S"' and "A'" modulators and coaxial cable lengths in the high-frequency channels. (RG 6/U cable contributes a i)hase shift of 0.27inch at 3600 kc.) Consequently, with the apparatus under test re- placed by a coaxial strap, a balance indication on the phase and transmission difference indicators may be obtained which shifts less than 0.1 degree in phase and 0.02 db in transmission when the master oscillator frequency is varied over its entire band. Because of the frequency independence of the system zero and the auto- matic frequency control of the slave oscillator, the master oscillator may be swept through the entire frequency band for rapid appraisal of the network performance by observation of the phase and transmission difference indicators. The component chassis of the set are mounted in a specially designed con- sole, shown in Fig. 4, which places all controls within easy reach of the operator. This console houses as much apparatus as three 6-foot relay racks within a floor space equal to that occupied by a 5-foot laboratory l)ench. Though not visible, a full bay of apparatus is mounted behind the central meter panel. Easily movable partitions and covers permit accessibility to all units, thus expediting maintenance. Some of the signiticant design considerations are discussed separately under the following headings: (1) Master Oscillator (2) Slave Oscillator (3) Modulators (4) Phase and Transmission Detector (5) Phase-shifter Master Oscill.\tor As indicated in Fig. 2 and Fig. 5C, the master oscillator is of the hetero- dyne type. It employs 15,000 and 11,400-14,950 kc local oscillators. A high degree of frequency stability has been achieved through special oscil- lator circuit design. A motion picture film type scale, 300 inches in length, calibrated every 10 kc, and further subdivided every 2 kc, covers the entire frequency range 50-3600 kc without band-switching. A 0-10 kc inter- polation dial with 100-cycle divisions, which operates on the fixed local oscillator frequency, is used to interpolate between adjacent 2 kc graduations on the main film scale. By oscilloscopic comparison with a 10 kc standard of frequency, the oscillator can be set within 50 cycles of any desired fre- 226 BELL S YSTEM TECH NIC A L JOURNA L quency in its band. An A.V.C. circuit maintains the output power at six db above one milliwatt. Fill. 4 — The assembled phase and transmission measuring system. Slave Oscillator To make possible the operation of the measuring attenuator, phase-shifter, and phase and transmission difference detectors at constant frequency, the inputs to the "S" and "X" channel modulators from the master and slave (A) CB) i. ii (C) Caf i93KC) I J-'Ji"'- SLAVE OSCILLATOR (AUTOMATIC FREQUENCY CONTROL CIRCUIT ) . Fig. 5 — Master ami slave oscillators. MEASURING SYSTEM FOR V I DFX) 227 oscillators must always dififer in frequency by a constant amount. This difference is maintained at 31 kc by the control of the master oscillator over the slave oscillator frequency. \'ery briefly, the scheme consists in applying the fixed local oscillator fre- ijuency, /, of the master oscillator, to an automatic frequency control rircuit which produces an output frequency/ + 31 kc. / + 31 kc is then modulated with variable local oscillator frequency, f — F, oi the master oscillator, resulting in an output of frequency F + 31 kc. Frequency F, formed by modulation of/ and/ — F, is the master oscillator frequency. In the automatic control circuit, frequency / is compared with that of a controlled oscillator, by detecting their difference in a modulator. The nature of the control is such, that any deviation of this difference from 31 kc causes the frequency of the controlled oscillator to change in the direction which eliminates the deviation. While it is simpler to compare / and the controlled oscillator frequency directly, in the slave oscillator the compari- son is made between the outputs of tripler circuits energized from the latter frequencies. In this way more complete isolation is realized between / and the controlled frequency than would be afforded with only buffer am- plifiers. Because of the triphng, it follows that the oscillator must be controlled according to the departure of the difference between the tripler circuit frequencies from 93 kc. This, however, has the advantage of avoiding the generation of 31 kc anywhere in the automatic frequency control circuit, which could, by spurious modulation, cause the / + 31 kc output to be contaminated with small traces of frequency/. The necessity for exceptional purity of / + 31 kc output arises in the measurement of high losses where minute amounts of F at the /^ 4- 31 kc input to ''.S" and "X" modulators may produce appreciable error. Owing to phase tracking requirements between "5" and "X" inter- mediate frequency channels, and to the frequency dependence of the phase- shifter calibration, it is necessary to maintain the intermediate frequenc\- as closely as possible to the precise value, 31,000 cycles. The permissible deviation from the correct value has been limited to ±1 cycle. This pre- cise control is maintained in the presence of 10 kc changes in /, which may occur when the setting of the 0-10 kc interpolation dial of the master oscillator is varied in the course of measurement. Figures 5A and 5B illustrate the automatic frequency control and hetero- dyne circuits of the slave oscillator. The frequency of oscillator 10 in Fig. 5 A is controlled by the reactance tubes 11 and 12. Reactance tube 12 is actuated by direct voltage from frequency discriminator 16, so that it controls oscillator 10 according to frequency error. Frequency error is the difference between the input frequency to discriminator 16 from am- plifier 9, and 93 kc, the frequency of zero voltage output from the dis- M EASU RING SYSTEM FOR VIDEO 227 oscillators must always differ in frequency by a constant amount. This difference is maintained at 31 kc by the control of the master oscillator over the slave oscillator frequency. \'ery briefly, the scheme consists in applying the fixed local oscillator fre- quency, /, of the master oscillator, to an automatic frequency control circuit which produces an output frequency/ + 31 kc. / + 31 kc is then [nodulated with variable local oscillator frequency, f — F, oi the master oscillator, resulting in an output of frequency F + 31 kc. Frequency /•", formed by modulation of / and/ — F, is the master oscillator frequency. In the automatic control circuit, frequency / is compared with that of ;i controlled oscillator, by detecting their difference in a modulator. The nature of the control is such, that any deviation of this difference from 31 kc causes the frequency of the controlled oscillator to change in the direction wliich eliminates the deviation. While it is simpler to compare / and the controlled oscillator frequency directly, in the slave oscillator the compari- son is made between the outputs of tripler circuits energized from the latter frequencies. In this way more complete isolation is realized between / and the controlled frequency than would be afforded with only buffer am- plifiers. Because of the tripling, it follows that the oscillator must be controlled according to the departure of the difference between the tripler circuit frequencies from 93 kc. This, however, has the advantage of avoiding the generation of 31 kc anywhere in the automatic frequency control circuit, which could, by spurious modulation, cause the / -f- 31 kc output to be contaminated with small traces of frequency/. The necessity for exceptional purity of / -+- 31 kc output arises in the measurement of high losses where minute amounts of F at the F -f- 31 kc input to ".S"' and "X" modulators may produce appreciable error. Owing to phase tracking requirements between "5" and "X" inter- mediate frequency channels, and to the frequency dependence of the phase- shifter calibration, it is necessary to maintain the intermediate frequency- as closely as possible to the precise value, 31,000 cycles. The permissible deviation from the correct value has been limited to ±1 cycle. This pre- cise control is maintained in the presence of 10 kc changes in /, which may occur when the setting of the 0-10 kc interpolation dial of the master oscillator is varied in the course of measurement. Figures 5A and 5B illustrate the automatic frequency control and hetero- dyne circuits of the slave oscillator. The frequency of oscillator 10 in Fig. 5A is controlled by the reactance tubes 11 and 12. Reactance tube 12 is actuated by direct voltage from frequency discriminator 16, so that it controls oscillator 10 according to frequency error. Frequency error is the difference between the input frequency to discriminator 16 from am- plifier 9, and 93 kc, the frequency of zero voltage output from the dis- 228 BELL SYSTEM TECHNICA L JOURNAL criminator. The voltage from phase discriminator 15 controls oscillator 10 according to the difference of phase between the input from stage 9, and an input of reference phase from amplifier 5. This difference of phase is pro- portional to the time integral of the frequency error. The gross effect, therefore, is to control the oscillator 10 according to the controller law, proportional to frequency error -\- lime integral of frequency error, or, in the terminology of feedback regulators, proportional + integral controls When in equilibrium, the system operates with a static phase difference between the phase discriminator inputs, a condition which can exist only when these inputs are of equal frequency. The system is thus endowed with the property zero frequency error, and the frequency at the output of modulator 8 is maintained in exact equality with crystal oscillator 2 frequency. Con- sequently the intermediate frequency difference between input, /, and controlled oscillator 10 is held precisely at the value 31,000 cycles. Automatic frequency control circuits of the phase sensitive type have been previously described ' ' . The system of combined phase and frequency sensitive control in the slave oscillator is superior to those which use only phase or frequency sensitive control. In a control circuit which uses only a phase discriminator and associated reactance tube, the controlled oscillator may lock-in at either of two sideband frequencies. These are/ -|- 31 kc, and/ - 31 kc. Opera- tion is at upper sideband when control stabilizes on the positive slope of the phase discriminator output voltage curve in Fig. 5A, and at lower sideband if control is along the negative slope. Thus an ambiguity of sideband exists, though the attribute of zero frequency error is retained. When only a frequency discriminator and reactance tube are used, lock-in is possible at only one of the two sideband frequencies, determined by the poling of the frequency discriminator output voltage. A frequency error, however, is present. The combination in Fig. 5 of the two systems operating jointly utilizes the phase sensitive discriminator to insure close control of oscillator fre- quency, and the polarizing property of the coarser frequency discriminator to eliminate the possibility of synchronization at the undesired sideband. The joint system of phase and frequency sensitive automatic control has the further virtue of possessing a far greater degree of stability than is obtainable with the phase discriminator loop acting alone. In the heterodyne circuit of Fig. 5B,/ + 31 kc from the automatic fre- quency control circuit is modulated with/ - F, the variable local oscillator frequency of the master oscillator. The frequency at the output of the heterodyne circuit is /«^ + 31 kc, and this is modulated with frequency F in the "5" and "X" modulators to produce the constant intermediate fre- quency, 31 kc, in the measurement portion of the set. MKASIR/XG SVSrK\f FOR VIDEO 229 TiiE Modulators The difficulties of precise measurement over a wide frequency band es- sentially are concentrated in the modulator. With the precision to which measurement must be made, effects ordinarily of small concern assume im- portance. The following discussion is valid for any modulator, though the specific example of the vacuum tube is used. It is the function of the modulator to convert linearly changes in ampli- tude and phase from the input frequency F to the output frequency 31 kc. The linear range of conversion is limited by overload at the high-level limit and by noise at the low-level limit. Let the input x to 'a modulator consist of two frequencies Fi and F-i. In the ideal square law modulator^ perfect linearity results between changes in the input signal Fi and the output signal F2 — Fi . The output filter rejects all frequencies but F2 — Fi . In actual tubes the plate current is (1) /p = Co + fll-V + fls-V- + flS-V + OiX -\- •■•. The effect of the term OiX^ and higher even-order terms is to contribute output currents of frequency Fo — Fi which do not vary linearly with the input. ^ In addition to this the effect of remodulation in plate, screen and suppressor circuits is that the coefficients a-i , a^ etc. are not independent of the input x and so contribute to the distortion. Further, in presence of modulation of higher than second order, thed-c. term in even-order modula- tion will cause distortion if cathode bias is used. Removal of d-c. degenera- tion using fixed bias eliminates this effect. The high-level limit may be defined as the signal value for which the total error due to overload equals the desired limits of modulator performance. The lowest input level into the modulator which may be tolerated, and hence the lower limit of loss which can be measured, is determined by the effective signal-to-noise ratio at the modulator output. If no amplification exists preceding the modulator the input grid noise is usually limiting. The signal-to-noise ratio of the signal Fi and a noise band centered on Fi is unaffected by the modulation process as only the modulated portion of the noise band passes through the output filter. Yet for a noise band centered on the intermediate frequency F2 — Fi for which the output filter is trans- parent the modulator acts as a straight amplifier; hence the effective signal- to-noise ratio is degraded approximately by the ratio of amplifier gain to conversion gain of the modulator. The low-level limit may be defined as the signal value for which the error due to noise equals the desired modulator performance limit. For example for a noise error of 0.01 db, a signal-to-noise ratio of 1000 to 1 or 60 db is required. 230 BELL SYSTEM TECHNICAL JOURNAL To obtain maximum signal-to-noise ratio, a tube must be chosen to have the lowest product of noise multiplied by the ratio of amplifier to conversion gain, the latter requirement being in conflict to overload requirements. When inputs below the low-level Hmit are to be utilized a preamplifier ahead of the modulator tube is required. This ampUfier also contains a noiseband centered on F2 — Fi . If the ampUfier is selective and rejects this noise band or if an F2 — Fi rejection filter is inserted ahead of the modulator tube the resultant new low-level Umit is determined by the signal- to-noise ratio of the preamplifier at the signal frequency Fi only. Dynamic range is defined as the useful range of a modulator limited by the high-level limit on one end and by the low-level limit on the other. The dynamic range of a number of pentodes was determined. It was found experimentally that differences in dynamic range between pentodes of different power ratings, such as 6AK5, 6AC7, 6AG7, 6L6, 829B, are small. A dynamic range of 30-36 db can be reaUzed with a 6AK5 for a .01 db linearity requirement. The 6AK5 was the most suitable tube of those investigated considering all other requirements of the circuit such as band width, available signal levels, etc. Buffer amplifiers are required ahead of the modulator tube to prevent crosstalk between measuring and reference modulator through common paths. These buffer ampHfiers are of conventional video ampUfier design, with phase and gain characteristics closely controlled to the order of 0.01 db and 0.1 degree. The Phase and Transmission Detector In the null type of phase measurement an initial circuit zero is made. When the circuit is rebalanced with the apparatus under test inserted, the phase detector must be able to verify that the same phase relationship has been reestablished as existed when the initial circuit zero balance was made. Bridge circuits yield high sensitivity and a high degree of independence of input voltage amplitudes. In Fig. 6 a four-arm resistance phase bridge is shown, which has two inputs E\ and Eo , and two outputs Es and Ed corresponding to the vectorial sum and difference of the input voltages Ei and E2 . As derived in the appendix, for the equal arm bridge, the amplitudes of the voltages Es and Ed are equal for phase angles of ^ = 7r/2 + nir, where n is any integer, regardless of the amplitudes of Ei and £2 • Thus equality of I £s I and [ £d | is convenient to define the circuit phase zero. Equality of \ Es\ and \ Ed\ by itself does not distinguish between 90° and 270° phase shifts. This ambiguity can be resolved with a detection circuit which responds to both the amount and the sign of the difference | £s | — I Ed I and by making provision for the introduction of a smaU increase A

) is derived in the ai)pendix and plotted for several values Ai/- in Fig. 8. The phase bridge essentially converts the measurement of phase into the measurement of voltage diflference. \'acuum tube diodes are used as dif- 20 LOG -r^ Fis;. 7 — Deflection response of the phase bridge. 10 15 20 26 JO 35 40 45 50 55 60 DEPARTURE,!", OF E, AND Ej FROM QUADRATURE Fig. 8 — Phase error A\t for unequal inputs. ferential rectifiers with a high resistance load consisting of hermetically sealed carbon deposited resistors closely matched for value and temperature coefficient and specially mounted to minimize temperature diflFerentials. The dififerential output of the rectifiers is amplified in a feedback stabilized MEASURfXG SySTK.\f FOR VIDEO 233 d-c. ami)lirier which luis adjusluble gain to adjust scale factors on the indicator. The phase detection circuit is energized from the output of the phase bridge and the almost identical transmission detection circuit is energized from the inputs to the phase bridge. Thus both phase and transmission are measured simultaneously. Each indicator (Fig. 9) has three sca\es, fine ( — 5° to +5°; —1 db to +1 db), coarse (-90° to +90°; -10 db to +3 db), and null balance. The fine and coarse scales are linear while the null balance scale has maximum Fig. 9 — Phase and transmission indicators. sensitivity in the neighborhood of the center zero and greatly reduced sensitivity at each end. X'aristor shunts across the indicators compress the null scale for large deflections. Colored {)ilot lights at the ends of the indicator scales, operated by the scale switch, indicate the scale in use directly. TiiK PiiASK Shifter The phase-shifter employs a four-quadrant variable sine condenser. It has two linearly subdivided scales — coarse 0-360° on a cylinder and fine 234 BELL SYSTEM TECHNICAL JOURNAL 0-10° on a dial. The line dial is connected through reduction gearing to the shaft of the sine condenser. The construction of a phase-shifter which has a sufficiently linear correspondence of electrical phase-shift and me- chanical displacement of a shaft is not practical. Instead a movable index for the fine scale permits correction of the deviation from linearity. The index position is controlled by a corrector which is fastened to the condenser shaft. As the corrector is rigidly associated with the sine condenser posi- tion and not with the scales this permits shifting the linear scales inde- lU— (J]>tical cam of phase shifter. pendently without affecting the correction. The correction curve (Fig. 10) is printed on a photographic negative which is placed on a transparent Incite drum and projected optically as an index (Fig. 11) adjacent to the line dial of the phase-shifter. The calibration curve is obtained by marking the correction at each calibrating point on a ])iece of cellulose acetate placed on the Incite drum. The correction point is projected on the screen ad- jacent to the fine scale during the calibration and problems arising from divergence or misalignment of the light beam are thus avoided. Since the index is projected upon a surface coplanar with the dial, no parallax exists. MEASl'KfA'G SYSTEM FOR VIDEO 235 The pluise-shiftcr's deviation from linearity is sulliciently small that no correction is needed on the coarse scale. Both dials can be moved with respect to their shafts by releasinj^ friction clutches. Thus the measuring system phase zero can be established by an initial balance of the phase-shifter and restoration of the coarse and fine scales to zero. This is onl}- possible because the scales are linear. Thus no zero readings have to be subtracted from the measurement readings and the need for a separate zero setting phase-shifter is avoided. The phase-shifter is calibrated by a method of substitution. As dis- cussed previously the phase-indicator indicates balance uniquely in mul- 11 — Phase shifltT scales and projected index. tiples of 360° phase-shift. E.xact sub-multiples of 360° can be generated and used to calibrate the phase-shifter. For example (Fig. 12), to establish an exact 180° phase-shift the standard phase-shifter is set to an arbitrar}- starting point. With the switches in the position shown a null is obtained on the indicator by adjusting the auxiliary phase-shifter. The network of nominally 180° phase-shift is inserted and a null obtained on the indicator by adjusting the standard phase-shifter. Now the network is removed and the null reestablished by adjustment of the auxiliary phase-shifter. The 180° network again is inserted and a null obtained by adjustment of the standard phase-shifter, which now has been moved through twice the actual phase-shift of the 236 BELL SYSTEM TECHNICAL JOURNAL nominal 180° network. The amount the standard phase-shifter failed to return to the original starting point indicates the residual error of the 180° network. The 180° network is adjusted accordingly and the procedure repeated until an error is no longer discernible. Thus the 180° point on the phase-shifter scale can be determined. In similar fashion, by combination of the 180°, 90° and 60° networks, calibration points in multiples of 30° are obtained. The equivalent of a 10° network is obtained by use of the ±5° scale on the indicator and scale factor adjustment. Interpolation to 1° is then made using the scale divisions on the indicator. Calibration to an absolute accuracy of ±0.1° was found adequate for use in the measuring system. Much higher accuracy could be obtained if the need arose. There appears to be no inherent frequency limitation in this calibration method. AUXILIARY SIGNAL OSCILLATOR PHASE SHIFTER 0-360° CONTINUOUS LI 11 J STANDARD PHASE SHIFTER 0-360° CONTINUOUS PHASE INDICATOR Fig. 12 — Phase shifter calibration circuit. Conclusion The design efifort has been directed toward achieving laboratory precision in measurement and at the same time maintaining the speed necessary for production testing of transmission networks. The measurement of phase-shift is unambiguous with respect to quad- rants and the measurements of insertion phase-shift and loss or gain are independent of each other. The entire frequency range is covered without band switching by use of a heterodyne signal oscillator and the system zero is independent of measurement frequency. Detector tuning is eliminated through the use of frequency conversion, employing a beating oscillator automatically controlled in frequency by the signal oscillator. Phase-shift and transmission may be read directly, without auxiliary computations. ]fEASr KING SYSTEM lOli VIDEO lil from tlic (liuls of the phase-shifter and attenuator or from the scales of the incHcalors. ACKNOWI.KDC.MENT Acknowled^mient is due nienihers of the j,^roui)S sui)ervised by Mr. E. P. Felch (electrical) and Mr. W. J. Means (mechanical) for contributions to the design. APPENDIX The Ph.ase Discriminator Bridge The general [ihase relationship of the discriminator is shown in Fig. 13a. """7 •jEi / (ir -k^^ \ L.A (a) lb) Fig. 13 — Phase bridge vector relationships. Using complex vectorial notation, (4) Es = (1 •2)(£ie^'^' + E^J"') (5) Eu = (1 '2){E,e''' - eJ""-) Hence for

^ Vp Vr ^ I. SIGNALffU COLLECTOR LOAD -BASE T Fig. 1— Schematic of transistor showing circuit for ampUfication of an_a-c. signal and the conventional directions for current flow. Normally h and W are positive, h and Vc negative. The explanation of the action of the transistor depends on the nature of the current flowing from the emitter. It is well known that in semi-con- ductors there are two ways by which the electrons can carry electricity which differ in the signs of the effective mobile charges.'^ The negative carriers are excess electrons which are free to move and are denoted by the term conduction electrons or simply electrons. They have energies in the conduction band of the crystal. The positive carriers are missing or defect "electrons" and are denoted by the term "holes". They represent unoccupied energy states in the uppermost normally tilled band of the crystal. The conductivity is called n- or p-type depending on whether the mobile charges normally in excess in the material under equilibrium conditions are electrons (negative carriers) or holes (positive carriers). The germanium used in the transistor is n-type with about 5 X 10^^ conduc- tion electrons per c.c; or about one electron per 10^ atoms. Transistor ac- tion depends on the fact that the current from the emitter is composed in PRINCIPLES OF TRANSISTOR ACTION 243 large part of holes; that is of carriers of opposite sign to those normally in excess in the body of the semi-conductor. The collector is biased in the reverse, or negative direction. Current flowing in the <2;ermanium toward the collector point j)rovides an electric Fig. 2 — Microphotograph of a cutaway model of a transistor field which is in such a direction as to attract the holes flowing from the emitter. When the emitter and collector are placed in close proximity, a large part of the hole current from the emitter will flow to the collector and into the collector circuit. The nature of the collector contact is such as to provide a high resistance barrier to the flow of electrons from the metal to the semi-conductor, but there is little impediment to the flow of holes into 244 BELL SYSTEM TECHNICAL JOURNAL / -/ io"' / / / t 1 1 — ,,^' ''°} / / -"vi 1 ^.' .-'' ^ ^^ / / ;«i X*" / / >J >>> / / / t • f / ^/^ • y ^ ■o^ / / / h-/ \ \, ,^ ov^ ^ ^ / s k ^ ^^ /' / / N D-'^ ^ \ \ \ \ \ r "^ -- O Q. ^■ N r \ \ V \ \ \ ^ ^ ' ^^0 7^ / \ \ -N \ \ > 1 7\ r^ V / / / \, \ \ \ N \ \ > \ \ N ^ 1 \ \ i s, ^ > ^ N \ S •"l^ ^V \ \ w '■^ >» ^•v^ \ X --. ^ 1 ^ ^-1^-X ^ N 'x X N \ r °^ "-X .^ '\ \ \ \ ^"^x "* -x^ ^>» V \ *^ '^ ^ v\ V ^ q ^ (/I -a tr rt w C a. o i < c _j > — i o oH U SnOA Nl 3a- (^ rRlSCIPLES or transistor actios 245 the contact. This theory explains how the change in collector current might l)e as large as but not how it can be larger than the change in emitter current. The fact that the collector current may actually change more than the emitter current is believed to result from an alteration of the space charge in the barrier layer at the collector by the hole current flowing into the junction. The increase in density of space charge and in field strength makes it easier for electrons to llow out from the collector, so that there is an increase in electron current. It is better to think of the hole current from the emitter as modifying the current-voltage characteristic of the collector, rather than as sim{)ly adding to the current flowing to the collector. In Section III we discuss the nature of the conductivity in germanium, and in Section I\' the theory' of the current-voltage characteristic of a ger- manium-point contact. In the latter section we attempt to show why the emitter current is composed of carriers of opposite sign to those normally in excess in the body of germanium. Section \' is concerned with some aspects of the theory- of transistor action. A complete quantitative theory is not yet available. There is evidence that the rectifying barrier in germanium is internal and occurs at the free surface, independent of the metal contact.^' ^^ The bar- rier contains what Schottky and Spenke^^ call an inversion region; that is a change of conductivity type. The outermost part of the barrier next to the surface is p-tj-pe. The p-type region is xQvy thin, of the order of 10~^ cm in thickness, .^n important question is whether there is a sufficient density of holes in this region to provide appreciable lateral conductivity along the surface. Some evidence bearing on this point is described below. Transistor action was first discovered on a germanium surface which was subjected to an anodic oxidation treatment in a glycol borate solution after it had been ground and etched in the usual way for diodes. Much of the early work was done on surfaces which were oxidized by heating in air. In both cases the oxide is washed off and plays no direct role. Some of these surfaces were tested for surface conductivity by potential probe tests. Surface conductivities, on a unit area basis, of the order of .0005 to .002 mhos were found.- The value of .0005 represents about the lower limit of detection possible by the method used. It is inferred that the observed surface conductivity is that of the p-type layer, although there has been no direct proof of this. In later work it was found that the oxidation treatment is not essential for transistor action. Good transistors can be made with surfaces prepared in the usual way for high-back-voltage rectifiers provided that the collector point is electrically formed. Such surfaces exhibit no measurable surface conductivity. One question that may be asked is whether the holes flow from the emitter to the collector mainly in the surface layer or whether they flow 246 BELL SYSTEM TECHNICAL JOURNAL through the body of the germanium. The early experiments suggested flow along the surface. W. Shockley proposed a modified arrangement in which in effect the emitter and collector are on opposite sides of a thin slab, so that the holes flow directly across through the semi-conductor. Inde- pendently, J. N. Shive made, by grinding and etching, a piece of germanium in the form of a thin flat wedge. ^^ Point contacts were placed directly opposite each other on the two opposite faces where the thickness of the wedge was about .01 cm. A third large area contact w^as made to the base of the wedge. When the two points were connected as emitter and collec- tor, and the collector was electrically formed, transistor action was obtained which was comparable to that found with the original arrangement. There is no doubt that in this case the holes are flowing directly through the n- type germanium from the emitter to the collector. With two points close together on a plane surface holes may flow either through the surface layer or through the body of the semi-conductor. Still later, at the suggestion of W. Shockley, J. R. Haynes-" further es- tablished that holes flow into the body of the germanium. A block of germanium was made in the form of a thin slab and large area electrodes were placed at the two ends. Emitter and collector electrodes were placed at variable separations on one face of the slab. The field acting between these electrodes could be varied by passing currents along the length of the slab. The collector was biased in the reverse direction so that a small d-c. current was drawn into the collector. A signal introduced at the emitter in the form of a pulse was detected at a slightly later time in the collector circuit. From the way the time interval, of the order of a few microseconds, depends on the field, the mobility and sign of the carriers were determined. It was found that the carriers are positively charged, and that the mobility is the same as that of holes in bulk germanium (1700 cmVvolt sec) . These experiments clarify the nature of the excess conductivity observed in the forward direction in high-back-voltage germanium rectifiers which has been investigated by R. Bray, K. Lark-Horovitz, and R. N. Smith^i and by Bray.^^ These authors attributed the excess conductivity to the strong electric field which exists in the vicinity of the point contact. Bray has made direct experimental tests to observe the relation between con- ductivity and field strength. We believe that the excess conductivity arises from holes injected into the germanium at the contact. Holes are introduced because of the nature of the barrier layer rather than as a direct result of the electric field. This has been demonstrated by an experunent of E. J. Ryder and W. Shockley .^^ A thin slab of germanium was cut in the form of a pie-shaped wedge and electrodes placed at the narrow and wide boundaries of the wedge. When a current is passed between the electrodes, PRINCIPLES OF TRAXSfSPOR ACTION 247 the field strength is large at the narrow end of the wedge and small near the o[)posite electrode. An excess conductivity was observed when the nariow end was made j)ositive; none when the wide end was positive. The magnitude of the current flow was the same in both cases. Holes injected at the narrow end lower the resistivity in the region which con- tributes most to the over-all resistance. When the current is in the oppo- site direction, any holes injected enter in a region of low field and do not have sufficient life-time to be drawn down to the narrow end and so do not alter the resistance ver\^ much. With some surface treatments, the excess conductivity resulting from hole injection may be enhanced by a surface conductivity as discussed above. The experimental procedure used during the present investigation is of interest. Current voltage characteristics of a given point contact were displayed on a d-c. oscilloscope.-^ The change or modulation of this char- acteristic produced by a signal impressed on a neighboring electrode or point contact could be easily observ^ed. Since the input impedance of the scope was 10 megohms and the gain of the amplifiers such that the lower limit of sensitivity was of the order of a millivolt, the oscilloscope was also used as a xers' high impedance voltmeter for probe measurements. Means were included for matching the potential to be measured with an adjustable d-c. potential the value of which could be read on a meter. A micromanipulator designed by W. L. Bond was used to adjust the positions of the contact points. II — Some Transistor Ch.\racteristics The static characteristics of the transistor are completely specified by four variables which may be taken as the emitter and collector currents, le and Ic, and the corresponding voltages, Ve and Vc. As shown in the schematic diagram of Fig. 1, the conventional directions for current flow are taken as positive into the germanium and the terminal voltages are relative to the base electrode. Thus /« and Ve are normally positive, Ic and Vc negative. There is a functional relation between the four variables such that if two are specified the other two are determined. Any pair may be taken as the independent variables. As the transistor is essentially a current operated device, it is more in accord with the physics involved to choose the currents rather than the voltages. All fields in the semi-conductor outside of the space charge regions immediately surrounding the point contacts are determined by the currents, and it is the current flowing from the emitter which controls the current voltage characteristic of the collector. The voltages are single-valued functions of the currents but, because of inherent feedback, the currents may be double-valued functions of the voltages. 248 BELL SYSTEM TECHNICAL JOURNAL In reference 1, the characteristics of an experimental transistor were shown by giving the constant voltage contours on a plot in which the independent variables le and Ic are plotted along the coordinate axes. In the following we give further characteristics, and show in a general way how they depend on the spacing between the points, on the tempera- ture, and on the frequency. The data were taken mainly on experimental setups on a laboratory bench, and are not to be taken as necessarily typical of the characteristics of finished units. They do indicate in a general way the type of results which can be obtained. Characteristics of units made in pilot production have been given elsewhere.^ The data plotted in reference 1 were taken on a transitor made with phos- phor bronze points on a surface which was oxidized and on which potential probe tests gave evidence for considerable surface conductivity. The col- lector resistance is small in units prepared in this way. In Fig. 3 are shown the characteristics of a unit^^ in which the surface was prepared in a differ- ent manner. The surface was ground and etched in the usual way", but was not subjected to the oxidation treatment. Phosphor bronze contact points made from 5 mil wire were used. The collector was electrically formed by passing large currents in the reverse direction. This reduced the resistance of the collector in the reverse direction, improving the transis- tor action. However, it remained considerably higher than that of the collector on the oxidized surface. While there are many ways of plotting the data, we have chosen to give the collector voltage, Vc, as a function of the collector current, Ic, with the emitter current, le, taken as a parameter. This plot shows in a direct manner the mfluence of the emitter current on the current-voltage char- acteristic of the collector. The curve corresponding to /« = 0 is just the normal reverse characteristic of the collector as a rectifier. The other curves show how the characteristic shifts to the right, corresponding to larger collector currents, with increase in emitter current. It may be noted that the change in collector current for fixed collector voltage is larger than the change in emitter current. The current amplification factor, a, defined by a = — (5/c/<9/e)vv = const. (2-1) is between 2 and 3 throughout most of the plot. The dotted lines on Fig. 3 correspond to constant values of the emitter voltage, Ve- By interpolating between the contours, all four variables corresponding to a given operating point may be obtained. The Ve con- tours reach a maximum for /« about 0.7 ma. and have a negative slope beyond. To the left of the maximum, V, increases with L as one follows along a line corresponding to Vc = const. To the right, Ve decreases as PRINCIPLES OF TRANSISTOR ACTION 249 le increases, corresponding to a negative input admittance. For given values of I',, and ]',., there are two possible operating points. Thus for W = (^-l and Vc = —20 one may have /« = 0.3 ma, Ic = —1.1 ma or /, = 1.0, Ic = -2.7. The negative resistance and instahiUly result from the effect of the col- lector current on the emitter current.^ The collector current lowers the potential of the surface in the vicinity of the emitter and increases the efTective bias on the emitter by an equivalent amount. This potential drop is RfIc, where Rp is a feedback resistance which may depend on the currents flowing. The effective bias on the emitter is then Ve — RfU, and we may write 7, = J{Ve - RfIc), (2.2) where the function gives the forward characteristic of the emitter point. In some cases Rp is approximately constant over the operating range; in other cases Rp decreases with increasing h as the conductivity of the ger- manium in the vicinity of the points increases with forward current. In- crease of le by a change of Ve increases the magnitude of Ic, which by the feedback still further increases le. InstabiUty may result. Some conse- quences will be discussed further in connection with the a-c. characteristics. Also shown on Fig. 3 is a load line corresponding to a battery voltage of — 100 in the output circuit and a load, Rl, of 40,000 ohms, the equation of the line being Vc= - 100 - 40 X 10 Uc. (2.3) The load is an approximate match to the collector resistance, as given by the slope of the solid lines. If operated between the points Pi and P2, the output voltage is 8.0 volts r.m.s. and the output current is 0.20 ma. The corresponding values at the input are 0.07 and 0.18, so that the over- all power gain is Gain '-' 8 X 0.20/(0.07 X 0.18) -^ 125, (2.4) which is about 21 db. This is the available gain for a generator with an impedance of 400 ohms, which is an approximate match for the input impedance. We turn next to the equations for the a-c. characteristics. For small deviations from an operating point, we may write AVe = Ru Me + i^l2 A/„ (2.5) AVc = ^12 Me + R22 Mc, (2.6) in which we have taken the currents as the independent variables and the directions of currents and voltages as in Fig. 1. The differentials represent 250 BELL SYSTEM TECHNICAL JOURNAL small changes from the operating point, and may be small a-c. signals. The coefficients are defined by: i?U= (aFe/a/e)/,= const.. (2.7) Rn = (aFe/a/c)/. = const.. (2.8) R2I = {dVc/dIe)l, = const.. (2.9) R22= {dVjdh)l,=. const. (2.10) These coefl5cients are all positive and have the dimensions of resistances. They are functions of the d-c. bias currents, /« and Ic, which define the operating point. For le = 0.75 ma and /, = —2 ma the coefficients of the unit of Fig. 3 have the following approximate values: Rn = 800 ohms, Rn = 300, ^2 in i?2i = 100,000, ^ ■ i?22 = 40,000. Equation (2.5) gives the emitter characteristic. The coefficient Rn is the input resistance for a fixed collector current (open circuit for a-c). To a close approximation, Rn is independent of h, and is just the forward resistance of the emitter point when a current le is flowing. The coefficient Rn is the feedback or base resistance, and is equal to Rf as defined by Eq. (2.2) in case Rf is a constant. Both Rn and Rn are of the order of a few hundred ohms, Ru usually being smaller than Rn- Equation (2.6) depends mainly on the collector and on the flow of holes from the emitter to the coUector. The ratio R-n/Rn is just the current am- plification factor a as defined by Eq. (2.1). Thus we may write: AFc = i?22 (aA/e + Mc). (2.12) The coefficient i?22 is the collector resistance for fixed emitter current (open circuit for a-c), and is the order of 10,000-50,000 ohms. Except in the range of large h and small h, the value of R^a. is relatively independent of Ic. The factor a generally is small when h is small compared with h, and increases with h, approaching a constant value the order of 1 to 4 when Ic is several times /«. The a-c. power gain with the circuit of Fig. 1 depends on the operating point (the d-c bias currents) and on the load impedance. The positive feedback represented by R^ increases the a\-ailable gain, and it is possible to get very large power gains by operating near a point of instability. In giving the gain under such conditions, the impedance of the input generator should be specified. Alternatively, one can give the gain which would exist with no feedback. The maximum a\ailable gain neglecting feed- back, obtained when the load R^ is equal to the collector resistance R^, PRINCIPLES OF TRANSISTOR ACTION 251 and the impedance of the generator is equal to the emitter resistance, Ru, is: Gain = a^R22/4.Riu (2.13) which is the ratio of the collector to the emitter resistance multipUed by 1/4 the square of the current amplification factor. This gives the a-c. power delivered to the load divided by the a-c. power fed into the tran- sistor. Substituting the values Usted above (Eqs. (2.11)) for the unit whose characteristics are shown in Fig. 3 gives a gain of about 80 times (or 19 db) for the operating point Pq. This is to be compared with the gain of 21 db estimated above for operation between Pi and P^. The difference of 2 db represents the increase in gain by feedback, which was omitted in Eq. (2.13). Equations (2.5) and (2.6) may be solved to express the currents as func- tions of the voltages, giving A7e= FnAF.-f 7i2AF, (2.14) Ale = Yn AVe + F22 AF. (2.15) where Fii = R22/D, F12 = -Rn/D , . F12 = -i?2i/£>, F22 = RxxlD ^ ■ ^ and D is the determinant of the coefficients D = Rn R22 - R12 R21. (2.17) The admittances, Fn and F22, are negative if D is negative, and the tran- sistor is then unstable if the terminals are short-circuited for a-c. currents. Stability can be attained if there is sufiicient impedance in the mput and output circuits exterior to the transistor. Feedback and instability are increased by adding resistance in series with the base electrode. Further discussion of this subject would carry us too far into circuit theory and applications. From the standpoint of transistor design, it is desirable to keep the feedback resistance, Rn, as small as possible. Variation with Spacing One of the important parameters affecting the operation of the transistor is the spacing between the point electrodes. Measurements to investigate this effect have been made on a number of germanium surfaces. Tests were made with use of a micro-manipulator to adjust the i^ositions of the points. The germanium was generally in the form of a slab from .05 to 0.20 cm'thick, the lower surface of which was rhodium plated to form a low resistant contact, and the upper plane surface ground and etched, or other- 252 BELL SYSTEM TECHNICAL JOURNAL wise treated to give a surface suitable for transistor action. The collector point was usually kept foed, since it is more critical, and the emitter point moved. Measurements were made with formed collector points. Most of the data have been obtained on surfaces oxidized as described below. As expected, the emitter current has less and less influence on the collec- tor as the separation-^, s, is increased. This is shown by a decrease in i?2i, or a, with s. The effect of the collector current on the emitter, represented by the feedback resistance R12, also decreases with increase in 5. The other coefficients, Ru and R^i, are but little influenced by spacing. Figures 100 ^ 80 O -. 60 \ i 0 POSITION A A POSITION B \ \ f N \ K- - N [ 1 - \ ^^ - (; ^ ^ ^*~^-^ 0.010 0.015 0.020 SEPARATION IN CENTIMETERS Fig. 4 — Dependence of feedback resistance Rvi on electrode separation for two differ- ent parts A and B, of the same germanium surface. The surface had been oxidized by heating in air. 4, 5 and 6 illustrate the variation of Rn and a with the separation. Shown are results for two different collector points A and B on different parts of the same germanium surface-^. In making the measurements, the bias currents were kept fixed as the spacing was varied. For collector A, le = 1.0 ma and Ic = 3.8 ma; for collector B, /<, = l.U ma and /c = 4.0 ma. The values of Rn and R22 were about 300 and 10,000, respectively, in both cases. Figure 5 shows that a decreases approximately exponentiaUy with s for separations from .005 cm to .030 cm, the rate of decrease being about the same in all cases. Extrapolating down to 5 = 0 mdicates that a further PR[\'CIPLES or TRAXSISTOR ACTION 253 increase of only about 2S i)er cent in a could be obtained by decreasing the spacing below .005 cm. Figure 6 shows that the decrease of a with distance is dependent on the germanium sample used. Curve 1 is similar to the results in Fig. 5. Curve o \- < u. 3 1.0 a B 0-8 < Z 0.6 LU cr a. ^ 0.4 \ 0 POSITION A A POSITION B 1 r^ ^^^^ - \ - X^' ^^^ - N ^\ O.OtO 0.015 0.020 SEPARATION IN CENTIMETERS 0.030 Fig. 5 — Dependence of current amplificat'on factor a on electrode separation. Po sitions A and B as in Fig. 4. H 1,0 < U 0.9 . t 0.8 n ? 0./ < 1- 06 ^ 111 cr O.S ir ■) u 0.4 0, . 2 0.010 0.015 0.020 SEPARATION IN CENTIMETERS Fig. 6 — Dependence of current amplification factor a on separation for germanium surfaces from two ditTerent melts, 1 and 2. 2 is for a germanium slice with the same surface treatment but from a differ- ent melt. Figure 4 shows the corresponding results for R^^- There is an approxi- mate inverse relationship between R12 and s. Another way to illustrate the decreased intluence of the emitter on the collector with increase in spacing is to plot the collector characteristic for fixed emitter current at different spacings. Figure 7 is such a plot for a 254 BELL SYSTEM TECS MICA L JOURNAL / / i\ / / / / / / / I §/ / / / / / / / / / A / f / / / 7 / f / r / / / / f\ r^ / / / / y / / 0^ / / y ,y y ,^ Xl y y X y^ y^ ^ ^ tion. 0 0 4 0,8 1.2 1.6 20 24 2.8 3.2 3 6 4 0 4.4 4.8 5.2 5 6 6.0 6.4 6 8 -Ic IN MILLIAMPERES Fig. 7 — Collector characteristic Y c vs /« for fixed /, but variable distance of separa 0.005 0.025 0.010 0.015 0.020 SEPARATION IN CENTIMETERS Fig. 8 — Emitter current /« vs separation for fixed Ic and V c 0.030 different surface which was ground Hat, etched, and then oxidized at 500°C in moist air for one hour. The resultant oxide film was washed off.^'' The emitter current /« was kept constant at 1.0 ma. PRINCIPLES OF TILlNSrSTOR ACTION 255 Data taken on the same surface have been plotted in other ways. As the spacing increases, more emitter current is required to produce the same change in collector current. The fraction of the emitter current which is 0.005 0.010 0.015 0.020 SEPARATION IN CENTIMETERS 0.030 Fig. 9-— The factor t; is the ratio of the emitter current extrajiolatcd to .? = 0 to that at electrode separation .v retjuired to give the same collector current, /, and voltage, !'«. Plot shows variation of g with s for different /,. The factor is independent of I'c over the range plotted. effective at the collector decreases with spacin^^ It is of interest to kee}) Vc and /, fi.xed by varying /, as s is changed and to plot the values of le so obtained as a function of 5. Such a plot is shown in Fig. 9. The collec- 256 BELL'JYSTEM2TECHNICAL JOURNAL ■ 1.00 \ • Ic = -2 MILLIAMPERES \ X Ic=-4 II 0 Ic--6 II D Ic =-8 II \ \ \ 0 75 0.70 \ \ 0.65 \ 0.60 \ 0.55 \ g 0.50 0.45 \ \ \ • 0.35 \ 0.30 0.25 ^ \ o\ 0.15 K . \ \^ 0.10 x\ • 0.05 0 0 2 4 6 8 10 12 14 16 16 20 22 24 S/lJ/3 X10-2 Fig. 10— The factor g (Fig. 9) plotted as a function of j//c'", with ^ in cm. and h in amps. tor voltage, Vr, is fixed at —15 volts. Curves are shown for /, = —3, — 4, —6, and —8 ma. We may define a geometrical factor, g, as the ratio of /« extrapolated to zero spacing to the value of le at the separation s: g(s) = (/e(0)/7e(^)) V,./,=const. (2.18) It is to be expected that g{s) will depend on /,, as it is the collector current which provides the field which draws the holes into the collector. For the PRINCIPLES OF TRANSISTOR ACTION 257 same reason, it is expected that g(s) will be relatively independent of Vc- This was indeed found to be true in this particular case and the values r,- = —5, —10, and —15 were used in Figure 9 which gives a plot of g versus 5 for several values of h. The dotted lines give the extrapolation to 5 = 0. As expected, g increases with h for a fixed s. The different cur\^es can be brought into approximate agreement by taking s/Ic as the independent variable, and this is done in Fig. 10. As will be discussed in Section \', such a relation is to be expected if g depends on the transit time for the holes. \\\RIATION WITH TEMPERATURE Only a limited amount of data has been obtained on the variation of transistor characteristics with temperatures. It is known that the reverse < 1.56 < (.0^ 2 0.5 § -50 -40 -30 -20 -10 0 10 20 30 ^ TEMPERATURE IN DEGREES CENTIGRADE Fig. 11— Current amplification factor a vs. temperature for two experimental units A and B. characteristic of the germanium diode varies rapidly with temperature, particularly in the case of units with high reverse resistance. In the tran- sistor the collector is electrically formed in such a way as to have relatively low reverse resistance, and its characteristic is much less dependent on temperature. Both 7^22 and Rn decrease with increase in T, R^z usually decreasing more rapidly than Ru. The feedback resistance, Rn, is rela- tively independent of temi)erature. The current multiplication factor, a, increases with temperature, but the change is not extremely rapid. Figure 11 gives a plot of a versus T for two experunental units. The d-c. bias currents are kept fixed as the temperature is varied. The over-all change in a from -50°C to +50°C is only about 50 per cent. The increase in a with T results in an increase in power gain with temperature. This may be nullified by a decrease in the ratio Rii/Rn, so that the over-all gain at fixed bias current may have a negative temperature coefficient. 25^ BELL SYSTEM TECHNICAL JOURNAL Variation with Frequency Equations (2.5) and (2.6) may be used to describe the a-c. characteristics at liigh frequencies if the coefficients are replaced by general impedances. Thus if we use the small letters ie, Ve, ic, Vc to denote the amplitude and phase of small a-c. signals about a given operating point, we may write Ve = Zii ie + Znic, (2.19) Vc = ^21 ie + ^22^c. (2.20) 0.2 10^ I06 10' FREQUENCY IN CYCLES PER SECOND Fig. 12 — Current amplification factor a vs. frequency Measurements of A. J. Rack and others,-^ have shown that the over-all power gain drops off between 1 and 10 mc,' sec and few units have positive gain above 10 mc/sec. The measurements showed further that the fre- quency variation is confined almost entirely to Z21 or a. The other coeffi- cients, Zu, Zn and Z22, are real and independent of frequency, at least up to 10 mc/sec. Figure 12 gives a plot of a versus frequency for an experi- mental unit. Associated with the drop in amplitude is a phase shift which varies approximately linearly with the frequency. A phase shift in Z21 of 90° occurs at a frequency of about 4 mc/'sec, corresponding to a delay of about 5 X 10"* seconds. Estimates of transit time for the holes to flow from the emitter to the collector, to be made in Section V, are of the same order. These results suggest that the frequency limitation is associated with transit time rather than electrode capacities. Because of the difference PRINCIPLES OF TRANSISTOR ACTION 259 in transit times for holes following difTerent paths there is a drop in amplitude rather than simply a phase shift. Ill — Electrical Conductivity of Germanium Germanium, like carbon and silicon, is an element of the fourth group of the periodic table, with the same crystal structure as diamond. Each germanium atom has four near neighbors in a tetrahedral configuration with which it forms covalent bonds. The specific gravity is about 5..S5 and the melting point 958°C. The conductivity at room temperature may be either n or p tyjje, de- pending on the nature and concentration of impurities. Scaff, Theuerer, and Schumacher^^ have shown that group III elements, with one less valence electron, give p-type conductivity; group V elements, wdth one more va- lence electron, give n-type conductivity. This applies to both germanium , -CONDUCTION BAND ' *"~--FEBMI LEVEL Eg i Ea ACCEPTORS-, i_ 1 i _ y/////////////////////////}i'////y////////y^^^^ ' ^-FILLED BAND Fig. 13 — Schematic energy level diagram for germanium showing filled and conduction bands and donor and acceptor levels. and silicon. There is evidence that both acceptor (p-type) and donor (n-type) impurities are substitutional'''. A schematic energy level diagram'^ which shows the allowed energy levels for the valence electrons in a semi-conductor like germanium is given in Fig. 13. There is a continuous band of levels, the filled band, normally occupied by the electrons in the valence bonds; an energy gap. Eg, in which there are no levels of the ideal crystal; and then another continuous band of levels, the conduction band, normally unoccupied. There are just sufficient levels in the filled band to accomodate the four valence electrons per atom. The acceptor impurity levels, which lie just above the filled band, and the donor levels, just below the conduction band, correspond to electrons local- ized about the impurity atoms. Donors are normally neutral, but become positively charged by excitation of an electron to the conduction band, an energy Ed being required. Acceptors, normally neutral, are negatively ionized by excitation of an electron from the filled band, an energy Ea 260 BELL SYSTEM TECHNICAL JOURNAL being required. Both Ed and Ea are so small in germanium that practi- cally all donors and acceptors are ionized at room temperature. If only donors are present, the concentration of conduction electrons is equal to the concentration of donors, and the conductivity is n-type. If only ac- ceptors are present, the concentration of missing electrons, or holes, is equal to that of the acceptors, and the conductivity is p-type. It is possible to have both donor and acceptor type impurities present in the same crystal. In this case, electrons will be transferred from the donor levels to the lower lying acceptor levels. The conductivity t\T3e then depends on which is in excess, and the concentration of carriers is equal to the difference between the concentrations of donors and acceptors. It is probable that impurities of both types are present in high-back-voltage germanium. The relative numbers in solid solution can be changed by heat treatment, thus changing the conductivity and even the conductivity type.^^ The material used in rectifiers and transistors has a concentration of conduction electrons of the order of lO^Vc.c, which is about one for each 5 X 10^ atoms. The conductivity depends on the concentrations and mobilities of the carriers: Let /x^ and jXh be the mobilities, expressed in crnVvolt sec, and We and fih the concentrations (number/cm^) of the electrons and holes respec- tively. If both t^'pes of carriers are present, the conductivity, in mhos/cm, is a = iieCfie + nhenh, (3.1) where e is the electronic charge in coulombs (1.6 X 10~^'). Except for relatively high concentrations ('~ lO^Vcni^ or larger), or at low temperatures, the mobilities in germanium are determined mainly by lattice scattering and so should be approximately the same in different samples. Approximate values, estimated from Hall and resistivity data obtained at Purdue University^^ and at the Bell Laboratories^^ are: MA - 5 X 10«r-3''2, (3.2) M. = 7.5 X 10«r-='/2 (cmVvolt sec), (3.3) in which T is the absolute temperature. There is a considerable spread among the different measurements, possibly arising from inhomogeneity of the samples. The temperature variation is as indicated by theory. These equations give ma '^ 1000 and Mc ^^ 1500 cmVvolt sec at room tem- perature. The resistivity of high-back-voltage germanium varies from about 1 to 30 ohm cm, corresponding to values of ih between 1.5 X 10'^ and 4 X lO'Vcm'. At high temperatures electrons may be thermally excited from the tilled band to the conduction band, an energy Eo being required. Both the ex- PRJXCIPLES OF TRAXSISTOR ACTION 261 cited electron and the hole left behind contribute to the conductivity. The conductivities of all samples approach the same limitinj^ values re- gardless of impurity concentration, given by an equation of the form CT = (^^exp {-Ea'2kT), (34) where k is Boltzmann's constant. P'or germanium, a^ is about 3.3 X 10'* mhos cm and Ea about 0.75 ev. The exponential factor comes from the variation of concentration with temperature. Statistical theory^-' indicates that ite and Uh depend on tem- perature as n.= CeT'"exp(-^,/kT) (3.5a) nH = C,r''~expi-

,/kT) (3.5b) where ^pe is the energ>^ difference between the bottom of the conduction band and the Fermi level and (ph is the difference between the Fermi level and the top of the filled band. The position of the Fermi level depends on the impurity concentration and on temperature. The theorv chives Ce^Ch^ 2{2Trmk/h-y'- ~ 5 X 10'^ (3.6) where m is an effective mass for the electrons (or holes) and h is Plank's constant. The numerical value is obtained by using the ordinar>^ electron mass for m. The product neUk is independent of the position of the Fermi level, and thus of impurity concentration, and depends only on the temperature. From Eqs. (3.5a) and (3.5b) iieiih = CeChT^ exp { — Ea/kT). (3.7) In the intrinsic range, we may set ;;, = n,, = ;;, and find, using (3.1), (3.2), and {?>.i), an expression of the form (3.4) for a with .3) X W, a difference of a factor of 3.6. A similar discrepancy for silicon appears to be related to a varia- tion of Ea with temperature. With an empirical value of CJOh = 25 X KP' X 3.6- - 3 X UF, (3.9) Eq. (3.7) gives Heiik ~ lO-'/cm*' (3.10) 262 BELL SYSTEM TECHNICAL JOURNAL when evaluated for room temperature. Thus for «« '^ lO^Vcm', corre- sponding to high-back-voltage germanium, Hh is the order of 10'^. The equilibrium concentration of holes is small. Below the intrinsic temperature range, rie is approximately constant and Uh varies as nh = {CeCnT^/ue) exp (-Eo/kT). (3.11) IV — Theory of the Diode Characteristic Characteristics of metal point-germanium contacts include high forward currents, as large as 5 to 10 ma at 1 volt, small reverse currents, correspond- 1 FORWARD / (EXPANDED/ SCALE) / / / / / / ^ ( /rever; >E \ \ -180 -<60 -140 -120 -100 -80 -60 -40 -20 0 <.0 APPLIED VOLTAGE Fig. 14 — Current-voltage characteristic of high-back-voltage germanium rectifier- Note that the voltage scale in the forward direction has been expanded by a factor of 20. ing to resistances as high as one megohm or more at reverse voltages up to 30 volts, and the ability to withstand large voltages in the reverse direction without breakdown. A considerable variation of rectifier characteristics is found with changes in preparation and impurity content of the germa- nium, surface treatment, electrical power or forming treatment of the con- tacts, and other factors. A typical d-c. characteristic of a germanium rectifier^^ is illustrated in Fig. 14. The forward voltages are indicated on an expanded scale. The forward current at one volt bias is about 3.5 ma and the differential resist- ance is about 200 ohms. The reverse current at 30 volts is about .02 ma PRINCIPLES OF TRANSISTOR ACTION 263 and the differential resistance about 5 X 10^ ohms. The ratio of the for- ward to the reverse current at one volt bias is about 500. At a reverse voltage of about 160 the differential resistance drops to zero, and with further increase in current the voltage across the unit drops. The nature of this negative resistance portion of the curv-e is not completely under- stood, but it is believed to be associated with thermal effects. Successive l)oints along the curve correspond to increasingly higher temperatures of the contact. The peak value of the reverse voltage varies among different units. Values of more than 100 volts are not difficult to obtain. Theories of rectification as developed by Mott,='*' Schottky," and others^ have not been successful in explaining the high-back-voltage characteristic in a quantitative way. In the following we give an outline of the theory and its application to germanium. It is believed that the high forward currents can now be explained in terms of a flow of holes. The type of barrier which gives a flow of carriers of conductivity type opposite to that of the base material is discussed. It is possible that a hole current also plays an important role in the reverse direction. The Space-Charge L.4.yer According to the Mott-Schottky theor>^ rectification results from a potential barrier at the contact which impedes the flow of electrons between the metal and the semi-conductor. A schematic energy level diagram of the barrier region, drawn roughly to scale for germanium, is given in Fig. 15. There is a rise in the electrostatic potential energy of an electron at the surface relative to the interior which results from a space charge layer in the serai-conductor next to the metal contact. The space charge arises from positively ionized donors, that is from the same impurity centers which give the conduction electrons in the body of the semi-conductor. In the interior, the space-charge of the donors is neutralized by the space charge of the conduction electrons which are present in equal numbers. Electrons are drained out of the space-charge layer near the surface, leaving the immobile donor ions. The space charge layer may be a result of the metal-semi-conductor con- tact, in which case the positive charge in the layer is compensated by an induced charge of opposite sign on the metal surface. Alternatively, the charge in the layer may be compensated by a surface cliarge density of electrons trapped in surface states on the semi-conductor.* It is believed, for reasons to be discussed below, that the latter situation applies to high- back-voltage germanium, and that a space-charge layer exists at the free surface, independent of the metal contact. The height of the conduction band above the Fermi level at the surface, ' of the surface states. 264 BELL SYSTEM TECHNICAL JOURNAL That the space-charge layer which gives the rectifying barrier in ger- manium arises from surface states, is indicated by the following: (1) Characteristics of germanium-point contacts do not depend on the work function of the metal, as would be expected if the space-charge layer were determined by the metal contact. (2) There is little difference in contact potential between different samples of germanium with varying impurity concentration. Benzer^^ Fig. 15 — Schematic energj- level diagram of barrier layer at germanium surface show- ing inversion layer of p-type conductivity. found less than 0.1-volt difference between samples ranging from n-type with 2.6 X 10'^ carriers cm^ to p-type with 6.4 X 10'* carriers cm^. This is much less than the difference of the order of the energy gap, 0.75 volts, which would exist if there were no surface effects. (3) Benzer^" has observed the characteristics of contacts formed from two crystals of germanium. He finds that in both directions the charac- teristic is similar to the reverse characteristic of one of the cr^^stals in con- tact with a highly conducting metal-like germanium crystal. PRISX'IPLKS OF TRAXSISTOR ACTfOX 265 (4) One of the authors^' has observed a change in contact potential with Hght similar to that expected for a barrier layer at the free surface. Prior to Benzer's experiments, Meyerhof^ had shown that the contact l^otential difference measured between different metals and silicon showed little correlation with rectification, and that the contact potential differ- ence between n- and p-type silicon surfaces was small. There is thus evidence that the barrier layers in both germanium and silicon are internal and occur at the free surface". In the development of the mathematical theory of the space-charge layer at a rectifier contact, Schottky and Spenke'^ point out the possibility of a change in conductivity type between the surface and the interior if the potential rise is sutlficiently large. The conductivity is p-type if the Fermi level is closest to the tilled band, n-type if it is closest to the conduction band. In the illustration (Fig. 15), the potential rise is so large that the tilled band is raised up to a position close to the Fermi level at the surface. This situation is believed to apply to germanium. There is then a thin layer near the surface whose conductivity is p-type, superimposed on the n-type conductivity in the interior. Schottky and Spenke call the layer of opposite conductivity type an inversion region. Referring to Eqs. (3.5a and 3.5b) for the concentrations, it can be seen that since Ce and Ci, are of the same order of magnitude, the conductivity type depends on whether if,- is larger or smaller than ipi,. The conductivity is n-t}q3e when iPe < 1/2 Ea, ipk > 1/2 Eo, (4.1) and is p-t>'pe when the reverse situation applies. The maximum resistiv- ity occurs at the position where the conductivity type changes and ^e ~ 1/2 Ea, (4.3) or if the overall potential rise, ^pt,, is greater than 1/2 Ea - /m,o), (4.33) where fihb is the hole concentration at the semi-conductor boundary of the space charge layer and fiho is the normal concentration. For iihb -^ 5 X 10'« and ;/^o ~ 10'^, Vb is about 0.35 volts. PRINCIPLES OF TRANSISTOR ACTION 273 The increased conductivity caused by hole emission accounts not only for the large forward currents, but also for the relatively small dependence of spreading resistance on contact area. At a small distance from the con- tact, the concentrations and voltages are independent of contact area. The voltage drop within this small distance is a small part of the total and does not vary rapidly with current. We have assumed that the electron current, !«, at the contact is negligi- ble compared with the hole current, Ih. An estimate of the electron cur- rent can be obtained as follows: From the diode theory, le = (enebVaAc/-i) exp (—{fb — eVb)/kT), (4.34) since the electron concentration at the semi-conductor boundary of the space-charge layer is Ueb and the height of the barrier with the voltage applied is (fb — eVb- For simplicity we assume that both neb and Hkb are large compared wdth Ueo so that w^e may replace Ueb by Uhb without appre- ciable error. The latter can be obtained from the value of xj/ at the contact: lA = Ih/ia (4.35) Expressing ip in terms of fihb, we find Hhb = Ih/SkTfXha (4.36) Using (4.33) for Vb, and (3.5b) for Uko we find after some reduction, /. = Ih'/Icrit, (4.37) where _ 256 Ch (kTnh) T , ,, V f . Icrit = exp { — 1) can be accounted for along these lines. As discussed in Section II, there is an influence of collector current on emitter current of the nature of a positive feedback. The collector current lowers the potential of the surface in the vicinity of the emitter by an amount V = pIJl-KS (5.6) The feedback resistance Rp as used in Eq. (2.2) is Rr = p/lirs (5.7) For p = 10 ohm cm and s = .005 cm, the value of Rp is about 300 ohms, which is of the observed order of magnitude. It may be expected that Rp will decrease as p decreases with increase in emitter current. The calculations made in this section confirm the general picture which has been given of the way the transistor operates. \T — Conclusions Our discussion has been confined to the transistor in which two point contacts are placed in close proximity on one face of a germanium block. It is apparent that the principles can be applied to other, geometrical designs and to other semi-conductors. Some prehminary work has shown that tran- sistor action can be obtained with silicon and undoubtedly other semi-con- ductors can be used. Since the initial discovery, many groups in the Bell Laboratories have contributed to the progress that has been made. This work includes investigation of the physical phenomena involved and the properties of the materials used, transistor design, and measurements of characteristics and circuit applications. A number of transistors have been made for ex- perimental use in a pilot production. Obviously no attempt has been made 276 BELL SYSTEM TECHNICAL JOURNAL to describe all of this work, some of which has been reported on in other publications^ In a device as new as the transistor, various problems remain to be solved. A reduction in noise and an increase in the frequency limit are desirable While much progress has been made toward making units with reproducible characteristics, further improvement in this regard is also desirable. It is apparent from reading this article that we have received a large amount of aid and assistance from other members of the Laboratories staff, for which we are grateful. We particularly wish to acknowledge our debt to Ralph Bown, Director of Research, who has given us a great deal of encouragement and aid from the inception of the work and to William Shockley, who has made numerous suggestions which have aided in clarify- ing the phenomena involved. References 1. J. Bardeen and W. H. Brattain, Phys. Rev., 74, 230 (1948). 2. W. H. Brattain and J. Bardeen, Phys. Rev., 74, 231 (1948). 3. W. Shockley and G. L. Pearson, Phys. Rev., 74, 232 (1948). 4. This paper was presented in part at the Chicago meeting of the American Physical Society, Nov. 26, 27, 1948. W. Shockley and the authors presented a paper on "The Electronic Theory of the Transistor" at the Berkeley meeting of the National Academy of Sciences, Nov. 15-17, 1948. A talk was given by one of the authors (W. H. B.) at the National Electronics Conference at Chicago, Nov. 4, 1948. A number of talks have been given at local meetings by J. A. Becker and other members of the Bell Laboratories Staff, as well as by the authors. 5. Properties and characteristics of the transistor are given by J. A. Becker and J. N. Shive in Elec. Eng. 68, 215 (1949). A coaxial form of transistor is described by W. E. Kock and R. L. Wallace, Jr. in Elec. Eng. 68, 222 (1949). See also "The Transistor, A Crystal Triode," D. G. F. and F. H. R., Electronics, September (1948) and a series of articles by S. Young White in Audio Eng., .\ugust through December, (1948). 6. H. C. Torrey and C. A. Whitmer, Crvstal Rectifiers, McGraw-Hill, New York (1948). 7. J. H. ScafT and R. S. Ohl, Bell Svstem Tech. Jour. 26, 1 (1947). 8. W. E. Meyerhof, Phvs. Rev., 71, 727 (1947). 9. J. Bardeen, Phys. Rev., 71, 717 (1947). 10. W. H. Brattain and W. Shockley, Phys. Rev., 72, p. 345(L) (1947). 11. W. H. Brattain, Phys. Rev., 72, 345(L) (1947). 12. R. B. Gibney, formerly of Bell Telephone Laboratories, now at Los .\lamos Scientific Laboratory, worked on chemical problems for the semi-conductor group, and the authors are grateful to him for a number of valuable ideas and for considerable assistance. 13. J. H. Scafif and H. C. Theuerer "Preparation of High Back Voltage Germanium Rectifiers" NDRC 14-555, Oct. 24, 1945^See reference 6, Chap. 12. 14. The surface treatment is described in reference 6, p. 369. 15. The transistor whose characteristics are given in Fig. 3 is one of an experimented pilot production which is under the general direction of J. A. Morton. 16. See, for example, A. H. Wilson Semi-Conductors and Metals, Cambridge University Press, London (1939) or F. Seitz, The Modern Theory of Solids, McGraw-Hill Book Co., Inc., New York, N.Y., (1940), Sec. 68. 17. The nature of the barrier is discussed in Sec. IV. 18. W. Schottky and E. Spenke, Wiss. Verof. Siemens-Werke, 18, 225 (1939). 19. J. N. Shive, Phys. Rev. 75, 689 (1949). 20. J. R. Haynes, and W. Shockley, Phvs. Rev. 75, 691 (1949). 21. R. Bray, K. Lark-Horovitz and R. N. Smith, Phys. Rev.. 72, 530 (1948). 22. R. Bray, Phys. Rev., 74, 1218 (1948). 23. E. J. Ryder and W. Shockley, Phys. Rev. 75, 310 (1949). PRINCIPLES or TKANSISTOR ACTIO.X 277 24. This instrument was designed and built by H. R. Moore, who aided the authors a great deal in connection with instrumentation and circuit problems. 25. The surface had been o.xidized, and ])otential probe measurements (ref. (2)) gave evidence for considerat)le surface conductivity. 26. Measured between centers of the contact areas. 27. Potential probe measurements on the same surface, given in reference (2), gave evidence of surface conductivity. 28. Unpublished data. 29. J. H. ScalY, H. C. Theuerer, and E. E. Schumacher, "P-type and N-type Silicon and the Formation of the Photovoltaic Barrier in Silicon" (in publication). 30. G. L. Pearson and J. Bardeen, PZ/ys. Rev. March 1, 1949. 31. See, for example, reference 6, Chap. 3. 32. K. Lark-Horovitz, A. E. Middleton, E. P. Miller, and I. Walerstein, Pliys. Rev. 69, 258 (1946). 33. Hall and resistivity data at the Bell Laboratories were obtained by G. L. Pearson on samples furnished bv J. H. Scat^" and H. C. Theuerer. Recent hall measure- ments of G. L. Pearson'on single crystals of n- and p-type germanium give values of 2600 and 1700 cm-/volt sec. for electrons and holes, respectively at room tem- perature. The latter value has been confirmed by J. R. Haynes by measurements of the drift velocity of holes injected into n-type germanium. These values are higher, particularlv for electrons, than earlier measurements on polycrystalline samples. Use of the new values will modify some of the numerical estimates made herein, but the orders of magnitude, which are all that are significant, will not be affected. W. Ringer and H. Welker, Zeits. f. Naturforschung, 1, 20 (1948) give a value of 2000 cmVvolt sec. for high resistivity w-type germanium. 34. See R. H. Fowler, Statistical Mechanics, 2nd ed., Cambridge University Press, London (1936). 35. From unpublished data of K. M. Olsen. 36. N. F. Mott, Proc. Ro\: Soc, 171A, 27 (1939). 37. W. Schottkv, Zeits. f. Phys., 113, 367 (1939), Pliys. Zeits., 41, p. 570 (1940), Zeits f. Pliys., 118, p. 539 (1942). Also see reference 18. 38. See reference 6, Chap. 4. 39. S. Benzer, Progress Report, Contract No. W-36-039-SC-32020, Purdue Umversity, Sept. 1-Nov. 30. 1946. 40. S. Benzer, Phys. Rev., 71, 141 (1947). 41. Further evidence that the barrier is internal comes from some unpubbshed experi- ments of J. R. Haynes with the transistor. Using a fi.xed collector point, and keeping a fixed distance between emitter and collector, he varied the material used for the emitter point. He used semi-conductors as well as metals for the emitter point. While the impedance of the emitter point varied, it was found that equivalent emitter currents give changes in current at the collector of the same order for all materials used. It is believed that in all cases a large part of the for- ward current consists of holes. 42. The space charge of the holes in the inversion region of the barrier layer is neglected for simphcity. 43. Reference 6, Chap. 4. 44. S. Benzer "Temperature Dependence of High Voltage Germanium Rectifier D.C. Characteristics," N.D.R.C. 14-579, Purdue Univ., October 31, 1945. See refer- ence 6, p. 376. 45. See, for example, E. H. Kennard, Kinetic Theory of Gases, McGraw-Hill, Inc., New- York, N. Y. (1938) p. 63. 46. Reference 6, p. 377. 47. Obtained by plotting log ,? versus s^/Ic- This plot is not a straight line, but has an ujoward curvature corresponding to an increase in t with separation. The value given is a rough average, corresponding to s^Ic the order of 10^' cm^, amp. 48. One mav expect that the mobility will depend on field strength when the drift veloc- itv is as large as or is larger than thermal velocity. Since ours is a borderline case, the calculation using the low field mobility should be correct at least as to order of magnitude. Lightning Current Observations in Buried Cable By H. M. TRUEBLOOD and E. D. SUNDE Results are given of observations of lightning currents, voltages, and charges in a buried cable over most of three lightning seasons. These are compared with theoretical expectations. Data regarding the incidence of lightning strokes to ground, as observed with automatic recording equipment, are also reported, together with comparisons with similar data published previously. Introduction • LIGHTNING currents in buried telephone cable are of considerable ' importance in that they may cause excessive voltages between the cable sheath and the conductors with resultant insulation failure, and may also cause severe damage by crushing the cable or fusing holes in the sheath. The incidence of lightning strokes to buried cable, the resulting voltages, and lightning trouble expectancy, have therefore been subjects of theoreti- cal, experimental, and field studies, which, together with operating experience, have pointed the way to improvements in the design of com- munication cable to minimize its liability to lightning damage, and in the application of remedial measures where excessive lightning trouble has been experienced.^ The territory around Atlanta, Georgia, has appeared to be particularly severe with respect to these lightning hazards, because of high earth re- sistivity and high thunderstorm rate. Buried cables initially installed in this territory were accordingly provided with protective measures in the form of increased core-sheath insulation and shield wires buried with the cable. In spite of these measures, however, a substantially higher rate of lightning damage than anticipated was experienced, as a result of which a new design was adopted for the transcontinental coaxial cable westward from Atlanta. In this cable, the lead sheath was insulated from an out- side corrugated 10-mil copper shield by a 100-mil layer of thermoplastic insulation intended to prevent the entrance of lightning currents into the sheath and thereby to minimize voltages between the sheath and the cable conductors. Simulative tests with surge currents, believed to have a wave shape representative of lightning stroke current, had indicated satisfactory agree- ment between measured and calculated voltages between sheath and cable conductors. It appeared, therefore, that the departure from predicted * References are listed at end of paper. 278 LIGHTNING CURRENTS IN BURIED CABLE 279 lightning trouble expectancy in the earlier cables was due to one or more of the following conditions: a higher rate of occurrence of lightning strokes during thunderstorms, higher stroke currents than in other parts of the country, a longer duration of the lightning currents than assumed, or a higher incidence of strokes to buried cable than predicted theoretically. The observations described here, the larger part of which have extended over a period of three lightnmg seasons, were intended to secure informa- tion on these points. The data forming the principal subject of this paper were obtained from a section of the coaxial cable mentioned above, which for a number of reasons was particularly suitable for the purpose. I. Theoretical Expectations 1 .0 General As mentioned above, the observations were made on a cable route through territory of high thunderstorm rate and high earth resistivity, both of which facihtate measurements of currents along the cable. As a result of the high thunderstorm rate, the incidence of strokes to ground is high, and because of the high earth resistivity, the number of strokes to ground near the cable which flash to latter is also high. Another result of the high earth resistivity is that the attenuation of current along the cable is relatively low, so that currents and voltages may be observed at appre- ciable distances along the cable from the points of the lightning strokes. The rate of attenuation is, furthermore, smaller the longer the duration of the lightnmg current, that is, the longer the time to half-value. Since lightning trouble experience in this territory indicated the possibility of currents of rather long durations, this was an additional factor favorable to the purposes of the tests, although, like the others, it increases the liability of cables to Hghtning damage. Though the relationships of the various factors mentioned above to earth resistivity and to lightning current wave shape have been dealt with in detail in the study^ referred to above, they are briefly reviewed here to facilitate comparisons with and discussions of the observations. 1.1 Incidence of Strokes to Buried Cable The current in a lightning stroke to ground spreads in all directions from the point where it enters the earth. If a cable is in the vicinity, it will provide a low resistance path, so that much of the current will flow to the cable and in both directions along the sheath to remote points. The cur- rent in the ground between the lightning channel and the cable may give rise to a voltage drop along the surface of the earth sufficient to exceed the breakdown gradient of the soil, particularly when the earth resistivity is 280 BELL SYSTEM TECHNICAL JOURNAL high. The lightning stroke will then arc directly to the cable from the point where it enters the ground, often at the base of a tree. Furrows exceeding 100 feet in length have been found along the ground path of such an arc. For a crest current / in the lightning stroke, the arcing distance in meters is given by^ r = k{Jpyi^ (1) where / is in kiloamperes, p is the earth resistivity in meter-ohms and k is a constant depending on the surface breakdown gradient of the soil. Low resistivity soil, up to p = 100 meter-ohms, has an average breakdown gradient of about 2500 volts/cm, and the corresponding value of k is about ,08. For high resistivity soil, p = 1000 meter-ohms and up, the average breakdown gradient is about 5000 volts/cm and k = .047. Thus, for an earth resistivity of 2000 meter-ohms, and / = 100 ka, r = 21 meters or 70 feet. The number of strokes arcing to a cable of length ^ may conveniently be expressed as N = Ifsn (2) where n is the number of strokes to ground per unit area, f is an equivalent arcing distance, and Ifs an equivalent area near the cable within which the cable is assumed to attract all lightning strokes. In obtaining r, the number of strokes arcing to the cable from various distances r as given by (1), depending on the current in the strokes, must be evaluated. This number and the equivalent arcing distance will thus depend on the crest current distribution of lightning strokes. For the distribution curve shown by Curve 1 in Fig. 1, the effective distance in meters is^ r^ .36p'^^whenp < 100 meter-ohms ^ . 22p'/2 when p > 1000 meter-ohms ^^ ' Thus, for soil having an average resistivity near the surface (to a depth of at least 10 meters) of 2000 meter-ohms, r = 10 meters = ZZ feet. A cable will thus collect direct lightning strokes for an eflfective distance f to either side of it, and when the incidence of strokes to ground per unit area is known, the number of strokes to a cable of a given length may read- ily be calculated. The incidence of strokes to ground has been estimated, on the average, as about 2.4 per square mile for each 10 thunderstorm days, and the corresponding expectancy- of strokes to a buried cable, per 100 miles of length, is about 2.1 for 10 thunderstorm days when the earth re- sistivity is 1000 meter-ohms and 3.0 when it is 2000 meter-ohms. The distribution of the crest currents in direct strokes to a cable may be LIGHTNING CURRENTS IN BURIED CABLE 281 obtained by use of Cun-e 2 in Fig. 1, which is a theoretical curve derived from Curve 1. Thus for a total of 2.1 for 10 thunderstorm days, the inci- dence of strokes exceeding 60 ka is 2.1-0.2 = 0.42. In Fig. 2 are shown crest current distribution curves for cable currents due to direct strokes obtained in this manner, together with distribution curves for currents due to both direct strokes and strokes to ground not arcing to the cable. The latter curves may be obtained by methods similar to those used in evaluat- ing curves for the lightning trouble expectancy of buried cable, which are 220 200 \ \ s s N s s \ s s LU IT S s s Q. 160 < O 140 2 N s V s S 2 s s s s \ ■s - y- X. s s \ a. N s \ s o s s S \ \h 60 111 a. 's \ \ \ O 40 \ S \ S V 20 0 1. ± _L ± _L _L s ^ 0.01 0.02 0.05 0.1 0.2 0.5 1.0 2 3 4 5 6 8 10 20 30 50 tOO PERCENTAGE OF LIGHTNING STROKES IN WHICH CURRENT EXCEEDS ORDINATE Fig. 1 — Distribution of crest currents in lightning strokes. Curve 1. Currents in strokes to transmission line ground structures, based on 4410 measurements, 2721 in U. S. and 1689 in Europe. Curve 2. Currents in strokes to buried structures derived from Curve 1. shown in Fig. 3 for cable having a dielectric strength of 2 kv between the sheath and the cable conductors. ^ The latter curves may, in fact, be used to find the incidence of cable currents of various crest values due to direct strokes and strokes to ground, by calculating the cable currents required to produce 2 kv between the sheath and the core corresponding to various sheath resistances shown in Fig. 3. Thus for a sheath resistance of 2 ohms per mile and an earth resistivity of 1000 meter-ohms, this current is 14.2 ka (see Section 1.3) and for a sheath resistance of 1 ohm, it is 28.4 ka, etc., as plotted in Fig. 2 for an earth resistivity of 1000 meter-ohms. From the above it follows that a verification of the distribution curves in Fig. 2 by observations of lightning currents in buried cable would apply 282 BELL SYSTEM TECHNICAL JOURNAL equally well to the curves in Fig. 3, which have been used to evaluate the liability of such cable to lightning damage. 100 95 90 75 70 2 60 55 45 •J 40 35 30 25 20 \ \ EA IN RTH RESISTIVITY METER-OHMS \ \ \ 1000 \ \ \ \ \ \ \ V \ \ \ \ \ V \ \ \ \ \ \ V \s V \ \ N 1 < \ \ '\, \ \fc- DIRECT STROKES AND \-''\ STROKES TO GROUND \ '^ h\ \ \ \ V^^ \ \, \ \ \ V A DIRECT STROKES-- \ \ — *A- A \ \ \ "v. X \ \ \ \ ""N.^ 'V V ^. \ \ \ \ , \ 0.3 0.4 0.6 0.8 1.0 1.5 2 3 4 NUMBER OF CURRENTS EXCEEDING ORDINATE 5 6 7 8 9 10 Fig. 2 — Incidence of currents exceeding ordinate per 100 miles and 10 thunderstorm days. 1 .2 Propagation of Currents A long Cable The current entering the cable at or near the stroke point, depending on whether a direct stroke or a nearby stroke to ground is involved, is at- tenuated as it travels along the sheath towards remote points. The cur- rent leaving the sheath must flow through the adjacent soil, and the leak- LIGHTNING CURRENTS IN BURIED CABLE 283 age current is therefore smaller the higher the soil resistivity. Thus the current will travel farther the higher the earth resistivity. It has been shown elsewhere'- ^ that a sinusoidal current would be propagated as I(x) = 7(0) e-"" (4) - 2 cr 2 K5 5 1.0 o 0.9 2 0.8 (T 0.7 ^ 0.6 ^0.5 _l 0.4 O 0.2 / / / J / / / _> J / / / / / / / / / / f /'', / SHEATH RESISTANCE IN OHMS PER MILE = 2 / r / / / / / / / / / / 1 A 0.6// / / / / / / r/°' V / / / * // ' /o r 25 / / / / / / / // / / / // / / / / ^ // / f / / / / / / / / J f / / / / / f / r / / / / / / / / 1 MILE = 1.61 KM / / » / / 1 / / 1 / / 300 500 1000 2000 3000 EARTH RESISTIVITY IN METER-OHMS 10,000 Fig. 3 — Theoretical lightning trouble expectancy curves showing number of times insulation failures due to excessive voltages would be expected per 100 miles for 10 thun- derstorm days, for cables having sheath resistances as'indicated on curves and 2000 volts insulation between core and sheath. Dashed line represents full size cable. where 7(0) is the current in the sheath in one direction from the stroke point, I{x) the current at the distance x, and the propagation constant F per meter is given by: r = y/ii>iv/2p (5) where w = lirf, v = inductivity of the earth = 1.257-10"^ henry/meter, and p is the earth resistivity in meter-ohms. Let it be assumed that the current at the distance x has been evaluated for a given earth resistivity p and radian frequenc}^ co. If the earth re- 284 BELL SYSTEM TECHNICAL JOURNAL sistivity is increased by a factor k, or if co is decreased by the same factor, it is evident from (4) and (5) that the same current will be obtained at the distance xi = k^i'^x. Thus, if the earth resistivity is increased by yfe = 4, Xi = 2x and the current attenuation will be half as great as before. This rule applies to surge-currents of a given wave-shape as well, since they may be regarded as composed of sinusoidal components, each of which would > 0.6 0.5 0.3 / \ X= DISTANCE FROM POINT OF TRANSMISSION IN METERS /) = EARTH RESISTIVITY IN METER-OHMS \ ' \ X \, / 25"*^ ■^- X ^ / ^ 100 / ^ / r ^.^ ""400 / ^ 40 50 60 TIME IN MICROSECONDS Fig. 4 — Attenuation and distortion of surge current transmitted along a buried con- ductor. travel farther by the factor ^^/^. Furthermore, it follows by the same rea- soning that for surge-currents of congruent wave-shapes, but different dura- tions, the rate of attenuation is inversely proportional to the square root of the duration. That is to say, let in one case the current reach its crest value at the stroke point in 5 microseconds and its half-value, in 65 micro- seconds, and let the crest current at a given distance x have a certain value. Then, if in another case the current reaches the same stroke-point crest LIGnTNING CURRENTS IN BURIED CABLE 285 value in 20 microseconds, with half-value in 260 microseconds, the Same crest current as that found before at .v is obtained at twice the distance .v. This follows because all component frecjuencies of the hrst surge are related to corresponding components of the second surge by the same factor, viz. 4. 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 DISTANCE ALONG CABLE (FROM STROKE POINT) IN KILOFEET Fig. 5 — Variation in crest current in cable and in voltage between sheath and copper shield for stroke currents having various times to half-value as indicated on curves, for an earth resistivity of 1000 meter-ohms. In Fig. 4 are shown curves^ from which the attenuation may be obtained for a surge-current reaching its crest value in 5 microseconds and its half- value in 65 microseconds, as shown by the curve for .r^/p = 0. The crest current has attenuated by 50 per cent when x^'p = 25 or .v = 5p''"-^. Thus, for an earth resistivity of 1000 meter-ohms, .v = 160 meters =» 525 feet when the time to half-value of the current at the stroke point is 65 micro- seconds, while X = 1050 feet when the current at the stroke point reaches its half-value in 260 microseconds. In Fig. 5 are shown crest current 286 BELL SYSTEM TECtlNlCAL JOURNAL attenuation curves obtained in this manner, together with similar curves for the vohage, between the sheath and the core conductor of an ordinary cable, or between the copper shield and the lead sheath of the cable on which these observations were made. 1.3 Crest Values and Attenuation of Voltages The current along the copper shield produces a voltage between this shield and the lead sheath, due to the resistance drop along the shield from the stroke point to a point sufficiently remote for the current in the shield to have become negligible. This voltage is proportional to the unit-length resistance of the copper shield. From the considerations of the precedmg section, it follows that the voltage will be proportional to the square root of the earth resistivity and, if the wave-shape remains congruent but the duration of the current is changed, that it will be proportional to the square root of the duration or of the time to half -value. These two propositions follow from the fact that the voltage is proportional to the distance traveled by the current before it is attenuated to a given value. The crest voltage between the sheath and the cable conductors of an ordinary cable, or between the copper shield and the insulated lead sheath of a cable of the type on which these observations were made, is given by the following expression for a current reaching its half-value in 65 micro- seconds: V = 2.25 JRp'i^ (6) where F is in volts and / is the crest current in kiloamperes, R the resist- ance per mile of the outer envelope (in this case the copper shield), and p the earth resistivity in meter-ohms. This formula follows from expressions given in the paper referred to previously, which also contains curves from which the voltage attenuation along the cable shown in Fig. 5 may be obtained. For a resistance of .7 ohm/mile, which is that of the copper shield, and p = 1000 meter-ohms, the crest voltage for a current of 1 ka would thus be 50 volts; and, for a crest current of 200 ka, 10,000 volts. If the dielectric strength of the thermoplastic insulation exceeds 10 kv, the liability of such cable to lightning damage would thus be small, unless the time to half-value of the current substantially exceeds 65 microseconds. II. Experimental Installation 2.0 General From the preceding discussion it is seen that a lightning current dropping to half-value in some 50 to 75 microseconds, which is of the wave-shape ordmarily assumed, would attenuate to half its crest value in 500 to 1000 feet when the earth resistivity is from 1000 to 2000 meter-ohms. With LIGHTNING CURRENTS IN BURIED CABLE 287 15 «0 Z2 i2 off 25 9- •-^ UJO SUJ ■og -I •*3 O fa 28S BELL SYSTEM TECHNICAL JOURNAL test points along the cable at intervals of about 2300 feet, as employed in the observations, it should thus be possible to secure substantial indications at a number of points along the test section, although closer spacings would of course be desirable. INSTRUMENT I 1 i I I PLAN MARKER POST ELECTROLYSIS// WIRES GUARD POST o SURGE VOLTMETER- BOX ELEVATION SURGE INTEGRATOR ■'mmmmm//mmmm^m//^m''4 TT recorders: — crest-current -f-vJ SURGE -FRONT •E^^^^^g- j:.. il i v^/m// ^^JsL Fig, 7 — Typical instrument arrangement at splice points about 2300 feet apart. In Fig. 6 is shown the 21.5-mile test section and in Fig. 7 a typical installa- tion at every second splice in the cable. At these points the lead sheath is accessible through a lead gas-pressure pipe extending to a marker post,^ an arrangement which was utilized in making measurements of voltage between the sheath and the outside copper shield. At the same points an LIGnTNING CURRENTS //V BURIED CABLE 289 insulated wire is installed along the outside of the copper shield for meas- urement of voltage drop m the copper jacket, in connection with routine corrosion surveys. This facilitated measurements of the charge transferred along the shield by lightning currents, as described in the following. Mag- netic link instruments intended to measure the steepness of the wave front were also employed, but lacked the sensitivity required for accurate meas- urements and are not discussed further here. In addition to these measurements of current, charge, and voltage, in- volving the cable structure, observations were also made of the incidence of strokes to ground as described later in this paper. 2.1 Crest Ctirrent Measurements To measure crest currents in the cable, magnetic links' were mounted at distances of 1.6, 5.7, 12.7, and 36.4 inches from the center of the cable, to cover a current range from 1 to 220 ka. The readmgs on these magnetic links indicated the total current in the cable, that is, the sum of the currents in the copper shield, the lead sheath, and inside cable conductors. 2.2 Measurements of Charge These measurements were made by means of surge integrators* In principle this instrument consists of a shunt Ro across which is connected a coil of inductance L. When a surge current 7o(/) passes through the shunt, the current 7(0 in the coil is given by; l'^-^ = Ro hit) lit) = J j' hit) dt = ^ QoiO where Qoit) is the charge which has passed through the shunt up to the tune t. By measuring the crest value / of the current /(/), the total charge may be obtained from the relation: This relation is always valid if the surge current rises to a peak value and then decays uniformly. The relation should provide a good approxima- tion to the total charge conveyed by natural lightning strokes, even if there are small oscillations. The shunt i^o consisted of about 26 feet of the copper shield over the cable, 290 BELL SYSTEM TECHNICAL JOURNAL which had a resistance of about 3.5 milliohms. The inductance L consisted of two coils connected in series, each containing a magnetic link. The larger of these coils had 187 turns of copper wire, the smaller 50 turns. The larger coil provided the greater sensitivity, on account of the more intense magnetization of the link. The relation between the current / in the coil and the deflection on the magnetic link meter^ used to measure the intensity of magnetization was obtained by calibration with direct current. The inductance of the two coils in series was about 700 microhenries and the d-c resistance about .39 ohms. The time constant of the coils L/R was thus about 1800 microseconds, which is large compared to the duration of the main surge of a lightning stroke, which may last for 100 to 500 mi- croseconds. The instrument will thus effectively integrate the main surge, but will not record the charge which may be caused by a small tail current of much longer duration. The measurements of charge were made mainly to determine the time to half-value of the currents. The theoretical curves in Part I and else- where in this paper are based on a current of the form: J{t) = 1.157 (e~°' -r'') where a = .013 • 10*, 6 = .5 • 10^ for a current reaching its crest value / in about 5 microseconds and decaying to its half-value in 65 microseconds. If a = R/L = .00056-10® for the surge integrator, the total charge re- corded for a current of the above wave shape is Q = /1. 15 r^— - r^ = /•83-10-« for a current decaying to its half-value in 65 microseconds. The relation- ship between Q/J and the time to half-value, /1/2, is as follows for currents reaching their half- values in 65, 130, 260, and 520 microseconds: Q/J: 83 160 295 540 microseconds /1/2 : 65 130 260 520 microseconds From a curve of Q/J versus /1/2, the time to half-value may be obtained from the observed ratio of charge to crest current. The values given later on, in Table I, were obtained in this manner. If a triangular wave shape had been assumed, the times to half-value would have been Q/J and there- fore somewhat longer. 2.3 Measurements oj Voltage Between Sheath and Copper Shield These measurements were made by means of a magnetic link voltmeter consisting of a solenoid of inductance L containing the magnetic link and a LIGHTNING CURRENTS IN BURIED CABLE 291 series resistance, R. When a constant voltage E is suddenly applied the current through the coil is / = I (1 - e-^'^'^^^) If the time constant L/R is small in comparison with the time to crest value of a variable voltage to be measured, there will be no material delay be- tween the crest value of the voltage and that of the current in the coil. The applied voltages may therefore be obtained by multiplying the coil current as obtained from the magnetic link reading by the series resistance, provided the latter is much greater than the impedance of the circuit to which the instrument is connected. Since the voltmeter in the present case was designed to measure the voltage between the sheath and the cop- per shield, and the surge impedance of this test circuit is less than 3 ohms, comparatively low values of series resistance could be used. Three sepa- rate solenoids and series resistances were used, to provide three voltage ranges, from 0 to 1 .5 kv, 0 to 4 and 0 to 9 kv. 2.4 Magnetic Link Calibrations When several magnetic links, which have been exposed to the same field, are inserted in the magnetic link meter, considerable differences in the de- flections may be observed due to variations in the material of the links. For this reason, all links used in this installation were placed in a magnetic field of 257 gauss and were then classified accordmg to their response in the magnetic link meter. This field was such as to produce deflections in the most useful part of the meter range, centering around mid-scale. By this method the magnetic links used in the installation were divided into four classes, in accordance with the ratio of the deflection observed on the magnetic link meter for the link in question to the average deflection for all links. The maximum deviation from the average in each class was about ± 3 per cent. Instruments of the same t>T3e at all installations were provided with links of the same class, to minimize inaccuracies. 2.5 Observations of Incidence of Strokes to Ground To obtain data on the incidence of strokes to ground, observations were made at one location within the test section, by means of an automatically operated magnetic wire recorder arranged to record thunder picked up by a microphone. The recorder was provided with a triggermg arrangement which put it in operation on the approach of a thunderstorm, by action of the voltage induced in an antenna by lightning current. The induced voltage from a given lightning stroke was also made to record itself upon the magnetic wire; and, from the delay between this indication and the 292 BELL SYSTEM TECHNICAL JOURNAL recorded thunder, the distance to the lightning stroke could be determined upon play-back of the wire record. In this manner the number of strokes to ground within areas of various radii around the observation point could be ascertained, and thus the incidence of strokes to ground. These ob- servations were made during the 1947 and 1948 lightning seasons. III. Results of Observations 3.0 General From the preceding discussion of theoretical expectations and of the ex- perimental arrangement, it is evident that considerable attenuation would take place between the stroke point and the nearest test points on either side, for a stroke midway between the latter. Accurate evaluation of the maximum current, voltage, and charge, and of the current wave-shape, would thus be rather difficult for strokes nearly midway between test points, since these quantities would have to be evaluated by extrapolation from the obser\'ations at the points along the cable. Such extrapolation is rendered more accurate by employing the theoretical attenuation curves given in Fig. 5. This has been done for the currents, by trial and error, until the current wave-shape derived at the stroke point approximately coincided with that assumed for the attenuation curve used in the extrapo- lation. These obser^-ations involving the cable structure extended over the greater part of three lightning seasons, and included a total of 108 thunder- storm days, 35 in 1946, 38 in 1947 and 35 in 1948. The average number of thunderstorm days per year as recorded by the Atlanta Weather Bureau is 49, which compares with about 60 given on the U. S. Weather Bureau map.^ The more significant data are tabulated in Table I. In the following, the observations made of currents, voltages and charges along the cable are discussed for a number of the more important strokes and compared with theoretical expectations. This is followed by a dis- cussion of the observed incidence of cable currents of substantial magni- tude and of the incidence of strokes to ground observed at one location along the route and at a second point in the northern part of the country. 3.1 Wave-Shapes and Attenuation of Currents In Fig. 8 is shown the distribution along the cable of the crest currents, the crest voltages, and the charges, for the most severe direct stroke meas- sured, which had a crest value of 70 ka and a total charge of 11 coulombs. This stroke occurred to a 35-foot antenna connected to the cable and used in oscillographic observations of induced voltages due to strokes to ground, as another means of securing data on stroke currents. At this same point LIGHTNING CIRREXTS I.\ BURIED CABLE 293 Table I Summary of Currents Exceeding 10 KA Stroke \o. • Year Date Extrapolated Crest Cur- rent (Kilo- amperes) Extrapolated Max. Charge (Coulombs) Derived Time to Half-value (Micro- seconds) Shown In 1 2 3 4 5 6 1946 35 Thunder- storm Days April 7 June 21 June 21 Aug. 3 Aug. 3 Aug. 25 30 20 15 16 16 50 7 4 14 3.6 4 11.2 430 170 950 190 240 190 Fig. 11 Fig. 9 7 8 9 10 1947 38 Thunder- storm Days May — July 28 Aug. 5 Aug. 18 20 14 14 10 12 4 3.8 6.4 580 240 180 620 11 12 13 14 15 1948 35 Thunder- storm Days April 19 to 23 July 14 July 28 July 28 Aug. 4 20 15 70** 50 12 8 7.6 11.2 8.8 12 370 530 130 140 1000 Fig. 8 Fig. 10 * For location of strokes, see Fig. 6. ** Measured at stroke point. lO 35 5 25 M 20 '\ \ <^ ^-,^ ■^ y k: ^ ^ V >^ >ac D tr 10 CD Si ?2 UJ ^ 2 t) O^ 65432101 23456 DISTANCE ALONG CABLE (FROM ESTIMATED STROKE POINT) INKILOFEET Fig. 8 — Variation in crest current, voltage, and charge along cable for max. observed stroke current of 70 ka, having a time to half-value of 130 microseconds. Dashed curve shows theoretical variation of current for this time to half-value and an earth resistivity of 1200 meter-ohms. Variation in voltage between sheath and copper shield indicates breakdown between sheath and copper shield near stroke point. 294 BELL SYSTEM BECHNICAL JOURNAL simulative surge tests had been made three years before to obtain data on voltages in the cable due to surge currents, and tests had also been made of the dielectric strength of the thermoplastic insulation between the sheath and the copper shield. These latter tests disclosed low dielectric strength in the thermoplastic insulation at the location of the antenna referred to above, a defect which was repaired at the time. The voltage curve in Fig. 24 H v- STR DKE C )F AU G. 25, 1946 22 20 1 \ 1 \ 18 / / \ 16 / \\ 1 \ 14 i \ ^ ii \\ V v 12 / — *>- / \ 10 8 6 / ^ THEORETICAL VARIATION — r K / / / X -^ \ L s 4 2 0 y W ^ ^ \ -^ q" j;^ N \ "^ cr' ^^ C\ Y ^^ ■^ 5 16 q 12 o LU 10 ^ 7654321 01 234567 DISTANCE ALONG CABLE (FROM ESTIMATED STROKE POINT) IN KILOFEET Fig. 9 — Variation in crest current, voltage, and charge along cable for an extrapolated stroke current of 50 ka, having an estimated time to half-value of 190 microseconds. Dashed curve shows theoretical variation of current for this time to half-value and an earth resistivity of 1700 meter-ohms. 8 indicates that breakdown of the thermoplastic insulation occurred as a result of excessive voltage between the sheath and the copper jacket, but no other damage to the cable resulted. In Fig. 8 is also shown the theoretical variation in crest current along the cable, for a uniform earth of 1200 meter- ohms resistivity, for a stroke current reaching its half-value m 130 micro- seconds, as obtained from the crest current and charge at the stroke point. In Fig. 9 are shown similar curves for an extrapolated stroke current of 50 ka and 190 microseconds to half -value, together with the theoretical LIGHTNING CURRENTS IN BURIED CABLE 295 attenuation curve for the current for 1700 meter-ohms, which appears to provide a fairly satisfactory check on the extrapolation. The maximum observ^ed voltage obtained l^y extrapolation is about 8 kv, which com- pares with 5.6 calculated as outlined in Section 1.3. The higher observed voltage may be due to a long duration tail current of small value, which mav increase the voltage appreciably because of its slow rate of attenuation along the cable. 5 O -z z LU < 5§ STROKE OF JULY 28, 1948 r j \ 1 \ > 1 7 \ y \ THEORETICAL VARIATION .<(* / / l< / \ 4' _^ ^ \ \ \ \, ^ oj ^ / / ^ ^'^ < V .Q^ \ \3 8 A D 1 v^ >^ *-— i ::& ' — " "- 26 24 22 20 18 W CD I 6 _i O 14 ^ Z '2o UJ 10 (J) < 8 5 6 4 9876543210123456 DISTANCE ALONG CABLE (FROM ESTIMATED STROKE POINT) IN KILOFEET Fig. 10 — Variation in crest current, voltage, and charge along cable for an extrapolated stroke current of 50 ka, having a time to half-value of 140 microseconds. Dashed curve shows theoretical variation of current for an earth resistivitj' of 1200 meter-ohms. In Fig. 10 are shown similar curves for an extrapolated stroke current of 50 ka, reaching its half-value in 140 microseconds, together with theoretical attenuation curve for the current, for an earth resistivity of 1200 meter- ohms. The maximum extrapolated voltage is in this case about 3.5 kv, as compared with 4.1 calculated for 1200 meter-ohms. '• ■ The curves in Fig. 11 are for a fairly low extrapolated current, 16 ka, reaching its half- value in 190 microseconds. Again satisfactory^ agreement between observed and calculated attenuation is obtained. The maximum observ^ed voltage of 1.5 kv in this case agrees with that calculated for an earth resistivity of 1200 meter-ohms. 296 BELL SYSTEM TECHNICAL JOURNAL Some of the other observations, not reproduced here, were less consistent than those shown, probably due to the combined effects of more than one stroke; but they permitted fairly satisfactory determinations of crest cur- rents and times to half- value. From Table I it appears that the minimum duration to half-value is about 130 microseconds, the maximum about 1000 and that the average for the three most severe strokes is about 150 microseconds. In general the dura- tion appears to be longer the lower the crest currents, the average for cur- rents of 20 ka and less being about 500 microseconds to half-value. 5 o < > o o Z Z I- iiJ 3 \ STROKE OF AUG. 3, 1946 \ \ / \ THEORETICAL VARIATION // / A ^I ^ \ r ^ ^ ^ --^ V ....^^^ 6- o- ^^^ H- ^ D~' ^ ^^ """^^ -O -n 6 ^ O _j 654321 01 234567 DISTANCE ALONG CABLE (FROM ESTIMATED STROKE POINT) IN KILOFEET Fig. 11 — Variation in crest current, voltage, and charge along cable for an extrapolated stroke current of 16 ka, having a time to half-value of 190 microseconds. Dashed curve shows theoretical variation of current for this time to half-value and an earth resistivity of 1200 meter-ohms. In the above derivations a simple wave-shape was assumed, although actually it is likely to be rather complex with many fluctuations. The use of an equivalent simple wave-shape is, however, permissible in evaluating the liability to lightning damage, since statistical results rather than the wave-shapes of individual currents are of main importance. Regarding the cause for the long duration of the currents, it appears to be inherent in meteorological rather than geological conditions, as for the wave-shapes of lightning currents in general. The significance of meteoro- logical conditions is also indicated by the observations discussed in Section 3.3. In the case of strokes to the cable, the latter provides a path of very low impedance compared to that of the lightning channel, so that the wave- shape is determined by the impedance of the lightning channel and the distribution of charge along the leader and in the cloud. This is also true UGIITXIXG CCRRKXTS IX BURIED CABLE 297 for strokes to ground not arcing to the cable, at least during the time re- quired for the tip of the channel to propagate from the earth towards the cloud, which may be of the order of 50 to 100 microseconds, depending on the height of the cloud. During this inver\-al ionization of the soil around tlie base of the channel provides a path in the earth of low impedance com- pared to that of the channel, as shown in the paper referred to previously. It is possible of course that, durmg later stages of the discharge, the resis- tivity of the earth to some extent may limit the current, as the impedance of the completely ionized channel will then be lower and that in the earth higher because of the lower current in the earth with resultant decrease in ionization. This, however, would tend to reduce the current and thereby decrease rather than increase the time to half-value, and at the same time it would tend to cause a long duration tail current of low magnitude. 3.2 hicidence of Cable Currents of Various Crest Values In Fig. 12 is shown the number of observed currents exceeding various crest values, together with curves of the crest current distribution expected on the basis of the theoretical cur\'es given in Fig. 2. The latter curves, together with those in Fig. 3, are based on an incidence of strokes to ground of 2.4 per square mile for 10 thunderstorm days, a value derived from the rate of strokes to transmission line ground structures, as outlined in the paper referred to previously. Although the observations appear to be in fairly satisfactory agreement with theoretical expectations, a total of 15 currents is hardly sufficient as a check of the theoretical curves, particularly since the latter presume a uniform earth structure. The intersections of the theoretical curves (Fig. 12) with the axis of ab- scissae indicate that from ti\-e to seven of the currents were due to direct strokes. Actually, visual evidence of direct strokes was found in but two cases, in which the strokes occurred to and damaged test equipment. This does not preclude the possibility of additional direct strokes, as evidence thereof in the absence of cable damage may easily escape detection. ?>.2> Incidence of Strokes to Ground In Fig. 13 is shown the incidence of strokes to ground observed during 1947 and 1948 from one point within the test section, by the method de- scribed in Section 2.5. In the same ligure are shown the results of smiilar observations by the same method, made at one location in New Jersey during 1948 for purposes of comparison. Published data obtained from direct visual and aural obser\-ations at one locality in Massachusetts^ are also shown in the same figure. As shown by the curves in Fig. 13, the observed or apparent incidence of strokes to ground dmiinishes as the radius of the obser\-ation area increases, 298 BELL SYSTEM TECHNICAL JOURNAL for the reason that more of the remote than of the near strokes of low in- tensity escape observation. To find the actual incidence, a curve of the apparent incidence versus the radius of the observation area may be extra- polated to an area of zero radius. On account of the comparatively few observations for small radii, however, such extrapolation is rather inaccu- 75 70 65 60 55 UJ Q. ■s. < 45 O 40 z (- ? 35 3 30 UJ 25 20 15 \ EARTH RESISTIVITY IN METER-OHMS \ \ \ 1000 2000 O OBSERVED NUMBERS OF CURRENT \ \ V \ \ > VI \ \ ' \ \ \ \ N ^DIRECT STROKES AND -' STROKES TO GROUND \ \ \ \ 1 k \ \ s > \ s S DIRECT STROKES- A- A •n "^^ \ \ \ -^.. . -^ \ \ \ V \ \ \ 1.5 2 3 4 5 6 7 8 9 10 NUMBER OF CURRENTS EXCEEDING ORDINATE Fig. 12 — Comparison between observed and theoretically expected number of cur- rents exceeding given crest values during 108 thunderstorm days in a 21.5-mile section. rate. Improved accuracy is ob tamed by using theoretical expectancy curves in the extrapolation as indicated by the curves in the figure. These curves are derived on the assumption that the proportion of currents exceeding a given crest value / is given by P (7) = e~ ^ where ^ is a constant — a relation in substantial agreement with observations' — and that the energy in the electromagnetic wave from the stroke current, as well as that in the sound wave due to the thunder, is proportional to P/r^, where r LIGIITNIXG CURRRXTS IX BURIED CABLE 299 is the distance to the stroke. If the trigger arrangement in the apparatus mentioned in Section 2.5 is sufficiently sensitive to be operated by strokes so remote that the thunder cannot be distinguished above noise on the wire record, then the energy in the sound wave would be controlling, in the sense that it would determine the making of a usable record. On the other hand if the triggering should occur only for strokes of such substantial intensity 5.0 3 4 5 6 7 RADIUS OF OBSERVATION AREA IN MILES Fig. 13. — Apparent incidence of strokes to ground, per square mile per ten thunder storm days, as a function of radius of observation area. that some of the more remote strokes of low intensity would not be recorded, then the energy in the electromagnetic wave would be controlling. Simi- larly, in the case of direct visual-aural observation, the light radiation from the stroke may be assumed proportional to /-/''"• If, for any of these methods the energy density is taken as ii = cP/r^ where c is a constant and the minimum energy required for observation is tio, then only stroke currents in excess of / = {iio/cy^r would be observed. The observed or 300 BELL SYSTEM TECHNICAL JOURNAL apparent number of strokes to ground within an area of radius r would then be, for an actual incidence n of strokes to ground and with P{I) = Jo \l-e-"' (1 + ar)] '0 where a = kiuo/cY'^. The apparent incidence of strokes to ground Ua = Na/irr"^ is accordingly Ua = jAr^ [1 - e-"' (1 + ar)] By choice of a proper value of a in the latter expression a theoretical curve, varying in substantially the same manner with r as a given observed curve, may be obtained. The actual incidence is next obtained by taking a value of n such that the two curves substantially coincide. This value of n also corresponds to the incidence given by the theoretical curve for r = 0, i.e. the value that would be expected if a sufficient number of ob- servations were available for small values of r to permit extrapolation of the observed curves to r = 0. The value of n obtained in the above manner is about 2.8 for the New Jersey and Massachusetts and 1947 Georgia ob- servations. The latter extended over the last half of the lightning season, while the 1948 Georgia observations, which indicate a higher incidence, extended over the entire season. The comparison, shown in the figure, between the observations in Georgia during the first and second halves of the 1948 season, indicates that the incidence during the first half is about 50 per cent greater than during the second half. A similar comparison of the New Jersey observations, not shown in the figure, mdicates the opposite trend, i.e. a somewhat smaller incidence during the first half. The diflfer- ence, however, is less marked than in the Georgia case. There is reason to believe that this change in Georgia with the advance of the season is due to a change in the character of the lightning storms. During several years the more severe lightning damage on cable routes in this territory has occurred during early-season thunderstorms, which ordi- narily are of the "frontal" type extending over fairly wide areas where hot and cold masses of air come together. These storms appear to be of greater extent, duration, and severity than the "convection" type of storm, ordi- narily experienced later in the season, which occur more frequently as the result of local air convection currents but are of more limited extent and duration than storms of the frontal typeJ UGHTMXG CURRENTS IX BURIED CABLE 301 As mentioned before, the theoretical expectancy of lightning damage and of strokes to the cable discussed in this pa])er has been based on an incidence of 2.4 strokes per square mile for 10 thunderstorm days, a value derived from magnetic link obsers-ations of the rate of stroke occurrence to the aerial supporting structures of transmission lines, on the assumptions that they attract lightning strokes in accordance with laws established from laborator\^ observations on small-scale models, and that, the average height of the ground wires is 70 feet above the earth or adjacent trees. ^ If this lieight had been taken as 60 feet instead the incidence would have been 2.8, in substantial agreement with that obtained from our observations for northern territor}' — in the main the territor}^ traversed by the transmission lines from which the data were obtained. The cur\'es shown in Fig. 13 include substantial areas and a rather large amount of data and should, therefore, be fairly representative. Thus a radius of four miles corresponds to an observation area of 50 square miles. Within this area a total of 342 strokes was recorded during 1948 at the ob- ser\-ation point in the test section near Atlanta. One of the storms during this period, in which the antenna was struck, passed directly over the ob- servation point and provided a considerable amount of the data. How- ever, even if the observations during this storm were omitted, the total for the season would have been reduced by less than 10 per cent, while the ob- ser\'ations during July, August, and September would have been reduced about 20 per cent and would have been slightly lower than in the same 1947 period. The data thus indicate that the yearly incidence per square mile of strokes to ground is about 2.8 per 10 thunderstorm days in northern parts of the country, but may be as high as 4.3 in those southern parts where more severe types of thunderstorms occur. Considering, however, both the 1947 and 1948 observations in Georgia, it appears that an incidence of 3.7 would be a reasonable expectation for an entire season. With this incidence, rather than 2.4 as assumed in Fig. 12, Curves 1 and 2 in that figure would approximately correspond to earth resistivities of 500 and 1000 meter-ohms, respectively. Conclusions The observations indicate that the duration of lightning currents in the southern territory under observation is substantially longer than the aver- age ordinarily assumed. The time to half-value of intense currents, which are of main importance as regards Uability to lightning damage, is of the order of 150 microseconds, and for lower currents even larger. This, together with the higher incidence of strokes to ground and the high earth resistivity, would appear to account for the high rate of lightning damage experienced in this territory- in cables of earlier design than the copper- 302 BELL SYSTEM TECHNICAL JOURNAL jacketed cable upon which measurements were made. The incidence of cable currents of various intensities, their rate of attenuation, and the resultant voltages appear to be in satisfactory agreement with theoretical expectations. Acknowledgements The field observations were made possible by the cooperation of the Long Lines Department of the A. T. and T. Company, and the Southern Bell Telephone and Telegraph Company, both in the installation and the opera- tion of the equipment. The observations were conducted by our associate Mr. D. W. Bodle, who was also responsible for the design of the automatic recording equipment used to measure the rate of strokes to ground, and who suggested, from some of the observ-ations discussed here, the greater in- tensity of early-season storms. References 1. E. D. Sunde: "Lightning Protection of Buried ToU Cable," B. S. T. J., Vol. 24, Apri^ 1945. 2. E. D. Sunde: "Earth Conduction Effects in Transmission Systems," D. Van Nostrand Company, Inc., New York, London, Toronto, 1949. 3. C. M. Foust and H. P. Kuehni: "The Surge-crest Ammeter," General Electric Review, Vol. 35, December 1932. 4. C. F. Wagner and G. D. McCann: "New Instruments for Recording Lightning Cur- rents," Trans. A. L E. E., Vol. 58, 1939. 5. W. H. Alexander: "The Distribution of Thunderstorms in the United States^l904- \9iy'— Monthly Weather Review, Vol. 63, 1935. 6. J. H. Hagenguth: "Photographic Study of Lightning," Trans. A. L E. E., Vol. 66, 1947. 7. F. A. Berry, E. Bollay and N. R. Beers: "Handbook of Meteorolog>%" McGraw-Hill Book Company, Inc., New York and London, 1945. The Electrostatic Field in Vacuum Tubes With Arbitrarily Spaced Elements By W. R. BENNETT and L. C. PETERSON VACUUM tubes with close spacing between electrodes have become of increasing importance in recent years. The higher transconductances and lower electron transit times thus obtained combine with other features to raise both the frequency and band width at which the tube may operate satisfactorily as an amplifier. Specific designs have been discussed in papers by E. D. McArthur and E. F. Peterson^ and by Kremlin, Hall and Shatford^. The important contributions to structural technique made by E. V. Neher have been described in the Radiation Laboratory Series^. Further im- portant advances in the art have been recently announced by J. A. Morton and R. M. Ryder of the Bell Laboratories at the recent I.R.E. Electronics Conference held at Cornell University in June, 1948. The material of the present paper represents work done by the authors over a decade ago, and naturally there has been considerable publication on related topics in the intervening years. It has been suggested by our colleagues, however, that some of the results are not available in the technical literature and are of sufiicient utility to warrant a belated publication. These results have to do with the variation of the electric intensity, amplification factor, and current density which takes place along the cathode surface because of the nearby grid wires. We shall deal mainly with the approximate solution which neglects the efifect of space charge. The correction required to take account of space charge is in general relatively small as shown by both qualitative argument and experimental data in an early paper by R. W. King'^. More recent theoretical work^^ extending into the high frequency realm has confirmed the minor nature of the modification needed. The problem is thereby re- duced to one of finding solutions of Laplace's equation which reduce to con- stant values on the cathode, grid, and anode surfaces. The original work on this problem was done by Maxwell^ who calculated the electrostatic screening effect of a wire grating between conducting planes long before the vacuum tube was invented. All subsequent work has followed the methods outlined by Maxwell. In particular he suggested the replacement of the conducting planes by an infinite series of images of the grid wires and described an appropriate solution in series for the case of finite size wires. The useful approximation obtained when the diameter of the grid wires is 303 304 BELL SYSTEM TECHNICAL JOURNAL assumed small compared to their spacing was discussed in detail only for the case of large distances between the grating and each of the conducting planes. Figure 1 shows the assumed geometry of the grid, anode, and cathode. End effects are neglected. The origin is taken at the center of one of the grid wires which have radius c, and the X-axis is along the grid plane. The spacing of the wires between centers is a, the distance from grid to anode is do, and that from grid to cathode is di. No restrictions are placed on the sizes of a, d^, and di. Above the anode and below the cathode is shown a doubly infinite set of images which may be inserted to replace the conducting planes of the anode and cathode. By symmetry the potential from the -t-qo o o o -q o o o o d2 fy r^*l ' G G G X >IU-2C d| -q o o o o hq o -q o o o o Fig. 1 — Array of images for production of equipotential surfaces in planar triode. array of charges there shown must be constant for all x when y = d^ and also for all x when y = —di. The double periodicity of the array suggests immediately an application of elliptic functions. The solution of the sym- metrical case was actually stated in terms of the elliptic function sn z by F. Noether". The extension to the non-symmetrical case shown in Fig. 2 is fairly obvious. One of the authors worked out such a solution in terms of Jacobi's Theta functions in 1935, but abandoned any plans for publishing his analysis in view of the excellent treatment appearing shortly after that time in the Proceedings of the Royal Society by Rosenhead and Daymond®, who appUed Theta functions to both tetrodes and triodes, and both cylindri- cal and planar tube structures for the case of fine grid wires. Some of their formulas were later included in a book by Strutt''. Methods of calculating ELECTROSTATIC FIELD IN VACCIM TUBES 305 the case of thick grid wires in terms of exi)ansions in series of elli[)tic functions were discussed by Knight, Rowland and McMullen**-'". The prol^lem of a finite number of grid wires was treated by Barkas". More recently tubes with close spacing between grid and cathode, but with anode and grid as- sumed far apart, have been analyzed in terms of elementary functions by 76 ^ X-O GIVES \ // = I04 2C*I k X d, y d2 c s \ N \ \ \ \ 1=0. \ \ \ a Mo-- (4 27rd2 of O 1- 5? \o.5 \ \ a LOG 2^c (J ^'^ < u. — -~. \\ u u- 44 ^^0.64 -J a. ^ in 1.0 V 36 1- Y ?-"" 32 \ -.^^ ^ 28 \ \^ .^ 24 \ \ 20 ^ ■ — 0 0.05 O.iO 0.(5 0.20 0.25 0.30 0.35 0.40 0.45 0.50 DISTANCE ALONG CATHODE, X/g Fig. 2— Variation of amplification factor along the cathode surface of a triode. Fremlin'2. A solution based on the Schwartz-ChristofTel transformation has been given by Heme" for the case of grid wires of finite size and approxi- mately circular in shape. Since the derivations have been adequately covered in the references cited, we merely state the final formula here and indicate how it may be verified as correct. Let V{x, y) represent the potential function corresponding to 306 BELL SYSTEM BECHNICAL JOURNAL Fig. 1, the planar triode with fine grid wires. The potential of the cathode is set equal to zero. Then in the space between anode and cathode, AV{x, y) = [2t d^{y - d^)/a + {d, + d,)f{x, y)]V, + [B{y + ^0 - 27r d,y/a - d,f{x, y)]Vj, , (1) where f{x, y) = fn di [7r(a; -\- iy — 2i d2)/a] (2) t?i [ir{x + iy)/a\ A = (di-\- d,)B - 27r dl/a (3) adiiliri di/a) B = (n TCt?i'(0) (4) Here we have used Jacobi's notation for the ??i-function, as explained by Whittaker and Watson , rather than the Tannery-Molk notation used by Rosenhead and Daymond. We write i?i(7rz) for their di{z). In our notation t?i(2) = 2 f; (_)V^"+i'==^*' sin {In + 1)2 (5) where the parameter r in the above formulas is given by: r = 2i{di + d2)/a (6) By t?i(z) is meant the derivative with respect to z: t?i'(s) = 2 E {-nin + l).«"+^/«*' cos (2« + 1) z (7) n— 0 Verification of the solution is straightforward. The resulting V{x, y) is seen to be the real part of a function which is analytic in the complex variable X + iy except for logarithmic singularities at the points where the Theta functions vanish. Hence V{x, y) satisfies Laplace's equation in two dimen- sions in the region excluding the singular points. Since the zeros of t?i(z) occur at z = mir + wttt, where m and n take on all positive and negative values as well as zero, the singular points of the solution are at X -\- iy ■= ma + 2in{d\ -}- d^ — 2idi 1 (8) and X -\- iy = ma + 2in{dx + c?o) J which coincide with the centers of the image circles of Fig. 1. The logarith- mic singularities represent fine charges with the first set arising from a i?i- function in the numerator, yielding a positive charge, and the second set from the «?i-function in the denominator giving a negative sign. The equipotential curves are approximately circular in the neighborhood of the ELECTROSTATIC FIELD IN VACUUM TUBES 307 charges and hence l^.v, y) gives a constant potential on the surface of each grid wire if the radius of the grid wire is small compared with the spacing. Wq may show by direct substitution that V{x, y) becomes equal to Vj, at all points of the anode and equal to zero at all points of the cathode. On the anode we have y = di which, when substituted in the expression for /(•^) y), gives the logarithm of the absolute value of the ratio of conjugate complex quantities, and hence /(:f, | ^2 2r->l k \ d, c_ _ J_ a 14 Vg = -2 VOLTS Vp = 100 VOLTS / ' • / ^ / / VOLTS "unit of distance 1 1 1 0 0.05 OJO 015 0.20 0.25 0.30 0.35 0.40 0.45 0 50 DISTANCE ALONG CATHODE, X/^ Fig. 3 — Variation of cathode field strength in a triode. Some calculated curves for \x and Eg are shown in Figs. 2 and 3, Figure 2 shows the amplification factor as a function of the distance along the cathode with the ratio of grid-cathode separation to grid wire spacing as a parameter. The ratio of grid-anode separation to grid wire spacing is held constant at five. Only half the grid spacing interval is included since the curves are symmetrical. The increase in ^i-variation as the grid-cathode separation becomes small is clearly demonstrated. For negligible ju-variation we must select d\la of the order of 2 or greater. 310 BELL SYSTEM TECHNICAL JOURNAL Figure 3 shows the variation in field strength along the cathode for the typical operating point, Vg = —2 and Vp = 100 volts. It is to be noted that for d\/a less than 0.6, the electric field actually changes sign as we move from a point immediately below a grid wire to the midpoint between two grid wires. In other words a part of the cathode will not emit at all in these cases while the remainder emits in a non-uniform manner. In the rather extreme case of di/a = 0.4 only about a quarter of the cathode is emitting. It is worth noting how relatively rapid the "shadow" or "island" formation increases between di/a = 0.64 and 0.5 as compared to the increase in the interval from 0.5 to 0.4. If the equation for m is solved for C{x) and the result substituted back in the expression for E^ at the cathode we find : ''' d, + (d, + d,)/^. ^'^^ where here of course m varies with x. This is identical with the expression derived by Benham^® from Maxwell's approximate solution except that in the latter case m was a constant. Our colleague, Mr. L. R. Walker, has pointed out that the equation follows directly from the assumption of small grid wires without explicit solution for the potential function. Since the charge density Cc on the cathode is proportional to the field strength (the factor of proportionality in MKS units is the dielectric constant e of vacuum or 9.854 X 10"^" farads/meter), Maxwell's capacity coefficients Cg„ and Cpc may be calculated from O-j = eEy = —{CgeVg + CpcVp) (25) The minus sign is used here because we are taking the ratio of charge to voltage at the negative plate of the condenser consisting of cathode, grid and anode surfaces. Hence di + (^1 + <^)/m C = ^-l^ (27) ^' dr + (d, + d,)/^. ^"-'^ Since m is variable, an integration is required to determine the total capaci- tance. From the periodicity of m with grid spacing it is possible to express the result in terms of the average values of Cgc and Cpc over an interval of length a along a direction parallel to the grid plane and multiply these values by the total area of cathode surface. ELECTROSTATIC FIELD IN VACUUM TUBES ill Equation (24) may be interpreted in a number of different ways of which we shall mention the following two: 1. The "equivalent voltage" Vg + VJn does not act at the grid but at a distance D from the cathode, where D = di+ {d, + d^la (28) Roth the equivalent voltage and distance vary along the cathode surface. 2. The "equivalent voltage" V, = (V, + Fp/m)/[1 + (1 + dMW] (29) acts in the grid plane and varies with distance along the cathode surface. As far as the cold tube is concerned the two formulas are equivalent at the cathode, but not at the grid. When the tube is heated and complete space charge is present, the two formulas also differ at the cathode. The current density in the presence of space charge is, according to (28) and Child's law: / = K(V, + VM^'^/D' (30) while, from (29), / = KVl^'/dl (31) In both, K"^ = 32 e^e/Sl m, where e/m is the ratio of electronic charge to mass. The value of current given by (31) is [1 + (1 + c?2M)/m]''^ times as large as that given by (30). li ti >> 1 + d^/di the two values are nearly the same. In tubes with close grid-to-cathode spacing the inequality may not be fulfilled. As to which viewpoint is more accurate, we note that Ferris and North in their papers "■ ^^ on input loading adopted the latter, and that at high frequencies where electron transit time must be considered the second viewpoint is preferable because of the more accurate representation of effects at the grid. For a more complete discussion see Reference 19. Figure 4 shows curves of relative current density as a function of distance along the cathode as computed from Eq. (31). The transconductance for unit area of cathode surface as computed from the same equation is given by: = 3.512(F, + Vp/ixY'^idi/Dy^ micromhos. (32) The resulting variation with distance along the cathode is shown in Fig. 5. Defining the figure of merit If at a point x along the cathode as the ratio 312 BELL SYSTEM TECHNICAL JOURNAL between the transconductance dl/dVg and the sum of Cgc and Cpc at this point, we find from (30) M = (u/syndi + {di + ^2)/m]-^V(m + 1) m where / = el /me, e/m = 1.77 X 10" coulombs/kg. From (31), we find on the other hand i/ = (47/3 d^yWip + 1) (34) 0 0.05 0.10 0.15 0 20 0.25 0.30 0.35 0 40 0.45 0.50 DISTANCE ALONG CATHODE. 'Va Fig. 4 — Variation of current density in a triode. Both formulas indicate that for a cathode capable of supplying a given cur- rent density the only means of improvement lies in decreasing the cathode- grid spacing. The impro\-ement is extremely slow; doubling the figure of merit requires an eight-fold decrease in spacing. We again emphasize that the calculated current densities and figures of merit are functions of x, the distance along the cathode. The total current between the two grid wires is found from (30) to be It = 2K (V„ -f VJnf^ dx/D" (35) ELECTROSTATIC El ELD IN VACUUM TUBES 313 0 0.05 0.10 0.(5 0.20 0.25 0.30 0.35 0.40 0.45 0.50 DISTANCE ALONG CATHODE, X/3 Fig. 5— Variation of transconductance along the cathode surface of a triode. ral2 while, from (31), 2K /""'' ^^^ d^ 4 ^^^^^ "^ ^^^/^^ '^ ^ + d,/dd]"' dx (36) where .To is given by Vg + VJix{x,) = 0 (37) On the basis of several reasonable assumptions it may be shown that both (35) and (36) lead to an approximate 5/2 power law instead of 3/2 power law. Such a law has actually been observed in cases where shadow forma- tion was suspected. We wish to express our appreciation to Messrs. R. K. Potter, J. A. Morton, 314 BELL SYSTEM TECHNICAL JOURNAL and R. M. Ryder for their encouragement, and to Miss M. C. Packer for aid in the numerical computations. References 1. E. D. McArthur and E. F. Peterson, "The Lighthouse Tube; A Pioneer Ultra-High- Frequency Development," Proc. Nat. Electronic Conference, Chicago, Oct. 1944, Vol. I, pp. 38-47. 2. J. H. Fremlin, R. N. Hall, and P. A. Shatford, "Triode Amplification Factors," Electr. Comm., Vol. 23 (1946), pp. 426-435. 3. Hamilton, Knipp, and Kuper, "Klystrons and Microwave Triodes, Radiation Labora- tory Series, New York, 1948, p. 153. 4. J. C. Maxwell, "A Treatise on Electricity and Magnetism," Vol. 1, pp. 310-316. 5. Riemann- Weber, "Differentialgleichungen der Physik," Vol. 2, p. 311. 6. L. Rosenhead and S. D, Daymond, "The Distribution of Potential in Some Thermionic Tubes," Proc. Roy. Soc, Vol. 161 (1937), pp. 382-405. 7. M. J. O. Strutt, "Moderne Mehrgitter-Elektronenrohren," Berlin, 1940, S. 154. 8. R. C. Knight, Proc. London Math. Soc. (2) Vol. 39 (1935\ pp. 272-281. 9. R. C. J. Howland and B. \V. McMullen, "Potential Functions Related to Groups of Circular Cylinders," Proc. Cambr. Phil. Soc, Vol. 32 (1936), pp. 402-415. 10. R. C. Knight and B. W. McMullen, "The Potential of a Screen of Circular Wires between two Conducting Planes," Phil. Mag. Ser. 7, Vol. 24, 1937, pp. 35-47. 11. Barkas, "Conjugate Potential Functions and the Problem of the Finite Grid," Phys. Rev., Vol. 49 (1936^, pp. 627-630. 12. J. H. Fremlin, "Calculation of Triode Constants," Phil. Mag. Ser. 7, Vol. 27 (1939), pp. 709-741; also Electr. Comm., Vol. 18 (1939), pp. 33-49. 13. H. Heme, "Valve Amplification Factor," Wireless Engineer, Vol. 21 (1944), pp. 59- 64. 14. Whittaker and Watson, "Modern Analysis," Third Edition, Cambridge (1940), Chapter XXI, p. 462. 15. R. W. King, "Thermionic Vacuum Tubes," Bdl System Technical Journal, Vol. H (1923), pp. 31-100. 16. W. E. Benham, "A Contribution to Tube and Amplifier Theory," Proc. L R. E., Vol. 26 (1938), pp. 1093-1170. 17. W. R. Ferris, "Input Resistance of Vacuum Tubes at Ultra-High Frequencies," Proc. I. R. E., Vol. 24 (1936), pp. 82-105. 18. D. O. North, "Analysis of the EtTects of Space Charge on Grid Impedance," Proc. I. R. K, Vol. 24 (1936), pp. 108-136. 19. F. B. Llewellyn and L. C. Peterson, "Vacuum Tube Networks," Proc. L R. E., Vol. 32 (1944), pp. 144-166. Transconductance as a Criterion of Electron Tube Performance By T. SLONCZEWSKI eUANTITATIVE evaluation of electron tube performance has assumed added importance with the increasing extension of electronics into the fields of measurement and control. Simplification of the process of selection of suitable tube types and operating conditions from the general data avail- able is of considerable value to all engineers concerned with electronics circuit design. The conventional procedure involves analysis of the plate current-grid voltage characteristics. The simpler method presented herem supplies the same information from an analysis of the transconductance-grid voltage characteristics. These are usually supplied by the manufacturer or can be obtained readily by measurement^ The method presented herein has been employed successfully for a number of years in the development of electronic measuring apparatus by a group of engineers who attended lectures on the subject given by the author. It applies chiefly to pentodes, where the internal plate impedance is high with respect to the load impedance. Its merit resides in the comparative brevity of the formulae, the ease of computation and the facility in obtaming the data from which the computations are made. It allows one to form a pre- liminary judgment of the performance of a tube from a brief glance at the characteristics furnished by the manufacturer better and faster than any other method known to the author. It should prove of value to the instruc- tor teaching electron tube theory. In the interest of simplicity some of the subscripts m, p, c and g appended usually to symbols for transconductance, plate current and grid voltages are deleted below. The scope of the discussion is so limited that no confusion may arise from this omission. The formulae are expressed in terms of amplitudes of voltage and current, capital letters being used for their symbols. All values are in peak volts. Levels are in decibels. The g-e characteristic is introduced into the problem by starting with the general expression for the plate current where v is the voltage measured from the bias point Ec , where the deriva- tives are taken, and utilizing the definition of the transconductance > Radio Engineers' Handbook, F. E. Terman, McGraw-Hill, 1943, p. 961. 315 316 BELL SYSTEM TECHNICAL JOURS A L di de (2) Inserting (2) into (1) and calling G the value of g at Ec we obtain ^= j_^ gde + Gv + --v +--V -\--~v ■■- (3) The first term of this expression is the space current of the tube at no load, that is when v = 0. On the g-e diagram, Fig. 1, it represents the area under the curve from the tube cut off C to the tube bias Ec . The second term represents the function of the tube as an amplifier. The third term repre- sents the second-order modulation current. The latter is responsible for the objectionable generation of a second harmonic in an amplifier and the useful presence of the second harmonic in the frequency doubler, the direct current in a rectifier and the sidebands in a modulator. The fourth and higher terms represent, in general, undesirable effects of modulation. They are usually smaller than the first two and, since their effects are additive, the first three terms of expression (3) may be studied profitably disregarding the others. If necessary, the effects of the higher- order terms may be added later. The Idealized Parabolic Pentode If, over a certain range of grid biases Ca to cb , the effect of the fourth and higher terms of series (3) is negligible the g-e characteristic will be a straight line. Herein lies one of the advantages of the method, for a straight portion of a curve can be easily selected by inspection and checked with a straight edge. It is thus possible to select easily such a tube and operating point that third and higher-order modulation products are absent in the output. There is no such simple method of verifying whether a current characteristic is parabolic. That there are tubes having approximately straight portions of g-e characteristics can be verified by inspection of (Fig. 1) where the characteristic of the 6AG7 is given. Since a portion of the g-e characteristic is a straight line, the third term coefficient — may be replaced by the ratio -^ where Ae is an arbitrary inter- ne Ae val of grid voltage and Ag the corresponding change in g. In many of the following computations it will be advantageous to use for Ae the total excur- sion of the grid voltage. On the basis of the simplifying assumption of a paraboUc pentode it is possible to derive the simple formulae given below which cover the perform- ance of the tube as an amplifier, rectifier and modulator. ELECTRON TUBE PERFORMANCE 317 The Parabolic Pentode Amplifier Over the straight portion of the g-e curve the following relations hold for an input v = P cos pi. The fundamental current is Ip = GP where P is the grid swing. The second harmonic current is f2p 1 Ag^ 4 Ae TYPE 6AG7 PLATE VOLTS 300 SCREEN VOLTS 300 GRID-BIAS VOLTS 10.6 GRID VOLTAGE, e Fig. 1 — g-e characteristic having principally second order modulation over part of the range. To find the level H2 of the second harmonic below the fundamental, prolong the straight portion of the characteristic down to the virtual cutoff £0 (Fig. 1). Then H, = 20 log "f 20 log ^Sr^" + 12. For the case when all of the parabolic characteristic is used P = ^ and F2 20 log ^ + 18. If it is desired to express Ih in terms of the output current ip and the space current I'o the following approximate formula may be used: Hi = 20 log J + 18. 315 BELL :iySTEM TECHNICAL JOURNAL This expression neglects the area under the characteristic on the left of the Une AEq , Fig. 1. The formula is useful in selecting a tube for closer consideration. When a tube is used as a preamplifier in a wave analyzer an error of meas- urement may occur if two input frequencies intermodulate in the amplifier to produce a current of the same frequency as the one being measured. For instance, the fundamental R cos rl and the second harmonic W cos wt may intermodulate to form the third harmonic. If Ir-w is the disturbing current and Ip the wanted output, then T F — Ft, R W 20 log ^ = 20 log ^' ^^° + 6 - 20 log ^ - 20 log ^ The Rectifier The portion of the plate current resulting from the rectification of a signal P cos pt is 4 Ae If several frequencies were present, Pi cos />/, Pz cos Pzt and so on, I^ = l^{Pl + Pl-h--') 4 Ae Thus Ide is proportional to the square of the root-mean-square voltage input. This property of the parabolic tube of measuring the root-mean-square voltage is often useful in the measurement field. If Ae is the paraboUc range of the tube and Ag the corresponding change in g, the largest possible rectified current obeying the root-mean-square Ae law will obtain for an amplitude P = — . Then /max = Ts^g^e. The Frequency Doubler The second harmonic is given, as before, by 4 Ae The largest possible output current is /max = Tff^gAe. ELECTRON TUBE PERFORMANCE 319 In general, the level of the undesirable fundamental will be higher than the harmonic by 7/2 = 20 log /-" = log ^^-^ + 12. lip r For the maximum current case this reduces to Hi = 20 log — + 18. The Modulator When two inputs P cos pt and Q cos ql are applied to the grid, the output is ^ = ^0 + 1 X^ (^' + Q^) + \ ^ (^' cos 2pt + (f coilqO 4 Ae 4 Ac t x^ _ i y + ^ ^ ^Q cos (/, + 5)/ + i ^ PQ cos (/> - q)l The last two terms represent the sidebands. In the case of a detector the available supply of the carrier voltage Q is Ae copious. Putting ^ = y the well known result is obtained Ip+g = I^g = lAgP. The conversion transconductance is To formulate filtering requirements the signal and carrier leaks must be found. The signal leak is The carrier leak is 20 log ^ =20 log ^ + 12. ^p±g Ag Iq ^r. , G . ^^ , Ae 20 log y-^ = 20 log - + 20 log ^ + 6. ■i p±g ^t" r^ In the design of a heterodyne oscillator a generous supply of both input voltages is easily available and maximum output current is desirable This Ae occurs when F = Q = — . Then 4 Ip-9 = -^AgAe. 320 BELL SYSTEM TECHMCAL JOURNAL Whether P equals Q or not, the unwanted products are = LV) log p^ 20 log /^ = 20 log ^ + 20 log /^ = 20 log ^^-^" + 6 20 log 1^ = 20 log ^ - 6 20 log /^ = 201ogf -6 ip_3 -L 20 log ^' = 0. TYPE 6AK5 PLATE VOLTS 150 SCREEN VOLTS 150 GRID-BIAS VOLTS 3.6 GRID VOLTAGE, 6 Fig. 2 — g-e characteristic exhibiting third order modulation. Third Order Modul.a.tion When we do take into consideration the fourth term of equation (3), that IS - — ^ t)^ the g-e characteristic will no longer be a straight line but a 0 de- parabola. It turns out that all of the computations of the second-order effects as shown are so little affected that no corrections are necessary. The 6 56^ add new types of modulation products to the output. These are usually objectionable. A typical g-e characteristic with third-order modulation present is shown on Fig. 2. The curvature is usually concave upward. Instead of measuring the derivative ~ needed for the computations, it is more practical to scale ae'' presence of third-order modulation caused by the term - j^ i^ will, however. ELECTRON TUBE TEKFORMANCE 321 off the amount S by which the characteristic sags in the middle of the interval ^e (Fig. 2). The plate current is then given by the expression 1 A? 2 , 45 3 /,v Single Frequency Input When the signal input to an amplifier consists of a single frequency v = P cos pt, the output current is given by i = io -\-^P^ + \g + -^pnp cos pi Ae L (Ae)2 The second-order effects consisting of the rectified current and second harmonic are seen to be unaffected by the presence of third-order modulation. However, the first-order effect, the fundamental output, ceases to be linear and a new product, the third harmonic, appears. The change in fundamental output is expressible as a loading effect on transconductance. The effective transconductance of the tube is. Ge = G+S ©■ Expressed in db's the non-linearity effect is approximately When P is large it is convenient to select Ae = 2P and get Ge = G-h^, 201og?-' = 2.15|. S is positive when the g-e curve is concave upward. The third harmonic content of the output is H, = 20 log /^^ = 20 log ^ + 40 log ^'+ 10. If we select Ae = 2P the second term drops out Hi = 20 log ^ + 22. If the curve is concave upward the third harmonic increases the peak value 322 BELL SYSTEM TECHNICAL JOURNAL of the wave. If it is concave downward the peak value of the wave decreases. In the case of a two-frequency input v = P cos pt -\- Q cos qt the second- order products are again unaffected. The third-order products will be: ^'^ " 3 (M^ ^ There are situations in the design of detectors where the /,±2p current may- be disturbing. For example, in a wave analyzer modulation stage when measuring the second harmonic Pi cos {2p)l the desired second-order product is Iq-iip)- This is, however, of the same frequency as the third-order product Iq-2p generated by the intermodulations of the strong fundamental Pi cos pt and the carrier Q cos ql. The level of the wanted product with respect to the unwanted one is given by 20 log ^^> = 201og^ + 20 log ^^ +20^-6. If there are two interfering inputs R cos rt and W cos wt they may, together with the carrier Q cos qt, form an objectionable product iri,w±q of the same frequency as the wanted product ip-q. The level of this disturbing product with respect to the wanted product is then given by 201og^ = 20 log ^^ -f 20 log ^' + 20 log ^- 12. When two input frequencies are present in the input, the effective trans- conductance of the tube becomes The amplifier gain depends on the level of all of the components of the input. Fourth Order Modulation 1 d^g 4 If the fourth term of equation (3) is absent, but the fifth term ^aTz^ is present, fourth-order modulation will occur. The g-e characteristic will be a cubic with an inflection point at Ee . iwl .^. ^ + 6. 1 •* * "^ 1 -H 1 I* o o o O a. **> CO 1 — ' 1 r^ 1 " o o o O 1 ^ 1 CJ + 5y O) Ss 0. Si + < 1 \ He joined the Engineering Department, Western Electric Company, in 1922; Bell Telephone Laboratories, 1925-. He has been concerned with voice frequency repeater and carrier terminal developments. Daxiel Leed, B.S., College of the City of New York, 1941; Kollsman Instrument Company, 1941-43. Federal Telephone and Radio Corpora- tion, 1943-44. Corps of Engineers, Los Alamos Laboratories of the Man- hattan District, 1944-46. Bell Telephone Laboratories, 1946-. Mr. Leed is engaged in circuit development for phase and transmission measure- ment systems, particularly in the field of automatic frequency control. D. B. Penick, University of Texas, B.S. in Electrical Engineering, 1923, B.A., 1924; Columbia University, M.A., 1927. Engineering Department, Western Electric Company, 1924-25. Bell Telephone Laboratories, 1925-. Mr. Penick has been engaged in the development of carrier telephone sys- tems. Liss C. Petersox, Chalmers Technical University, Gothenburg, 1921 ; Technical Universities of Charlottenburg and Dresden, 1921-23. American Telephone and Telegraph Company, 1925-30; Bell Telephone Laboratories, 1930-. Mr. Peterson has recently been concerned with the theory of hearing. P. W. Rounds, A.B., Harvard University, 1929. Bell Telephone Labora- tories, 1929-. Mr. Rounds has been engaged in the design of transmission networks. C. W. Schramm, B.S. in Electrical Engineering, Armour Institute 334 BELL SYSTEM TECHNICAL JOURNAL (now Illinois Institute) of Technology, 1927. Illinois Bell Telephone Company, 1927-29. Bell Telephone Laboratories, 1929-. Mr. Schramm has been concerned with the development of carrier telephone systems for both message and program use. During the war his attention was directed to the design of radar test equipment. T. Slonczewski, B.S. in Electrical Engineering, Cooper Union Institute of Technology, 1926. Bell Telephone Laboratories, 1926-. Mr. Slonczew- ski has been engaged in the development of electrical measuring apparatus. F. E. Stehlik, B.E.E., 1933, M.E.E., 1935, Polytechnic Institute of Brooklyn. Bell Telephone Laboratories, 1936-. Mr. Stehlik was engaged in the design of crystal filters and is now concerned with the development of high-frequency networks. E. D. SuNDE, B.S., Haugesund, Norway, 1922; E.E., Darmstadt, Ger- many, 1926. American Telephone and Telegraph Company, 1927-33; Bell Telephone Laboratories, 1933-. Mr. Sunde has been engaged in studies of interference in telephone circuits from power lines and railway electrification and is now concerned with studies of protection of the tele- phone plant against lightning damage. H. M. Trueblood, B.S., Earlham College, 1902; B.S., Haverford College, 1903; Mass. Inst. Technology, 1908-09; Ph.D. (physics). Harvard Uni- versity, 1913; Field Officer, U. S. Coast and Geodetic Survey, 1903-08; Instructor and Assistant Professor in Electrical Engineering, University of Pennsylvania, 1914-17; U. S. Naval Experimental Station, New London, Connecticut, 1917-19. American Telephone and Telegraph Company, Department of Development and Research, 1919-34; Bell Telephone Laboratories, 1934-. At present. Assistant Director of Transmission Engineering. Most of Dr. Trueblood's work has been on interference with communication systems from natural and other sources, with work on radar and radar testing equipment during World War II. Anthony J, Wier, L.L.B., New Jersey Law School, 1935. New York Telephone Company, Plant Maintenance; and Western Electric Company, Installation and Equipment Engineering; 1914-28. Bell Telephone Laboratories, 1928-. Mr. Wier has been engaged in development work on toll telephone and telegraph equipment since 1928. I ot =t I VOLUME XXVIII JULY, 1949 NO. 3 THE BELL SYSTEM ^*% TECHNICAL JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION Editorial Note regarding Semiconductors 335 Hole Injection in Germanium — Quantitative Studies and Filamentary Transistors W. Shockley, G. L. Pearson, J. R. Haynes 344 Some Circuit Aspects of the Transistor R. M. Ryder and R. J. Kircher 367 Theory of Transient Phenomena in the Transport of Holes in an Excess Semiconductor Conyers Herring 401 On the Theory of the A-C. Impedance of a Contact Rectifier J. Bardeen 428 The Theory of p-n Jimctions in Semiconductors and p-n Junction Transistors W, Shockley 435 Band Width and Transmission Performance C. B. Feldman and W. R. Bennett 490 Abstracts of Technical Articles by Bell System Authors 596 Contributors to this Issue 599 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 50^ per copy |J.5Q per Year THE BELL SYSTEM TECHNICAL JOURNAL Published quarterly by the American Telephone and Telegraph Company 195 Broadway, New York, N. Y. EDITORS R. W. i:ing J. O. Perrine EDITORIAL BOARD C. F. Craig O. E. Buckley O. B. BlackweU M. J. KeUy H. S. Osborne A. B. Clark J. J. PilUod F. J. Feely SUBSCRIPTIONS Subscriptions are accepted at $1.50 per year. Single copies are 50 cents each. The foreign postage is 35 cents per year or 9 cents per copy. Copyright, 1949 American Telephone and Telegraph Company PRINTEDINU. S. A. The Bell System Technical Journal Tr)/. XXVIff Jiilw I<^40 \o. 3 Editorial Note regarding Semiconductors ALL but one of the i)apers that comprise this issue discuss practical ap- >- lications of semiconductors and touch upon their properties as em- ployed in rectifying devices, detectors, and in a new amplifying unit — the so-called transistor. These semiconductor papers all relate to one another and present, as a whole, a current but well developed account of the be- havior and uses of these very promising additions to today's vast array of electrical applicances. Because semiconductors are relative newcomers, few engineers have as yet had occasion to become familiar with their characteristics and the reasons for their somewhat unexpected performance. Accordingly, it seems appro- priate to preface the present group of papers with a brief introductory note devoted to the nature of the physical phenomena encountered. The semiconductors of interest in the communications art are electronic rather than ionic conductors, and include copper oxide, various other oxides, selenium, germanium and silicon. Being electronic conductors, the constitu- ent atoms remain in fixed positions. They may lose or gain electrons during the conduction process but the structure of the conductor as a whole and its chemical composition are not affected. Basic to the theory of these semiconductors is the idea that electrons can carry current in two distinguishable and distinctly different ways : one being called "excess conduction," "conduction by excess electrons," or simply "conduction by electrons;" and the other being called "deficit conduction" or "conduction by holes." The possibility that these two processes may be simultaneously and separably active in a semiconductor affords a basis for explaining transistor action. We shall confine our attention to the behavior of electrons within the sili- con and germanium type of crystal lattice, and especially as it is modified by minute amounts of suitably chosen ini])urities.' ' There has been very marked development in the understanding of semiconductors since 1940. This understanding is an outgrowth of the research and development program on crystal rectifiers undertaken in connection with the radar program during the war and continued in several laboratories thereafter. Some of the wartime work was carried out in the Radiation Laboratory of M.I.T., which ojjerated under the supervision of the Xational Defense Research Committee. The Radiation Lal)oratories Series volume "Crystal Recti- fiers" by H. C. Torrey and C. A. Uhilmer reports this program and mentions in particular as chief contrit)utors to cr>-stal research and development in England: the General Electric 335 336 HEI.L SYSTEM TECIIXICAL JOLKXAL Silicon and germanium form what are called "covalent crystals," the atoms being held together by "electron-pair bonds" formed by the valence electrons. The covalent bond in the hydrogen molecule is the simplest elec- tron-pair bond. Figure 1 represents two hydrogen atoms and a hydrogen molecule.-' Each atom consists of a proton and one electron. The proton weighs approximately 2,000 times as much as the electron and is a relatively immobile particle about which the electron moves in its orbit or quantum mechanical wave function. In an isolated atom, this wave function has spherical symmetry and the electronic charge is distributed on the average as a diffuse sphere centered about the proton. When the two atoms are brought close together, interaction between the wave functions of the two electrons takes place and the electronic cloud of each becomes modified, as suggested in the figure. The result is to produce an extra accumulation of charge between the two protons which acts to bind them together. Accord- ing to quantum mechanical laws associated with the "Pauli exclusion prin- ciple," the bond is especially stable when it contains precisely two electrons. It is weakened considerably by removal of one electron and is not greatly strengthened by the addition of a third electron. This special stability of the electron-pair bond or covalent bond is a fundamental fact of chemistry which is now quite well understood on the basis of wave mechanics. The elements carbon, silicon and germanium have the common feature of being tetravalent. iVlthough they possess respectively 6, 14 and 32 electrons all together, in each case only four of these are able to enter into chemical reactions. The remaining electrons are closely bound to the nucleus produc- ing a stable "ionic core" having a net charge of +4 units. This core can be regarded as completely inactive so far as electronic processes in chemical reactions and in semiconductors are concerned. Each of these atoms tends to form covalent or electron-pair bonds with four other atoms. This tendency is completely satisfied in the diamond lat- tice which is the crystalline form of all three elements. The lattice, shown in Fig. 2, is a cubic arrangement and may be regarded as made up of four in- terpenetrating simple cubic lattices like the one formed by the atoms on the four corners of the cube shown. In this structure each typical atom is sur- rounded by four neighbors regularly placed about it, with which it forms four Company, British Thompson-Houston Ltd., Telecommunications Research Establishment and Oxford University; and in the United States: the Bell Telephone Laboratories, West- inghouse Research L'aborator\-, General Electric Company, Sylvania Electric Products, Inc., and the E.L duPont deXemours and Company. It is also pointed out that the crystal groui)S at the University of Pcnns\lvania and Purdue Universit\-, who operated under N.D.R.C. contracts, were responsible for much fundamental work. -The figures in this introduction and the text associated with them, like the following pa])er on "Hole Injection in Germanium", follow closely the presentation in a book en- titled "Holes and Electrons, an Introduction to the Physics of Transistors" now under prejniration by \V. .Sliockley. SEMJCOXDl CrORS .^M covalent bonds. These neighbors are arranged on the corners of a regular tetrahedron in conformity witli the known chemical behavior of the tetra- hedral carbon atom.^ For purposes of discussion of conductivity in these crystals, we shall represent the three-dimensional array in two dimensions as is shown in Fig. 3, indicating that each carbon atom forms an electron- l)air bond with four neighbors. On the basis of this valence bond structure we can intuitively see why diamond should be an insulator. Although it contains a large number of ELECTRON-.^- '.'.:;• •'•' -/PROTON . • '.'•■*• '.'T "**.•'.' ■*rV'.** ''■,*'•• . • -, . •*.•,*•'•,/•'.*.•'*•'/.■•■.'-•■., ' ' ■ *-•••'•* rr».- •','** • *. .* • '^•■••V-.y.;".';.'; •'.'• . " ■".••■ .••.••.■••'•.■•■'•■ ."^v >*'•.• • ••'•'••••■. DOTS REPRESENT ■.' ■. . ELECTRON CHARGE \:?^^0:-:' DENSITIES TWO H ATOMS 1 A = I ANGSTROM = 10"® CM Fig. 1. electrons, as does a metal, the covalent bond is a quite different structure from the metallic bond. In an ideally perfect diamond crystal, each valence bond would contain its two electrons; therefore, every electron would be tightly bound and thus unable to enter into the conduction process. Conductivity can be produced in diamond, however, in a number of ways, all of which involve destroying the perfection of the valence bond structure. 3 Long before the arrangement of atoms in the diamond crystal was estabUshed b>- X-rays, the organic chemists had coiickided that carlwn formed four bonds at the Ictra- hedral angles— a truly remarkat)le result of inductive reasoning based on observations of the ojitical properties of solutions of organic compounds. 338 BELL SYSTEM TECIIMCAL JOl'RXAL Thus, if higli-eiit'rg>- particles or quanta of radiation fall upon the crystal, they can break the bonds. Conductivity in diamond induced by bombard- ment in this way has recently received considerable prominence in connec- tion with "crystal counters" which have been used to detect nuclear parti- cles and in experiments on conductivity induced by electron bombardment. An electron ejected from a bond constitutes a localized negative charge in the crystal and, since initially the bond structure was electrically neutral, the electron as it departs from its point of liberation leaves behind an equal, localized positive charge. Such a migratory electron, because it represents 5^~V Fig. 2. an excess over and above that required to complete the bond structure in its neighborhood, is called an "excess electron." Since it cannot enter any of the completed bonds in the lattice, it moves about in the crystal in a ran- dom manner under the influence of thermal agitation. If an electric field is applied, it tends to drift in the direction of the applied force and to carry a current. This illustrates conduction by excess electrons (referred to simply as conduction by electrons) and, as we shall see, is to be distinguished from the other process whereby an electron deficit enables conduction to be ef- fected by "holes." Such a hole, constituting a net, localized, positive charge in the crystal, moves from place to place by a reciprocal motion of electrons in the valence bonds; and, under the influence of an electric field, its random motion ac- SKMnOXDI ( IONS 339 quires a systematic drift, riu'refore it also can contribute to the current; in other words, current can llow as well by virtue of a deficit of electrons as by an excess of them. In an illuminated and bombarded diamond crystal the electrons and holes, produced in pairs by the excitation, will of course drift in opposite directions under the intkience of a held; the electron, being negative, drifts in the op- + 4 IN CORE OF CARBON ATOM~-- m VALENCE ELECTRONS EACH ATOM, WITH THE CHARGE OF ITS SHARE OF VALENCE-BOND ELECTRONS, IS ELECTRICALLY NEUTRAL. (b) (a) ELECTRON PAIR BONDS (C) PLANE DIAGRAM OF DIAMOND LATTICE WITH BONDS REPRESENTED BY LINES Fig. 3. FOUR VALENCE BONDS posite direction from the applied held, but its current is in the direction of the lield. In the case of the hole, the recij)rocal electron motions are once more opposite to the direction of the held (on the average). As a consequence, the net result is that the hole drifts in a direction to increase the current repre- sented by the electrons. If the source of bombardment or illumination is removed, the conductivity dies away and the crystal will return to its nor- mal state. This can occur by the recombination of holes and electrons. Whenever an electron drops into a hole, both the hole and the electron dis- .UO BELL SYSTEM TECH MCA L JOI'RXAL appear and the bond structure becomes comj^letc, the excess energy being given up to the atoms in the form of thermal vibrations.' If the temperature is sufficiently elevated, spontaneous breaking of some fraction of the covalent bonds by agitation will occur producing electrons and holes in equal numbers. In a diamond this effect would occur at such high temperatures that it would not be observed. However, it plays a major role in silicon and germanium at temperatures well within the range of in- vestigation in the laboratory. On the basis of quantum mechanical theory, it is found that a very high degree of symmetry exists between the behavior of electrons and the behavior K -- EXCESS ELECTRON FROM ARSENIC ATOM ^ V sJ TsiT Tc5c=- ^^ ■"■---.^ARSENIC ATOM HAS ^K 7EXCESS + CHARGE / ELECTRIC FIELD / / V + 5-4 = + 1 +6-5=0 +4-4=0 N-TYPE SILICON _6_ (ARSENIC DONORS) ^^^ ■ --''■!■ '.■'.■ '•^■'v^^^V-.'' '' CHARGED FREE NEUTRAL ARSENIC ATOM ARSENIC ATOM SILICON ATOM IN SILICON CRYSTAL IN CRYSTAL Fig. 4. of holes. One may think of the hole as moving through the crystal as a posi- tively charged particle with much the same attributes as a free electron ex- cept for the sign of its charge. Impurity Semiconductors: Donors .\nd Acceptors If the only cases of conductivity open lo investigation were like those dis- cussed above, for which electrons and holes are present in equal numb rs, the problem of interpreting the data would be very difficult. Fortunately, in the semiconductors silicon and germanium, there are cases in which con- ductivity is due to excess electrons only or to holes only.^ ^ The process of recombination ma\' actually be much more complicated and ma>- involve intermediate stages in which the hole or the electron is trapped. ' The behavior of silicon with impurities of the sorts discussed here was investigated by Scaff, Theurer, and Schumacher. Their work was stimulated by the development of silicon detectors for microwave use by R. S. Ohl, also of Bell Telephone Laboratories, in the prewar years. .SKM/CU.XDl CJURS 341 If (he conductivity of the sample is due to excess electrons it is called n-lype, since the current carriers act like negative charges; if due to holes, it is called p-lype, since the carriers act like positive charges. Either tyi>e of conduction can be produced at will by admixture of a suit- able "impurity," a donor such as arsenic yielding an excess of free electrons, while an acceptor like boron causes an electron deficit or a surplus of positive holes. The reason why arsenic and boron serve in these opposite capacities comes readily to hand. The arsenic atom has five valence electrons surrounding a core having a net charge of +5 units and, when introduced (e.g. in silicon) as a low-fraction impurity, it is believed that each arsenic atom displaces one of the silicon atoms from its regular site and forms four covalent bonds with the nighbor- ing silicon atoms, thus using four of its five valence electrons (see Fig. 4). The extra electron cannot fit into these four bonds and is free to move about the crystal. This excess electron therefore constitutes a mobile localized negative charge. The arsenic atom, on the other hand, is an immobile local- ized positive charge, since its core, with a charge of +5 units, is not neutral- ized by its share (—4) of the charge in the valence bonds. Its net charge, therefore, just balances that of the excess electron it sets free in the crystal. Thus arsenic impurity atoms add excess electrons but do not disturb the over-all electrical neutrality of the crystal. The negative electrons, however, are somewhat attracted by the positive arsenic atoms and at low tempera- tures become weakly bound to them. At room temperature, thermal agita- tion shakes them ofif so that they become free. To produce a p-type semiconductor we choose an added impurity, such as boron, having three valence electrons and therefore not enough to com- plete the valence bond structure surrounding it. The hole in one of the bonds to the boron atom can be filled by an electron from an adjacent bond, and when this occurs the hole migrates away to the bond which just gave up one of its electrons. The boron atom thus becomes an immobile localized nega- tive charge. Because of the symmetry between the behavior of holes and electrons, we can describe the situation by saying that the negative boron atom attracts the positively charged hole but that thermal agitation shakes the latter ofif at room temperature so that it is free to wander about and con- tribute to the conductivity. Because of their valencies, phosphorous and antimony, as well as arsenic are in the donor class while aluminum, gallium and indium are additional examples of the acceptor class. It is beyond the scope of this prefatory note to describe how, b\' measure- ments of conductivity and the Hall efifect as inlluenced by the amount of added donor or acceptor, it has been possible to determine the concentration of electrons and holes, as well as to fix the energies needed to remove an elec- tron from a donor, a hole from an acceptor, and to break a covalent bond 342 BKLL SYSTEM Jj-XJJMCAL JOl R.WIL between lattice atoms. In samples of germanium of such purity that the amount of added donor or accei)tor was too small to determine by conven- tional chemical methods, the conductivity was still controlled by the proc- esses outlined above. And it is interesting to note that a portion of this investigation was carried out with the aid of radioactive antimony alloyed with the germanium, the radioactive property making possible an accurate count of antimony atoms, though present only in extremely attenuated amounts. The semiconductor papers in this issue of the Journal will explain how these simple facts of electron exchange give rise to rectifying and amplify- ing properties. Semiconductor Rectifiers and Amplifiers A contact between a metal and semiconductor may act as a rectitier, the contact resistance being high for one direction of current flow and low for the opposite. Rectification results from the presence in the semiconductor adjacent to the interface of a potential barrier or hill which the current carriers, electrons or holes, must surmount in order to flow across the junc- tion. The direction of easy flow is that in which the carriers flow from the semiconductor to the metal. An applied voltage which produces a current flow in this direction reduces the height of the potential hill and allows the carriers to flow more easily to the metal. When the voltage is applied in the opposite direction the height of the barrier which the carriers must surmount in going from the metal into the semiconductor is unchanged, to a first ap- proximation, and the resistance of the contact remains high. A p-type semi- conductor is positive, an »-type negative, relative to the metal, in the direc- tion of easy flow. Rectifying contacts can also be made between two semiconductors of op- posite conductivity types. The direction of easy flow is again that for which the /'-type is positive, the «-type negative. The rectifying boundary may separate two regions with different conductivity characteristics within the same crystal. In some contact rectifiers it is necessary to consider the flow of both types of carriers, electrons and holes, even though one type is overwhelmingly in excess under equilibrium conditions. .\n example is the germanium point contact rectifier such as the 400 tyi)e varistor. The germanium used is n- type and the normal concentration of holes is small compared to the con- centration of conduction electrons. Nevertheless, a large part of the current in the forward direction consists of holes flowing away from the contact rather than electrons flowing in. The flow increases the concentration of holes in the vicinity of the contact and there is a corresponding increase in the concentration of electrons to compensate for the space charge of the holes. This increase in concentration of carriers increases the conductivity SKMfCOXDrCTORS ,U3 of the germanium. The lH)les iiitrochiced in this way gradually combine with electrons and disappear so that at large distjuices the current consists largely of electrons. Similar effects occur at )i-p boundaries in germanium; the current in the forward direction consists in part of holes flowing from the p-iypo region into the ;/-type region and electrons tlowing from the «-type region into lhc/>-type region. The alteration of concentration of carriers and conductivity by current tlow may be used to produce amplification in a number of ways. In the type-A transistor two point contacts are placed in close proximity on the upper face of a small block of ii-iype germanium. A large area low resistance contact on the base is the third element of the triode. Each point, when con- nected separately with the base electrode, has characteristics similar to those of the rectifier. When operated as an amplifier, one point, called the emitter, is biased in the forward direction so that a large i)art of the current consists of holes flowing away from the contact. The second point, called the collector, is biased in the reverse direction. In the absence of the emitter, the current consists largely of electrons flowing from the collector point to the base electrode. When the two points are in close proximity there is a mutual in- fluence which makes amplification possible. The collector current produces a field which attracts the positively charged holes flowing from the emitter, so that a large part of the emitter current flows to the collector and into the collector circuit. It has been found that rectifying boundaries between n- and p-type germanium may be used both as emitters and collectors, so that it is possible to make transistors without point contacts. The following five papers are concerned with the behaviors of holes and electrons in semiconductors, with particular emphasis upon rectifying junc- tions and transistors. The first paper "Hole Injection in Germanium" de- scribes new experiments on the behavior of holes and shows how their numbers and velocities may be measured and how they may be used to modulate the conductivity in the "filamentary transistor." The second paper "Some Circuit Aspects of the Transistor" describes the characteristics and equivalent circuits for the transistor. "Theory of Transient Phenomena in the Transport of Holes in an Excess Semiconductor" describes in mathe- matical terms a number of the processes encountered in the first paper and brings out interesting features of the nature of an advancing wave front of holes. "The Theory of Rectifier Impedances at High Freciuencies" analyzes the behavior of metal-semiconductor rectifiers for high frequencies for the case in which the current is carried by one type of carrier only. As mentioned above, in rectifiers formed from p-ii junctions, currents of both holes and of electrons must be considered. Such rectifiers and related subjects are dealt with in "The Theory of p-n Junctions in Semiconductors and p-n Junction Transistors." Hole Injection in Germanium — Quantitative Studies and Filamentary Transistors- By W. SHOCKLEY, G. L. PEARSON and J. R. HAYNES Holes injected by an emitter i^oint into lliin single-crystal filaments of german- ium can he detected by collector jioints. From studies of transient phenomena the drift velocity and lifetimes (as long as 140 microseconds) can be directly observed and the mobility measured. Hole concentrations and hole currents are measured in terms of the modulation of the conductivity ])roduced by their presence. Filamentarv transistors utilizing this modulation of conductivity are described. 1. Introduction THE iiwention of the transistor by J. Bardeen andW. H. Brattain'' -■ =' has given great stimulus to research on the interaction of holes and elec- trons in semiconductors. The techniques discussed in this paper for investi- gating the behavior of holes in w-type germanium were devised in part to aid in analyzing the emitter current in transistors. The early experiments sug- gested that the hole flow from the emitter to the collector took place in a surface layer.^' - The possibility that transistors could also be produced by hole flow directly through w-type material was proposed in connection with the p-n-p transistor.^ Quite independently, J. N. Shive^ obtained evidence for hole flow through the body of »-type germanium by making a transistor with points on opposite sides of a thin germanium specimen. Such hole flow is also involved in the coa.xial transistor of W. E. Kock and R. L. Wallace." Further evidence for hole injection into the body of »-type germanium under conditions of high fields was obtained by E. J. Ryder.' In keeping with these facts it is concluded^ that with two points close together on a plane surface, as in the type-A transistor , holes may flow either in a surface layer or through the body of the germanium. For surface flow to be large, special surface treatments appear to be necessary; such treatments were not employed in the experiments described in this paper and the results are consistent with the interpretation that the hole current from the emitter point flows in the interior. The experiments described in this paper, in addition to any practical implications, serve to put the action of emitter points on a quantitative basis and to open up a new area of research on conduction processes in semicon- * It is planned to incorporate this material in a book entitled "Holes and Electrons, an Introduction to the Physics of Transistors" currently being written by VV. Shockley. This book is to cover much of the material planned for the "Quantum Physics of Solids" series which was discontinued because of the war. 344 HOLE ISJECnOS IX GERM AM CM 345 (luctors. It is worth while at the outset to contrast some of the new aspects of these experiments with the earlier experimental status of the bulk proj)- erties of semiconductors. I'rior to the invention of the transistor, inferences about the behaviors of holes and electrons were made from measurements of conductivity and Hail efTecl. l-'or both of these effects, under essentially steady state conditions, measurements were made of such cjuantities as len<^ths. currents, voltages and magnetic fields. 'J'he measurement of time was not involved, except indirectly in the calibration of the instruments. .Nevertheless, on the basis of these data, definite mental pictures were formed of the motions of holes and electrons describing in particular their drift velocity in electric fields and the transverse thrust exerted upon them by magnetic fields. The new experiments show that something actually does drift in the semiconductor with the predicted drift velocity and does behave as though it had a plus or minus charge, just as expected for holes and electrons. In addition, experiments described elsewhere show that the effect of sidewise thrust by a magnetic field actually is observed in terms of the concentration of holes and electrons near one side of a filament of germanium. We shall discuss here evidence that holes are actually introduced into //-type germanium by the forward current of an emitter point and show how the numbers and lifetimes of the holes can be inferred from the data. We shall refer to this important process as "hole injection." Discussions of the reasons why an emitter should emit holes are given for point contacts by J. Hardeen and W. H. Brattain ' " and for p-n junctions elsewhere in this journal. There are other possible ways in which semiconductor amplifiers can be made without the use of hole injection into «-type material or electron injection into />-type material.* In this paper, however, our remarks will be restricted to semiconductors which have only one type of carrier present in appreciable proportions under conditions of thermal equilibrium; for such cases the theoretical considerations are simplified and are apparently in good agreement with the e.xperiments. 2. The Measurement of Density and Current of Injected Holes The experiment in its semiquantitative form is relatively simple and is shown in Fig. 1.^" A rod of «-type germanium is subjected to a longitudinal electric field E applied by a battery Bi. Collector and emitter point contacts are made to the germanium with the aid of a micromanipulator. The col- lector point is biased like a collector in a type-A transistor by the battery B>, and the signal obtained across the load resistor R is applied to the input of an oscilloscope. At time /i the switch in the emitter circuit is closed so that a forward current, i)roduced by the battery B-^, tlows through the emitter point. At /s the switch is opened. The voltage wave at the collector, as * For example see references 1 and 1 1 . 346 BELL SYSTEM TECHNICAL JOCKS A L observed on the oscilloscope, has the wave form shown in part (b) of the figure. These data are interpreted as follows: When the emitter circuit is closed, the electrons in the emitter wire start to flow away from the germanium (i.e. positive current flows into the germanium). These electrons are furnished by an electron flow in tlie germanium towards the point of contact. The flow in the germanium may be either by the excess electron process or by the hole process. In Fig. 2 we illustrate these two possibilities. At first -^83 B, 1 T ■xxKWsVS^VS^VVH^yXyCAVl- u J n-TYPE GERMANIUM TO CATHODE-RAY OSCILLOSCOPE : s CLOSES (HOLES INJECTED AT 6) ARRIVAL OF HOLES AT C s OPENS DEPARTURE OF HOLES AT C > 1 1 LU < (- _l 0 > \r 1 'J i\ If Rd i^ 0 f (b) ti ta "t.3 t4 TIME,t >■ Fig. 1 — Experiment to investigate the behavior of holes injected into «-type germanium (a) Experimental arrangement. (b) Signal on oscilloscope showing delay in hole arrival at h in respect to closing S at /i and delay in hole departure at ti in respect to opening S at /s . glance it might appear that the difference between these two processes is unimportant since the net result in both cases is a transfer of electrons from the germanium to the emitter point. There is, however, an important differ- ence, one which makes several forms of transistor action possible. In the case of the hole process an electron is transferred from the valence band struc- ture to the metal. After this the hole moves deeper into the germanium. As a result the electronic structure of the germanium is modified in the neigh- borhood of tlie emitter point by the presence of the injected holes. Under the influence of the electric field !>, the injected holes drift toward HOLE INJECTION IN GERMANIUM 347 Ihc collector i)oinl with velocity y.,>E, where Mp is the mobility of a hole, and thus tra\-erse the distance /. to the collector point in a time L/npE. When they arrive at the collector point, they increase its reverse current and pro- duce the si^Mial shown at /n . There are two important differences between the signal produced at k and that produced at k . The signal at /i , which is in a sense a pickup signal, would l)e produced even if no hole injection occurred. We shall illustrate this by considering the case of a piece of ohmic material substituted for the METAL SEMICONDUCTOR ELECTRON GAS MOTION OF HOLE Fig. 2 — Electron flow to the metal may be jjroduced by an excess electron moving toward the metal or by bonding electrons jumping (dashed arrows) successively into a hole thus displacing the hole deeper into the semiconductor. germanium. Conventional circuit theory applies to such a case; however, in order to contrast this purely ohmic case with that of hole injection, we shall also give a description of the conventional theory of signal transmission in terms of the motion of the carriers. According to conventional circuit theory, the addition of the current I, would simply produce an added IR drop due to current flow in the segment of the specimen to the right of the collector. This voltage drop is denoted as hRa in part (b), Ra representing the proper combination of resistances to take into account the way in which /< divides in the two branches. This signal will be transmitted from the emitter to the collector with practically the speed of light — the ordinary theory of signal 34S BELL S\yrEM TECIIMCAL JOURS AL transmission along a conductor being applicable to it. This high speed of transmission does not, of course, imply a correspondingly high velocity of motion of the current carriers. In fact the rapidity of signal transmission has nothing to do with the speed of the carriers and comes about as follows: If the ohmic material is an electronic conductor, then the withdrawal of a few electrons by the emitter current produces a local positive charge. This positive charge produces an electric field which progresses with the speed of light and exerts a force on adjoining electrons so that they move in to neutralize the space charge. The net result is that electrons in all parts of the specimen start to drift practically instantaneously. They flow into the specimen from the end terminals to replace the electrons flowing out at the emitter point and no appreciable change in density of electrons occurs any- where within the specimen.* The distinction between the process just described and that occurring when holes are injected into germanium is of great importance in under- standing many effects connected with transistor action. One way of sum- marizing the situation is as follows: In a sample having carriers of one type only, electrons for example, it is impossible to alter the density of carriers by trying to inject or extract carriers of the same type. The reason. is, as described above (or proved in the footnote), that such changes would be accompanied by an unbalanced space charge in the sample and such an unbalance is self-annihilating and does not occur.f When holes are injected into ;/-type germanium, they also tend to set up a space charge. Once more this space charge is quickly neutralized by an electron flow. In this case, however, the neutralized state is not the normal thermal equilibrium state. Instead the number of current carriers present has been increased by the injected holes and by an equal number of electrons drawn in to neutralize the holes. The total number of electrons in the speci- men will thus be increased, the extra electrons coming in from the metal terminals which complete the circuit with the emitter point. The presence of the holes and the neutralizing electrons near the emitter point modify the conductivity. As we shall show below, this modification of conductivity may be so great that it can be used to measure hole densities and also to give power gain in modified forms of the transistor. We shall summarize this situation as follows: /;/• a semiconductor conlainiug substantially only one type of current carrier, it is impossible to increase the total carrier co)icentralion by * This is a descrii)tion in words of the result ordinarily expressed in terms of the dielectric relaxation time obtained as follows: V-/ = —p,I = aE= —aV^, V"* = — 47rpA = p /a so that p = po exp [- (47r injecting carriers of the same type; hoarier, such i)icr eases can he produced bv injecting the opposite type since the space charge of the latter can he neutralized by an increased co)icenlration of the type normally present. Thus \vc conclude that the existence of tu'o processes of electronic conduction in semiconductors, corresponding respectively to positive and negative mobile charges, is a major feature in several forms of transistor action. In terms of the description given al)ove, the experiment of Figure 1 is readily interpreted. The instantaneous rise at ti is simply the ohmic contribu- tion due to the changing total currents in the right branch when the emitter current starts to flow. After this, there is a time lag until the holes injected into the germanium drift down the specimen and arrive at the collector. When the current is turned off at /s , a similar sequence of events occurs. The measured values of the time lag of ti — /•.. , the field E and the distance L can be used to determine the mobility of the holes. The fact that holes, rather than electrons, are involved is at once evident from the polarity of the effect; the disturbance produced by the emitter point flows in the di- rection of E, as if it were due to positive charges; if the electric field is re- versed, the signal produced at /-j is entirely lacking. The values obtained by this means are found to be in good agreement with those predicted from the Hall effect and conductivity data. The Hall mobility values obtained on single crystal filaments of ;/- and p-type germanium'* are IJL„ = 1700 cm sec per volt/cm Hn — 2600 cm/ sec per volt cm The agreement between Hall effect mobility and drift mobilitv, as was pointed out at the beginning of this section, is a very gratifying confirmation of the general theoretical picture of holes drifting in the direction of the electric field. We shall next consider a more quantitative embodiment of the experi- ment just considered. In Fig. 3, we show the experimental arrangement. In this case it is essential in order to obtain large eft'ects that the cross-section of the germanium filament be small. A thin piece of germanium is cemented to a glass backing plate and is then ground to the desired thickness. After this the undesired portions are removed by sandblasting while the desired portions are protected by suitable jigs consisting of wires, scotch tape, metal plates, etc. After the sandblasting, the surface of the germanium is etched. In this way specimens smaller than 0.01 X 0.01 cm in cross-section have been produced. The ends of the filament are usually made very wide so as to simplify the problem of making contacts. Under experimental conditions, a battery like Bi, of Figure 1 applies a ''sweeping" field in the filament so that any holes injected by the emitter 350 BELL SYSTEM TECILXICAL JOURNAL current are s\vej)t along the filament from left to right. In the small filaments used for these experiments, the resulting concentration of holes is so high that large changes in conductivity are produced to the right of the emitter point and, as we shall describe below, these changes can be measured and the results used to determine the hole current at the emitter jioint. In order to treat this situation quantitatively, we introduce a ciuantity 7 defined as follows: 7 — the fraction of the emitter ctirreut carried by holes. Accordingly, a current yh of holes flows to the right from e and produces a hole density, denoted by p, which is neutralized by an equal added electron density. A fraction (1 — 7)/^ of electrons flows to the left; these electrons do not, however, produce any increased electron density to the left of the emitter since they are of the sign normally {^resent in the //-type material. The presence of the holes to the right in the filament increases the con- ductivity a (as shown in Fig. 3c) both because of their own presence and the presence of the added electrons drawn in to neutralize the space charge of the holes. The mobility of electrons is greater than the mobility of holes, the ratio being'^ h = /x„ Vj) =1.5 for germanium and the electrons are always more numerous than the holes* // = ;/o + p, (2.1) where ;?o is the concentration of electrons which would be present to neu- tralize the donors if p were equal to zero; consequently, the current carried by electrons is greater than the current carried by holes. The concentration of holes diminishes to the right due to the fact that holes may recombine with electrons as they flow along the filament. From this experiment the value of 7 and the lifetime of a hole in the filament can be determined. The measurements are made with the aid of the two probe points I\ and P-i . The conductance of the filament between these points is obtained by measuring the voltage difference AV and dividing it into the current //, -|-- T( , no current being drawn by the probes them- selves. The necessary formulae for calculating hole density and hole current, * The notation used in the eriuations is as follows: n, p, »o = respectively density of electrons, of holes, of electrons when no holes are injected. Nd and Na are the densities of donors and acceptors, assumed ionized so that uo = Nd — .Va. Ii, h, Ic are as shown on Figs. 3 and 9. {E used for the probe collector in Figures 1 and 8 does not enter the equations.) q = \q\ is the charge on the electron, used to be consistent with Ref. 4, where e is used for 2.718 • • •. HOLE INJECTION IN GERMANIUM 351 1'^ ©+0"_W © ©"©_©_© ©_©"©_®"®'^® (a) if+ib f ih a ylt 1+) = DONOR + = HOLE - = EXCESS ELECTRON ~-~-n„= Nh-N, = DENSITY OF DONORS ° ° MINUS ACCEPTORS «ro=qb//pno " — ff^ q//pP + qbApn = q//p(p+b(no + p)) = ^°^'^^'^F'?ro) 1(c) '-In=CURRENT CARRIED BY HOLES = (I^ + I b) r (d) 0 f X, DISTANCE THROUGH SEMICONDUCTOR^ *■ Fig. 3— Method of measuring hole densities and hole currents. (a) Distribution of holes, electrons and donors. Acceptors, which may be present, are omitted for simplicity, the excess of donor density Na over acceptor density .Vo being «o . (b) To the right of the emitter the added hole density p is compensated by an equal increase in electron concentration. (c) The conductivity is the sum of hole and electron conductivities. (d) The total current h + /« to the right of the emitter is carried by /p and /„ in the ratio of the hole to the electron conductivity. 352 lil'-l.l. SYSTEM TPXIIXICAr. JOrRXAL s'.owu on the Figure, are derived as follows: Normal conductivity o-n = (/M""ii , {'2.2) conductivity with holes present a = i]n„n + (/tx,,p = (lH„(n, + p) ^ qn.p = cr„[l + (1 + b-'){p //o)l. (2.3) The conductance, G = (A + /.)/Al', between Pi and Pi is proportional to the local conductivity, and hence to ! + (!+/> '){p II,), so that a measurement of the conductance gives a measurement of p/nn . Letting G and Go be the conductances between the points with and without hole injection, we have ^- = - = 1 + (1 + b-')(p/no) (2.4) tro (To or p _ '-? ! p = 1.0 A 0.1 O 1.0 1.0 — 0.8 ^Vi sr"!"^ -7 n = 0.6 0.6 0.5 \ '\.'-' 1 \ A ^ 0.3 '\ 1 1 1 1 \ \ c 0-2 H 1 i\ 1 1 N. Z < — t ! ^ k 1 1 1 1 A \ s^Ip/lf FOR n-TYPE ^7 — 1 — "■ V °- ^ ^r. t — ^ — 1 — , 1 \, M 0.08 \l \^ \ s 0.06 \ \ c ) 0.05 I/) I 1 k s 0.04 0.03 ^1 ^ 1 =^1 6 \ ^In/If FOR p-TYPE . -- " 1 c 1 K 1 1 \ 1 1 0.01 1 1 1 ' 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 TRANSIT TIME,7-t, IN MICROSECONDS Fig. 4 — E.xlrapohiticii of measured hole and electron curreiUs to zero transit linic in order to determine 7. comes at the left edge.t The conclusion reached from this i)lot is that for the case of the //-type sample, the value of 7 is substantially unity, — all the t It the lifetime of a hole is t, then the hole current at the points is I„ = ylt exp ( — //V) where / is the transit lime to a point midway between the points, say a distance L from the emitter. If the electric field is E = Al'/AA, then the transit time / = /,A/.,7i,,Ar. Hence if In /,„ as determined from the ratio of conductivities is plotted afi;ainst / = LM. fi„AV a straight line with intercej)! In 7/, and sloi)e - 1 r should he ohiained. 354 BELL SYSTEM TECHNICAL JOURNAL emitter current is holes. For the opposite case in which electrons are injected into />-type material,^' the corresponding value of /„,'/t extrapolates to 0.6 indicating that for this case 60% of the current is carried by electrons and 40% by holes. For these particular specimens the lifetimes are found to be 0.9 and 0.41 microseconds respectively. There is a body of evidence, some of which we discuss below, that holes combine with electrons chiefly on the surface of the filament. 3. The Influenxe of Hole Density ox Point Contacts The presence of holes near a collector point causes an increase in its reverse current ; in fact the amplification in a type-A transistor is due to the modulation of the collector current by the holes in the emitter current. The influence of hole density upon collector current has been studied in con- nection with experiments similar to those of Fig. 3. After the hole current and the hole density are measured, a reverse bias of 20 to 40 volts is applied. The reverse current is found to be a linear function of the hole density. Figure 5 shows typical plots of such data. Different collector points, as shown, have quite different resistances. However, once data like that of Fig. 5 have been obtained for a given point, the currents can then be used as a measure of hole density. This experimental procedure for determining hole density is simpler than that involved in using the two points and much better adapted to studies of transient phenomena. It is necessary in em- ploying this technique to keep the current drawn by the collector point somewhat smaller than /b + /« ; otherwise the disturbance in the current flow due to the collector current is too great and the sample of the hole current is not representative. Experiments have shown, however, that this condition is readily achieved and that the collector current may be satis- factorily used as a measure of hole density. The hole density also affects the resistance of a point at low voltage. Studies of this effect have also been made in connection with the experi- ment of Fig. 3. After the hole density has been determined from measure- ments of AV and /& -f- /« , a small additional voltage (0.015 volts) was ap- plied between Pi and P2 and the current flowing externally between Pi and P2 was measured. From these data a differential conductance, for small cur- rents, is obtained for the two points Pi and P2 in series. As is shown in Fig. 6, this conductance is seen to be a linear function of the hole concen- tration. The conductance of a point contact arises in part from electron flow and in part from hole flow. l""rom experiments using magnetic fields^, it has been estimated that under equilibrium conditions the two contributions to the conductance may be comparable. In connection with Fig. 6 it should be noted that the hole concentration on the abscissa is the average hole iioi.ii iwiEcrios i.\ c;F.R\iA.\ir \f 355 concentraticju throughout the entire cross section; the hole concentration may be much less near the surface due to recombination on the surface. Techniques of the sort described above can be used to measure the })roi)- erties of collector points. If a collector ])oint is placed l)et\veen the emitter and 7^ in Fig. >■?, then the hole current extracted by the collector can be determined in terms of the hole current past I\ and 7^. . By these means an "intrinsic a" for the collector point can be determined. The intrinsic a is 0.04 0.06 0.08 0.10 0.12 0. HOLE DENSITY IN TERMS OF p/p 0.16 0.18 Fig. 5 — Dependence of collector current Ic upon average hole density being swcj)! \>y collector point. Collector biased 20 volts reverse. defined as the ratio of change in collector current per unit change in hole current actually arriving at the collector. 4. Studies of Tr.ansiext Piienomen.\ 'I'he technique of using a collector point to measure hole concentrations has been employed in a number of experiments similar to those described in connection with Fig. 1. These experiments give information concerning hole lifetimes, hole mobilities, diffusion and conductivity modulation. One of the methods employed to measure hole lifetime involves the meas- urement of the increase in collector current, produced by the arrival of the leading edge of the hole pulse, as a function of the transit time of the holes from emitter to collector. This time is varied by varying the distance be- tween the emitter and the collector jxjints. 356 BELL SYSTEM TECHNICAL JOURNAL In Fig. 7 we show a plot, obtained in this way, from a sample of germa- nium having dimensions 1.0 X .05 X .08 cm. It is seen that the increase in collector current due to hole arrival decays exponentially with a time con- stant of 18 microseconds. This time constant increases as the dimensions of the germanium sample are increased so that a time constant of 140 micro- seconds was measured, using a sample having dimensions 2.5 X .55 X .50 a 13 O o 0.05 ma D A 0.2 D 0.5 y y y y y y / y^ ^ y ^ -■.y X °^ / y V <^. y K f 0.3 0.4 0.5 0.6 0.7 HOLE DENSITY IN TERMS OF p/n^ Fig. 6 — Conduclance of P\ and Pi of Fig. 3 in series as a function (;f p!n» , showing that conductance depends on hcle concentration but not on currents in filament. For each value of /< the hole density was varied hv varying h + /< from .038 to 0.78 ma. cm. Since the holes injected into the interior of this sample can diffuse to the surface and recombine in about 100 microseconds, the process may still be largely one of surface recombination. In any event, it may be concluded that the lifetime in the bulk material used must be at least 140 microseconds. Making use of the electron density determined from other measurements, we conclude that the recombination cross section must be less than 10 cm-. This cross section, which is less than 1 400 the area of a germanium //()/./•: ISJECTIOX fX GERM AM! M .S37 ;itom, mav be so smuU because a liDle-elertroii i)air has (litVuully in satisfying:; in the crystal the conditions somewhat analogous to conservation of energy and momentum which hinder recombination of electrons and positive ions in a gas discharge. Thus it has been pointed out that a hole-electron pair will have a lowest energy state in which the two current carriers behave some- thing like the proton and electron of a hydrogen atom/' Such a bound pair are called an excitoii and the energy given up by their recombination is the "exciton energv." In order to recombine they must radiate this energy in 10 ,0 DIFFERENT COLLECTOR POINTS z ^ 3 a. \ m •>^ C % 3 Z \ o \ \ \. - \ (t 0.8 D O - °\ V° 0.6 a. O0.5 H - \ O0.3 - c \\ Z \ \ ujO.2 If) < tr o - \ \ 0 10 20 30 40 50 60 70 80 90 100 110 120 TRANSIT TIME IN MICROSECONDS Fig. 7 — Tlie decay of injected holes in a sample of «-iype germanium. the form of a light quanta (photon) or a quantum of thermal vibration of the crystal lattice (phonon). The recombination time for the photon recombina- tion process can be estimated from the optical constants for germanium and the theory of radiation density using the principle of detailed balancing, which states that under equilibrium conditions the production of hole electron pairs by photon absorption equals the rate of recombination with photon emission; the lifetime obtained in this way is about 1 second at room temperature indicating that the photon process is unimportant.^^ As has been pointed out by A. \V. Lawson,'" the highest energy jihonon will have 358 BELL SYSTEM rECHNICAL JOURNAL insufficient energy to carry away the "excilon energy" oi a hole-electron pair and, therefore, the release of energy will require the cooperation of several phonons with a correspondingly small transition probability. When a square pulse of holes is injected in an experiment like that of Fig. 1, the leading and trailing edges of the current at the collector point are deformed for several reasons. Due to the high local fields at the emitter point, some of the holes actually start their paths in the wrong direction — i.e. away from the collector; these lines of flow later bend forward so that those holes also pass by the collector point but with a longer transit time than holes which initially started towards the collector. A spread in transit times of this sort is probably largely responsible for the loss of gain at high fre- quencies in transistors. For the experiments described below, however, this effect is negligible compared to two others which we shall now describe; On top of the systematic drift of holes in the electric field, there is super- imposed a random spreading as a result of their thermal motion. This would cause a sharp pulse of holes to become spread so that after drifting for a time td the hole concentration would extend over a distance proportional to -y/Dtd where D, the diffusion constant for holes, = kTup/q = 45 cm-/sec. As a result of this effect, the leading and trailing edges of the square wave of emission current become spread out w'hen they arrive at the collector. This is shown in Fig. 8, curve A for the leading edge and B for the trailing edge. The points are 10 microsecond marker intervals traced from an oscillo- scope, the time being measured from the instant at which the emitter current starts. For A and B the emitter current was so small compared to the current /& that the holes produced a negligible modulation of conductiv- ity and each hole moved in essentially the same electric field. It is to be observed that the wave shapes are nearly symmetrical in time about the half rise point and that the .1 and B waves are identical except for sign. This is just the result to be expected from diffusion. Furthermore, analysis shows that the spread in arrival time is in good quantitative agreement with the theoretical wave shape using the diffusion constant appropriate for holes. For this case the mid-point of the rise, corresponding to the crossing point of the curves, gives the average arrival time and has been used to obtain an accurate measure of the mobility. Curves C and D correspond to conditions in which the emitter current was relatively large — two thirds of the base current. High imjiedance sources are used so that h is constant and h is a good flat topped wave. For the currents used in this experiment, the conductivity is appreciably modulated by the presence of holes. This accounts for the shape of curve C, correspond- ing to the arrival of holes at the collector. It is seen that this curve is not symmetrical but is much more gradual towards later times. The reason for //()/./•: i.x.ijx ii().\ i.\ c,j:i\ i\ (,i:i<\i i\ii \f m,\ Figure 0(1)I, whidi shows the rurrciit and xoltagc relationships in a form equivalent to those ust-il in coniUHtion with the type-A transistor. The point ./ in Im^. '•) represents a jioint in tiie lilamenl near the emiller point. The current from llie emitter point will Ije determined by the differ- ence l)elween its voltage I', and that of the surrounding semiconfluclor, namely the voltage at ./. Thus we can write /. =/,(l'e - Vj). (5.1) l'\)r small a-c \ariations, /< , r, and v,, , this equation leads to the rela- tionshij) /« = i'c't — Vj)fe , (5.2) whcre/( is the derivative oi l\ in respect to ils argument. Letting /", = \/Rf this equation becomes t\ — v., = RJ, . (5.3) This relationship is correctly represented by the R^ branch of the equivalent circuit. The voltage at /, under the assumed operating conditions with /« positive and much less that h , will be —IbRb where Rb is the resistance from the base to an imaginary equipotential surface passing through J and Vh = 0, corresponding to grounded base operation. This leads to vj = —Rkib = +RbL + Rhic , (5.4) since //, + i, + /,. = 0. This relationship is obviously satisfied by the Rb branch of the equivalent circuit. We now come to the collector branch which we have represented as a resistance Re and a parallel current generator* acif . (This circuit is equivalent to another in which the parallel current generator is replaced by a series voltage generator acRcif ■) We must show that this part of the equivalent circuit represents correctly the effect of injecting holes into the right arm of the hiament. We shall suppose that there is negligible recombination so that the hole current injected at the emitter point flows through the entire filament. (We consider recombination in the next section.) The current /c in the collector branch thus contains a component —yf( = I,, of hole current (minus because of the algebraic convention that positive /c(= —lb — /«) flows to the left). The added hole and electron concentrations lower the resistance and Re changes to Re + 8Re , where 8Rc is negative. The current voltage relationship for this branch of the filament then becomes Vc - I'./ = (Re + dRe)Ic . • (5.5) * ae in the equivalent circuit dilTers from a = — ((ilr/<^l ,)v,- by the relationship r^, = f> + in — \)(K\,/R,-), equivalent to eriuation f6.8). 362 ni'J.I. SV.ST/':\f TECIIXICAL JOURXAL Our i)r()l)leni is to reex{)ress this relationship in tcrnis of the small a-c coni- |)om'nts and show that it rcducos to the relationship 'i'c — V.I = RMc + occit) (5.6) corresponding to the equivalent circuit. For small emitter current the analysis is carried out conveniently as follows: The ratio of hole current to the total current is —-^I./Ic. The ratio {Re + bR^/Rc corresponds to Ga/G discussed in connection with Fig. 3. The ratio of hole current to total current is given in (2.6) in terms of Ga/G and may be rewritten as _yh^ 1 - (gp/G) _ -8Ro . Ic \ + b {\ + b)Re' ^''•^^ giving bRc = Rc{\ + b)y Lh. (5.7) (Since Ic is negative and /« is positive this equation shows that bRc is nega- tive, i.e., the conductivity has been increased by the hole current.) Putting this value of Re + bRc into the equation for T^ — V j gives ]'. - Vj = (Re + bRc)Ic = RcUc+ (1 + b)yll .(5.8) If we consider small a-c variations in the currents and voltages, this reduces to the equation given by the equivalent circuit with «.-(! + bh. (5.9) The data of Section 2 indicate that for holes injected into ;z-type germanium 7 = 1, and since b — 1.5 we obtain ae = 2.5. The quantity Vj can be eliminated by using Vj = Rbii^ + ie) in equation (5.3) for Ve and the small signal form of (5.8) for zv leading to the pair of equations lu = {Re + Rb)i. + Rbic (5.10) Vc = {Rb + aeRe)ie + {Re + Rb)ic . (5.11) These equations are formally identical with those for the equivalent circuits of the type-A transistor. It should be emphasized that although hole injection into //-type germa- nium plays a role in both the type-A and the particular form of filamentary transistor shown in Fig. 9, there are differences in the principles of operation. One imj)ortant feature of the type-A is the high impedance of the rectifying collector contact which, however, does not impede hole liow and another important feature is the current amplification occurring at the collector contact. Neither of these features is present in the filamentary type shown. Instead, the high impedance at the collector terminal arises from the small iioi.i-: ixjEcriox i\ i,i:i<\i.\\ir\[ ?,m cross-section of the filament. The modulation of the output current takes place through the change in body conductivity due to the presence of the added holes, a change which appears to be unimportant in the type-A transistor. In the filamentary type, current amplification is produced by the extra electrons whose presence is required to neutralize the space charge of the holes. Current amplification in the type-A transistor is, probably, also produced by the space charge of the holes'* but the details of the mechanism are not as easily understood. 6. Effkcts Associated with Transit Time Two important effects arise from the fact that a finite transit time is required for holes to traverse the Re side of the filament: during this time the holes recombine with electrons and the modulation effect is attenuated for this reason; also the modulation of the conductivity of the filament at any instant is the result of the emitter current over a previous interval and for this reason there will be a loss of modulation when the period of the a-c signal is comparable with the transit time or less. For the small signal theory, the effect of transit time is readily worked out in analytic terms. We shall give a derivation based on the assumption that the lifetime of a hole before it combines with an electron is Tp . According to this assumption, the fraction of the holes injected at instant ti which are still uncombined at time Aj is exp[— {k — /i) Vp]. This means that the effect in the filament at any instant /« is the average, weighted by this factor, of all the contributions prior to k back to time /j — n where Tt is the transit time; holes injected prior to h — n have passed out of the filament by time k • If the emitter current is represented by ieoe"^ , the effective average emitter current is i.eff(/2) = /.c f'^' e'"''-'''--''"'^ dh/rt. (6.1) The term dti/rt is chosen so that a true average is obtained since the sum of all the dti intervals add up to rj . The integral is readily evaluated and gives . f,\ • iu,t. 1 — exp [— i(jjTt — {rt/Tp)] . -s l(j}Tt -\- [Tt/Tp) The result so far as the equivalent circuit is concerned is that obtained by taking ae as* ue = t(1 + b)^, (6.3) * The derivation of equations (5.10) and (5.11), describing the equivalent circuit, shows that hole injection enters only through the term bRJc in (5.8). This term leads only to aeRcit = (1 + b)yRcit in (5.11) and should be replaced by (1 + b)yRcit cff = (1 + b)y0Rcii leading to (6.3). MA BELL SYSTEM TECJLMCAL JOLKXAL where ^ ^ 1 - oxpl-uor. -(r^A^ _ (6.4) liOTt + {Tl/Tp) ti represents the effect of recoml)inati()ii and transit anj^le, w-/ , in reducing the gain. We shall consider two limiting cases of this expression. First if cor^ is very small, the new factor becomes 0 = (r, tM\ - €-''''")■ (6.5) If Tt is much larger than t,, , so that the holes recombine before traversing the filament, then the exponential is negligible and (3 becomes simply Tp ti . This means that the effectiveness of the holes is reduced by the ratio of their effective distance of travel to the entire length of the filament, i.e., TpTt is the ratio of distance travelled in one lifetime to the entire length of the fila- ment. Essentially the holes modulate only the fraction of the filament which they penetrate. The transit time depends on the field in the filament which is I Fr — Vj \/Lc, the absolute value being used since Vc is negative. The transit time is thus Tt = Lr [np I 1'^ — Vj I Lj = Lc/fJ-p I Vc — Vj\. (6.6) For very small emitter currents Vc — Vj = RcVr/{R, + Rb) so that Tt = lI(R.-+ Rh) ^JipRr\V,\ (6.7) and Tt is inversely proportional to ]',: . For large values of I',. , Tt approaches zero and 0 approaches unity. The dependence of li upon Vc has been investi- gated by measuring a and plotting it as a function of |l "'Vc\ as shown in Fig. 10. The value of a = -(df,. dJ,)vc (6.8) is readily found from the ecjuivalenl circuit, using equation (5.11), lo l)e ' Rb UeRc , , For the particular structure investigated, the values of Ri, and R, , obtained at /< = 0, were in the ratio 1:4. The value of a obtained by extrapolating the data to 1 F, I = ^c is 2.2; the value given by the formula for this case with /3 = 1 , is a = n.2 + 0.8 X 2.5 X 7, (6.10) from which we lind 7 = 1.0, in agreement with the result of I-'ig. 4 that HOLE l.WfKCIlOX l\ (il.KMAMI \l .U)5 substantially all of the emitter current is carried by iioles. The theoretical curve shown on the ['"if^ure is 10/1 Vc I 0.2 + O.S X 2.5 X |F,,/1()|(1 - e'"'""-') This corres])()n(ls to r, ^ 10 ^ lJ:{Rc + Rb) T \V \ T a R\V \ ' (6.11) (6.12) from which it was conckulcd that for the particular bridge studied t,, was 0.2 microseconds. >- 1.5 Ol (u M M II 1.0 <3 Va' -EXPERIMENTAL POINTS \ \ ^ -v^ / , -THEORETICAL ^ 1 i 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 00 '/Vc IN VOLTS-' Fig. 10 — a versus l/d'ci showing agreement with the theory for the \alue of /^. If Ti is much shorter than r^ , then the holes penetrate the whole filament and /3 becomes ^ = 1 — exp( — iajT<) — lUT ( I'l sin (cor- in I'i^. 2. One point, called the collector, Fig. 1— Cutaway view of transistor. EMITTER COLLECTOR Fig. 2- -Transistor mechanism. is a rectitier biased stiongly in the low-conducting direction. It therefore has a rectifying barrier in the germanium near it, which causes the collector impedance to be high. However, the collector can be influenced by the 370 BELL SYSTEM TECHNICAL JOURNAL emitter if the latter is arranged to emit anomalous charge carriers, that is, carriers of the sign not normally present in the interior of the material. Equivai.kntt Circuits As has been explained by liardecn, Bratlain, and Shockley, many features of the transistor are nicely explained by this picture of its action; but, for present purposes of circuit analysis, we shall now take the purely empirical EQUIVALENT CIRCUIT Equations /■l'g21 + t2Cg22 + Z^) = 0 Circuit determinant A = C^u + Zg)(^i2 + Zl) — %-^-ii Input impedance Zu = '%u — Output impedance Z22 = ^22 — Operating power gain Go = ^RgRl ?22 + Zl ?ii + Zg A 1 (Zg + Zl)% Insertion power gain G\ = Fig. 3 — Synopsis of general four-pole — imjiedancc analysis. view and regard the transistor as a black box whose performance is to be determined by electrical measurements on its terminals. A picture of a black box is shown in Fig. 3 along with the equations de- scribing it. The performance is completely characterized if one knows the voltage and current at each of the two pairs of terminals. Now, of these four variables, only two are independent since, if any two are fixed, the other two are determined. One can therefore describe the network in terms of any two variables and, since there are six possible ways to choose a pair of variables from a set of four, there are sk ways of describing the network. To recall what is done for electron tubes is helpful. In the case of a triode SOME CIRCllT ASl'ECrS ()!■ THE TRAXSISTOH -VI the voltages on grid and plate are usually taken as independent variables; the grid and j^late currents are taken as functions of the voltages. It be- comes natural, then, to measure tubes with regulated power supplies having low impedances to keep the voltages constant, and one is then naturally led to describe tubes in terms of admittances. Now the trouble with this scheme for transistors is that many of them oscillate when connected to low im- pedances, that is, many transistors are short-circuit unstable. To avoid this dilBculty it is convenient to measure with high impedances in the leads; the analytical counterpart is to regard the currents as independent variables, leading naturally to a description of the transistor in terms of impedances, as shown in the figure. This description by open-circuit impedances happens to be a good one for many purposes, but there is nothing final or unique about it. In fact at high frequencies one of the other descriptions becomes more convenient. By interpreting the '^ equations as circuit equations, one is led directly to the first equivalent circuit of Fig. 4. A little consideration shows why the •g's are called open-circuit impedances. For example, if the second mesh is open-circuited, then the equation say that %i is the ratio of input voltage to input current, that is, the input open-circuit impedance; while '^21 is the ratio of output voltage to input current, that is, the open-circuit forward transimpedance. Sunilarly '^12 is the open-circuit feedback transimpedance and "^22 is the open-circuit output impedance. Most of the subsequent dis- cussion is concerned with low frequencies, where the unpedances reduce to resistances. This equivalent circuit for small signals is only one of many possibilities. Another, which is in fact more frequently used, is shown on Fig. 4. It con- sists of a T of resistors, each of which is associated with one of the transistor leads, and a voltage generator in series with the collector lead whose ratio to the emitter current is also of the dimensions of a resistance. The elements of this equivalent circuit are related to the former one by a simple sub- traction. The other equivalent circuit on Fig. 4 is obtained by converting the series voltage generator to the equivalent shunt current generator, whose ratio to the emitter current is now a dimensionless constant which we shall call a . These circuits, as well as all the other numerous possibilities, are equiva- lent in the sense that they all give exactly the same performance for any external connection of the unit. These three, however, are particularly well- behaved in that usually none of the circuit elements is negative; they are readily accessible to measurement; the association of the various circuit elements with corresponding regions withiii the transistor appears to have some physical significance; and, finally, the parameters are not too dread- fully dependent on the exact operating point used. 372 BELl, SYSTEM TECIISICM. JOIRXAL In the choice among various equivalent circuits, it appears that tlie oji- timum of convenience is also the one which most closely approaches the underlying physical situation. In agreemg to use the hlack box approach we have resolutely ignored the physical details, but here they are presenting themselves in a new way, having sneaked in the back door after we barred the front. Now, however, having chosen an equivalent circuit, we shall continue pursuing the circuit analysis in resolute ignorance of the physics. In what follows various equivalent circuits may be used, depending on the convenience of the moment. Figs. 4 — Some equivalent circuits. CONSTANT- CURRENT GENERATOR HIGH- IMPEDANCE VOLTMETER Principle of measurement method. The principle of a method used for rapid measurement of the transistor impedances is shown in Fig. 5, illustrating the measurement of forward transimpedance. A pair of terminals of the transistor is driven by a small alternating current of a few thousand cycles from a high impedance gen- erator; the voltage developed is read by a high-impedance voltmeter. By calibrating the meter directly in ohms, one can read off the open circuit resistances of the unit as rapidly as one can switch and read meters. Average values found by this method for the Type A transistor are shown on Fig. 6, together with data on the direct-current operating point. Since SOME CIKCL rr ASI'KCTS Oi THE IKASSISlUR Hi development is still at an early stage, there are considerable variations between units. Sinc;le Stage Amplifiers. SrAiiii.irv, Electron' Tube Analogy An amplifier can be built in a straightforward manner by using the emitter as input electrode and collector as output electrode, the base being common to the two circuits. This amplifier is therefore called the grounded base amplifier. Figure 7 shows a schematic circuit using the average parameters just mentioned, working between 500 ohms and 20,000 ohms. The ampli- fier has an operating power gain of 17 db, power output Class A 10 milli- watts, noise figure at 1000 cycles 60 db with a variation inversely with fre- quency, and frequency response down 3 db at 5 megacycles. Type A Transistor D.C. Operating Point: /< = 0.6 ma Ic = — 2 ma Circuit Parameters: r, = 240 ohms Yc = 19000 ohms ^u = 530 ohms %i = 34000 ohms F, = 0.7 F Vc = -40F ri, = 290 ohms r,n = 34000 ohms %2 = 290 ohms 'g.., = 19000 ohms Fig. 6 — Equivalent circuit parameter values. Some comments are in order on how this amplifier compares with an electron tube amplifier. First of all, the amplifying function and the manner of analyzing it from the circuit point of view are very similar, even though the internal mechanisms are markedly different. Secondly, there are quali- tative differences in circuit behavior, which are set forth on Fig. 8. The base resistance rb acts as a positive feedback element which, under adverse conditions, can cause the circuit to oscillate. A necessary condition for stability is that the circuit determinant shall be positive, and this can be written as follows: ^ < 1 + ^ + ?^ (1) ■^c Rb Rc Here the quantity r,„ is the net mutual resistance of the transistor, and the capital R's are the total resistances in the corresponding leads, internal and external. One can see several features, as follows: 1. If Rb = 0, the circuit can be stable. 2. If Rb > 0, as usual, the circuit can be stable if the emitter and collector lead resistances are large enough or if rm is not too large. In other words, resistance in the base lead tends toward instability if rn, is large; resistance in emitter or collector leads tends toward stability. 374 BELL SYSTEM TECH SIC A I. JOURJ^AL In the grounded base circuit the property of low base resistance is im- portant, since the backward transmission depends directly on this property. In circuit terms, the base impedance is the feedback impedance in the grounded base circuit, and its value helps to set a limit on the stable gain which can be realized. TUBE ANALOGY -GROUNDED GRID f EQUIVALENT CIRCUIT Equations: ii{Ro + r,+ ;■(,) + 12 rb = vg h{n + r,„) + i2(rb + ro+RL) =0 Circuit determinant A = (i?G + »-£ + rb){RL + ^c + rb) - rb{rb + r„^) > 0 for stability Input impedance i?n = r^ -\- Tb — Output impedance Rn = fe -{- n — bin + rj RL + rc + /-fc fbin + rm) RG + r, + n Operating power gain Go = ■^Rg -(--^•y Typical values: For Rg = oOO", Rt = 20,000' Then Rn = 280'", /eo. = 9600- Go = 17'^'' Fig. 7 — Synopsis of grounded base ami)lilier. The grounded base circuit has [)roperties which are strongly reminiscent of the grounded grid electron triode amplifier in that both have low input impedance, high output impedance, and no change of signal polarity in transmission. The analogy was pointed out by Shockley. That this similarity is no coincidence can be seen by comparing the third equivalent circuit SOME ciRcnr asi'kcts or run transistor ilS above with the triode equivalent circuit of F. B. Llewellyn and L. C. I'eter- son^ in Fig. 9. Both circuits have the same topological form, and have similar impedance levels if the triode is considered to be operating in the frequency range of some tens of megacycles. The most important difference concerns the quantity a, a current ampUfication factor which, for the tran- sistor, mav be considerably greater than unity; while the analogous quantity V Can be stable if: X Re Ri Re R's include resistive elements both internal and external to the transistor. Fig. 8— Stability TRANSISTOR a If a > 1 USUALLY Fig. 9 — Transistor-electron tube analogy. for the triode is close to unity for usual conditions. Another difference, of less importance, is the fact that the tube quantities analogous to re and n, are capacitative reactances; their ratio, however, is like the ratio of r^ to rb in magnitude. One of the first consequences of this transistor-tube analogy is the sugges- tion that different transistor connections analogous to the different electron triode connections may be interesting.^ The analogy makes emitter analogous ^"Vacuum Tube Networks," F. B. Llewellyn and L. C. Peterson, Fror. LR.E.. March 1944, page 159, Fig. 13. ^Loc. cit. 376 BELL SYSTEM TECHNICAL JOURNAL to cathode, base to grid, and collector to plate; the conventional or grounded cathode tube connection is therefore analogous to the grounded emitter connection of a transistor, shown on Fig. 10. It is found that when a = \ the analogy is fairly close, in that the transistor has comparatively high- TUBE ANALOGY- GROUNDED CATHODE EQUIVALENT CIRCUIT Equations: Circuil determinant: Input impedance Output impedance Operating Gain hiRc + rh + /-,) + h}\ = I'G ■ h{r, - Ym) + HRl + /-. + ;-c - r,n) = 0 A = (/?G + rt + }\){Rl + i\ + Tc - /-„,) + r,{r,„ - r.) > 0 for stability Ru = n + r,+ re + r, — Ym + Rl + r^ + re - r„ r,(rm — r,) Gf = 4 "•'<^') '•(*} Ra + Tb-^r, Backward Operating Gain Gr = 4 R Typical values: For Ra = 500'^, R, = 20000"'. Then Rn = 21 00-, R-^-i = -6900-, Gy = ' 24'"', Gh= - 19'"' Kig. 10 — Synoi)sis of grounded emitter amplifier. input impedance, high-output imjK'dance, and changes signal polarity in transmission. When a > 1, as is usual, the analogy becomes less close, and feedback effects tend to become large and obnoxious; the open-circuit output impedance is usually negative. This behavior is readily under- SOME CJRCl IT ASPKCrS Ui THE TR.WSISTOU Mi standable from stability considerations, since the base lead is now one of the signal terminals and, as before mentioned, putting resistance in the base lead tends toward instability if a is enough greater than unity. The etTect is so severe that often it is worth while to add resistance in the collector lead, thereby reducing (/ to the neighborhood of unity, and simultaneously reducing the amplifier to a state of greater tractability. Another feature of the grounded emitter amplifier is that the base re- sistance rb is usually negligible, in contrast to its pronounced effect on the reverse transmission of the grounded base amplifier. The role of feedback element is taken over here by the emitter resistance r,. These considerations have important effects on the properties of cascaded amplifiers and will be reverted to later. For numerical comparison we might work the grounded emitter ampliticr between the same two terminations as the grounded base amplifier above, namely from 500 into 20,000 ohms. It would then have a gain of about 24 db, an improvement of 7 db over the grounded base, with about the same power output and noise figure. This improvement is obtained at greater risk of oscillation; in fact the output impedance of this amplifier is negative. The remaining tube connection — the cathode follower or grounded plate — is analogous to the grounded collector connection (Fig. 11); again, when a = 1 the analogy is fairly close, in that the transistor has high-input im- jiedance, low-output impedance, and no change of polarity in transmission. In fact when a = 1 the device is usable in very much the same manner as the cathode follower. The power output is lower than the other connections because the output electrode (the emitter) does not carry much direct current. However, when we make a greater than 1 the effect is even more pro- nounced than it was in the grounded emitter case. As a increases from 1 , the grounded collector amplifier rapidly loses its resemblance to the cathode follower and begins to transmit in both directions as a bilateral element. When (2 = 2, the operating gains in the two directions are the same; and for a > 2 the transmission is actually greater in the "backward" direction. Another curious feature is that, while the "forward" transmission is still without change in signal polarity, the "reverse" transmission inverts the signal polarity. In any device which is supposed to give gain in both directions, naturally stability must be a controlling consideration. This amplifier is of course still subject to the aforementioned stability condition (1) and it is found that with care one can actually get power gains in both directions of trans- mission without instability, i.e. a simi)le bilateral amplifier is present. One numerical example may suffice. Assume a transistor liaving the properties 378 BELL SYSTEM TECHNICAL JOURNAL Fe = 250 ohms, rb = 250 ohms, To = 20,000 ohms, rn, = 40,000 ohms, so that a = 2 and both base and emitter resistances r,, and rb are negUgible. tube analogy - grounded plate (cathode follower) EQUIVALENT CIRCUIT Equations: Circuit determinant Input impedance Output impedance Operating Gain ii{Ra + rb + re) + h{rc — fm) = iig i\ re + «2(i?L -]rri + rc — r„^ = vl ^ = {Ro+ n+ Tc){Ri. + r, + r, - ;-.„) + ;>(;•„, - ;v) > 0 for stability Rn == rb + re + rdrm — re) /?22 = I't + re — rm + Rl + re + re - r, Teirm — re) Ro + rb + re Backward Operating Gain Gr = ^ R Typical values: = (1 - o)^G^ For Rg = 20000", Rl = 10000" Then Rn = -41000" i^j, = -7600" Gp = IS-** Gh = \i'"' Fig. 11 — Synopsis of grounded collector amplifier Working l)et\veen 20,000-ohm terminations, such an umplilier should have 6 db power gain in both directions and should still be stable even if one of its terminations changes 50% in the unfavorable direction. SOME CIKCL ir AS/'J-X TS ()/■ THE fKANSLSTUK 379 The grounded emitter connection can also exhibit bilateral properties. Recapitulating these three single-stage amplihers, we see that when a = 1 their properties are close enough to the analogous electron tube arrange- ments to be easily remembered; but that, when a is different from 1, their properties begin to diverge from their tube counterparts. Some of these circuits will perform in a simple manner functions which are impossible to the analogous tube connections, although of course the functions could be accomplished by using more tubes or more complicated circuits. Frequency Response So far the analysis of transistors has been given only for the resistive case, appropriate at low frequencies. When the frequency is raised, reactive components appear and the situation becomes more complicated, although of course still subject to the same general methods of analysis. One might expect that smce semiconductmg diodes work at microwave frequencies, so also would semiconducting triodes. For the Type A transistor, this hope is blasted because of the essentially different nature of the mecha- nism, mvolving as it does the physical transport of charge carriers over ap- preciable distances. For certain features of the transistor, however, the analogy does hold. For example, the emitter by itself is a diode; and, in keeping with this fact, its open-circuit impedance does not change much with frequency in the range in which we shall be interested. For most en- gineering purposes the open-circuit input impedance of a Type A transistor may be regarded as a resistance independent of frequency. Such deviations as occur are small and entirely similar to what take place in an analogous diode. The same situation holds with respect to the base resistance rb and the collector resistance re, that is, they act as one might expect of a diode. The base resistance is substantially constant with frequency; the collector re- sistance has associated with it a slight amount of capacitance, mostly due to the case, leads, and wiring external to the unit, which gives a variation of properties with frequency in high-impedance circuits. The analogous capacitance on the emitter side is negligible because of the lower value of emitter impedance. One has, therefore, the T of resistors in the equivalent circuit substantially constant with frequency. The dominant factor governing frequency response of the transistor is therefore largely expressed as a variation of the net mutual impedance r.u or, one may say as well, in the factor a which is the ratio of rm to ro. Measurements of r^ as a function of frequency encounter the practical difficulty that it is impossible to present to the transistor over a wide fre- quency range an impedance high compared to the collector impedance. It is, however, quite easy to present to the collector a relatively low impedance 380 BELL SYSTEM TECHNICAL JOURNAL (75 ohms), which is constant over the frequency range of interest. Con- currently it is relatively simple to present to the emitter a high impedance, 00 — I MIX 25-40 MC HIGH R DETECTOR ^O SCOPE LOW R Fig. 12 — Sweeper for measuring frequency response. uj -2 Z -4 < z O -6 If = -1.2 MA. . s. ■v. \ N ^^^ ?-. ■ 13^l[]~~ ^ ' ' - \ \, N \ N^ ^ "^ ^ \ \, \ s. \ \, \ \\ x\ \, \ \ \ X' K ^\\ \^ \ \ f;. \ \ \\ V \ \ s \ \ \ \ \ \ Xt- \ \ \ " \ > \ \ \ \ N \ \ \ \ 3 4 5 6789 10 FREQUENCY IN CYCLES PER SECOND Fig. 13 — Alpha versus frequenc\-. 20 X10« that is, to drive it witli a constant current generator. Under these conditions the insertion power gain of the transistor is appro.ximatcly a-, where the current amplification factor a is the ratio of increment in collector current to SOME CIRCLIT ASPECTS OE THE TluXSfSTUK 381 increment in emitter current at constant collector voltage.^ The quantities a and a are usually nearly the same. An oscilloscopic presentation of a versus frequency is possible and is a great convenience since many units can be measured quickly and variation with operating point observed directly. The sweep frequency generator built for this purpose is diagrammed in Fig. 12. It presents on an oscillo- scope the magnitude of a as a function of frequency from 0 to 15 megacycles. Means are also available for making point-by-point plots which are more accurate, though much slower. 100 -^ ^^ ■--- --> 90 \ s, , ->^ 80 \ N . "< \ I , V S^ ' 60 N N P-TYPE \ ^'^ 50 40 30 n-typeS, s. \ 1 k V < K N. X N i 1 s \ \ ( S i ^ 10 0 I i 2 3 4 5 6 7 8 9 10 n 12 13 CUT-OFF FREQUENCY, f^, IN MEGACYCLES PER SECOND Fig. 14 — Cut-off frequency statistics. A set of curves of current amplitication factor a versus frequency, as obtained with this apparatus, is shown in Fig. 13. The cutofT shape is a little sharper than that of a single R-C circuit but less so than that of a pair, one of which is shunt-peaked enough to make the combination tlat. The ap- parent high-frequency asymi)tote varies in different units from 7 to 11 db per octave. The phase shift associated with this curve has been found to be related to the amplitude in the same way as if the characteristic were that of a ''mini- » .\ctually, a = (c)Ic/3I<)vc is only one of a set of four circuit parameters h,j whose relationship to I, and Vc is the same as that of the Z's to I^ and Ic, and which furnish an alternative circuit representation of the transistor. The other three h's can Ije measured in a similar manner but are of less interest. 382 BELL SYSTEM TECHNICAL JOURNAL mum phase" passive circuit.'^ Accordingly the phase shift, like the amplitude variation, is also intermediate between a single R-C interstage and the flat compensated pair of interstages. When variations between curve shapes are not too large, the shape can be characterized by a single parameter which w^e take as the cutoff frequency fc. Cutoff is defined as the frequency where the magnitude of a- is halved. Some statistical data on cutofif frequency of different units made of N-type and P-type germanium are plotted in Fig. 14. The P-material is somewhat (0 10 ?, 5 u. 3 If = -1.2 MA. i^ 1 1 / / r 1 1 / / / 1 / \/ .4 / rV y /I 1 / / f 1 1 / / / / 1 / / / \ / / / / A '^/ / / c- ?1 '/ / / > y / / y / / '-"■ -''' „ 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 COLLECTOR VOLTAGE, Vj Fig. 15 — Cut-off frequency versus collector voltage. better, in keeping with the fact that the active charge carriers producing the transistor effect in it are electrons having greater mobility than the holes which are active in N-type germanium. As one changes the operating point of the transistor the frequency re- sponse curve changes in such a way that the shape remains sensibly constant on a logarithmic frequency scale, but the scale changes. The cutofif frequency is usually roughly proportional to the collector voltage, with only minor dependence on the other operating parameter, as shown in Fig. 15 unit AS62. * "Network Analysis and Feedback Amplifier Design," H. W. Bode, D. Van Nostrand Publishing Co., 1945. SOME CIRCr/T ASPECTS OF THE TAM V.S/STOA' 383 16 r- »5 - 14 - 13 - 12 - 11 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 - (a) N-TYPE • • • • J \ \ L £ 1 m a 13 tu a: u. 12 - IL 9 8 7 6 5 4 3 2 t 0 - • • ••• • • (b) P-TYPE • - ^ • - • • • - • • • _ • • •• • • • - •• - - • - ••• r • - • - • - • • 1 1 , 1 1 1 1 1 1 2 3 4 POINT SPACING IN MILS Fig. 16 — Cut-off frequency versus point spacing. 3S4 BELL SYSTEM TECHNICAL JOCRXAL Other types of variations of cutoff frequency with collector voltage are exhibited by some transistors. That frequency cutoff is affected by the spacing between points of the transistor is shown in Fig. 16, which gives some support to the idea that the cutofif frequency might vary inversely as point spacing, other things being equal. However, one has only to look at the graph to see that other things are not equal for, at any given point spacing, the cutofif frequencies of different units vary by almost an order of magnitude. It is, however, clear that point spacing is one of the important factors. In recapitulation of the measurements of frequency behavior, it appears possible to build Type A transistors with frequency cutofifs well above 10 megacycles. At the present time, the factors determining the frequency behavior are not yet under good control. Cascade Amplifiers Many cascading possibilities exist, since any connection of the transistor might be used in combination with other connections, as well as involving all the parameter variations which might be made on each single stage. Some of the more elementary possibilities will be mentioned. Since feedback in each unit greatly complicates the situation, the essential features of the amplifiers may become clearer by discusing an idealized case where feedback is absent or greatly reduced. For similar reasons, the preliminary discussion is confined to frequencies low enough so that the equivalent circuits are purely resistive. Perhaps the most straightforward cascade amplifier is the iterated grounded-base cascade, outUned in Fig, 17. Neglecting feedback, the in- sertion power gain is nearly equal to the current amplification factor a squared. For the Type A transistor this amounts to some 5 db per stage. For most uses this could be regarded as unpractically low, but it might be pointed out that the tube analog (grounded grid cascade) is even worse; for when a = 1 the maximum insertion gain is 0 db per stage. Both am- plifiers of course can be made practical by interstage transformers (Fig. 18). For the Type A transistor, the matched gain without feedback rises to about 15 db per stage, which still compares favorably in magnitude with most grounded-grid tubes. When feedback is considered by allowing rb to return to its usual value of a few hundred ohms, the question of stability becomes important. The nominal Type A transistor is still stable when the cascade interstages are matched, the gain rising to about 21 db per stage. For many units having more than the usual amount of feedback, the interstages cannot be matched without violating the stability condition and therefore encountering os- SUM/-: ClRCl IT A.Sl'KCTS Oi Till'. TR.WSISTDK 5H5i dilations; but one can normally count on stable gains of 15 to 20 db per stage, the transformers being perhaps somewhat mismatched. Interesting possibilities for a good cascade amplifier with more gain than the grounded base cascade are offered by the grounded emitter connection, incidentally, this gain advantage is also enjoyed by the grounded cathode or conventional tube connection, so that one would exi>ect it to apply here from the electron tube analogy; but in transistors the feature that a may be greater than 1 brings in complications having no simple analogy for tubes. Without feed back (<-j^i2 = 0) : Iterative impedance Rg = fJvsi . -^l = fivu Circuit determinant A = (^u + 9^22)^ R21 \^ Insertion Power Gain Gj = Nominal Tj'pe A Gain = S""* Fig. 17 — Synojisis of grounded laase cascade. The iterated grounded emitter cascade without feedback (that is, emitter resistance Vo = 0) is unstable for the nominal Type A transistor, but can be stabilized in many ways of which we shall mention only one. The equivalent circuit of Fig. 19 shows an added resistor which may be thought of as ad- justing the value of the collector resistance, and tends to make the unit more stable. When this resistor is adjusted to make the total collector resistance Re about equal to the net mutual resistance fm, thus reducing the effective value of a to the neighborhood of unity, then the cascade am- {)lifier becomes stable, its gain being sensitive to the e.xact value chosen for the adjusting resistor. A numerical calculation for the grounded emitter 386 BELL SYSTEM TECHNICAL JOURNAL amplifier using the nominal Type A transistor adjusted in this way gives the following results: Assuming an adjusted value of collector resistance of 36000 ohms to be satisfactory for stability, then the iterative input impedance is 2300 ohms, output impedance 4000 ohms, and insertion gain about 21 db per stage with- out transformers. Three-stage stable amplifiers having power gains of about 55 db have been operated. Without feed back i^ln = 0): Iterative impedance Rg = ^22/\ + Re- r„,) = 0 Circuit delerininant A = {Rq + n + r,){Ri^ -\- r , -V Re - r„) > 0 for stability Without feed back (r, = 0) Iterative impedance Rq = Re — >'m , Rl = fb Circuit determinant A = {rb + Re — rnY Insertion Power Gain Gi = ( '- -Y \n + Re - r,nj Nominal Type A Gain with Re = 36000" without feed back 23'"" with ft normal 21'"' Fig. 19 — Synopsis of grounded emitter cascade. base Stage followed by a grounded collector and accordingly has the tube analog grounded-grid, cathode follower, from which one would expect that the terminating impedances would be low and the interstage impedance high. This amplifier matched a 600-ohm line to better than 10% and had 16 db insertion gain, with a bandwidth of about a megacycle. An adaptation for video purposes was made to obtain over a band from 100 cycles to ,S.5 megacycles, an insertion gain of 20 db in a 75-ohm coaxial line. 388 BELL SYSTEM TECILMCAL JUlRSAL . I EQUIVALENT CIRCUIT Kc|uati()iis: Terminalions: ii%i + h{%2 + Zn) = 0 Zn = Zj- = -%, + i(?u + ?-a)(l + \/l ^ Z,, = Zo = -?u + hC^n + %,)i\ + Vl - .v) Circuit cietcrminanl A = ^(?u + ?i2)-(l — v + Vl — 3') i %i 2 ;2 Insertion Power Gain G; = —-■ , : Fig. 20 — Sj'nopsis of iterated cascade of four-poles. EQUIVALENT CIRCUITS V:i ^^^TJl (-) (2) Equations: /i(?ii + Zq) + i'l'^vilf = ''g h%xlf^ iA—^zA = 0 ■Zl Fig. 21 — Four-i)olo willi ideal transformer. SOME ClRCLir ASPECTS 01' THE TR.l.XS/STOR 3S9 The foregoing amplifiers both have rather low output powers because of the fact tliat the emitter, a low-current electrode, is the output electrode. A wav of improving this situation has been suggested in the second amplifier schematic shown in Fig. 22. The first stage is a grounded emitter and the second a grounded collector transistor, the latter operating in what we have called the "backward" direction so that the output electrode is the base and the power level is improved. This amplifier can be stabilized by negative feedback obtainable by inserting a resistor in the first stage emitter lead. These e.xamples emphasize that one can cascade unlike stages and that feedback can be used to stabilize performance, just as with electron tubes. These amplifiers can be further cascaded to obtain more gain. Other pos- sibilities worthy of mention include modifying the design of the first stage (a) (b) Fig. 22 — Non-iterative cascade amplifiers. of an iterative amplifier to obtain good noise figure, or of the last stage for greater power output. Band P.ass Amplifiers Bandpass amplifiers require a few remarks before concluding the small- signal discussion. The design within the band may be carried out by the methods previously discussed; but frequently attention must also be paid to properties outside the band, to an extent unusual with tubes. The reason, of course, is connected with that Dr. Jekyll and Mr. Hyde of transistors, a (or a) greater than 1. Wlien a transistor may be short-circuit unstable, then oscillations may result from the practise usual with electron tube amplifiers of letting the impedances outside the band fall to low values. For the same reason design of power leads requires more care than usual. The problems encountered are somewhat similar to those of tube amplifiers with feedback in that one must pay attention to characteristics far outside the useful band. In the case of transistors, one may have to exercise design care to avoid oscillations even when the gain of the amplifier is less than unity. Large Signal Analysis Large signals are those which involve considerable excursions over the electrical characteristics of the device and cannot be regarded as small 390 BELL SYSTEM TECHNICAL JOURNAL changes near an assumed operating point. For their general study a most convenient tool is provided by the set of static characteristics of the unit. Since most analyses begin with the static characteristics, perhaps some excuse is needed for the unorthodox approach which has delayed them to tliis point. Two reasons may be cited: First, the small-signal behavior is in a sense simpler, being capable of discussion by the familiar linear methods of circuit theory. Second, the small-signal behavior has brought out some features, notably short-circuit instability, which have a bearing on certain features of the static characteristics, on the methods of measuring them, and on the particular manner of expressing them. A set of characteristics representative of Type A transistor performance is shown in Fig. 23, consisting of four plots, one of each of the electrode voltages against each of the currents with the other current as parameter. Contrary to electron tube practise, rather than the voltages we take the currents as the independent variables. This choice avoids the experimental difficulty that the short-circuit unstable transistors might oscillate if we were to attempt to hold the electrode voltages constant, as well as the con- comitant analytical trouble that in that case the voltage-dependent char- acteristics become double-valued. The relationship of these characteristics to the open-circuit impedances is direct and quickly shown. Suppose the voltages are expressed formally as functions of the currents: V6 = fl (l6, Ic) (2) Vc = f2 (le, Ic) Differentiating, and identifying the differentials as small-signal variables, we get immediately the equations for the open-circuit resistances: . dh , . dii (3) . afo , . dh Accordingly, the open-circuit resistances are the slopes of these static characteristics. The reactive components do not appear because our as- sumptions (2) were not sufficiently general to take them into account or, in other words, the reactive information is not contained in the static char- acteristics. Just as there are five other pairs of small signal parameters which could have been chosen, so there are five other ways in which the static character- istics could have been expressed. Often these other ways are convenient for special purposes or are closely connected with particular large signal circuits. 0.8 0.4 .5yi it-by-point r has been le six pairs y two-pole resistance : a thermal esults as a sual region able if the -0.8 -20 -30 -40 -O.b 0 om a tran- by means the class of illustrated f a Type A y that part slope, and mpt to get distortion, where the zorrespond- load region imitter im- nomenon is t is a minor s it can no ter current in electron t overloatis om heating overloading ssion. to overload to choosing sOisf^ M ^ 1 f/^ r INPUT CHARACTERISTIC SLOPE =JfM \ ^ ' / ^A ^ ^ / ^ y / / FEED BACK CHARACTERISTIC SLOPE =J!|2 Ic = o ff // "^^^^^ ^ Ih // J ' y 11 If ^ 1 FORWARD CHARACTERISTIC SLOPE =J!j, -^ ^ 1 ./ 7, ^ r/ —1-4 / / / / o/ / / n 1 / 7 is ' 1 / 1 1 °//" / / ' 1 1 ° 1 7 y / / — 7 / A y OUT PUT / 1 / // CHARACTERISTIC / / / // SLOPE =J?22 / ' / Fig, 23 — Static characteristics. .SOMK CIRCl IT ASI'I'X rs Ul HIE I RASSISTOK .Wt Measurement of the characteristics can be by conventional point-by-point plots or by oscilloscope presentation. An oscilloscopic curve tracer has been built which can show any of the four cliaracteristics for any of the six pairs of independent {parameters of the Type A transistor, as well as any two-pole characteristic which might be of interest (such as a negative resistance characteristic). Occasionally the static characteristics are affected by effects of a thermal nature such that an oscilloscope trace does not give the same results as a slow point-by-j)oint plot. These thermal efTects are small in the usual region of operation of the Type A transistor but may become appreciable if the unit is heated by excessive power dissipation in it. Power Output and Distortion The problem of obtaining good ''undistorted" power output from a tran- sistor at low frequencies is one which is conveniently discussed by means of the static characteristics. Analytically this question belongs to the class of slightly non-linear problems but, for descriptive purposes, it is illustrated by the curves of Fig. 24. The family of collector characteristics of a Type A transistor is shown. The region of linear operation is substantially that part of the plot where the curves are uniformly spaced, have constant slope, and lie within the permitted power dissipation of the unit. In driving a Type A transistor harder and harder in an attempt to get greater power output, one may encounter four types of overload distortion, analogous to the types found in tubes. 1. One may drive the emitter negative into the cutoff region where the collector current fails to respond to changes in emitter potential, correspond- ing to grid cut-off in a tube. 2. One may drive the emitter positive into an emitter overload region where non-linear distortion may be encountered because the emitter im- pedance changes with its voltage. The corresponding tube phenomenon is positive grid distortion. For both tubes and transistors this effect is a minor one which may be actually beneficial in practical cases. 3. The collector may be driven down to low potential where it can no longer draw the current required to follow the impressed emitter current variations. This distortion corresponds to plate "bottoming" in electron tubes. 4. The collector may be driven up to high currents where it overloads because of the non-linear voltage response in that region arising from heating effects. This effect has practical consequences something like the overloading of electron tubes which may arise from insuflicient cathode emission. In other words, either emitter or collector may be driven into overload or cut-off and the problem of getting good power output reduces to choosing 390 changes ne convenient Since mc excuse is n this point, a sense sim of circuit t features, n( features of and on the A set of is shown ir voltages ag Contrary t currents as difficulty t were to att comitant a acteristics 1 The rela is direct ar as function Different we get imn Accordin characteris sumptions in other wc acteristics. Just as t have been « istics coulc for special circuits. .SOMK CIRCriT ASPECTS Of HIE TR.WSLSTOR 391 Measurement of the characteristics can be by conventional ]Joint-by-point plots or by oscilloscope presentation. An oscilloscopic curve tracer has been built whicii can show any of the four characteristics for any of the six pairs of independent parameters of the Type A transistor, as well as any two-pole characteristic wliich might be of interest (such as a negative resistance characteristic). Occasionally the static characteristics are affected by effects of a thermal nature such that an oscilloscope trace does not give the same results as a slow point-by-])oint plot. These thermal effects are small in the usual region of operation of the Type A transistor but may become appreciable if the unit is heated by excessive power dissipation in it. Power Output and Distortion The problem of obtaining good "undistorted" power output from a tran- sistor at low frequencies is one which is conveniently discussed by means of the static characteristics. Analytically this question belongs to the class of slightly non-linear problems but, for descriptive purposes, it is illustrated by the curves of Fig. 24. The family of collector characteristics of a Type A transistor is shown. The region of linear operation is substantially that part of the plot where the curves are uniformly spaced, have constant slope, and lie within the permitted power dissipation of the unit. In driving a Type A transistor harder and harder in an attempt to get greater power output, one may encounter four types of overload distortion, analogous to the types found in tubes. 1. One may drive the emitter negative into the cutoff region where the collector current fails to respond to changes in emitter potential, correspond- ing to grid cut-off in a tube. 2. One may drive the emitter positive into an emitter overload region where non-linear distortion may be encountered because the emitter im- pedance changes with its voltage. The corresponding tube phenomenon is positive grid distortion. For both tubes and transistors this effect is a minor one which may be actually beneficial in practical cases. 3. The collector may be driven down to low potential where it can no longer draw the current required to follow the impressed emitter current variations. This distortion corresponds to plate "bottoming" in electron tubes. 4. The collector may be driven up to high currents where it overlixids because of the non-linear voltage response in that region arising from heating effects. This effect has practical consequences something like the overloading of electron tubes which may arise from insufficient cathode emission. In other words, either emitter or collector may be driven into overload or cut-off and the problem of getting good jM)wer output reduces to choosing i^)l HELL SYSTEM TECILXICAL JOLRSAL an operating point and load impedance such as to avoid these non-linear effects as long as possible. Reverting to Fig. 24, since one wants as large a product of AV- AI as possible, the problem may be thought of in geometrical terms as appro.ximately that of constructing the largest possible rectangle such that a load line extending diagonally across the corners of this rectangle COLLECTOR CURRENT, Ic, IN MILLIAMPERES -25 -20 -15 -10 -5 Fig. 24— Collector power output plot. lies within the "linear" region of operation. The slope of this line gives the load impedance required, its intercept the collector supply voltage (for resistance coupling), and the sides of the rectangle give the extreme values of voltage and current. The center of the rectangle is approximately the quiescent or small-signal operating point. Under optimum conditions of load impedance and operating point, .S'().1/A; (7A'(77/" ASrF.CTS or THE TK.WSISTOR .V).^ one ()l)tains power ct'ticitMuics (oniparahlf to ( 'lass A flcclron tultt- (»|H'ratic)ii, that is, 20 to .^5''^' cnu inu \- with a lew percent harmonic distortion. As contrasted to recommendations lor good low-level j^ain for the Type A transistor, the optimum conditions for power output have usually involved lower load impedances and higher currents. Representative values may be: load impedance, 5000 ohms; collector current, —8 milliami)eres at —35 volts bias; emitter current, 3 milliamperes; power output, 60 milliwatts, with distortion less than ten percent. One complication of the power transistor is that, when the optimum load imi)edance is low, the operating point gets nearer to the region where the transistor may tend to oscillate if it happens to be one of the kind which is short-circuit unstable. A saving circumstance here is available in that Fig. 25 — Some power transistors. added resistance in the emitter lead tends to promote stability, so that the transistor may be stabilized by operating out of a higher generator im- pedance, possibly at some cost in reduced gain. A corollary aspect of the same phenomenon is that the input impedance of a high-power transistor may become very low or even negative. Higher power output from the transistor can also be obtained by in- creasing the permissible collector dissipation. This has been accomplished by using a thin wafer of germanium directly soldered to a copper base equipped with suitable fins to facilitate the removal of heat generated in the vicinity of the collector point. An increase in allowable dissipation from 200 to 600 milliwatts has been thereby obtained. Output powers of ap- j)ro.ximately 200 milliwatts at a conversion ef^ciency of ?% have been realized. The photograph of Fig. 25 shows on the left the type A transistor, in the center the power version of this unit, and on the right is shown a double 3<>4 BELL SYSTEM TECHNICAL JOIRXAL ended type of power transistor using t^\-o gerniaiuum wafers witli a, common radiator for push-pull applications. Other Largic-Sigxal Applications The static characteristics can be used for calculations of many large- signal circuits of which only a few examples can be given here. The first is a tickler feedback oscillator of Fig. 26, which uses the grounded-base circuit with a resonant circuit in the collector lead, transformer-coupled back to the emitter. Other circuits making use of the special possibilities of the transistor include an oscillator with anti-resonant circuit in the base lead, or with a :b) r X -^wy^ (c) Fig. 26 — Transistor oscillators. series resonant circuit from collector to emitter. Some of these circuits make use of the short-circuit instability peculiar to the transistor and accordingly would not work with electron tubes. Noise A discussion of small-signal amplifiers would be incomplete without some mention of the limiting factor of noise. The noise has been left to the last, however, because its discussion complicates the circuits slightly, and perhai^s because it is not well to present too early an aspect of per- formance which is at the moment so much inferior to electron tubes. On the circuit representation of noise as well as signal much work has been done by L. C. Peterson.^ It turns out that in the general four-terminal network in which we are interested, a complete noise representation for ' "Signal and Noise in .Microwave Tetrode," Proc. LR.E., Nov. 1947, pp. 1264-1272. .SY)i//; ciKcri r asi'ECTS of the traxsisioh .VJ.S circuit purposes may be obtained by adding two noise generators to thr equivalent circuit of four signal parameters, as shown in Fig. 27. These noise representations are on an entirely similar basis to the signal representations. Just as four elements in any independent configuration suffice for signal description, so two noise generators in either series or shunt in any convenient independent locations can be added to account for the noise. All these representations give the same signal and noise be- havior for any external connections. Still, some may be better than others in corresponding to the actual physics of the transistor; presumably the EQUIVALENT CIRCUIT Equations: hiZg + ?ii) + iSvi = ''■'a © -Vi ii%A + /2(?- + 2,) = © .v. Circled ® signs indicate addition with attention to any correlations which may exist between noise generators or mean square additions if no correlation exists. Noise Figure F = 1 + 1 _ _ /%, + Zo\ 4 kTBRa Fig. 27 — Synopsis of general four-pole, including noise. better representations will show particularly simple behavior, for example, in their dependence upon the d-c operating point of the transistor. The usual choice puts noise voltage generators in series with the emitter and collector leads, as shown. If the two noise generators were truly independent physical sources of noise, their outputs would be expected to show no correlation and their noise power contributions would be simply additive. This independence is not usually the case for the Type A transistor. By adding the noise outputs and comparing the power in the sum to that in the separate components, correlation coefficients ranging from — .8 to +.4 have been found. From this the conclusion can be drawn that the physical sources of noise in the network do not act in series with the leads but at least to some extent arise elsewhere 396 HEI.I. SYSTEM TECllXICM. JOIRXAL in the transistor and contribute (onclalcd noise oulpul to both tlie genera- tors of the circuit rej^rescntation. The transistor noise is of two types. One is a rushing sound somewhat similar qualitatively to thermal resistance noise; the other is a frying or rough sound which occurs erratically, usually in the noisier units. The noise 70 If = 0.65 MA. Q 2 Ic= 2.0 MA. <^ 60' _J o > V k ^ "^ .2 I 50 Z 1- _l O 40 O a. u 2 30 "^ ^ SLO = E -1.1 ^ ^ .^o -. > ^^>-. ^>X 5 O tr LL ">> SLOPE -1.1 ^^ --- "^^ "^ iJj 20 o z ^ -30 .L .. - 1 1.. _L 1 ._. 1 1 1 _L- 1 1 .. 1 _L. _1_ 1 40 60 80 100 200 400 600 1000 2000 4000 FREQUENCY IN CYCLES PER SECOND Fig. 28 — Transistor noise versus frequency. 10,000 20,000 power per unit bandwidth varies almost exactly inversely with frequency as shown in Fig. 2S, being in this respect reminiscent of contact noise. Since the noise dependence on frequency is known, its level may be given as noise voltage per unit bandwidth at a reference frequency (1000 cycles). The collector noise usually dominates as far as practical effects on the output are concerned. Representative values are about 100 microvolts per cycle at 1000 cvcles for the collector, and one or two microvolts for the emitter. SUMK CIRCL iT A.SJ'KCT.S Uh THE TR.WSISIVR Ml The noise voltages depend mainly on the collector direct voltage as shown in Fig. 29. While they do vary with the other operating parameter at constant collector voltage, such variations rarely exceed 10 db, which is much less than the variations with collector voltage. More important than the actual level of the noise is its relation to thermal resistance noise, which is the ultimate limit to amplification. This relation- ship is conveniently expressed by means of the noise figure, or number of times noisier than amplified thermal noise in the output of the amplifier. Fig. 29 — Transistor noise versus operating point. A representative noise ligure for the Type A transistor at 100() cycles is 60 db, with individual units ranging from 50 to 70 db. Noise figure formulas for the three single-stage connections are given in Fig. 30. The noise performance of the three connections would usually not be ver} different if it were not for stability considerations, which may render unusable the generator impedance which would give optimum per- formance. Mainly, on account of stability, the grounded base connection may be said to give the best noise performance, with the grounded emitter running a close second. The noise figure of any device depends upon the generator impedance out of which it works but does not depend upon the load. Accordingly, there exists an optimum generator impedance which gives the best noise •WS HEI.I. SYSTEM TECHNICAL JOURNAL Equivalent Circuit /b /^ ^r mLf "c i ■Zl Grounded Base F = 1 + 4kTBR Grounded Emitter 1 f— ^. /Z„ + ^. + ztVI /•' = 1 + ■ikTBR Grounded Collector + Z,n + r.,,V _ _ Forward F = 1 -\- Backward F = 1 + AkTBRr 1 Fig. 30 — Noise figure formulas. SOMK CIRCUIT ASPECTS OF THE TRANSISTOR W) figure of which the unit is capable. This optimum source impedance is best for signal-to-noise performance, not for signal performance alone; hence, as is well known for vacuum tubes, it is usually not a mitch for the unit, and in general both the resistive and reactive components of impedance may be mismatched to the unit. For the transistor at low frequencies in the grounded-base connection, reactive effects are negligible and the emitter noise generator may usually be neglected. Under these conditions the optimum noise figure is obtained from a generator of impedance equal to the open-circuit input resistance of the transistor (not the actual working input resistance, which may be quite difTerenl). The best operating point for low noise is usually obtained at a moderate collector voltage (20 volts) and a small emitter current (0.5 ma.). SUMiL^RY A tentative evaluation of the Type A transistor may be made on the basis of presently available information. Before making it, we should say that a comparison with the field of electron tubes is obviously unfair — there are many against one, and a little one at that. Furthermore the little one is a baby not only in size but in length of time under development. It is only natural that the full possibilities are not yet apparent. With these reservations, we can make the following statements about the present Type A transistor: Gain: the transistor figure of about 17 db per stage is somewhat low compared to 30 or 40 db obtainable from audio tubes. When the band- width is taken into consideration the gain-band product of the transistor is good but, since the excess bandwidth cannot be exchanged for gain, this number is in this case illusory for narrow-band amplifiers. For video amplifiers the comparison is more favorable. Stability considerations differ from the electron tube in such a way as to be likely to give more trouble at low frequencies. At video frequencies this difference is less marked if we play fair by comparing with a triode tube instead of a pentode. The latter is of course better shielded than the transistor. Frequency response appears to be practical up to 10 megacycles or more. Power output efficiency of around 30%, Class A, seems fully comparable to an electron tube, so that a comparison between the two can be based on input d-c power. Noise figure of 60 db at 1000 cycles is much worse than that of a good electron tube, which can come close to 0 db. In view of the frequency de- pendence which brings the transistor noise figure down to 30 db at a mega- cycle, the comparison at video frequencies is less unfavorable, particularly if some developmental improvement can be made. 400 BELL SYSTEM TECHNICAL JOURNAL So far on most counts the comparison is not too favorable but, as we said before, it isn't fair to the baby. In addition there are a number of other considerations which are secondary from the point of view of pure technique but may be dominant from other points of view. Among favorable factors here are: small size; low power drain; no standby power, but instant re- sponse when needed; low heating effect when used in large numbers; and ruggedness. The life of transistors should be fairly long on the basis of diode per- formance, but the device is too new to permit definite statement. The mechanical simplicity might well lead one to hope for low cost, but no production figures are as yet available. In fine, even if Type A transistor performance does not excel all electron tubes, it is still good enough for many applications and will be considerably better in the future. Acknowledgement This survey is based on the work of many people, only a few of whom have been mentioned in the text. The examples of circuits have not been numerous or exhaustive, but rather have been used to illustrate the methods adopted; these are general enough to be adapted to the solution of many particular problems. Theory of Transient Phenomena in the Transport of Holes in an Excess Semiconductor By CONYERS HERRING An analysis is given of ihe transienl behavior of ihc density of holes Hh in an excess semiconductor as a function of time / and of position .v with respect to the electrode from which they arc being injected. When the geometry is one-dimen- sional, an exact solution for the function >th(x, I) can be constructed, provided certain simplifying assumptions are fulfilled, of which the most important are that there be no appreciable traijjjing of holes or electrons and that diffusion be negligi- ble. An attemi)t is made to estimate the range of conditions over which the neglect of ditYusion will be justified. A few applications of the theory to possible experiments are discussed. A variety of experiments have been performed, and others are planned, whicli involve measurement of transient or steady-state phenomena due to the drift of positive holes along a specimen of «-type semiconductor after I hey have been introduced at an injection electrode or emitter} These phe- nomena are presumably a result of the interplay of drift, space-charge, re- combination, and diffusion effects. This paper seeks to relate these effects to the phenomena, and its principal contribution is an explicit calculation of the transient phenomena outside the range of small-signal theory, for cases where the geometry is one-dimensional and where certain simplifying as- sumptions, notably the neglect of diffusion, are justified. Removal of some of these simplifying assumptions and a more careful development of the theory will be necessary in certain applications. Section 1 discusses the physical assumptions and boundary conditions involved in setting the problem up. Section 2 contains calculations of the (Hstribution of holes along the length of the semiconductor at various times, for the mathematically simplest case where recombination and diffusion are ignored and all currents are held constant after the start of the injection. This simple case illustrates the method of attack to be used in the more general calculations of Section 4, and it is hoped that this sketching of basic ideas will enable the hasty reader to pass on to Section 6 without going ' llxpL-riments of this sort have been undertaken with the objective of testing and extending the theoretical interpretation of transistor action proposed by J. Bardeen and W. H. Brallain, Phys. Rev., 75, 1208 (1949), especially as regards the role of volume transport of holes, a role first suggested by J. N. Shive, Pliys. Rev., 75, 689 (1949). Ex- ami)les of the tvpe of experiment discussed in the present paper have been described by: |. R. Havnes arid W. Shocklev, Plivs. Rev., 75, 691 (1949) (transient effects); W. Shockley, (;. !.. Pearson, .M. Sparks and W. H. Brattain, in a paper presented at the Cambridge Meeting of the American I'h\sical Society, June 16-18, 1949 (steady-state transport); \V. Shockley, (). L. Pearson, and J, R. Hayncs, Bel! .Sys. Tech. Jour., this issue (steady- state and transient effects). 401 4()2 BELL SYSTEM TECHNICAL JOURNAL through the mathematical details of Sections 3, 4, and 5. Section 3 contains the complete differential equations of the problem, including diffusion and recombination, and Section 4 gives the solution when only the diffusion terms are neglected. Section 5 contains some order-of-magnitude estimates regarding diffusion effects. Section 6 summarizes the capabilities of the theory so far developed, presents some obvious generalizations, and dis- cusses an interesting shock wave phenomenon which occurs whenever the injected hole current is quickly decreased. 1. Basic Assumptions and Boundary Conditions Consider the «-type semiconducting specimen shown in Fig. 1, having electrodes at its two ends, x = —a and x = b, respectively, and an injection electrode system at x = 0 somewhere in between. Let a current of density ja per unit area enter at the left-hand end, and let a current of density je be injected at x = 0. To make the problem strictly one-dimensional, it will be Fig. 1 — Idealized experiment on hole transport in one dimension. supposed that this injection takes place uniformly over the plane cross- section of the specimen at x = 0, instead of taking place at isolated points of the surface, as is usually the case in experiments. This idealization will presumably be justified if the thickness of the specimen is small compared with lengths in the .^-direction which are significant in the experiment and if the injected positive holes are able to spread themselves uniformly over the cross-section before appreciable recombination has taken place. Unless otherwise stated, it will be supposed that je consists entirely of positive holes, i.e., that the number of electrons withdrawn from the speci- men by the electrode at re = 0 is negligible compared with the number of holes injected. The currents ja a,ndje need not be constant in time, although most of the analysis to be given below will assume them constant after the time of initiation of ;"« . One can set up differential equations for the variation with x and time of the electron density, Ue , and the hole density, iii, . These equations will in the general case involve migration due to electrostatic fields, diffusion, re- combination, trapping, and thermal release of electrons and holes from traps. It will be assumed, however, that trapping and thermal release from traps can be neglected, or, more precisely stated, that creation of mobile EXCESS SEMICONDUCTOR HOLE TRANSPORT 4()3 holes and electrons occurs only at the electrodes, and that the disappearance of mobile holes and electrons is caused only by mechanisms which cause holes and eleclrons to disai)i)ear in equal numbers at essentially the same time and i)hice. If this assumi)tion is valid, the charge density due to im- purity centers will never differ from its equilibrium value by an amount comparable with the density due to free electrons. This assumption can be expected to be reasonably good for an n-iype impurity semiconductor in which the number of donor levels is very much greater than the number of acceptor levels and for which, at the operating temperature, practically all the donor levels have been thermally ionized, while thermal excitation of electrons from the normally full band has not yet become appreciable. As has just been mentioned, the differential equations for the behavior of the electron and hole densities involve migration under the influence of the local electric field E{x, t). This field is in turn influenced by the space charge due to any inequality between the hole density iih and the electron excess {rie - »o), where wo is the normal electron density. If the difference {nh - He + no) were comparable with Uh or fie , the problem would be very complicated. Fortunately, however, this difference cannot have an appreci- able value over an appreciable range of x, on the scale of typical experiments. For example, if (»/, - iie + m) were IQ-^ of Wo for a range Ax of 1/x, and if no is 10^^ cm-3, then the difference in field strength on the two sides of Ax- would be about 2000v/cm, a field which would outweigh all other fields in the problem and rapidly neutralize the space charge. Moreover, the time required for the evening out of any such abnormally high space charge would be very short, of the order of magnitude of the resistivity of the specimen expressed in absolute electrostatic units (1 sec. = 9 X 10' ^ 12 cm). Thus it will be quite legitimate to assume {nu - th + wo) = 0 in all equations of the problem except Poisson's equation which determines the field E, and so fie can be eliminated from the conduction-diffusion equations for holes and electrons. These two equations can then be used, as is shown below, to determine the two unknown functions fih and E, Poisson's equation being discarded as unnecessary. The boundary conditions for these differential equations consist of two parts, the conditions at / = 0 and those at and to the left oi x = 0. In most of the applications to be considered, the injection current j^ will be assumed to commence at / = 0. Thus, initially, the specimen will be free of holes and, at / = 0+, will have a field Ea = jjco in the region -a < .t < 0, and a field Eo = jb/ao in the region 0 < .r < b, where ao is the normal conductivity of the specimen and jb = ja + je is the total current density to the right of :c = 0. The boundary condition at .t = 0 is determined by the magnitudes of the electronic and hole contributions to the injection current je . If no electrons are withdrawn by the electrode at x = 0, then the electron cur- rents just to the left and just to the right of x = 0 must be equal, and the 404 BELL. SYSTEM TECIIMC A L JOURNAL hole current densities on the two sides must ditTer by 7, ; if a part oi jc is due to withdrawal of electrons, then the electronic current will have a cor- responding discontinuity. If /„ is positive, i.e., flows from left to right in the specimen, the current can be assumed to be practically entirely electronic over most of the range from -(/ to 0; i.e., as .v becomes negative the hole current must rapidly approach zero and the electron current must rapidly api)roach7„ . In fact, if diffusion is ignored the electron and hole currents must have these limiting values for any negative .v. The preceding discussion and the mathematics to follow have been couched in purely one-dimensional language, i.e., have been formulated as if the electron and hole densities were functions of x alone, independent of y and a, and as if the semiconductor extended to infinity in the y- and s-direc- tions. However, it is easy to see at each stage that practically the same equations can be written for transport of holes along a narrow filament whose thickness is small compared with the linear scale of the phenomena along its length, even when the density of holes is not uniform over the cross-section of the filament. If the density of holes is uniform over the cross-section, all the equations will of course hold as written. However, recent work- has suggested that holes recombine with electrons so rapidly at the surface that the density of holes may be much smaller near the surface than in the center of the cross-section. In such case all the equations of this memorandum must be interpreted as applying to the mean value, nh{x), of the density of holes, «fc(.r, y, s), averaged over the cross-section of the filament; also, the rate of recombination of holes and electrons must be set equal to some function of fih , as yet not reliably known, instead of to a constant times the product of electron and hole densities. This will of course alter most of the quantitative predictions of Section 4, but will not require any change in the method of calculation. 2. FORMUL.\TIOX .\XD SOLUTIOX OF THE PROBLEM WITH XeGLECT OF Diffusion* and Recombix.\tiox For this case the electron and hole currents can each be equated to the product of tield strength E by particle density 11 by mobility m- and the continuity equations are dl dx dl dx -U. Sulil and \V. Shockley, paper Qll prcscnlcd at the Washington Meeting of the .\merican Physical Society, .April 29, 1Q49; see also Shockley, Pearson, Sparks and Hrat- tain, reference 1. EXCESi; SKMJCO.XDL CTOK IIOI.I-: IKA.WSl'URT 41)5 Since the neutrality condition requires -— = — -", subtractingf 1 )anclf2)and at ot integralinj^ f^ives the equation of conservation of total current: Eififii, + M/i"/') — J(l)c = const, indep. of .v where of course 7 = ^6 = (j„ + /,) when 0 < .v < 6 and when conditions are such that all currents flow from left to right. Putting the neutrality con- dition ;/, = Hh + "(1 , into the equation gives the following relation between E and »;. : E[(iJic + Hh)nn + Me«nl = j/c (3) This can ])e used to eliminate cither E or m, from (1). Tf E is eliminated we have d>lh _ _ HelJ-h'hj dNn _ _y( \ ^'^ (a\ dt e[{ne + fJih)nh + He nof dx dx where V(tih) is an abbreviation for the coefficient shown. If, instead, «/, is eliminated from (1) a similar equation results: ^ = |f-F(£)££ (5) dt J at ox where V{E) = cE-iihficno/j = E^JLh(E/Eo) (6) where Eo = j/ « 7',, , so that E is never greatly dilTerenl from E^, , (4) (or (5)) and (6) indicate that ///, (or E) is propagated with the constant velocity £0 ma , as is of course to be expected. More interesting is the case where 7,. and ja are comparable, so that V departs significantly from con- stancy. It is tempting to sui)i)ose that, for this case also, the curve of ni, against x at any time / can l)e constructed by taking the graj)h of ;/;. against .V at / = 0 and moving each point of the curve horizontally to the right a 406 BELL SYSTEM TECHNICAL JOURNAL distance V{nh)l. One can, in fact, easily verify that this construction gives a solution of (4), by writing (4) in the form /dnh\ \cVji 'II,, (^_!!h\ whence it is obvious that the function «./.(.t, /) defined implicitly by x(n,, , t) = x{n, , 0) + V(nH)t satisfies (4) for any form of the arbitrary function x(nh , 0), and that, con- versely, any solution of (4) must be of this form. Assuming, as in the preceding, that all currents flow from left to right, the boundary conditions a,t t = 0^ are : fih = Oiorx < 0 and x- > 0 (8) or, equivalently, E = Ea= ia/o-o for x <0] . E = E,= {ja^-je)/ o) The boundary conditions at a; = 0 are, for ; > 0, wa = 0 or, equivalently, E = E^iox x = Qr (10) and nh = iihi or, equivalently, E = Ei , for x = 0+ (11) where £i and nia are given by the requirement of continuity of electronic current, i.e., EaUolJ-e = -El(«0 + «/a)Me whence, using the relation (3) between Ei and rihi and expressing £„ as ja/noeiJLe fihi = no (12) or, alternatively, £i = £o I 1 - iaMfc _ J r_Gf^±MO ie 1 (13) L Mft (7a + Je)J According to (12), Uhi is small when j<, is small; and, by (13), £i is only slightly below Eo for this case. As^e mcreases, riu increases and £i decreases, EXCESS SEMICONDUCTOR HOLE TKANSFORT W7 and (12) and (13) would make iihi infinite and £i zero when je/ja = Mft/Me . This merely means that the assumptions made in this section, in particular the neglect of diffusion and recombination or the assumption that.no elec- trons are taken out by the injection electrode, must fail to be valid before jc gets as large as tihjjiic ■ It will, in fact, be shown in Section 5 how the presence of enormous concentration gradients makes it essential to consider the effects of diffusion near x = 0 whenje becomes large. Putting the boundary conditions (8), (9), (10), and (11) into the wave- t n C "hi UJ r i i-a El uo' UJ La E, / / 1 t>0 Fig. 2 — Schematic variation of hole density nh and electric field E with distance x from injection electrode and time / after the start of the injected current, in the approximation neglecting diffusion and recombination. propagation construction described above gives the solution shown schemati- cally in Fig. 2. An infinitesimal instant after / = 0, Uh is zero everywhere except in an infinitesimal interval at x = 0, where it rises to a maximum value Hhx given by (12). This is shown schematically in the upper left dia- gram of Fig. 2. The corresponding plot of £, shown in the upper right, dips down to El , which is less than either £„ or Eo , in this infinitesimal interval. After a finite time has elapsed, the curves of iiu and E against .v are simply those obtained by moving each point of the right-hand portions of these / = 0+ curv'es a distance Vl horizontally to the right, as shown in the bottom two sketches. Here V depends on the ordinate in each diagram, taking on its maximum value Fquu when ii,a = 0 or E = Eo . Since T' is proportional to E^, the curve in the lower right diagram is a parabola in the range be- 40S BELL SYSTEM TECHNICAL JOIRXAL tweeii the front and the rear of the transient disturbance; this parabola, if continued, would have its vertex at the origin. After a sufficiently long time a steady state will be reached in which the field for positive x has the uni- form value El and the density of holes the uniform value nu • It is possible to measure ;/;, as a function of / for fixed x by using a closely spaced pair of probes to measure the potential gradient E, and convcxting E to Hh by (3); alternatively, the current to a single negatively biased probe can be used as a measure of ;;/. , if calibrated by the two-probe method. The t rihi Fig. 3— Schematic variation of hole density «/, with time / after the start of the injected current, at some given distance downstream from the injection electrode, in the approxi- mation neglecting dilTusion and recombination. portion of this curve of ih, against / for which 0 < ;/a < uni is given, in the present approximation, by where , s x[(iie -\- fih)nh -^ Heno] e t = x/V{nh) = / • I -N HelXhrnKja -\- Je) = /^[l + (1 + txi,/tie)nh/n^- tf = x/EoHh (14) (15) is the time of arrival of the front of the disturbance. This curve is a parabola, as shown in Fig. 3; if continued, the parabola would have its vertex on the negative Hh axis, as shown. The rear of the disturbance, at which un becomes constant and equal to hhi , arrives at a time Ir given by inserting Uhi from (12) into (14): Ih = If/[\ - (1 + tJi./fJih)jenja + je)]' (16) KXCKS.S SKMICONDl'CTOK llO/J: rKA.XSI'ORT M)'> Xole that the velocity of advance of the rear of the disturbance is less than that with which the holes drift in the steady-state field £i . In other words, wave velocity and i)arlicle \-elocity must be distinguished in phenomena of this sort, although they happen to coincide at the front of the disturbance. The discussion just given has been based on the assumption that /'„ and /, are independent of time, and that they both flow from left to right in Kig. 1. Time changes in the currents are easily taken into account in the n,, con- struction of iMg. 2: according to (4), it is merely necessary to move the various points of the curve of iii,. against x to the right with the variable velocity V{iih , l) instead of the constant velocity V{nk); in addition, ni,i will in general not be a constant, so that the part of the curve for small x will no longer be a horizontal line. As for the restriction that the currents all flow from left to right, only a change of notation is needed to make all formulas apply to the case where all currents flow from right to left; and the case where part oije flows to the right and part to the left can, obviously, occur only under conditions where the assumptions of this section are not fultilled, i.e., can occur only if electrons are removed at .v = 0 or if both diffusion and recombination are important. For, if diffusion is negligible, the existence of a potential maximum at .v = 0 implies a convergence of electrons from both sides onto the plane x = 0, and recombination alone cannot annihilate electrons at a finite rate in an infinitesimal volume. Mention has already been made of the fact that equations such as (12) and (13) give an infinite density of holes when je/ja = Mft/Me , and are non- sensical for larger values of je/ja ■ It is easy to see why any theory which neglects diffusion must break down for values of je/ja of this size and larger if no electrons are removed by the injection electrode. If je/ja is too large, any positive field just to the right of the injection plane .v = 0 will cause more electrons to flow in the negative .v-direction than can be carried off by the current ja which flows in the region of negative x. This difficulty cannot be eliminated by making the field smaller in the region of small l)ositive .V, since making the field smaller requires a higher density of holes to carry the hole current /, ; and this in turn requires a higher density of electrons to preserve electrical neutrality. Thus, though it may be possible to realize experimental conditions under which the approximations of this section are valid for moderate values oi jjja , increase of >//« above the critical value will always result in the building up of an enormously high density of holes and electrons near .v = 0, and one must then consider diffusive transport and possibly other phenomena such as breakdown ot the assumption that no electrons are removed by the injection electrode. It will be shown below that the effect of recombination on the curves of ;//, against x at various times / can be taken into account by using a geometri- cal construction similar to that of Fig. 2 except that, instead of moving the 410 BELL SYSTEM T ECU NIC A L JOURNAL various points of the curve horizontally to the right with increasing time, one must move them along a family of decreasing curves (cf. Figs. 4, 5, and 6). The effect of diffusion can be described roughly as a migration of each point from one of these curves to another. 3. Complete Differential Equations of the Problem As was mentioned in Section 1 , the transport of electrons and holes along a narrow filament can be described by one-dimensional equations even if recombination at the surface of the filament causes the distribution of electrons and holes to be non-uniform over its cross-section. In the equations to follow, Uk and Ug will be understood to refer to averages, over the cross- section, of the hole and electron densities, respectively; the electrostatic field E can always be assumed uniform over the cross-section of the filament, if the latter is thin. The as yet uncertain influence of the surface on the rate of recombination of electrons and holes can be allowed for by writing the recombination rate as «oi?(»;,/«o)/'r particles per unit volume per unit time, where i? is a function which is asymptotically w/./»o as its argument — K), and where r is the recombination time for small hole densities. For pure volume recombination, R — nutie/iiQ = {nh/n^{\ -\- nh/no), while a con- ceivable extreme of surface recombination would he R = Uh/jio . Using this function, the continuity equations for electrons and holes can then be written, with inclusion of recombination and diffusion terms ^ = -5- {EiJ-hm) - —R[-]-^ ^[Dh ^] (17) at ox T \iio/ ox \ dx / — - = - (£MeWj - —R\-] + ^[De-^] (18) ot ox T \iio/ ox \ ox J where the Z)'s are the diffusion constants, related to the mobilities /x by the Einstein relation D/y. = kT/e (19) Using the neutrality condition Ue = «o + f^h , subtracting (17) from (18) and integrating gives the equation of constancy of current, the generali- zation of (3) : E[Qie + fih)nh + Me«o] + — (pe - mO t^ = JiO/e. (20) e ax Solving for E gives y — kT(jjie - ma) -^ ^= r, , ^ , 1 (21) EXCESS SEMICONDUCTOR HOU TRANSPORT 411 which can be substituted into (17) to give a differential equation for nu alone: > r (jihUh 1 _ ^ ^ ("A dill, dJ .1^ e d: («o + 2nh) ~ ox , kT d (22) The first term on the right represents drift, the second recombination, and the third diffusion. This holds whether j is constant in time or not. How- ever, as the remainder of this memorandum will be devoted to the case where the currents involved are held constant after their initiation, it will be convenient to simplify the notation by introducing a current-dependent scale for x and writing the equation in terms of the dimensionless variables V = nu/uo , s = t/r, ^ = x/EqUht In terms of these (22) becomes simply XeHoile/jilh-i ds [r + (1 -f fXh/iJ.e)v . - R{v) + (D'l (1 + 2v) ^ d^ Ll + (1 +M;yiu>J (23) (24) where R{v) = j^(l + v) for pure volume recombination, ox = v for a surface recombination uninfluenced by the electron density, and where / = {kTe (J-ello/nhT) \\I2 (25) (TQ{kT/e HhrY Numerically the characteristic field is, at 300°K, with fih = 1700 cm-/v sec, {kT/eiiHTf = 3.90 (T/lfjLs)~"^ volts/cm (26) Note that the importance of the diffusion term in (24) goes down in- versely as the square of the current density used and inversely as the square of the recombination time; this is because an increase in the distance the holes travel decreases the distance they diffuse by decreasing the concen- tration gradient, and also makes a given diffusion distance less serious by comparison with the total distance traveled. Note also that, if /Xe = Ma , the last term of (24) reduces simply to ( - I — ^ , but that, if ^u,. ^ m, , the diffusion term is not a simple second derivative. ' G. L. Pearson, paper Q9 presented at the Washington Meeting of the American Physical Society, April 29, 1949. 412 BELL SYSTEM TECILMCAL JOIRSAL 4. Solution- In-cludixc. Rkcombi.vatiox hut Neglectixg Diffusion It is plausible to expect by analogy with Fig. 2 that (24) can be solved, neglecting the last term, by a similar construction in which the curve of /?/, against .v at time / is derived from that at time 0 l)y moving each point to the right along a descending curve, instead of along a horizontal line as be- fore. To show that this is indeed the case, and at the same time to show that the diffusion term cannot so easily be taken into account, let (24) be written, omitting its last term, as *: = -*w % - KM where * is just the translation into dimensionless variables of the velocity V encountered in (4). This can ])e converted into a differential equation for ^ l)y writing and multiplving through by ( — J \dv/ (dA\ A = R(^)(~) +*W (27) [dp ds dw a U + [ * = 1/[1 + (1 + Hh/t^e)v\\ R = v{\ ^ v) into (29) and (30). The integrations are easily carried out and give 1 + (4 + Hh/(J'e)vo ^ = + In 1 + (1 + yih/iJ-^n + ^, In [same with v instead of I'ol 1 + (1 + Hh/lJ-e)V0 1 + J^o = In 1 +V0 - In 1 +P (31a) (32a) EXCESS SEMICONDUCTOR llOIJ'. TRANSPORT 415 For the case of a surface recombination unintluenced by electron concen- tration one obtains similarly, with R — v. — [same with v instead of V(\ (31b) 5 = in -" (32b) V When ^Xe = 3^/, 2, as for germanium, (31a) and (31b) become respectively 5/2 v^ 9 1 + 5V3 + 5j'o/3 ^ 1 + Sv,/Z "^ 4 " 1 + j/o ] and — [same with v instead of j'ol (33a) r 1 VQ + 51/0/3 ' 1 + 5t'o/3j — [same with v instead of vo] (33b) These can also be written, using (32a) and (32b), 5/2 5/2 5 r(l + 5.o/3)(l + u)! ^ - ' + 1 + 5.0/3 ~ 1 + 5./3 + 4 ^ L(l + 5»'/3)(l + .o)J ^-^ ^ and /p -\- 3/5\ 1 1 ,^^^, ^ - 5 + In , ' - ) - , , , ,^ + , , r n 34b) \vo + 3/3/ 1 + :)v/3 1 + 5^3 Figures 5a and 5b show as a full curve the plot of eq. (33a) for the case j/o = oc , and the full curve in Fig. 6 shows in the same way the plot of (33b) for j/o = °o . Changing pq of course merely shifts either curve hori- zontally. Note the very sharp increase of v for small |, which shows up in pronounced manner on the expanded scale of Fig. 5b. The corresponding values of s, computed from (32a) or (34a), are marked on the curve of Fig. 5; the corresponding marks on the curve of Fig. 6 also represent values of s at intervals of 0.2, but are not labeled with absolute values because (32b) is infinite for I'd = 00 . For large ^, v becomes very small and it becomes legitimate to expand the logarithms. The first few terms of the resulting asymptotic expression for ^ are, for vo = °^ and the recombination function leading to (31a), ^ ~ (--2 - 1) In (1 + //;, M.) - (1 + Me/MO - In P -i- (3 -f 2^Mn/^Jie)v (35a) 416 BEI.I. SVSTI-:\I TKCIIXICAL JOIRXAL Vl.40 (a) Vl.60 \ 2.00V \ \ \ \ "-.^ "-^^ 2. X^ ^^ 2.40 ^^ .60 7- |^i^3^60 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 C |C II (b) 1 VP.4 \ N yO.6 >^0.8 ^ VALL 1.0 es OF , 5- t T 1.2 1 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Fig. 5 — Steady-state curve of hole density »a against distance .v, for the case of ideal volume recomhination (recombination rate = «/,«e/rWo), and asymptotic approximations to this curve. EXCESS SKM/COXDICTOK HOLE TK.WSl'UKT while, for the recombination function leadinj^ to (.^Ih), t ^ - hid + M/^ )"-■>- 1 - ''1" + 2(1 + ix, nu)v 4i; (35b) In Figs. 5a and 6 the lower dotted curve represents the sum of the terms of (35a) or (35b) respectively as far as the term in In v. in this approximation the dei)endence of v on ^ is exponential. An exponential behavior of this sort is assumed in the small-signal theory of the modulation of the resistance of a filament of semiconductor by hole injection.' The upper dotted curve 2.0 Fig. 6 — Steady-state curve of hole density Uh against distance .v. fur the case of ideal surface recombination (recombination rate = «/,/r), and asymptotic appro.ximations lo this curve. in Figs. 5a and 6 is a plot of (35a) or (35b), respectively, with the linear term included. It will be seen that in both figures the simple exponential apjiroximation is already quite far off when v =^ )ih )in = 0.1, though it improves rapidly for smaller v. Figure 7 shows a sample plot of v against ^ for the case of ideal volume recombination (eqs. (31a) etc.), for the numerical conditions s ^ 1, I'l = 0.3 (cf Fig. 4). According to (12), whose validity at ^ = 0 is unimpaired by the occurrence of recombination, this value of j-i imi)lies /,. /, = 6.^. The left- ' \V. Shocklev. O. L. Pearson, and J. R. Haynes, Bell Sys. Tech. Jour., this issue. 418 BELL SYSTEM TECHNICAL JOURNAL hand portion of this curve is simply traced from Fig. 5, with a horizontal shift suflicient to give an intercept at i' = 0.3; the right-hand portion was constructed by placing the paper for Fig. 7 over that for Fig. 5, shifting horizontally until the point corresponding to one of the values of 5 marked on Fig. 5 lay on the axis of ordinates of Fig. 7, marking the position of the point labeled with one plus this value of s, and repeating. 0.30 \ \ 0.26 \ " \ \ 0.22 0.20 \ \ \ 0.18 -|^O0..6 II \ ^ \ \ \, 0.12 \ \^ N k \ V 0.06 \ 0.04 \ 0 0.4 4 = Eo/^hT" Fig. 7 — Variation of hole density nn with distance x at time t = t assuming nh\ = 0.3 «o recombination rate = nhnt/rno, and neglecting diCfusion. Figure 8 shows sample plots of v against 5 for the same case of ideal volume recombination, with vi = 0.3, for ^ = 0.5 and ^ = 1.0. Curves for a different vx would start out exactly the same, but rise higher. The rising portion of the curve for ^ = 0.5, for example, was constructed from the curve of Fig. 5a by locating various points (^, v) on the latter curve and associating with the v value of each such point a value of 5 equal to the difference of the 5 values marked on the curve of Fig. 5a for the two points abscissae ^ and (^ — 0.5). As Fig. 5a was prepared entirely by slide rule, the accuracy is not all that can be desired; the individual computed points are shown to give an idea of the magnitude of the computational errors. EXCESS SEMICONDUCTOR HOLE TRANSPORT 419 For convenience in future calculations the equations will be appended which correspond to (31) to (34) when, instead of »/, , the field E is used as dependent variable in the differential equations. In terms of the dimension- less variable € = E/Eo = 1 1 + ^1+ Mft/Me) (36) and the parameter eo corresponding to v = ^'o , the equations are, for ideal volume recombmation (eqs. (31a) etc.), t = (1 + Me/M/-)eo - -2 In ( 1 + - eo ) + In (1 o) ~ [same with e instead of eo] while, for the recombmation function leading to eqs. (31b) etc., 1 -60 (37a) (38a) ^ = eo - e + In 5 = In 1 - e fl ~ 1 eo 1 1 _e (37b) (3Sb) The electrostatic potential U is U = — / E dx = — EluhT I € d^. In the steady state the relation between e and ^ is given by (37) with eo (Me + Mk) je set equal to ei which, by (13), is 1 fJ-h (ji + je) For this case U = -£oM/.Tret - I idt = —ElfihT] t(jj.l/fxl — 1) — e"(l + \ij\i})ll (39a) ln(l-e)-^ln(l-l-^-%)l + Ma \ M. /J const. 420 BELL SYSTEM TECIIMCAL JOi KXAL for ideal volume recombination; while, for the assumptions leading to eqs. (31b), etc. the relation is U = -EIh>, t[( - e- 2 - In (1 - e)] + const. (39b) Thus, in the steady state, the difference in potential between any two points to the right of .v = 0 can be obtained by finding the values of e for these two points by (37), and then using these to evaluate the difference in the values of (39) at the two points. To the left of .v = 0, of course, E is constant and equal to / 100 5 80 o z 60 o Z 40 O I (0 Z 20 O 1- o z D 10 " 8 6 4 2 1 - - 1 / - 1 1 1 / (tF DIVERGENCE DIFFUSION FLUX DIVERGENCE DRIFT FLUX 1 / - / - / - / / - / J 1 / / / - / / - y J / - / / - / / / 4f DIFFUSION FLUX DRIFT FLUX , 1 ...L. ^ 1 1 1 0.02 0.04 0.06 0.1 0.4 0.6 1.0 Fig. 9 — Asymptotic magnitude of diffusion terms in the steady-state flux of holes, whenj// is large. from (29) or (31). For the case of ideal volume recombination with m-; m/. — 3/2 this gives, if the diffusion efTects are not too large, [diffusion current drift current div. diffusion current div. drift current '\ (I + .)(1 + 2.)(1 +^vf (42) (43) These functions are plotted in Fig. 9. From this ligure one can estimate roughly when diffusion will begin to have serious effects other than a slight rounding of the leading and trailing ends of the transient. Vov example, if 422 BELL SYSTEM TECHNICAL JOURNAL it is desired that the ratio (43) be less than about 0.1 in the steady state for values of v as high as 0.3, the upper curve of Fig. 9 shows that the current density used must be large enough to make ( - j < ryy. i-C-, j ^ 11-7 /, where / is given by (25) and (26). An approximate evaluation of (40) and (41) in the transient region can be performed by graphical or numerical differentiation of a curve such as that of Fig. 7. For example, a rough calculation based on Fig. 7 gives, in the middle of the transient portion (^ = 0.75), div. diffusion curren div. drift current ■'l-tO' More important and also more difficult to estimate is the effect of difi"usion in rounding off the front and rear edges of the transient. Various ways can be devised to estimate a rough upper limit to the amount of rounding off to be expected. One such is to compute what the diffusive flux just behind the front of the advancing disturbance would be if the distribution of holes were the same as in the absence of diffusion. Under conditions where diffusion is not too serious the time integral of this diffusive flux between any two times can be equated to the increase in rounding of the front, as measured by the area between an ideal curve such as that of Fig. 7 and the actual curve of v against ^ for the same time s. The integration cannot be extended back to time zero, however, since the integral of the flux diverges logarith- mically. The fact that the diffusive flux is actually finite instead of mfinite of course arises from the fact that at small times the concentration gradient a short distance behind the front can no longer be approximated by the gradient which would obtain in the absence of diffusion, but instead is very much less. This suggests that an upper limit to the total diffusive flux passing into the region of the front from time 0 to time 5 can be obtained by taking the flux computed as described above between the times Sa and s, and adding to it the total number of holes which have left the injection electrode be- tween time 0 and time 5o . Since this gives an upper limit for any Sq , one may use the minimum of the resulting sum as So is varied. The results of some sample calculations of this sort are shown in Fig. 10, which refers to the same time, currents, and recombination function as Fig. 7, viz., s = 1/t = l.O, je/ja = 2/13, ideal volume recombination. The full curve is the transient portion of Fig. 7 replotted on a larger scale. The lower dotted curve is a curve drawn in by hand in such a way as to make the area between it and the full curve equal the upper limit computed in the manner just described, for the case; = 1007. The upper curve was drawn EXCESS SEMICONDUCTOR HOLE TKANSrORT AU similarly for 7 = 31.6/. Since tiic true curve of v against ^ must lie between the dotted curve and the full curve in each case, it can be concluded that for times and current ratios of this order the difTusionless theory of Section 4 gives a useful appro.ximation to the transient when 7 >, 100/. At the other end, it seems likely that for / < 10/ the theory of Section 4 has no quantita- tive utility at all in the transient region. When diiTusion effects are sufficiently great, account must also be taken of the fact that the boundary conditions at the injection electrode {x = 0) 0.09 0.07 0.03 0.02 0.01 \ \ \ k \ V \ \ \ -"!- \ \ •^ 1 '31.6 \ K 1 ''100 ■\ "^■-- _ .,__ 0.8 4 = Eoy"h7" Fig. 10 — Approximate magnitudes of the rounding of the front by diffusion for various values of y/y, for the case < = T,jjja = 2/13, ideal volume recombination. Ordinate is proportional to hole density, abscissa to distance from injection electrode. take a different form from those in the absence of diffusion. In the absence of diffusion and with the assumption that only holes are injected at x = 0, the current just to the right of x = 0 must consist of a contribution je from holes and a contribution ja from electrons, while the current just to the left of .V = 0 is purely electronic and of magnitude ja . This implies, as we have seen in Section 2, that the hole density be discontinuous, with the value «/.i given by (12) just to the right of x = 0, and the value zero just to the left. But if diffusion is allowed, the hole density must be continuous. For the 424 HKLI. SYSTKM TKCIIMCAI. JOl RXAL idealized case wliere holes are injected on the plane .v = 0 and no electrons are removed there, the equations to be satisfied are A (^^^ - Dn {^-^ + (».M. + nnn,)E- = jje (46) (44) (45) where subscripts + and — refer to conditions just to the right of x = 0 and just to the left, respectively. Using the neutrality condition tie = «o + "a these are three equations for the five unknowns ( — ) , E± ,)ih .To complete \ dx /i the determination of these quantities the differential equation (22) must be solved and the boundary conditions imposed that ;/;, -^ 0 as .v -^ ± ^. Actually the problem of estimating conditions at .v = 0 may not be quite as formidable as the preceding paragraph suggests, at least if the diffusion parameter J /J is reasonably small and ii jc/Ja is also not too large. For such cases the "upstream diffusion" of holes into the region of negative x will probably reach a steady state in a very short time. Solutions of the steady state differential equation in this region have been obtained numerically by W. van Roosbroeck (unpublished). Such solutions will give one relation be- tween )ih and I — - J ; another relation, in the form of a fairly narrow range of limits, is provided by the fact that ( — -^' I will under these conditions be m. « ( — ) , being in fact probably somewhere between zero and the value for the diffusionless case with the same value of Uhi . Of course, if the mathematical solution for this one-dimensional idealiza- tion is to be applied to a case where holes are injected into a filament by a I)ointed electrode on its boundary, little meaning can be attached to vari- ations in the ;//. of the mathematical solution within a range of .v values smaller than the diameter of the filament. 6. SUM.MAKV AND DiSCUSSIO.V There are three principal factors which limit the range of conditions within which the present theory provides a useful approximation to the transient behavior of )ii. as a function of / and .v. These are diffusion, trap- ])ing, and departure from one-dimensional geometry. If the geometry is sufficiently nearly one-dimensional and trapping is negligible, the discussion EXc/'iss sK.\ti(0.\nrcr()M ik >/./■: TK.wsroRT 4>.> ot" Sfi tioii 5 shows llial tin- theory nl Set tioii 1, with its iic^lot I (tf ditfusifjii, will givr ;i useful a|)|)r()\iniatir)n to the truth whciK'xcr the ticid in which the holes migrate is suniciently strong ^e.g., strong enough to make the current (lensit\- / >, 100 ./, where J is given by (25) and (26). The obtaining of "sufficiently strong" I'lelds without excessive heating or other undesirable effects is facilitated by the use of specimens with as long a recombination time r as possible, and by the use of specimens of low conductivity. How- ever, it is hard to sa_\- how low the conductivity can be made without danger that the "no trapping" assumption will break down, since for this assumption to be valid the density of hole traps must be « the density of donors. The numerical predictions of the theory depend u{)on the way in which the rate of recombination is assumed to depend upon the concentrations of electrons and holes, i.e., upon the form of the function R(p) introduced in (17) and (18). The full curves of Figs. 5 and 6 give the steady-state depend- ence of Hh on .V for two simple assumptions regarding R(v), the dependence corresponding to an}' given boundary value iihi at .v = 0 being simply ob- tained by a suitable horizontal shift of the curve plotted. When the currents are held constant after their initiation, the auxiliary time scale in these figures can be used to construct the transient disturbance at any time, by the methods described in connection with the examples of Figs. 7 and 8. These results should hold for a plane-parallel arrangement of electrodes or, to a good approximation, for electrodes placed along the length of a narrow- filament, provided the hh appearing in the equations is interpreted as a cross- sectional average of the hole density and provided the other assumptions given in Section 1 are fulfilled. It is easy to see, however, that practically the same equations apply to cases of cylindrical or spherical geometry, in the approximation where diffusion is neglected. For, in these cases, the d \ d original equations (17) and (18) merely have Z^,(' ' ') replaced by ~ 7~ (r • • • ) 1 a . . , or ~ — (r- • • •); if the diffusion terms are neglected the solution is the same r^ or as before with .v replaced by r-/2 (cylindrical case) or r^/i (spherical case) and with j replaced by I lird (cylindrical case, d = thickness of sample, / = total current) or by / Aw (spherical case). However, it may be difficult to realize experimentally conditions approximating cylindrical or spherical geometry which satisfy the requirement that diffusion effects be small. .\nother generalization which is easily made is the removal of the assump- tion that no electrons are withdrawn by the electrode at .v = 0. As far as conditions to the right of .v = 0 are concerned (Fig. 1), the only change required in the diffusionless theory is to interpret y^ as the current density leaving the emitter electrode in the form of holes, rather than as the total current from the emitter electrode, and to interpret ja as the sum of the 426 BELL SYSTEM TECHNICAL JOURNAL current leaving the emitter electrode in the form of electrons and any current to the left oix = 0. It should also be clear that the entire analysis of this paper, though it has for definiteness been formulated for the case where holes are injected into an excess semiconductor, applies just as well to any case where electrons can be injected into a defect semiconductor. For the latter case it is merely necessary to interchange the subscripts e and h in the formulas. Though the types of experiments discussed in this paper have to date only been reported for «-type germanium, the occurrence of similar phenomena in /'-type speci- mens is indicated by the successful use of such specimens in transistors.^ An interesting and possibly quite useful phenomenon should occur when, after establishment of a steady state, the current je is suddenly turned off. There will result a transient disturbance propagated in the direction of in- Fig. 11 — Schematic variation of hole density m with distance x, illustrating formation of a sliock wave by quickly decreasing j^ to zero, for the case where j = je + ja is kept constant. (a) Immediately after reduction of 7e to zero. (b) Later time. creasing x, which is very much like a shock wave in a gas. This, the most interesting feature of the phenomenon, will occur regardless of whether ja remains constant when je is cut off; however, the simplest example for il- lustrative purposes is the case where ja is increased by the amount je at the instant when the latter is cut off, so thatj remains constant. For this case, illustrated in Fig. 11, the values of Uh ahead of the advancing front will remain the same at each point as in the previous steady state. Just behind the front, Uh must drop abruptly to zero. If j/J is large, where J is given by (25), the drop will be extremely sharp. For the change in the form of the front with time is compounded out of diffusion and propagation with variable velocity along descending curves, as shown schematically in Fig. 4. Since the latter propagation involves a more rapid motion to the right, the smaller Hh , it tends to steepen the front, and this steepening must continue until ' W. G. Pfann and J. H. Scaff, paper presented at the Cambridge Meeting of the Ameri- can Physical Society, June 16-18, 1949. EXCESS SKM /CONDI '(TON IIOI.E TRAXS/'OKT 427 the diffusive spreading becomes sufficient to counterbalance it. It is not necessary, for tlie production of a steep front of this kind, tliat the decrease of 7e to zero be brought about with corresponding rapidity; even a gradual decrease of jc will lead to a front which becomes steeper as it advances, and if the decrease of y,, is not too gradual a "shock front" will have devel- oped after a short distance. The order of magnitude of the "shock front thickness" can be estimated by finding the value of the time A/ for which the diflfusion distance A.vo = {2D A /) " equals the difference Axv between the drift distances of the holes at the top and bottom of the front, i.e., A.Vk = [V{0) — V(nu)]M, where V is given by (4) and ri/, is the height of the front. For this value of A/, Axo = 2D/[V(0) - V(n,)] (48) and this is presumably of the order of magnitude of the thickness of the front. If D is interpreted as D^ = kTmJe, which is good enough for the present purpose, this gives 2kT 1 L 1 + Kl + W^e)J Of course, this extremely sharp front can be realized only when the condi- tions of one-dimensional geometry are accurately fulfilled. When the geome- try is made sufficiently ideal, observation of the thickness of the "shock front" can provide a valuable check on the validity of the basic assumptions of the theory such as the neglect of trapping.^ The author would like to express his indebtedness to many of his col- leagues, and especially to J. Bardeen, J. R. Haynes, and W. van Roosbroeck, for many illuminating discussions of the topics covered in this paper. ^ The accompanying paper by W. Shockley, G. L. Pearson and J. R. Haynes describes some observations of tliis shock wjve effect, though under conditions where y i)e of carrier takes part in conduction. 428 Fig. 1 — Equivalent circuit for contact rectifier. The parallel components k and C repre- sent the barrier layer itself and R,, represents the resistance of the hody of the semicon- ductor. ///, % 1.. i \ i k o I LL o > 1X1 z \ ) © © © © © i^ 1 eV(x) 1 1 eVm 1 1 1 1 1 1 ^ ^ ^ FILLED BAND 1 LJJ ^- X - © © ©^ i_ — ^^m^m^^^^%^%;<^ © © ©-«- ACCEPTORS 1 FERMI u LEVEL © © ° ©©©©©©© TAL — -SEMICONDUCTOR — -»- io EXHAUSTION REGION ^y "-TOTAL DISTANCE INTO SEMICONDUCTOR, X *■ Fig. 2 — Schematic eneig\' le\el diagram of p-lype semiconductor in conlaci wilii a metal. The diagram is plotted ujjside down from the usual way in onler to show the energ\- of holes increasing upward. The energ_\' of electrons increases tlownward. The lower diagram gives the density of charge in the harrier la\cr. In the body of the semiconductor the s])ace charge of the holes is compensated b)- the space charge of the negati\el\ iharged acce])tor ions. Holes are drained out of the barrier la_\er bv the electric tield, leaving the negative space charge of the acceptors. The rise in electrostatic potential in the barrier region re- sults from this negative sjiace charge together with the comjiensating jjositive charge on the metal. The capacitance of the barrier layer is api)roximalel\' that of a parallel l)latc condenser with plate sejjaration (. 429 430 BEI.I. SYSTEM TECILXfCAL JOlliNAL being occupied by electrons; those below the Fermi level are most likely unoccupied. Holes are depleted from the barrier layer, leaving the negative space charge of the acceptors. This negative space charge, together with the compensating positive charge on the metal, gives the potential energy barrier which impedes the flow of holes from semiconductor to metal. The thickness of the barrier layer may vary from 10^ to 10"^ cm, depending on the materials forming the contact. In drawing the diagram of Fig. 2 it has been assumed for simplicity that the concentration of acceptors is uniform over the region of interest. In the main body of the semiconductor only a few of the acceptors are charged. Throughout a large part of the barrier layer practically all acceptors are negatively charged and there are very few holes in the filled band. This part of the barrier layer has been called by Schottky the exhaustion region and is in our case a region of uniform space charge, as shown in the lower diagram of Fig. 2. The transition zone in which the concentration of holes is decreasing and the concentration of charged acceptors is increasing is called the reserve region. In thermal equilibrium, with no applied volatge, the potential drop across the barrier layer, Vm, may be a fraction of a volt. If a voltage is applied in such a direction as to make the semiconductor positive relative to the metal, the effective height of the barrier is reduced and holes flow more easily from the semiconductor to the metal. This is the direction of easy flow. If a voltage is applied in the opposite dhection the height of the barrier is increased for holes going from semiconductor to metal and remains un- changed, to a first approximation, for holes going from metal to semiconduc- tor (actually electrons going from the filled band of the semiconductor to the metal). This is the reverse or high resistance direction. If a voltage is applied in the reverse direction, and equilibrium is estab- lished, the thickness of the exhaustion layer mcreases. The reserve region keeps the same form but moves outward from the metal. A forward voltage decreases the thickness of the space charge layer. The change in charge density corresponding to a small reverse voltage is shown schematically by the curve marked hQ in the lower diagram of Fig. 2. The maximum of 5() occurs where the total charge density is changing most rapidly with distance. If / is the distance from the metal to this maxi- mum, the effective capacitance C, is approximately that of a parallel plate condenser with plate separation I and with the dielectric constant of the medium equal to that of the semiconductor. The capacitance decreases as / increases with a d-c. bias applied in the reverse direction and the capaci- tance increases with forward bias. Schottky'' has shown that information * Walter Schottky, Zeitsf. Fliys. 118. 53^ [WAD. .i-c'. I \i rh:i)A\( !■: or co.xt.k r ri-atiiiek 4.m about the coiKCiUratiuns of donors and acceptors can be obtained from the \ariation of capacitance with bias. In the equivalent circuit of Fij^. 1 the capacitance C is in parallel with the difTerential resistance, R, of the contact, and the parallel components are in series with the resistance R, of the body of the semiconductor. Spenke showed that R and C are independent of frequency if the frequency is low enough so that the charge density is in equilil)rium during the course of a cycle. If the applied voltage is suddenly changed, it will take time for the charges to adjust to new equilibrium values. The time constant for the readjustment of charge of the carriers (holes in this case) is x-p Air, where p is the resistiv- ity (in e.s.u.) of the body of the semiconductor and k is the dielectric constant, and is ^ 10^^" sec. for a resistivity of 100 ohm cm. Even if a larger value of p is used, corresponding to a point in the reserve layer, the relaxation time for the carriers is very short.^ A m.uch longer time may be required for readjustment of charge on the donor or acceptor ions, giving a varia- tion of R and C at lower frequencies. If the barrier is nonuniform over the contact area, so that much of the current flows through low-resistance patches, the equivalent circuit may consist of a number of circuits like those of Fig. 1 in parallel. In this case, if an attempt is made to represent the contact by a single circuit of this form, it will be found that R and C vary with frequency. The derivation of the current voltage characteristic for the general case of a time dependent applied voltage follows. The total current per unit area is the sum of contributions from conduction, diffusion, and displace- ment currents: /(/) = (tE - eDidn/dx) + iK/4Tr)(dE/dt), (1) where )i{x,() — concentration of holes; a = n{x,t)eij, is the conductivity; e = magnitude of electronic charge; IJL = mobility of holes; D = p.kT/e = difTusion coefficient; V(x,l) = electrical potential; E(x,l) — — dV{x,i)/dx = electric lield strength. The coordinate .v extends into the semiconductor from the junction. F^qua- lion (1) may be written in the form /(/) = neti(-dV/dx) - pikT{dn/dx) - {K/4Tr)(JfV/dxdl) (1') *> Another limit is the transit time of carriers through the barrier layer. This time is generally shorter than the relaxation time of the semiconductor. 432 BELL SYSTEM TECHNICAL JOURNAL The potential V is determined from the charge density, q, by Poisson's equation d'^V/dx^ = -4Trq/K. (2) Since the charge density may be expressed in terms of n{x,l) and the density of fixed charge, these two equations may be used to determine n and V when /(/) is specified. Spenke eliminates the potential V between (1) and (2) and gets a rather complicated equation for n. We prefer to deal with Eq. (1) directly, to treat the potential V{x,l) as a known function, and to solve for the concentration, n{x,l). The plane x = 0 is taken at the interface betw-een metal and semiconduc- tor and the plane .v — xi just beyond the barrier layer in the semiconductor. It is assumed that F = 0 at .r = .vi. Under thermal equilibrium conditions, with no current flowing, the hole concentration in the barrier layer varies as exp (—eV/kT), taking the values: ft = Ho at X = xi (3a) n = iim = iio exp (—eVm/kT) at x = 0, (3b) where no is the equilibrium concentration in the body of the semiconductor and Vm is the height of the potential barrier. We suppose that the boundary conditions (3a) and (3b) also hold when a current is flowing and when there is an additional voltage, Va, across the barrier layer. Our procedure is to solve Eq. (1) for n{x,l), with V{x,l) assumed known, and then to determine /(/) in such a way that the boundary conditions are satisfied. The solution of Eq. (1) which satisfies (3b) is: „(,, ,) . ,, exp[-.(F - K)/m --L I' (/ + ^1 g^ expHV - V)/kT] dx' (4) The prime indicates that the variable is .v' rather than x. At .v = 0, V is the sum of Vm and the applied potential, Va'. V = Va-h Vmat X = 0 (5) The current /(/) is determined in such a way that (3a) is satisfied. Setting X — xi, using (3a), and solving the resulting equation for /(/), we get: nkT[noexp(eVa/kT) - noexp{eV/kT)] - [ '^ ^ exp{eV'/kT) dx' ^/ N _ Jo 47r ax ot I \xp{eV'/kJ)dx' (6) AC. IMPEDANCE OF CON T ACT RECTlinER 433 Provided tliuL the barrier heiglit, F„ + Va, is as much as several times kT/e,^ the integrand in both integrals is largest near x = 0 and drops rapidly with increase in x. Where the integrand is large we inay write to a sufficient ap- proximation: F = F„ + F„. - Fx, (7) where /*' is the field in the semiconductor at the interface. The approxima- tion (7) may be used if kT/eF is small compared with the thickness of the barrier layer. The value of d-V/dxdl is nearly constant over the important part of the integration and may be replaced by its value at x = 0 and taken out of the integral. The upper limit .ti may be replaced by x without ap- I)reciable error, so that we get finally: where and /(/) = Im{Q){l - exp [-eVJkT]) + dQ/dt, (8) Im{Q) = (4we M Q He/K) cxp [-eVJkT] (9) Q = kF/4w (10) is the surface charge density at the metal interface. The current Im{Q) has a simple interpretation; it is just the conduction current in the semiconductor at the interface resulting from the field F. In equilibrium, this conduction current is balanced by a diffusion current of equal magnitude and opposite sign. A voltage Va applied in the reverse direction reduces the diffusion current at the interface as compared with the conduction current by the factor exp [—eVa/kJ]. The current dQ/dl is the displacement current at the interface. Actually, the diffusion theory as given above is not complete. The Schottky effect, the lowering of the barrier by the image force, has been neglected. There may be appreciable tunneling through the barrier. There may be a patch field resulting from nonuniformity of the barrier. If the variations in the patch fields are not too large, the modification of current resulting from these factors depends only on the field at the metal and not on the form of the barrier at some distance from the metal. Thus we may expect the form (8) to be generally valid if /„ (Q) is considered to be a general function of Q. Equation (10) is also of the form to be expected from the diode theory.^ In the latter case, I,„{Q) is the thermionic emission current from metal to semiconductor. If the current is varying in time it is the instantaneous value of Q at ' The value of kT/e at room temperature is .025 volts. 434 BELL SYSTEM TECIIXICM. JOVRXAL time / wliicli is to Ix' used in E(j. (10). Al high fre(}uencies, the charge at the interface need not he in phase with the applied voltage. Tf the frequency is low enough so that the charges maintain their equilibrium values during the course of a cycle, Q will be in phase with V and the parallel capacitance for unit area is simply: C = dQ/dV. (12) The barrier layer may be represented by this caj)acitance in parallel with the d-c. differential resistance, R. Both R and C may depend on the d-c. bias current flowing. Variations of R and C with frequency at moderate frequencies may result from large scale nonuniformities of the barrier such that the patch fields extend over a large fraction of the thickness of the barrier layer or from charge relaxation times associated with acceptors, donors or trapped carriers. At low frequencies, drift of ions may be involved. Attempts which have been made to determine the variation of resistivity in the barrier layer from impedance data are invalid. It is not correct to tike the impedance of an element of thickness dx to be dx/\a{x) + C/'co/v. Vtt)] and integrate over dx to obtain the impedance of the layer. This procedure omits terms arising from diffusion and changes of concentration in time. It is possible to obtain an integral of Eq. (1') if both sides are divided by nen. Integrating over x from .v = 0 to .v = .vi, and using the boundary condi- tions (3a), (3b) and (5), we get K. = f lO + i^Mi^'V/"'^') ,,,. (13) Jo ne^L which means that the integral of the conduction current over the conduc- tivity gives the applied voltage. This is consistent with the representation of the barrier by a resistance and capacitance in parallel. Acknowledgments The author is indebted to W. Shockley, \\'. II. Hrattain, and P. Debye for stimulating discussions and suggestions. The Theory of (y-n Junctions in Semiconductors and p-n Junction Transistors By W. SHOCKLEY In a single crystal of semiconciuclor the inii>urity concentration may vary from />-ty])e to «-tyi)e producing a mechanically continuous rectifying junction. The theory of potential distribution and rectification ior p-n junctions isdeveloi)ed with emphasis on germanium. The currents across the junction arc carried In' the diffusion of holes in ;/-t\pc material and electrons in /)-type material, re- sulting in an admittance for a simple case varying as (1 + uot,,Y'~ where Xp is the lifetime of a hole in the //-region. Contact potentials across p-n junctions, carrj-- ing no current, may develop when hole or electron injection occurs. The principles and theory of a p-n-p transistor are described. Table of Contents 1. Introduction 2. Potential Distribution and Capacity of Transition Region 2.1 Introduction and Definitions 2.2 Potential Distribution in the Transition Region 2.3 The Transition-Region Capacity 2.4 The Abrupt Transition ^. General Conclusions Concerning the Junction Characteristic 4. Treatment of Particular Models 4.1 Introduction and Assumptions 4.2 Solution for Hole Flow into the H-Region 4.3 D-C. Formulae 4.4 Total Admittance 4.5 Admittance Due to Hole Flow in a Retarding Field 4.6 The Effect of a Region of High Rate of Generation 4.7 Patch Effect in p-n Junctions 4.8 Final Comments 5. Internal Contact Potentials 6. p-n-p Transistors Vppendix I A Theorem on Junction Resistance Vppendix II Admittance in a Retarding Field \l)pendix III Admittance for Two Layers \l)pendix IV Time Constant for the Capacity of the Transition Region Vppendix V The Effect of Surface Recombination \ppendix VI The Effect of Trapjiing upon the Diffusion Process \l)pendix VII Solutions of the Space Charge Equation Vppendix VIII List of Symbols 1. Introduction AS TS well known, silicon and germanium may be either «-type or /)-type semiconductors, depending on which of the concentrations Nd of donors or A^^ of acceptors, is the larger. If, in a single samjile, there is a transition from one type to the other, a rectifying photosensitive p-n junction is formed.^ The theory of such junctions is in contrast to those ' I'"or a review of work on silicon and germanium during the war see H. C. Torrey and C. A. VVhitmcr, Crystal Rectifiers, McGraw-Hill Book Company, Inc., New York (1948). P-n junctions were investigated before the war at Hell Telephone Laboratories by R. S. Ohl. Work on p-n junctions in germanium has been published b\' the grouj) at Purdue 435 436 BELL SYSTEM TECHNICAL JOURNAL of ordinary rectifying junctions because, on both sides of the junction, both electron flow and hole flow must be considered. In fact, a major portion of the hole current may persist into the w-type region and vice- versa. In later sections we show how this feature has a number of inter- esting consequences, which we shall describe briefly in this introduction. A p-n junction may act as an emitter in the transistor sense, since it can inject hole current into n-type material. The a-c. impedance of a p-n junc- tion may exhibit a frequency dependence characterized by this diffusion of holes and of electrons. For high frequencies the admittance varies ap- proximately as {iuy^ and has comparable real and imaginary parts. When a p-n junction makes contact to a piece of «-type material containing a high concentration of injected holes, it acts like a semipermeable membrane and tends to come to a potential which corresponds to the hole concentration. Although some results can be derived which are valid for all p-n junctions, the diversity of possible situations is so great and the solution of the equa- tions so involved that it is necessary to illustrate them by using a number of special cases as examples. In general we shall consider cases in which the semiconductor may be classified into three parts, as shown in Fig. 1. The meaning of the transition region will become clearer in later sections; in general it extends far enough to either side of the point at which Nd — Na = 0 so that the value of ] A^d — Na \ at its boundaries is not much smaller than in the low resistance parts of the specimen. As stated above, appreciable hole currents may flow into the //-region beyond the transition region. For this reason, the rectification process is not restricted to the transition region alone. We shall use the word junclion to include all the material near the transition region in which significant contributions to the rectification process occur. It has been found that various techniques may be employed to make nonrectifying metallic contacts to the germanium; when this is properly done, the resistance measured between the metal terminals in a suitably proportioned specimen is due almost entirely to the rectifying junction up to current densities of 10~^ amp/cm^. directed by K. Lark-Horovitz: S. Renzer, Pli\s. Rev. 72, 1267 (1947); M. Becker and H. Y. Fan, Pliys. Rev. 75, 1631 (1949); and H. Y. Fan, Pliys. Rev. 75. 1631 (1949). Similar junctions occur in lead sultide according to L. Sosnowski, J. Slarkiewicz and 0. Simpson, Nalure 159, 818 (1947), L. Sosnowski, Pliys. Rev. 72, 641 (1947), and L. Sosnowski, B. \V. Socle and J. Starkiewirz, Nalure 160. 471 (1947). The theory described here has been discussed in connection wiih photoelectric effects in p-n junctions by F. S. Goucher_. Meeting of the American Physical Society, Cleveland, March 10-12, 1949 and by W. Shocklcy, G. L. Pearson and M. Sparks. Pliys. Rev. 76, 180 (1949). Fcr a general review of ccnductivitv in p- and n-lvpe silicon see G. L. Pearson and J. Bardeen, Plivs. Rev. 75, 865 (1949), and I. H. Scaff.'H. C. Theucrer and F. F. Schumacher, Jl. of Metals, 185, 383 (1949) and W. G. Pfann and J. H. Scaff, .//. of Melals, 185, 3S9 (1949). The latter two papers also discuss photo-voltaic barriers. The most recent and thorough theory for frequency effects in metal semiconductor rectifiers is given elsewhere in this issue (J. Bardeen, Bell Sys. Tech. Jl., July 1949). p-n JUNCTIONS IN SEMICONDUCTORS 437 Even for distributions of impurities as simple as those shown in part (b) there are two distinctly diflferent types of behavior of the electrostatic po- tential in the transition region, each of which may be either rectifying or nonrectifying. The requirement that the junction be rectifying can be stated in terms of the current distribution, certain cases of which are shown in (c). The total current, from left to right, is /, the hole and electron currents being 0 Xb DISTANCE THROUGH SAMPLE, X *- Fig. 1 — The p-n junction. (a) Schematic view of specimen, showing non-rectifying end contacts and convention for polarities of current and voltage. (b) Distribution of donors and acceptors. (c) Three possible current distributions. Ip and /„ , with I = Ip-\- In- Well away from the junction in the p-type material, substantially all of the current is carried by holes and Ip = I; similarly, deep in the «-type material In = I and I p = 0. In general in a nonrectifying junction, the hole current does not penetrate the ;z-type ma- terial appreciably whereas in the rectifying junction it does. Under some conditions the major flow across the junction will consist of holes; such 438 BKLL SYSTEM TECHNICAL JOURNAL cases are advantageous as emitters in transistor applications using «-type material for the base. Where the hole current flows in relatively low resistance n-type material, it is governed by the diffusion equation and the concentration falls off as exp(— x/Lp) where Lp is the diffusion length: L = VD^>. Here D is the diffusion constant for holes and r^ their mean lifetime. The lifetime may be controlled either by surface recombination or volume re- combination. Surface recombination is important if the specimen has a narrow cross-section. Under a-c. conditions, the diffusion current acquires a reactive component corresponding to a capacity. In addition, a capacitative current is required to produce the changing potential distribution in the transition region itself. In the following sections we shall consider the behavior of the junction analytically, treating first the potential distribution in the transition region and the charges required change the voltage across it in a pseudo-equilibrium case. We shall then consider d-c. rectification and a-c. admittance. 2. Potential Distribution and Capacity or Transition Region 2.1 Introduction and Definitions We shall suppose in this treatment that all donors and acceptors are ionized (a good apj^roximation for Ge at room temperature) so that w-e have to deal with four densities as follows: n = density of electrons in conduction band p = density of holes in valence-bond band Nd = density of donors Xa = density of accej^tors The total charge density is p= q(p- n + Nd - No), (2.1) where q is the electronic charge. We shall measure electrostatic potential \p in the crystal, as shown in Fig. 2, from such a point, approximately mid- way in the energy gap, that if the Fermi level tp is equal to i/', the concentra- tions of holes and electrons are equal to the concentration »,• = pi char- 2 H. Suhl and W. Shocklcy Pliys. Rev. 75 1617 (1949). ^ A difference in effective masses for holes and electrons will cause a shift of \p from the midiKiint between the l)ands. p-n JINCTIO.XS l\ SEMlCONDfCTOKS 439 •I- o^ = y p = ni_e kT [a] INTRINSIC (b) p-TYPE q (v^-^) + + + + q(^^-yn) n = n, e ;,j (C) n-TYPE WITH INJECTED HOLES >?: Fig. 2 — Electrostatic potential i/-, Fermi level

«-. - ^ ^t^' 7 ^. ^ - "^ ■«>^ ^ - Fi oT xi \ ^ 1^ "^ -> - -- ^ ■^ ^ ■^ "*5^ ^eo/^f. ., N ""^^ I ^ ^ — ^^^ ■«»^ "^ -le k ', - ^ > \ <. - ■-»> - >-. *. P [- "^ s — \ "^^ - ■s^ o in "A - oso^ "^ - '- J s t •^ ^ -6; ^ K \ \ !^ , 0)1 ^ •«« ^ ^ ^ - ^ A \ > \ v •N. - - ^- ^ r ^ ^ "^ ^>^ I ^^ ^^^ - "-■n^ ^^ \ > k.^^ - iioT" ■^ -^ \ V > ^ - ^- ^ ^ Cj V V - V s, ^ )/ ^ ^ |?s^ ^r^ -^o. - J ■^ - 3/ X- ^^ V ,•> ""^"^ \ \ ~^ ^^ \ \ ^ - -Si rv^ '\ "--: - :^ ■■~~^^ s <: \ *% ^ - ->. s > N ^ > ==^ \ \ "^ V \ - ^^ ^^ ■"I \ ^ - t>iO L,^ ■^ -...^ \ \ "* .^ - ^- C^ t ^ \ "^ - \ ^^ \ k > ^ "^ ^ ^ ^ ^ ^ > s ^ '^~»» ^ > - cii^/ ' 'vT; ^ \ " vt^: ^^S ^« -^^ " ^c ^ - ^ > \ k *». - J. ■ _i 1 1 1 j- -L 1 ^ 1 1 N 'kj -L 1 1 "- 2 ? E do 6 oS° °- ■ 2 9. §o oo o oHR 9- SilOA Nl ' ^i^i '"lVliN3iOd OliViSObiDgig ° ° o O Fig. 3 — Solutions for tlic boundaries of the space-charge region. 444 BELL SYSTEM TECHNICAL JOURNAL becomes a family of straight lines with "a" as a parameter. (Only a = 10 cm~ is shown for Si, all the other lines being for Go.) The half thickness Xm {— iim/a) of the space-charge region is also shown. Solutions are obtained when these lines cross the curves iim = fii exp (q\pm/kT), which are shown for room temperature. The condition that the intersection lie well to the right on the curve is equivalent to K » 1. For two Si samples cut from a melt, a was determined from measurements of conductivity and was about 10^^ to 10^^ cm"*. For these, the space change region has a half-width Xm of more than 10~ cm. For other temperatures, the curves can be ap- propriately translated.^ In Fig. 4(a) we show the limiting potential shapes: ax = Im sinh ^ for K « 1 (2.23) yp = (^Pm/2)(- {x/xmY + 3(x/xm)) for K » 1 (2.24) In Fig. 4(b) the charge densities are shown. For the space-charge case, I Nd — Na I is greater than n or p. For a higher potential rise, i.e. larger i/'„, , the discrepancy would be greater and Nd — Na would be unneutralized except near Xm . 2.3 The Transilion-Region Capacity When the voltage across the junction is changing, a flow of holes and electrons is required to alter the space charge in the transition region. We shall calculate the charge distribution in the transition region with the aid of a pseudo-equilibrium model in which the following processes are imagined to be prevented: (1) hole and electron recombination, (2) electron flow across the /^-region contact at Xa (Fig. 1), (3) hole flow across the «-region boundary at Xb . Under these conditions holes which flow in across Xa must remain in the specimen. If a potential 5^ is applied at the p end, then holes will flow into the specimen until ipp has increased by 5.^ so that the holes insiie are in equilibrium with the contact which applies the potential. Since the speci- men as a whole remains neutral, an equal electron flow will occur at Xb . When the specimen arrives at its pseudo-equilibrium steady-state, the potential distribution will be modified in the transition region and the num- ber of holes in this region will be different from the number present under conditions of true thermal equilibrium. The added number of holes is pro- portional to bif for small values of bip and thus acts like the charge on a con- denser. Our problem in this section is to calculate how this charge depends * Unpublished data of W. H. Brat tain and G. L. Pearson. ' The effect of unionized donors and acceptors can also be included by letting Wi include the properly weighted donor states and pi, the acceptor states. p-n JUNCTIONS IN SEMICONDUCTORS 445 upon 5(^ for various types of transition regions and to express the result as a capacity. The justitkation for this pseudo-equilibrium treatment is as follows: Under actual a-c. conditions the potential drop in the p- and n-regions them- selves are small because of their high conductivity so that most of the po- 5 I --^ ^ --^ K « 1, NEUTRAL CASE - ~~^. ^ / X 3 / * K » 1 , SPACE CHARGE CASE "'(xm = 2LaSinh^ K2= ^^T sunh '^^"^ =10-') 3q0m kT ^ ^k /, / o / / / •3- / 7 / / f / (a) • 'V / «^ ^ ^ -5 _ X.m X-m FOR BOTH CASES-, ._i r. -140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 DISTANCE, X, IN UNITS OF '-a=-^ Fig. 4 — Electrostatic potential and densities for p-n junctions. tential drop occurs across the transition region. On the />-side of the transi- tion region a large supply of holes is available to modify the potential and the fact that a current is flowing across the junction disturbs their concen- tration negligibly; the electrons on the «-side are similarly situated. Hence the distribution of holes and electrons in the transition region will be much the same as for the pseudo-equilibrium case. The question of how the hole 446 BELL SYSTEM TECHNICAL JOURNAL current required to change the potential distribution in the transition region is related to other hole currents is discussed in Section 4.1. Under our assumptions, after the voltage V is applied, a steady state is reached involving no current hence Vipj, = V-region no matter what dis- turbances occur and \}/ — ip,, tends to remain constant in the //-region. The Capacity for Ihc Xeiilnil Case K « / For the neutral case, we calculate the total number of holes, P, between Xc and .V6 as a function of 8ip. The charge of these holes is qP and the effective capacity is q dP d dip. As explained above, we are really interested in the change in number of holes in the transition region. However, the value of P is relatively insensitive to the location of the limits .Xa and .Vb so long as they lie in regions where the conductivity approaches the maximum values in the P-ii jrxcTioxs rx sK.\f[coxDrcrnRs 447 p- aiul //-regions. In the following calculations, we shall consitler a unit area of the junction so that values of P and of capacity are on a unit area bases. The value oi P is obtained by integrating p dx making use of the neu- trality condition to establish the functional relationship between p and .v. The neutrality condition can be written as ax = 2,u sinh ^^^--^'^ = 2//, sinh u (2.32) where ii = ij{\p — <^i ) kT and p = me'^"'"^^"'' = me-'' (2.33) ;/ = »i e " (2.34) so that the value of /-• can be obtained by changing variables from .v to u: P = p (Ix = / p{2ni/a) cosh u dn J X,, '' »,, (2.35) = («l/fl) ["' [1 + e"--"] du = (nl/a)[u, - «« + (r""" - r'"^)/2] J Ua For the cases of practical interest, the value of p at .v = Xa , denoted by pa , and the value of 11 at x = Xb , denoted by iib , will both be large compared to ;/i . Consequently, we conclude that Ha = —ln{pa/)ii) and Ub = In nb/ni are both larger than unity in absolute value but probably less than twenty for a reasonable variation of impurity between Xa and Xb . (For example for a change in potential of 0.2 volts such as would occur between p- and //-type germanium, //„ and iib would each be about 4 in magnitude.) Hence we ob- tain for P, P = (nl/2a)(2iub - Ua) + (A, ;';/i)' - (ni/ubf) ^ pl/2a + Oh/a)(t(b — Ua) (2.36) where we have neglected (ni/nb)' which is «1 and the negligible compared to Kb — Ua ■ The term pa/2a is simply the integrated acceptor-miiuis-donor density in the /^-region, as may be seen as follows: [ (Xa ^ X,) dx = f (-a.v) dx = axl/2 = pl/2a. (2.37) The second term in (2.36) is essentially the sum of the holes of the right of .V = 0 plus the electrons to the left of .v = 0, whose charge is also com- 448 BELL SYSTEM TECHNICAL JOURNAL p3nsated by holes. The total number of holes can be expressed in terms of 8

La so that Xm/Lo > Lj Ld = 1/2 K. In Section 4.4 w^e shall compare these capacities with that due to dilusion of holes and electrons beyond the transition region. 2.4 The Abrupt Transition For completeness we shall consider the case in which the impurity con- centration changes abruptly from pp to Un at .t = 0. For this case the po- tential in the space-charge layer will be of the parabolic type discussed by Schottky, the potentials varying as ^ = {2ir/K)q pp(x - Xpf + constant, x < 0 (2.49) ^ = -(27r/K)(/ Unix - Xnf + cottstant, .V > 0 (2.50) 150 BELL SYSTEM TECH X ILAL JUL' RS A L where Xp < 0 and .v„ > 0 are the ends of the sj)ace-charge layer in the p- and ^-regions. The gradient of potential at x = 0 must be equal for the two layers leading to — ppXj, = HnXn (2.51) so that if the total width of the space charge layers is IT = .v„ — .v^, , it follows that Xp -= —)iJV/(nn + pp) and .v„ = pp\V/(nn + pp). (2.52) The potential difference across the layer, which is xph — ^a is h - ^a = {2Trq/K){ppx\ + n,/n) = [2Trq ppUn/Kipp + ;/,0]n'' (2.53) If pp » )in this reduces to 4'b — ^a = 2Trq UnW /k (2.54) the formula given by Schottky, which should be appreciable in this case, for which all the voltage drop occurs in the //-region. The capacity for the abrupt transition will be C = K/iwW (2.55) where \V is obtained by solving (2.53). For this case (l/Cf should plot as a straight line versus \pb — 4^a '• - = ISiriPp + n,)/KqPp uM, - iA„). (2.56) 3. Oeneral Conclusions Concerning the Junction Characteristic In this section we shall consider direct current flow through the junction and shall derive the results quoted in Fig. 1 relating the current distribution to the characteristics of the junction. We shall suppose that holes and elec- trons are thermally generated in pairs at a rate g and recombine at a rate rup so that the net rate of generation j)er unit volume is (net rate of generation) = ,(,' — nip, » wliich \-anishes at equilibrum. Obviously, g = ni'i . If relatively small con- centrations bp and bn of holes and electrons are present in e.xcess of the equilibrium values, the net rate of generation is 5j, = 5/, ^ g - rill + bii)(p -f bp) = -nibp - rpbii (3.1) Tliis is equivalent to sa\-iiig thai excess holes in an //-tyj)C semiconductor, f>->i jiwcnoxs /.v sE.\fuo\i)rcT()Rs 451 and excess electrons in a /'-lyi'c semicoiiducLor, respectively, have lifetimes 7-p aiul r„ given hy dp = —bpTp = —rn8p or T J, = \ rii ^ p g (3.2) and 5/'/ = —b)r T„ = —rpbii or r„ = 1 r/? = ;/ ,(f. (3.3) We shall have occasion to use this interpretation later. (We later consider the moditkations required when surface recombination occurs, Section 4.2, Appendix V, and the effect of a localized region of high recombination rate, Section 4.6, Ai){)endix III.) In principle, the steady-state solution can be obtained in terms of the three potentials yp, ifp and ^„ . These must satisfy three simultaneous or- dinary diflferential equations, which we shall derive. As discussed in Section 2, we consider all donors and acceptors to be ionized so that Poisson's equa- tion becomes d^ = _1^ (^., + n^e"'"-''"''' - n,e"^-'^''''') (3.4) dx^ K an equation in which the unknowns are the three functions s?„ , s^- Ji'id \p. The total current density, from left to right, is I ^ r,-^ h= -qn p 'Z*^''' + bn "^p dx dx (^.5) The elimination of p and // by equation (2.4) results in an equation in- volving the three unknown functions and /. The divergence of hole current, equal to the net rate of generation of holes, is dip ^ _ r_^ (^_^\ _ ^ # (i I (^'fpl dx "^^^IkTydxJ kTdxdx^dx-'j (3.6) = gig - rnp) = qg{\ - e"'^'"-'"'"'), with p in the second term given by (2.4) so that (3.6) is also an equation for the three unknown functions. The equation for dl,dx can be derived from the last two and adds nothing new. These three equations can be used to solve for d'xp r/.v", d'^pp dx' and dip,, dx in terms of lower derivatives and /. They thus constitute a set of equations sufticient to solve the problem pro- vided that physically meaningful boundary conditions are imposed. We shall not, however, deal directly with these equations; the main reason for deriving them was to show that the problem in question is, in principle, completely formulated. Instead of attempting to solve the equations, we shall discuss certain general features of the solutions for ^pp and (p„ , using 452 BELL SYSTEM TECHNICAL JOURNAL approximate methods, and in this way bring out the essential features of the theory of rectilication. 0 Xt) DISTANCE THROUGH SAMPLE, X- > Fig. 5 — Potential and current distributions for forward current in p-n junctions. (a) j)-n junction under equilibrium conditions. (b) Division of current between holes and electrons. (c1 Distribution of potentials for forward current flow showing how the potential 5^ applied at Xa changes ipp, .6) we obtain I = Ip{Xa) - Ip{Xb) = -dip = gq\ {e'^'^p-^'^^"'' - 1) dx. •I Xa From (3.8) we conclude that if g is increased indefinitely for a specified current /, then cpp — IRo . We shall next give an approximate treatment for the case in which b^pj (J for junction), the value of ipp - IRo is given in Appendix I. 454 BELL SYSTEM TECH MCA L JOCRXAL the total voltage drop. For this purpose we treat ipp — ^n as constant over a range of integration from x = —L„ to x = -\-Lp obtaining = I.ie^"'^^"''' - IJ where /. = gq{L,. + L,) (3.12) is the current density corresponding to the total rate of generation of hole- electron pairs in a volume L,, + Lp thick. We next consider 5^«p + 5^r„ shown in Fig. 5c, where, as the subscript R implies, these are thought of as resistive terms and are given by the integrals 8ip, Rp + Bf„ = / Ip dx/qfip + / /„ dx/qubn. Jxa •'0 Jza •'0 The denominators are both approximately qtxip + bn) which occurs in the integral for Ro . Furthermore, for most of the first range Ip = I and for most of the second /„ = /. Near .v = 0, /;, or In must be at least 1/2. Hence it is evident that dtpHp + Vk« cannot be much less than IRo . We shall repre- sent it by IRi where Ro < 2Ri < 2Ro . In terms of Ry and h , the relationship between current and voltage becomes dip = 8^Rp + 5sr«„ + 5v-y = i?i/ + — In ( 1 + - j . (3.13) This corresponds to an ideal rectifier in series with a resistance Ri . The junction will, therefore, be a good rectifier if the second term represents a much higher resistance. We shall compare the two resistances for the case corresponding to A' « 1. For this case, we have p = —ax and n = +o.v except in the narrow range \x\ < La = ni/a. The integral Rq can be approximated by integrating dx/(T for .V outside of the range ±La using the approximation ±a.v for p and n and approximating the integral from —L„ to +/.„ by 2/.<,,V (in- trinsic). This procedure gives Ri = / dx/q^xax -\ ° + / dx/qubax (3.14) = ■^(l-\'\)\nix,/La) qHUi \ b/ where it is supposed that -Xa = xt and that In {xb/L^ is large compared to 2f(j) _(- 2 -(- 1 'b). The evaluation of /,p and /.„ for use in /., is more involved /' ;/ jr.WJ'lO.XS I.\ .S/:.\lU().\J)l i TORS 455 since T,, aiul r„ arc both functions of .v. W'c shall obtain an approximate self- consistent (lifTusion length by assuming that the holes diffuse, on the average, to just such a depth, A,, , that in uniform material of the type found at /.;, , their ditTusion length would also be L,, . At a depth L,, , the value of // is rp > 6 X 10"' cm. This is much greater than most transition regions. The major drop in ipp must occur to the right of the transition region. This follows from our assumptions: First, we may neglect the IR drop in the ^-region; hence (pp is substantially constant from :v = Xa to x = Xrp • Second, the decrease in ipp is much less in the transition region than in the «-region; this follows from two considerations: the resistance for hole flow is lower in "C. Wagner, Pliys. Zeils. 32, 641-645 (1931). p-n JUNCTIONS IN SEMICONDUCTORS 457 the transition region than in the n-region; the effective length of flow in the //-region, being Lp , is greater than the width of the transition region. Con- sequently, the variation of cpp shown in Fig. 6(c) is seen to be reasonable. Similar considerations apply to -region and ;/-region are hirge compared to the diffusion lengths, we may assume the current at Xa to be substantially Ip only and that at xi , /„ only. The total current entering at Xa can be accounted for as doing three things: (1) neutralizing the electron current flowing into the />-region across Xrp , (2) contributing to the charge in the transition region (this corresponds to the capacity discussed in Section 2) and (3) contributing a current flow to the right across .vr„ . We have selected the hole current for analysis because the hole has a positive charge and the connection between the algebra and the physical picture is simplified. For the same reason, the text emphasizes forward current, although the equations are equally applicable to reverse currents. Nothing essential is left out by this process; since the sample as a whole remains uncharged, the current / is the same for all values of .v and if //, is known, then I n — I — I p is also determined. 4.2 Solution fur Hole Floic into the n-region We shall calculate first the hole current I pixm) flowing across .Vm . It is readily evaluated as follows: The value of p{xT„) is given by pKXTn) = nte (4.1j where p,, is the hole concentration in the /^-region for thermal equilibrium. If we apply a small a-c. signal superimposed on a d-c. bias so that 6(/? = ro + ric""' (4.2) where vi is an a-c. signal, assumed so small that linear theory may be em- ployed (i.e. Ti « kT q), then Pixm) = (^e^^'^^'Od + {qv,/kT)e''''). We resolve this density into a d-c. component po and an a-c. comi)onent p\ iwt e : piXrn) = pn + pV + pi <''"' (4.3) where Po = Pnie"'""'-!) (4.4) />, = {qPnih/kT)e'">"\ (4.5) So long as P(xt„) « it,, , tlie normal concentration of electrons in the /) <; .icxcrioxs i\ si:\ii(0\i)rcroRs 459 // ifj^'ion, the lit\'tiim' r ,, and diffusion roiistaiil J) l<>|- a hole will be sul> stantiall}- unaltered by V- Ai)plicalion of llie hole cuncnl e(|ualion lo the hole density p(.\\ I) i^ix'es I„=-qD^^. (4.6) Combining this with the reconil)ination e(|uation ^ = ^"_Zli _ i ^^ = Pn-^P ^ jj^ll (4.7) dt Tp q dx Tj, a.r^ leads to (he solution P = P,^ + /,0 6*^^"~^"^'^^' + ^^gi./ + (.r«-x)a+/Aj(^""°'"- 1) ; = Ip.{e''""' - I) and the a-c. part is similarly obtained from (4.5) for p\ : Tpiixr.) ={qPnU^/L,)\e""""'""]{\ + /a;r,y %-, e''^" = (Gp + iSph.e'"' ^Apv.e'"' where . I ^, is called the admittance (i)er unit area) for holes diffusing into the //-region; its real and imaginary parts are the conductance and suscept- (4.11) (4.12) 460 bELL SYSTEM TECnNlCAL JOURNAL ance. For cjiTp small, the real term Gp is simply conductance per cm^ of a layer Lp cm thick with hole conduction corresponding to the density pn -\- to ; it is also the differenliil conductance obtained by differentiating (4.11) in respect to tq . For the case of zero bias this establishes the result quoted in Section 1 that the voltage drop is due to hole flow in the /^-region where the hole conductivity is low. In this section we have treated Tp as arising from body recombination. In a sample whose y and z dimensions are comparable to Lp or L„ , surface recombination may play a dominant role. However, as we show in Appendix V, the theory given here may still apply provided appropriate values for Tp and T„ are used. 4.3 D-C. Formulae The total direct hole current flowing in at Xa is /^o plus the current re- quired to recombine with electrons in the />-region. This latter current is, of course, equal to the electron current flowing into the ^-region. This electron current, denoted by /,o or I,.o{xtp), is obtained by the same procedure as that leading to (4.11) for /^o except that bD replaces D and the subscripts of L and t are now n. Combining the two currents leads to the total direct current: /o = / pO + /no = iqL) (j^ + ^"^ ie'-'""' - 1) (4.13) for the direct current per unit area for applied potential tq . The algebraic signs are such that / > 0 corresponds to current from the /^-region to the w-region in the specimen; tq > 0 corresponds to a plus potential applied to the p-end. The ratio of hole current to electron current across the transi- tion re^iion is /pO _ Pn Ln _ pp \/bD7 (4.14) 7„o Lp blip biin -s/Dtp P . /bitn _ . /^ ^n 'V pp V „ antl «„ ^ thermal ecjuihurium value of p and n, as- sumed constant throughout .Vv,,); n,. and />p = similar values for x < Xtp', Lp^ dilTusion length ^^y/ Dt^ for holes in »-rcgion; r^ ^ lifetime of hole in «-region be- fore recombination; b = electron moijilily/hole mobilily; L,. and t„ similar in (|uantities for electrons in /^-region; cr„ = qiihn,, and Op = quPp are the conductivities of the two regions. p-n JUNCTIONS IN SEMICONDUCTORS 461 sistivity. We can also say that the hole current depends only on the ?;-type material and vice versa. For a p-ii junction emitter in a transistor with an w-type base, it is thus advantageous to use high conductivity /'-type ma- terial so as to suppress an unwanted electron current. For comparison with experiment, it is advantageous to express the values of pn and lip in terms of the conductivities (r„ and (Xp . If the conductivity of the intrinsic material is written as ai = qnn,(\ + b), (4.15) then, if pn « «« and iip « pp , we find qnPn = 6(7-/(l + bfan (4.16) qnbup = bcVil -^b)\p. (4.17) Using these equations, we may rewrite (4.11) and a corresponding equation for electron current into the />-region so as to express their dependence ond-c. bias tq and the properties of the regions: /.c(tv) = ,,,^; , ■ - («""'" - 1) (1 + bfcTnLp q -G,o-(.'-'^-^-l) ^^-^^^ ^ Ip.ie""'"'' - 1) (1 + b)'apLn q _r kT .n.onr .X (4.19) ^ lUe'"^'''' - 1). The values of G^o and Co (which are readily seen to be the values of the low- frequency, low-voltage (j'o < kT/q) conductances) and the saturation reverse currents are given by , 2 The expression for direct current then becomes wo = !-;. + Co! (■-)!/••'" -11 ^^^^^ 462 BELL SYSTEM TECHMCA L JOLRXAL 4.4 Tola! Admil lance In order to calculate the alternating current, we must include the capacity of the transition region, discussed in Section 2. Denoting this by Cr , we then find for the total alternating current. la. - {G, + iS„ + Gn + iSn + icoC r) V, - Avx (4.23) where G„ and S,, are similar to G„ and .5"^ hut apply to electron current into the /j-region. The value of the hole and electron admittances can be ex- pressed as A, = G, + iS, = (1 + io:r,f"G,,e'"'^"'' (4.24) A^ = Gn + iSn = (1 + io^ryGn.e'''"^ (4.25) For low frequencies, such that w is much less then l/Xp , we can expand Gp + iSp as follows: Gp + iSp = Gp.e'"'"' + i^irp/DGp^e"'''"^ (4.26) Hence (rp/2)G;:o(^''°'^^ behaves like a capacity. It is instructive to interpret this capacity for the case of zero bias, iq — 0, for which we find: Cp = TpGpo/2 = TpqpniJi/2Lp = q pnLp/lkT. (4.27) The last formula, obtained by noting that t^m = qTpD/kT = gLp^kT, has a simple interpretation: qpnLp is the total charge of holes in a layer Lp thick. For a small change in voltage v, this density should change by a fraction qv/kT so that the change in charge divided by the change in v is (q/kT){qpnLp) which differs from Cp only by a factor of 2, which arises from the nature of the diffusion equation. This capacity can be compared with Ct neut. , discussed in Section 2, (see equation (2.39) and text for (2.42)) for germanium at room temperature as follows: Cp _ q' pn Lp kTa _ pnLpU (4.28) Ct neut. 2kT IQq" n'i lOui For a structure like Fig. 6(c), the excess of donors over acceptors reaches its maximum value, equal to ;/„ , at Xm leading to ;?„ = a.Vm . Consequently a = Hn/XTn ■ Substituting this value for a in (4.28) and noting that pniin = tfi gives ^T iifUt. 2\)XTn (4.29) As discussed at the beginning of this section, /.;, = 6 X 10 cm for holes p-u JUNCTIOXS I.\ SEMI CONDUCTORS 463 in germanium. Hence if the transition region is 6 X 10^ cm thick, the diffusion capacity Cp will dominate the capacitative term m the admittance. Although Ap simulates a conductance and capacitance in parallel at low frequencies, its liigh-frequency behav or is quite different. In Fig. 7 the z 1 III 1 1 1 (fl ;^ .-^ D . CORRESPONDS T( Gpo AND a»rp = . x' ^ ^ < .^ ^ <« ^ x* ' 1^ ^^' G -X-r^" ~ ^^' ^^ - ^ .^ y' < '^ ^- ' i- - (c) ^ ^- ,x s^ z - r'S ^ '^^ .^ —^^ >^ OJ u z - G ^' .-' ^^ ^'^ (dl -y- 5 10^ / / C ■7^- x-^ <^ .y -^ ;^ a " - / -/ ir^ r''' 4- — , 7^ »" u. r — y '^Z z' , /" y O (0 - • y f S .y -•/ X ^' ■/ / z 2 10 • /' >« — / Q. — A 1 y • ;^ 5 y -G- ___ c 3, ' / > - y' ' /^ J / < z < 5 • ■* — I* • / - y -G- -A -«^ ^ Q Z < ' • '- ^10-' ^ 1 _1_-L 1 ,.1 1 L. _i-_i ill — 1 _i_-i 1 J .J-S 1 1- 1 1 1 10-' 102 103 lOS- CcTr Fig. 7 — Real, G, and imaginary. 5, components of admittance for hole flow into n-region. (a) 10^.1 p /Gpo = 10'(1 + /Wp^''- corresponding to uniform w-region. (b) ICF X Formula of Apiicndi.x III, corresponding to layer of high recombination rate in front of n-region. This causes G to exceed 5 at higher frequencies than for (a). (c) 10 X Equation (4.33), corresponding to a retarding field in the w-region, with U = Lp/VTo. (d) Equation (4.33) with Lr = Lp/lO. behavior of (1 + io^Tpf'^ = Ap/Gpo , is shown. For high frequencies Gp and Sp are equal: Gp = Sp = \/tp/2 Gpo V CO — bcTi vo (1 + bY Xm - Xp„ , tt will be much less than t 4.5 Admit lance Due to Hole Flow in a Retarding Field In Appendix II we treat the case in which a potential gradient, due to changing concentration for example, is present in the n- and ^-regions. This tends to prevent holes from diffusing deep in the //-region and for this reason the w-region acts partly like a storage tank for holes under a-c. conditions, thus enhancing Sp compared to Gp in Ap. If the electric field is -d\p/dx = kT/qLr , where Lr is the distance required for an increase of kT/q of potential (i.e. a factor of e increase in ;;„), then the value of Ap is I \ J. /T ^ {2Lr/Lp){\ + io^Tp) .^2-) Ap = [qi.p./Lp] 1 ^ [1 _^ (1 +i^rp){2LJLpn" ^ ^ For wTp > 1, this admittance is largely reactive provided lErLp is sutTi- ciently small. The dependence of Ap upon w is shown in Fig. 7 for two values of Lr/Lp ■ The i)lot shows the real and imaginary parts of Ap/[2qt.pnULl] = 1 + [1 + (1 + ,w^)(2VL;?F^ ^^■^''^^ for Lp/Lr ^ 10^'" and Lp/Lr = 10, the two curves being relatively displaced vertically by one decade. The second value imi)lies that the field keeps the holes back so that they penetrate only yo their possible diffusion length in \w field. It is seen that for this case the storage effect is very jjronounced and the susceptance ^" is much larger than G for high frequencies. The function (1 + io^TpY'', discussed earlier, corresponds to the limiting case of (4.32) for Lr = ■^- . '3 In Appendix IV ;in ;uial\-tic treatment of Cr is given. p-n JU.\CTIO.\.S l.\ .SKMlLUM)rcmRS 465 //) 'flic Effect oj a Rcgitvi of Iliiih Rule of Gntcralioii Thore is cvideiux tl\at iniporlCclions, such as surfaces and cracks, add materially to ihe rale of ^a'neration and recombination of holes and elec- trons. If there is a localized region of high recombination rate in the transi- tion region, there will be a pronounced modification of the admittance char- acteristics. In Fig. 8(a) such a layer is represented at .v = 0. In Fig. 8(b) the customary plot of ^p and v?,, versus .v is shown. If we neglect the efTect of the series resistance terms denoted by Rx in Section 3, the change h

per unit area. We sui)pose, further- more, that the layer is so thin that // and p are uniform throughout the layer. The net rate of generation is thus .?'< r.np = ^Jl - e"^'"-^'"'"'''] (4.34) 466 BELL SYSTEM TECHNICAL JOURNAL since for equilibrium conditions the rates balance so that raiti = ga . The net hole current recombining in the layer per unit area is thus lX - Xm , it flows into the «-region in amount Ipoi^^)- This leads to / = InoM + hoM + IrM. (4.36) In other words the layer of high recombination acts like a rectifier in parallel with /„o(M + IpoM. The frequency characteristic of IrM, however, will be independent of frequency and will contribute a pure con- ductance to the admittance of the junction. If the layer is considered to have finite width, however, it will exhibit frequency effects just as does Ip in the w-region. In Appendk III, we treat a case in which the layer is a part of the j7-region itself but has a recombina- tion time different from the main layer. If the time is shorter, a large amount of the hole current may recombine in this layer. For high frequencies, the current may not penetrate the layer, in which case the admittance for hole current is determined by the thin layer rather than by the whole «-type region. A case of this sort is shown in Fig. 7. In this case the thickness of the layer is I of its diffusion length and in it the lifetime of a hole n is i the value Tp in the main body of the w-region. The hole current will thus be restricted to this layer when the diffusion distance -y/D/u is less than the layer thickness (i) y/Drt; this corresponds to cot/ > 9 or oiTp > 81. The presence of the high rate of combination in the layer is evidenced by the tendency of G to be greater than S at high frequencies. If the layer were infinitely thin, as discussed above, it would simply add a constant conduct- ance to the admittance. 4.7 Patch Effect in p-n Junctions If there are localized regions of high recombination rate, a "patch effect" may be produced in an n-p junction. As an extreme example, suppose the value of ga for the layer just considered is allowed to become very large; then the recombination resistance may become small compared to R\ in Section 3 and the junction will become substantially ohmic. If the region of high rate of recombination is relatively small compared to the area of the rest of the junction, then the behavior of the junction as a whole may be re- garded as being that due to two junctions in parallel. Over most of the area, p-n JUNCTIONS IN SEMICONDUCTORS 467 the currents will flow as if the patch were not present so that one compo- nent of the current will be that due to the uniform junction. In addition there will be current due to recombination and generation in the patch. The series resistance to the patch will be relatively high due to the constric- tion of the current paths. On the other hand, the value of /r(50) associated with the patch may be very high. Hence the current due to the patch will be that of a low impedance ideal rectifier in series with a high resistance; and if the ratio of impedances is high enough, such a series combination amounts essentially to an ohmic leakage path. Thus patches in the p-n junction will tend to introduce leakage paths and destroy saturation in the reverse direction. An extreme example of a region of high rate of recombination would be a particle of metal making a non-rectifying contact to both p- and «-type germanium. Since holes and electrons are essentially instantly combined in a metal, the boundary condition at the metal surface would be equality of (fp and (fn . This would mean that near the metal particle, ipp and (r„ could not differ by 8(p, the condition required, over some parts of the junction at least, in order for ideal rectification to occur. A common source of imperfection in p-n junctions arises from dirt or fragments on the surface which overlap the junction. Even if these do not actually constitute a short circuit across the junction, they may furnish patches of the sort discussed here and modify the junction characteristic. 4.8 Final Comments Another possible cause for frequency effects may be found in the trapping of holes or electrons. ^^ When an added hole concentration is introduced into an n-region, a certain fraction of the holes will be captured by acceptors and later re-emitted or else recombined with electrons while trapped. Investiga- tion of this process is given in Appendix \T. One interesting result is that the trapping of holes in a uniform ;z-region cannot produce an effective suscep- tance (i.e. iwC) in excess of the conductance, as can a retarding field. Finally it should be remarked that important and significant variations of the conductivity in the p- and w-regions may be produced by hole or electron injection. Under these conditions, when the hole concentration approaches w„ , i/' — ipn will vary. Under these conditions Ri may be appreci- ably altered. These factors favor the p-n junction as a rectifier since they lead to a reduction of series resistance under conditions of forward bias and thus tend to improve the rectification ratio. '■' Frequency dependent effects in Cu^O rectifiers have been explained in this way by J. Bardeen and W. H. Brattain, personal communication. 468 BFJ.I. SYSTEM TKCIIXICAL JOCRXAL 5. InTKKXAL CoMAl I I'lMI.MlAI.S The theory of p-ii junctions presented above has interesting consequences when applied to the distribution of potential between two semiconductors t i 53- 1 1 1 p-TYPE 1 ^ 1 n-TYPE 1 ^^S^ -N ^-^.. 1 t (b) (c) HORIZONTAL DISTANCE THROUGH SAMPLE, X- J2 VERTICAL DISTANCE THROUGH SAMPLE , L) • Fip: 9 — Internal contact jjotcntials showing how presence of iniecled contact potential across /j. holes jiroduces a under conditions of hole or electron injection. In Fig. 9 we illustrate an X-shaped structure. A forward current llows across the junction Pi and out of branch .Vi . If the distance across the intersection is com{)arable with or small compared to the diffusion length for holes, a potential difference should l)e measured between A and A'l; . The reason for this is that holes /> « JL \CTlO\S l.\ SKMICO.XDl CTORS 469 tlow easily into I'^ since the i)otential distribulioii there favors their en- trance. Since, however, A is open-circuited this hole How biases J2 in the forward (Hrection; since J-i is high resistance, an a|)preciable bias is developed before the counter current equals the inward hole llow and a steady state is reached. Xo similar effect occurs in the branch X^ ; consequently P2 will be found to be floating (open-circuited) at a more positive jxjtential than Xo . Parts (b) to (e) describe this reasoning in more comj^lete terms. We suppose that the /^-regions are more highly conducting than the ;'-regions so that the current across Ji , shown in (b), is mainly holes. The [)otentials s?p and ip„ along the .v-axis will be similar to those of Figs. 5 and 6; (c) shows this situation and indicates that the difTusion length for electrons in the /^-region is less than for holes in the ^-region. Along the y axis (f,, and (fn vary as shown in (e), the reasoning being as follows: At the origin of coordi- nates ipp and (fn have the same values as for (c). The transverse hole current (d) has a small positive component at y = 0 since, as mentioned above, P» tends to absorb holes and thus increase difTusion along the plus y-axis. Since the net transverse current is zero, /„ = —Ip in (d). The

0 in (e). For y < 0, the reasoning is the same as that used in Sections 3 and 4 and we conclude that (f,, is essentially constant. Hence, a difference in the Fermi levels at 7^2 and X2 will result. In Fig. 10 we show a structure for which we can make quantitative calcu- lations of the variations of tpp and tp,, ■ We assume for this case that the forward current from Pi to .V does not {produce an apj^reciable voltage drop, i.e. change in \j/ and s?„ , in region T. This will be a good approximation if the dimensions are suitably proportioned. We shall next solve for the steady- state distribution of p subject to the indicated boundary conditions assuming that p is a function of .v only. As we have discussed in Section 4.1, when p is small compared to ;/ in the //-region, we can write In keeping with the treatment in the next section of this structure as a transistor, the terminals are designated emitter, collector and base, the po- tentials with respect to the base being ^p^ and s^, . The contact to .V or the base is such that (5.2) p.=^Pnc'"'"' x=-Viv (5.3) 470 HELL SYSTEM TECHNICAL JOURNAL The function p{x) which satisfies these boundary conditions and the equation = 0 (5.4) is ^(^) =P-+ ^o"^w n^^ ^«^^ (^/^-) + , \ f ^K , sinh (x/L,) 2 cosh (w/Lp) 2 sinh {w/Lp) (5.5) ^EMITTER, e COLLECTOR, C 5° = ^f + Vtj i" 5 -J A 1 ^J^p__ 1 V 1 \ V3 = V3 +.93 (b) -w 0 vv DISTANCE THROUGH SAMPLE X- ^b Fig. 10 — Model used for calculation of internal contact potential and to illustrate p-n-p transistor. (a) Semiconductor with two p-n junctions and ohmic metal contacts. (b) Quasi Fermi levels showing internal contact potential between h and c. which gives rise to a hole current across Ji into Pi of amount dp I Ip = -qD doc x="ij 2Z-, (11/ I ID IS) I = 2X ^^' ~ ^^^ ^°^^ -J- ^ {2pn - pi - po) tanh — [(/>! - Pn) (coth ~ - tanh 1^^ ^p)J - (A' - A,.) { coth ~ + tanh ~ )] OSP /kT - 1] Lsinh {2w/Lp) tanh {2w/L = csch (2w/Ap)/po(v?J - coth (2wLp)Ipo((Pc) > (5.6) p-n JUNCTIONS IN SEMICONDUCTORS 471 where, by I ^o {>p), we mean the hole current which would flow in the forward direction across either /i or J2 if uninfluenced by the other (i.e. the function of (4.11) or (4.18) and (4.20).) The equation shows that a fraction csch {2ic/L,^ of the current Ipo (<^«), which would be injected by ^, on Pi in the absence of Jy , flows into Pn . The conductance of P2 across Jo is increased by the factor coth(2te'/Z,p). The current into Po carried by electrons wiU be unaffected by Ji and can be denoted by -Ino{ kT/q, the exponential approximation may be used for Ipo in both terms: ^c = ^.- (kT/q) In cosh {2w/Lp), (5.9) so that, if (2w/Lp) is the order of unity, /Lp) large, we get ^. = i-type contacts to /'-type material with in- jected electrons) are reminiscent of probes in gas discharges which tend to become charged negative in respect to the space around them because they catch electrons more easily than positive ions. The situation may also be compared with that producing thermal e.m.f.'s; in fact a "concentration temperature" of the semiconductor with injected holes can be defined by finding the temperature for which np = n\{T). We conclude that, in the 472 BELL SYSTEM TECIIMCAL JOLRSAL absence of thermal equilibrium, different i)otentials depending on the niiture of the contact are, in general, the rule rather than the exception. 'Ilie bias developed on /\> or c will change its conductance. If we suppose that (^c and ipi, are held constant, then the current flowing into c is obtained by the same reasoning that led to (5.7) and is /e((^,., ip,) = InfM + coth ^ Ip(Uc) - csch j^ Ip„M. (5.13) For an infniitesimal change in s^, from the value which makes Ic{(Pc , ft) vanish, the admittance to c is readily found from (4.18) and (4.19) to be l~) = l'ni){pc is in the reverse direction, and the 474 BELL SYSTEM TECHNICAL JOURNAL input impedance will be low, the power gain formula a R22/Rn will yield power gain even when a is less than unity. In certain ways the structure of Fig. 10 resembles a vacuum tube. In Fig. 11, we show the energy band diagram, with energies of holes plotted upwards so as to be in accord with the convention for voltages, (a) shows the thermal equilibrium distribution and (b) the distribution under operating condi- tions. It is seen that the potential hill, which holes must climb in reaching the collector, has been reduced by ip^ . The w-region represents in a sense the grid region in a vacuum tube, in which the potential and hence plate current, is varied by the charge on the grid wires. Here the potential in the w-region is varied by the voltage applied between base and emitter. In both cases one current is controlled by another. In the vacuum tube the current which charges the grid wires controls the space current. Because the grid is negative to the cathode, the electrons involved in the space current are kept away from the grid while at the same time the electrons m the grid are kept out of the space by the work function of the grid (provided that the grid does not become overheated.) In Fig. 11, the electrons flowing into the base control the hole current from emitter to collector. In this case the controlled and controlling currents flow in the same space but in different directions because of the opposite signs of their charges. As this discussion suggests, it may be advantageous to operate the p-n-p transistor like a grounded cathode vacuum tube, with the emitter grounded and the input applied to the base. The p-n-p transistor has the interesting feature of being calculable to a high degree. One can consider such questions as the relative ratios of width to length of the «-region and the effect of altering impurity contents and scaling the structure to operate in different frequency ranges. However, we shall not pursue these questions of possible applications further here. Acknowledgment The writer is indebted to a number of his colleagues for stimulating dis- cussions and encouragement, in particular to H. R. Moore, G. L. Pearson and M. Sparks, whose experhnental work, to be described in later publications, suggested development of the theory along the lines presented above. He is also indebted to J. Bardeen, P. Debye, G. Wannier and W. van Roosbroeck for theoretical comments and suggestions and especially to the last and to Mrs. G. V. Smith for valuable assistance in preparing the manuscript. " Physical Principles Involved in Transistor Action, J. Bardeen and W. H. Brattain, Phys. Rev. 75, 1208 (1949). p-n JUNCTIONS IN SEMICONDUCTORS 475 APPENDIX I A Theorem on Junction Resistance We shall here prove that the junction resistance is never less than the value obtained by integrating the local resistivity l/qii(P + bn). This is accom- plished by analyzing the following equation which we shall discuss before giving the derivation: 75^ = i r (i + ^P) dx + qg r {^, - ^rdie'^'^-'-'^'-Ddx, 9M J^a \ P on J Jxa the meaning of the symbols being that shown in Fig. 5. This expression is valid even if large disturbances in p and n from their equilibrium values occur. The second integral is positive since the integrand is never negative. It may be very large if (^p — (pn » kT/q in some regions. If, in the first in- tegral, we consider that Ip and In may be varied subject to the restraint Ip-^ In = I, we may readily prove that the first integrand takes on a mini- mum value when T P^ AT ^'^^ Iv = — T— ^ and /„ = p -\- bn p -{- bn For this minimum condition, the first integral becomes / lb dx/qix{p + In) = I Re where i?o is simply the integrated local resistivity. If / does divide in this way, the second integral is zero, a result which we can see as follows: Ip = — qnp dipp/dx In — ~ qiibn difn/dx d(pp/dx _ Ip/p d(pn/dx In/bn Hence, if the current divides in the ratio of p to bn, then d Fq . The equation discussed above is derived as follows: We suppose that SJ'p(-Vo) = ^n{Xa) = ^a -/; ./r.VC77().V.S' /.V Sr.MKOXDCi'rORS 477 The corresiKJiuliu^ currcuL c\alualL'(l at .v = 0 where p = pi e.\i)(/co/) = {pn(/Vi'kT)exp{iu}l) is given by / = -V d.v f/.v = - 0 corresponds to the ;/-region. We shall suppose that the potential in the layer is uniform with value \pi whereas in the ;i-region it has value i/'2 . The lifetimes of holes will be taken ri and r-^ in the two layers. The solutions for pi and po are evidently pi = pic + (-1 e + B e ) e p'l = pio -\- C e -^x+iut X < 0 .V > 0 where a = (1 + /cor,)''''/V£>ri ^ (1 + i^rxf'-fU ^ = (1 + /coT,)''7VZ)r2 ^ (1 + /a;ro)''7^2. The boundary condition for continuity of /p , recjuired to avoid singularit}- in difp/dx, is 478 BELL SYSTEM TECHNICAL JOURNAL and, for continuity of hole current, is dpi/dx = dpi/dx. Expressing these in terms of A, B, C, a and (3 for the a-c. components yields: A + B = Ce"^'^'"^''^"''' = CF a{A - B) = ^C so that A ^ {F + /3/a)C/2. B = {F - /3/a)C/2. Hence the ratio - [dp/dx\/p atx = -c is dlnp _ aiAe^"" - Be'"") ^ ajFa s'nh ac + ^ cosh ac) ~ ~~Q^ (A e+"'= + B e-"") Fa cosh ac -\- ^ sinh ac Since at x= —c, the a-c. component of pi is {qvi/ kT) pioe^" , the admittance is = -gPdp/dx ^ ^.^j^p^^/kTX-d In p/dx) / /. \/. , • M/2 F« sinh ac + /3 cosh ac = (gM/'io/li)(l + tcon) ^, cosh ac + ^ sinh ac- . For c -^ 0, this transforms into iqUiPio/Li) (1 + io:nf'^/Fa = {qix{p,,/F)/U){\ + ^corz)'" which agrees with Section 4, since pw/F then corresponds to />„ . If c/Li and F are not large, an appreciable amount of recombination takes place for x > Q for low frequencies. Dispersive effects will then occur corresponding to t2 . The a-c. will not penetrate to x = 0, however, if ciui/Df'^ » 1 and the dispersive effects will then be determined by n . The frequency-dependent part of the admittance, Fa sinh ac -\- fi cosh ac (1 + iuiTl) Fa cosh ac + /3 sinh ac ' has been computed and is shown in Fig. 7 for r^ = t2 , F = 1, ri = 7^/9 and c/Li = |. For these values about half the hole current reaches a; = 0 for low frequencies. As the time constant for diffusion through the layer is Tp/81, as discussed in Section 4.6, the layer will act as a largely frequency-inde- pendent admittance well above the point for corp = 1. This is reflected in the behavior of the curves of Fig. 7 and, for frequencies in the V^/ range, it is seen that G is larger than S by about 50% of the low-frequency value of G; this split of G + iS into {^)Co plus approximately (|)Go (1 + io:TpY'^ corre- sponds to the fact that about half the holes are absorbed in layer 1 for the assumed conditions. p-n JUNCTIONS IN SEMICONDUCTORS 479 APPENDIX R' Time Constant for the Capacity of the Transition Region For this case we shall consider the case of holes in an a-c. field with po- tential X xe \ ^ = ^f'^ + - where the d-c. retarding field is kT/qLr and the a-c. field is kT/qU where \/L\ is considered small for the linear theory presented here. The expression for the current of holes is We shall obtain a solution for p by letting p = poe -{- p\[e - e \e , while neglecting recombination in this region so that p must satisfy the con- dition p = —d (hole current)/a.v leading to the differential equation There are three separate exponential dependencies of the variables leading to three equations (neglecting terms of order (l/Li) ) d[poJi-Poj{\ = 0 The first equation is satisfied by the equilibrium distribution and the second by />1 = —pQ D/ioJ L\Lr and the last by 1 + Vl 4- ^ii^n/D '^= 2L. It is evident that dispersive effects set in when CO = D/aC 480 BELL SYSTEM TECIIXICAL JOCKS AL This corresponds to the result used in (4.31) in which (.Vr„ — .Vrp)/10was used for L, . For smaller values of oo the current may be calculated and put in simi)le form by expanding 7 up to terms including co". The resulting ex- pression for the current is / = —iicici pi) L.{Lr'L\)e"^ This is interpreted as follows: The a-c. voltage across a layer L, thick is # = (kT'c]) {L,'U)e'^' and, if we consider plus voltage as producing a field from left to right, then the a-c. voltage across /.,. is V = —5\}/. Substituting this for (/,,. /.i)e.\p(/w/) gives I = iojc/po Lr(q/kT)V Here qpoLr is the total charge in the layer Lr , (qV/kT) is an average frac- tional change in this charge for V so that (qpoL,) (qV/kT) H- F is a capacity. APPENDIX V The Effect of Surface Recombin.\tiox In this appendix we shall consider the effect of surface recombination upon the characteristics of the p-n junction. As for Section 4 we shall illustrate the theory for the case of holes diffusing into «-type material. For sim- plicity we shall treat a square cross-section bounded by y = iw, z = ±w, the current flow being along +-^". We shall denote the a-c. component of p as pi = pi {x, y, z, I) At .V = 0, the edge of the //-region, we shall suppose that tfp and \}/ are inde- pendent of y and z so that we shall have pi(0,y,z,l) = pwe"' = {pnqvi/kT)e"^ by reasoning similar to that used for equation (4.5). The boundary condi- tion at the surface will be — D -p^ = spi for V — +w dy ^ This states that the recombination per unit area is spi and is equal to the diffusion to the surface —Ddpi'dy. Similar boundary conditions hold for the other surfaces. Ky standard procedures involving separation of variables we may verif\' that the solution satisfying the boundary conditions is pi = Z^ a,je ' cos Pi y cos ^jS i".;=0 J>-u JL'\CTI().\S l.\ SKMICOXDICroKS 4SI where the eigenvalues jJ^ are determined by the boundary condition l3iu' tan iSrcH' = sb/D = x- We use dt = iSrw for brevity later, liecause of the symmetry of the boundary conditions it is not necessary to include sine functions in the sum. The value of a,j is given by a.j = (1 + iu>T,j)'/iDT,if where r,/ is the lifetime of a hole in the eigenfunction cos jSyy cos /3;S; i.e. Tij is the lifetime which makes p = exp {-I /Tij) cos (3iy cos jSys, a function which satishes the surface boundary conditions, a solution of the equation dp/dl = DV-p - p/t = -Difi + l^'j)p - p/t where to simplify the subsequent expressions we have omitted the subscript p from r. This equation leads to Tij T The coefficients an are readily found since the cos ji,y functions form an orthogonal set (as may be veritied by integrating by parts and using the boundary conditions). The values are an/pK = 4[sin 0, sin dA'9>ej[l + (1/26,) sin 2d,\-[l + {l/2dj) sin 26,] The current corresponding to this solution is /i = -qD j I (dp'dx) dydz integrated over the cross section at .v = 0. This gives /i = qDpice"^ Z aij{a,j/pioj{-i-d\ '6fij) sin 6i sin 6; Substituting for a,; and inserting pu) = p,^qv\ kT, we obtain an exjiression for the admittance -1^ = /i/l'i exp(/a)/): o 4 sin- 6i sin- dj where the sum plays the role formerly taken by (I + iuil)^''/y/DT in equation (4.12); the factor Aiv is the area of the junction. We shall analyze the formula for the case in which recombination on the 482 BELL SYSTEM TECHNICAL JOURNAL surface is smaller than diflfusion to the surface so that x is not large. The values of di, over which the sum is to be taken, may be estimated as follows: in each interval of di of the form iit to (n + (^))7r, di tan Oi varies from 0 to oo, giving one solution to di tan di = x- For x small, the solutions are approxi- mately ^0 - sin do - tan ^o - Vx ^1 - 7r + x/t^; — sin di = tan di = x/t dn - nw + x/nr; (-1)" sin dn - tan dn - x/nrr From this we see that the terms in the sum are as follows: aoo-4x /x -i = «'oo anO-2(x/«7r)V(«7r)^ = ocn(P-x/nTr .4/4 4 8 From this it is evident that unless x is large, the series converges very rapidly. (This conclusion is not altered when the increase in anm with ^„^m is considered.) Thus the dominant term in the admittance is Aw^qupo (1 + iwTQoy'^/\^DTm where 1/roo = 2 ^1^ {dl) + 1/r This expression is valid only for sw/D small so that dl = sw/D. The term s/iw/l) represents the rate of decay due to holes recombining on the surface, s having the dimensions of velocity. For co » 1/too , the admittance becomes 4-d'^qfjLpo{io)''Dy^-, the same value as given in equation (4.12) for large co and an area -iw^. The conclusion from this appendix is that for x small, the effect of surface recombination is simply to modify the effective value of t and otherwise leave the theory of Section 4 unaltered. For very large values of x, it is necessary to consider higher terms in the sum and several values of r will be unportant. Under these conditions the p-n JUNCTIONS IN SEMICONDUCTORS 483 approximation is that, at x = 0, pi is independent of .v and y may become a poor one, especially for forward currents, because the transverse currents to the edges will be important. Under these conditions the role of surface re- combination will give rise to patch cfTects of the sort discussed in Section 4. APPENDIX VI The Effect of Trapping Upon the Diffusion Process In this appendix we shall investigate the effect of the trapping of holes upon the impedance. We denote the density of mobile holes in the valence- bond band by p and the density of holes trapped in acceptors by pa . For thermal equilibrium at room temperature there will be an equilibrium ratio, called a, for pc/p. For germanium a — IQr* and for silicon a — 0.1 to 0.2. We shall consider four processes which occur at rates (per particle per unit time) as follows: Vr direct recombination of a hole with an electron (free or bound to a donor) vt trapping of a hole by an acceptor Vra recombination of a hole trapped on an acceptor Ve ex'citation of a trapped hole into the valence-bond band. Under equilibrium conditions as many holes are being trapped (rate pvt) as are being excited (paVe) '. hence v t = ave . We shall study solutions of the customary form for the a-c. components: , . iut—yx pi = pioc Pla = Plaoe These must satisfy the equations pl = D\ pi — {v t + Vr)pl + Vepla Pla = Vtpl — {Ve -f Vra) pla These lead readily to the equation for 7: Dy = io} -]r Vr -\- Vt — VeVi/iioi -\- Ve -\- Vrc) = '" 1 + 7 I ^^2^1 2^ \ -'r Vr -\- Vt {Ve + Vra) -r 00 )\ 1 (ve + Vra) + W /("e + Oj From this equation we can directly reach the important conclusion that the trapping process can never lead to a capacitative term larger than the resistive term. This result is obtained by analyzing the complex phase of 7, the admittance being proportional to 7. In particular, we find that the real term in Dy is always positive, as may be seen from inspection, so that the complex phase angle of 7 is less than 45°. The form reduces to a simple expression if Ve and v t are very large com- 484 BELL SYSTEM TELllMCALJOlRSAL jmred to v,-, Vm iuul w , a situation wliich insures local equilibrium between p and pa . Under these conditions we obtain Dy' = ico[\ -\- a] -j- V r -\- OiVra Dividing by (1 + ex) gives pVr + PaVra [D/(l + a) It' = [Dp/ip + pJh" - 'CO + P + pa The interpretation is that the holes diffuse as if their diffusion constant were reduced by the fraction of the time p/(p + pa) they are free to move and recombine with a properly weighted average of u,- and Vra • APPENDIX VII Solutions of the Space Charge Equation \\"c shall I'lrst show that the space charge equation (2.11) has a unique solution for the one dimensional case. For simplicity we write (2.11) in the form ^J^ = sinh u - fix) (A7.1) ax- to which it can be readily reduced. We shall deal with the case for which f = fa for .V < .v„ (A7.2) / = fh for .V > Xh > Xa (A7.3) so that the interval (.r„ , Xb) is bounded by semi-infinite blocks of uniform semiconductor. We shall require that u be finite at .v = ± cc . This boundary condition requires that for large values of | .v | w = Ua + AaC^'"'' x-^ -^ (A7.4) u = Ub + Abe~'"''' a; ^ + «) (A7.5) where sinh Ua = fa , sinh Ub = fh 7„ = I (cosh Ma)"" I , 7'- = 1 (cosh UhY'' I (If the opposite signs of the7's were present, the boundary conditions would not be satisfied.) The exponential solutions are valid for | u — Ua \ or I « — 2^6 I « 1. For larger values, however, solutions exist which are ob- tained by integrating (A7.1) to larger or smaller values of x. For these extended solutions the values of nix, Aa) and u'{x, .1„) (= du V/.v) p-n jrXCTIOXS /.V SKMhOXnrCJORS 485 are monotonically iiKTeasin<^ luiulions of .1,, . This may be seen by con- sidering -v = .v„ . For .1 a sufficiently small, the value of Ji{xa , Ag) and u'{xa , Aj are given simply by (A7.4). For larger values of .la, an exact integral will be required. It is evident, however, that all solutions of the form (A7.4) are related simply by translation for .v < Xn . Hence increasing An is simply equivalent to integrating (A7.1) to larger values of .v and it is evident that this increases n and u' monotonically. It may be verit'ied that for a sulliciently large .!« the solution becomes infinite at .v„ so that i((x a, Aa) u'{xa , Aa) both vary monotonically and continuously from — ^- to -{- x as .la varies from negative to positive values. \\'e shall refer to this property of u(xa , A„), ii'{xa , A a) as Pi . We next wish to show that h{xi , .!„), u'{xi , .4a) has the property Fi for \alues of .Vi > .v„ . To prove this we note that if for any .vi , m(.Vi , .4a) and /<'(.vi , A a) are finite, the solution may be integrated somewhat further to obtain «(.V2 , .4a), n'{x-> , .4a) for .V2 > Xi . From equation (.^.7.1) it is evident that an increase in either «(.Vi , a) or w'(.Vi , a) will result in an increase in d'-u dx- in the interval Xi < x < .Vo so that u and u' at x^ are monotonically increasing functions of u and n' at Xi . Hence if ti and u' at .Vi have the I)roperty Fi , so do u and u' at .v; . By extending this argument we conclude that H and ;/' at any value of .i- have the property Pi . (A rigorous proof can easily be completed along these lines provided that |/(.v) ] is finite.) Similarly it may be shown, starting from (.•\7.5), that u{x, Ab) is a mono- tonically increasing function of .1,, and h'{x, .1?,) is a monotonically decreasing function of Ai, . In order to have a solution satisfying (.^7.4) and (.A7.5) we must have, for any selected point .v, n{x, Ac) = h{x, Ab) (A7.6) ;/'(.v, A,) = u'(x, Ab) (A7.7) Xow as the equation u(x, .!„) = n{x, Ab) varies from — --c to + x , ;/(.v, .4^) varies from — x to + x and u'(x, Ab) varies from + --c to — x , monotoni- cally and continuously. Hence there is one and only one solution of (.'\7.1) satisfying (.^7.4) and (.\7.5). In order to verify that the solutions discussed in Section 2 are correct for large and for small A', we show schematically in Fig. Al the solution for a representative A' as a dashed line together with the curve u = Uoiy) = sinh" y. In terms of 7^0 , equation (2.16) becomes —r — -— (sinh H — sinh ih). (A7.8) dy- A- 486 BELL SYSTEM TECHNICAL JOURNAL From the symmetry of the equation, it is evident that u must be an odd function of y and hence that the solution must pass through the origin. The boundary condition in this case will be that u — > Uo for y — > ± co so that there will be no space charge far from the junction. We can conveniently use the origin as the point at which the solution from y = + ^ joins that from •y = — 00 ; from symmetry, this requires merely that u = 0 when y = 0. U AND ao ao=sLnh'y ^ — ■-^-:r^ / • / • > y ^-^= A; [sLnh u-slnh uqI d y2 k2 L J Fig. Al — Behavior of the solution of Equation (2.16) or (A7.8). For large negative y,u ^ smh~^ y and duldy = 1/cosh «o so that duldy is small. It is at once evident that, for large values of K, u must lie above «o so that the integral (sinh u - sinh Wo) dy = j- (A7.9) will be large enough to make the solution u{y) pass through the origin. If w — Wo > 2 over the region of largest diflference, the space charge will be largely uncompensated and the solution will correspond to that used in equation (2.18). On the other hand, as iT -^ 0, the requirement that u{y) pass through the origin leads to the conclusion that w — wo must be small for all values of y. The possibility that u oscillates about wo need not be con- sidered since it may readily be seen that, if for any negative value of y, say ji , both n{y\) and u'{yi) are less than no{y\) and u'{yi), then u{y) and u'iy) are progressively less than wo(y) and wo(y) as y increases from Vi to 0. Hence, if for negative y the u curve goes below the Wo curve, it cannot pass through the origin. p-n JUNCTIONS IN SEMICONDUCTORS 487 APPENDIX VIII List of Symbols (Numbers in parentheses refer to equations) a = {Nd - Na)/x (2.14) A = admittance per unit area of junction (4.23) Ap = component of A due to hole flow into ^-region (4.12) (4.24) An = component of A due to electron flow into />-region (4.25) .1 T = component of A due to varying charge distribution in transition region A also used as a constant coefficient in various appendices b = ratio of electron mobility to hole mobility b = symbol for base in Sections 5 and 6 B constant coefficient in various expansions in appendices c = symbol for collector in Section 6; a length in Appendix III C = capacity per unit area C„ , Cp (4.25) (4.27) as for ^„ , ^p Cr (2.42) (2.45) (2.56) as ior At D = diffusion constant for holes {bD is the diffusion constant for electrons) e = 2.718... / see Appendix 7 g — rate of generation of hole-electron pairs per unit volume (3.1) G = conductance per unit area of junction Gn , Gp as for /I's i = V^ / = current density In , Ip = current densities due to electrons and holes (2.5) (2.6) (4.10) /no,/po /pi (4.11) (4.12) (4.18) (4.19) Is , Ins , Ips saturation reverse current densities (4.11) (4.18) (4.21) /r see text with (4.35) / = subscript in Section 3 for junction Fig. 5 equation (3.11) k = Boltzmann's constant K = space charge parameter (2.17) L — length I-a = nil a (2.15) Lo = Debye length (2.12) Ln ,Lp= diffusion lengths for electron in /^-region and holes in «-region (4.8) Lr = length required for potential increase of kT/q in region of constant field (4.32) Appendices II and IV Li corresponds to a-c. field, Appendbc IV n — density of electrons 4KH BELL SYSTEM TEC/LXICAL JOl K\AL ii„ , lip = equilibrium densities of electrons in ;/- and />-regions f) = density of holes p„ , p,, = equilibrium densities of holes in //- and /"-regions pt) = d-c. component of non-equilibrium hole density (4.3) pi e.\p(/co/) == a-c. component of non-equilibrium hole density (4.3) P = total number per unit area of holes in specimen (2.35) (/ = electronic charge {q = \q \) Q — qP = total charge per unit area (2.39) r = recombination coefficient for holes and electrons (3.1) R — resistance of unit area Ka = resistance of unit area obtained by integrating conductivity (3.10), Append i.\ I R\ = effective series resistance, discussed in connection with (3.13) .y = rate of recombination per unit area of surface per unit hole density. Appendix V S ^ susceptance per unit area (imaginary i)art of admittance) S,, , S„ , 5ras for .fs. / = time T = temperature in °K T = subscript for transition region I, = cpP/kT (2.9), q{4^ - = (l:,.'lectric constant ^ = mobility of a hole (6/x — mobility of electron) V = rates of recombination etc., Appendix \'I p = charge density (2.1) a = conductivity I p-n Jl'XCTIOXS IX SKMICOXDICTOKS 4H9 a, ^ (diuluctivity of intrinsic niatcrinl (4.15) fT„ = conductivity of //-region = (jbnii,, a^ = conductivity of /^-region = t/np^ T = time T„ , Tp = life times of electrons in /^-region and holes in //-region {M) {}.}>) (4.7) 7 7 = relaxation time of transition region, Ap{)endix IV sT, v?,, , sf„ = Fermi level and quasi Fermi levels (2.2) (2.4) hi- = applied voltage across specimen in forward direction, Section 2.3, (4.2) X = sii\ D in Appendix \' xp = electrostatic potential (2.2) w = circular freciuency of a-c. (4.2) Band Width and Transmission Performance By C. B. FELDMAN and W. R. BENNETT In modern communication theory band width plays an important role as a transmission parameter. The authors discuss the significance of signal band width and frequency occupancy in relation to other transmission factors such as power, noise, interference, and overall performance for certain specific multiplex systems under assumed operating conditions. The intent of the paper is to show how such problems may be attacked rather than to find an unequivocallj' best system. The scope of the paper is described by the following table of Headings and Captions. I. INTRODUCTION Fig. 1. Outline of multiplex transmission methods 1. Non-simultaneous Load Advantage in FDM Table I. Non-Simultaneous Multiplex Load Advantage 2. Instantaneous Companding Advantage in Time Division Fig. 2. Performance of an experimental instantaneous compandor 3. Non-simultaneous Load Advantage in Pulse Transmission Fig. 3. Quantizing noise in each channel when PCM is applied to an FDM group 4. Signal Band Width and Frequency Occupancy 5. Regeneration and Re-Shaping 6. The Radio Repeater Fig. 4. Arrangement of two-way two-frequency repeater of television type showing spacing of bands and antenna discrimination Fig. 5. Discrimination of I.F. and R.F. circuits in television type repeater II. BAND WIDTH CHARACTERISTICS Fig. 6. Basic pulse shape and its spectrum Fig. 7. Marginal condition in reception of AM pulses and an FM wave in presence of noise. Fig. 8. Time allotments in Pulse Position Modulation Fig. 9. PPM-AM; fluctuation noise. Relations between band width, power, and signal-to-noise ratio. Fig. 10. PPM-AM; CW and similar system interference. Relations between band width and signal-to-interference ratio. Fig. 11. PPM-FM; fluctuation noise Fig. 12. PPM-FM; CW and similar system interference Fig. 13. PAM-FM; fluctuation noise Fig. 14. PAM-FM; CW and similar system interference Fig. 15. PCM-AM; peak interference Fig. 16. PCM-FM; fluctualion noise Fig. 17. PCM-FM; CW and similar system interference Quantized PPM Fig. 18. Comparison of quantize 1 PAM with quantized PPM Fig. 19. FD.M-F.\1; fluctuation noise Fig. 20. FDM-FM; CW interference III. BAND WIDTH AND POWER TABLES Table II. Optimum Band Widths for Minimum Power for Message Tvpe Circuits Table III. Optimum Band Widths for Minimum Power for Program Type Cir- cuits 490 HANI) w nrrii am) rRASSMissiox I'EKiouMAycE vn Table IV. Miiiinium Bund Widths and Corresponding Tower Reciuiremenls for Message Tj'pe Circuits Table V. Minimum Band Widths and Corresponding Power Requirements for Program Type Circuits IV. FREQUENCY OCCUPANCY TABLES FOR RADIO RELAY \. Antenna Characteristics Fig. 2\. Directional selectivity of microwave antenna Fig. 22. Simplified route patterns for study of selectivity required in congested localities Table VL True Frequency Occupancy of Various Message Grade Radio Relay Systems for Congested Routes Table VIL True Frequency Occupancy of Various Program Type Radio Relay Systems for Congested Routes 2. Conclusions as to Radio Table VIIL Comparisons of Band Width and Frequency Occupancy for Systems of Equal Ruggedness V. MORE ABOUT THE NON-SIMULTANEOUS LOAD ADVANTAGE Fig. 23. Theoretical possibilities of exploiting non-simultaneous load advantage by an elastic PLM-AM system VI. OVERLOAD DISTORTION AND NOISE THRESHOLD Fig. 24. Noise threshold and overload ceiling in frequency divided PCM groups Fig. 25. Overload characteristics of multirepeater systems VII. PULSES, SPECTRA, AND FILTERS Fig. 26. Typical pulses and their spectra 1. Pulses for PPM 2. Pulses for PAM 3. Pulses for PCM 4. Optimum Distribution of Selectivity Between Transmitting and Receiving Filters Fig. 27. Crossfire between frequency divided pulse groups 5. Delay Line Balancing VIIL TRANSMISSION OVER METALLIC CIRCUITS Fig. 28. Variation of circuit length with number of repeater sections in an AM system with fixed power capacity and noise figure Fig. 29. Optimum number of repeater sections and maximum circuit length for metallic AM system with fixed power capacity and noise figure^ Fig. 30. Optimum number of repeater sections and maximum circuit length for metallic FM system with limiting only at end of system Fig. 3L Optimum number of repeater sections and maximum circuit length for metallic PPM-AM system with reshaping at every repeater Fig. 32. Optimum number of repeater sections and maximum circuit length for metallic FM system with limiting at every repeater Fig. 33. Relation between circuit length, power, and number of repeaters m radio relay systems IX. CONCLUSIONS X. APPENDICES Appendix I. Noise in PCM Circuits Fig. 34. Stepping and sampling an audio wave Fig. 3.5. Variation of quantizing noise with samp'ing frequency Appendix II. Interference Between Two Frequency Modulated Waves Fig. 36. Geometric solution for resultant phase of two frequency modulated waves Appendix III. PCM for Band Width Reduction 492 BELL SYSTEM TEX TIXK \ I /. JOl 'A'.V. 1 L Appendix IV'. Siii)i)k'iiunlai>- Details (jf Dcrivalidii nf Hand VVidlli Curvi-s Appendix V. Sani])iinK a Hand of Frequencies Displaced from Zero J'"ij;. .^7. Minininin sani|)ling freriueiic>" tor Iiand (il ^vidth W List of Frkquicntlv Usku Symbols B = radio signal hand width in megacycles. (Not to he confused with frequency oc- cu])ancyj. h = hase or radix of PCM system. j3 = peak-to-peak frequenc\- swing of FM systems in megacjxles. Eh = width of baschand (video band) in megacycles. /, = repetition or sampling frequency in megacycles. A' = load rating factor (amplitude ratio), log = logarithm to hase 10. In = logarithm to hase e. X = number of channels in a multiplex system. ;; = number of digits in a PCM system or number of spans in a multirepeater system. T = wanted carrier amplitude. }'„ = mean fluctuation noise power per megacycle. {) = interfering carrier amplitude. .V = span length in miles. U = band spacing factor. I. Introduction CARRIER systems for the transmission of many telephone channels on a single metallic circuit have grown to be very important in the telephone network. Since the development of the coaxial cable system in which 480 channels are transmitted in a 2-mc baseband, advances in high frequency techniques, including the war-accelerated microwave art, have inspired efforts to utilize the broad band capabilities of high transmission frequencies. Some of the efforts have related to the wave-guide conductor but mainl}' they relate to radio relay transmission. As a consequence of these efforts a considerable number of new multiplex methods for use at microwave frequencies have been devised. All of these methods employ bandwidth more liberally than the 4 kc per channel rate associated with single sideband carrier systems, in return for which various transmission advantages are obtained. Theoretically, transmission advantages can be sacrificed to permit bandwidth reduction but the transmission requirements then become very severe. Bandwidth as a transmission parameter has grown to a prominent position in modern communication theory as set forth by Shannon et al.'' -■ '' The liberal use of bandwidth, employed in an effective way, operates to j)ermit higher noise and distortion within a system and, in the case of radio relay systems, o})erates to permit higher interfering signals from other radio systems. When all the frequency space necessary to avoid mutual inter- ■ C. E. Shannon, ".\ Mathematical Theory of Communication," Bell Svs. Tech. Jl., Vol. 27, pp. 379-423, 623-654, Julv-Oct. 194cS.' 2B. M. Oliver, J. R. Pierce and C. K. Shannon, "The Philodophv of PCM," Proc. /. R. E., Vol. 36 (1948), i)p. 1324-l.i^l. * C. E. Shannon, "Communication in the Presence of Noise," J'roc. LR.E., Vol. 37 (1949), pp. 10-21. BAM) Winill AM) TRAXSMISSIOX I'l'.KIOKM A \CK 4'>3 ference l)et\vct'n systems in a congested area is taken into account, certain wide-band methods, less \ulneral)le to interference, ma}' he as or more etlicient in the use of frecjuency space than other narrower hand muUipiex methods. The principal pur|)()se of (liis pai)er is to examine, for various systems, the rehitions governing the exchange l)etween freciuency space and transmission advantages. It will be shown that the preferred multiplex method depends in jjart upon : 1. 'J1ie grade of facility required; low-grade and high-grade channels lead to difTerent preferences. These preferences also are influenced Ijy the length of circuit. 2. The nature of the transmission obstacle over which advantage is sought. These obstacles may be: (a) intrasystem distortion (phase distor- tion, overload distortion, etc.) and noise; (b) intersystem interference as between similar radio systems or between different tyj)es of radio systems, operating on the same frequency. Other factors beside the transmission considerations discussed here are likely to be involved in a practical multiplex application; hence the system preferences arrived at in this study may not be the controlling factors in practice. Before a detailed analysis is undertaken, it may be helpful to e.xamine and comment upon the chart shown in Fig. 1. All of the multiplex methods shown here have been studied sufficiently to i)ermit their approximate evaluation with the aid of some theoretical considerations and subject to certain qualifications as pointed out from time to time. Variations and combinations of these are possible,* some of which will be discussed later. In addition to the two general classifications of frequency and time division there is a third type based on carrier phase discrimination. A familiar example is the quadrature carrier system,^ which is capable of yielding two channels for each double sideband width. In another form^ each of X channels is modulated simultaneously on A 2 carriers with a different set of carrier phases provided for each channel. Time division multiplex may be regarded as a kind of phase discrimination in which the signal is modulated on harmonic carriers so i)hase(l as to balance out excej)! during the channel sami)ling time intervals. In true phase discrimination. ' .\ loiuprelR-nsisc lislinii ;uul (lisiussiim ol' \ari()us romhinaliuns will ho louiul in a recciU paper by V. D. Laiulon, "Theoretical .Viialysis of Various Systems of .Miihipk-.\ Transmission" K.C.A. Reviav, vol. IX, numbers 2 and 3, June-Sept. 1948, pp. 287-351, 438-482. ^ H. Nyquist, "Certain Topics in 'lY-legrajjli Transmission 'I'lieoiA," A.I.E.F.. Iraiis., .\pril, 1928, pp. 617-644. « \V. R. Bennett, "Time Division .Multiplex Svstems," Bell Svs. Tech. Jl. Vol. 20, pp. 199-221, April. 1941. ■194 BELL SYSTEM TECHNICAL JOURNAL cnz < tu- ccz i-o a UJ UJ 5 F3 mo - ZZw, o^<'" UjQ cdQ^ »<-J- zz^im. <^zu. DOoQ 00 ct LLI -i < t- z 1- UJ 1 SIMU ADVA OF CO DVAN < OO-: Z J-" fa MAM) WlDl'll AM) I KAXSMISSIOX I'l.KIOKM A.XCE 495 however, there need be no separation of channels in eitlier time or freciuency, and a homodyne detection process is required at the receiver for channel selection. The necessarj- precision of instrumentation seems in general more diHicult to achieve than with either frequency or time division, and only minor i)rospects appear for exchange of bandwidth for transmission advantage. Of the systems tabulated, the frequency division method (FDM) witli single-sideband suppresscd-carrier transmission is the only method in which bandwidth cannot be traded for some transmission advantage." This system will be used as a standard of comparison. The PAM method, with transmission by AM pulses, can trade upon bandwidth only as a means for reducing interchannel crosstalk. In the other pulse systems, as well as all systems using FM, bandwidth may be expended to gain advantage over noise, intersystem interference, and, generally speaking, intrasystem distor- tion and noise. N0N-SIMULT.A.NEOUS Lo.\D Advant.vge in FDM The non-simultaneous load advantage pertaining to frequency division multiplex refers to the fact that the channel sidebands rarely add to an instantaneous value even approaching the value N times the peak value for one channel. This means that the required peak capacity of a relay system transmitting the N channels increases slowly with N. Current toll trans- mission practice provides for relative power capacity roughly as follows.^ Table I NON-SlMUrTANEOUS MULTIPLEX LOAD ADVANTAGE N Required Relative Power Capacity Advantage 1 10 100 500 1000 Odb +6db +9db + 13 db + 16 db 0 20 - 6 = 14 db 40 - 9 = 31 dl) 54 - 13 = 41 db 60 - 16 = 44 db To emphasize the strikingly large non-simultaneous load advantage statisti- cally obtainable with conversational speech we may examine Table I and note, for instance, that the capacity of a lOOO-channcl system is completely ' We have in mind here a system such as Type K or L in which a mininuun separation of adjacent channels in frequency is used. It is true that by spreadins; the channels far apart in frequency, a reduction in cross-modulation falUng in individual channels could be obtained, hut the resulting amount of im[)rovemenl is minor comi)ared with that otTered 1)\' a corresj)on(ling band increase in tlie other systems. ** B. I). Ilolbrook and J. T. Dixon, 'M.oad Rating 'rheor\- for Multichannel .\mpliliers," Bell Sys. Tech. JL, Vol. 18, pp. 624-644, October, 1939. ' The values in the table come from curve C, Fig. 7, taking the single cliannel sine wave power capacity as +9.5 dbm. 496 BELL SYSTEM TECHNICAL JOURNAL used up by peak instantaneous voltage when 994 channels are disconnected and 6 carry full-load tones. If a group of carrier channels in frequency-division multiplex were trans- lated to microwave frequencies, the overload distortion affecting the trans- mission would be predominantly of the third-order class. To a first approxi- mation the third order distortion follows a cube law and may be predicted from the single-frequency compression. We assume here that the power capacity of the repeater is the output at which the single frequency com- pression occurring through the ccmplete system does not exceed 1 db.^ This criterion applies roughly to systems of several hundred channels capacity, and to present transmission standards. Instantaneous Companding Advantage in Time Division In time-division systems, as ordinarily understood and known in the current literature, each channel successively is provided with its full-load capacity, and thus a non-simultaneous load advantage does not accrue. However, because of the sampling process, instantaneous compression may be applied at the transmitting terminal before noise and distortion are en- countered; when complementary expansion is applied at the receiving term- inal the noise is suppressed. The expanded samples derived at the receiving terminal then bear an improved relation to noise, particularly in the case of weak samples. Such an instantaneous companding process applied without sampling to a continuous speech wave requires a greatly increased trans- mission band between compressor and expandor but, in a time division system, no more bandwidth is needed to transmit the speech samples after they have been compressed than before. An instantaneous compandor currently being used experimentally to handle 12 channels in time division has the noise performance characteristics shown'" in Fig. 2. It is shown as applied to a telephone system in which the channel noise power (unweighted) would be 45 db down from the power of a sine wave which employs the full load capacity provided for the "loudest talker". Abrupt overloading is assumed to take place when peak amplitudes exceed that of the full-load tone. The location, at —7.5 on the load scale, for the power representing the very loud talker (one in a thousand) conforms approximately to current practice. The speech volumes, referred to the point of zero db transmission level, are shown for the sake of completeness. ' In a multi-repeater system the compression accumulates. This means that each re- peater must be restricted to operate approximately 10 log n db below the 1 db compression point of one repeater, (n denotes the number of repeaters.) See Section VI. '" Use of the same curve to represent the performance with tone or speech implies an independence of wave form which is not rigorously valid. Calculations based on speech- like signals have indicated that the curve for tone loading is a good approximation when average power is used as the criterion in the manner shown. 1 BAND Ml mil AND TRANSMISSION PERFORMANCE Wl The compression and expansion result in a uniform improvement of 26 db for weak signals including the "very weak talker" and a lesser improvement for stronger signals. The noise power in the absence of si)eech is 71 db SPEECH VOLUME IN VOLUME UNITS (vu) -33 -10 +7 (very weak) (average) (very loud) ■+■ -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 TONE POWER IN DECIBELS (FROM FULL LOAD) ALSO average power OF CONTINUOUS SPEECH IN DECIBELS (from POWER OF FULL-LOAD TONE) Fig. 2— Performance of an experimental instantaneous compandor. below the power of the full load sine wave which the channel is designed to handle. The performance is substantially equivalent for telephone purposes to a 71-db circuit without companding, in spite of the fact that for all except the very weak talkers the average noise is greater than with the 71-db circuit. 498 BELL SYSTEM TECHNICAL JOIRXAL The noise is increased (over llie 71-db value) by the compandor only in the presence of speech and then only in proportion (roughly) to the amplitude, and so becomes masked by the speech. The masking is sufficient to make impairment of medium and loud speech imperceptible provided that the ratio of speech power to noise power is greater than about 22 db. Under these conditions we are justified in defining the equivalent signal-to-noise ratio in terms of the low level noise. For compandors with a more drastic characteristic, yielding more low- level improvement, the high-level noise increase is enhanced and the limit to this enhancement is controlled by the "uncompanded" signal-to-noise ratio (the ratio without companding.) Thus the amount of low-level im- provement that is permissible from the standpoint of high-level performance is determined by the uncompanded signal-to-noise ratio. Experiments have shown that the permissible low-level improvement increases several db for each db increase in uncompanded signal-to-noise ratio. Another way of putting it is that the value of the equivalent signal-to-noise ratio in the speech channel determines the amount of compandor advantage which may be invoked to attain that ratio, and that the permissible compandor contribu- tion increases nearly as fast as the equivalent signal-to-noise ratio. The uncompanded signal-to-noise ratio is thus required to increase only slightly. For the 45 db uncompanded signal-to-noise ratio of Fig. 2 the compandor could have been designed to yield more than the 26 db low-level improvement shown without impairing high-level performance. In the time-division systems of message grade with which we will deal later, a 22 db compandor advantage is assumed." In the quantized systems included in the PCM headmg the instantaneous compandor advantage applies to the granularity, or quantizing, noise in the same way as to the common kinds of noise which plague other systems. The compandor of Fig. 2 was actually used in an experimental PCM system.^- A discussion of quantizing noise appears in Appendix I and a more compre- hensive treatment appeared in the Bell System Technical Journal recently.'^ In transmitting frequency divided groups of channels by pulse methods" " This is the maximum compandor advantage permissible for a circuit equivalent to 57 db signal-to-noise ratio. We will use this figure in connection with power requirements for circuits whose signal-to-noise ratio is intended to be equivalent to 60 dl) but since we pre- sume that interference or crosstalk ma\- be present in an amount equal to noise and since the compandor acts on interference as on noise we must protect against high level impair- ment on the basis that the noise is 3 db greater. 12 L. A. :Meacham and E. Peterson, ".An Experimental Multichannel Pulse Code Modu- lation Svstem of Toll Qualitv", Bell Sys. Tech. JL, Vol. 27, pp. 1-43, Jan. 1948. " \V.' R. Bennett, "Spectra of Quantized Signals," Bell Sys. Tech. Jl. Vol. 27, pp. 446- 472, July, 1948. '^ If the group occupies a frequency range extending from zero to Fb, the minimum sampling rate is well known to be 2Fb. If the group range does not start at zero frequency the minimum sampling rate is not twice the highest frequency of the group but lies be- tween two and four times the width of the band depending on the location of the band. This matter is treated in Appendix V. B.WP W I Dill AM) /'A'.IA.S 1//.V.S70.V I'l'.KlOKM AWE V)') the instaiilauooiis mmpamior advantage is substaiitiall}- zero because, at full system load, compaucling aclually increases the total noise. In time division the noise is increased at full load by comi>an(ling but, as discussed earlier, this is permissible because full load occurs only with loud talkers who mask the noise. In a frequency divided group transmitted by pulse methods nearly- full load may be produced when a number of loud talkers are momentarily active; the weak talkers then enjoy no improvement due to companding but may, on the contrary, suffer some degradation. NON-SIMUI.T.VXEOUS LOAD AdVANTACK IN PULSK TRANSMISSION Transmission of a freciuency-divided group by i)ulse methods does, how- ever, permit the realization of a portion of the non-simultaneous load ad- iTi 60 ^^ UJ CD LU a Z 56 O a. m ^ 52 O z O 50 -J V / y / y — ^ / Z 48 (J y / g '^'° a 3 44 1 1 1 1 1 1 1 _l — _l_ " 1 2 4 6 8 10 20 40 60 80 100 200 400 600 1000 NUMBER OF CHANNELS IN FREQUENCY-DIVISION MULTIPLEX Fig. 3 — Quantizing noise in each channel when PCM (128 equal steps) is applied to an FDM group. vantage in lieu of the instantaneous compandor advantage realizable in time division. A pulse system, designed to carry N channels in time-division with a certain full load signal-to-noise ratio (without companding) may also be used to carry N channels in frequency division. These iY channels in frequency division may be treated as a single channel .Y times wider than the time-divided channels and sampled X times faster. The ratio of the full-load signal in this wide band to the noise in the wide band turns out to be the same as the corresponding ratio in each of the narrow, time-divided channels. This fact makes the transmission of large groups of frequency- divided channels advantageous compared to small groups. For instance, in a 100-channel frequency-divided group, a single channel would have avail- able a total load capacity which is 9 db less than that for the multiplex group. This comes from Table I, which shows that 100 channels require 500 BELL SYSTEM TECHNICAL JOURNAL 9 db more range than one channel. This makes the signal-to-noise ratio in a channel 9 db lower than if all of the entire load capacity were devoted to that channel. However, in the 100-channel group a single channel receives only 1% of the noise power in the entire band, so a 20 db improvement accrues on this score. The net improvement is 20 - 9 = 11 db. Applied to a 128 step PCM system in which the full-load signal-to-noise ratio is 45 db'^ (7 digits binary PCM), the full-load signal-to-noise ratio in one channel of a 100-channel group thus becomes 45 + 11 = 56 db. With smaller groups than 100 channels the signal-to-noise ratio falls to 45 db while for larger it reaches 59 db, as shown in Fig. 3. Better results than these are obtainable with time division and instantaneous companding, as shown in Fig. 2, but these results may have significance in relation to the transmission of tele- vision by a pulse method, such as PCM. If a 128-step system were used, a large frequency-divided group of telephone channels filling the television band could be substituted for television when desired. A more powerful application of the non-simultaneous load advantage in time division will be discussed later in Section V. Signal Band Width and Frequency Occupancy We define signal band width as the width of the signal spectrum (or more realistically as that portion of the signal spectrum which must be preserved in order to make the signal sufficiently undistorted). Frequency occupancy is greater than signal bandwidth in two respects: First, the frequency range accepted by the receiving filter at the end of each span must be greater than the signal band for reasons of filter imperfec- tion. In all of the pulse or FM systems it would be advantageous from the circuit point of view to make the receiving filter much wider in order thereby to reduce the phase distortion over a small central frequency range occupied by the signal band. The assigned frequency space must include the entire band accepted by the receiving filter. Our comparisons will assume that the filters make use of an appropriate amount of refinement to conserve fre- quency space. Second, frequency occupancy must include the multiplication of assign- ments made necessary to avoid interference between converging or inter- secting radio relay routes, between the two directions of a single route, or between a main route and a spur. Our procedure in evaluating these systems will be to plot for each system certain curves relating power, signal bandwidth and channel signal-to-noise ratio or signal-to-interference ratio for various associated transmission 1^ Appendix I shows that the quantizing noise power at the minimum sampling fre- quency is the same for wide and narrow signal bands. This illustrates the general prin- ciple used here. BAND W lirni AM) TRANSMISSION I'ERFORMANCE 501 conditions. From these curves and other pertinent data we will prepare tables which show the significant frequency occupancy for various radio relay conditions. Such tables will be made for two grades of transmission facilities and for the extremes of signal bandwidth, one corresponding to minimum power and the other to minimum bandwidth. The minimum power condition prevails when the bandwidth has been increased, and the power reduced, to the point where any further increase of bandwidth would require an increase of power to prevent noise from "breaking" either the pulse sheer or the FM limiters.'^ The minimum bandwidth condition occurs when any further band limitation operates to impair the signal too much, assuming that the power is ample to override noise. Regeneration and Re-siiaping Two distinct classes of relay operation exist, one applying to the quantized systems (PCM) and the other applying to non-quantized systems. When the transmitted signal is intended to convey a continuous range of values (amplitude, time or frequency) noise and distortion accumulate as the signal progresses from repeater to repeater over a relay route. If, however, a range of values is represented by a discrete (quantized) value, a signal may suffer displacement within the boundaries of that range without altering the in- formation conveyed by the signal. If, therefore, in one span of the relay route the displacement is confined to those boundaries the signal may be regenerated and re-transmitted as good as new. No accumulation of noise and distortion need occur, therefore, as the signal traverses span after span. The most common application of regenerative repeatering is in printing telegraphy where the signal is either a mark or space and, if correctly deter- mined, may be re-transmitted afresh. In all of the non-quantized systems the repeaters must have low distortion so that a signal may be conveyed through a large number of them (say 133 for a 4000-mile circuit made up of 30-mile spans) without too much mutila- tion. In spite of good repeater design a signal passing through such a large number of repeaters will accumulate considerable noise, interference from other systems, and distortion characterizing the repeater design limitations. In non-quantized systems there is no escaping accumulation of this sort. In pulse systems, for instance, phase distortion, common in flat band repeaters, may result in tails and the like, while cumulative frequency discrimination (band narrowing), characterizing simple forms of linear phase repeaters, results in cumulative broadening of the pulses. In the former case the tails "^ In this connection it is of interest to mention that if the objective were a very low grade circuit the power required to prevent breaking might be higher than that required by a method having no improvement threshold, and no power saving could be accomplished by the bandwidth exchange principle. For circuits of telephone grade this situation does not occur. 5()2 BELL SVSLKM TELIIMCAL JOURNAL may eventually grow large enough to break the sheer (if the system employs such a device) while in the latter case the reduced pulse slope and the spread- ing out of time bounds may also bring about transmission disaster. In both cases these growing distortions successively reduce the margin that it is necessary to provide for noise and interference. To circumvent such effects, the pulses may be reshaped at all or some repeaters. Reshaping consists of measuring the information conveyed by the pulse (in the time or amplitude dimension) and sending out a new pulse of standard shape pos- sessing that measured characteristic in time or amplitude. This process is distinctly different from regeneration as practiced in quantized systems; in general, reshaping can only be counted upon to confine the rate of accumula- tion of noise, interference and crosstalk to that of power addition from span to span. In FM systems any distortion which results in amplitude "modulation" of the FM wave may be treated with limiting at each repeater to prevent such amplitude variation from accumulating and breaking the hmiter. Like pulse reshaping, this measure does not stop the accumulation of disturbance to the intelligence. Certain kinds of distortion may be combated by double FM.'^ Reshaping (or, in the case of FM, limiting) may be employed to conserve power in the systems having an improvement threshold. Without reshap- ing, the minimum repeater power is the marginal'^ value for the total noise accumulated from all spans. If reshaping is practiced at each repeater the power need be marginal for the noise from only one span. More bandwidth must be used, then, in exchange for the lower power; and, while this in turn increases the marginal power, the result is a net power saving. Tables II and III of Section III illustrate this point and Section VIII illustrates its application to metallic circuits. The Radio Repeater Repeaters for relaying television signals must achieve low distortion and we will take a current design and assume that such a repeater represents a basis for discussing the transmission of multiplex telephony by non-quantiz- ing methods. This repeater employs, in the two-way api)lication, four antennas and two frequencies as shown in Fig. 4. It is proposed to transmit 5-mc video television signals by FM in bands spaced 40 mc. The repeater employs double detection and the band separation is effected mainly by the '^ Leland E. Thompson, "A Microwave Relay System," Proc. I.R.E., Vol. 34, Decem- ber, 1946, pp. 936-942. '* By marginal power is meant that power which just safely exceeds the improvement threshold power. For a given noise level, minimum power is achieved when the bandwidth imjirovement factor yields the required signal-to-noisc ratio iu the channel with the power that is marginal for that bandwidth. HAM) Winril IA/> /7\.I.V.S1//.S-.SY(J.V I'KR/ORMA.VCK 50.S selectivity foUowiiif:; conversion lo iiUer-nu-diiite fa'(|uen(y. Microwave receiving fillers afford enough selectivity to divert alternate hands into their correct frequency-converting units without disturbing the other bands; and microwave combijiing filters serve in the transmitting side of the repeater to BACK TO BACK(OVER_tOODB) FRONT TO BACK (75 DB) ^2 4040 I 1 3960 I 1 H h 1 1 3880 ! 1 BAND- COMBINING FILTERS 4000 3920 I I ! 3840 I 1 1 H I I BAND- RECEIVING FILTERS y FRONT TO BACK (75DB) p;g_ 4 — Arrangement of Iwo-way Iwo-freciuency repeater of television type showing spacing of bands and antenna discrimination. bring the bands into the common antenna with small loss and small mutual disturbance.^^ The combining and separating processes are made easier by the interleaving of transmitting with receiving frequencies. Interleaving 19 W D Lewis and L. C. Tillotson, ".V Non-rellecting Branching Filter for Micro- waves," Bell Sys. Tech. J I., Vol. 27, pp. 83-95, January, 1948. 504 BELL SYSTEM TECHNICAL JOURNAL also eases the intermediate-frequency selectivity requirement when the side- to-side antenna loss is greater than the span loss (repeater gain). The stagger between input and output frequencies in one direction is employed to permit frequency selectivity to augment the back-to-back ratio in separat- ing the high-power output from the low-power input. Figure 5 further describes the repeater in relation to the frequency plan of Fig. 4, showing how the three kinds of antenna discrimination are em- ployed. The back-to-back (BB) and side-to-side (SS) antenna ratios at- tenuate the high-power transmitted band adjacent to a low-power received transmitters: east-west(bb) west-east (ss) 1 , receivers: east-west -10 - a. O-20 -25 3950 3960 3970 3980 3990 RADIO FREQUENCY I -75 DB SUBJECT TO _. DIFF. FADING TRANSMITTERS: EAST-WEST(BB) WEST-EAST (SS) JL 1 TOTAL I-F AMPLIFIER 1 R-F AMPLIFIER g. MODULATOR ■TOTAL OF 4000 MC BRANCH OF 1 REPEATER 4020 4030 4040 4050 N MEGACYCLES PER SECOND WEST-EAST (FB) (by OPPOSITE ANTENNA) Fig. 5 — Discrimination of I.F. and R.F. circuits in television type repeater, ANTENNA RATIOS'. BB= BACK TO BACK SS= SIDE TO SIDE FB= FRONT TO BACK band to a level comparable with the receiving level. The front-to-back ratio (FB) separates bands at the same frequency (which are always similar in power level except for the disparity produced by differential fading of the signals received from the two directions). For systems in which the signal is highly susceptible to adjacent band crossfire and phase distortion, the signal band must be confined to a certain fraction of the 40-mc band spacing in order to be carried through many such repeaters. With signals of more rugged characteristics, frequency occu- pancy can be reduced because a larger segment of the pass-band shown in Fig. 5 can be utilized by the signal and/or because the bands can be more closely spaced. With quantized systems and particularly with binary PCM BAND WIDTH AND TRANSMISSION PERFORMANCE 505 this reduction of frequency occupancy can be carried very much further when regeneration is practiced. We will later assign, for each modulation method, a value for the fraction of the band spacing to which the signal bandwidth must be confined. The frequency occupancy is then given by the product of the following three factors: 1. The reciprocal of the above factor, to be called U. 2. The signal bandwidth for the specified number of channels. 3. The number of frequencies required to operate in the assumed radio situation. Inspection of Fig. 4 shows two sources of common frequency interference between East- and Westbound signals: (1) backward radiation from one transmitter into the beam of the other and (2) backward reception of the signal intended for the oppositely directed receiving antenna. Both kinds of interference are suppressed by the front-to-back ratio (assumed as 75 db) compared with the desired signal. The effects occur at every repeater and in a 4000-mi. system with 30-mi. spans, addition of the interfering power contributed by the individual repeaters gives two "equivalent" single sources 75 — 10 log 133 = 54 db down. With no fading the two effects combine to form one source 51 db down. The receiver crosstalk is, however, subject to differential fading in adjacent spans. On the assumption that simultaneous deep differential fades on more than one pair of such spans are extremely rare, we base our estimates on a severe fading condition at only one repeater (or a number of less severe differential fades distributed simultaneously over the system having an equivalent effect.) The total interference from the other repeaters is then virtually the same as for 133 non-fading repeaters, while the fading repeater contributes an amount greater than the receiver crosstalk at one non-fading repeater by the depth of the fade expressed in db. We take this differential fade to be 30 db. The total interference from repeater crosstalk is, then, the result of a source 51 db down together with another source 75 — 30 = 45 db down. The combined interference is accordingly 44 db down. It is thus necessary that any long distance system using the repeater plan of Fig. 4 must be operable in the continuous presence of interference 44 db down. Under these assumptions, systems in which bandwidth is not or cannot be expended to gain tolerance to interference 44 db down do not allow any frequency to be used more than once in one repeater and therefore a repeater plan must be used which employs four frequencies. The four-frequency plan would be necessary in the case of conventional frequency division multiplex because such a system cannot tolerate crosstalk only 44 db down. 506 BELL SYSTEM TECILMCAL JOURNAL In all of the systems in which bandwidth may be exchanged for tolerance to interference we restrict, in our comparison tables, the minimum band- widths to those which provide at least the 44 db tolerance demanded in the two-frequency plan. RECTANGULAR ENVELOPE THROUGH GAUSSIAN FILTER ORIGINAL DURATION TIME X R-F BANDWIDTH BETWEEN ONE NEPER POINTS = 2 / ^ "^ \ / / N \, / \ / ' ORIGINAL DURATION TIME,T ^ \ r *i / 1 1 1 1 1 1 \ ^^ -1.4T -1.2T -T -0.8T -0.6T -0.4T -0.2T 0 0.2T 0.4T 0.6T 0.8T T 1.2T 1.4T TIME IN TERMS OF T 1 5 O -^ "^ ^V \ '^ 10 UJ ^ V \ a 15 a \, \, z \ s. \ O >- O 25 \ N LU a. \ i ^^ — ^ \ ! / UJ 2 35 ^ ! / ] / 40 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 J_ 1.1 TTTTTTTTTT FREQUENCY IN TERMS OF T (fROM MIDBAND) 1.3 T 1.4 T Fig. 6 — Basic pulse shape (approximately sinusoidal) and its spectrum. II. B.vxD Width C'h.\k.\cteristics The type of pulse assumed in the various pulse transmission systems is shown on Fig. 6. As pulse 4 of Fig. 26, it is further discussed in Section VII. The si)ectral density or distribution of energy vs. frequency associated with such a pulse is also shown. It is evident from this curve that omission of frequencies beyond a baseband width \'T can result in distortion or tails of BAND W IDl II AM) TRAXSM ISS/ON PERFORMANCE 507 only a few per cent of the pulse height. We deline signal bandwidth for the pulse systems studied here as \/T, or 2/T in the r-f medium, i.e., double sideband is assumed in all of the AM pulse cases. In assuming double sideband, we bow to the obvious difficulty of dealing circuit-wise with single sideband and its pulse demodulation problem. In the P^M systems we define the radio signal bandwidth as the peak-to- peak frequency swing, ^3, plus two times the baseband width, 2Fh. We shall consider individually the following types of systems where the meaning of the symbols is explained in Fig. 1. 1. PPM-AM 3. PAM-AM 5. PCM-AM 7. FDM 2. PPM-FM 4. PAM-FM 6. PCM-FM 8. FDM-FM The source of disturbance may be either fluctuation noise, a constant-fre- quency interfering wave (CW), or a similar but independent system operat- ing on the same frequency allocation. CW^ interference may fall anywhere within the radio signal band. Interference from echoes, which is a special case of similar system interference, is not treated. In certain cases echoes such as might be produced by multiple reflections in waveguide connections to the top of radio towers may be more detrimental than independent system interference of the same amplitude. We assume that such echoes are sup- pressed sufficiently by good design. Our first set of curves, Figs. 9-20, exhibits quantitatively the audio signal- to-noise and audio signal-to-interference ratios which can be obtained with increased radio bandwidth in the various systems. Audio signal is taken to be the power of a test tone which fully loads one channel. Audio noise is expressed as the total noise power in the channel. Audio interference is expressed as the power of aU of the extraneous frequencies produced in a channel by the assumed interfering signal. The term "radio bandwidth" is intended to mean double-sideband width and does not imply that the transmission is necessarily by radio. Two of the systems, PAM-AM and FDM, are omitted from this study because, as has been pointed out earlier, they do not provide a significant basis of exchange of bandwidth for suppres- sion of noise and interference. The other systems possess this trading prop- erty in varying degree as illustrated b}' the curves. The FDM and PAM-AM systems are entered in Table I\' and discussed under Section III. For comparison with the following curves it may be of interest to note here that for 1000 4-kc message channels in FDM, the bandwidth (single side- band) is 4 mc, and the received power for a 60-db audio signal-to-noise ratio must be —77 dbw.-" This is the power in a sine wave which employs 2" Throughout this paper wo shall use tlie alilneviation "(llnv" for power expressed in decibels relative to one watt. 508 BELL SYSTEM TECHNICAL JOURNAL the full load capacity in accordance with Table I, at a point where the noise power is — 189 dbw per cycle of bandwidth (15 db noise figure, NF). In most of these curves, plotted for 1000 message channels, the bandwidth scale runs to hundreds of megacycles. We do not mean to imply that the microwave transmission medium can be relied upon to transmit faithfully such wide-band signals or that circuit techniques for producing them are available. As suggested by Fig. 4, the 1000-channel system might be divided into several groups of fewer channels to avoid frequency selective transmis- sion difficulties or circuit limitations. The total frequency occupancy is not altered by such a division, while the required power per group is reduced in proportion to the number of channels.-^ Curves are shown of audio signal-to-noise ratio as a function of radio signal bandwidth at constant power and at marginal power. Audio signal- to-interference ratios are plotted against radio bandwidth for marginal ratio of radio signal power to interfering signal power. By "marginal power", we mean the radio signal power which just safely exceeds the threshold below which noise or interference causes system failure. In the case of fluctuation noise, any further increment of bandwidth from this point is untenable without an increase in radio signal power. Points on the marginal power curves show as abscissa the bandwidth at which minimum radio power is required to obtain the audio signal-to-noise ratio given by the ordinate. In calculating these curves, we have specified the marginal condition as occurring when the peak disturbance is actually 3 db below the theoretical value which just breaks the system. These relations are shown graphically in Fig. 7. We have in this paper followed the accepted practice of ignoring all fluctuation noise peaks exceeding the rms voltage by more than 12 db. Radio signal power is taken as the power averaged over a cycle of the high frequency in the FM wave, or, in the AM pulse case, over a cycle of the high frequency when the pulse is maximum. A curve is included in Fig. 9 showing marginal AM radio pulse power values for various bandwidths of fluctuation noise and a similar curve for FM is shown in Fig. 13. A noise fig- ure of 15 db-- is assumed for the receiver. W^e have taken the noise band- width as equal to the signal bandwidth throughout. This equality cannot be quite attained in actual systems because of the departure of physical fil- ters from ideal characteristics. In practice an allowance for frequency instability would also have to be included. The relation of the PPM pulse to channel allotment time is shown in Fig. 8. Pulses in channels adjacent in time can just touch when full load signals are impressed on each. The slicer operates at half the pulse height which, for the assumed pulse shape, is also the point of maximum slope. The width of the ^' These statements are not exactly true for FDM and FDM-FM, where multiplex load rating is used in the design. ^ This means that tlie noise power is 189 db below a watt per cycle of bandwidth. BAND WIDTU AND TtUNSM ISSION PERFORMANCE 509 values: THEORETICAL ASSUMED PRACTICAL ■-6DB -9DB 12 DB (MAX. PEAK) Fig. 7 — Marginal condition in reception of AM pulses and an FM wave in presence oC noise. Pulse case applies to PCM only if binary. 510 BELL SYSTEM TECIIXICAL JOi'RXAL pulse is inversely proportional to the signal bandwidth. The time available for modulating the pulse position is equal to the channel time minus the pulse duration. The combination of these factors leads to the PPM "slicer 80 75 70 65 _i LU 5 55 O lU O50 z ^45 4-^ JiL Fig. 8 — Time allotments in pulse position modulation 2f -■ J Nfr --- ^ - — ^-^ /^ ,-^ -/ y •^ NfJ^-"^ -^ / ■ 9~ -^^ / '^ ^ "^ 'X - f] .9^ / ^ 1 ^ r- / / / ' A^ 3" 1 / p^ u ^ / ^-^^ / __^ ^^ ^ -^ I L^ r ,.o>> P^"^ u ^ ^^-^^ 1 p ■3^ y^ ^ ^^^^^^ ^ -- ^ 1 '■ 1 I 1 ^ ^ 5 65 5 :2o| \ 1 — vj — h- 1 1 1 — I ij--^;;:]^^ — t — 1 1^1 ""^° ' ?^r^^~^^^^f^^ '' I 0 ^^^ ^ 90 0- -5 -==^ 95 > UJ O tOOuj 10 20 40 60 60 100 200 400 600 1000 2000 4000 10,000 °- RADIO SIGNAL BANDWIDTH IN MEGACYCLES PER SECOND Fig. 9 — PPM-.\M; performance with respect to tiuctuation noise. Relations between bandwidth, power, and audio signal-to-noisc ratio for 1000 4-kc channels. advantage" which, when applied to the r-/ pulse-to-noise ratio, gives the full load audio tone-to-noise ratio in each channel. Details of this calculation and others pertaining to various pulse systems are included in Appendix I\'. BA.\D W IDI II AM) TRAXSMISS/OX PERFORM ANCE 511 Fig. 9— PPM-AM, Fluctuation Noise The curves of Fig. 9 were computed from the sheer advantage derived in Appendi.x I\'. The asymptotic slope of the constant power curves of ^ db per octave of bandwidth reflects the 6 db advantage due to the two-fold greater pulse slope (sheer advantage) diminished by the 3 db increase of noise accepted by the two-fold wider band. In the marginal power case 80 / 75 / 70 10 / ^65 O y / -V y_ Q 60 z / / / ^ 55 g ./ < 50 cr ^ ^V^ / u45 z UJ f / ^ a. 40 LL / ,^>' A UJ 35 1- z / / ' 1- / y / Z (J) _ / / / / 5 '^ // / 10 // 5 0 /, 1 1 1 1 ... 1 _^_ 0 ; ?o ^ RAC 10 i )I0 s >o e IGN 0 1 AL 30 2 BANDW 00 4 DTH IN 00 6 MEG/ 00 \CY( 1C 00 2C .S PER ; )00 4C 3ECOND )00 10.C Fig. 10— PPM-x\M; performance with respect to C\V and similar system interference for 1000 4-kc channels with ratio of pulse to interference marginal. Relations between bandwidth and audio signal-to-interference ratio. the power is increased with bandwidth so the sHcer advantage is preserved and the slope is 6 db per octave. The sharp reduction of signal-to-noise ratio with reduced bandwidth appearing at the left end of the curves arises from immobilizing the pulse position as the widened pulse uses up more of the total channel time allot- ment. According to the definition of bandwidth used here and the plan of Fig. 8, no modulation is possible when the bandwidth is IjT and T is half of the channel time. For 1000 channels the channel time is 0.125 microseconds and T = 0.0625, which makes the audio signal-to-noise ratio zero at 32 mc. 512 BELL SYSTEM TECHNICAL JOURNAL Fig. 10— PPM-AM, CW and Similar System Interference The curve of Fig. 10 for marginal ratio of pulse power to CW interfering power has the same shape as the corresponding curve of Fig. 9 for marginal power over tluctuation noise. There is a shift of 9 db in ordinates, however, because the peak factor of the CW interference is 9 db less than that of fluctuation noise. The interference from similar systems follows a different law because of the "exposure factor" arising from the finite probability that the interfering pulse does not overlap the wanted pulse. A straightforward probabiUty calculation taking into account the distribution of pulse voltages in an interfering system occupying the same radio frequency band yields the curve shown on the assumption that the repetition frequencies are asyn- chronous. As the bandwidth is increased the pulses become shorter and their coincidences less frequent, leading asymptotically to a 9 db per octave slope instead of the 6 db per octave of the CW interference. Fig. 11— PPM-FM, Fluctuation Noise In the transmission of PPM by FM there are two sources of advantage over noise. One is the ordinary FM advantage and the other is the sheer advantage of PPM acting on the noise remaining in the FM output. There are, likewise, two separate conditions for system failure; one a breaking of the limiter and the other a breaking of the sheer. A certain amount of radio power will result in marginal operation of the limiter for a certain frequency swing. The corresponding deviation ratio is the quotient of the frequency swing and the baseband width; this ratio is maximum when the baseband is least. Except in the region near the minimum PPM band, advantage accrues faster with bandwidth in FM than in PPM. It is apparent, there- fore, that most of the radio bandwidth should be devoted to FM advantage. The optimum proportioning occurs when the baseband width has a small value but not so small as to invoke an unsurmountable penalty by not providing for any position modulation. Mathematical analysis given in Appendix IV shows that the optimum baseband for the pulse position modu- lation varies with radio bandwidth in the manner shown in Fig. 11 by curve 1. Curve 2 shows the audio signal-to-noise ratio vs. radio bandwidth when the baseband width follows curve 1 and the FM limiter is marginal. It is of interest to compare curve 2 with the poorer performance of the dashed curve 3 which is calculated for the case in which both the FM limiter and the PPM slicer are marginal. The baseband width for the double marginal condition follows curve 4. Curves 5 and 6 show audio signal-to-noise ratio vs. band- width for constant radio power and optimum baseband. The curv^e of marginal amount of radio power is not given in Fig. 11, but is the same as the one given later in Fig. 13. BAND WIDTU AND TIOINSMISSION PERl'ORMANCE 513 Fig. 12— PPM-FM, CW and Similar System Interference The curve showing interference from a similar system of lower jiower was based on a calculation of the beat spectrum between two FM waves, both frequency modulated over the same r-f range (j3 mc) by 8 mc. The phase dillerence between the 8-mc modulating frequencies was assumed to vary, giving rise to various beat spectra. The power in those beat components accepted by a band zero to Fb was averaged over all 8-mc i)hase differences 80 / y y/ ./ // / 4 ^ / / #/ y' ^^5 ^ y y CD 60 O i f .^^ y Q 55 z / / / / f ^'A^ -^'^S^ V g 5 45 UJ —R\ / / f\ f l'>^ / / / ■f- O z 6 35 1 ® / .f 7 / / • / -■ -,r\ / < 30 z 1 ( v/.. g 1 / '®// ( / / BA SEBAND WIDTH TH DETECTORS MARGINAL) D < 15 / V-CSy~^ t // • ■|MUM \ND WIDTH - 1 // { — oT" -■' ~~ OPT BASEB/ 0 1 1 1 L L I 1 1, 1 20 m 40 60 80 100 200 400 600 1000 2000 4000 RADIO SIGNAL BANDWIDTH IN MEGACYCLES PER SECOND 10,000 Fig. 11— PPM-FM; performance with respect to fluctuation noise. Relations between bandwidth, power, and audio signal-to-noise ratio for 1000 -i-kc channels. and this average power was taken as a measure of the interference to which the baseband signal is subjected. As outlined here, this procedure is valid for interference between idle PAM-FM systems in which the FM waves are frequency modulated as assumed above. For the PPM-FM case in which we are here interested, we take the position suggested by the transient viewpoint that the effect of interference from spaced pulses will not be much different because of their spacing and so we apply the sheer advantage possessed by the wanted system to the total interference calculated above and obtam the curve shown. At the left hand end where the pulse spacing is only slight 514 BELL SYSTEM TECHNICAL JOURNAL and the sinusoidal frequency modulation is nearly the correct representation, the above procedure is not subject to much suspicion. The validity of the right-hand portion of the curve is upheld by the fact that it is about 12 db lower than the marginal fluctuation noise curve of Fig. 11. If the wide swing FM interfering wave had a sjjectrum much like fluctuation noise of the same i)ower as the FM wave the difference would be 9 db. 10 20 400 600 1000 N MEGACYCLES 2000 4000 PER SECOND 10,000 40 60 80 100 200 RADIO SIGNAL BANDWIDTH Fig. 12 — PPM-FM; performance with respect to CW and similar system interference for 1000 4 kc channels with ratio of FM wave to interference marginal. Relations between bandwidth and audio signal-to-interference ratio v.'hen baseband is optimum for suppres- sion of fluctuation noise. It has been explicitly assumed that both systems are idle, but we see no reason to believe that if either or both were normally active the interference would be significantly different for our purposes. Audible interference from a CW wave is caused by a disturbance to the frequency of the FM wave. Let us first assume that the CW frequency lies near the middle of the frequency swing range. No disturbance to the FM wave occurs as its frequency passes through coiitcidence with that of the CW but, as the frequencies diverge, the magnitude of the disturbance as well as BAND II //;/•// AND TRANSM ISSIOS PJIKIORM A \CE 515 tlie frequency of tlic disturbance increases linearly, 'i'he baseband lilter is excited only during the time the difference is less than I''t>. Thus, the disturbance results from a series of perturbations to the otherwise smooth frequency variation of the I"'M wave. The time during which tliese pertur- ])ations can atTect the baseband filter is short compared with the shortest |)ulse the baseband hlter can pass, e.xcept when the baseband width is greater than half of the swing. This occurs at the extreme left-hand end of the curve. We have not attempted to calculate the response to these transients except to note that the response is a j)ulse which extends roughly 2T from its point of origin, peaking somewhere near the center of this interval. If we assume that the PPM pulses are closely spaced (e = 0) so that they result in a wave frecjuency modulated by 8 mc, there are two such evenly spaced disturbance pulses per cycle of modulation (two per 2T interval) and there- fore there is an almost continuous disturbance wave in the base band filter output whose amplitude does not greatly exceed its RMS value. We have accordingly calculated the power sum of all the extraneous frequencies j)assed by the baseband fdter, assuming the FM wave to be sinusoidally modulated. The location of the CW frequency giving greatest interference power was used in these calculations excej)t in the wide band cases where the worst frequency appeared to be near the edge of the band. Here the transient viewpoint indicated that the resulting interference in the ba.se- band would be greater if the CW frequency were nearer the center. If the trailing edge is used to measure the time of the pulse, the princij)al disturbance of this time arises from the perturbation produced at the leading edge of the same pulse, and so the calculation for close-spaced i)ulses is not greatly in error when applied to wider-spaced pulses. If the leading edge were used the worst CW frequency for widely spaced pulses would be one differing from the rest frequency by Fb and the interference would be worse, we think, than that arising from the frequency worst for trailing edge opera- tion. It has been explicitly assumed that the system is idle, but we see no reason to believe that the interference would be significantly different with normal activity. Fig. 13— PAM-FM, Fluctu.atiox Noise Fluctuation noise in a PAM-FM system produces the sloped noise spec- trum characteristic of FM in the output of the frequency detector. The noise power per cycle is zero at zero frequency and increases with the square of the frequency. The baseband filter accepts only the portion of the spectrum between zero and Fi,. If instantaneous sampling of the signal values is used, all noise frequencies in this range are equally effective as causes of errors. Use of a channel gate of maximum permissible duration 516 BELL SYSTEM TECHNICAL JOURNAL consistent with a satisfactory margin over crosstalk from adjacent channels furnishes a practical method of discriminating against the influence of noise components near the top of the baseband where the noise spectrum is strongest. The exact shape of the gate is not very critical. ^ The curves have been calculated for a rectangular gate coincident with the channel allotment time, which is just possible without crosstalk in the case of non- 80 y /• y. / FOR > MARGINAL 1 y y / /, I POWER V / / y / * f —^ of y / / / / / 60 ~v / #^ / 4 0 _i 55 lU (0 C 7 z45 / y -HT A ,_40 < a. / < ^ ^ -H 2T K- BASEBAND PULSES CUNMODULATED) O -,n '^/ / f / / / 2 JU / / / / V 25 _i < // / / / ^-, CO // ' \^ o ii> a 1 / t^ ^ < '^ 1 / ne>^-5 ^^^ 1 1 1 \^ — v\f^^- ?^ f v^^-- 0 r" j^ ^ -15 1 1 > 1 1 _^ 1 , 1 ^ -80 ^ O -85 "^ -90 ^ -95 10 20 40 60 80 100 200 400 600 1000 2000 4000 10,000 RADIO SIGNAL BANDWIDTH IN MEGACYCLES PER SECOND Fig. 13 — PAM-FM; performance with respect to fluctuation nuse. Relations between bandwidth, power, and audio signa!-to-noise ratio for 1000 4-kc channels. overlapping sinusoidal pulses. A somewhat shorter rectangular gate or a gate of sinusoidal shape leads to very nearly the same results. The ad- vantage of gating as compared to instantaneous sampling is approximately 8 db. Calculation of the gated noise is a straightforward process if based on the concept of the FM noise spectrum acting as signal on a product demodu- lator in which the carrier consists of the harmonics of the gating function. Each harmonic demodulates the spectrum centered about the harmonic BAND WIDTH AND TRANSMISSION PERFORMANCE 517 frequency and contributes audio power proportional to the product of harmonic power and spectral density. The channel 6 Iter accepts only the demodulated noise falling in the audio signal range. The marginal power curve has been drawn for a 3 db ratio of peak carrier to peak interference or 12 db ratio of mean carrier power to mean fluctuation noise power. Curves for specific amounts of received power are included as 80 • 75 70 I/) -I , / . ^' .v'i m 65 o AoA / z 4\ ' ,,' /■ >>" 55 O I- / ^1 /:¥, / /^3 >^ 'm 2 r. ^ Y IL a. / / / w ^- y 0135 1 / / #7 / t f / z / ' ^// --' r^ — ,/ V a Ii5 D < 4 '^ / / — 10 / / 5 0 1 1 /,/ f— 1 1 1 _j_ 1 1 1 1 _L 20 400 600 1000 2000 4000 IN MEGACYCLES PER SECOND 10,001 40 60 80100 200 RADIO SIGNAL BANDWIDTH Fig. 14 — P.\M-FM; perfcrmance with respect to CW and similar system interference for 1000 4 kc ciiannels with ratio of FM wave to interference marginal. Relations between bandwidth and audio signal-to-interference ratio. well as the curve of marginal received power vs. radio signal bandwidth for a receiver with 15 db noise figure. Fig. 14— PAM-FM, CW and Similar System Interference CW interference can be calculated conveniently by assuming all channels idle and determining at what frequency within the radio signal band a CW component of fixed power produces ma.ximum disturbance of an audio chan- nel. This worst possible amount of disturbance is then assumed not to be much affected by the various channel loading conditions existing during normal operation of the system. When all channels are idle, the transmitted 518 BELL SYSTEM TECHXICAL JOCRXAL carrier of a PAM-FM system using sinusoidal pulses assumes the particularly simple form of an FM wave modulated by a sinusoidal signal having fre- quency Nfr (8 mc for 1000 channels) and total frequency swing 13/2. Hence the rigorous steady state solution for interference between CW and sinu- soidally modulated FM was calculated and the interfering components falling in the baseband range selected. The gating function was then applied to these components in the same way as described above for fluctuation noise, and the resulting products falling in the audio channel range evaluated. The signal- to-interference ratio was expressed as the ratio of rms signal power received from a full-load channel test tone to the rms value of the audio interference. A range of frequency locations for the CW interference was investigated for each radio signal bandwidth and the one giving maximum audio interference used for the point on the curve. The worst position of the CW was usually found to be near the extremities of the idle channel frequency swing. Cur\'es are shown for a rectangular gate of maximum duration and for instantaneous sampling. When the source of interference is a similar system, we assume that the midband frequencies differ only slightly. With both systems idle, we have two sinusoidally modulated FM waves which are identical except for (1) a small variable difference between mean carrier frequencies and (2) a variable phase shift between the two modulating frequencies. The interference falling in the baseband consists of steady state components which are approximately harmonics of the channel slot frequency Nfr- As is charac- teristic of FM interference, the amplitude at the mth harmonic contains a factor proportional to m; and the component near zero frequency, the approximate zeroth harmonic, is very small. If we gate this interference with a rectangular gate of duration 1/Nfr, we find that the gated output vanishes for input components at Nfr, 2Nfr, ... , because these frequencies are located at the infinite loss points of the aperture admittance. The gate would transmit the zeroth harmonic, but this component tends toward zero amplitude. Our conclusion is that two idle PAM-FM systems accurately lined up to occupy the same frequency range are balanced against inter- ference from each other when a rectangular channel gate of full channel allotment time is used. The balance tends to disappear as the channels are loaded because the interference then spreads throughout the base band instead of being concentrated at the blind spots of the aperture. Thus, for the first time in our consideration of pulsed systems, we are obliged to take account of channel loading conditions.-' 23 A wave could be frequency modulated about a central frequency by P.\M pulses of plus and minus sign and an idle system would thus consist of a wave of constant frequency. The weaker of two such idle systems aligned in frequency would produce no (or very little) interference in the other, using either channel gating or instantaneous sampling. The susceptibility to CW interference would be greater than in the biased modulation assumed above, however. H.l.M) W 11)111 AM) T/ino>oO(oO''iOino«no - 1 / ^ -)o <0 / ^ > m Zt UJ (J) J m C ■ y* >/^^ >k - r; "- n lo ^ ui c '^m ^"^ \' \ y />y II v^ y // CO S \ \ > >r y.^ y O ~ T^Q ^ \-^ ^x /^ ^''^ _^ J^v ^^-^ // < .^ V yy< V " ^^ ^^ y.^ ^ ^ ^-^^ ^ ^ y - ^ ^D ^^ '>'^ — (0 ,. ' (0 -^^ '^ s^ (\J ' ^< l""^ - ■^ — II J3 <0 <\J o o iC - u 1- 1 -- i Z 10 i «, :: ■ ^ ^^^ - rOl >-. ^ p<^ r5>^=. ro^^ "^ - CMI >^ > ^'^ ^ ^^*>» OJ^^ - _ _ A p^^is^ ^-Vj s*. v^ N, C ( M 0) 1 J c • is • II II c II c M - - 0 O lil o 2 oO 2^ AUDIO SIGNAL-TO-NOISE RATIO IN DECIBELS Fig. 15 — PCM-AM; performance with respect to number of steps. Relations between bandwidth and audio signal-to-noise ratio and between bandwidth and required pulse-to- interference ratio for 1000 4-kc channels. permitted by the system, versus bandwidth for three different audio signal- to-noise ratios. The lower boundary of the curves corresponds to binary PCM while the left-hand boundary corresponds to the other extreme, namely BAND 11 //)/// AND TRANSMISSION PERFORMANCE 521 quantized PAM having the number of steps necessary to yield the specified signal-to-noise ratio. The quantized PAM bandwidth of 16 mc assumes the use of overlajiping sinusoidal pulses as in binary PCM. Actually, such an overlap would be hazardous in the higher base systems; and quantized PAM, like unquantized PAM, should perhaps be assigned more time per pulse but not as much as 2T because regeneration could be employed to prevent accumulation of interchannel crosstalk. The tables presented later do not include the bandwidth increase that would follow such an increase in time per channel. The curves at the right in Fig. 15 are terminated at 16 mc corresponding to one pulse per channel. In accordance with the principles of Appendix III more than one channel per pulse can be transmitted, theoretically. To include such a hypothetical case of less than one digit per channel, the cur\-es could have been e.xtended upward to the left. The 39 db signal-to- noise ratio curve would have reached an ordinate of 81 db at 8 mc on the bandwidth axis. It is of interest to compare the audio signal-to-noise ratio of unquantized PAM with that of quantized PAM for the interference ratios demanded by quantized PAM. In the case of marginal C\V interference the audio noise (evaluating the audio disturbance as noise of equivalent power) turns out to be the same as the quantizing noise and so, in a circuit of more than one span, quantized PAM is advantageous from a transmission point of view. With fluctuation noise the unquantized PAM audio noise would be 9 db lower than the quantizing noise and so, in a circuit of more than 9 spans of equal loss, the quantized PAM would be preferred. Fig. 16— PCM-FM, Fluctuation Noise Here FM advantage is employed to permit operation in the presence of more noise than is possible with AM. It seems more illuminating to explain these curv-es by checking their correctness rather than by deriving them. In all cases, a baseband signal-to-noise ratio giving the same margin over noise peaks as for AM (Fig. 15) is obtained by FM advantage. For the solid curves the FM limiter is assumed to be marginal (12 db radio signal-to-noise ratio), and for the dashed curves the radio signal-to-noise ratio is assumed to be the same as the marginal requirement for binary PCM-AM (18 db). The FM advantage with respect to an FM wave of the same power as in the peak AM pulse is, in db 20 log |-^ -f 4.8 = 20 log (^1 - 2) + -^-8 However, the FM power is greater than the peak AM pulse power by 10 522 BELL SYSTEM TECHNICAL JOURNAL Q 50 i2 40 O 1 1 / / '// 1 1 // 1 // ■ / / // '///// / / 1! 1 ^ ////// / / // // // 1 '///// AM) TNAXSMISSIO.X I'l.KiORM A \CE >2S and this musl jusL make up for the difiference between the 12 db (or 18 dbj FM wave-to-noise ratio and the pulse-to-noise ratios of 18 db -f 20 log {h — 1) required in the AM case. Substituting values from the curves will show that this is so. These curves show a minimum bandwidth for an optimum PCM base. 'I'his is to be e.xpectcd since two different rates of exchange Ijetween ])and- width and advantage are involved. One is the advantage growing out of PCM of reduced base while the other is the conventional FM advantage. An analogous situation was found in PPM-FM. It is of interest to examine the PCM-FM situation when the FM circuit is as tolerant of noise as the most tolerant AM case, namely when the r-f signal-to-noise ratio is 18 db. The optimum PCM base is octonary and the corresponding minimum bandwidth (as we define it) is actually 20% less than for binary AM. This apparent advantage of PCM-FM is not ob- tained when tolerance to C\V and similar systems is considered. Figure 17, which follows, shows that when allowance is made for a 9 db r-f signal-to- interference ratio (as in binary PCM-AM), the minimum FM bandwidth is greater by about 30% than for binary AM and the optimum base is ternary or quaternary. If the 3 db interference tolerance possible in FM is required, it is obtained, as shown in Fig. 17, with ternary PCM-FM, at a cost of ap- proximately twice the bandwidth required in binary PCM-AM, which has a tolerance of 9 db. We should point out here that binary PCM transmitted by single sideband and detected by a local carrier has a tolerance of 3 db and requires half the bandwidth shown in Fig. 16. PCM-FM requires a bandwidth 3.8 times that of single sideband binary PCM for the same 3 db tolerance. Fig. 17 — PCM-FM, CW and Similar System Interference In PCM, sequences of several pulses of the same amplitude may occur. The FM signal then consists of a steady frequency. A steady beat frequency persisting for several pulse periods will be produced by CW interference." If this beat frequency is Fh the maximum interfering amplitude will be pro- duced. The amplitude is {Q/P) Fb while the step interval is (3/(b - 1). To confine the interference to a half step (with 3 db margin) requires that /3/(6 - 1) ^ 2(Q/P) V2 Fb For Q/F = 0.707, ^^ 2(b - \)Fb ^* The general solution of the problem of frecjuency error produced by superimposing a sine wave on an unmodulated carrier is given in Appendix II. 524 BELL SYSTEM TECHNICAL JOURNAL 1 1 1 1 1 / ' 1 1 / Mh 7 / / or' "^1 n 7 or^ ' ill ^// / / / (0/ l\ / / / 1 / o'l ' 7/// / / ^ = 0.0 1 056 / 'i 'iiii / / ' 45 D B / / ^11 ^, ^11 i // / / / 1 1 ^1 1 y / J, / / 18.5 ih/ / / h // 27 assunu'S tluil llic peak intcrlVrencc is iiiar,L!;iiuil lor all hamiwiillhs (<^ dh down). 'riiel)an(l\vi(lthrati()istlicii2( 1 + ' ). It was i)rcvii)usly found thai in PCM-KM, with peak interference M db down, llie radio bandwidth nuist be {b + 1 )/''/,. Since llie radio bandwidth of PC'M-AM is 2F„, the bandwidth ratio is {b + l)/2. C^omparing these bandwidth ratios we see that the PPM bandwidth required to operate in the presence of marginal interference is nearly two times that required in PC^M-FM. Furthermore, this PPM bandwidth ratio applies to marginal fluctuation noise wdiereas in PCM-FM a more favorable result was obtained. Fig. 19— FDM-FM, Fluctu.atioiN Noise When a group of channels in frequency division is transmitted by fre- quency modulation, the addition of channel voltages is translated to an addition of instantaneous frequency shift. The non-simultaneous load advantage applicable to a multichannel amplifier for frequency divided channels thus becomes an advantage in reduction of total frequency swing as compared to the sum of the individual peak frequency swings of the channels. The numerical db increments versus number of channels listed in Table I should, how^ever, be modified for the following reason: The tluctuation noise spectrum in the output of an FM detector is not uniform with frequency, and hence the noise is unequally distributed among the channels. In order to obtain the same noise in all channels it is necessary to taper the signal levels in such a way that the full load frequency swing pro- duced by one channel is proportional to the frequency of the channel. The frequency swing corresponding to full load in the top channel is therefore a larger part of the maximum instantaneous swing required for the group than the swings corresponding to lower channels. The result is, in effect, phase modulation. The multiplex addition factors for tapered level channels have not been determined experimentally. We have assumed here a 3 db reduction in the power capacity values listed in Table I. These reduced values then give the incremental capacity referred to full load on the top channel. Curves are shown for 100, 500 and 1(X)0 channels. On account of the multiplex addition factor, it is not possible to obtain results for other numbers of channels from one curve by simply changing the frequency scale. The derivation of these curves is straightforward but leads to an expres- sion for the required bandwidth as a root of a cubic equation. As in the case of Fig. 16 we shall discuss the FDM-FM curves by checking them numerically. We have assumed that the channels are tapered in level and that we have, in fact, phase modulation with its consequent flat base- band noise distribution. To check the 60-db point on the 1000-channel 528 BELL SYSTEM TECHNICAL JOURNAL curve, \vc calculate that the uoise in llie entire baseband must be 43 db below the power in a sine wave whicli employs the full system load capac- ity. This ligure comes from reducing the 60 db full load channel ratio by 30 db because of the lOOO-fold greater baseband width and increasing it l)y the amount, 16 — 3 = 13 db, by which the full system load must ex- ceed full load in the top channel. Thus 60 - 30 + 13 = 43 db. An FM advantage of 31 db must be obtained to permit the marginal r-f signal- so 75 70 '!^ 65 m m O 60 5 50 < o ^7 35 < :? 30 / / / / y / / / / / / / / / / 1/ ^/ / / / / fl '< / / / f^ ¥ / / r / ^ ^/ / / / / / / / / / / / / 1 / 1 / 1 / 1 1 I 1 1 1 1 1 1 1 ^ 1 2 4 6 8 10 20 40 60 100 200 400 600 1000 RADIO SIGNAL BANDWIDTH IN MEGACYCLES PER SECOND Fig. 19 — FDM-FM; performance with respect to fluctuation noise. Relations between Ijandwidth and audio signal-to-noise ratio for marginal power; 4-kc channels. to-noise ratio of 12 db to satisfy the above requirement, (43 — 12 = 31). We get this advantage in part by phase modulation gain given by 20 log — — 6 db. This gain is referred to 100% modulation AM whose un- modulated carrier power is the same as the FM wave power. This means that the reference is a system in which the FDM baseband appears as upper and lower sidebands which, when demodulated, yield a baseband BAND WIDTH AND TRANSMISSION PEIUORMANCE 529 signal-to-noise ratio equal to the ratio of unmodulated carrier power to the noise power in the double width radio band. Since we keep the FM power marginal for all bandwidths an additional banflwidth imjirovcment » of 10 log -=r accrues. Substituting H = 92 mc, F,, = 4 mc, and li = 2Fb 92 — 8 = 84 mc, will show that the above gains total 31 db. / y^ / y y ^ ^ /" ^ r,n > Y y / -^ i<-> y -^ y ^ o^ y y / < y / /^ •^A Y y y y ^ y ^ ^ V y y y y y — ^ y y ^ y y /- y y y ^ ' / 1 -- / 1 _^ _j_ 1 1 1 1 1 1 1 _L_ _L 75 70 UJ 65 u 2 - 55 O < 50 a z UJ cr 40 LU LL S35 z T 30 O ^^25 < z O 20 in Q 15 Q D < 10 10 20 40 60 100 200 400 600 1000 2000 4000 10,000 RADIO SIGNAL BANDWIDTH IN MEGACYCLES PER SECOND Fig. 20 — FDM-FM; performance with respect to CW interference. Relations between Ixandwidth and audio signal-to-interference ratio for marginal ratio of FM wave to inter- ference; 4-kc channels. Fig. 20— FDM-FM, CW Ixterferknck The disturbance produced by CW is most readil}- e\-aluated when all channels are idle for then we have only the frequency error produced by a sine wave of relatively small amplitude superimposed on the steady sinusoi- dal carrier wave. To a first approximation (see Appendi.x II) the error has a frequency equal to the difTerence between the carrier and CW frequencies and an amplitude equal to this frequency difTerence multiplied by the ratio of the CW to the carrier amiilitude. The error thus increases linearly with the frequency of the channel in which it falls but, since the channel levels 530 BELL SYSTEM TECHNICAL JOURNAL are also tapered in the same way, the signal-to-interference ratio is inde- pendent of the frequency of the disturbing CW. Varying the CW frequency only changes the number of the channel into which the interference falls. Loading the channels distributes the interference over several channels instead of concentrating it in one, but we have plotted in Fig. 20 the more severe case in which all channels are idle. We have not undertaken to compute curves for similar system inter- ference in the case of FDM-FM, but estimates for two extreme conditions can be made. In the case of low index FM systems the carrier frequency component of the spectrum is not affected by the modulating signal and the FM wave is, in fact, like an AM wave with the carrier displaced 90 degrees in phase. A similar interfering FM wave combines with the wanted FM to produce frequency or amplitude variations and does this cyclically as the r-J phase between the systems varies. When the phases are appropriate for the production of frequency variation, crosstalk appears in the wanted reception at a level lower by the ratio of FM wave amplitude. Averaging over all r-/ phases should reduce the crosstalk by 3 db. The actual amount of interference received in a channel is less than would be predicted from replacement of the interfering FM wave by fluctuation noise of the same mean power spread over the r-f band, because the bulk of the interfering power is contained in the carrier component located at a frequency which does no harm. Increase of the frequency swing in both systems produces significant reduction in crosstalk when the carrier amplitude diminishes appreciably and important higher order sidebands appear, i.e. when the interfering system has its spectrum spread out more or less uniformly, like noise. Systems designed for wide swings under full load may, however, operate with only a few channels active; in such cases the low index situation may exist and the received interference will be down approximately by the ratio of the FM waves, without the benefit of FM advantage. While in this situation the bulk of the interfering power is again contained in the harmless carrier, the received interference is concentrated in a few channels and is greater than if the interfering wave power were spread, like noise, thinly over the r-f band, which in this case is many times wider than the band occupied by the low index signal. For such adverse loading conditions, the curve for similar system interference, while starting at the left above the corresj)onding noise curve of Fig. 19, may actually cross over and finally approach it from the lower side. In the case of systems of very wide swing such as are involved in Table II we regard the interfering system as equivalent, under all common load con- ditions, to noise spread uniformly over the bandwidth and having the same power as the interfering wave. The entry in Table II is obtained by reading BAND WIDTH AXD TKANSMISSIOy PERFORMANCE 531 the curve of Fig. 19 at 69 db audio signal-to-noise ratio which would be appropriate to yield 60 db when the "noise" is marginal at 3 db below the FM wave power instead of 12 db. A different procedure is required for the narrow band entries of Table I\'. Here the emphasis on conservation of bandwidth leads to a two-frequency repeater plan with tolerance of similar system interference 44 db down. A 60 db audio signal-to-inter- ference ratio can be met under these conditions with moderate swings for which the equivalent noise representation of the interference is not valid. Tlie result is considerably influenced by the channel loading and we have no impeccable method of calculating the necessary bandwidth. We estimate that a bandwidth of 22.5 mc, with jS = 14.5 mc, will satisfy the require- ments for all except unusually adverse loading conditions. III. B.A.ND Width .a.nd Power Tables The information contained in the curves of Fig. 9-20 has been used in preparing Tables II and III, which show what can be done with the various systems when bandwidth is used freely. The prime objective studied here is the conservation of peak transmitted power. In Table II the audio channel must meet message circuit requirements-^ while, in Table III, a much better grade of performance— more than sufficient for transmission of high fidelity musical programs— is stipulated. We have prepared Ta- ble III (as well as Table \) on the basis of replacing the 1000 4-kc. message channels of Table II with 250 16-kc. channels. Since we have available established load rating theory only for message circuits, we have omitted FDM and FDM-FM from Table III (and Table V) . The values of Table II are based on a nominal 60 db ratio of signal-to-noise, but it is assumed per- missible to meet this in the pulse systems by using 22 db of instantaneous companding so that only 38 db signal-to-noise ratio is actually required within the compandor. The PCM systems provide for a 39 db circuit within the compandor, corresponding to 6 binary digits, 3 quaternary digits, 2 octonary digits or one 64-ary digit. No allowance is made for the accu- mulation of quantizing noise arising when several PCM links are connected in tandem at voice frequency. In practice, 7 binary digits might be used. This would provide for several links and would permit slightly more com- panding. Table III assumes a 75 db signal-to-noise ratio and no compand- ing and is referred to here as a "program" circuit. We use such a high-grade circuit to illustrate more emphatically how the system preferences depend 25 We do not pretend to deal fully with the involved matter of system requirements distinguishing between kinds of noise and interference or crosstalk that appear in message channels. We merely assume that the power of the separate types of disturbances con- sidered must be individually 60 db below that of a full load test tone under the worst specified transmission condition. 532 BELL SYSTEM TECHNICAL JOURNAL a o ea Ni 30N3a3da31NI Ol m m ^V3d IVNIObVlN ' UJ n pr. — n n 2 5^ :; :•: ■n ■-.•; •/•■.■ OuJ < UJ "■ U-ki La-:^ HOC i^ 1 1 L o 1 f 1 O o o O D O o o o o o a^ 1- ,o<« O o O D o o5 1 1 1 1 L O 1 L o 1 1 O lO o o T\ O m o m Ol "^z ~ - o - O 6 ^^ n I So :; c 1 ;•: 55 QZ 1 1 1 1 L o 1 1 1 o o o o o o o o o o o Z~ o o o D o in O lO < o o o D J1 (M m 00 (M — * 10 2 Q. 10 CCI- UJI- •'.'•■. '.'•I'V^r ti tii o5 1 1 1 L o 1 1 1 o lO O o lO O lO O o lO o I '^z " -■ o -■ 6 -■ d (O ^^ UJ OJ I IT" __ u. (- '1* r< I"'*' %•' O 9o £j| p.-.V •■>"• a. UJ 5 55 QZ 1 1 1 1 L o 1 1 1 o o o o o o o o o o o z o D O lO o in < o o o D lO nj lO (M D CO (\J Z <£) vO n n to O o o tr 1- Ul H 6 d o o o5 1 1 1 1 L o 1 1 1 o vO o o o o lO o o m o Q-z ^ -■ o - 6 d I 1- •*••'.*/•"'.* :•■•"•'•'■•'■.■•■. 9u QZ •:*•■:•.••••.■•■•'.■ 1 1 1 1 L o r 1 1 o o o o o o o o o o o o o o 3 O lO O in o o D lO (^J in fM (D (\J •=■ CD 11 <0 CO CO o ro ro n (D 2 5 2 5 2 UJ < u. a u. 1- 1 1 1 2 5 2 5 5 « Q. Q. < o Q. Q. a u. BAXD WIDTH AXD TRANSMISSION PERFORMANCE 533 in (O in a 1 1 ^ n n 1 1 1 1 1 1 1 7 1 1 1 •}:; ■■::■■ 1 .-./ , 1 : , 1 O o o o o o o o o o o o o o o o O o O o o o o o o «o o ::{:• t i_ 1 L o 1 , 1 o o o >o o o in o o >n o in o o in o o in o o in o in (M in N in (M vn N in (M OJ — (M fU rj o o o o o o o o l_ i_ 1 1_ 1 r: 1 1 r 1 '^ M0 [i-}'^ I 1 1 L O 1 i_ o 1 O o o o o o o o o o o o o o o o o O o o o o o n o o in o o in o o in o in o o o in o o o in o o in o o - o o O o o o 6 6 o 6 o 6 o 6 o 6 O o 6 d 1_ 1 ^ r 1 1 1 1 '^ :^:|f-:^// y>> 1 1 1 L o 1 i_ o o o o o o o o. o o o o o o o o o o o o o o m o in o o o o o m o in in O) at 0) c 5> a> 0) n n n n ,. cr > c < tr >- t V z < (T cr z a. (T z < < r < o 1 z J O (- ■9 - ■" o o o ^ X 7- •'-/«•; *••'•*,•••' ONE E RGINAL BA 5 o ';;"..*;. •/,■ .*,•_•;•;.*. •/ *.'.' L o 1 1 1 1 1 o o o o o o o n o o o o o o < o o o o o o 5 o o d ^ o' "" o in ~~ (/) '-'•'. • '•'"',*'•'•'.*.'■' '.'l' ,' • .•".• cr 1- *.* 5$ 05 o O O O O o o o o o in t\J - OJ - lO C\J n I ^o Cl L^v: ii $5 Q Z L o 1 1 1 1 o O o o o o o o o o o o o o o o O O o o d o d -^ d >" 't in ^ en •.'.'.•. ' •'. in CTH '*'*•'•* z CL 5| o5 °-z L o 1 1 1 L o 1 O O o o o O o < c in (\J - I 2 mc, and indefinitely great signal power would be required as we approach this limit. In PAM-AM we have assumed that pulses in adjacent channels just touch, thereby setting the bandwidth at 32 mc. Smaller bandwidths could be used if the pulses were allowed to overlap. This would reduce the allowable duration of the channel gate and deprive the system of some of its tolerance to similar system interference as well as noise. The maximum pulse power required for 100% modulation is tabulated. If instantaneous sampling were used this would be 6 db above the unmodulated pulse power which is, in turn, 38 db above the mean total fluctuation power accumulated in a 32-mc band from 133 spans. We have reduced the value of power thus computed by 1.7 db to allow for a calculated improvement in signal-to-noise ratio obtainable by gating at the channel input with a time function of the same shape as the signal pulse. The FM systems listed are of two kinds. The first is a relatively narrow- band type in which advantages such as relative immunity to gain fluctua- tion and amplitude non-linearity are sought with small increment in band- width over AM. Since these objectives are not sufficient in themselves to fix the actual bandwidth needed, an arbitrary additional requirement has 2« We do not here entertain the idea of using certain exchange methods to permit use of less band width than the conventional minimum of 4 kc per channel, but rather to use modest amounts of additional band width. Appendix III discusses briefly a band reduc- tion principle. Table IV Minimum Band Widths and Corresponding Power Requirements for Message Type Circuits 133 30-mi spans, 1000 4-kc channels, 15 db NF. SYSTEM S/N IN DB BAND WIDTH IN MC POWER TOLERABLE INTERFERENCE RATIO IN DB SPAN LOSS IN DB CW SIMILAR SYSTEM 850 PPM-AM 38 100 r 200 [- •".'•• V 46 4|(') 75 50 0 - 100 0 1 '■'■'■'■r 850 PPM-FM 38 100 50 - 200 - 49 39(') 75 100 0 0 PAM-AM 38 300 200 44 40(^) 75 100 r - 50 0 - 100 0 - NARROW BAND 38 300 200 r 46 23(2) 75 100 r 5 u. 50 0 - 100 0 - ':'•'•'■ WIDE BAND 38 20 9(2) 75 50 - 100 - 0 0 1.2 PDM 60 100 - 200 - 76 60 75 50 0 - 100 0 - FDM-FM 60 100 50 0 I^^'f^l 200 100 0 I;r;^;;1 68 44 75 538 BA\D Ml DTI/ AXD TRAXSMISSION PERFORMANCE 539 Table TV— Concluded SYSTEM S/N IN DB BAND WIDTH IN MC POWER IM WATTS TOLERABLE INTERFERENCE RATIO IN DB SPAN LOSS IN DB CW SIMILAR SYSTEM Binary 39 100,- 200r- 9 9 85 ^:-}-X 50 - 100 - 2 < o 0 ••■■'•'•■': 0 0.25 QUATERNARY 39 too - 200 - 18.5 18.5 85 Q. 50 100 0 0 S 1.1 64-ARY 39 100 200 45 45 85 50 - 100 - m. 0 I;^;"l 0 QUATERNARY 39 150 100 9 9 85 - 200 - 50 - '.'•':'.*. 100 - 5 u. 2 0 0 0.076 OCTONARY 39 100 - 200 - 18.5 18.5 85 a 50 0 - 100 0 0.25 64-ARY 39 lOOr 50 - 0 m 200 c 100 0 1^"^ 45 45 85 REGENERATE AT EVERY REPEATER IN PCM (0 INTERFERING SYSTEM IDLE; ACTIVITY HAS SMALL EFFECT. (2) ASSUMING 12.5% CHANNEL ACTIVITY. Table V Minimum Band Widths and Corresponding Power Requirements tor Program Type Circuits 133 30-nii spans, 250 16-kc channels, 15 db NF. SYSTEM S/N IN DB BAND WIDTH POWER TOLERABLE INTERFERENCE RATIO IN DB SPAN LOSS IN DB IN MC CW SIMILAR SYSTEM 4.3 X 10® 200 - 10 - PPM-AM 75 100 0 F^'1 5 0 - 83 780' 75 200 100 4.: X 10® PPM-FM 75 100 0 \;:}}:i\ 5 0 - 86 76(^) 75 200 10 1.4 xl 36 PAM-AM 75 100 0 rrrji 5 0 - 81 77(1) 75 5 u. 1 < a NARROW BAND 75 200 100 0 l^;:l 10 5 0 1.4 X 1 36 83 60(2) 75 200 1 270 10 8.8 .•.••.• 1 WIDE BAND 75 100 0 - 5 0 - 25 9(2) 75 5 < 1 2 o Q. BINARY 75 200^ lOr- 9 9 85 100 0 - 5 0 0.5 200 10 QUATERNARY 75 100 5 18.5 18.5 85 0 0 IT^ lOr 200 p 5 1 o Q. QUATERNARY 75 100 0 - 5 0 0.15 9 9 85 OCTONARY 75 200 too r- 10 5 r 18.5 18.5 85 - 0 0 REGENERATE AT EVERY REPEATER IN PCM (1) INTERFERING SYSTEM IDLE; ACTIVITY HAS SMALL EFFECT. (2) ASSUMING 12.5% CHANNEL ACTIVITY. 540 BAND WIDTH AND TRANSMISSION PERFORMANCE 541 been imposed that tlie transmitted power in the FM system should be equal to that required in the corresponding AM system. The resulting values of bandwidth are found to be reasonable ones for low index FM. An exception is made in the case of FDM-FM systems of message type where it is found tliat a net economy of frequency occupancy can be obtained by increasing the frequency swing sufficiently to tolerate similar system inter- ference 44 db dow^n. This enables the two-frequency repeater plan of Fig. 4, as discussed in Section I, to be used and substantially reduces the frequency occupancy over that of a lower index FM system more vulnerable to antenna crosstalk and therefore requiring more frequency assignments. We have estimated that a radio signal bandwidth of 22.5 mc achieves the required 44 db tolerance. The second type of FM is a wdde-band system designed for specified tolerances of interference from similar systems. Data on this second type will be used later in our study of inter-route interference, where it will appear that ruggedness is a more important criterion of frequency occupancy than the minimum bandwidth needed for transmission. No curves have been furnished to determine the FDM entries, since there is no variation with radio bandwidth to consider. The band required is merely the number of channels multiplied by the width of a channel. The power required for message charmels is determined by calculating the amount of powder in one channel to give a 60 db margin over mean fluctuation noise power in a 4-kc band and applying the multiplex addition factor of Table I. Similar system interference is simply linear crosstalk and must be, we say, 60 db down. CW interference referred to maximum system power must be down an additional amount equal to the multiplex addition factor of Table I in order to meet 60 db suppression in the disturbed channel. Since the two-frequency plan of Fig. 4 does not suppress interference between the two directions of a single route by 60 db, we must use twice as many fre- quency assignments as there shown. This duplication will appear in Table VI. FDM is the only system of Table IV for which such duplication is necessary, since the others do not require more than 44 db suppression. In the program type systems of Table V, however, the first four listed would need duplicated frequency assignments. The PCM-AM systems of course do not use any smaller bandwidths than those given in Tables II and III and would, therefore, be expected to show disadvantageously in a bandwidth comparison with the other systems. On the other hand they make, relatively, a good showing in power requirements and in tolerance to CW and similar system interference. In the next section we shall show that economy of bandwidth may, in fact, be illusory because of the greater susceptibility to intra- and inter- system interference associated with narrow band methods. It is not the bandwidth actually needed for transmission that is important, but the 542 BELL SYSTEM TECHNICAL JOURNAL tightness with which the bands may be packed into the frequency range without mutual interference. Because of the ruggedness of low-base PCM, neighboring frequency bands can actually be allowed to overlap. Introduc- tion of the proper spacing factors for satisfactory separation in frequency between adjacent bands causes the PCM system to overtake the other methods in effective utilization of frequency space, especially when inter- secting routes are involved. The PCM-FM systems listed in Tables IV and V are of the second class listed above, in which equivalent ruggedness rather than equivalent power as compared to AM is the criterion. Thus, binary PCM-AM is compared with a PCM-FM system having the same 9-db tolerance of interference. The curves of Fig. 17 show that, with such a tolerance, the minimum PCM- FM bandwidth is secured when the base is either three or four. We choose the quaternary case here because the signal-to-noise ratios obtainable co- incide with those of the binary. Likewise, either octonary or hexary PCM-FM furnishes the optimum base for the 18.5 db tolerance possessed by quaternary PCM-AM and, of the two, octonary is more suitable for our tabu- lation. In determining the power required to override fluctuation noise in a PCM-FM system designed for a specified tolerance of similar system inter- ference, we must make sure that both the limiter and sheer are protected against breaking. The values of repeater power capacity shown in Tables IV and V will satisfy the noise requirements on a 133-span non-regenerated circuit with 75 db loss on all spans. For spans of 60 db free space loss the tabulated power thus provides for 15 db fades simultaneously on all spans, or for 13 db simultaneous fades of 25 db, or for a single fade of 36 db. PCM systems employing regeneration on every span must be powered for the deepest fade that is likely to be encountered. We have arbitrarily taken this to be 25 db making the span loss 85 db. This is probably not a sufficient allowance for some situations but will serve for illustrative purposes. If regeneration were not practiced the power would be 25 — 15 = 10 db lower from the fading allowance standpoint but would have to be increased 10 log 133 = 21 db for noise accumulation, so regeneration results in a power saving of 1 1 db. If, with regeneration, we were to protect each span against the deepest single fade (36 db) permitted by the power provided for non-regenerative operation, the power advantage of regeneration would disappear (36 — 15 — 21 = Odb). In general, when the power without regeneration protects against simul- taneous fades upwards of just a few db there is little or no power advantage in regeneration if we then protect each span against the deepest single fade permitted when regeneration is not practiced. This is true even for large numbers of spans. There remain, however, important advantages for BAND WIDTH AND TRANSMISSION PERFORMANCE 543 regeneration in preventing accumulation of disturbances that are not much affected by the distribution of fading. 1\'. Frequency Occupancy Tables For Radio Relay The frequency space occupancy for a single two-way route is, according to principles laid down in the introduction, a frequency block 2U times the signal bandwidth.-^ Our problem, as stated in the introduction, is to ex- amine the situations arising when a number of 1000-channel routes converge toward a terminal city, assuming all of the routes to be of the same kind. We will determine the number of times the above frequency blocks must be repeated in the spectrum in order to keep interference within tolerable bounds. The sum of these blocks then really defines the frequency occu- pancy and determines the space which must be allocated or, conversely, determines the number of routes a given allocation will accommodate. We will use the tolerable ratios of similar system interference taken from Tables IV and V, together with appropriate antenna directivity, to deter- mine the number of these blocks. Antenna Characteristics The directional discrimination afforded by the antennas is obviously an important factor in frequency economy. For our present study, we employ an antenna having a directional pattern slightly superior to that of the 4000-mc shielded lens antenna in use on the New York-Boston radio relay circuit. Figure 21 shows the assumed directional characteristic omitting "nulls" between the minor lobes. Of importance also are the nearby dis- crimination characteristics of the antennas as given in Fig. 4. The situations arising at a point where a number of routes converge (or cross) or where a route is equipped with a spur connection are variations of that occurring at a single repeater point. In fact, the situation in which two routes converge from approximately opposite directions occurs at every repeater point in a straight route, while a repeater point at which the route bends sharply is like a terminal point at which two routes converge at a small angle. The crosstalk in our assumed two-frequency long distance repeater system has been estimated in Section I (under "The Radio Repeater") and was found equivalent to a single source of similar system interference 44 db down. A system which possesses just enough tolerance to withstand the accumulated crosstalk on a long straight repeater system is not capable of meeting another such system at an angle unless additional frequencies are 2' In the case of verv tender systems, such as FDM, the factor 2U is replaced by 4U be- cause a four-frequency plan is needed for a two-way repeater. U is the band spacing factor discussed under "The Radio Repeater." 544 BELL SYSTEM TECHNICAL JOURNAL invoked. The FDM-FM system of "minimum band width" listed in Ta- ble IV is intended to possess this 44 db tolerance. To illustrate the require- ment of new frequencies let us take the case of several FDM-FM routes 20 10 0 10 20 110 rt; ^7^ 30 1 __ /20 ^ !^ Li 40 \ - 50 60 \ UJ j \ h60 / \uij / Ji^70 / R ?^ 70 \\ 80 / / (/x^ ^A 80 yV^ /:5^ \_t2\ 90 100 \ ~ j too ^ ri"^ '/\ ^ V>>rM<(\ ^ jH~~j 110 IT^\\ "^^ aR/ 120 ■h-ZL 70 LJ --jL 60 A- w \>\ 130 140 150 160 170 180 170 160 150 140 130 Fig. 21 — Assumed directional selectivity of 10 ft. x 10 ft. antenna at 4000 mc. 5 6 Fig. 22 — Simplified route pattern for studj' of selectivity required in congested localities. converging in the manner suggested by any of the diagrams of Fig. 22. A critical situation occurs in regard to the frequencies used for receiving at the point toward which the systems converge. If the same receiving BAND WIDTH AND TRANSMISSION PERFORMANCE 545 frequency were to be used on two or more routes, receiving directional dis- crimination amounting to 75 db would have to be secured: 1. Required interference ratio 44 db from Table IV FDM-FM 2. Allowance for repeater crosstalk 1 (51 db down) 3. Differential fading allowance^* 30 75 db The repeater crosstalk is here taken to be equivalent to one source 51 db down which is the value corresponding to no differential fading on adjacent spans as calculated in Section I. It will be remembered that allowance was made for a single differential fade of 30 db occurring somewhere along the route. Here we assume that this differential fade may occur between two of the converging paths and we demand that the receiving directional dis- crimination shall protect the system against such an occurrence. In this case the required directional discrimination turns out to be equal to the 75 db front-back ratio from which the 44 db figure was obtained. This is manifestly impossible with the assumed directivity characteristic^* and the angles involved. Therefore, different receiving frequencies are required on each route. These same frequencies may be used for transmitting at the junction, provided the disposition of terminals is such as to provide enough directional discrimination and physical separation to permit operation at the low received level in the face of the high transmitted level on the same frequency. The interference path loss plus antenna discrimination must be, for the case involving the longest span: 1. Required interference ratio 44 db (FDM-FM, Table IV) 2. Allowance for repeater crosstalk 1 3. Free space span loss 60 4. Fading allowance 25 130 db We continue our discussion of the converging routes of Fig. 22 by assum ing that: 1. Conditions encountered elsewhere on the routes do not restrict the freedom to switch the frequencies among the routes. 2. The disposition of terminals at the junction is such that inter- terminal interference is not a controlling factor. Under the above assumptions the directional discrimination of the terminal ^^ This differential fading allowance corresponds to a fade of 25 db below free space on one route and a 5 db increase over free space on the other. ^' The use of perpendicular polarizations cannot, we assume, be counted on to give further discrimination when the directional discrimination is already 40 or more decibels. 1 2 3 4 5 A B C A B B C A B C (C A B C A) 546 BELL SYSTEM TECHNICAL JOURNAL antennas alone determines the number of different frequencies. The same receiving frequency may be used at the junction on two routes separated by an angle sufficient to yield the required antenna discrimination. All inter- vening routes must employ different frequencies. The frequencies so deter- mined may be used for transmitting from the junction if they are staggered with respect to the receiving frequencies. Take, for instance, the five-route plan shown in Fig. 22. Suppose the directional discrimination needs to be 60 db for a particular system. The directional pattern shows that this requirement is met at 85 degrees. Thus, routes 1 and 4 may use frequency A, say. Routes 2 and 3 then must use different frequencies, B and C. Thus, we have: Route Trans. Freq. Rec. Freq. or While the treatment of the route congestion problem outlined above is oversimplified it enables us to make a broad survey having some significance. Table \T for 1000 4-kc message channels and Table \TI for 250 16-kc "program" channels were derived on the above basis. The decibel figures at the head of each column are the allowable interference ratios from Tables IV and V increased by 30 db for differential fading.^" A single source of mterference of the values given in the table is supposed to degrade the circuit to the minimum requirements for a long circuit. In regenerative PCM there is no accumulation of degradation due to interference occurring on various spans. In non-quantized systems such degradations are cumulative. However, when protection to the above values is provided, with no allowance of 30 db for differential fading of the desired and interfering signals, the oc- currence of simultaneous additional degradations is extremely unlikely. Pro- tection against this severe fading at one point alternately protects against the simultaneous occurrence of several less severe fades. The values of repeater power capacity shown in the table will satisfy the noise requirements on a 133-span transcontinental nonregenerated circuit with 15 db fades simultaneously on all spans. This is equivalent to pro- viding for 13 simultaneous fades of 25 db or, statistically, for the fading that is not likely to be exceeded except during a small fraction of the time. The PCM systems employing regeneration are powered for 25 db fades on any or all spans.'^ The free space span loss is 60 db. In computing the frequency occupancy we take cognizance of the fact ^•^ The FDM-FM entrj' provides 1 db additional allowance for repeater crosstalk as mentioned before. Repeater crosstalk is negligible in the other systems. ^' No such distinction was made in Tables II and III. There, provision was made for 75 db span loss in all cases. Truk Frequency Otc Table VI OF Various Message Grade Radio Relay System 133 30-rai spans, 1000 4-kc channels, 15 db NF t Congested Routes 5 1 J5 SYSTEM PCM-AM g 5 5 i 2 5° "w PCM-AM PCM-FM Z ^ ^5 (39 db) "uITobT"' ag ZQ cco] zs" i 7600 irrm 2 90" 2000 1000 0 7600 ETrm •X\i !'.'•'•'• 3 45° 760C FTm 2000 r- •>;'.•■ 1000 0 - 4 36° f:?:?! 760C 1 1000 0 - 5 30° 4160 -T^/.,^ l:T^^^ rrrrrf :>;v:- ■0 2000 1000 0 - =:S 6 25.7° 3840 4990 307C 3450 7600 ITTTT] rrm 2000 1000 0 " .v.* . POWER REQUIRED 2^ IN WATTS 1 - 0 4.3x10^ 4.3 X 10^ 1.4 XlO® 1.4X10® 19 iSl 548 BAND WIDTH AND TRANSMISSION PERFORMANCE 549 Table VII Concluded q9 <2 1000 0 SYSTEM PCM-FM 100 0 1000 0 1000 0 2000r 1000 0 ct < — zen <2 < n O -t O-" r?:?! p:^ 2*^ SI 2 0 r 1^- ■::! PCM -AM (/) BINARY QUATERNARY i^ (39 DB) (48.5 DB) 9^ ZUJ ^ SINGLE REQUENCY 2y UJ -1 cr < -REQUENCY AL POLARIZ. N SINGLE REQUENCY REOUENCIES AL POLARIZ. FREQUENCY AL POLARIZ. u. a. D D u. u. D J Q ~o Q -Q 200 r 100 0 fiS •:"•■• •/■•■• •^S F^ 200 100 0 ■■■:■ '•'.•'•'■'■ i ii: FA^>M 200 100 •;.;/: 1 0 mi •'.'.'.' v^-V 200p 100 0 '■':::• :g: m •p- 200 100 r p^ 0 v.':':" •ij.\ m .••:•;•;. ^:"-;; 200 100 [- TV":: ■'•'.'•': ■■!•■.: 0 >:•• m 550 BELL SYSTEM TECHNICAL JOURNAL 1 to -I o< tr I O LU iT) CL 9 >> O tJ ^ S o a (U w H > > tn '^ m o (U (H >- g o <: _Q wO -o l^f^ 1> 2^ O" c« W -ID Iz; -JS pq t« t« o 2 Q ^ 5 U. 0^ H 5 cvj iZ *T U 55 J I UJ -i ■^ UJ BAND WIDTH AND TRANSMISSION PERFORMANCE 551 These factors were multiplied by the product of bandwidth and number of frequencies to obtain the dotted bars in Table \T. In regard to tlic "program grade" of circuit we must be more liberal in our allowance for guard space. Our estimates for the band spacing factor are: System of Table VII Factor U PPM-AM 4 PPM-FM 4 PAM-AM 4 PAM-FM (narrow band) 4 PAM-FM (wide band) 3 PCM-AM (quaternary) 0.9 PCM- AM (binary) 0.75 PCM-FM (octonary) 1 • 5 PCM-FM (quaternary) 1 . 5 These factors were used to compute the dotted bars in Table VII. If transmission on two polarizations can be accomplished with mutual cross-fire suppressed to a sufficient degree, half of the channels could be transmitted by each polarization, on the same frequency, thus halving the frequency occupancy. A probably unattainable cross-fire ratio seems neces- sary to meet the requirements in the non-regenerative systems, if we remem- ber that the interference produced by cross-fire accumulates from span to span; but a suppression likely to be attainable, of the order of 15-20 db, makes this frequency saving feasible in the rugged systems such as binary PCM-AM or PCM-FM. The tables show entries for binary and quaternary PCM-AM, assuming dual polarization transmission. If antennas could be improved to insure nearby discrimination ratios adequate to allow use of the same frequency in and out and west and east, the single-route occupancy would be halved again; with such a one-frequency repeater plan the occupancy in a congested area is not, however, always halved. Whenever the frequency requirements, as determined by the terminal antenna directivity, result in two or more frequencies. A, B, C . . . etc., there is no saving accruing from a one-frequency repeater plan, because two-frequency routes can be accommodated with no additional frequencies by suitably switching frequencies. It is only in the case of a system so rugged that the terminal antenna directivity permits a single frequency, A, to be used that the occupancy is reduced and it is then halved. Witli PCM of low base this is a possibility and the tables include entries for this case. As to achieving antenna characteristics suitable for one-frequency opera- tion, it may be noted that reflection from a heavy rainfall in front of the 552 BELL SYSTEM TECHNICAL JOURNAL antennas limits the attainable side-to-side ratio.^^ Reflection from aircraft may also impose a practical limitation. Spacing the antennas laterally (on two towers) would achieve freedom from these limitations. Another way of coping with the antenna discrunination obstacle is to use short spans in congested areas. This reduces the discrimination requirements particularly because fading is reduced by shortening the spans. Conclusions as to Radio Of the systems included in Table VI we find that, for six routes, binary PCM-AM, even without the potential frequency economy of dual polariza- tion and/or single-frequency repeaters, has come close to being the most economical of frequency space; quaternary PCM-AM shows a slight advan- tage (which would be lost if the route spacing were less than fifteen degrees). Even without dual polarization or single-frequency repeaters, the binary PCM-AM occupancy is less, for more than 3 routes, than the occupancy required by FDM whose hand width is 4 kc per channel. There is here an excellent illustration of the possibility of a net saving in frequency space through the use of tough wide-band systems. The power requirements also favor the low-base PCM systems. It should be noted, in particular, that the linearity requirements with FDM demand that the tabulated power of 80 watts be a very light load on the repeaters. Inspection of Table VII brings out the effectiveness of the coding prin- ciple if very high-grade channels are required. Only with PCM (of low base, as shown) are the occupancy and power requirements both within the practical reabn. The non-PCM methods that achieve small occupancy, comparable with that of low-base PCM, all require colossal amounts of power. When the power requirement is reduced and the ruggedness in- creased by use of band width, the occupancy becomes^ in turn, colossal. This is illustrated by the two entries for PAM-FM. As route congestion increases without limit, any type of system that permits exchange between bandwidth and ruggedness will always achieve the minimum occupancy when bandwidth has been used to secure the degree of ruggedness that avoids multiplying the frequency assignments. Our studies have shown that, with the assumptions made, this result is valid for channels of message grade when the congestion has reached a degree that is by no means fantastic. We have accordingly prepared Table VIII in which the dotted bars show the bandwidths (taken from Fig. 9-19) of the various systems when their interference tolerances are alike and have values of 18.5, 9, and 3 db.'^ While these systems, having the same tolerance, all ^ Measurements made at the BTL radio laboratory at Holmdel, N. J. indicate that this Umit to side-to-side ratio is of the order of 85 to 90 db. ^ The AM pulse systems are here assumed to achieve the 6 db increase in tolerance by suppressing the carrier. BAND WIDTH AND TRANSMISSION PERFORMANCE 553 fare alike in respect to frequency requirements imposed by antenna direc- tivity, the bandwidth figures do not adequately reflect the merits of the systems. This is because the band-spacing factors are different and, in addition, only the regenerative systems can be expected to achieve the halving of occupancy accruing from dual polarization and from one-fre- quency routes. The crosshatched bars of Table VHI include the effect of multiplying by the estimated band-spacing factors shown beneath the bars. These band spacing factors are in some cases smaller than those previously tabulated for the less rugged systems of Tables IV and V. Only the PCM methods are shown for the case of very high-grade channels, since the non- PCM methods are so strikingly less effective here. These conclusions depend for validity on the assumptions made and par- ticularly on those concerning antermas, route disposition and fading, and apply when the converging systems are of the same kind. In a real situation, departures from the assumed conditions could markedly affect the conclu- sions. For instance, the meritorious showing of PCM in respect to efl&cient utilization of frequency space in the face of route congestion depends heavily on the assumption that all routes in the occupied space employ PCM. Any routes employing a modulation method that is highly vulnerable to interference like some of the narrower bandwidth methods would have to employ higher power to operate in the face of interference from the PCM routes. This higher power, concentrated in a narrower band, could destroy the PCM routes. In some cases it would obviously be impossible to assign values of power which permit the two kinds of routes to share the same frequency band. Our calculations should be taken to illustrate the factors involved and the philosophy by which such problems may be approached rather than to find an unequivocally best system. V. More About The Non-Simultaneous Load Advantage The transmission advantage enjoyed by multiplexing many single side- band telephone channels in frequency division, discussed in the introduction, stems from several factors: 1. During the busiest period, only a small percentage (of the order of 12 to 15%) of the channels are actually transmitting speech (''talk spurts") at one time, on the average. 2. There are only a few loud talkers; the remaining ones range downward to a volume 35 to 40 db lower. 3. In the addition of the sideband voltages representing the talkers actu- ally producing talk spurts, only a fraction of the grand maximum occurs often enough to be significant. With frequency division all of these factors jointly contribute in a natural 554 BELL SYSTEM TECHNICAL JOURNAL and automatic manner to the low peak load ratings given in Table I. In time division, complicated instrumentation is needed to obtain such a low load rating (in time, now, not power capacity) and the saving is in bandwidth (time). Savings accruing from item (1) above are theoretically obtainable in all time-division systems (and, in fact, in nonmultiplexed multipair cable transmission systems) by having automatic devices which skip the channels that are momentarily inactive and which advise the receiver of the skipping. It is possible also to benefit from items (2) and (3) above in systems which transmit a time interval to represent an amplitude. The amplitudes may be sent as absolute magnitudes together with a polarity indication. If this is done the channel time allotments actually required in a given multiplex frame appear piled up end to end, and many more channels can be handled than if provision were made for full amplitude on all. PPM is one such system, and PCM is another if the code symbols containing fewest digits are used to represent the smallest absolute magnitudes. The use of instantaneous companding, which tends to make all talkers contribute equally to the system load, reduces the advantage represented by (2) above, but does not basically affect (1) which represents a substantial part of the total multiplex advantage. It is illuminating to compute the performance of a pulse length modula- tion system (PLM) employing the elastic time allotment and assuming that the load ratings of Table I apply. We imagine a system working on the principles illustrated in Fig. 23. There we assign a time T{= 2/B) to each inactive channel. Active channels whose absolute amplitudes are described by /, are assigned t -^ ZT and those that are negative are preceded by a 2T pulse to designate that they are negative. If the interference is no greater than marginal (9 db down) the receiver can distinguish between (a), the T intervals which count off the channels that are skipped and (b) , the 7.T polarity indications and (c), the 2)T minimum signal intervals. The frame time of 125 microseconds must include the sum of these intervals plus A7o where /o is the time shift for a full-load tone in a single channel which gives the required signal-to-noise ratio for the bandwidth B{ = 2/T). The load rating factor is K, expressed as an amplitude ratio. The rela- tions used to plot the two curves of Fig. 23 are shown in the insert. Little or no instantaneous companding could be used to advantage so that a signal-to-interference ratio of 50 to 60 db would be required and for 1000 channels the bandwidth would be between 30 and 50 mc, which is some two or three-fold less than in binary PCM- AM, both systems being equally tolerant to a single source of CW interference. The elastic principle could presumably be applied to PCM also to achieve a several-fold bandwidth reduction, but no experience has been obtained with any of these elastic systems. While this paper has avoided for the most part questions of instru- BAND WIDTH AND TRAXSMISSIOX PERFORM A\XE 555 mentation, it should be pointed out that the ehistic schemes tend to become complex apparatus-wise. If one chooses to discount this on the grounds 4 6 8 10 20 40 60 80 100 20C RADIO SIGNAL BANDWIDTH IN MEGACYCLES PER SECOND 400 r I ' I ^1 FRAME TIME = 125//S- I 8(-) 5|6I 7(+) X-T" l|2]3l 4(+) rur\ rO" _ — , ;T1t1t; •t4 + 3T iT;T; t7 + 3T ; 2T ; ts + ^T l tg + ST , 21 , N = NUMBER OF CHANNELS a - NUMBER OF ACTIVE CHANNELS IIO(-) to = '^^-i ^^17 TT tc SLICER ADVANTAGE = 20 LOGio ^ ^-3DB I :RF PULSE TO INTERFERENCE RATIO + 20 LOG ,o -3- rr tc -3 CALCULATED for: RFRATI0=9DB (MARGINAL) N = 1000 K - 6.3 (16 DB) N = 100 K = 2.83 (9DB) A- 1 N 8 Fig. 23— Theoretical possibilities of exploiting non-simultaneous load advantage by an elastic PLM-AM system. Ihat future developments may resolve the complexity, there remains the objection that any system designed to take advantage of the multiplex load rating counts heavily on being used almost exclusively for conversational 556 BELL SYSTEM TECHNICAL JOURNAL speech under present operating procedures. The extensive use of telephone channels for nontelephone purposes is thus curtailed. VI. Overload Distortion and Noise Threshold In designing a microwave system for a large number of channels the power required to override noise may exceed the power capacity of available ampli- fiers. Also, the bandwidth may exceed the limit imposed by microwave transmission phenomena or circuit techniques. In either case, the remedy is to divide the channels into several groups of fewer channels and trans- mit the groups in adjacent narrower bands spaced by the proper factor U, and separable with filters for individual amplification, reshaping or regenera- tion. The power requirement falls ofif linearly with bandwidth. The filter problem for AM pulse transmission is considered in Section \TI. The total frequency occupancy is no greater for this division since the same percentage "guard band" is involved if, in both cases, the neighboring, foreign signals are of the same kind as the wanted signals. In case the neighboring signals are of a different kind, the multiple band arrangement is in fact likely to represent a smaller occupancy because the occupancy is in general more sharply defined when made up of several narrower bands. When considering a multiple group arrangement, it may be economical to provide for a substantial amount of common amplification prior to separa- tion into the several bands which receive individual treatment. The non- linearity of the common amplifier then sets a limit to the common amplifica- tion. Experiments bearing on this overload limit were made with the PCM equipment described by Meacham and Peterson.^- Two- and eight-fre- quency groups were employed and the amplifier load was increased until the effects of distortion began to appear. The distortion was measured in terms of the maximum amount of CW interference which, when added to the amplifier output, resulted in no audible effect in the PCM channels. The right-hand part of Fig. 24 plots the results. For eight bands (six of which were not pulsed but were left on as unmodulated carriers) it is seen that the margin provided against CW interference begins to shrink rapidly when the single group load is 20 db below the output at which 1 db compres- sion occurs. The margin is completely used up (the channels begin to show noise) when the load is 13 db higher. The left-hand part of Fig. 24 plots the manner in which the low level limitation (noise) was found to appear. Margin against CW interference shows a reduction for a pulse-to-noise ratio of 28 db and is completely used up at a ratio of 18 db. The overload occurred in the 4000-mc power amplifier associated with the repeater, and the noise originated in the receiver. In non-reshaping amplitude-modulated systems, the effect of compression " Loc. cit. BAND WIDTH AND TRANSMISSION PERFORMANCE 557 occurring in the repeaters is cumulative. In microwave repeaters second- order distortion products fall outside the band and third-order distortion is likely to be predominant. We assume in what follows that the distortion arises solely from a cubic term. \\'hen the low-level gains of the repeaters are maintained equal to the preceding span losses, it can be shown that the single-frequency compression characteristic at the end of n spans is approxi- O[f,UJ40 ,9 30 \^0 10 ■4 2 0 / / / / / / / ^ IDBj /' y -2 -4 -6 -8 10 /X /^ /^ / / / *-Po -10 -8 -6-4-2 0 2 •" INPUT TO RADIO REPEATER IN DECIBELS NOISE THRESHOLD CHARACTERISTIC __ / ^'^ / / / OVERLOAD CEILING CHARACTERISTIC .^^^ , ^ X 2 pulsedX GROUPS \ ONLY \ DECREASING 2 PULSED GROUPsV PLUS 6 UNPULSED \^ \ POWER \ Po— 15 40 30 25 20 15 10 5 INPUT OF ONE GROUP IN DECIBELS (BELOW Pq) 20 25 30 35 RATIO OF PULSE POWER TO NOISE POWER IN RECEIVING FILTER (DECIBELS) Fig. 24 — Noise threshold and overload ceiling in frequency divided PCM groups, mately the same as for one span but occurs at a power level 10 log n db lower. This approximation becomes more exact as the over-all distortion involved becomes less (as by lower input power). Fig. 25 shows a third order compression curve for one span and the resulting curves, obtained graphically, for 2, 4 and 10 spans. Examination of these curves shows that the curves are substantially the same as for the single span but displaced, 3, 6 and 10 db respectively. This is illustrated by line A, which intersects all of the curves at the same compression value (1.7 db). The points 558 BELL SYSTEM TECHNICAL JOURNAL marking the intersections are seen to be displaced from the intersection with the cur\^e for one span by approximately 3, 6 and 10 db. If the phase of the repeaters is as linear as it must be in pulse systems, this single frequency characteristic can be applied for the entire signal band as if it resulted from a single source of third-order distortion. The effect of this distortion is / / / yr/ ^^ 'A ^ A^^ ^ 10 - -- NOOF SPANS LOW-LEVEL GAIN = SPAN LOSS ^ y ^ / / 20 16 14 12 10 8 6 4 INPUT IN DECIBELS DOWN Fig. 25 — Overload characteristics of multi-repeater systems. serious in multiband PCM repeaters, as illustrated by the measurements on Fig. 24, but is, generally speaking, less important in single band pulse repeaters. For instance^* PPM-AM pulses and binary PCM-AM pulses might operate on the flat part of curve 10 (Fig. 25). With PCM of higher ''' On the grounds that pulse slicers themselves include the compression function to a high degree, one might not see the harm of compression in repeaters. If all of the com- pression occurred after the noise had been acquired there would be no fundamental com- pression penalty in slicing pulse systems. The penalty comes about because as the pulses progress from span to span they shrink and become more vulnerable to noise. BAM) W IDTII AM) TRAXSM I.SSIOX PEKIVRM AXCE 559 base as well as with PAM, llic repeater loading would have to be sharply reduced, however. More power on all spans could be obtained by making the repeater gain greater than the span loss. This very quickly defeats its jjurpose, however, because the excess low-level gain raises low-level noise between pulses to a high level status as it progresses from span to span. Reshaping of AM pulses (and of course regeneration in PCM-AM) at all repeaters avoids the cumulative effect of compression by permittmg the repeater gain to be greater than the span loss by the amount of compression on one span. When a signal is transmitted by FM, the phase curve of the transmission circuit plays a role somewhat analogous to the amplitude characteristic of an AM system. The correspondence is not complete, however, for we find that modulation products arising from even-order phase distortion as well as from odd fall in the signal band even though the FM band is located in a frequency range ver>' high compared with the baseband width. For ampli- tude modulated signals in the baseband, we can replace the FM phase distortion effects by an equivalent non-linear baseband amplifier charac- teristic which has the same shape with respect to zero voltage input as the phase characteristic has with respect to the midband frequency of the FM range. If the distortion is small, the square and cube law approximations obtained by expanding the phase-shift function about the mid frequency may be applied as in conventional multichannel cross-modulation theory.*^ We shall not here attempt to discuss the accumulation of phase distortion in a multi-repeater FM system. VH. Pulses, Spectra, and Filters In this section, we will consider: (1) pulse shapes in relation to the par- ticular pulse modulation method employing them, (2) the shaping filters by which they may be obtained and (3) the transmitting and receiving filters employed in systems comprising a multiplicity of adjacent frequency bands each carr>'ing pulse signals. Column A of Fig. 26 shows various pulse shapes which can be approxi- mated (with the exception of shapes 8 and 9) by fairly simple circuits, both in the baseband and radio spectrum. Pulse 1 is an "unshaped" rectangular pulse. A good approximation to it can be obtained in wide-band circuits accommodating the extensive spectrum it possesses, i.e., in circuits having rise and decay times short compared with the duration To . Such a pulse when transmitted through Gaussian filters of the various widths shown in ^ W. R. Bennett, "Cross-Modulation in Multichannel Amplifiers" Bell System Technical Journal, Vol. 19, pp. 587-610, October, 1940. 560 BELL SYSTEM TECHNICAL JOURNAL column C emerges with smooth transitions as shown m 2, 3 and 4. These pulses rise and fall in a nearly sinusoidal manner. The width between half- amplitude points is To . Shortening the rectangular pulse ("curbing") and narrowing the shaping filter can be made to result in pulses 5 and 6 which have the same width between, say, 3% points (at /i) as pulse 4. (A) PULSE (B) SPECTRUM (C) FILTER -To 0 To 2To TIME— *• Fig. 26 — Typical pulses and their spectra. Pulses 5 and 6 are then shorter than 4 between half-amplitude points. If the half-amplitude width is made the same as in pulse 4 the width between lower amplitude points is greater than in pulse 4. This is illustrated in pulse 7. Gaussian filters^^ as defined here are naturally linear phase networks * Gaussian filters are networks whose transfer admittance follows the error law as a function of frequency. A decibel plot of a bandpass Gaussian filter is accordingly a para- bola in shape. BAND WIDTH AXD TRANS.UISSION PERFORMANCE 561 and we have assumed linear phase in computing pulses 2 to 7. A good approximation to the Gaussian filter can be obtained both as to phase and amplitude with a number of tuned circuits in tandem, coupled through buffers. A fair approximation can also be obtained by combining a 3- or 4-section maximally flat filter" with a tuned circuit through a buffer. Rectangular or near-rectangular shaping filters produce pulses with over- shoot as shown by pulses 8 to 1 1. The filter corresponding to pulses 8 and 9 is assumed to have rectangular shape and linear phase. Filters of this sort have no simple approximation in practice and are included for comparison with filters 10 and 11 which are made up of simple maximally flat networks. In pulses 9, 10 and 11 the "unshaped pulse" is assumed to be very narrow and of ampHtude sufficient to yield pulses of the heights shown. Let us now regard these pulses as received pulses and compare them in respect to shape for use in various kinds of pulse systems. PPM. In PPM the pulses may occupy any time position in the assigned inter\'al and so the tails of pulses 8 to 11 may "crosstalk" into time assigned to an adjacent channel. To allow guard tune for the train of tails or to design for satisfactory operation in the presence of the tails is uneconomical of frequency space. It follows that pulses which are more definitely bounded in time such as those obtained with Gaussian filters are more desirable and likely to be more economical of frequency space in general despite their wider spectrum. In PPM where the trailing (or leading) edge of a pulse is used to convey the information a flat top pulse such as pulse 2 is no better than one in which the flat portion is absent and the two transitions brought together.^* The latter pulse would, in fact, be superior since more time would then be avail- able for additional channels or for greater swing. We are thus led to conclude that one of the pulses in the 4 to 6 group is the preferred shape for PPM. We chose pulse 4 in our illustrative calcula- tions and defined bandwidth as 2/To , but pulses 5 or 6 would have given substantially the same results. PAM. In PAM the pulses occur at standardized, regular times so that if pulse 9 were used the accompanying tails, which disappear completely at instants To, 2To , etc., from the pulse peak, need not theoretically pro- duce crosstalk between channels if the channels are spaced To and the pulse ampHtudes are measured instantaneously at the time the nuUs occur. As a practical matter both the precise pulse shape and the instantaneous measure- " W. W. Mumford, "Maximallv-Flat Filters in Wave Guide," Bell Sys. Tech. //., Vol. 27, October, 1948, pp. 684-713. ' . 38 Such a pulse would look like pulse 4 if the latter were shrunk to occupy 0.6 of the time shown in the plot. The spectrum would accordingly be that of pulse 4 expanded by the factor 1.7 but would not include more significant band width than is necessary to form pulse 2 as shown. This deduction follows from the fact that the rise time of pulse 2 is the same with or without the flat top. 562 BELL SYSTEM TECHNICAL JOURNAL ments at precise instants are probably not realizable to a degree which would keep the crosstalk within tolerable limits, so that one of the smooth pulse shapes is preferred. Pulse 4 with a spacing of To is feasible from the sampling precision point of view but a spacing of 2 To provides margin against crosstalk arising from small imperfections in any realizable approxi- mation to the theoretical pulse. It is to be noted that, if an instantaneous sample is taken of a PAM pulse, the measured magnitude is affected directly by the instantaneous value of noise present in the entire band occupied by the pulse. No fre- quency selectivity can be applied afterward to remove the influence of any part of the noise band because the error, even though caused by wide-band components, is exactly the same as could have been produced by a uniquely determined wave wholly confined to the signal band itself. The best signal-to-noise ratio obtainable with instantaneous sampling is that asso- ciated with minimum bandwidth for the pulse (i.e., pulse 9) and the corre- sponding maximum stringency of synchronization requirements on the sampling and pulse distortion. The same signal-to-noise ratio can, however, be approached with a wider band provided that we allow a finite segment of the received pulse to enter the channel filter. An averaging out of higher- frequency disturbances produced by wide-band noise is thus attained. PCM. In PCM a short sample taken near the center of a pulse serves to determine correctly the presence or absence of a pulse even in the presence of interference at or near the breaking point of the sheer. Thus, pulse 4 may be used with a spacing of To , and if a gate pulse 25% of To is used, it need not be aligned with an inordinate precision to obtain good operation.^^ Greater tolerance in the matter of sampling would be obtained with pulse 2 but the frequency extravagance could scarcely be countenanced. As stated we assume pulse 4 in our PCM bandwidth curves but employ pulse 11 in Tables \T-\TII. Use of pulse 11 is a frequency conservation measure that seems feasible only with PCM and is attractive only with binary PCM. Optimum Distribution of Selectivity Between Tr.\nsmitting and Receiving Filters In a regenerative repeater system both the receiving and transmitting filters may be Gaussian without suffering cumulative narrowing of the system bandwidth since each span commences with a freshly shaped pulse. In this case, the transmitting filter of one repeater and the receiving filter of the succeeding repeater combine, as Ciaussian filters do, to make anotlier Gaussian filter. The resulting pulse may be one of the series 2 to 6 of Fig. 26. On the assumption that one of these shapes is desired and that the trans- '' This is the pulse shape approximated in the experimental system described by Meacham and Peterson (loc. cit.). ^. 1 A /; 11 7 /> 77/ .1 \D TRA NSMISSION PERFORM A NCR 563 mitting and receiving Ultcrs arc to be tlaussian, a problem arises as to how to divide the total selectivity (in column C) between them. If most of the pulse shaping is done at the transmitter, the Gaussian receiving filter must be extremely broad, with the result that discrimination between pulses in adjacent bands is poor and the bands must be spaced widely in order to keep cross-tire down. If, on the other hand, all of the shaping is done at the receiver the wide spectrum of the unshaped transmitted pulse spills over into neighboring bands unless the bands are widely spaced. Clearly, an optimum proportioning of selectivity exists and it is interesting and enlightening to analyze this problem. Such an analysis was made for j^ulses 4, 5 and 6. This analysis pertains only to crossfire and not to signal-to-noise ratio as influenced by curbing (shortening of the rectangular pulses) and by the division of selectivity between transmitting and receiving filters. Wide receiving filters accept more noise and narrow ones may prevent the transmitted pulse from attaining full height in the receiving filter output if curbing is used. If the curbing is pronounced, as in pulses 6 to 1 1 , amplifica- tion may have to follow the transmitting filter to establish the desired transmitted power level. For divisions of selectivity close to the optimum, the receiving filter selectivity appreciably reduces the transmitted pulse height in the case of pulse 5 and seriously reduces it in the case of pulse 6. Crossfire from a pulse in an unwanted band appears as a transient in the wanted band. In some circumstances, this transient has peaks which occur while the crossfiring pulse is rising and falling and has a minimum between which sometimes dips below the level fixed with the steady-state discrimina- tion to the crossfiring carrier. If the pulses in the crossfiring band are synchronized with those in the wanted band as they might be in PAM and PCM only the minimum, central, crossfire might be significant. If, as in PPM, the pulses cannot be synchronized, the peak crossfire is significant. Curves for two values of band separation are shown in Fig. 27, one appro- priate to yield minimum crossfire in the 25 to 35 clb range and the other to yield peak crossfire in that range. This is the range that is sufiicient for binary PCM. The steady state discrimination is also shown. \\c conclude from this study that pulses 4 or 5 are about equally good in respect to mini- mum central crossfire and that pulse 4 is slightly preferable in that the trough and the crest are more symmetrical. For PCM in which the pulse spacing is made equal to T,, this symmetry means that there is the same margin for misalignment of the gating pulse, as regards correctly inter- preting a space or a mark. Pulses 5 and 6 appear to be about equally good in respect to peak crossfire but both (and particularly pulse 6) incur a signal- to-noise penalty because the receiving filter does not permit the transmitted pulse to attain full height. In practice, the approximations to Gaussian filters have shown worse 564 BELL SYSTEM TECHNICAL JOURNAL LU 36 O 48 ^ \, PRODUCT OF Tq and BAND SPACING=2.8 \ ^v PULSE \ ■>- x' \ ^ .y \ \ 6-^ ->X ^"^ ,,''j; 'X \ \ 5 \ \6 \ \ \ \ \ \ 0.8 1.0 1.2 1.4 1.6 RATIO OF RECEIVING FILTER WIDTH TO TRANSMITTING FILTER WIDTH Fig. 27 — Crossfire between frequency divided pulse groups. BAXD WIDril AM) TRAXSMISSIOX PERFORM AXCE 5r)5 pulse crosshre than tlic curves predict. This is particularly so for cases in which the curves show crossfire 30 or more decibels down. Api)roximations usually possess less rapidly falling attenuation skirts and {wssess phase distortion, ])()th of which prevent realization of the calculated crossfire values. Because regeneration (or reshaping) permits the use of Gaussian receiving filters, it does not follow thai Jl a t-l op pcd filters are inferior as receiving filters. Calculations were made for maximally flat receiving filters of about the same overall complexity as was involved in the (iaussian ai)[)roximations. They showed that when the transmitted pulse has the shape 4 and the flat filter is scaled to transmit such a pulse without much distortion, values of peak crossfire of the order of 30 db can be obtained when the product of band spacing and To is 2.8. It was also found that the crossfire in that case consists of a single peak (not unlike the main pulse) nearly coincident in time with the crossfiring pulse. Our Gaussian approximations gave peak crossfire of this same order, for band spacing times T(, = 2.8. The maxi- mally flat receiving filter accepts roughly twice the noise power accepted by the optimum Gaussian filter, so the favor remains with the Gaussian filter and pulses 4 or 5. The main conclusion from all of this is that, if smooth pulses, like num- bers 4 or 5, are employed, band spacings of the order of 2.8 To (perhaps 2.5 To) can be used with crossfire entirely suitable for binar}' PCM, as well as for PPM systems with suflicient swing ratio. Larger spacings would be required for PCM using multi- valued digits, and for P.\]VI. As mentioned earlier, the use of pulses 10 or 11 spaced by To is possible in binary PCM, with small penalty, if very short accurately aligned gate pulses are used. The spectrum of these pulses is more sharply defined and includes a band only slightly wider than 1 To . Rectangular receiving filters of that width could be used side-by-side so that the band spacing would be only slightly greater than 1 To . This is the "theoretical mini- mum" and in telegraph parlance would be specified as a band spacing of twice the dot frecjuency. Pulse 10 results from transmitting a ver\' short pulse through a 4-section maximally flat filter whose response is shown in Column C. The phase distortion characteristics of such a filter produces asymmetr}- in the pulse. Pulse 11 is produced by the filter shown, assuming that the distortion is corrected. Most of the pulse shaping is assumed to reside in the transmit- ting filter. The assumed receiving filter is a 4-secti()n maximally flat filter, and therefore has the shape of the filter shown for pulse 10, but is about 309f wider than shown. When two such bands are spaced 1.5/ To the maxi- mum crossfire is about 26 db down. With shaping and receix'ing filters of reasonable comj)lexity a band spacing 566 BELL SYSTFAf rFXIIXICAL JOURNAL of 1.5/ To to 1.7/7'(i can be expected to have satisfactorily small crossfire for binary PCM. Pulse 1 1 and a spacing of 1 .5/To were assumed for binary PCM-AM in Tables \'I to \III. Figure 26 shows the enx'elopes of r.f. pulses produced by passing flat- topped r.f. pulses or r.f. spikes through r.f. hlters. These envelopes are the baseband shapes produced by wide-band envelope detectors. If baseband pulses are shaped by baseband filters the resulting pulses are the same as shown for pulses 1 to 7, but for pulses 8 to 11 the tails turn out to be over- shoots passing smoothly through zero instead of reaching zero cusp-wise. If these pulses are used to modulate the amplitude of a carrier in a product modulator, the cusps in the envelope are produced as shown, but if they are used to modulate the frequency of a carrier the baseband pulses produced by frequency detection retain their smooth transition through zero. In PAM-FM relatively wide gate pulses could be centered at time zero, Tq , 2To , etc., and the inter-pulse crosstalk would be partially balanced out by partial cancellation of positive and negative contributions. By the use of biases in the AM case a similar result could be obtained. Our tables, assuming pulse 4 spaced 2 To , do not reflect this possibiUty of operation. Delay Line B.\lancing Techniques have been developed^ • ■*" in which the received pulse train is spUt into two or more branches, after detection to the baseband, and recom- bined with suitable delays, attenuations and polarity reversals. Such a procedure is effective in reducing the pulse tails or hangover and its use has been especially valuable in experimental PAM and PAM-FM systems. While this device may be regarded as a kind of phase and amplitude equalizer (comprising as it does only linear, passive elements) the result may be a pulse shape slightly more desirable than those obtained from simple but "ideal" networks, shown in Fig. 26. Our judgment that pulses of shape 4, spaced 27^0 , should be used in PAM-FM may be slightly pessimistic if this kind of balancing is used. More significant reductions of inter-pulse interference may be sought by the method suggested by MacColH^ (which is more than "equalizing") but this method, like the PCM method of Appendix III soon makes pre- posterous demands on the transmission medium and upon the transmitted power. \ III. Transmission Over Metallic Circuits In radio relay transmission we have assumed a span length of 30 miles and have assumed span losses in keeping with the microwave antenna art "• V. D. Landon, loc. cit. ""W. D. Boothroyd and E. M. Creamer, Jr., "A Time Division Multiplexing System," Paper presented at winter general meeting, A.I.E.E., New York, Jan. 31, 1949. « U. S. Patent No. 2,056, 284 Oct. 6, 1936 issued to L. A. MacColl. AMA/) winiii \\n TKwsM issi<)\ rr:i 3 5 (20 DB PER Ml) 50 100 200 500 CIRCUIT LENGTH IN MILES Fig. 28 — Variation of circuit length with number of repeater sections in an AM system with fixed power capacity and noise figure. is favorable to all systems but is most favorable to PCM, which gets no credit for low loss spans. 3. In the case of wave guide, frequency space may be much less precious than in coaxial or radio relay transmission. 4. There is a possibility that many small repeaters should replace the few higher powered repeaters used in radio relay. These different considerations may lead to a different evaluation of the MAXIMUM CIRCUIT LENGTH , Lq.IM M'LES (FOR 1 DB PcR MILE") 200 400 600 1000 2000 4000 10,000 40,000 100,000 10 40 60 100 200 400 600 1000 2000 4000 10,000 OPTIMUM NUMBER OF SPANS , Pq Fig. 29 — Optimum number of repeater sections and maximum circuit length for metal- lic AM system with fixed power capacity and noise figure. MAXIMUM CIRCUIT LENGTH, Lo, IN MILES (FOR 1DB PER MILE) 100 200 400 600 1000 2000 4000 10,000 2 40,000 45 100,000 40 60 100 200 400 600 1000 2000 4000 10,000 OPTIMUM NUMBER OF SPANS, Ho Fig. 30— Optimum number of repeater sections and maximum circuit length for metal- lic FM system with limiting only at end of system. 570 BAM) WIDTH WD TRAXSM ISSIO.X PERFORMANCE MAXIMUM CIRCUIT LENGTH, Lq, IN MILES (FOR I DB PER MILE) 571 400 600 1000 2000 4000 10,000 40,000 100,000 40 60 100 200 400 600 1000 2000 4000 10,000 OPTIMUM NUMBER OF SPANS, Oo Fig. 31 — Optimum number of repeater sections and maximum circuit length for metal- lic PPM-AM system with reshaping at every repeater. 100 18 MAXIMUM CIRCUIT LENGTH , Lq.IN MILES (FOR I DB PER MILE) 200 400 600 1000 2000 4000 10,000 40,000 -r 100,000 40 60 10,000 100 200 400 600 1000 2000 OPTIMUM NUMBER OF SPANS , Hq Fig. 32— Optimum number of repeater sections and maximum circuit length for metal- lic FM system with limiting at every repeater. modulation methods discussed in this paper. We will not attempt to make such a re-evaluation here. 572 BELL SYSTEM TECHNICAL JOURNAL It is of interest now to return to radio relay transmission and examine the relations derived for metallic conductors, but now assuming that the span attenuation is that associated with an inverse ^-power of distance law (k — 2 for free space attenuation). If we use the symbol E to denote the excess power capacity (in decibels) of the repeater over that required for a unit span of, say, one mile, we get the relation lOy^ log L = E-\- (mk - x) log n (8) where x = 20, 10, 5, 3.33, 0 for the cases described by equations (1), (2), (3), (4), (5) respectively. The equation shows no optimum number of spans ^- 'H < CD •- UJ 20 Lu q: 5 O O IL y^ FREE SPACE ATTENUATION (k = 2) y y y y 1 y^ ^/(20-x)lOG 1 ' ^ Y y y y y y y am: X = 20 fm: X = 10 RESHAPED PPM; X = 5 LIMITED fm: X = 3.33 REGENERATED PCM: X = 0 y y' y y y y y '^ 1 y y 1 1 1 1 1 i_ \ 6 8 10 20 40 CIRCUIT LENGTH IN MILES 60 80 100 Fig. a — Relation between circuit length, power, and number of repeaters in radio relay systems. corresponding to a maximum circuit length. It also shows that when x is less than 10^ the circuit length can be increased indefinitely by adding spans although the spans become shorter with increased circuit length. When X = 10^ the circuit length can not be increased beyond the maximum single span, i.e., it depends solely upon E and is not affected by the number of spans. If X is greater than 10^ the circuit length again cannot be increased beyond the maximum single span and is reduced by employing more than one span. This last case does not occur for free space attenuation. In Fig. 33 is plotted the relationship between L, E and n for the free space attenuation law {k = 2). The curve passing through zero decibels excess power capacity at one mile circuit length applies to one span for any value BAND WIDTH LVD TRAXS.\nsSION PERFORMANCE 573 of X or to any number of spans when x = 20. In other words the maxi- mum circuit length for x = 20 is the length of span corresponding to the excess power capacity as noted above for .v = 10^. For all smaller values of -v any circuit length can be achieved with any value of excess power ca- pacity if a sufficient number of spans is employed. The number of spans required for a given circuit length is obtained by moving the curve downward until it intersects the desired length at the appropriate excess power ordinate, and equating (20 — x) log n to the downward shift in decibels. Notwithstanding the present radio outlook in which large towers and antennas seem indicated, it is of interest to imagine small repeaters powered for a one-mile span, say. Using FM with limiting at every repeater, a 100-mile circuit could be obtained with 250 repeaters spaced 0.4 miles. This result comes from Fig. 33 with excess power = zero db and x = 3.33. A difficulty with such a case might be multiple paths produced by one re- peater output overreaching into other spans. The inverse k power attenuation does not accurately describe propagation over long spans; fading then occurs and is greater for long spans than short spans. This introduces a term in the span loss similar to that of the metallic conductor case in which the span loss is proportional to span length. IX. Conclusions We have, in this paper, examined some of the relations governing the exchange of bandwidth for advantages in transmission that grow out of the liberal use of bandwidth. While we have not dealt specifically with the instrumentation involved in the application of the various exchange methods, we have taken cognizance of certain basic obstacles in circuit design such as overload distortion, phase distortion and discrimination characteristics of selective networks and the limitations of microwave antennas. Not having, in most cases, a wealth of experience bearing on the manner in which these obstacles affect the transmission problem, we have been obliged to estimate their effect in many cases. Considerable unreliability in these estimates would not, however, much affect the broad purpose of the paper. The economic factor that is involved in achieving reliable operation of apparatus has been largely ignored, although methods that seem to lead to fantastic instrumentation have not been given much attention. Ruggedness of the transmitted signal, which is obtained at the cost of increased bandividth can, properly handled, be made to conserve frequency occupancy in two ways: (a) ruggedness reduces the required "guard space" between one band and neighboring bands carrying other signals; (b) rugged- ness reduces the multiplication of frequency assignments necessary in congested radio route situations. For wave guide systems, the inter-route interference problem arising from 574 BELL SYSTEM TECHNICAL JOURNAL route congestion disappears but ruggedness is still a valuable feature. As to PCM, regeneration is an outstanding asset applicable also in wave guide transmission. In the case of very high-grade channels the unique advantage of PCM that stems from the coding principle is presumably valuable in any transmission medium. We have shown that, theoretically, PCM methods can achieve lower power requirements than any of the other methods con- sidered and can do so with considerably less frequency space. While this paper is primarily concerned with the transmission of multiplex telephony, it seems appropriate to dwell briefly on the transmission of television signals by radio relay. The repeater plan of Fig. 4 is capable of handling long distance transmission of a 5-mc (video) television signal (by FM). The frequency occupancy of a single two-way route is 80 mc. The occupancy for 1000 4-kc telephone channels is 72 mc from Table \T for binary PCM-AM with dual polarization. At this rate a 5-mc video tele- vision band would require 90 mc assuming that the 39 db signal-to-quantizing noise ratio is satisfactory for television.'*- Remembering that route conges- tion can lead to a greater occupancy than 80 mc in the FM case and per- haps to no increase over 90 mc in the PCM case, we conclude that on these assumptions PCM might be a desirable method for long television relay routes. In the event that a better signal-to-noise ratio is found necessary, binary PCM provides 6 db improvement for each additional digit. These conclusions relate to the transmission problem under the assumed conditions, and do not reflect the impact of many factors that may grow out of an application to a real situation. As has been said before, this paper should be taken to illustrate the way in which the transmission factors are interrelated, and the philosophy by which the problem is approached, rather than to find an unequivocally best system. In preparing this paper the authors have, of course, drawn on the general transmission background of the Bell Telephone Laboratories. Nourish- ment has come particularly from W. M. Goodall, A. L. Durkee, H. S. Black, D. H. Ring, J. C. Schelleng and F. B. Llewellyn in addition to those mentioned specifically in the paper. We wish specifically to thank Mr. R. K. Potter, whose broad transmission concepts were responsible for initiating the work. APPENDIX I Noise in PCM circuits In the transmission of speech by PCM the kinds of noise and distortion which are acquired by other systems in transmission are completely missing. "^W. M. Goodall, "Television by Pulse-Code Modulation." Paper presented at 1949 IRE National Convention, New York, March 9, 1949. BAX/) WIDTH AXD TRAXSM/SSIOX PERFORMANCE 573 Instead, a special kind of impairment is incurred at the terminals, because of the fact that the speech wave is transmitted by quantized amplitude samples of the wave. Transmitting; samples of a wave results in a received wave ha\dng no impairment, provided the samples are not subjected to time or amplitude distortion. In PCM, since the samples are telegraphed, their reception is inaccurate by the quantization imposed by the code. These errors in the samples constitute the sole inherent imj)airment in transmission. Strictly speaking, the transmission imj^airment in PCM is manifested only when a signal is being transmitted. An imaginary telephone circuit with the transmitting side completely devoid of any kind of signal, except that from tl:e talker, could be transmitted by PCM from coast to coast and would sound completely silent if the talker were silent. In any real situation, however, some background noise (room noise, breath noise or line noise) is always present in the subscriber's circuit. This background noise is usually comparable to or greater than the weak parts of weak speech. In order to transmit the speech of weak talkers the size of the discrete amphtude steps must be small with the result that at least a few steps are always brought into play by background noise. Being thus enabled to rule out the case of no signal we are able to ascribe a basic signal impairment to a PCM system. This impairment is, strictly speaking, a result of non-linear distortion inherent in the quantizing, but because of its very complex nature it behaves, and sounds, much like thermal noise and we have accordingly called it quantizing noise. A PCM circuit can be regarded as a source of noise whose rms value is simply related to the size of the quantizing step and the sampling frequency, as follows: In a low-pass band extending to approximately 40% of the sampling fre- quency the basic noise power is related to the power of a sine wave signal by Signal power -^ , peak-to-peak signal voltage , _ ,, -^^ = 20 logio ^ ^ — -f 3 db JNoise power step voltage This band of noise has an amplitude distribution somewhat different from thermal noise, and a spectral distribution which depends somewhat upon the spectral distribution of the signal and upon its amplitude and disposition with respect to the step boundaries. For a sine wave signal the noise spectrum is characterized by a number of prominent components rising above a diffuse background of numerous smaller components. The outstanding components may be either harmonics of the signal frequency or dilTerences between harmonics of the sampling frequency and harmonics of the signal frequency. The background thus consists of an array of various orders of cross-products between the signal and the sampling rate. When the amplitude of the signal is comparable to one step in the quantizing process, a few components may contain a substan- 576 BELL SYSTEM TECHNICAL JOURNAL tial portion of the total power in the distortion spectrum. If the signal is not only weak but has its frequency near the low edge of the band, the distor- tion spectrum has a decided downward slope on the frequency scale with a major part of the distortion power concentrated in the lower harmonics of the signal frequency. Similarly, a weak signal at the upper edge of the band may cause a few scattered difference products to be outstanding. Stronger signals with more centrally located frequencies give practically a uniform distribution of distortion power throughout the signal band. For all except the extreme cases of low ampUtudes and frequencies near the edges of the band, the weighting network used to evaluate the telephonic interfering effect of noise gives a reading equal to that obtained with a flat band of thermal noise of the same mean power. The exceptional cases show a spread in the readings which are sensitive to amplitude, frequency and disposition with respect to step boundaries. The spread is reduced when complex signal waves are applied. An operationally significant case is that in which the noise is produced by residual power hum in the equipment. In such a case, weighted noise readings range from approximately the value obtained for flat noise of the same mean power, to several db lower. Connecting even a short subscriber's loop to the input usually adds enough miscellaneous noise, if the steps are as small as they need to be, to remove the variability and to yield a reading within one db of the equivalent flat noise case. Thus, a PCM. system, like any other transmission system, possesses a noise source and experiments show that this noise combines by power addi- tion with that from another system connected to the input or output of the PCM system. In tandem connections of PCM systems in which successive quantizations may occur, the quantizing noise also adds like power, from sys- tem to system, and soon becomes almost indistinguishable from thermal noise. The quantizing noise consists of distortion products which maybe classified as two kinds. One class includes those products which would be produced by transmitting the wave through a transducer whose input-output charac- teristic is stepped like a staircase. If such a transducer were actually used the PCM process would be equivalent to sampling its output at a regular rate and transmitting the step designations by code. This sampling process, applied to the stepped transducer output, produces the other class of distor- tion (or noise) and is illustrated in Fig. 34. Let us consider the sampled value as the sum of the true value plus the step error, and focus attention on the step error which is responsible for the distortion. .\t minimum permis- sible sampling frequency (twice the highest signal frequency), the step errors in consecutive samples are practically unrelated to each other. The low- pass output filter passes most of the power in this sequence of random errors BAND WIDTH AND TRANSMISSION PERFORMANCE 577 when they occur at a frequency only twice the filter cutoff frequency. See A in Fig. 34. If the sampHng frequency were increased from the minimum permissible value, the consecutive step errors would still be unrelated to each other, and more and more of the step error spectrum (noise) would fall above the low-pass filter. This is shown in B. STEPPED F- TRANSDUCER OUTPUT -250/U.S (a) SAMPLING AT 8 KG ] (B) SAMPLING AT 64 KC — *\ U — 31^/iS (C) SAMPLING CONTINUOUS \^^^^^sNJ — vvMtmMV'''^\wwMM^ ^THE QUANTIZING NOISE CONSISTS OF THE RESPONSE OF A 3500-CYCLE LOW-PASS FILTER TO THE STEP ERROR Fig. 34 — Stepping and sampling an audio wave. Reduction of noise would occur in this way until the sampling frequency became so high that a considerable number of samples are taken while the wave crosses a step interval. Correlation between successive step errors then begins to be apparent. When the interval between samples becomes vanish- ingly small, the process is equivalent to transmitting the stepped transducer output directly. This case appears in C. In an alternate Hne of thinking, one may regard the stepped transducer 578 BELL SYSTEM TECHNICAL JOURNAL output as the signal wave plus a wide spectrum of distortion frequencies representing the effect of the steps. From this point of view, it is clear that only a high sampling frequency prevents lower sidebands associated with the sampling frequency and its harmonics from overlapping the signal band. Quantizing noise decreases with increase of sampling frequency at an initial rate of approximately 3 db per octave and continues until correlation of suc- cessive errors becomes appreciable. This occurs at a sampling frequency which is dependent upon the spectral distribution of the signal, being lower for signals having a predominately low-frequency spectral density. An increase of step size also reduces the lowest sampling frequency at which effects of correlation are observed. Figure 35 shows curves calculated for an D-m NOTE SIGNAL INPUT IS COMPLEX WAVE .^ WITH UNIFORM SPECTRAL DENSITY AND RANDOM PHASE ^ ^"^^ ^ 5 DIGITS (32 EQUAL STEPS) ^ ^^ ^ .tdigit: 5(128 EQUAL STEPS) 1 1 1 ^ 1 , 1 1 „.l..... 1 , _l- 6 8 10 20 40 60 80 100 200 SAMPLING FREQUENCY 400 600 1000 WIDTH OF SIGNAL BAND Fig. 35 — Variation of quantizing noise with sampling frequency. input consisting, in fact, of thermal noise. Such an input is a rough ap- proximation to a speech wave. The asymptotic values shown for five and seven digits represent the quan- tizing noise corresponding to transmission of the thermal noise signal through stepped transducers having 32 and 128 steps, respectively. The curves suggest that sampUng is a penalty such that 32-step granularity without sampling is about equivalents^ to 128-step granularity with sampling at the minimum rate. However, sending information which designates the irregu- lar instants of time at which the signal enters and leaves each step interval is far less efficient than designating the steps at the regular instants of the minimum sampling rate. "" The equivalence would he in terms of total noise power; the properties of the asymp- totic noise are different than were described earlier in this appendix, for sampling at the minimum rate. BAND WIDTH AND TRANSMISSION PERFORMANCE 579 APPENDIX II Interference between two frequency modulated waves This problem occurs so frequently in the present paper that its solution is ai)pcnded liere for reference. Figure 36 shows a geometric figure from whicli the phase of a two-component wave can be calculated. We write where P cos d -\- Q cos (f = R cos \l/ Ri = p2 _^ g.' ^ 2PQ cos (d - if) P sin 0 -{- Q s\n (f tan \p P cos 6 -\- Q cos

or ^/^— ^K4-\ (8) When the interference is from a similar system, we calculate the distribu- tion of the disturbance as follows. The probability that there is an inter- fering pulse present during slicing is the ratio of the pulse duration to the channel allotment, or p, = 2TNfr = '^-^\ (9) The interfering carrier will not, in general, be exactly synchronous with the wanted carrier, and hence the actual interference is a beat frequency with envelope having a voltage distribution calculable from the pulse shape. For a sinusoidal pulse of height A, the probability p{y) dy that the instantaneous magnitude of the interfering envelope is in the interval dy at }' is Since the relative phase of the two carriers drifts with time, the mean square interfering voltage is half the mean square interfering envelope, or En = i ] y P(y) dy - -^^ = -^^. (11) Hence and BAND WIDTH AXD TILiNS. MISS ION PERFORMANCE 583 For marginal interference E ^ 2 y/l A and This equation shows that S/I varies as (B/NfrY'^ for large bandvvidths giv- ing 9 db improvement per octave of bandwidth. The curves of Fig. 10 were plotted from equations (8) and (14). PPM-FM The pulse here is transmitted by a change in frequency from /j to /i + j3 and back again. The total frequency swing i3 corresponds to the pulse height E in the AM case. The frequency detector delivers a pulse of height jS to the baseband filter. Associated with the pulse is the error caused by noise or interference in the rf-band. In the case of fluctuation noise having mean power P„ per cycle in the rf-medium, a baseband filter of width Fb accepts the familiar triangular voltage distribution of noise with frequency resulting^^ in a mean square integrated magnitude expressed on a frequency scale as : El = Pn Fl/ZWc (15) where Wc is the mean carrier power. Then, on substituting /3 for E, and the above expression for E\ i^i the equation for A^: ^ = ^Ml (16) Taking the ratio of e- to A/^, ^^"^' - SP.Fl W/. /• ^ ' The radio signal bandwidth B is approximately equal to the frequency swing plus a sideband at each end or B = ^+2Fb (18) Using this relation to eliminate /3, we have For marginal operation of the FM limiter: Wc = kPnB (20) where we shall assume k = 16 in numerical calculations. ^ An elementary component of interference Q cos qt produces a frequency error (Q/P)f cos l-rrft where/ is the ditlerence between the interfering and carrier freciuencies. The cor- responding mean square frequency error \sPQ-/2F-. But Q-,'2 = P„(//and there are equal contributions from u|)per and lower sidebands centered around the carrier. Also replacing P^/2 by Wc, we get a mean square frequency error in band df at/ equal to P„pdf/Wc. In- tegrating over frequencies from 0 to Fb gives the above result. 584 BELL SYSTEM TECHNICAL JOURNAL Then (,/;V)' = f (0(l-2 0(^-|.^-lJ. (21) The signal-to-noise ratio is found to be maximum when To calculate the CW interference with an idle channel we assume that the carrier wave of the system is represented by Fi(0 = P cos [iTTJit + ^ (01 (23) where 7r,3 ( / + -4- sin TrFb A , 0 < / < ^ 4>i.t) = \ (24) ^(_/) = -0(0, 0 (^^ ± ^) = 0(0, m = 1, 2, . . . . (25) We have chosen (/) so that the phase is a continuous function of time with a derivative representing the correct frequency. This gives a sinusoi- dal change in the instantaneous frequency 4>'{t)/2-K starting with the value /^ at / = — =r , reaching the peak/i 4- /3 at / = 0, and subsiding to/i at / = Fh — . By making the wave repeat with frequency Nfr, we assume all channels Fh of the system are idle since all pulses are at their central positions. It seems reasonable to neglect the effect of variations in adjacent channel loading on CW interference in one channel. The interfering CW wave is represented by V^{t) = Q cos (27r/2/ -I- 6) (26) To a first approxunation the resulting error in frequency at the output of the frequency detector is : ^W = ^ J, ^i^ t27r(/i - /,)/ + 0(0 - e] (27) lirr at By straghtforward Fourier series expansion and differentiation: 5(0 = p i^6 Z (c + "^)^« cos {lirFic -I- n\)t - 6] (28) Jr n=— 00 BAND WIDTH AND TRANSMISSION PERFORMANCE 585 where: c = (/i -/2)//^6. X = Nfr/Ft>, y = ^/Ft, (29) An = 2\J,nx-y(y) " - [(-)" + sin (2;/X - >07r], (30) mr Ao = 2\y^yiy) + (1 - 2X) cos Try. (31) The function Jy (y) is Anger's function:''^ J^(y) = 1 f cos (vd - y sin 6) dd (32) T Jo It is equal to Jy(y), the more famiUar Bessel function of the first kind, only when V is an integer. The values of v = 2n\ — y appearing in this solution are in general not integers and hence the ordinary tables of Bessel functions are inapplicable. The baseband filter accepts the components of the error which have fre- quencies in the range : -Fb < Fb (c+ n\) < Fb (33) or -l±^{t - r) + d] (42) This gives 5(^) = ^ i sin [0(/) -4>{t-r) -d^ 2,r(/i - fM (43) 27rjr at We distinguish between the cases of overlapping and non-overlapping pulses. BAND WIDTH AND TILINSMISSION PERFORMANCE 587 If the pulses do not overlap, we take the origin of time midway between pulses and write <^(/) - 0(/ - t) T/3 -1 - ± {l - t) TTiS -1 -^<,<-r-i t\ • 2NJ, 2 Fi 1,2. FbT ^ T In bolh cases the right-liaiul members are even functions of t. Hence ^ < / < r - ^ i (45) Vb I sm [(^(/) - 0(; - r) - e] = S 7i,„ cos ImirXfrt (46) 588 BELL SYSTEM TECHNICAL JOURNAL A °° COS [ -^ 2 4 Jo L ' '"=1 2A' -| + E alwnimfr - f)\df (57) m=l J The summations stop at 2N because the baseband filter cuts off at f = Fb ^ 2Nfr. The contribution of the Oq term is negligible. Then . 2 WZTT 2N sm — - /,/2 1F„ = -^ Z —f- [« + f? + imfr - ff] df . tP- Wc m=i m^ Jo P^j T^Wcm=i \ 12m-/ N When N is large, the sum approaches Wn ^ ^' (59) and 590 BELL SYSTEM TECHNICAL JOURNAL on substituting Wc = kPnB, ^ = B - 2Fb, and Fb = 2Nfr. Equation (61) with ^ = 16 was used to obtain the marginal power curve of Fig. 13. The result may be compared with that given by Rauch^^ (See also Landon^) for a rectangular pulse and rectangular gate, which requires a higher value for Ff,. The two systems give the same signal-to-noise ratio when the rectangu- lar pulse and gate of Ranch's system endure for one half of the total channel allotment time. The value of Fb necessary for such a case was estimated by Ranch to be S.SN^fr. A calculation made as above, except that the gate was assumed sinuosoidal instead of rectangular, showed very nearly the same signal-to-noise ratio. The case of CW interference with all channels idle is represented by the r.f. wave: V{l) = P cos llivfit + J|^ sin Tr/^b/ j + (3 cos lirf-.L (62) When Q/ P is small, the detected frequency is /2)/ + ;^sin7rF5/ (63) f = /, + _cosxF./-2^-sm 27r(/i - Z^)/ -f AsiuTrFfe/ Itb -/i + ^cosTrFft/ - 5(/).. 4 By Fourier series expansion followed by differentiation, the error b{t) may be written as: m = 2 t^ (/, - /, + ^') J,.(x) cos 2, (/, - /, + '4') (, (64) where x = ^/2Fb. The baseband filter passes only those components of 5(/) which have frequencies in the range —Fb to Fb- Writing c = (/i — fo) Fb, we find: 5o(/) = ^ Z (c + l) /,„(.r) cos 2^Fb (c + |) I (65) where W2 is the largest integer which does not exceed 2(1 — c) and nii is the smallest integer which is not exceeded by —2(1 + c). The term integer is here understood to include zero and both positive and negative integers. The wave 5o(0 is next multiplied by the gating function G{t) and the com- ponents falling in the audio band —fr/2 to/r/2 selected. We find: GiOm =^ f: J: <^n(c + '^) JM) cos 2^[{2c + m)X + u]frt ^■L m=mi n= — oo \ ^ / (66) with a„ = a_„. * Loc. cit. ■** L. L. Rauch, Fluctuation Noise in Pulse-Height Multiplex Radio Links, Proc. I.R.E., Vol. 35, Nov. 1947, pp. 1192-1197. BAND WIDTH AND TRANSMISSION PERFORMANCE 591 For each value of ;;/, there is only one value of n satisfying the audio filter inequality, which may be written: -^ - (2c + nt)X < // < ^ - (2c + m)N (67) lIcMue iiiti'rference accepted by the channel filter is: /(/) = ^ Z ''" (c + "0 J'nix) COS 27r[(2c + m)X + n\Jrl (68) 11 m=mi \ " / The mean square value of interference is (0 = j^ Z^ al [c + I j Jl(x). (69) r / = The signal-to-interference ratio is 32 [/=^(0]"' or S/I = 2FtQ [t^ a\ [c + fj il(.v)' -1/2 (70) (71) When a rectangular gate of duration equal to the full channel allotment is used, we substitute fl„ = 2(sin mr/N)/n'K. We then find that the largest values of mean square interference occur when c is an odd multiple of one fourth. If we set c= -(2r-M)/4,r=0,±l,±2, it follows that if N is an even integer, n = {r -\r h ~ ni)N, sm ^ = sm (r + t — ni)Tr, sm rnr = 1. Substituting these values in the expression for S/I, we find 1-1/2 s/i = 2Fb Q [_m=T- [r+2 -1-1/2 m=r—l J (72) (73) (74) (75) (76) The value of r is to be chosen as the integer which makes S/I a minimum; i.e., we place the CW^ frequency at that part of the band where it does tJie most damage. The curve marked CW(Gate) of Fig. 14 was obtained in this way. 592 BELL SYSTEM TECHNICAL JOURNAL When instantaneous sampling is used instead of a gate, the value of c„ becomes a constant for all values of n of interest and is equal to a^. We then find: on r :^2(l-c) / ^A2 "1-1/2 Here c is to be selected to make S/I minimum for each value of x. The poorest values of S/I occur when Bessel functiojis of comparable order and argument appear in the summation, which means that c is in the neighbor- hood of —x/2. The corresponding difference between the CW and mid- carrier frequencies is one-fourth the peak-to-peak swing. To calculate the interference between two similar idle systems, we set (78) (79) V{t) = Pcos ilTrfit -f- Jr sin TrFbtj -f Q cos \2irf2t + Jr sin {-kF^I - d)\ The frequency error registered in the first system is then hit) = -^ ^ sin [27r(/i - U)t -1- X sin irFbl - x sin {irFbt - e)] 2-wP at = -^ - sin 2r{fi - f2)t + 2x sin - cos iirFbl - -j • cos 27r I /i - /2 + -y- ) t - yJ • It follows that the response of the baseband filter is m = ^^ E^ {c + I) /. (2. sin 0 cos [2.F. {c + 1) t - .n ^ . (80) The effects of the channel gate and filter are computed in the same way as for CW, giving the audio interference: cos ( 27r[(2c -f m)^ -\- n]frt - m -j (81) BAND WIDTH AXD TR.IXS.M fSSIOX PERIORM WTR 593 Since the two systems occupy the same frequency assignment, we assume that c is not greatly different from zero. Then for fixed 6: I\t) = (0 = ||'^E ^^rani^2.vsin0 (82) Since 6, the frame phase difference, is a random angle we average over its possible values by setting: d(t) AM = ~ jf ' Jl Ur sin 0 ^^e^ll J'M^- sin 4>) _rV + ^)(2.rr 7r(2w)!(w!)2 (g3) • 2F3(w + i w + ^; 2w + 1, m + 1, w + 1; -4x-'), w > 0 ^_„(.t) = ^m(.v) Noting further that for c = 0, mi = -2, m2 = 2, and n = -mN, we have then: ",2^2 I\l) = l^ [alAiix) + 4aL^2(x)l (84) and 5// = ^^ [alAiix) + 4aLv^2(-x)r'^ (85) For a sinusoidal pulse and rectangular gate of full channel allotment time in duration, c.v = 02.V = 0, /-(/) = 0, and S/I is infinite. If instantaneous sampling is used, Go = (In = diN (86) and S/I = ^tII;^ Ui(.r) + 4^,(.v)]-^'^ (87) The curve for similar system interference with instantaneous sampling, Fig. 14, w^as calculated from Eq. (87). For small values of x, we may use the ascending power series: -^iW = H' " ?^M ! '■' + 3-4.2'-y-2! •>■'-•■] (8« 594 BELL SYSTEM TECHNICAL JOURNAL For large values of x, the following asymptotic formula has been derived by Mr. S. O. Rice by use of the Mellin-Barnes contour integral representation and the method of steepest descents: TT xA, ,(.v) ~ In .V + In 32 + 7-2 ( ■-hi+ + 1 2m — -,) -(-)' Tl/l^-^°^0-^-+i) + (90) 7 = Euler's constant = 0.5772 . . . As X approaches zero, S/I approaches 2Pj'Q, which is to be expected because the frequency deviation of the unwanted carrier is represented by a pair of first order sidebands P/Q times as great as those on the wanted carrier. Averaging over the random carrier phase difference brings in a factor \/2, and averaging over all frame phases accounts for another. 4W as m — ag 3W '^Q Z< HCD Q-u. 2W < l^ w 2Z 0 W 2W 3W 4W 5W 6W 7W HIGHEST SIGNAL FREQUENCY IN TERMS OF BAND WIDTH, W Fig. 37 — Minimum sampling frequency for band of width W. APPENDIX V Sampling a band of frequencies displaced from zero It is often necessary to transmit a signal band which does not extend all the way down to zero frequency. For example, a group of channels in FDM may be based on a set of carrier frequencies remote from zero. When we consider the application of pulse methods to transmit such a signal, the ques- tion of what sampling rate is needed immediately arises. A band extending from /i to/i + 11' could of course be translated to the range 0 to W by stand- ard modulation techniques, sampled at a rate 211', and restored to the original range by an inverse translation at the receiver. The frequency shifting apparatus required includes modulators, carrier generators, band separating filters, and possibly amplifiers to make up the inevitable losses. BAND WIDTH AND TRANSMISSION PERFORMANCE 595 A direct sampling process which avoids shifting the band may therefore be preferred. A useful theorem for uniformly spaced samples is that the minimum sampling frequency is not in general twice the highest frequency in the band but is given by the formula: /. = 2.r(i + ^J, (1) where: fr — minimum sampling frequency 11' = width of band /> = highest frequency in band m = largest integer not exceeding /2/l'r The value of k in (1) varies between zero and unity. When the band is located between adjacent multiples of W, we have ^ = 0 and it follows that fr = 2TF no matter how high the frequency range of the signal may be. As k increases from zero to unity the sampling rate increases from 2IF to 2W (1 + -). The curve of minimum sampling rate versus the highest frequency in a band of constant width thus becomes a series of sawteeth of successively decreasing height as shown in Fig. 37. The highest samplmg rate is re- quired when m = I and k approaches unity. This is the case of a signal band lying between W — A/ and 2 IF — A/ with A/ small. The sampling rate needed is 2(2IF — A/) which approaches the value 4TF as A/ approaches zero. Actually when A/ = 0, we change to the case oi m = 2, k = 0, and fr = 2W. The next maxhnum on the curve is 3 IF, which is approached when fj nears 3W. The successive maxima decrease toward the limit 2TF as fi increases. The sampling theorem contained in Eq. (1) may be verified from steady state modulation theory by noting that the first order sidebands on harmonics of 2W do not overlap the signal when the equation is satisfied. Abstracts of Technical Articles by Bell System Authors The Transistor — A New Semiconductor Amplifier} J. A. Becker and J. N. Shri\t. This article describes the construction, characteristics, and behavior of the newly discovered device, the transistor. Used as a semicon- ductor amplifier, it works on an entirely different principle and is capable of performing the same tasks now done by the vacuum tube triode. A Review of Magnetic Materials? R. A. Chegwidden. Significant advances have been made within recent years in the development of new and better magnetic materials, and in the theories of magnetism. High permeability materials that may be classed as non-conductors, materials with greatly improved initial permeabilities, and permanent magnet alloys capable of storing four or five times as much energy as those obtainable ten years ago are now available. Descriptions of some of these developments are given. The paper gives compilations of data and curve sheets showing some of the typical characteristics of many of these materials. Ratio of Frequency Swing to Phase Swing in Phase- and Frequency-Modula- tion Systems Transmitting Speech} D. K. Gannett and W. R. Young. Computed and measured data are presented bearing on the relation be- tween the phase and the frequency swing in phase- and frequency-modula- tion systems when transmitting speech. The results were found to vary with different voices, with the microphone and circuit characteristics, and with the kind of volume regulation used. With a particular carbon microphone, it was found that a phase deviation of 10 radians corresponds to a frequency deviation of between 11 and 15 kc in a phase-modulation system, and be- tween 6 and 12 kc in a frequency-modulation system, depending on condi- tions. Design and Performance of Ethylene Diamine Tartrate Crystal Units.'^ J. P. Griffin and E. S. Pennell. A research program on synthetic crystals has resulted in the development and adoption of EDT for carrier telephone filters. Some unusual physical properties of the crystalline material give rise to novelty in the processing methods and mechanical design of the units. These properties include anisotropic expansion coetlicients, fragility, natural cleavages and water solubility. The electrical properties of EDT result in filters with wider pass bands and lower impedance levels than commerically obtainable with quartz. ^Electrical Engineering, v. 68, pp. 215-221, March 1949. 2 Metal Progress, November 1948. » Froc. I. R. E., V. 37, pp. 258-263, March 1949. * A. I. E. E. Transactions, v. 67, pt. 1, pp. 557-561, 1948. 596 ABSTRACTS OF TECHNICAL ARTICLES 597 Recent Improvements in Loading Apparatus for Telephone Cables} S. G. Hale, A. L. Quinlax, and J. E. Ranges. Through the use of improved ma- terials, manufacturing techniques, and designs, a series of exchange-area loading coils has been provided which is equivalent electrically to the super- seded types but requires one-third less copper and has considerably smaller overall dimensions. Similarly, 3-coil toll cable loading units have been pro- vided with a saving of one-half in both copper and core material, with a small sacrilice in electrical behavior as compared with superseded types. The reduced size of the new coils and units, together with improved assem- bly arrangements, made possible a 65 per cent saving in the volume and weight of the cases housing them. The Coaxial Transistor} Winston E. Kock and R. L. Wallace, Jr. The success of the earlier types of transistors led to the exploration of other forms of similar ampliliers, one of which is the coaxial transistor. A descrip- tion of its construction, characteristics, and many advantages is contained in this article. Paralleled-Resonator Filters? J. R. Pierce. This paper describes a class of microwave filters in which input and output waveguides are connected by a number of resonators, each coupled directly to both guides. Signal com- ponents of different frequencies can pass from the input to the output largely through different resonators. This type of filter is a realization of a lattice network. An experimental filter is described. .4 Broad-Band Microwave Relay System between Ne-iV York atid Boston} G. N. Thayer, A. A. Roetken, R. W. Friis, and A. L. Durkee. This paper describes the principal features of a broad-band microwave relay system which has recently been installed between New York and Boston. The system operates at frequencies around 4,000 Mc and provides two two-way channels, each accommodating a signal-frequency band extending from 30 cps to 4.5 Mc. Noise and distortion characteristics are satisfactory for the transmission of several hundred simultaneous telephone conversa- tions or a standard black-and-white television program. Growing Crystals of Ethylene Diamine Tartrate} A. C. Walker and G. T. KoHMAN. The need for a synthetic piezoelectric crystal to relieve the critical quartz supply situation has resulted in the development by the Bell Tele- phone Laboratories of a new organic salt crystal, ethylene diamine tartrate, which is being used in place of quartz in telephone circuits. This crystal is grown from a supersaturated aqueous solution of its salt by an entirely new method known as the constant temperature process. It ^A.I.E.E. Transactions, v. 67, pt. 1, pp. 385-392, 1948. 6 Electrical Engineering, v. 68, pp. 222-223, March 1949. ^ Proc. I. R. E., V. 37, pp. 152-155, February 1949. ^Proc. I. R. E.—Waves and Electrons Section, v. 37, pp. 183-188, February 1949. 'A.I.E.E. Transactions, v. 67, pt. 1, pp. 565-570, 1948. 598 BELL SYSTEM TECHNICAL JOURNAL differs from previous methods used for growing large single crystals from solution, in that the solution saturated at one temperature is continuously fed into a crystallizer tank maintained at a slightly lower temperature, thus providing the supersaturation condition necessary for crystal growth. Fur- ther, the solution is circulated in such a manner that the partially im- poverished mother liquor overflows from the growing tank back into the saturator where it is refortiiied and filtered. It is then heated and returned to the growing tank in such a way as to avoid the formation of undesirable crystal nuclei. The paper contains a description of the new method which is now in commercial operation, together with a general discussion of some of the important principles involved in the successful growth of large single crystals of water soluble salts. Crystal Fillers Using Ethylene Diamine Tartrate in Place of Quartz}'^ E. S. Willis. Ethylene diamine tartrate (EDT) crystal filters were developed to replace the earlier quartz type channel filters in the broad-band carrier telephone systems, because of the threatened scarcity of quartz. These new filters give performance comparable to that of the earlier design. The growth of the EDT crystals from seeds and their fabrication into crystal units for use in filters are covered in companion papers on "Design and Performance of Ethylene Diamine Tartrate Crystal Units" and "Growing Crystals of Ethylene Diamine Tartrate" in this same volume of the Transactions iM. /. E. E. Transactions, v. 67, pt. 1, pp. 552-556, 1948 Contributors to This Issue John Bardeex, University of Wisconsin, B.S. in E.E., 1928; M.S., 1930. Gulf Research and Development Corporation, 1930-33; Princeton University, 1933-35, Ph.D. in Math. Phys., 1936; Junior Fellow, Society of Fellows, Harvard University, 1935-38; Assistant Professor of Physics, University of Minnesota, 1938-41; Prin. Phys., Naval Ordnance Labora- tory, 1941-45. Bell Telephone Laboratories, 1945-. Dr. Bardeen is en- gaged in theoretical problems related to semiconductors. \V. R. Benxett, B.S., Oregon State College, 1925; A.M., Columbia University, 1928. Bell Telephone Laboratories, 1925-. Mr. Bennett has been active in the design and testmg of multichannel communication sys- tems, particularly with regard to modulation processes and the effects of nonlinear distortion. He is now engaged in research on various transmis- sion problems. C. B. Feldman, University of Minnesota, B.S. in Electrical Engineering, 1926; M.S., 1928. Bell Telephone Laboratories, 1928-. As Transmission Research Engineer, Mr. Feldman has charge of a group studying new trans- mission methods. He is a Fellow of the Institute of Radio Engineers. J. R. Hayxes, B.S. in Physics, University of Kentucky, 1930. Bell Telephone Laboratories, 1930-. Mr. Haynes is in the Physical Research Department, engaged in solid state studies. CoNYERS Herring, A.B., University of Kansas, 1933; Ph.D., Princeton University, 1937; National Research Fellow, Massachusetts Institute of Technolog>% 1937-39; Instructor in Mathematics and Research Associate in Mathematical Physics, Princeton University, 1939-40; Instructor in Physics, University of Missouri, 1940-41; Columbia University Division of War Research, 1941-45; Professor of Applied Mathematics, University of Texas, 1946. Bell Telephone Laboratories, 1945-. Dr. Herrmg has been engaged in theoretical problems in the tields of solid state physics and elec- tron emission. R. J. KiRCHER, B.S. in E.E., California Institute of Technology, 1929; M.S., Stevens Institute of Technology, 1941 . Bell Telephone Laboratories, 599 600 BELL SYSTEM TECHNICAL JOURNAL 1929-. Mr. Kircher was engaged in radar and counter measures projects during the war. Electronic Apparatus Development Department since 1944. Transistor Development Group since 1948. G. L. Pearson, A.B., Willamette University, 1926; M.A. in Physics, Stanford University, 1929. Bell Telephone Laboratories, 1929-. Mr. Pear- son is in the Physical Research Department where he has been engaged in the study of noise in electric circuits and the properties of electronic semi- conductors. Robert M. Ryder, Yale University, B.S. in Physics, 1937; Ph.D., 1940. Bell Telephone Laboratories, 1940-. Dr. Ryder joined the Laboratories to work on microwave amplifier circuits, and during most of the war was a member of a group engaged in studying the signal-to-noise performance of radars. In 1945 he transferred to the Electronic Development Depart- ment to work on microwave oscillator and amplifier tubes for radar and radio relay applications. He is now in a group engaged in the development of transistors. W. Shockley, B.Sc, California Institute of Technology, 1932;. Ph.D., Massachusetts Institute of Technology, 1936. Bell Telephone Laboratories, 1936-. Dr. Shockley's work in the Laboratories has been concerned with problems in solid state physics. voLLMH XXVIII OCTOBER, 1949 no. 4 Pubfic LtDrary nsas City, Mo. THE BELL SYSTEM TECHNICAL JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION Reactance Tube Modulation of Phase Shift Oscillators F. R. Dennis and E. P. Fetch 601 A Broad -Band Microwave Noise Source W. W. Mumford 608 Electronic Admittances of Parallel-Plane Electron Tubes at 4000 Megacycles Sloan D. Robertson 619 Passive Four-Pole Admittances of Microwave Triodes Sloan D. Robertson 647 Communication Theory of Secrecy Systems C. E. Shannon 656 The Design of Reactive Equalizers A. P. Brogle, Jr. 716 Abstracts of Technical Articles by Bell System Authors. .751 Contributors to this Issue 753 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 50 i per copy $1.50 per Year THE BELL SYSTEM TECHNICAL JOURNAL Published quarterly by the American Telephone and Telegraph Company 195 Broadway, New York, N. Y. EDITORS R. W. King J. O. Perrine EDITORIAL BOARD C. F. Craig H. S. Osborne J. J. Pilliod O. E. Buckley M. J. KeUy A. B. Clark F. J. Feely «■»»«■»»»■» SUBSCRIPTIONS Subscriptions are accepted at $1.50 per year. Single copies are 50 cents each. The foreign postage is 35 cents per year or 9 cents per copy. Copyright, 1949 American Telephone and Telegraph Company PRINTED IN U.S.A. The Bell System Technical Journal Vol. XXVIII October, 1949 No. 4 Reactance Tube Modulation of Phase Shift Oscillators By F. R. DENNIS and E. P. FELCH This paper describes a l)asic circuit for reactance tul)e modulation of phase shift oscillators. The design of suitable phase shift oscillators for freciuencies from audio through the ultra-high frequencies is discussed. E.xperimental performance data derived from several types of frequency modulated ])hase shift oscillators are presented. Introduction FREQUENCY modulation of oscillators is finding vvide-s})read use in such diverse fields as FM broadcasting, telemetering systems for guided missiles and measuring apparatus for observing transmission fre- quency characteristics on cathode ray tubes. Design objectives for such oscillators may be listed briefly as: 1. A wide range of frequency modulation or, alternatively, high modula- tion sensitivity. 2. A linear relationship between instantaneous values of modulation input voltage and frequency deviation. 3. Freedom from accompanying amplitude modulation. 4. Inherent center frequency stability. 5. Ease and stability of adjustment. 6. A minimum number of components, none of which should be critical. 7. Modulation by dc, audio, or video inputs. 8. Operation anywhere in the frequency spectrum from low audio fre- quencies through the ultra-high frequency region. The circuits described in this paper were developed in the course of an investigation of various frequency modulation circuits for use in visual trans- mission measuring systems. The oscillators had to be capable of linear modu- lation at 60 cycles over a ±3 megacycle band at 25 megacycles and over a ±9 megacycle band at 80 megacycles. Existing designs fell short of meeting the requirements with respect to several of the characteristics listed above. The reactance tube modulated phase shift oscillator circuit was found to perform satisfactorily in the transmission set and proved superior in many respects to all the other circuits tried. Tests of the circuit at other frequencies disclosed that the advantages were not peculiar to the frequency range and the following description is presented with the expectation that it may prove useful to others. 601 602 BELL SYSTEM TECHNICAL JOURNAL OSCILLATOR TUBE TUNED CIRCUIT AND 180 PHASE SHIFT NETWORK ±90° PHASE SHIFT NETWORK REACTANCE TUBE MODULATING INPUT Fig. 1 — Simplified schematic of conventional reactance tube modulated oscillator. OSCILLATOR TUBE 90° PHASE SHIFT NETWORK 90° PHASE SHIFT NETWORK REACTANCE TUBE MODULATING INPUT Fig. 2 — Simplified schematic of phase shift reactance tube modulated oscillator. OSCILLATOR TUBE + 90" PHASE SHIFT NETWORK t90° PHASE SHIFT NETWORK REACTANCE TUBE MODULATING INPUT TUBE CONNECTION NETWORKS OSCILLATION FREQ. LEADING (+90°) LAGGING (-90°] LEADING (+90°) DECREASES INCREASES B DECREASES LAGGING (-90°) INCREASES Fig. 3 — Direction of frequency deviation for increasing Gm of reactance tube. PHASE SHIFT OSCILLATORS 603 fH OSCILLATOR °) TUBE 2C REACTANCE TUBE fo=- 'T=^ '°~2-tT/LC' Yig. 4 — LC reactance tube modulated phase shift oscillator. OSCILLATOR TUBE r REACTANCE TUBE r r fo = 1.2 2TrRC Fig. 5 — RC reactance tube modulated phase shift oscillator. Circuit Description The theory and design of conventional reactance tube modulated oscil- lators has been discussed adequately in the literature' ■■'•^. A schematic in ' "Frequency Modulation" (book) by August Hund — McGraw-Hill, New York, 1942. Page 155. ^ "Automatic Tuning, Simijliticd Circuits and Design Practice," D. E. Foster, and S. W. Seelev. Proc. I. R. E., Vol. 25, 1937, page 289. 3 .\TC Systems— Wireless World, February 19, 1937, page 177. 604 BELL SYSTEM TECHNICAL JOURNAL REACTANCE TUBE + B Vk L (MEGACYCLES) WHERE K = DIELECTRIC CONSTANT. L= LENGTH OF LINE IN METERS. Fig. 6 — Transmission line reactance tube modulated oscillator. 36 34 33 u 32 5 29 28 ' I ^! !i 2.0 d > K 1.0 o -15 -13.5 -12.0 -10.5 -90 -7.5 3.0 -4.5 -3.0 REACTANCE TUBE GRID VOLTS Fig. 7 — Performance curves of typical LC reactance tube modulated phase shift oscillator. simplified form is shown in Fig. 1. The input and output of a vacuum tube amplifier are connected together by a tuned circuit and feedback network which introduces 180° phase shift at the undeviated frequency Fq . I'll ASK Slllir OSCILLATORS 605 2600 2400 2200 2000 O ^ 1400 JIZZZZ~Z. zzz~zzzz: 1.4 o > \- 1.0 o 8-7 -6 -5 -4 -3 REACTANCE TUBE GRID VOLTS Fig. 8— Performance curves of typical RC reactance tulie modulated phase shift oscillator. IU5 / / 95 / ^ / 90 / z ■' ' — - ^ ^ ■8 -7 -6 -5 -4 -3 REACTANCE TUBE GRID VOLTS Fig. 9 — Performance curves of typical transmission line reactance tul)e moduialei oscillator. 606 BELL SYSTEM TECHNICAL JOURNAL An auxiliary path contains the reactance tube fed from a 90° phase shift network connected as shown. The direction of frequency deviation is deter- mined by the sign of the 90° phase shift. The amount of the deviation is Fig. 10 — Construction of transmission line reactance tube modulated oscillator, (a) Tube side, (b) Line side. determined by the transconductance variation of the reactance tube, by the impedance across which the reactance tube is connected and by the loss in the 90° phase shift network. The linearity is a function of all of these factors. In general the frequency deviation may be increased by increasing PHASE SHIFT OSCILLATORS 607 the L/C ratio in the oscillator tuned circuit, but only at the expense of frequency stability. A simplified schematic of the reactance tube modulated phase shift oscillator is shown in Fig. 2. The mathematical theory of operation is anal- ogous to that of the conventional reactance tube modulated oscillator, and the same methods of analysis may be applied. The 90° phase shift network required in the reactance tube grid circuit is a portion of the feedback net- work and provides half of the 180° phase shift required for oscillation. In this circuit the reactance tube is tightly coupled into the oscillating circuit with minimum loss in the 90° phase shift network. Hence small values of L/C ratio may be employed with a consequent increase in the inherent fre- quency stability. In practice, oscillators comparable in stability to good nonmodulated oscillators may be realized. The direction of deviation is determined by whether the phase of the reactance tube grid voltage leads or lags the reactance tube plate current. The permutations of connections and signs of the 90° phase shift networks are shown on Fig. 3 with the correspond- ing directions of frequency deviation. The phase shift networks need not be of the LC lumped constant variety. For example, RC networks or sections of transmission line may be employed to particular advantage at the lower and higher frequencies respectively. A few of the many possible circuit configurations are shown in Figs. 4, 5, 6. Experimental Data Frequency deviation and output variation curves for some typical oscil- lators are shown in Figs, 7, 8, and 9. The oscillator of Fig. 9 which was built by Mr. D. Leed, is shown in Fig. 10. The transmission line is a section of RG59U cable with the shield removed, encased in a copper tube with a slot for bringing out the center tap of the line to the reactance tube grid. The tubes are 6J6's with both sections connected in parallel. Conclusion Frequency modulated phase shift oscillators of several types have been described. These offer interesting possibilities for applications over a wide range of frequencies wherever stable, simple frequency modulated oscillators are required. With respect to range, linearity, and freedom from amplitude modulation their performance, as shown, is superior to that of conventional circuits and is at least equal to that of the complex circuits employed in the most critical applications. A Broad -Band Microwave Noise Source By W. W. MUMFORD Measurements of the microwave noise power available from gaseous discharges, such as in an ordinary fluorescent lamp, show remarkable uniformity and sta- liilily. Such tubes are therefore suitable for a new type of standard noise source. Introduction A STANDARD noise source, such as a hot resistance or a temperature -^ ^ limited diode, has been used advantageously for making measurements of the noise figure of radio receivers in the short-wave and the ultra-short wave region. The use of such a tool eliminates the possible errors which are practically inescapable when using the large amounts of attenuation wdiich are needed for the determmation of the ratio of power levels encountered in measuring noise figures with a standard signal generator. For example, the power from a standard signal generator might be measurable and known accurately at a level of 40 db below a watt, whereas the noise power avail- able from a resistance might be 141 db below one watt.^ It is difficult to ascertain accurately power ratios of this magnitude, 10^°. Another advantage of using a standard noise source arises from the fact that ordinarily the bandwidth of the receiver need not be considered, thereby eliminating a time consuming measurement. This assumes, of course, that the bandwidth of the noise source is much greater than that of the amplifier under test. In the microwave region it is possible to match a resistive element to the waveguide over a wide enough band, but ordinary resistive materials will not stand the high temperatures (5000 degrees or more) needed to measure the noise figures encountered in practice. The noise diode is capable of furn- ishing adequate noise power, but one with wide bandwidth has yet to be developed. A good, stable, broadband microwave noise generator is needed. Another possible source of noise power consists of a gaseous discharge. - Before we examine the data which have led us to conclude that the gaseous discharge is a good, broad-band, stable microwave noise generator and pos- sibly a calculable noise standard, we review our definitions of noise figure ^ This figure, 141 db below one watt, assumes that the effective bandwidth is 2 mc. The resistance noise power available from a generator at 290° Kelvin is 204 db below one watt per cycle. «G. C. Southworth, Journal of the Franklin Inslilule, Vol. 239, ^U, pp. 285-298, April 1945. 608 BRUM) BAM) MU l«)\\ W !■: XOISI'l SOURCE 609 and gain,^ and discuss the factors involved in making noise ligure measure- nienls hv means of a noise source. Notes on Noise Figure Definition: The Noise Figure of a network, with a generator connected to its input terminals, is the ratio of the available signal-to-noise power ratio at the signal generator terminals (weighted by the network bandwidth) lo the available signal-to-noise power ratio at its output terminals. Definition: The G.\ix of a network is the ratio of the available signal power at the output terminals of the network to the available signal power at the output terminals of the signal generator. INPUT TERMINALS OUTPUT TERMINALS en = 4KTiRiB T,R NETWORK GENERATOR TERMINALS GAIN = G NOISE FIGURE - f Fig. 1 — Schematic diagram of generator, network and output power meter. These defmitions apply to a circuit consisting of a generator, a network and an output jKjwer meter as shown schematically in Fig. 1. The signal power available from the generator, having an open circuit voltage e and an internal resistance Ri , is: SA e 4Ri (1) The noise power available from the signal generator resistance, Ri , at ab solute temperature Ti , is AKTiRiB Pf/A — 4i?i = kTiB (2) where B is the effective bandwidth of the network, by which the generator noise is weighted in this case. 3 H. T. Friis, Froc. I. R. E., Vol. 32, # 17, pp. 419-422, July, 1944. 610 BELL SYSTEM TECHNICAL JOURNAL The weighted available signal-to-noise ratio at the generator terminals is: e PsA ^ m_ (3) Pna KT,B The network amplifies (or attenuates) the generator's signal power by the factor G, the gain of the network, so that the available signal power at the output terminals of the network is : Pso = G ^4_ (4) The network amplifies (or attenuates) the generator noise power by the same factor G, and also delivers noise power which originates within itself, Nn , so that the total available noise power at the output terminals of the network is: P^o = GkTiB + .¥,v (5) The available signal-to-noise ratio at the output terminals of the network is then : r— Pso^ ^ 4Ri (6) P^o GkT.B + Nu We now express the noise figure of the network, F, which by definition is the ratio of equation (3) to equation (6), thus, ^ GkT.B + Nu ,-N ^ = GkT^B ^^^ We should pause at this point to consider this equation further, for it leads us to a simpler definition of noise figure. Definition: The noise figure of a network is the ratio of the noise power output of that network to the noise power output which would exist if the network were noiseless. The temperature of the signal generator resistance is 290 degrees Kelvin. The choice of generator temperature of 290 degrees is an arbitrary one, which makes kTi = 4(10)--^ watts per cycle bandwidth; — lOlog^^i = 204 db below one watt per cycle. Putting Ti = 290 in equation (7) gives: Gk 290 B -{- Nf, . s Gk290B ^ ^ Rearranging (8) we have: iV^ = (F - l)Gk 290 B (9) BROAD-BAXD MICROWAVE NOISE SOURCE 611 Equation (9) will now be used to illustrate one method of measuring noise figures. In this method, the network output noise power is measured for two known values of the temperature of the generator resistance, Tn and Ti. When the generator is hot, the output noise power is, by equation (5): I\0H = GkT,B + A';v (10) When the generator is cool, the output noise power is: P:,oc = GkT.B + Ns (11) Calling the ratio of these two noise powers F: _ Pnoh _ GkT^B + Nn /J2') iW ~ GkT.B + Nn Substituting for Nn the value given in equation (9), we have for the noise figure: {m - 0 - K^ - 0 p _ .-- . (13) In practice Ti is often near enough to 290 degrees so that the second term in the numerator of equation (13) is negligible. Setting Ti equal to 290 degrees, equation (13) becomes: F = 1^ (14) The determination of noise figure by this method is independent of the gain of the network, the degree of mismatch and the bandwidth, provided that the band of the noise source is broad compared with the overall RF band of the network and the output power meter. The Noise Source The limitations at microwaves of a noise source such as a heated wdre will now be discussed. In particular we are interested in measuring amplifiers which have noise figures between 10 and 100 (10 db to 20 db) and band- widths up to 200 mc. If a hot wire could be matched to the impedance of a waveguide over a wide enough band, and raised to a temperature of 10 X 290 degrees our F factor would be (rearranging eq. 14): T 1 y = ??C> ^ , (,5) 612 BELL SYSTEM TECHNICAL JOURNAL and setting 2\ = 2900 degrees Kelvin F = 1.9 for F = 10 F = 1.09 for F = 100 Assuming that F can be read to within ±1% our accuracy in determining F would be within about ±1% for F = 10 but only within about ±10% for F = 100. If the noise source had a temperature of 40 X 290 degrees, our experimental errors would be reduced accordingly to about ±1/4% for F = 10 and ±2.5% for F = 100. Since metal wires will not stand such tem- peratures, we must look to something different for the noise source if these accuracies are to be achieved. In view of the foregoing considerations, the nonoscillating reflex klystron presented one possibility of a suitable microwave noise source. This, how- ever, was not exploited because the bandwidth was not wide enough. Another possibility was found to be an electrical gas discharge. This type of source was determined to generate noise at microwave lengths when the open end of the input-waveguide of a sensitive microwave receiver was directed toward various gaseous discharge tubes, including a 721 A TR tube containing water vapor and hydrogen, a neon light m a stroboscope, a mercury vapor rectifier and an ordinary fluorescent desk lamp. Of .these, the commercial fluorescent lamp appeared to lend itself most readily to mounting in a waveguide without the complication of the effects of the internal metal electrodes, so further tests were performed on it. Microwave Measurements A T-5, 6-watt, daylight fluorescent lamp,^ lighted from a d-c. source, was mounted with its axis parallel to the magnetic vector in a waveguide as illustrated in Fig. 2. The lamp itself was 9" long, with cathodes at each end. These could be isolated from the field in the 1" x 2" waveguide by enclosing the portion of the lamp which extended beyond the walls of the waveguide in cylindrical metal shields which formed waveguides beyond cutoff. Tlius, energy was kept from reaching the cathodes, and the noise source was effectively confined to that part of the discharge which appeared inside the main waveguide. A piston in back of the gaseous discharge tube served to tune out the susceptance and a trimming screw provided an additional adjustment. The conductance could be adjusted by varying the direct current. The admittance of the combination could be adjusted for an impedance ^ A commercial tluorescciU lam]) contains about two mm. of argon and si.x to ten microns of mercury gas. The argon merely facilitates the initiation of the discharge; the mercury furnishes the radiation which excites the fluorescent material. B ROAD-BAND MICROWAVE NOISE SOURCE 613 match at any operating frequency from MOi) mc U) 4500 mc. The admittance diagram when the circuit was adjusted for match at .^960 mc is shown in Fig. 3; the standing wave ratio was less than 2.9 db from v^70() to 4240 mc. At 3960 mc the conductance of the gaseous discharge varied cUrectly with the (Urect current, while the negative susceptance had a broad maximum of —j.bl I'o mhos at a current of 65 to 100 milhamperes, as shown in Fig. 4. These values are for the gaseous discharge; the susceptances of the enclosing glass tubing, the back piston and the holes in the sidewalls have been sub- tracted from the measured results. It is interesting to note that the discharge appears to be inductive. The waveguide circuit containing the gaseous discharge tube was con- nected to the input waveguide of a sensitive microwave receiver which was used as a relative noise power meter. The noise power available from the GASEOUS METAL DISCHARGE TUNING 1" X 2" SHIELD TUBE SCREW WAVEGUIDE Fig. 2 — Waveguide circuit for microwave noise generator using a gaseous discharge tube. gaseous discharge was substantially independent of the direct current from 40 ma to 140 ma. These data are plotted in Fig. 5, which gives 10 log ( ^ — 1 j versus direct current in milliamperes. The ordinate has been chosen so as to conform witli absolute measurements made subsequently. The r.m.s. deviation from the straight line which represents a probable coefficient of only —.003 db per milliampere was about ±.05 db. We do not claim to be able to achieve even this degree of accuracy with our present measuring equipment and hence do not place much coniidence in the numerical value of this coefficient. Actually the decrease in noise with increasing current may have been associated with a change in the ambient temperature rather than with the increased current density. At least it is in the right direction for this to be the case. The temperature coefficient of the noise from the discharge was found to be negative; when a piece of dry ice was held on the tubular shield of the circuit for a few minutes (long enough for frost to form on the brass) the output noise power of the discharge increased 0.6 db. The circuit was heated 614 BELL SYSTEM TECHNICAL JOURNAL on a hot plate and allowed to return to room temperature gradually, then cooled with an air stream and allowed to warm up gradually while the output noise and the temperature of the waveguide were being recorded. This re- vealed the temperature coefficient of — .055 db per degree centigrade. The data (plotted in Fig. 6) show an r.m.s. deviation of ±.114 db from this coefficient. ^,e. ALONG LINE /^ Fig. 3 — Admittance diagram of microwave noise generator. The ambient temperature of the waveguide circuit had very little efifect on the admittance of the gaseous discharge. As a check on variability with respect to time, two of these noise sources were compared, one against the other, at tive-minute intervals for 65 min- utes. During this time the waveguide temperature of source # 1 rose from 34° to 35.2° C and that of source # 2 rose from 33.7° to 36.1°. Each compari- son was corrected, according to the coefficient of — .055 db per degree C BROAD-BAM) M ICKDW AV E XOlSE SOL'RCE 615 and the observed temperature, to a common temperature of 34° C. Assum- ing that the noise figure of the microwave receiver was constant, source # 1 1.2 < .4 y /" y \< y /' 1 \ / y \ CONDUCTANCE /> / y y J Z' ^ y 1 < s. N -^ . ^ SUSCEPTAr^CE ^ 0 20 40 60 80 100 120 140 I, MILLIAMPERES. DC Fig. 4 — Admittance of the gaseous discharge at 3960 me as a function of the direct current in the discharge. 16.4 ,.^ 16.2 I l-|o; 16.0 § 15.8 o " 15.6 15.4 30 -0.003 DB/MA± 0.045 DB 1 "~"~~" 1 ~l 1 "^ •' 50 60 70 80 90 100 110 120 130 DISCHARGE CURRENT IN MILLIAMPERES Fig. 5 — The microwave noise power is practically independent of the discharge current. showed variations whose r.m.s. deviation was ± 0.11 db, while source % 2 had similar deviations of ±.092 db. Assuming on the other hand tliat source % 1 held constant and that the microwave measuring set varied with time, 616 BEIJ. SYSTE\r TECHNICAL JOURNAL source ^2 displayed r.m.s. deviations of ±.088 db. These variations are in fact comparable with the probable experimental error, and the proof that they actually exist at all still remains to be demonstrated. Of thirty-two different lamps, including 10 different types of fluorescent coatings such as used in the pink, red, gold, soft white, daylight, green, white, 4500° white, black light and blue, thirty-one^ were all within ±0.25 db of each other as was also a germicidal lamp with no fluorescent coating. Thus it appears that the source of the microwave noise energy lies chiefly in the gaseous discharge rather than in the fluorescent coating. 16.2 16.0 15.8 '^15.6 |o ^15.4 O -* 15.2 o 15.0 14.8 14.6 28 30 32 34 3G 38 40 42 44 46 48 50 52 54 WAVEGUIDE TEMPERATURE IN DEGREES CENTIGRADE Fig. 6 — The microwave noise power depends slightly upon the temperature of the waveguide circuit. V, -0.055 DB/°C ± 0.114 DB V ^ • \ ^V V N^ \, V,^ V > ^^ V \. s_ "' • \ ^ If this noise is tied up with the electron temperature of the discharge, we should expect the noise to be flat, or "white" noise. Corroborative evidence of this was observed when the spectrum of the noise was examined over the band from 3700 to 4500 mc at points 20 mc apart and no irregularities were found. The nature of the experiment was such that frequency bands of ex- cessive noise power would have been observed had they been present. Further tests should indicate whether or not a gradual change in noise with frequency exists. It appears, however, unlikely that such a slope exists at 4000 mc. Furthermore, since the level of the noise energy is so constant with respect to time, reproducible from tube to tube, practically mdependent of the current and only slightly affected by the ambient temperature, we might expect that it is being controlled or limited by some invariant physical property of the atoms and ions within the gaseous discharge. If this is the case, an absolute measurement of the noise power might lead us to some * One of the 32 lamps llickcred erratically. At times its noise was ^ db higher than the average. BROAD B.WI) \//CR(H\ \ Vli NOISE SOURCE 617 theoretical explanation which, when applied to the case in hand, would explain the observed results cjualitatively and cjuantitatively, thereby es- tablishing a new absolute standard noise source for microwave measure- ments. 'I'he micro\vave noise power from such a discharge tube was measured at 3950 mc in cooperation with Mr. C". 1"". Kdwards on his calibrated measuring set on two different occasions, 16 davs apart. The values obtained were T 15.86 (lb and In. 80 db respectively for 10 log ( t^ — 1).'' This places the temperature, 7\ in the neighborhood of 11,400 degrees Kelvin. It is believed that the absolute measurements are correct to within ±.25 db or better. Having determined the temperature of this noise source, we might ask, "If we should terminate our waveguide in a black body at 11,400 degrees, how much microwave noise power would we get from it?" The black body radiates with three polarizations, only one of which is propagated along the waveguide, and this available power is given by Nyquist:^ where /; = 6.61 (10)~^^ joule sec. k = 1.381 (10)--3 joule/deg. / = frequency in cycles per sec. B = bandwidth in cycles per sec. hf At 4000 mc, -■'- is, for T = 290 degrees, 6.6 (10)-" which is so small that the denominator of (16) can be replaced by -^. This gives us the familiar expression for thermal noise: PxA = kTB watts (17) In other words, thermal noise is black body radiation with but one polarization . Going one step further we might also ask the question, "If we should examine the radiation from this black body with an ojitical spectroscope, at what wavelength would we lind its maximum radiated energy?" The spec- trosco{)e detects radiation having three polarizations, and Planck's radia- tion law applies. From Wien's displacement law, the wavelength of maxi- mum radiation is given by the relation: K,r = 0.289 cm deg. (18) "The temperature of the waveguide was 32°C when these values were measured. ^ H. Nyquist, F/iys. Rev., Second Series, Vol. 32, pp. 110-113, July 1928. 618 BELL SYSTEM TECHNICAL JOURNAL Substituting T = 11,400 degrees, X,„ = 2535 (10)-« cm (19) Tliis is indeed an interesting result, since the mercury vapor discharge in the fluorescent lamp radiates most of its energy at X = 2536.52 (10)~* cm. The design of the lamp was guided by the effort to accentuate the radiation at this wavelength, and the manufacturers state that this has been achieved so that no other spectral line is excited to radiate more than two percent of the input power. ^ The conversion loss from dc to 2536 (10)~^ cm is only 2 or 3 db. The striking similarity between the black body and the mercury vapor discharge at these two wavelengths, 7.6 cm and 2536 (10)~^ cm, suggests the following hypothesis: Hypothesis: In a gaseous discharge which is radiating light energy sub- stantially monochromatically at a particular wavelength, X^ , the micro- wave noise energy is the same as that available from a black body which radiates its maximum energy at that wavelength. Applying this hypothesis to the case in hand, where X^ is 2536.52 (10)~^ cm, and using Wien's displacement law (eq. 18) we calculate the tempera- ture to be ^ = 2W2 = ''•'''' ^''^ (Jo - 0 = ''•'' *''' 10 log (25^) = 15.84 db (23) Since this calculated value is so close to the measured values of 15.8 db and 15.86 db, it will be assumed to be correct until proved otherwise. Conclusions A commercial fluorescent lamp is a reliable source of microwave noise energy. At 4000 mc its effective temperature is 11,394 degrees Kelvin which is convenient for measuring noise figures of 20 db or less. The noise power is practically independent of the fluorescent coating, the current density and only slightly affected by the room temperature. The lamp lends itself readily to a broad-band impedance match in the waveguide. 8 G. E. Tnman and R. N. Thaver, A. I. E. E. Transactions, Vol. 57, pp. 723-726, Dec. 1938. Electronic Admittances of Parallel-Plane Electron Tubes at 4000 Megacycles By SLOAN D. ROBERTSON This paper reports the results of some measurements of the electronic admit- tances of close-spaced parallel-plane diodes and "1553" triodes at a frefjuency of 4060 megacycles. These results reveal that the diode admittance and the input short-circuit admittance of the triode depart considerably from the values predicted \>y single-velocity theory. The triode transadmittance, however, is onl\- slightl\- lower in magnitude than the low-frequency value. THE high-frequency admittances of electron streams flowing between parallel-plane electrodes have stimulated considerable theoretical interest. Llewellyn^'--^'^ has given an analysis of the particular case in which all electrons in any plane perpendicular to the direction of flow are assumed to have identical velocities. In practice, this approximation gives a reason- ably accurate expression for electron stream admittances if the electrode spacing is relatively large, and if the frequency is not so high that the actual spread in electron velocities represents an appreciable fraction of the transit time. Others have treated various aspects of the general prob- lgni4,5.6,7,8.9,io_ Theoretical consideration has also been given to the problem of electron flow in which the electrons possess a MaxweUian velocity dis- tribution"'^-'^'-^*. There has been, however, no complete analysis of the microwave-frequency case which takes account of the MaxweUian velocities. In order to orient the present work properly with previous work let us consider briefly the parallel plane diode shown in Fig. 1, which shows three representative potential distribution curves. If only a relatively few elec- trons are available at the cathode, the potential distribution between elec- trodes will be approximately equal to the space-charge-free distribution indicated by curve a. If an ample supply of electrons is provided by the cathode and if all electrons leave the cathode with zero velocity, then the space charge is complete in accordance with Child's law, and the potential distribution follows curve b. If, on the other hand, the cathode is capable of supplying an ample supply of electrons, the electrons being emitted with a MaxweUian velocity distribution, the potential distribution wUl be rep- resented by a curve of the type shown by c. The cases shown by curves a and b can be treated by the Llewellyn analysis. With wide spacings and at lower frequencies the admittances obtained with distributions of the c type may be approximated by the results obtained by analysis of distribu- tions of the b type. With the very close spacings encountered in the Bell 619 620 BELL SYSTEM TECHNICAL JOURNAL Laboratories 1553 triode^^ the theoretical analysis no longer represents a valid approximation. Let us consider curve c in greater detail. The fact that electrons are emitted with a Maxwellian velocity distribution, instead of being emitted at zero velocity as in the Child's law or complete space charge case, means that more electrons are introduced in the space between the electrodes than can flow to the anode in accordance with Child's law. The surplus electrons depress the potential in front of the cathode to a value below that of the cathode. This potential minimum is indicated by Vm in the figure. Electrons which have insufficient energy to cross this barrier return to the cathode. In the space between the cathode and the potential minimum, electrons are found traveling with various velocities in both directions. Between the potential minimum and the anode, electrons travel in one direction only, Fig 1 — Potential distributions in a diode toward the anode, but with multiple velocities. With close spacings and higher frequencies the distance between the cathode and the potential minimum may be an appreciable part of the total cathode-anode spacing, with the result that the electrons returning to the cathode may absorb a substantial amount of power from the high-frequency held. This argument also applies to the cathode-grid region of a microwave triode such as the 1553. In order to increase the transconductance of the triode, it is desirable to locate the grid as close to the cathode as possible. The close spacing, however, leads to a greater loss of power to the returning electrons, which prevents a realization of the full benefits expected from the reduced spacing. All of these difficulties are a result of the Maxwellian veloc- ity distribution of the emitted electrons. In view of the importance of electron stream admittances in the design of microwave amplifiers and of the need for a better understanding of the performance of the 1553, a program was initiated to investigate some of ADMITTANCES OF PARALLEL-!' LANE ELECTRON TUBES 621 these effects experimentally. It seemed best to start this work with a study of the electron stream admittances of simple diodes, with the object of extending the measurements to the triode as the work progressed. Diodes Tlie diodes used in this work were identical in construction with the 155,^ triode, Inil for the substitution of a solid copper anode in place of the grid. Tn all cases the cathode-anode spacing was approximately 0.65 mil, and the area of the cathode was 0.164 square centimeters. With this spacing one would expect the potential minimum to be relatively close to the anode such that a considerable portion of the cathode-anode region would contain electrons moving in both directions. The potential distribution then would be something like that shown in Fig. 2. Fig. 2 — Electron motion in a close-spaced diode. The method used in measuring the microwave-frequency input admit- tances of diodes was based largely on a technique used by Mr. J. A. Morton, and will be described in some detail. In a typical amplifier, radio-frequency power is fed from a waveguide source to the cathode-grid input region of a 1553 triode through a waveguide- cavity transformer. A similar circuit can be used for measuring diode ad- mittances. The fundamental problem is to learn how to relate admittances measured with a standing wave detector located in the wa\cguide supply line to the equivalent two-terminal admittances located at the cathode- anode gap of the diode itself. In other words, we have to know the trans- formation-ratio between an admittance across the cathode-anode gap of the diode and the corresponding admittance which will be measured in the waveguide. Let us refer to the circuit in Fig. 3. The circuit shows an input trans- 622 BELL SYSTEM TECHNICAL JOURNAL mission line which, for example, may be a waveguide having a characteris- tic impedance Zoy, connected through an ideal transformer to an output line having a characteristic impedance Z„x. The output line is connected to the transformer at the point Xo, where Xo represents the gap terminals of the diode. Suppose for the moment that provision has been made for connect- ing the output line at the point in the circuit normally occupied by the cathode-anode planes of the diode. This can be done by means of the special testers shown in Fig. 4. In these testers the anode has been omitted and provision has been made for attaching a coaxial line across the gap between the cathode and anode planes. The diodes used in later tests were identical wdth the device of Fig. 4, except that the coaxial output fitting was replaced by a sheet copper anode. Referring again to Fig. 3, assume that the output line is shorted at point .To . If power is introduced in the input line at the left, a standing wave pattern in the input line will pass through a minimum at some point jo . r --Ax-- OUTPUT Fig. 3 — Equivalent circuit of diode measuring equipment. If the short circuit is now moved to the right by an increment Ax, the stand- ing wave minimum will move by an increment Ay. The relation between A.x; and Ay is given by the following equation: 1 , 27rAv cot 27rA.T

BoZqii A typical cot-cot plot is shown in Fig. 5, (2) (3) ADMITTANCES OF I'ARALLEL-PLANE ELECTRON TUBES 623 Now, assume that the right-liand transmission line is removed and that the diode gap is connected at the transformer terminals Xo . The normalized admittance referred to the point yo on the input line can be measured by a simple standing wave measurement. Represent this admittance by F^y . Fig. 4 — Coaxial tester. Let the unknown diode admittance be represented by F^ . Yx is then given by the following relation: ^ [Yu-o + jp\ (-i) Fx = Z OxW Hence, having determined yo , it is only necessary to measure the slope m and the intercept p on the cot-cot curve in order to relate Y:, to Y,,.g . The characteristic impedance of the output line Zqx used in obtaining the cot-cot plot must also be known. Since a coaxial is used for this line, its charac- teristic impedance is easily calculated. If no losses were associated with the transformer or the parts of the diode external to the actual cathode-anode region, such as the metal vacuum 624 BELL SYSTEM TECH MCA L JOURNAL envelope and certain ceramic details of the tube, the above measurements would give complete information regarding the circuit. Certain losses have been found, however. These are measured as follows: At the time when terminals .vo are shorted a standing wave measurement is made in the wave- cox IZ^ Ax Fig. 5 — Typical cotangent-cotangent plot. guide line at the left. From this measurement and the cot-cot data it is possible to compute an equivalent resistance in series with the gap caused by losses present in the circuit. This equivalent series resistance is given by R.= SWR (5) ADMITTANCES OF TARALLEL PLAN E ELECTRON TUBFJi 625 where SWR is the voltage staiidiiif^ wave ratio mentioned above. The deter- mination of a series loss resistance in this manner is quite analoj^ous to the short-circuit test used in determininjf the losses in a power transformer. There is one other factor in the cot cot technique which is worthy of mention. If, at the very beginning, the output line is terminated in Zo/ and if the transformer is adjusted so that the input line is matched, then tlie value of ;;/ will be unity and p will equal zero. It is then unnecessary to take a cot-cot curve. It is, however, still necessary to locate Vo by shorting the terminals at .to . Diode Admittance at 4060 Megacycles Electron stream admittance measurements with diodes were made in the following way: A coaxial tester was installed and the circuit was adjusted for a slope m of about one. This coaxial tester was then removed and re- placed by another in order to learn whether the slope obtained with one tester would be the same with another, supposedly identical, tester. This process was repeated several times, and the slope was found to vary no more than about 10% from one tester to the other. The procedure was then to replace the coaxial tester with a diode and make admittance measurements with the assumption that the slope would be the same for the diode as for the tester. This assumption was believed to be reasonable since the structure of the diode was identical with that of the tester except that an anode was substituted for the coaxial output connector. In either case all elements that were located inside the w^aveguide cavity were presumably identical. Electron stream measurements were made at a frequency of 4060 mega- cycles with a number of diodes over a wdde range of anode and heater voltages. In making these measurements, the radio-frequency power was kept at a relatively low level (0.2 milliwatt) in order that the measured admittances would be independent of the radio frequency voltage. Results for several diodes are shown in Figs. 6 through 13. The various symbols used in the figures are defined as follows: Vh = heater voltage Ih = heater current I'o = anode voltage (neglecting contact potentials) /o = anode current in ma Jo = anode current density in ma/cm- gQ = low-frequency diode conductance measured with an audio fre- quency bridge g = high-frecjuency diode conductance measured as described above b = high-frequency diode susceptance Rs = equivalent resistance in series with diode In computing the admittance of the electron stream it was necessary to 626 BELL SYSTEM TECHNICAL JOURNAL -1.0 -0.5 0 ANODE VOLTAGE, Vq Fig. 6 — .\dmittance of a diode. allow for the circuit and tube losses previously discussed. The equivalent series resistance Rs of the diode circuit was determined by biasing the tube negatively to the point where a further increase in bias failed to produce a ADMITTANCES OF PARALLEL-PLANE ELECTRON TUBES 627 perceptible change in the waveguide standing wave ratio. Under such conditions the electrons experienced a large retarding field at tlie cathode and did not emerge an appreciable distance into the cathode-anode region. Any resistance measured at tiiis time was due to the series loss and was not 0.5 1.0 ANODE VOLTAGE, Vq Fig. 7— Effect of heater voltage upon diode conductance. produced electronically. The diode series resistances varied from about 1.3 to 5.0 ohms with an average value around 3 ohms. Figure 6 shows the results of admittance measurements of a diode. As expected, the high-frequency conductance is considerably greater than the low-freqiiency value go . In fact g is seen to have a value of several thousand micromhos when the negative bias of the tube is such that no perceptible anode current flows. The susceptance b for large negative anode potentials 628 BELL SYSTEM TECHNICAL JOURNAL has a value of l.SO.OOO micromhos, which agrees fairly well with the value computed from the jfcomctrical capacitance. As anode current is drawn and a space charge conchtion i)revails, b drops to a value of 125,000 micromhos. Theoretical considerations would predict a drop of about 40% in the case of a single-velocity electron stream. This is somewhat greater than the drop exhibited in Fig. 6. n A \ 10 \ DIODE NO. SN61 \ \ \ v e \ , \ HEATER VOLTAGE, Vh / 7 \6.6 ^ \ \ y.A 1 5 \ o\ \ f5 V 4 sA \\ V \ ^. 3 2 V oS k^ V \, --. *^ "^ ^^^ ' ^■o 0 0 0.5 1.0 1. ANODE VOLTAGE, Vq Fig. 8 — Effect of heater voltage upon g/g^. Figures 7 and 8 show the effect of cathode temperature on go and the ratio g/gQ . The parameter used to represent the cathode temperature is the heater voltage Vh ■ As the heater voltage is raised the total conductance g increases. The ratio g/gn , however, decreases, particularly for low or negative anode voltages. This means that, with a given anode voltage, as the cathode tem- perature is raised, go increases more rapidly than g. If the curves of Fig. 8 are replotted in terms of Jo rather than F'o , the ratio g/go is relatively inde- pendent of Vh ■ This is shown in Fig. 9. The results of measurements on another diode are shown in Fig. 10. AD.\flTTANCF.S OF PARALLEL PLANE ELECTROS TUBES 629 These are very similar in all respects to those of the preceding figure. Tt is probable that the eathode-anode spacings of the two diodes of Figs. 6 and 10 were somewhat greater than the 0.65 mil for which they were designed. In both cases the capacitances measured at low frequency were somewhat low. HEATER VOLTAGE, Vh=8.2 DIODE NO. SN 61 "'T^^ 0 20 40 60 80 100 120 140 160 180 200 ANODE CURRENT DENSITY, Jq, IN MA/CM^ Fig. 9 — Variation of g/go with current density and heater voltage. In Fig. 11, results are shown for a third diode. In this case the susceptance at a large negative bias is in almost exact agreement with the value to be expected with the intended diode spacing of 0.65 mil. It is interesting to observe that, with this tube, b drops a greater amount as the current in- creases. Moreover, the ratio g/go is greater than that found with earlier diodes. In Fig. 12 data are shown for a diode having a very high value of go . From the standpoint of cathode activity this was the best tube that was 630 BELL SYSTEM TECHNICAL JOURNAL -3.0 400 < 10^ DIODE NO. SK 519 Wv^ 1 m = 0.64 Rs > \ Rs= 1.34 OHMS Vh = 4.55 volts o IbC too / / / / 80 70 / / yu / / -1 60 50 / 1^ Lv. 60 / / ;>^ ^ «40 O X O30 a. o Z20 HI o Z 15 / 1 r -'lo 5u j ^^ / / <^ / / / / / / < 20 15 10 / //^ 7 / / 1 Q < 10 g 1 ? / 1 1 1 1 1 / r^ , / / / 8 7 6 5 4 3 2 1.5 1 7 / / # / / / 5 / / / / [ 1 1 1 3 2 1.5 1 / J 1 1 1 1 1 1 i -2.5 ■1.5 0.5 -1.0 -0.5 0 ANODE VOLTAGE. Vq Fig. 10 — Admittance of a diode. tried. At maximum current the susceptance h dropped to 50% of the initial value. The data of Fig. 12 have been replotted in Fig. 13 in terms of the variable 126.v'/X/o', where x is the cathode-anode spacing. In the Llewellyn ADMITTANCES OF PARALLEL-PLANE ELECTRON TUBES 631 Fig. -1.0 -0.5 0 ANODE VOLTAGE , Vq 11 — Admittance of a diode. theory this variable is equal to the transit time. The solid curves in the figure are the theoretical results of the Llewellyn theory, whereas the broken curves present the corresponding experimental values. In the latter it should 632 BELL SYSTEM TECHNICAL JOURNAL -1.0 -0.5 0 ANODE VOLTAGE, Vo Fig. 12 — Admittance of a diode. be understood that the abscissa do not represent transit time. The curves do serve, however, to compare the theoretical diode resulting from a single- valued electron velocity assumption with the actual diode in which a Max- ADMITTANCES OT PA RA [.LEI. I' LAX E ELECTRON TUBES 633 wcUian velocity distribution prevails, in the experimental case it is prob- able that, for values of the abscissa greater than 6 or 7, the actual transit time is considerably greater than in the theoretical case. In fact, at a value of 11.4 the anode voltage was zero, the anode current being maintained by the thermal energy of the electrons. Z 3 o o 0 4> DIODE NO. SC189 Vh = 6.2 VOLTS THEORETICAL EXPERIMENTAL y 1 ; 1 1 1 1 / 1 f ;go go/ 1 1 1 / $ / f 1 ^ 1 1 1 1 ^v / / / / J go / <:? • L £§^ '^^^^ ,x' ^ =^ -— -^ >^, y go — 1 2 3 4 5 6^7 8 9 10 11 12 ' '3 AJo''^ Fig. 13— Comparison of thcorclical and experimental values of diode conductance and susceptance. Other diodes were tested, but they exhibited results substantially equiva- lent to those already disclosed. In a few cases anomalous results were ob- tained. With some diodes the capacitance with no electron flow did not approach the low-frequency value. These were rejected on the assumption that there was some mechanical imperfection in the tube which changed the calibration of the measuring equipment. 634 BELL SYSTEM TECHNICAL JOURNAL With the reaUzation that sufficient data are not available to define the phenomena in all detail, it is believed that certain general conclusions can be drawn. From the present work and that of Lavoo^^ and others^'''^*'^', it is ai)parent that the microwave conductance of a close-spaced diode is sub- stantially greater than the low-frequency value. The ratio g/go appears to increase as the spacing decreases. This increase will probably continue until the position of the potential minimum approaches the anode plane. The susceptance decreases with increasing current and appears to level off at high-current densities. The fmal value at a current density of 240 ma cm- varied between 0.5 and 0.9 of the initial value. For a given current density, the ratio g/go does not appear to vary ap- preciably as the cathode temperature is changed. An attempt was made to study the available diodes at 10,000 megacycles. It was found, however, that the value of Rs was so high at this frequency and that variations in tube conductance were so small in comparison with Rs that accurate results could not be obtained. V2Lj,2 V, LJ21 I I I, -* ^— I2 V, \ yii y22 JV2 Fig. 14 — Equivalent circuit of a triode. Four-Pole Admittances of a Triode A triode may be considered as an active linear four-pole transducer, and may be defined by the network of Fig. 14. It is apparent that yii is the input admittance with the output shorted, y22 is the output admittance with the input shorted, yn is the feedback admittance with the input shorted, yn is the transadmittance with the output shorted. The values of the parameters yn , y22 , yn , and y-n to be measured at the grid, cathode, and anode terminals differ from the values of the y admit- tance coefficients given by Llewellyn and Peterson^ who define yn as the admittance of the diode coinciding with the cathode and the fictitious equiva- lent grid plane, and y^o as the admittance between the equivalent grid plane and the anode, and finally y^i as the transadmittance between the two. The relations between the y admittance coefllcients of Llewellyn and Peterson and the coefficients measured by the author are given by Peterson.^ It turns out that, with a high-mu tube, such as the 1553 triode, the two sets of ADMITTAIVCES 01' IW RM.l.EI. PLAN E ELECTRON TUBES 635 coefficients (liller in the order of 10-20% over the useful operating range of current densities; so, for j)ractical considerations, the measure coefficients may he regarded as substantially equivalent to the coefficients referred to the fictitious grid plane. Xot that they will be equal to the theoretical values, but they may be regarded as being associated with the same geometry and will serve at least as a qualitative test of the validity of the theoretical values for the physical tube. In order to measure the four-pole [)arameters, the 1553 triode was mounted in a coaxial circuit of the type shown in Fig. 15. The grid-anode output circuit of the tube is seen to connect directly with the coaxial output line. The input circuit required a more careful design. Due to the size of the base of the tube it was necessary to taper the input coaxial as shown. In the early stages of this work, difficulty was experienced with higher order modes in the large diameter section of the input coaxial. It was believed that these modes were generated by the action of the parallel wire grid which lacked the CATHODE-^ 'ANODE INPUT LINE Fig. 15 — Detail of coa.xial mount for measuring four-pole admittances of a triode. radial symmetry appropriate to coaxial transmission. The difficulty was overcome by constricting the outer diameter of the coaxial line in the im- mediate vicinity of the grid of the tube, thus inhibiting generation of the higher order mode. Before measurements could be made it was necessary lirst to calibrate both the input and the output circuits in a manner similar to that used and described in connection with the diode measurements. The coaxial tester used for calibrating the input circuit was identical with that used for the diode work. For the output circuit a similar tester was use. As one might expect, the value of the cot-cot slope of the output circuit was close to unity. The value actually turned out to be 0.9. In the input circuit the slope was so great that it was difficult to measure, so that it was necessary to introduce a transformer in the coaxial input circuit to permit tuning. The complete apparatus necessary to measure Tn and yio is shown in Fig. 16. This equipment, save for the details already discussed, is quite con- ventional in every respect. 636 BELL SYSTEM TECHNICAL JOURNAL In order to measure yn , the output coaxial line was short-circuited at a point an integral number of half-wave-lengths from the grid-anode terminals of the tube. The admittance measured in the input line could then be used in computing ^'n . To measure J22 , the procedure was reversed, the input line being shorted, and the corresponding admittance being measured in the output line. In either case the normalized line admittances were measured by the standard procedure of determining the standing wave ratio in the line and locating the position of the standing wave minimum with respect to the equivalent terminals of the tube. The transfer admittances were measured with the equipment shown in Fig. 17. The equipment shown here has been fully described in a recent TRANSFORMER GRID PLANE Fig. 16 — Circuit connected for measuring input short-circuit admittance of a triode. paper-" and will be described only briefly here. The output of a signal oscil- lator is divided into two portions. One portion is applied to a balanced modulator where it is modulated by an audio-frequency signal. The sup- pressed-carrier, double-sideband signal from the modulator is applied to the input circuit of the triode. Probes are provided for sampling the voltages Vi and F2 at points an integral number of half wavelengths from the input and output gaps of the tube respectively. The other portion of the oscillator power is fed through a calibrated phase shifter and is applied to a crystal detector in the manner of a local oscillator of a double-detection receiver. The signal samples at V i and V 2 are then alternately applied to the crystal detector where they are demodulated by the action of the homodyne carrier. In each case the phase shifter is adjusted so that the audio signal disappears in the detector output. This occurs when the phase of the homodyne carrier ADMITTANCES OF PARALLEL-PLANE ELECTRON TUBES 637 is in quadrature with tlic signal sidebands. The difference in phase between the two adjustments of the phase shifter is equal to the phase between V i and V ni = slope of input cot-cot curve ni-2 — slope of output cot-cot curve Pi „ P2 Bqi = — ; Brn = 66 Wi ""' 66 W2 Zo = characteristic impedance of input and output coaxial lines. Rs was measured by shorting the output line, placing a large negative bias on the tube, and measuring the admittance of the input line. Then Rs = 66}niRe{Y{) (6) where the number 66 represents the characteristic impedance of the coaxial line used in calibrating the input circuit, corresponding to Zox: in Equation 4. Fortunately for simplicity, the series resistance in the output circuit (7) (8) /as negl igible. The computations are then as f( 3II0WS: y'n 66wi 1 y-11 = F2 66 m2 yu 66mi Fi YxR, + Jpi 66wi 640 BELL SYSTEM TECHNICAL JOURNAL 3'22 = T2 ti^2 + jpi] oonio In order to compute y-n , the following four-pole equations are used: h I I "2}'! 2 y'n y'n ,. -^1 I ^ 2J12 I 1 — — -r yn yn ,. ^2,1 ]>'21 I 2 ~ i- J22 >'22 ^ /2 1 1 1>'21 >'22 >'22 It follows that V,yn ^ Fl>'21 y2i^ . f; Referring to Fig. 19, one may write Fi = V[ - {l[ + F2>';2)i?s Combining (10) and (15) f: 1 Fi 1 - >'iii?s 3^21 can be evaluated by making use of the relation n ^ 4 + ^' >'22 >'22 Dividing (17) by V2 and rearranging terms }'22 722 r r' -1 where /2/F2 can be expressed as 1 1 F2 66m2Zo 66w2 (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) where Zo = 1- F(/F2 can be expressed in terms of 721 , nii , and ;;/o by using the relations: 1 1 _ . /66mi ^ 2 _ /66 66m2 (20) ADMITTANCES OF J'ARALLEL-FLANE ELECTRON TUBES 641 Solving (20) for ^17^2 and remembering that V1/V2 = rn , Tf = 721 A/ — V2 y nii (21) If (19) and (21) are substituted in (18), one finds 721 y mx\_ o6m2 ji^J By using (14) and (16), y-n can then be written as 1 ' / — r ^ J22 , /mi . I 721 ~ — 4/ — 1 + .^ 721 \ Vll L 66W..>'22 1 _ 1 — ynRs (23) Several 1553 triodes were available for study. Typical experimental results obtained with two of them are shown in Figs. 20, 21, and 22. The triode used in obtaining the data of Fig. 20 had input and output .spacings of 0.65 and 12 mils, respectively. The cathode and anode diameters were 180 mils. The grid opening was 250 mils and was wound with 0.3 mil tungsten wire at 1000 strands per inch. In the figures, Vg and Vp represent the d-c. grid and plate potentials, respectively. There are a number of interesting things to observe in Fig. 20. As with the diode, 611 for a large negative bias approaches the "cold" value computed from the capacitance. However, as anode current is drawn, bn drops rapidly to a much lower value than was the case for the diodes. The conductance ^u behaves somewhat like g for the diode. 622 is equal to the value computed from the grid-anode capacitance and is not appreciably influenced by the electron stream. ^22 was very low with a magnitude of slightly less than 1000 micromhos at maximum anode current. It is not shown in the figure. The transadmittance ^21 is worth considering. When the bias is several volts negative, y2i has a value of about 9000 micromhos. This is about 50 times as high as one would expect from a consideration of the electrostatic capacitance between the cathode and anode of the tube. This effect has been investigated more fully and is discussed in another paper.-^ As the tube starts to draw plate current, ^21 rises and reaches a maximum of about 40,000 micromhos. The low-frequency transconductance was measured and is plotted in the figure. It will be observed that the high-frequency transadmittance is only slightly lower than gm . This is in agreement with the theories of Llewellyn. The agreement appears reasonable when one remembers that, in the theo- retical analysis, the magnitude of the ratio yn/go is relatively independent of the transit time in the input space. Figure 21 shows the results of measurements on a triode identical with that of Fig. 20 except that the grid consists of a mesh of 0.3 mil tungsten wires wound at 550 strands per inch in both directions. It will be noted 642 BELL SYSTE.\[ TECHNICAL JOURNAL X103 -3.5 -3.0 -2.0 -1.5 -1.0 -0.5 GRID VOLTAGE , Vg Fig. 20 — Four-pole admiltances of a triodc having a parallel-wire grid. that y-n is much lower when this tube is biased beyond cutofif than in the pre- vious case. The electromagnetic coupling is therefore much less for the mesh grid. This has also been treated in the above reference.-^ With high negative bias the feedback admittance yx2 was substantially equal to y-n ADMITTANCES OT I'AlLiLLEL-PLANE ELECTRON TUBES 643 100 90 80 70 60 50 10 40 O I O 30 tr o 5 Z 20 < 10 9 8 7 6 5 - Il TRIODE NO. TS106 CROSS LATERAL GRID Rs= 3.40HMS Vh= 7.2 VOLTS Vp = 250 VOLTS \^f\r jV^y 12 v,.X 1 _brf_ -f — f^ — 1- o < 10 8 ^ z 6 Ol q: tr 5 D 3 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 GRID VOLTAGE , Vg Fig. 21 — Four pole admit tances of a Iriocle having a cross-lateral grid. but, as the current density increased, jn tended to decrease. The feedback admittance was ahvays lower for the mesh grid than for the parallel-wire crid. 644 BELL SYSTEM TECHNICAL JOURNAL The remaining parameters for the triode of Fig. 21 are very similar to those of Fig. 20. Figure 22 shows the variation of the phase of the transadmittances yn for the two triodes. The figure also shows the theoretical curve of the Llewellyn analysis for purposes of comparison. As in the case of Fig. 13 the abscissa do not represent transit time for the experimental values. The quantity x is equal to the cathode-grid spacing. It is of interest to compare the triode measurements with those of the diode. It was expected that gn for the triode should correspond with g for the diode. Within the limits of reasonable experimental accuracy this appears to be the case. For the triode at low frequencies go ^ gm- The triode ■~~~^ -\ 0 PARALLEL WIRE • CROSS-LATERAL GRID ^ \ N^HEORETICAL • • • • \^ • o 0 d \ s. \ \ V 5 (26 X 1/3 Fig. 22 — Phase of triode transadmittance. results indicate that the ratio gn/gm is quite comparable in magnitude with the corresponding ratio g/go for the diode. This was expected. The behavior of ill for the triode was unexpected. It was thought that, as the grid voltage was varied so that the input space changed from a condition of zero space charge to one of maximum space charge, hn would vary from its initial "cold" value to a value approaching 60% of the latter. This was not so. In the figures one observes that it drops to a much lower value. This effect has not been explained from a theoretical standpoint. There are several qualitative interpretations, but as yet no way of determinmg which of them is correct m a quantitative sense has been found. The observed phenomenon could, for example, be explained by an increase in the effective series resis- tance of the tube caused perhaps by an increase in the resistance of the ADMITTANCES OT PARALLEL-PLANE ELECTRON TUBES 645 cathode coating." Since the effect was not observed to such a marked degree in the case of the diodes, it seems probable that this is not the correct explanation. It is probable that the ol)serve(l variation in bn is a space charge effect. Tt is evident in examining the diode curves that tubes which possessed the higher values for ^o exhibited a greater variation in b. If maximum go can be taken as a measure of the cathode activity, we can then perhaps relate the variation in susceptance with cathode activity and hence with the loca- tion of the potential minimum. A shift in the position of the potential minimum, however, may produce two effects. It varies the transit time of the electrons and changes the degree of space charge in the input space. Either effect might account for the variation of bn- A clue to this effect might be discovered by making measurements on structures with different cathode-grid spacings. The following experiments were performed to determine the effect of plate voltage on the input admittance of the triode of Fig. 20. The plate and grid voltages were varied simultaneously in such a way that the sum of the direct currents to the grid and plate remained constant at 30 milliamperes cor- responding to a current density of 184 ma/cm^. The input admittance did not vary from the value shown for this same current density in Fig. 20 even though the plate voltage was varied from 250 volts to 40 volts. In a second experiment the plate potential was maintained at —90 volts with respect to the cathode and the grid potential was varied such that the direct grid cur- rent varied over a range of 0 to 10 milliamperes. Again the admittances were found to be equal to those of Fig. 20 for the corresponding total currents. These two experiments suggest that, for a given geometry, the value of bn is primarily a function of the total current density in the input circuit. Acknowledgement The author wishes to acknowledge his indebtedness to the late Mr. A. E. Bowen who contributed valuable advice during the course of these experi- ments, to Messrs. J. A. Morton and M. E. Hmes who provided the neces- sary tubes and testers which made this work possible, and to Mr. F. A. Braun who played an indispensable role in the taking and reduction of data. Rkferences 1. "Electron Inertia Effects," F. B. Llewellyn, Camljridge University Press. 2 "Equivalent Networks of Negative Grid Vacuum Tubes at Ultra-High Frequencies," F. B. Llewellyn, B. S. T. J., Vol. 15, i)p. 565-586, October 1936. 3. "Operation of Ultra-High Frec|ucncy Vacuum Tubes," F. B. Llewellyn, B. S. T. J., Vol. 14, pp. 632-665, October 1935. 4. "Theory of the Internal Action of Thermionic Systems at Moderately High Frequen- cies," VV. E. Benham, Phil Mag., Vol. 5, pp. 641-662, March 1928; and Vol. 11 pp. 457-517, Feb. 1931. 646 BELL SYSTEM TECHNICAL JOUliNAL 5. "Vacuum-Tube Networks," F. B. Llewellyn and L. C. Peterson, Proc. I. R. E., Vol. 32, no. 3, pp. 144-166, March 1944. 6 "Equivalent Circuits of Linear Active Four-Terminal Networks," L. C. Peterson, B. S. T. /., Vol. XXVII, No. 4, pp. 593-622, October 1948. 7. "Impedance Properties of Electron Streams," L. C. Peterson, B. S. T. J., Vol. 18, pp. 465-481, July 1939. 8. "Klystron and Microwave Triodes," Hamilton, Knijjp and Kuper, pp. 97-169, Radiation Laboratory Series, Vol. 7, McGraw-Hill, 1948. 9. "High-Freciuency Total Emission Loading in Diodes," Nicholas A. Begovich, Journal Applied Physics, Vol. 20, No. 5, pp. 457-461, May 1949. 10 "On the Velocity-Dependent Characteristics of High-Frequency Tubes," Julian K. Knijip, Journal Applied Physics, Vol. 20, No. 5, pp. 425-431, May 1949. 11. I. Langmuir, Phys. Rev., 21, pp. 419-435, 1923. 12. "Extension and Application of Langmuirs' Calculations on a Phase Diode with Max- wellian Velocity Distribution of the Electrons," A. Van Der Ziel, Philips Research Reports, Vol. 1, No. 2, pp. 97-118, January 1946. 13. "Extension of Langmuir's (?, 77) Tables for a Plane Diode with a Maxwellian Dis- tribution of the Electrons," P. H. J. A. Klegmen, Philips Research Reports, Vol. 1, No. 2, January 1946, pp. 81-96. 14. "Some Characteristics of Diodes with Oxide-Coated Cathodes," W. R. Ferris, R.C. A. Review, Vol. X, No. 1, pp. 134-149, March 1949. 15 "A Microwave Triode for Radio Relay," J. A. Morton, Bell Laboratories Record, Vol. XXVn, No. 5, May 1949. 16 "Transadmittance and Input Conductance of a Lighthouse Triode at 3000 Mega- cycles," Norman T. Lavoo, Proc. I. R. E., Vol. 35, No. 11, pp. 1248-1251, Novem- ber 1947. 17 "Total Emission Damping in Diodes," A. Van Der Ziel, Nature, Vol. 159, No. 4046, May 17, 1947, pp. 675-676, (52 mc). 18. "Total Emission Damping," J. Thomson, Nature, Vol. 161, No. 4100, pp. 847, May 29, 1948. 19. "Total Emission Damping with Space-Charge Limited Cathodes," C. N. Smyth, Nature, 157, 841, June 22, 1946. 20. "A Method of Measuring Phase at Microwave Frequencies," S. D. Robertson, B. S. T. J., Vol. XXVIII, No. 1, pp. 99-103, January 1949. 21. "Passive Four-Pole Admittance of Microwave Triodes," S. D. Robertson, this is- sue of the B. S. T. J. 22 "Total Emission Noise in Diodes," A. Van Der Ziel and A. Versnel, Nature, Vol. 159, No. 4045, pp. 640-641, May 10, 1947. 23. "Ultra-High-Frequency Oscillations by Means of Diodes," F. B. Llewellyn and .\. E. Bowen, B. S. T. J., Vol. 18, pp. 280-291, April 1939. Passive Four-Pole Admittances of Microwave Triodes By SLOAN D. ROBERTSON Measurements have been made of the passive, four-pole admittances of parallel- plane triodes over a wide range of calhode-to-grid and grid-to-piate spacings at a frequency of 4060 megacycles. Results are given for a parallel wire grid and a cross-lateral grid. The microwave transadmittances are found to be much higher than the values measured at low frc.iuencies. DURING the course of an experimental study of the active four-pole admittances^ of the 1553 close-spaced triode,'- a question arose as to whether the grid wires were introducing any appreciable inductance or resistance in the circuit used for measurement. It appeared necessary, therefore, to learn something of the passive four-pole parameters of the triode in order to separate the electronic from the passive admittances. It was generally believed that the electrostatic analyses of the passive admit- tances which have been successfully applied at the lower frequencies would no longer be valid with close-spaced structures at microwave frequencies. For example, it was considered possible that the grid wires themselves might possess an effective inductive reactance, so that the admittances between the grid and cathode or between the grid and anode might not be equal to the values computed from the electrostatic capacitances. Moreover, it was thought likely that energy might be transmitted from the cathode-grid region to the cathode-plate region or vice versa, not only by the medium of the electrostatic coupling, but also by means of an electromagnetic coupling through the grid. The measurements to be reported below indicate that the first of these conjectures was false, but that the second was true. In view of the lack of available information on these questions in general, it seemed highly desirable to employ the available measuring equipment, not only to determine the passive parameters of a triode having electrode spacings corresponding with those of the 1553, but to extend the scope of the measurements to include a wide range of electrode spacings in order that the results would be of more general interest. Although these measurements were in principle very simple, in practice the mechanical problem of achieving the desired degree of accuracy proved rather difficult. It was required that the cathode, grid, and anode planes be almost perfectly parallel and that the spacuigs between them be adjustable ^ S. D. Robertson, "Electronic Admittances of Parallel-Plane Electron Tubes at 4000 Megacycles," this issue of the B. S. T. J. ^ J. A. Morton, "A Microwave Triode for Radio Relav," Bell Laboratories Record, Vol. XXVII, No. 5, pp. 166-170, May 1^49. 647 648 BELL SYSTEM TECHNICAL JOURNAL to specific values with a high degree of precision. In order to equal the dimen- sional tolerances of the 1553 it was necessary that parallelism and spacing be accurate to 0.1 mil. A schematic diagram of the apparatus is shown in Fig. 1. A flat, circular disc having a 250-mil diameter aperture, across which the grid was stretched, was mounted upon the face of the hollow micrometer screw 7^ 1. The latter was mounted so that its face was accurately parallel with the end face of the central conductor of the input coaxial line in the upper part of the figure. By means of the micrometer ^ 1 the input spacing Si, which we shall con- sider as representing the cathode-grid spacing, could be adjusted. The cen- tral conductor of the coaxial line was insulated at d.c. from the outer con- ductor; hence it was possible to use an ohmmeter to indicate when the grid was just touching the coaxial face. The micrometer could then be backed away from the grid by any desired amount. The input coaxial was fitted with a standing wave detector in the form of a probe which could be moved along the line and placed at any arbitrary distance h from the grid. On the output side of the circuit, in the lower part of the figure, there was another coaxial line arranged so that its center conductor could be driven by micrometer ^ 2. The latter was insulated from the outer con- ductor of the coaxial by means of a condenser in order that an ohmmeter could be used to determine the position of the micrometer which caused the central conductor to just touch the grid. Spacing ^2 could then be adjusted. The output coaxial line was terminated in its characteristic impedance of 62 ohms. At a distance of X/2 from the grid a probe was located for samplmg the power in the output line. The diameter of the input coaxial conductor was 180 mils at the end. In the figure it will be noted that at a short distance from the end the diameter increased to a larger diameter (250 mils). Because of the required length of the central conductor, it was necessary to increase its size in this way for mechanical rigidity. The effect of this change in cross-section was computed and allowed for in the final results. The output coaxial conductor was relatively short, so that it was possible to assign a diameter of 180 mils for its entire length. The 180-mil diameter was selected to correspond with the diameters of the cathode and anode in the 1553 triode. The procedure for making the measurements was as follows: With a particular set of spacings Si and S-i the standing wave ratio in the input line was measured. This ratio, together with the measurement of the posi- tion of a standing wave minimum, permitted the calculation of an input admittance F to be made. Then with the standing wave detector probe placed at a distance h = X/2 from the grid, the ratio of the voltage at the input terminals of the triode to the voltage appearing at the output probe was measured both as to magnitude and phase as described in a recent FOUR-POLE ADMITTANCES OF MICROWAVE TRIODES 649 STANDING WAVE DETECTOR OHMMETER SPACINGS MICROMETER NO. 1 CONDENSER /^ j OHMMETER MICROMETER NO. 2 Fig. 1 — Apparatus for measuring passive admittances of triode. 650 BELL SYSTEM TECHNICAL JOURNAL paper .^ This quantity will be called 7. These measurements were sufficient for an evaluation of the four-pole parameters of the structure. All measure- ments were made at a frequency of 4063 megacycles. The equivalent circuit of the passive triode structure is shown in ¥\g. 2. The desired parameters are yn, yn, and y22. The following equations indi- INPUT V,| yi2 yi.-yi2 y22-y.2 1^2 >62W Fig. 2 — Equivalent passive circuit of a triode. Fig PARALLEL-WIRE GRID 3 — Types of grids used in the measurements. CROSS-LATERAL GRID cate the relation between these parameters and the measured quantities Y and 7 : ,„ = 1- + J>^^ « Y (1) 62^22 + 1 yi2 y22 7 1 + 1 62 V. (2) where the number 62 represents the output terminating impedance. For all cases to be described here the second term on the right side of Equation 1 is small in comparison with F. This is a result of the small values encountered for yi2 . To a good approximation yn is equal to the measured input ad- mittance F. This was verified by observing the variation m input admittance as the output spacing was varied while keeping the input spacing fixed. Only a slight variation in admittance was observed, which indicated that the fractional term in Equation 1 was small in comparison with F. Suppose, then, that for a given input and output spacing Si and .S'2 , ' "A Method of Measuring Phase at Microwave Frequencies," S. D. Robertson, Bell System Technical Journal, Vol. XXVIII, No. 1, pp. 99-103, January 1949. rOUK-POLE ADMITTANCES 01- MICROWAVE TRIODES 651 Y and 7 are known. J22 can readily be determined by readjusting the input spacing to equal the outj)ut spacing and measuring a second admittance V . A'22 will be aijproximately equal to this value. There is, then, sulTicient information to compute yi2 . ^' 0.1 a. O 0.09 ^ 0.08 t|[ 0.07 2 ^ 0.06 Q 0.06 f = 4060 MC \ \ \ \ \ k \ N \. N K N ^ N > '^^ ^ ) ^ 0 1 23456789 10 11 12 SPACING, S, OR S2,IN MILS Fig. 4 — Variation of passive input and output admittances with spacing. Two grids were used in this work. The first was a parallel wire grid of 0.3 mil tungsten wire wound at 1000 turns per inch. The second was also of 0.3 mil tungsten wound in a crisscross fashion at 550 turns per inch. Both grids are shown in Fig. 3. It will be noted that the cross-lateral grid has an aper- ture 220 mils in diameter. The values of ju and ^'22 were found to be almost purely capacitive and were the same for both types of grid. These values are shown in Fig. 4. A'u and y->i correspond to capacitances C'n and C22 , which agree surprisingly well with the calculated capacitances between the grid and cathode, and grid and plate planes, respectively. Figure 5 shows the experimentally 652 BELL SYSTEM TECHNICAL JOURNAL determined values of Cn and C22 plotted as a dashed curve. The theoretical values (neglecting fringing capacitance) are shown by the solid curve. Since fringing was neglected, it is not surprising that the measured capacitances should exceed the calculated values by the amount shown. (5 10 Q < ? 0 0 EXPERIMENTAL 1 1 \ i \ \ v \\ \\ \\ \\ \\ \\ V I ^, ^^^ "**T ' L ' *>- , \ ^ j> 10 12 13 01 23456789 SPACING, S, OR S2,IN MILS Fig. 5 — Comparison of theoretical and experimental values of input and output ca- pacitances. The magnitudes of yn for each grid over a range of values of Si and Si are shown in Figs. 6 and 7. It will be noted that, for a given set of spacings 5i and S-i , yu is much greater for the parallel wire grid than for the cross- lateral. This is the sort of result one would e.xpect if y^ resulted from electro- magnetic coupling through the grid, since the parallel wire grid would be expected to offer a better transmission path than the cross-lateral grid. It was not practicable with the equipment used in these experiments to measure the values of yu at low frequencies where yn would be determined by the cathode-plate capacitance. Data were available, however, for the low-frequency, cathode-plate capacitance of the standard, parallel-wire FOUR-POLE ADMITTANCES OF MICROWAVE TRIODES 653 f = ^.060 MC < \ ( \ -- i\ V i i\ S V ( \ ^ \ N \ I ^ \ N N ^ ^, ( \ \ \ N N ( [^ ^, ^ ^ ^' '^/ L^S ^0.5 ">>- \ \ \, "V ^, N^ K N ». N, ^ ) \ \, ( ks ^^ iO-^-s IN ) \ \, ^ k *"<. ) \^ \ s c ks "^ ) 1 \ < K. ^^ k^ [ r v^ ^ \ s ^N, \ ^ '^. \ N \ s s \ '^ \. ) <^ ~~- >^ ) N k s \ ^ ^ ) ^ '4 N V \ N, \ ^ "^^ ^ ^ ^ ^ > ^ \ ^ ^ N^ ^*> f-^. '*^ b^ ) ^^ K > ) 6 7 8 9 10 SPACING , S2,IN MILS Fig. 6 — Passive transadmittances of a triode having a parallel wire grid of 0.3 mil wire wound at 1000 turns per inch. 654 BELL SYSTEM TECHNICAL JOURNAL 5 6 7 6 9 10 SPACING, S2,IN MILS 13 14 \b 16 Fig. 7 — Passive transadmittances of a triode having a cross-lateral grid of 0.3 mil wire wound at 550 turns per inch. ^ttl'20 f = 4060 MC S > > c ( k iJ^:?-^-^ 1 01 2345670 SPACING, 52, IN MILS Fig. 8 — Phase of the transadmittance of the parallel-wire grid. VOUR-FOLE ADMITTANCES OT MICROWAVE TRIODES 655 grid, 1553 triode having input and output spacings of 0.5 and 12 mils respec- tively. The capacitances averaged about 0.008 /x/xf, which would correspond to a value of Vi2 of 0.0002 mho at 4060 megacycles. The latter is about 50 times lower than the measured 4060 megacycle value. Evidently, therefore, electromagnetic coupling plays a dominant role. Reciprocity should give a reasonable idea of the accuracy of these meas- urements. Thus, for Si = 0.001" and ^2 = 0.012'', one would expect the same yn as for the case where Si = 0.012" and S2 = 0.001". An examination of the data will indicate that the reciprocal differences are of the order of 10% in some cases. These differences may be partly the result of the change in line cross section encountered in going from the input to the output. That is to say, the two cases bemg compared are not quite reciprocal in geometr)'. Figure 8 shows the phase of yn for the parallel wire grid. Because of the low transmission through the grids there was not sufficient energy to deter- mine the transfer phases with any very great accuracy, particularly for wide spacings in the case of the parallel wire grid and for all spacings in the case of the cross-lateral. Consequently, Fig. 8 shows only those results which are believed to be reasonably accurate. The author wishes to acknowledge the contribution of Mr. F. A. Braun who ably assisted in this work. Communication Theory of Secrecy Systems* By C. E. SHANNON 1. Introduction and Suivimary THE problems of cryptography and secrecy systems furnish an interest- ing appUcation of communication theory.^ In this paper a theory of secrecy systems is developed. The approach is on a theoretical level and is intended to complement the treatment found in standard works on cryp- tography.- There, a detailed study is made of the many standard types of codes and ciphers, and of the ways of breaking them. We will be more con- cerned with the general mathematical structure and properties of secrecy systems. The treatment is limited in certain ways. First, there are three general types of secrecy system: (1) concealment systems, including such methods as invisible ink, concealing a message in an innocent text, or in a fake cover- ing cryptogram, or other methods in which the existence of the message is concealed from the enemy; (2) privacy systems, for example speech inver- sion, in which special equipment is required to recover the message; (3) "true" secrecy systems where the meaning of the message is concealed by cipher, code, etc., although its existence is not hidden, and the enemy is assumed to have any special equipment necessary to intercept and record the transmitted signal. We consider only the third type — concealment systems are primarily a psychological problem, and privacy systems a technological one. Secondly, the treatment is limited to the case of discrete information, where the message to be enciphered consists of a sequence of discrete sym- bols, each chosen from a finite set. These symbols may be letters in a lan- guage, words of a language, amplitude levels of a "quantized" speech or video signal, etc., but the main emphasis and thinking has been concerned with the case of letters. The paper is divided into three parts. The main results will now be briefly summarized. The first part deals with the basic mathematical structure of secrecy systems. As in communication theory a language is considered to * The material in this paper appeared originally in a confidential report "A Mathe- matical Theory of Crj^stography" dated Sept. 1, 1945, which has now been declassified. 1 Shannon, C. E., "A Mathematical Theory of Communication," Bell System Technical Journal, July 1948, p. 379; Oct. 1948, p. 623. 2 See, for example, H. F. Gaines, "Elementary Cryptanalysis," or M. Givierge, "Cours de Crj^tographie." 656 COMMUNICATION THEORY OF SECRFX'Y SYSTEMS 657 he represented by a stochastic process which produces a discrete sequence of s)-mbols in accordance with some system of probabilities. Associated with a language there is a certain parameter D which we call the redundancy of the language. D measures, in a sense, how much a text in the language can l)c reduced in length without losing any information. As a simple example, since u always follows q in English words, the u may be omitted without loss. Considerable reductions are possible in English due to the statistical struc- ture of the language, the high frequencies of certain letters or words, etc. Redundancy is of central importance in the study of secrecy systems. A secrecy system is defined abstractly as a set of transformations of one space (the set of possible messages) into a second space (the set of possible cryptograms). Each particular transformation of the set corresponds to enciphering with a particular key. The transformations are supposed rever- sible (non-singular) so that unique deciphering is possible when the key is known. Each key and therefore each transformation is assumed to have an a priori probability associated with it — the probability of choosing that key. Similarly each possible message is assumed to have an associated a priori probability, determined by the underlying stochastic process. These prob- abilities for the various keys and messages are actually the enemy crypt- analyst's a priori probabilities for the choices in question, and represent his a priori knowledge of the situation. To use the system a key is first selected and sent to the receiving point. The choice of a key determines a particular transformation in the set forming the system. Then a message is selected and the particular trans- formation corresponding to the selected key applied to this message to produce a cryptogram. This cryptogram is transmitted to the receiving point by a channel and may be intercepted by the "enemy*." At the receiving end the inverse of the particular transformation is applied to the cryptogram to recover the original message. If the enemy intercepts the cryptogram he can calculate from it the a posteriori probabilities of the various possible messages and keys which might have produced this cryptogram. This set of a posteriori probabilities constitutes his knowledge of the key and message after the interception. "Knowledge" is thus identified with a set of propositions having associated probabilities. The calculation of the a posteriori probabilities is the gen- eralized problem of cryptanalysis. As an example of these notions, in a simple substitution cipher with ran- dom key there are 26! transformations, corresponding to the 26! ways we *The word "enemy," stemming from military applications, is commonly used in cryiv tographic work to denote anyone who may intercept a cryptogram. 658 BELL SYSTEM TECHNICAL JOURNAL can substitute for 26 different letters. These are all equally likely and each therefore has an a priori probability 1/26!. If this is applied to "normal English" the cryptanalyst being assumed to have no knowledge of the message source other than that it is producing English text, the a priori probabilities of various messages of .V letters are merely their relative frequencies in normal English text. If the enemy intercepts N letters of cryptogram in this system his prob- abilities change. If N is large enough (say 50 letters) there is usually a single message of a posteriori probability nearly unity, while all others have a total probability nearly zero. Thus there is an essentially unique "solution" to the cryptogram. For .V smaller (say X = 15) there will usually be many messages and keys of comparable probability, with no single one nearly unity. In this case there are multiple "solutions" to the cryptogram. Considering a secrecy system to be represented in this way, as a set of transformations of one set of elements into another, there are two natural combining operations which produce a third system from two given systems. The hrst combining operation is called the product operation and cor- responds to enciphering the message with the first secrecy system R and enciphering the resulting cryptogram with the second system S, the keys for R and 5 being chosen independently. This total operation is a secrecy system whose transformations consist of all the products (in the usual sense of products of transformations) of transformations in S with transformations in R. The probabilities are the products of the probabilities for the two transformations . The second combining operation is "weighted addition." T = pR+ qS /' + (/=! It corresponds to making a preliminary choice as to whether system R or S is to be used with probabilities p and q, respectively. When this is done i? or 5 is used as originally defined. It is shown that secrecy systems with these two combining operations form essentially a "linear associative algebra" with a unit element, an algebraic variety that has been extensively studied by mathematicians. Among the many possible secrecy systems there is one type with many special properties. This type we call a "pure" system. A system is pure if all keys are equally likely and if for any three transformations T,, Tj, Ti-, in the set the product T.Tr'Tk is also a transformation in the set. That is enciphering, deciphering, and enciphering with any three keys must be equivalent to enciphering with some key. COMMUNICATION THEORY Oh SECRECY SYSTEMS 659 With a pure riphcr il is shown that aU keys arc essentially efiuivalent— tliey all lead to the same set of a posteriori probabilities. l''urthermore, when a ^'iven cryptogram is intercepted there is a set of messages that might have produced this cryptogram (a "residue class") and the a posteriori prob- abilities of messages in this class are proportional to the a priori proliabilities. All the information the enemy has obtained by intercepting the cryptogram is a specification of the residue class. Many of the common ciphers are pure systems, including simple substitution with random key. In this case the residue class consists of all messages with the same pattern of letter repeti- tions as the intercepted cryptogram. Two systems R and 5 are defined to be "similar" if tliere exists a fixed transformation A with an inverse, /1~', such that R = AS. If R and S are similar, a one-to-one correspondence between the resulting cryptograms can be set up leading to the same a posteriori probabilities. The two systems are crypt analytically the same. The second part of the paper deals with the problem of "theoretical secrecy." How secure is a system against cryptanalysis when the enemy has unlimited time and manpower available for the analysis of intercepted cryptograms? The problem is closely related to questions of communication in the presence of noise, and the concepts of entropy and equivocation developed for the communication problem find a direct application in this ]iart of cryptography. "Perfect Secrecy" is defined by requiring of a system that after a crypto- gram is intercepted by the enemy the a posteriori probabilities of this crypto- gram representing various messages be identically the same as the a priori probabilities of the same messages before the interception. It is shown that perfect secrecy is possible but requires, if the number of messages is hnite, the same number of possible keys. If the message is thought of as being constantly generated at a given "rate" R (to be defined later), key must be generated at the same or a greater rate. If a secrecy system with a finite key is used, and X letters of cryptogram intercepted, there will be, for the enemy, a certain set of messages with certain probabilities, that this cryptogram could represent. As A' increases the field usually narrows down until eventually there is a unique "solution" to the cryptogram; one message with probability essentially unity while all others are practically zero. A quantity //(A^) is defined, called the equivoca- tion, which measures in a statistical way how near the average cryptogram of N letters is to a unique solution; that is, how uncertain the enemy is of the original message after intercepting a cryptogram of A' letters. \'arious properties of the equivocation are deduced— for example, the equivocation 660 BELL SYSTEM TECHNICAL JOURNAL of the key never increases with increasing N . This equivocation is a theo- retical secrecy index — theoretical in that it allows the enemy unlimited time to analyse the cryptogram. The function H{X) for a certain idealized type of cipher called the random cipher is determined. With certain modifications this function can be applied to many cases of practical interest. This gives a way of calculating approxi- mately how much intercepted material is required to obtain a solution to a secrecy system. It appears from this analysis that with ordinary languages and the usual types of ciphers (not codes) this "unicity distance" is approxi- mately H{K)/D. Here H{K) is a number measuring the "size" of the key space. If all keys are a priori equally likely H{K) is the logarithm of the number of possible keys. D is the redundancy of the language and measures the amount of "statistical constraint" imposed by the language. In simple substitution with random key H{K) is logio 261 or about 20 and D (in decimal digits per letter) is about .7 for English. Thus unicity occurs at about 30 letters. It is possible to construct secrecy systems with a finite key for certain "languages" in which the equivocation does not approach zero as i\' ^ oo. In this case, no matter how much material is intercepted, the enemy still does not obtain a unique solution to the cipher but is left with many alter- natives, all of reasonable probability. Such systems we call ideal systems. It is possible in any language to approximate such behavior — i.e., to make the approach to zero of H{y) recede out to arbitrarily large T. Hov»'ever, such systems have a number of drawbacks, such as complexity and sensi- tivity to errors in transmission of the cryptogram. The third part of the paper is concerned with "practical secrecy." Two systems with the same key size may both be uniquely solvable when X letters have been intercepted, but differ greatly in the amount of labor required to effect this solution. An analysis of the basic weaknesses of sec- recy systems is made. This leads to methods for constructing systems which will require a large amount of work to solve. Finally, a certain incompat- ibility among the various desirable qualities of secrecy systems is discussed. PART I MATHEMATICAL STRUCTURE OF SECRECY SYSTEMS 2. Secrecy Systems As a first step in the mathematical analysis of cryptography, it is neces- sary to idealize the situation suitably, and to define in a mathematically acceptable way what we shall mean by a secrecy system. A "schematic" diagram of a general secrecy system is shown in Fig. 1. At the transmitting COMMUNICATION THEORY OF SECRECY SYSTEMS 661 end there are two information sources — a message source and a key source. The key source produces a particular key from among those which are possible in the system. This key is transmitted by some means, supposedly not interceptible, for example by messenger, to the receiving end. The message source produces a message (the "clear") which is enciphered and the resulting cryptogram sent to the receiving end by a possibly inter- ceptible means, for example radio. At the receiving end the cryptogram and key are combined in the decipherer to recover the message. ENEMY CRYPTANALYST E CRYPTOGRAM MESSAGE SOURCE MESSAGE ENCIPHERER DECIPHERER MESSAGE M E ' E M KEY KEY K KEY SOURCE Fig. 1 — ^Schematic of a general secrecy system. Evidently the encipherer performs a functional operation. If M is the message, K the key, and E the enciphered message, or cryptogram, we have E = f(M, K) that is £ is a function of M and K. It is preferable to think of this, however, not as a function of tw^o variables but as a (one parameter) family of opera- tions or transformations, and to write it E = TiM. The transformation Ti applied to message M produces cryptogram E. The index / corresponds to the particular key being used. We will assume, in general, that there are only a finite number of possible keys, and that each has an associated probability pi . Thus the key source is represented by a statistical process or device which chooses one from the set of transformations Ti , Ti, • • • , T,,, with the respective probabilities pi , p2 , ■ ■ ■ , pm • Similarly we will generally assume a finite number of possible messages Mi , M2 , • • • , Mn with associated a priori probabilities qi , q-i , ■ ■ ■ , qn . The possible messages, for example, might be the possible sequences of English letters all of length N, and the associated probabilities are then 662 BELL SYSTEM TECHNICAL JOURNAL the relative frequencies of occurrence of these sequences in normal English text. At the receiving end it must be possible to recover M, knowing E and K. Thus the transformations Tt in the family must have unique inverses Ti such that r,T7 = /, the identity transformation. Thus: M = TTE. At any rate this inverse must exist uniquely for every E which can be obtained from an M with key i. Hence we arrive at the definition: A secrecy system is a family of uniquely reversible transformations Ti of a set of possible mssages into a set of cryptograms, the transformation Ti having an associated probability pi. Conversely any set of entities of this type will be called a "secrecy system." The set of possible messages will be called, for convenience, the "message space" and the set of possible cryptograms the "cryptogram space." Two secrecy systems will be the same if they consist of the same set of transformations Ti , with the same message and cryptogram space (range and domain) and the same probabilities for the keys. A secrecy system can be visualized mechanically as a machine with one or more controls on it. A sequence of letters, the message, is fed into the input of the machine and a second series emerges at the output. The par- ticular setting of the controls corresponds to the particular key being used. Some statistical method must be prescribed for choosing the key from all the possible ones. To make the problem mathematically tractable we shall assume that the enemy knows the system being used. That is, he knows the family of trans- formations Ti , and the probabilities of choosing various keys. It might be objected that this assumption is unrealistic, in that the cryptanalyst often does not know what system was used or the probabilities in question. There are two answers to this objection: 1. The restriction is much weaker than appears at first, due to our broad definition of what constitutes a secrecy system. Suppose a cryptog- rapher intercepts a message and does not know whether a substitution, transposition, or Vigenere type cipher was used. He can consider the message as being enciphered by a system in which part of the key is the specification of which of these types was used, the next part being the particular key for that type. These three different possibilities are assigned probabilities according to his best estimates of the a priori probabilities of the encipherer using the respective types of cipher. 2. The assumption is actually the one ordinarily used in cryptographic studies. It is pessimistic and hence safe, but in the long run realistic, since one must expect his system to be found out eventually. Thus, cn]f}fr\n'.i'rfox tiieory or si'ARrx v .svsTi':.]rs 663 even when an entirely new system is devised, so that the enemy cannot assign any a priori probability to it without discovering it himself, one must still live with the expectation of his eventual knowledge. The situation is similar to that occurring in the theory of games* where it is assumed that the opponent "finds out" the strategy of play being used. In both cases the assumption serves to delineate sharply the opponent's knowledge. A second possible objection to our definition of secrecy systems is that no account is taken of the common practice of inserting nulls in a message and the use of multiple substitutes. In such cases there is not a unique crypto- gram for a given message and key, but the encipherer can choose at will from among a number of different cryptograms. This situation could be handled, but would only add complexity at the present stage, without sub- stantially altering any of the basic results. If the messages are produced by a MarkofT process of the type described in (') to represent an information source, the probabilities of various mes- sages are determined by the structure of the Markoflf process. For the present, however, we wish to take a more general view of the situation and regard the messages as merely an abstract set of entities with associated prob- abilities, not necessarily composed of a sequence of letters and not neces- sarily produced by a Markoff process. It should be emphasized that throughout the paper a secrecy system means not one, but a set of many transformations. After the key is chosen only one of these transformations is used and one might be led from this to define a secrecy system as a single transformation on a language. The enemy, however, does not know what key was chosen and the "might have been" keys are as important for him as the actual one. Indeed it is only the existence of these other possibilities that gives the system any secrecy. Since the secrecy is our primary interest, we are forced to the rather elabor- ate concept of a secrecy system defined above. This type of situation, where possibilities are as important as actualities, occurs frequently in games of strategy. The course of a chess game is largely controlled by threats which are not carried out. Somewhat similar is the "virtual existence" of unrealized imputations in the theory of games. It may be noted that a single operation on a language forms a degenerate type of secrecy system under our definition — a system with only one key of unit probability. Such a system has no secrecy — the cryi)tanalyst finds the message by applying the inverse of this transformation, the only one in the system, to the intercepted cryptogram. The decipherer and cryptanalyst in this case possess the same information. In general, the only difference be- tween the decipherer's knowledge and the enemy cryptanalyst 's knowledge ^See von Neumann and Morgenstern "The Theory of Games," Princeton 1947, 664 BELL SYSTEM TECHNICAL JOURNAL is that the decipherer knows the particular key being used, while the crypt- analyst knows only the a priori probabilities of the various keys in the set. The process of deciphering is that of applying the inverse of the particular transformation used in enciphering to the cryptogram. The process of crypt- analysis is that of attempting to determine the message (or the particular key) given only the cryptogram and the a priori probabilities of various keys and messages. There are a number of difficult epistemological questions connected with the theory of secrecy, or in fact with any theory which involves questions of probability (particularly a priori probabilities, Bayes' theorem, etc.) when applied to a physical situation. Treated abstractly, probability theory can be put on a rigorous logical basis with the modern measure theory ap- proach.^'^ As applied to a physical situation, however, especially when "subjective" probabilities and unrepeatable experiments are concerned, there are many questions of logical validity. For example, in the approach to secrecy made here, a priori probabilities of various keys and messages are assumed known by the enemy cryptographer — how can one determine operationally if his estimates are correct, on the basis of his knowledge of the situation? One can construct artificial cryptographic situations of the "urn and die" type in which the a priori probabilities have a definite unambiguous meaning and the idealization used here is certainly appropriate. In other situations that one can imagine, for example an intercepted communication between Martian invaders, the a priori probabilities would probably be so uncertain as to be devoid of significance. Most practical cryptographic situations lie somewhere between these limits. A cryptanalyst might be willing to classify the possible messages into the categories "reasonable," "possible but un- likely" and "unreasonable," but feel that finer subdivision was meaningless. Fortunately, in practical situations, only extreme errors in a priori prob- abilities of keys and messages cause significant errors in the important parameters. This is because of the exponential behavior of the number of messages and cryptograms, and the logarithmic measures employed. 3. Representation of Systems A secrecy system as defined above can be represented in various ways. One which is convenient for illustrative purposes is a line diagram, as in Figs. 2 and 4. The possible messages are represented by points at the left and the possible cryptograms by points at the right. If a certain key, say key 1, transforms message M^ into cryptogram £4 then Mo and £4 are connected ■• See J. L. Doob, "Probabilitv as Pleasure," Annals of Malli. Slat., v. 12, 1941, pp. 206-214. ^ A. Kolmogoroff, "Grundbegriffe der Wahrscheinlichkeits rcchnung," Ergebnisse der Mathematic, v. 2, No. 3 (Berlin 1933). COMMUNICATION TUEORY OF SECRECY SYSTEMS 005 by a line labeled 1, etc. From each possible message there must be exactly one line emerging for each different key. If tlie same is true for each cryptogram, we will say that the system is dosed. A more common way of describing a system is by stating tlie operation one performs on the message for an arbitrary key to obtain the cryptogram. Similarly, one detines implicitly the probabilities for various keys by de- scribing how a key is chosen or what we know of the enemy's habits of key choice. The {probabilities for messages are implicitly determined by stating our a priori knowledge of the enemy's language habits, the tactical situation (which will influence the probable content of the message) and any special information we may have regarding the cryptogram. CLOSED SYSTEM NOT CLOSED Fig. 2 — Line drawings for simple systems. 4. Some Examples of Secrecy Systems In this section a number of examples of ciphers will be given. These will often be referred to in the remainder of the paper for illustrative purposes. 1. Simple Substitution Cipher. In this cipher each letter of the message is replaced by a fixed substitute, usually also a letter. Thus the message, M = minhmsnii- • • where nii , nh , • • • are the successive letters becomes: E = 61^263^4 • • • = f{nii)J{nh)f{mz)f{mi) ■ ■ ■ where the function /(w) is a function with an inverse. The key is a permuta- tion of the alphabet (when the substitutes are letters) e.g. A' G U A C D T B F H R S L M QV Y Z W I E J O K N P. The first letter A is the substitute for .1 , G is the substitute for B, etc. 666 BELL SYSTEM TECHNICAL JOURNAL 2. Transposition {Fixed Period d). The message is divided into groups of length d and a permutation appHed to the first group, the same permutation to the second group, etc. The per- mutation is the key and can be represented by a permutation of the first d integers. Thus, for d = 5, we might have 2 3 1 5 4 as the permutation. This means that: tn\ ni-i Ws nii ws me mi m^ m^ Wio • • ■ becomes m« niz nil Ws rui m^ m%m^ mi(\ nig ■ ■ ■ . Sequential application of two or more transpositions will be called compound transposition. If the periods are (/i , ^2 , • • •, d,, it is clear that the result is a transposition of period d, where d is the least common multiple of di , d2 , • • • , ds . 3. Vigenere, arid Variations. In the Vigenere cipher the key consists of a series of d letters. These are written repeatedly below the message and the two added modulo 26 (con- sidering the alphabet numbered from .1 = 0 to Z = 25. Thus ei — tUi -\- ki (mod 26) where ki is of period d in the index /. For example, with the key G A Hy we obtain message N 0 W I ST HE-- repeated key GAHGAHGA--- cryptogram TOD 0 S AN E -■■ The Vigenere of period 1 is called the Caesar cipher. It is a simple substi- tution in which each letter of M is advanced a fixed amount in the alphabet. This amount is the key, which may be any number from 0 to 25. The so- called Beaufort and Variant Beaufort are similar to the Vigenere, and en- cipher by the equations ei = k^— nii (mod 26) and ei = nii — ki (mod 26) respectively. The Beaufort of period one is called the reversed Caesar cipher. The application of two or more Vigeneres in sequence will be called the compound Vigenere. It has the equation ei = nti -\- ki+ li-\- --• -\- Si (mod 26) COMMUNICATION TIIPIORY OF SRCRKCV SYSTEMS 667 where k, , I, , ■ ••, 5, in general have different perifxls. The period of their sum, ^, + /. + ■ • • +5, as in compound transposition, is the least common multiple of the indi\ idual {)eriods. When the X'igenere is used with an unlimited key, never repeating, we have the \'ernam system,^ with ei = nti + ki (mod 26) the ki being chosen at random and independently among 0, 1, • ■ ■, 25. If the key is a meaningful text we have the "running key" cipher. 4. Digram, Trigram, and N-gram substitution. Rather than substitute for letters one can substitute for digrams, tri- grams, etc. General digram substitution requires a key consisting of a per- mutation of the 26- digrams. It can be represented by a table in which the row corresponds to the lirst letter of the digram and the column to the second letter, entries in the table being the substitutes (usually also digrams). 5. Single Mixed Alphabet Vigenere. This is a simple substitution followed by a Vigenere. Ci = finii) + ki nii = f~'^(ei — ki) The "inverse" of this system is a Vigenere followed by simple substitution ei = ginii + ki) tUi = g^^iei) — ki 6. Matrix System? One method of ?/-gram substitution is to operate on successive «-grams with a matrix having an inverse. The letters are assumed numbered from 0 to 25, making them elements of an algebraic ring. From the »-gram wi w> • • • w„ of message, the matrix a^ gives an «-gram of cryptogram Ci = ^ aijnij / = \, • ' • , n 7 = 1 •> G. S. Vernam, "Cipher Printing Telegraph Systems for Secret Wire and Radio Tele- graphic Communications," Journal American Institute of Electrical Engineers, v. XLV, pp. 109-115, 1926. ' See L. S. Hill, "Cryptography in an Algebraic Alphabet," American Matli. Monthly, V. 36, No. 6, 1, 1929, pp. 306-312; also "Concerning Certain Linear Transformation Apparatus of Cryptography," v. 38, No. 3, 1931, pp. 135-154. 668 BELL SYSTEM TECHNICAL JOURNAL The matrix a,7 is the key, and deciphering is performed with the inverse matrix. The inverse matrix will exist if and only if the determinant | 0,7 | has an inverse element in the ring. 7. The Playfair Cipher. This is a particular type of digram substitution governed by a mixed 25 letter alphabet written in a 5 x 5 square. (The letter / is often droj)ped in cryptographic work — it is very infrequent, and when it occurs can be re- placed by /.) Suppose the key square is as shown below: L Z Q C P A G N 0 U R D MI F K Y H V S X B T E W The substitute for a digram AC, for example, is the pair of letters at the other corners of the rectangle defined by .1 and C, i.e., LO, the L taken first since it is above .1 . If the digram letters are on a horizontal line as RI, one uses the letters to their right DF; RF becomes DR. If the letters are on a vertical line, the letters below them are used. Thus PS becomes UW. If the letters are the same nulls may be used to separate them or one may be omitted, etc. 8. Multiple Mi.xed Alphabet Substitution. In this cipher there are a set of d simple substitutions which are used in sequence. If the period d is four mi m-i niz mi W5 m^ • • • becomes fi(mi) /2(m2) fsinis) fi{nh) fi{m^) /.(we) • • • 9. Autokey Cipher. A Vigenere type system in which either the message itself or the resulting cryptogram is used for the "key" is called an autokey cipher. The encipher- ment is started with a "priming key" (which is the entire key in our sense) and continued with the message or cryptogram displaced by the length of the priming key as indicated below, where the priming key is COMET. The message used as "key": Message 5 ENDS U P P L I E S ••• Key COMETS ENDS UP •■■ Cryptogram USZHLMTCOA Y H (<)\l Ml MC \ri(>.\ lllh.OKV Oh' SECRECY SYSTEMS (;69 rho (Tyi)t()^nim used as "key":** Messa^ro s E .V I) s r r r L I e s ■■■ Key C 0 ME '^_^'J Z H L O II ■ CrypK.trram F s'YllL O II O S T S ••• 10. I-'raclioiial Ciphers. In these, each letter is first enciphered into two or more letters or num- bers and these symbols are somehow mixed (e.g. by transposition). The result may then be retranslated into the original alphabet. Thus, using a mixed 25-lettcr alphabet for the key, we may translate letters into two-digit (juinary numbers by the table: 0 12 3 4 0 L Z Q C P 1 A G N O U 2 R D M I F 3 K Y H V S. 4X5 TEW Thus B becomes 41. After the resulting series of numbers is transposed in some way they are taken in pairs and translated back into letters. 11. Codes. In codes words (or sometimes syllables) are replaced by substitute letter groups. Sometimes a cipher of one kind or another is applied to the result. 5. Valuations of Secrecy Systems There are a number of different criteria that should be applied in esti- mating the value of a proposed secrecy system. The most important of these are: 1. AmoHiil of Secrecy. There are some systems that are perfect— the enemy is no better off after intercepting any amount of material than before. Other systems, although giving him some information, do not yield a unique "solution" to intercepted cryptograms. Among the uniquely solvable systems, there are wide varia- tions in the amount of labor required to effect this solution and in the amount of material that must be intercej)ted to make the solution unique. 8 This system is trivial from the secrecy slandpoint since, with the excci)li()n of the first d letters, the enemy is in possession of the entire "key." 670 BELL SYSTEM TEL II MCA L JOURXAL 1. Size of Key. The kcv must he transmitted In- ii()ii-in1('rcrptil)lc means from transmit- ting to receivinf^ points. Sometimes it must be memorized. It is therefore desirable to ha\-c the ke\- as small as possible. 3. Comphwily of liticiphcring and Deciphering Operations. Enciphering^ and deciphering; should, of course, be as simple as j)ossil)le. If they are done manuall>-, complexity leads to loss of time, errors, etc. If done mechanically, complexity leads to large expensive machines. 4. I^ropagaiion of Errors. In certain types of ciphers an error of one letter in enci{)hering or trans- mission leads to a large number of errors in the deciphered text. The errors are spread out by the deciphering operation, causing the loss of much in- formation and frequent need for repetition of the cryptogram. It is naturally desirable to minimize this error expansion. 5. Expansion of Message. In some t\-pes of secrec}- systems the size of the message is increased by the enciphering process. This undesirable effect may be seen in systems where one attempts to swamp out message statistics by the addition of many nulls, or where multiple substitutes are used. It also occurs in many "concealment" types of systems (which are not usually secrecy systems in the sense of our definition). 6. The Algebr.a. of Secrecy Systems If we have two secrecy systems T and R we can often combine them in various ways to form a new secrec>' system .V. If T and R ha\-e the same domain (message space) we may form a kind of "weighted sum," S = pT + qR where p -\- q "^ 1- This operation consists of first making a preliminary choice with probabilities p and q determining which of 7' and R is used, 'lliis choice is part of the key of .V. .\fter this is determined T or 7? is used as originally defined. The total ke\' of .V must specif}- which of T and R is used and which key of T (or R) is used. If T consists of the transformations 7\ , • • ■ , T,,, with i)robabilities />] , • ■ • , />,„ and 7^ consists of 7?i . • • • , Ri with probal)ilities (/i , • •• , (/a- then S = pT + ^7? consists of the transformations T^ , T^ , • ■ •, T„, , 7?i , • • •, 7?;. with probabilities ppi , ppi > • • -, ppm , qq\ , qqi ^ • - -, qqi^ respectiveh-. More generally we can form the sum of a number of systems. COMMi'NICATlOX THEORY OF SECRECY SYSTEMS 671 We note that any system T can be written as a sum of fixed operations T - P,T^ + p{r. + • • • + p,nl\n T, beinji; a defniile enciphering operation of T corres[)onding to key choice /, which has probability p,. A second way of combining two secrecy systems is by taking the "prod- uct," shown schematically in Fig. 3. Suppose T and R are two systems and the domain (kmguage space) of R can be identified with the range (crypto- gram space) of T. Then we can apply first 7' to our language and then R I 1 I 1 I 1 t, — T — * — R » » •— ■ R"' -•— T" I > ' ' ' Fig. 3 — Product of two systems S = RT. to the result of this enciphering process. This gives a resultant operation 5" which we write as a product S = RT The key for .S" consists of both keys of 7' and R which are assumed chosen according to their original probabilities and independently. Thus, if the m keys of T are chosen with probabilities pi p2 ■ • • pm and the n keys of R have probabilities / / / pi p2 • ■ ■ Pn , then .S' has at most mn keys with probabilities pipj . In many cases some of the product transformaions RiTj will be the same and can be grouped to- gether, adding their probabilities. Product encipherment is often used; for example, one follows a substi- tution by a transposition or a transposition by a Vigenere, or applies a code to the text and enciphers the result by substitution, transposition, frac- tionation, etc. 672 BELL SYSTEM TECHNICAL JOURNAL It may be noted that multiplication is not in general commutative, (we do not always have RS = SR), although in special cases, such as substitu- tion and transposition, it is. Since it represents an operation it is definition- ally associative. That is RiST) = (RS)T = RST. Furthermore we have the laws pip'T + q'R) + qS = pp'T + pq'R + qS (weighted associative law for addition) T{pR + qS) = pTR + qTS (pR + qS)T = pRT + qST (right and left hand distributive laws) and piT + p.2T + p,R = (Pi + p-2)T + p:ji It should be emphasized that these combining operations of addition and multiplication apply to secrecy systems as a whole. The product of two systems TR should not be confused with the product of the transformations in the systems TiRj , which also appears often in this work. The former TR is a secrecy system, i.e., a set of transformations with associated prob- abilities; the latter is a particular transformation. Further the sum of two systems pR -\- qT is a. system— the sum of two transformations is not de- fined. The systems T and R may commute without the individual Ti and Rj commuting, e.g., if i^ is a Beaufort system of a given period, all keys equally likely, RiRj dp RjRi in general, but of course RR does not depend on its order; actually RR = V the Vigenere of the same period with random key. On the other hand, if the individual Ti and Rj of two systems T and R commute, then the sys- tems commute. A system whose M and E spaces can be identified, a \'ery common case as when letter sequences are transformed into letter sequences, may be termed endomorphic. An endomorphic system T may be raised to a power T" . A secrecy system T whose product with itself is equal to T, i.e., for wliich TT = T, will be called idempotent. For example, simple substitution, transposition of period p, Vigenere of period p (all with each key equally likely) are idempotent. COMMUNICATION THEORY OF SECRECY SYSTEMS 673 The set of all endomorphic secrecy systems defined in a fixed message space constitutes an "algebraic variety," that is, a kind of algebra, using the operations of addition and multiplication. In fact, the properties of addition and multiplication which we have discussed may be summarized as follows: The set of endomorphic ciphers ivith the same message space and the two com- bining operations of weighted addition and multiplication form a linear associ- ative algebra idth a unit element, apart from the fad that the coefficients in a weighted addition must be non-negative and sum to unity. The combining operations give us ways of constructing many new types of secrecy systems from certain ones, such as the examples given. We may also use them to describe the situation facing a cryptanalyst when attempt- ing to solve a cryptogram of unknown type. He is, in fact, solving a secrecy system of the type T = p,A + p.B^ •■■ + prS + p'X Z /» = 1 where the ^, -B, ■ ■ ■ , S are known types of ciphers, with the pi their a priori probabilities in this situation, and p'X corresponds to the possibility of a completely new^ unknown type of cipher. 7. Pure and Mixed Ciphers Certain types of ciphers, such as the simple substitution, the transposi- tion of a given period, the Vigenere of a given period, the mixed alphabet Vigenere, etc. (all with each key equally likely) have a certain homogeneity with respect to key. Whatever the key, the enciphering, deciphering and decrypting processes are essentially the same. This may be contrasted wath the cipher pS+ qT where S is a simple substitution and T a transposition of a given period. In this case the entire system changes for enciphering, deciphering and de- cryptment, depending on whether the substitution or transposition is used. The cause of the homogeneity in these systems stems from the group {property— we notice that, in the above examples of homogeneous ciphers, the product TiTj of any two transformations in the set is equal to a third transformation Tf,- in the set. On the other hand TiSj does not equal any transformation in the cipher pS + qT which contains only substitutions and transpositions, no products. We might define a "pure" cipher, then, as one whose Ti form a group. This, however, would be too restrictive since it requires that the E space 674 BELL SYSTEM TECHNICAL JOURNAL be the same as the M space, i.e. that the system be endomorphic. The fractional transposition is as homogeneous as the ordinary transposition without being endomorphic. The proper definition is the following: A cipher T is pure if for every T, , Tj , Tf, there is a Ts such that TiT'^Tk = T, and every key is equally likely. Otherwise the cipher is mixed. The systems of Fig. 2 are mixed. Fig. 4 is pure if all keys are equally likely. Theorem 1: In a pure cipher the operations TJ Tj which transform the message space into itself form a group whose order is m, the number of diferent keys. For T-j'TkT-^Ti = / so that each element has an inverse. The associative law is true since these are operations, and the group property follows from TtTjTtTi = TZ'nT~kTi = T^'Ti using our assumption that T~i Tj ^ T~s Tk for some s. The operation T~i Tj means, of course, enciphering the message with key j and then deciphering with key i which brings us back to the message space. If T is endomorphic, i.e. the Ti themselves transform the space ^m into itself (as is the case with most ciphers, where both the message space and the cryptogram space consist of sequences of letters), and the Ti are a group and equally likely, then T is pure, since TiTfTk = TiTr = Ts . Theorem 2: The product of two pure ciphers which commute is pure. For if T and R commute TiRj = RiTm for every i,j with suitable /, m, and TiRj{TkRl) TmRn — TiRjR i TI TmRn = RuRv Ru'TrT\ Tt = RhTg. The commutation condition is not necessary, however, for the product to be a pure cipher. A system with only one key, i.e., a single definite operation Tx , is pure since the only choice of indices is TiT~iTx = Ti . Thus the expansion of a general cipher into a sum of such simple trans- formations also exhibits it as a sum of pure ciphers. An examination of the example of a pure cipher shown in Fig. 4 discloses COMMUNICATION rilEORY OF SECRECY SYSTEMS 675 certain properties. The messages fall into certain subsets which we will call residue classes, and the possible cryi)tograms are divided into corresponding residue classes. There is at least one line from each message in a class to each cryptogram in the corresponding class, and no line between classes which do not correspond. The number of messages in a class is a divisor of the total number of keys. The number of lines "in parallel" from a message M to a cryptogram in the corresponding class is equal to the number of keys divided by the number of messages in the class containing the message (or cryptogram). It is shown in the appendix that these hold in general for pure ciphers. Summarized formally, we have: CRYPTOGRAM RESIDUE CLASSES ] '' PURE SYSTEM Fig. 4 — Pure system. Theorem 3: In a pure system the messages can be divided into a set of "residue classes" Ci , C2 , ■ ■ ■ , Cs and the cryptograms into a corresponding set of residue classes Ci , C2 , • ■ ■ ,Ca with the following properties: (1) The message residue classes are mutually exclusive and col- lectively contain all possible messages. Similarly for the cryptogram residue classes. (2) Enciphering any message in d with any key produces a cryptogram in d . Deciphering any cryptogram in C,- zvith any key leads to a message in d . (3) The number of messages in d , say ^ tlin)u,ii;li application of the operation A. This is a device that is frequent 1\- used in practical cryptanalysis. As a trivial examjjle, simj)le substitution where the substitutes are not letters but arbitrary symbols is similar to simple substitution using letter substitutes. A second example is the Caesar and the reversed Caesar type ciphers. The latter is sometimes broken by first transforming into a Caesar t>'pe. This can be done by reversing the alphabet in the cryptogram. The \igenere, Beaufort and Variant Beaufort are all similar, when the key is random. The "autokey" cipher (with the message used as "key") primed with the key Ki A'o • • • Kd is similar to a \'igenere type with the key alter- nately added and subtracted Mod 26. The transformation .1 in this case is that of "deciphering" the autokey with a series of d A's for the priming key. PART II THEORETICAL SECRECY 9. Introduction We now consider problems connected with the "theoretical secrecy" of a system. How immune is a system to cryptanalysis when the cryptanalyst has unlimited time and manpower available for the analysis of crypto- grams? Does a cryptogram have a unique solution (even though it may require an impractical amount of work to find it) and if not how many rea- sonable solutions does it have? How much text in a given system must be in- tercepted before the solution becomes unique? Are there systems which never become unique in solution no matter how much enciphered text is inter- cepted? Are there systems for which no information whatever is given to the enemy no matter how much text is intercepted? In the analysis of these problems the concepts of entropy, redundancy and the like developed in "A Mathematical Theory of Communication" (hereafter referred to as MTC) will find a wide application. 10. Perfect Secrecy Let us suppose the possible messages are finite in number M] , • • ■ , M„ and have a priori probabilities P(Mi), ■ • ■, P(M„), and that these are en- ciphered into the possible cryptograms Ei , •••,£,„ by E = TiM. The cryptanalyst intercepts a particular E and can then calculate, in principle at least, the a posteriori probabilities for the various messages, Pb{M). It is natural to define perjecl secrecy by the condition that, for all E the a posteriori probabilities are equal to the a priori probabilities inde- pendently of the values of these. In this case, intercepting the message has 680 BELL SYSTEM TECHNICAL JOURNAL given the cryptaiialyst no information.^ Any action of his which depends on the information contained in the cryptogram cannot be altered, for all of his probabilities as to what the cryptogram contains remain unchanged. On the other hand, if the condition is not satisfied there will exist situations in which the enemy has certain a priori probabilities, and certain key and message choices may occur for which the enemy's probabilities do change. This in turn may afifect his actions and thus perfect secrecy has not been obtained. Hence the definition given is necessarily required by our intuitive ideas of what perfect secrecy should mean. A necessary and sufficient condition for perfect secrecy can be found as follows: We have by Bayes' theorem PMI) = ''-^^^ in which: P{M) = a priori probability of message M. Pm{E) = conditional probability of cryptogram E if message M is chosen, i.e. the sum of the probabilities of all keys which pro- duce cryptogram E from message M. P{E) = probability of obtaining cryptogram E from any cause.' Pe{M) = a posteriori probability of message M if cryptogram E is intercepted. For perfect secrecy Pe{M) must equal P{M) for all E and all M. Hence either P{M) = 0, a solution that must be excluded since we demand the equality independent of the values of P{M), or P^(E) = P{E) for every M and E. Conversely if Pm{E) = P(E) then Pe(M) = P{M) and we have perfect secrecy. Thus we have the result: Theorem 6: A necessary and sufUcient condHion for perfect secrecy is thai P„{E) = P{E) for all M and E. That is, Pm{E) must be independent of M. Stated another way, the total probability of all keys that transform Mi ' A purist niiglit. object tfiat the enemy lias obtained some information in that he Icnows a message was sent. This may be answered by having among the messages a "biank" corresponding to "no message." If no message is originated the blank is enciphered and sent as a cryptogram. Then even this modicum of remaining information is efiminated. COMMUNICATION THEORY OF SECRECY SYSTEMS 681 into a given cryptogram E is equal to that of all keys transforming Mj into the same E, for all 1/, , Mj and E. Now there must be as many E's as there are M's since, for a fixed i, Ti gives a one-to-one correspondence between all the If' s and some of the E's. For perfect secrecy Pm{E) = P{E) 5^ 0 for any of these £'s and any M. Hence there is at least one key transforming any M into any of these jE's. But all the keys from a fixed M to diflferent £'s must be different, and therefore the number of (liferent keys is at least as great as the number of M's. It is possible to obtain perfect secrecy with only this number of keys, as Fig. 5 — Perfect system. one shows by the following example: Let the Mi be numbered 1 to « and the Ei the same, and using n keys let TiMj = Es where s = i -\- j (Mod n). In this case we see that Pe(M) = - = P(E) and we have perfect secrecy. x\n example is shown in Fig. 5 with 5 = i -{- j - 1 (Mod 5). Perfect systems in which the number of cryptograms, the number of messages, and the number of keys are all equal are characterized by the properties that (1) each M is connected to each E by exactly one line, (2) all keys are equally likely. Thus the matrix representation of the system is a "Latin square." In MTC it was shown that information may be conveniently measured by means of entropy. If we have a set of possibilities with probabilities Pi , p2, ' ■ ■ , pn , the entropy H is given by : H - -Zpilog/'i. 682 BELL SYSTEM TECHNICAL JOURNAL In a secrecy system there are two statistical choices involved, that of the message and of the key. We may measure the amount of information pro- duced when a message is chosen by H(M) : H{M) = - S P{M) log P{M), the summation being over all possible messages. Similarly, there is an un- certainty associated with the choice of key given by : H{K) = - Z P{K) log P{K). In perfect systems of the type described above, the amount of informa- tion in the message is at most log n (occurring when all messages are equi- probable). This information can be concealed completely only if the key un- certainty is at least log n. This is the first example of a general principle which will appear frequently: that there is a limit to what we can obtain with a given uncertainty in key — the amount of uncertainty we can intro- duce into the solution cannot be greater than the key uncertainty. The situation is somewhat more complicated if the number of messages is infinite. Suppose, for example, that they are generated as infinite se- quences of letters by a suitable Markoff process. It is clear that no finite key will give perfect secrecy. We suppose, then, that the key source generates key in the same manner, that is, as an infinite sequence of symbols. Suppose further that only a certain length of key Lk is needed to encipher and de- cipher a length Lm of message. Let the logarithm of the number of letters in the message alphabet be Rm and that for the key alpiiabet be Rk ■ Then, from the finite case, it is evident that perfect secrecy requires RmLm ^ RrLk . This type of perfect secrecy is realized by the Vernam system. These results have been deduced on the basis of unknown or arbitrary a priori probabilities for the messages. The key required for perfect secrecy depends then on the total number of possible messages. One would expect that, if the message space has fixed known statistics, so that it has a definite mean rate R of generating information, in the sense of MTC, then the amount of key needed could be reduced on the average in just this ratio ^— , and this is indeed true. In fact the message can be Rm passed through a transducer which eliminates the redundancy and reduces the expected length in just this ratio, and then a Vernam system may be applied to the result. Evidently the amount of key used per letter of message D is statistically reduced by a factor -^— and in this case the key source and Rm information source are just matched — a bit of key completely conceals a COMMUNICATION TUF.ORV Ol- .SI-ICRFa Y SYSTEMS 683 bit of message information. It is easily shown also, by the methods used in MTC, that this is the best that can be done. Perfect secrecy systems have a place in the practical picture — they may be used cither where the greatest importance is attached to comi)lete secrecy — e.g., correspondence between the highest levels of command, or in cases where the number of possible messages is small. Thus, to take an extreme example, if only two messages "yes" or "no" were anticipated, a perfect system would be in order, with perhaps the transformation table: M A' A B 0 1 1 0 yes no The disadvantage of perfect systems for large correspondence systems is, of course, the equivalent amount of key that must be sent. In succeeding sections we consider what can be achieved with smaller key size, in par- ticular with fmite keys. 11. Equivocation Let us suppose that a simple substitution cipher has been used on English text and that we intercept a certain amount, iV letters, of the enciphered text. For .V fairly large, more than say 50 letters, there is nearly always a unique solution to the cipher; i.e., a single good English sequence which transforms into the intercepted material by a simple substitution. With a smaller N, however, the chance of more than one solution is greater; with JV = 15 there will generally be quite a number of possible fragments of text that would fit, while with N = 8 a good fraction (of the order of 1/8) of all reasonable English sequences of that length are possible, since there is seldom more than one repeated letter in the 8. With N = 1 any letter is clearly possible and has the same a posteriori probability as its a priori probability. For one letter the system is perfect. This happens generally with solvable ciphers. Before any material is intercepted we can imagine the a priori probabilities attached to the vari- ous possible messages, and also to the various keys. As material is inter- cepted, the cryptanalyst calculates the a posteriori probabilities; and as .V increases the probabilities of certain messages increase, and, of most, de- crease, until finally only one is left, which has a probability nearly one, while the total probability of all others is nearly zero. This calculation can actually be carried out for very simple systems. Table I shows the a posteriori probabilities for a Caesar type cipher applied to English text, with the key chosen at random from the 26 possibilities. To enable the use of standard letter, digram and trigram frequenc}- tables, the 6g4 BELL SYSTEM TECHNICAL JOURNAL text has been started at a random point (by opening a book and putting a pencil down at random on the page). The message selected in this way begins "creases to . . ." starting inside the word increases. If the message were known to start a sentence a different set of probabilities must be used, corresponding to the frequencies of letters, digrams, etc., at the beginning of sentences. TABLE I A Posteriori Probabilities for a Caesar TjT^e Cryptogram Decipherments A' = 1 N=2 N = 3 N = i C R E A S .028 .0377 .1111 .3673 D S F B T .038 .0314 E T G C U .131 .0881 F U H D V .029 .0189 G V I E W .020 H W J F X .053 .0063 I X K G Y .063 .0126 J Y L H Z .001 K Z M I A .004 L A N J B .034 .1321 .2500 MB 0 K C .025 .0222 N C P L D .071 .1195 0 D Q M E .080 .0377 P E R N F .020 .0818 .4389 .6327 Q F S 0 G .001 R G T P H .068 .0126 S H U Q I .061 .0881 .0056 T I V R J .105 .2830 .1667 U J WS K .025 V K X T L .009 W L YUM .015 .0056 X MZ V N .002 Y N A WO .020 Z 0 B X P .001 A P C Y Q .082 .0503 B Q D Z R .014 U (decimal digits) 1.2425 .9686 .6034 .285 N = 5 The Caesar with random key is a pure cipher and the particular key chosen does not affect the a posteriori probabilities. To determine these we need merely hst the possible decipherments by all keys and calculate their a priori probabilities. The a posteriori probabilities are these divided by their sum. These possible decipherments are found by the standard process of "running down the alphabet" from the message and are listed at the left. These form the residue class for the message. For one intercepted letter the a posteriori probabilities are equal to the a priori probabilities for letters'" and are shown in the column headed A' = 1. For two intercepted letters the probabilities are those for digrams adjusted to sum to unit}- and these are shown in the column X = 2. '» The probabilities for this table were taken from frequency tables given by Fletcher Pratt in a book "Secret and Urgent" published by Blue Ribbon Books, New York, 1939. Although not complete, they are sufficient for present purposes. COMMUNICATION THEORY OF SECRECY SYSTEMS 685 Trigram frequencies have also been tabulated and these are shown in the column .V = 3. For four- and five-letter sequences probabilities were ob- tained by multiplication from trigram frecjuencies since, roughly, piijkl) = p(ijk)pM. Note that at three letters the field has narrowed down to four messages of fairly high probability, the others being small in comparison. At four there are two possibilities and at five just one, the correct decipherment. In principle this could be carried out with any system but, unless the key is very small, the number of possibilities is so large that the work involved prohibits the actual calculation. This set of a posleriori probabilities describes how the cryptanalyst's knowledge of the message and key gradually becomes more precise as enciphered material is obtained. This description, however, is much too involved and difficult to obtain for our purposes. What is desired is a sim- plified description of this approach to uniqueness of the possible solutions. A similar situation arises in communication theory when a transmitted signal is perturbed by noise. It is necessary to set up a suitable measure of the uncertainty of what was actually transmitted knowing only the per- turbed version given by the received signal. In MTC it was shown that a natural mathematical measure of this uncertainty is the conditional en- tropy of the transmitted signal when the received signal is known. This conditional entropy was called, for convenience, the equivocation. From the point of view of the cryptanalyst, a secrecy system is almost identical with a noisy communication system. The message (transmitted signal) is operated on by a statistical element, the enciphering system, with its statistically chosen key. The result of this operation is the cryptogram (analogous to the perturbed signal) which is available for analysis. The chief differences in the two cases are: first, that the operation of the en- ciphering transformation is generally of a more complex nature than the perturbing noise in a channel; and, second, the key for a secrecy system is usually chosen from a finite set of possibilities while the noise in a channel is more often continually introduced, in effect chosen from an infinite set. With these considerations in mind it is natural to use the equivocation as a theoretical secrecy index. It may be noted that there are two signifi- cant equivocations, that of the key and that of the message. These will be denoted by Ue{K) and IIe(M) respectively. They are given by: He{K) = E PiE, K) log P^iK) E.K Ee{M) = Z P{E, M) log Pe{K) 686 BELL SYSTEM TECHNICAL JOURNAL in which E, M and K are the cryptogram, message and key and P{E, K) is the probability of key K and cryptogram E Pe{K) is the a posteriori probabiUty of key K if cryptogram E is intercepted P{E, M) and Pe{M) are the similar probabilities for message instead of key. The summation in He{K) is over all possible cryptograms of a certain length (say iV letters) and over all keys. For He{M) the summation is over all messages and cryptograms of length .V. Thus He{K) and He{M) are both functions of .Y, the number of intercepted letters. This will sometimes be indicated explicitly by writing He{K, N) and He{M, N). Note that these are "total" equivocations; i.e., we do not divide by N to obtain the equiv- ocation rate which was used m MTC. The same general arguments used to justify the equivocation as a measure of uncertainty in communication theory apply here as well. We note that zero equivocation requires that one message (or key) have unit prob- ability, all others zero, corresponding to complete knowledge. Considered as a function of N, the gradual decrease of equivocation corresponds to increasing knowledge of the original key or message. The two equivocation curves, plotted as functions of N, will be called the equivocation charac- teristics of the secrecy system in question. The values of He{K, N) and He{M, X) for the Caesar type cryptogram considered above have been calculated and are given in the last row of Table I. He(K, X) and He(M, X) are equal in this case and are given in decimal digits (i.e. the logarithmic base 10 is used in the calculation). It should be noted that the equivocation here is for a particular cr>'ptogram, the summation being only over M (or K), not over E. In general the sum- mation would be over all possible intercepted cr>'ptograms of length .V and would give the average uncertainty. The computational difficulties are prohibitive for this general calculation. 12. Properties of Equivocation Equivocation may be shown to have a number of interesting properties, most of which fit into our intuitive picture of how such a quantity should behave. We will first show that the equivocation of key or of a fixed part of a message decreases when more enciphered material is intercepted. Theorem 7: The equivocation of key He(K, X) is a non-increasing function of X. The equivocation of the first A letters of the message is a non-increasing function of the number N which have been inter- cepted. If X letters have been intercepted, the equivocation of the first X letters of message is less than or equal to that of the key. These may be written: COMMUNICATION THEORY OF SECRECY SYSTEMS 687 //k(A', S) < //k(/v, A^) s > n, IIk{M,S) < nK{M,N) S > N (If lor first .1 letters of text) //k(m, .V) < n,(K, .V) The qualification regarding .1 letters in the second result of the theorem is so that the equivocation will not be calculated with respect to the amount of message that has been intercepted. If it is, the message equivocation may (and usually does) increase for a time, due merely to the fact that more letters stand for a larger possible range of messages. The results of the theorem are what we might hope from a good secrecy index, since we would hardly expect to be worse ofi on the average after intercepting additional material than before. The fact that they can be proved gives further justi- lication to our use of the equivocation measure. The results of this theorem are a consequence of certain properties of con- ditional entropy proved in MTC. Thus, to show the first or second state- ments of Theorem 7, we have for any chance events A and B H{B) > Ha{B). If we identify B with the key (knowing the first S letters of cryptogram) and .1 with the remaining N — S letters we obtain the iirst result. Similarly identifying B with the message gives the second result. The last result fol- lows from He{M) < He{K, M) = He{K) + He.k{M) and the fact that He.k(M) = 0 since K and E uniquely determine M. Since the message and key are chosen independently we have: H(M, K) = H{M) + H{K). P'urthermore, H{M, K) = H(E, K) - //(£) + ^^^(A'), the first equality resulting from the fact that knowledge of M and A' or of E and A' is equivalent to knowledge of all three. Combining these two we obtain a formula for the equivocation of key: He{K) = H{M) + H{K) - H{E). In particular, if H(M) = H(E) then the equivocation of key, He(K), is equal to the a priori uncertainty of key, II (K). This occurs in the perfect systems described above. A formula for the equivocation of message can be found by similar means. We have: H(M, E) = H{E) + He{M) = H{M) + Hm{E) He{M) = H{M) + H^t{E) - H{E). 688 BELL SYSTEM TECHNICAL JOURNAL Tf wc have a product system S = TR, it is to be expected that the second enciphering process will not decrease the equivocation of message. That this is actually true can be shown as follows: Let M, Ex , Ei be the message and the first and second encipherments, respectively. Then Pe,e,{M) = Pe,{M). Consequently He,e,{M) - He,{M). Since, for any chance variables, x, y, z, H^yiz) < Hy{z), we have the desired result, He,{M) > He,{M). Theorem 8: The equivocation in message of a prodiid system S = TR is not less than that when only R is used. Suppose now we have a system T which can be written as a weighted sum of several systems R, S, ■ ■ ■, U T ^ PiR+ PoS + ■■■ -\- p,nU E Z'. = 1 and that systems R, S, ■ ■ ■ , U have equivocations Hi, H^, Hi, ■ ■ ■ , Hm. Theorem 9: The equivocation H of a weighted sum of systems is bounded by the inequalities E prHi < II <1L p^ii^ - Z />- log p^ ■ These are best limits possible. The H's may be equivocations either of key or message. The upper limit is achieved, for example, in strongly ideal systems (to be described later) where the decomposition is into the simple transforma- tions of the system. The lower limit is achieved if all the systems R, S, ■ ■ ■ , U go to completely different cryptogram spaces. This theorem is also proved by the general inequalities governing equivocation, Ha{B) < H{B) < H{A) + H4B). We identify .1 with the particular system being used and B with the key or message. There is a similar theorem for weighted sums of languages. For this we identify A with the particular language. Theorem 10: Suppose a system can be applied to languages Li , L>, ■ ■ ■ , Lm and has equivocation characteristics Hi, H-y, ■ ■ ■ , H,,, . When applied to the iveighted sum JZ P^P^^ '/"' equivocation H is bounded by Z P^H^ < li <11 P^H^ -HP^ log />. . CCi.l/.l/r.\7t.l/7(>.V rilEUKV OF SECRECY SYSTEMS 689 These liniils arc llie best possible and the equkvcalioiis in ques- tion can be either for key or message. The total redundancy Ds for A' letters of message is defined by Ds = log G - IKM) where G is the total number of messages of length .V and H{M) is the un- certainty in choosing one of these. In a secrecy system where the total number of possible cryptograms is equal to the number of possible messages of length .V, II{E) < log G. Consequently IIe(K) = ^(A') + II{M) - H(E) > H(K) - [log G - H(M)]. Hence H{K) - He(K) < Dy . This shows that, in a closed system, for example, the decrease in equivoca- tion of key after .V letters have been intercepted is not greater than the redundancy of X letters of the language. In such systems, which comprise the majority of ciphers, it is only the existence of redundancy in the original messages that makes a solution possible. Now suppose we have a pure system. Let the different residue classes of messages be Ci , d , C3 , ■ ■ ■ ,Cr , and the corresponding set of residue classes of cryptograms he Ci , Co , ■ ■ ■, Cr. The probability of each E in C[ is the same: P{E) = E a member of C, where >p, is the number of dififerent messages in C, . Thus we have H(E) = -L ., '^'^ log ^ - -2: Pia log ^^ Substituting in our equation for He(K) we obtain: Theorem 11: For a pure cipher H,{K) = //(A) + H(M) + £ P(C,) log ^^ . This result can be used to compute He{K) in certain cases of interest. 690 BELL SYSTEM TECHNICAL JOURNAL 13. Equivocation for Simple Substitution on a Two Letter Language We will now calculate the equivocation in key or message when simple substitution is applied to a two letter language, with probabilities p and q for 0 and 1, and successive letters chosen independently. We have He{M) = He{K) = -E F{E)PE{K)\ogPE{K) The probability that E contains exactly 5 O's in a particular permutation is: \ \ ^ \ O \ \, 5 -1.2 < 2 \ \ \ \ ^ o u o \ s N 1 \ vj ^p=73,S-'/3 z 2" \ ^ ^ I \ ^^ ■^ II •' "z \ ^^v^ -^ 5C u \ ^ \. P'y8,c|=Ve 0 ■-- 0 2 4 6 S 10 12 14- 16 18 20 NUMBER OF LETTERS, N Fig. 6 — Equivocation for simple substitution on two-letter language. and the a posteriori probabilities of the identity and inverting substitutions (the only two in the system) are respectively: P^(0) = P'q^ {p'q^-o + qop^-') PM = pff->qs (psqN-s _^ qopff-') There are ( " ) terms for each .y and hence HM<, N) ?(*■>•.«- log p»qff-t (^p'qN-s _|_ q'pN-i>f COMMLMCATIOy THEORY OF SECRECY SYSTEMS 691 For /> = 3, 9 = I, and for p = \, q = ^, He(K, N) has been calculated and is shown in Fig. 6. 14. The Equivoc.vtiox Cilvkacteristic for a "Ra.vdom" Cipher In the preceding section we have calculated the equivocation charac- teristic for a simple substitution applied to a two-letter language. This is about the simplest type of cipher and the simplest language structure pos- sible, yet alread}' the formulas are so involved as to be nearly useless. What are we to do with cases of practical interest, say the involved transforma- tions of a fractional transposition system applied to EngHsh with its ex- tremely comple.x statistical structure? This complexity itself suggests a method of approach. Sufficiently complicated problems can frequently be solved statistically. To facilitate this we define the notion of a "random" cipher. We make the following assumptions: 1. The number of possible messages of length N is T = 2'^"'"', thus Ro = log2 G, where G is the number of letters in the alphabet. The number of possible cr}^ptograms of length N is also assumed to be T. 2. The possible messages of length A' can be divided into two groups: one group of high and fairly uniform a priori probability, the second group of negligibly small total probability. The high probability group will contain S = 2^^^ messages, where R = H{M)/X, that is, R is the entropy of the message source per letter. 3. The deciphering operation can be thought of as a series of lines, as in Figs. 2 and 4, leading back from each E to various M's. We assume k different equiprobable keys so there will be k lines leading back from each E. For the random cipher we suppose that the lines from each E go back to a random selection of the possible messages. Actually, then, a random cipher is a whole ensemble of ciphers and the equivoca- tion is the average equivocation for this ensemble. The equivocation of key is defined by IIe(K) = E P(E)Pe{K) log Pe(K). The probability that exactly m lines go back from a particular E to the high probability group of messages is C) ©■(' - ¥>' If a cryptogram with m such lines is intercepted the equivocation is log m. TYlT The probability of such a cryptogram is — — , since it can be produced by 692 BELL SYSTEM TECHNICAL JOURNAL T m keys from high probability messages each with probability — . Hence the equivocation is: We wish to find a simple approximation to this when k is large. If the expected value of m, namely m — Sk/T, is » 1, the variation of log m over the range where the binomial distribution assumes large values will be small, and we can replace log m by log m. This can now be factored out of the summation, which then reduces to m. Hence, in this condition, He{K) = log ^ = log ^ - log r + log k He{K) = H{K) - DN, where D is the redundancy per letter of the original language {D = Ds/N). If m is small compared to the large k, the binomial distribution can be approximated by a Poisson distribution: /A .m k—m , '■' A p q = — - ml where X = — . Hence 4 00 * wi HeiK) - z- e~^ ^ —. m log m. A 2 ml If we replace mhy m -\- 1, we obtain: 00 X"* He{K) = e-^ y — , log (m + 1). ■^ ml This may be used in the region where X is near unity. For X <$C 1, the only important term in the series is that for m = 1; omitting the others we have: Hb(K) = e-^\ log 2 = X log 2 = 2-A-^yfe log 2 . To summarize: He(K), considered as a function of A'', the number of intercepted letters, starts off at H(K) when AT" = 0. It decreases linearly H(K) with a slope —D out to the neighborhood of TV = . After a short transition region, Hb{K) follows an exponential with "half life" distance COMMUNICATION THEORY OF SECRECY SYSTEMS 693 — if D is measured in bits per letter. This hcliavior is shown in Fig. 7, to- gether with the approximating curves. By a similar argument the equivocation of message can be calculated. It is He{M) = RoX for RoN « He{K) He(M) = He(K) for RoN » He{K) He{M) = He(K) - cp(N) for RoN ^ He{K) where 2 the He(K) line. After a rounded transition it follows the He(K) curve down. It will be seen from Fig. 7 that the equivocation curves approach zero rather sharply. Thus we may, with but little ambiguity, speak of a point at which the solution becomes unique. This number of letters will be called the unicity distance. For the random cipher it is approximately H{K)/D. 15. Application to Standard Ciphers Most of the standard ciphers involve rather complicated enciphering and deciphering operations. Furthermore, the statistical structure of natural languages is extremely involved. It is therefore reasonable to assume that the formulas derived for the random cipher may be applied in such cases. It is necessary, however, to apply certain corrections in some cases. The main points to be observed are the following: 694 BELL SYSTEM TECHNICAL JOURNAL 1. We assumed for the random cipher that the possible decipherments of a cryptogram are a random selection from the possible messages. While not strictly true in ordinary systems, this becomes more nearly the case as the complexity of the enciphering operations and of the language structure increases. With a transposition cipher it is clear that letter frequencies are preserved under decipherment operations. This means that the possible decipherments are chosen from a more limited group, not the entire message space, and the formula should be changed. In place of Rq one uses Ri the entropy rate for a language with independent letters but with the regular letter frequencies. In some other cases a definite tendency toward returning the decipherments to high probability messages can be seen. If there is no clear tendency of this sort, and the system is fairly complicated, then it is reasonable to use the random cipher analysis. 2. In many cases the complete key is not used in enciphering short mes- sages. For example, in a simple substitution, only fairly long messages will contain all letters of the alphabet and thus involve the complete key. Obviously the random assumption does not hold for small ^V in such a case, since all the keys which differ only in the letters not yet appearing in the cryptogram lead back to the same message and are not randomly distrib- uted. This error is easily corrected to a good approximation by the use of a "key appearance characteristic." One uses, at a particular N, the effective amount of key that may be expected with that length of cryptogram. For most ciphers, this is easily estimated. 3. There are certain "end effects" due to the definite starting of the message which produce a discrepancy from the random characteristics. If we take a random starting point in English text, the first letter (when we do not observe the preceding letters) has a possibility of being any letter with the ordinary letter probabilities. The next letter is more completely specified since we then have digram frequencies. This decrease in choice value continues for some time. The effect of this on the curve is that the straight line part is displaced, and approached by a curve depending on how much the statistical structure of the language is spread out over adja- cent letters. As a first approximation the curve can be corrected by shifting the line over to the half redundancy point — i.e., the number of letters where the language redundancy is half its final value. If account is taken of these three effects, reasonable estimates of the equivocation characteristic and unicity point can be made. The calcula- tion can be done graphically as indicated in Fig. 8. One draws the key appearance characteristic and the total redundancy curve Z>.v (which is usually sufficiently well represented by the line ND^). The difference be- tween these out to the neighborhood of their intersection is He(M). With a simple substitution cipher applied to English, this calculation gave the COMMUNICATION THEORY OF SECRECY SYSTEMS 695 curves shown in Fig. ^). The key apj)earance characteristic in this case was estimated by countin (siioia) H 0 1 2 3 4 5 1.41 1.24 .97 .60 .28 0 1.41 1.25 .98 .54 .15 .03 698 BELL SYSTEM TECHNICAL JOURNAL as simple substitution will be approximately as shown in Fig. 10. The Vigenere, Playfair and Fractional cases are more likely to follow the the- oretical formulas for random ciphers than simple substitution and trans- position. The reason for this is that they are more complex and give better mixing characteristics to the messages on which they operate. The mixed alphabet Vigenere (each of d alphabets mixed independently and used sequentially) has a key size, H(K) = J log 26! = 26.3^ and its unicity point should be at about 53d letters. These conclusions can also be put to a rough experimental test with the Caesar type cipher. In the particular cryptogram analyzed in Table I, section 11, the function (He(K, A^) has been calculated and is given below, together with the values for a random cipher. N H (observed) H (calculated) The agreement is seen to be quite good, especially when we remember that the observed H should actually be the average of many different cryp- tograms, and that D for the larger values of iV is only roughly estimated. It appears then that the random cipher analysis can be used to estimate equivocation characteristics and the unicity distance for the ordinary types of ciphers. 16. Validity of a Cryptogram Solution The equivocation formulas are relevant to questions which sometimes arise in cryptographic work regarding the validity of an alleged solution to a cryptogram. In the history of cryptography there have been many cryptograms, or possible cryptograms, where clever analysts have found a "solution." It involved, however, such a complex process, or the material was so meager that the question arose as to whether the cryptanalyst had "read a solution" into the cryptogram. See, for example, the Bacon-Shake- speare ciphers and the "Roger Bacon" manuscript.^" In general we may say that if a proposed system and key solves a crypto- gram for a length of material considerably greater than the unicity distance the solution is trustworthy. If the material is of the same order or shorter than the unicity distance the solution is highly suspicious. This effect of redundancy in gradually producing a unique solution to a cipher can be thought of in another way which is helpful. The redundancy is essentially a series of conditions on the letters of the message, which •0 See Fletcher Pratt, loc. cU. COMMUNICATIOS TIIEORV OF SFAKKCV SYSTE.\[S 699 insure that it be statistically reasonable. These consistency conditions pro- duce corresponding consistency conditions in the cryptogram. The key gives a certain amount of freedom to the cryptogram but, as more and more letters are intercepted, the consistency conditions use up the freedom al- lowed by the key. Eventually there is only one message and key which satishes all the conditions and we have a unique solution. In the random cipher the consistency conditions are, in a sense "orthogonal" to the "grain of the key" and have their full effect in eliminating messages and keys as rapidly as possible. This is the usual case. However, by proper design it is possible to "line up" the redundancy of the language with the "grain of the key" in such a way that the consistency conditions are automatically satisfied and IIe(K) does not approach zero. These "ideal" systems, which will be considered in the next section, are of such a nature that the trans- formations Ti all induce the same probabilities in the E space. 17. Ideal Secrecy Systems, We have seen that perfect secrecy requires an infinite amount of key if we allow messages of unlimited length. With a finite key size, the equivoca- tion of key and message generally approaches zero, but not necessarily so. In fact it is possible for He(K) to remain constant at its initial value H{K). Then, no matter how much material is intercepted, there is not a unique solution but many of comparable probability. We will define an "ideal" system as one in which He(K) and He(M) do not approach zero as .V — ^ co , A "strongly ideal" system is one in which He{K) remains constant at H{K). An example is a simple substitution on an artificial language in which all letters are equiprobable and successive letters independently chosen. It is easily seen that He(K) — H(K) and He(M) rises linearly along a line of slope log G (w^here G is the number of letters in the alphabet) until it strikes the line H{K), after which it remains constant at this value. With natural languages it is in general possible to approximate the ideal characteristic — the unicity point can be made to occur for as large A as is desired. The complexity of the system needed usually goes up rapidly when we attempt to do this, however. It is not always possible to attain actually the ideal characteristic with any system of finite complexity. To approximate the ideal equiv^ocation, one may first operate on the message with a transducer which removes all redundancies. After this almost any simple ciphering system — substitution, transposition, \'igenere, etc., is satisfactory. The more elaborate the transducer and the nearer the output is to the desired form, the more closely will the secrecy system ap- proximate the ideal characteristic. 700 BELL SYSTEM TECHNICAL JOURNAL Theorem 12: A necessary and sufficient condition that T be strongly ideal is that, for any two keys, TJ Tj is a measure preserving transforma- tion of the message space into itself. This is true since the a posteriori probabiHty of each key is equal to its a priori probability if and only if this condition is satisfied. 18. Ex.\MPLEs OF Ideal Secrecy Systems Suppose our language consists of a sequence of letters all chosen inde- pendenth' and with equal probabilities. Then the redundancy is zero, and from a result of section 12, He(K) = H(K). We obtain the result Theorem 13: If all letters are equally likely and independent any closed cipher is strongly ideal. The equivocation of message will rise along the key appearance char- acteristic which will usually approach H(K), although in some cases it does not. In the cases of w-gram substitution, transposition, Vigenere, and variations, fractional, etc., we have strongly ideal systems for this simple language with He{M) -^ H(K) as iV -^ . Ideal secrecy systems sufifer from a number of disadvantages. 1. The system must be closely matched to the language. This requires an extensive study of the structure of the language by the designer. Also a change in statistical structure or a selection from the set of possible mes- sages, as in the case of probable words (words expected in this particular cryptogram), renders the system vulnerable to analysis. 2. The structure of natural languages is extremely comphcated, and this implies a complexity of the transformations required to eliminate redun- dancy. Thus any machine to perform this operation must necessarily be quite involved, at least in the direction of information storage, since a "dictionary" of magnitude greater than that of an ordinary^ dictionary is to be expected. 3. In general, the transformations required introduce a bad propagation of error characteristic. Error in transmission of a single letter produces a region of changes near it of size comparable to the length of statistical effects in the original language. 19. Further Remarks on Equivocation and Redundancy We have taken the redundancy of "normal English" to be about .7 deci- mal digits per letter or a redundancy of 50%. This is on the assumption that word divisions were omitted. It is an approximate figure based on sta- tistical structure extending over about 8 letters, and assumes the text to be of an ordinary type, such as newspaper writing, literary work, etc. We may note here a method of roughly estimating this number that is of some cryptographic interest. COMMUNICATION THEORY OF SECRECY SYSTEMS 701 A running key ciplier is a Vernani type system where, in place of a random sequence of letters, the key is a meaningful text. Now it is known that run- ning key ciphers can usually be solved uniquely. This shows that English can be reduced by a factor of two to one and implies a redundancy of at least 50%. This figure cannot be increased very much, however, for a number of reasons, unless long range "meaning" structure of English is considered. The running key cipher can be easily improved to lead to ciphering systems which could not be soK^ed without the key. If one uses in place of one English text, about 4 different texts as key, adding them all to the message, a sufficient amount of key has been introduced to produce a high positive equivocation. Another method would be to use, say, every 10th letter of the text as key. The intermediate letters are omitted and cannot be used at any other point of the message. This has much the same effect, since these spaced letters are nearly independent. The fact that the vowels in a passage can be omitted without essential loss suggests a simple way of greatly improving almost any ciphering system. First delete all vowels, or as much of the message as possible without run- ning the risk of multiple reconstructions, and then encipher the residue. Since this reduces the redundancy by a factor of perhaps 3 or 4 to 1, the unicity point will be moved out by this factor. This is one way of approach- ing ideal systems — using the decipherer's knowledge of English as part of the deciphering system. 20. Distribution of Equivocation A more complete description of a secrecy system applied to a language than is afforded by the equivocation characteristics can be found by giving the dislribulion of equivocation. For N intercepted letters we consider the fraction of cryptograms for which the equivocation (for these particular £'s, not the mean He{M)) lies between certain limits. This gives a density distribution function P(He(M), N) dHE(M) for the probability that for N letters H lies between the limits H and H -{- (III. The mean equivocation we have previously studied is the mean of this distribution. The function P{He(M), N) can be thought of as plotted along a third dimension, normal to the paper, on the IIe(M), X plane. If the language is pure, with a small influence range, and the cipher is pure, the function will usually be a ridge in this plane whose highest point follows approximately the mean IIe(M), at least until near the unicity point. In this case, or when the conditions are nearly verified, the mean curve gives a reasonably complete picture of the system. 702 BELL SYSTEM TECHNICAL JOURNAL On the other hand, if the language is not pure, but made up of a set of pure components having different equivocation curves with the system, then the total dis- tribution will usually be made up of a series of ridges. There will be one for each Li weighted in accordance with its pi. The mean equivocation char- acteristic will be a line somewhere in the midst of these ridges and may not give a very complete picture of the situation. This is shown in Fig. 11. A similar effect occurs if the system is not pure but made up of several systems with different // curves. The effect of mixing pure languages which are near to one another in sta- tistical structure is to increase the width of the ridge. Near the unicity i'-a p(H,Nr Fig. 11 — Distribution of equivocation with a mixed language L = \L\ ■{- \L>. point this tends to raise the mean equivocation, since equivocation cannot become negative and the spreading is chiefly in the positive direction. We expect, therefore, that in this region the calculations based on the random cipher should be somewhat low. PART III PRACTICAL SECRECY 21. The Work Cila.racteristic After the unicity point has been passed in intercepted material there will usually be a unique solution to the cryptogram. The problem of isolating this single solution of high probability is the problem of cryptanalysis. In the region before the unicity point we may say that the problem of crj^pt- analysis is that of isolating all the possible solutions of high probability (compared to the remainder) and determining their various probabilities. CUMMLMLATIO.\ THEORY OF SECRFXV SYSt'EMS 703 Although it is always possible in principle to determine these solutions (by trial of each possible key for example), difTerent enciphering systems show a wide variation in the amount of work required. The average amount of work to determine the key for a cryptograni of X letters, W{N), measured say in man hours, may be called the work characteristic of the system. This average is taken over all messages and all keys with their appropriate prob- abilities. The function Tr(A') is a measure of the amount of "practical secrecy" afiforded by the system. For a simple substitution on English the work and equivocation char- acteristics would be somewhat as shown in Fig. 12. The dotted portion of Fig. 12 — Tyjiical work and equivocation characteristics. the curve is in the range where there are numerous possible solutions and these must all be determined. In the solid portion after the unicity point only one solution exists in general, but if only the minimum necessary data are given a great deal of work must be done to isolate it. As more material is available the work rapidly decreases toward some asymptotic value — where the additional data no longer reduces the labor. Essentially the behavior shown in Fig. 12 can be expected with any type of secrecy system where the equivocation approaches zero. The scale of man hours required, however, will differ greatly with different types of ciphers, even when the He{M) curves are about the same. A Vigenere or compound Vigenere, for e.xample, with the same key size would have a 704 BELL SYSTEM TECHNICAL JOURNAL much better (i.e., much higher) work characteristic. A good practical secrecy system is one in which the W{N) curve remains sufficiently high, out to the number of letters one expects to transmit with the key, to prevent the enemy from actually carrying out the solution, or to delay it to such an extent that the information is then obsolete. We will consider in the following sections ways of keeping the function W{N) large, even though He{K) may be practically zero. This is essentially a "max min" type of problem as is always the case when we have a battle of wits.^^ In designing a good cipher we must maximize the minimum amount of work the enemy must do to break it. It is not enough merely to be sure none of the standard methods of cryptanalysis work — we must be sure that no method whatever will break the system easily. This, in fact, has been the weakness of many systems; designed to resist all the known methods of solution, they later gave rise to new cryptanalytic techniques which rendered them vulnerable to analysis. The problem of good cipher design is essentially one of finding difficult problems, subject to certain other conditions. This is a rather unusual situa- tion, since one is ordinarily seeking the simple and easily soluble problems in a field. How can we ever be sure that a system which is not ideal and therefore has a unique solution for sufficiently large N will require a large amount of work to break with every method of analysis? There are two approaches to this problem; (1) We can study the possible methods of solution available to the cryptanalyst and attempt to describe them in sufficiently general terms to cover any methods he might use. We then construct our system to resist this "general" method of solution. (2) We may construct our cipher in such a way that breaking it is equivalent to (or requires at some point in the process) the solution of some problem known to be laborious. Thus, if we could show that solving a certain system requires at least as much work as solving a system of simultaneous equations in a large number of unknowns, of a complex type, then we would have a lower bound of sorts for the work characteristic. The next three sections are aimed at these general problems. It is difficult to define the pertinent ideas involved with sufficient precision to obtain results in the form of mathematical theorems, but it is believed that the conclusions, in the form of general principles, are correct. " See von Neumann and Morgenstern, loc. cil. The situation between the cipher de- signer and crj'ptanalyst can be thought of as a "game" of a very simple structure; a zero- sum two-person game with complete information, and just two "moves." The cipher designer chooses a system for his "move." Then the cryptanalyst is informed of this choice and chooses a method of analysis. The "value" of the play is the average work re- quired to break a cryptogram in the system by the method chosen. COMMUNICATION THEORY OF SECRECY SYSTEMS )05 22. Generalities on the Solution of Cryptograms After the unicity distance has been exceeded in intercepted material, any system can be solved in principle by merely trying each possible key until the unique solution is obtained — i.e., a deciphered message which "makes sense" in the original language. A simple calculation shows that this method of solution (which we may call complete trial and error) is totally impractical except when the key is absurdly small. Suppose, for example, we have a key of 26! possibilities or about 26.3 decimal digits, the same size as in simple substitution on English. This is, by any significant measure, a small key. It can be written on a small slip of paper, or memorized in a few minutes. It could be registered on 27 switches, each having ten positions, or on 88 two-position switches. Suppose further, to give the cryptanalyst every possible advantage, that he constructs an electronic device to try keys at the rate of one each micro- second (perhaps automatically selecting from the results by a x" test for statistical significance). He may expect to reach the right key about half way through, and after an elapsed time of about 2 X 1026/2 X 60^ X 24 X 365 X 106 or 3 X lO^^ years. In other words, even with a small key complete trial and error will never be used in solving cryptograms, except in the trivial case where the key is extremely small, e.g., the Caesar with only 26 possibilities, or 1.4 digits. The trial and error which is used so commonly in cryptography is of a different sort, or is augmented by other means. If one had a secrecy system which required complete trial and error it would be extremely safe. Such a system would result, it appears, if the meaningful original messages, all say of 1000 letters, were a random selection from the set of all sequences of 1000 letters. If any of the simple ciphers were applied to this type of language it seems that little improvement over complete trial and error would be possible. The methods of cryptanalysis actually used often involve a great deal of trial and error, but in a different way. First, the trials progress from more probable to less probable hypotheses, and, second, each trial disposes of a large group of keys, not a single one. Thus the key space may be divided into say 10 subsets, each containing about the same number of keys. By at most 10 trials one determines which subset is the correct one. This subset is then divided into several secondary subsets and the process repeated. With the same key size (26! = 2 X lO-^) we would expect about 26 X 5 or 130 trials as compared to lO^" by complete trial and error. The possibility of choosing the most likely of the subsets first for test would improve this result even more. If the divisions were into two compartments (the best way to 706 BELL SYSTEM TECHNICAL JOURNAL minimize the number of trials) only 88 trials would be required. Whereas complete trial and error requires trials to the order of the number of keys, this subdividing trial and error requires only trials to the order of the key size in bits. This remains true even when the different keys have different probabilities. The proper procedure, then, to minimize the expected number of trials is to divide the key space into subsets of equiprobability. When the proper subset is determined, this is again subdivided into equiprobabiUty subsets. If this process can be continued the number of trials expected when each division is into two subsets will be ; H{K) If each test has S possible results and each of these corresponds to the key being in one of S equiprobability subsets, then H{K) h = log S trials will be expected. The intuitive signiticance of these results should be noted. In the two-compartment test with equiprobability, each test yields one bit of information as to the key. If the subsets have very different prob- abilities, as in testing a single key in complete trial and error, only a small amount of information is obtained from the test. Thus with 26! equiprobable keys, a test of one yields only r26! - 1 , 26! - 1 , 1 , 1 1 - L"^6!- ^"^ ^l6r- + 2-6! ^^^ 2-6"!] or about 10^25 \^[^^ ^f information. Dividing into S equiprobability subsets maximizes the information obtained from each trial at log S, and the ex- pected number of trials is the total information to be obtained, that is H(K), divided by this amount. The question here is similar to various coin weighing problems that have been circulated recently. A typical example is the following: It is known that one coin in 27 is counterfeit, and slightly lighter than the rest. A chemist's balance is available and the counterfeit coin is to be isolated by a series of weighings. WTiat is the least number of weighings required to do this? The correct answer is 3, obtained by first dividing the coins into three groups of 9 each. Two of these are compared on the balance. The three possible results determine the set of 9 containing the counterfeit. This set is then divided into 3 subsets of 3 each and the process continued. The set of coins corre- sponds to the set of keys, the counterfeit coin to the correct key, and the weighing procedure to a trial or test. The original uncertainty is log2 27 COMMUNICATION TIIKORV ()!■ SJ-XRECV SVSTIiMS 707 bits, and each trial yields logo 3 hits of information; thus, when there is no "diophantine trouble," log2 27/log2 3 or 3 trials are sufficient. This method of solution is feasible only if the key space can be divided into a small number of subsets, with a simi)le method of determining the subset to which the correct key belongs. One does not need to assume a complete key in order to apply a consistency test and determine if the assumption is justilied^an assumption on a part of the key (or as to whether the key is in some large section of the key space) can be tested. In other words it is possible to solve for the key bit by bit. The possibility of this method of analysis is the crucial weakness of most ciphering systems. For example, in simple substitution, an assumption on a single letter can be checked against its frequency, variety of contact, doubles or reversals, etc. In determining a single letter the key space is reduced by 1.4 decimal digits from the original 26. The same effect is seen in all the elementary types of ciphers. In the Vigenere, the assumption of two or three letters of the key is easily checked by deciphering at other points with this fragment and noting whether clear emerges. The com- pound Vigenere is much better from this point of view, if we assume a fairly large number of component periods, producing a repetition rate larger than will be intercepted. In this case as many key letters are used in en- ciphering each letter as there are periods. Although this is only a fraction of the entire key, at least a fair number of letters must be assumed before a consistency check can be applied. Our first conclusion then, regarding practical small key cipher design, is that a considerable amount of key should be used in enciphering each small element of the message. 23. Statistical Methods It is possible to solve many kinds of ciphers by statistical analysis. Consider again simple substitution. The first thing a cryptanalyst does with an intercepted cryptogram is to make a frequency count. If the cryptogram contains, say, 200 letters it is safe to assume that few, if any, of the letters are out of their frequency groups, this being a division into 4 sets of well defined frequency limits. The logarithm of the number of keys within this limitation may be calculated as log 2! 9! 9! 6! = 14.28 and the simple frequency count thus reduces the key uncertaint}' by 12 decimal digits, a tremendous gain. In general, a statistical attack proceeds as follows: A certain statistic is measured on the intercepted cryptogram E. This statistic is such that for all reasonable messages M it assumes about the same value, Sk, the value 708 BELL SYSTEM TECHNICAL JOURNAL depending only on the particular key K that was used. The value thus ob- tained serves to limit the possible keys to those which would give values of S in the neighborhood of that observed. A statistic which does not depend on K or which varies as much with M as with K is not of value in limiting K. Thus, in transposition ciphers, the frequency count of letters gives no information about K — every K leaves this statistic the same. Hence one can make no use of a frequency count in breaking transposition ciphers. More precisely one can ascribe a ''solving power" to a given statistic S. For each value of S there will be a conditional equivocation of the key HaiK), the equivocation when S has its particular value, and that is all that is known concerning the key. The weighted mean of these values T.P{S) Hs{K) gives the mean equivocation of the key when S is known, P(S) being the a priori probability of the particular value S. The key size H{K), less this mean equivocation, measures the "solving power" of the statistic S. In a strongly ideal cipher all statistics of the cryptogram are independent of the particular key used. This is the measure preserving property of TjT^^ on the E space or TJ^Tk on the M space mentioned above. There are good and poor statistics, just as there are good and poor methods of trial and error. Indeed the trial and error testing of an hypothesis is is a type of statistic, and what was said above regarding the best types of trials holds generally. A good statistic for solving a system must have the following properties: 1. It must be simple to measure. 2. It must depend more on the key than on the message if it is meant to solve for the key. The variation with M should not mask its variation with K. 3. The values of the statistic that can be "resolved" in spite of the "fuzziness" produced by variation in M should divide the key space into a number of subsets of comparable probability, with the statistic specifying the one in which the correct key lies. The statistic should give us sizeable information about the key, not a tiny fraction of a bit. 4. The information it gives must be simple and usable. Thus the subsets in which the statistic locates the key must be of a simple nature in the key space. Frequency count for simple substitution is an example of a very good statistic. Two methods (other than recourse to ideal systems) suggest themselves for frustrating a statistical analysis. These we may call the methods of difusion and confusion. In the method of diffusion the statistical structure of M which leads to its redundancy is "dissipated" into long range sta- COMMUNICATIOX TUEORV 01 SECRECY SYSTEMS 709 tistics— i.e., into statistical structure involving long combinations of letters in the cryptogram. The effect here is that the enemy must intercept a tre- mendous amount of material to tie down this structure, since the structure is evident only in blocks of very small individual probability. Furthermore, even when he has sufficient material, the analytical work required is much greater since the redundancy has been diffused over a large number of individual statistics. An example of diffusion of statistics is operating on a message M = Wi , nh , Mz , ■ ■ ■ with an "averaging" operation, e.g. s yn = ^ nin+i (mod 26), adding 5 successive letters of the message to get a letter >-„ . One can show that the redundacy of the y sequence is the same as that of the m sequence, but the structure has been dissipated. Thus the letter frequencies in y will be more nearly equal than in m, the digram frequencies also more nearly equal, etc. Indeed any reversible operation which produces one letter out for each letter in and does not have an infinite "memory" has an output with the same redundancy as the input. The statistics can never be eliminated without compression, but they can be spread out. The method of confusion is to make the relation between the simple statistics of E and the simple description oi K a. very complex and involved one. In the case of simple substitution, it is easy to describe the limitation of A' imposed by the letter frequencies of E. If the connection is very in- volved and confused the enemy may still be able to evaluate a statistic Si , say, which limits the key to a region of the key space. This limitation, however, is to some complex region R in the space, perhaps "folded over" many times, and he has a difficult time making use of it. A second statistic ^2 limits K still further to R^ , hence it lies in the intersection region; but this does not help much because it is so difficult to determine just what the intersection is. To be more precise let us suppose the key space has certain "natural co- ordinates" ki , kt, • ■ ■ , kp which he wishes to determine. He measures, let us say, a set of statistics s^ , So , ■ ■ ■ , Sn and these are sufficient to determine the ki . However, in the method of confusion, the equations connecting these sets of variables are involved and complex. We have, say, fi{ki ,h, • • -, kp) = si hih ,h, ■■■, kp) = S2 fn(kl , ki , ■ • • , kp) = Sn , 710 BELL SYSTEM TECHNICAL JOURNAL and all the ft involve all the ki . The cryptographer must solve this system simultaneously — a difficult job. In the simple (not confused) cases the func- tions involve only a small number of the k^ — or at least some of these do. One first solves the simpler equations, evaluating some of the ki and sub- stitutes these in the more complicated equations. The conclusion here is that for a good ciphering system steps should be taken either to diffuse or confuse the redundancy (or both). 24. The Probable Word Method One of the most powerful tools for breaking ciphers is the use of probable words. The probable words may be words or phrases expected in the par- ticular message due to its source, or they may merely be common words or syllables which occur in any text in the language, such as the, and, Hon, that, and the like in English. In general, the probable word method is used as follows: Assuming a probable word to be at some point in the clear, the key or a part of the key is determined. This is used to decipher other parts of the cryptogram and provide a consistency test. If the other parts come out in the clear, the assumption is justified. There are few of the classical type ciphers that use a small key and can resist long under a probable word analysis. From a consideration of this method we can frame a test of ciphers which might be called the acid test. It applies only to ciphers with a small key (less than, say, 50 decimal digits), applied to natural languages, and not using the ideal method of gaining se- crecy. The acid test is this: How difficult is it to determine the key or a part of the key knowing a small sample of message and corresponding crypto- gram? Any system in which this is easy cannot be very resistant, for the cryptanalyst can always make use of probable words, combined with trial and error, until a consistent solution is obtained. The conditions on the size of the key make the amount of trial and error small, and the condition about ideal systems is necessary, since these auto- matically give consistency checks. The existence of probable words and phrases is implied by the assumption of natural languages. Note that the requirement of difficult solution under these conditions is not, by itself, contradictory to the requirements that enciphering and deciphering be simple processes. Using functional notation we have for enciphering E = f{K, M) and for deciphering M = g(K, E). COMMUNICATION TUEORY OF SECIULCY SYSTEMS 711 Both of these may be simple operations on their arguments without the third equation A' = h{M, E) being simple. We may also point out that in investigating a new type of ciphering sys- tem one of the best methods of attack is to consider how the key could be determined if a sulllcient amount of M and E were given. The principle of confusion can be (and must be) used to create diOiculties for the cr>'ptanalyst using probable word techniques. Given (or assuming) M = mi , m-i , • ■ • , nis and E = ei , 62 , • ■ • , Cs the cryptanalyst can set up equations for the different key elements k] , ko , ■ ■ ■ , kr (namely the en- ciphering equations). ei = fi(mi ,m-i,---, nis \ki , ■■■ , kr) €2 = Mtni , m-i , ■ ■ ■ , Ms ; ki , ■ • • , kr) es = fs(nii , nh , ■ ■ ■ , m, ; ki , ■ • • , kr) All is known, we assume, except the ki . Each of these equations should therefore be complex in the ki , and involve many of them. Otherwise the enemy can solve the simple ones and then the more complex ones by sub- stitution. From the point of view of increasing confusion, it is desirable to have the fi involve several rm , especially if these are not adjacent and hence less correlated. This introduces the undesirable feature of error propagation, however, for then each Ci will generally aflfect several w, in deciphering, and an error will spread to all these. We conclude that much of the key should be used in an involved manner in obtaining any cryptogram letter from the message to keep the work characteristic high. Further a dependence on several uncorrelated w.- is desirable, if some propagation of error can be tolerated. We are led by all three of the arguments of these sections to consider "mi.xing transforma- tions." 25. Mixing Transformations A notion that has proved valuable in certain branches of probability theory is the concept of a mixing transj'ormation. Suppose we have a prob- ability or measure space fi and a measure preserving transformation F of the space into itself, that is, a transformation such that the measure of a 712 BELL SYSTEM TECHNICAL JOURNAL transformed region FR is equal to the measure of the initial region R. The transformation is called mixing if for any function defined over the space and any region R the integral of the function over the region F"R approaches, as w -^ 00 , the integral of the function over the entire space J2 multiplied by the volume of R. This means that any initial region R is mixed with uniform density throughout the entire space if F is applied a large number of times. In general, F"R becomes a region consisting of a large number of thin filaments spread throughout fi. As n increases the filaments become finer and their density more constant. A mixing transformation in this precise sense can occur only in a space with an infinite number of points, for in a finite point space the transforma- tion must be periodic. Speaking loosely, however, we can think of a mixing transformation as one which distributes any reasonably cohesive region in the space fairly uniformly over the entire space. If the first region could be described in simple terms, the second would require very complex ones. In cryptography we can think of all the possible messages of length A'^ as the space fi and the high probability messages as the region R. This latter group has a certain fairly simple statistical structure. If a mixing transforma- tion were applied, the high probability messages would be scattered evenly throughout the space. Good mixing transformations are often formed by repeated products of two simple non-commuting operations. Hopf- has shown, for example, that pastry dough can be mixed by such a sequence of operations. The dough is first rolled out into a thin slab, then folded over, then rolled, and then folded again, etc. In a good mixing transformation of a space with natural coordinates Xi , X2 , • ■ • , Xs the point Xi is carried by the transformation into a point Xi , with Xi = fi(X} , X2 , ■ ■ ■ , Xs) i = 1,2, ■ ■ • , S and the functions/,- are complicated, involving all the variables in a ''sensi- tive" way. A small variation of any one, X3 , say, changes all the Xi con- siderably. If X3 passes through its range of possible variation the point Xi traces a long winding path around the space. Various methods of mixing applicable to statistical sequences of the type found in natural languages can be devised. One which looks fairly good is to follow a preliminary transposition by a sequence of alternating substi- tutions and simple linear operations, adding adjacent letters mod 26 for example. Thus we might take ^ E. Hopf, "On Causality, Statistics and Probability," Journal of Math, and Physics, V. 13, pp. 51-102, 1934. COMMUNICATION THEORY OF SECRECY SYSTEMS 713 F = LSLSLT where T is a transposition, L is a linear operation, and 5 is a substitution. 26. Ciphers of the Type TkFSj Suppose that F is a good mixing transformation that can be applied to sequences of letters, and that Tk and Sj are any two simple families of trans- formations, i.e., two simple ciphers, which may be the same. For concrete- ness we may think of them as both simple substitutions. It appears that the cipher TFS will be a very good secrecy system from the standpoint of its work characteristic. In the first place it is clear on reviewing our arguments about statistical methods that no simple sta- tistics will give information about the key — ^any significant statistics derived from E must be of a highly involved and very sensitive type — the re- dundancy has been both diffused and confused by the mixing transformation F. Also probable words lead to a complex system of equations involving all parts of the key (when the mix is good), which must be solved simultane- ously. It is interesting to note that if the cipher T is omitted the remainmg system is similar to S and thus no stronger. The enemy merely "unmixes" the cryptogram by application of F-^ and then solves. If .S* is omitted the remaining system is much stronger than T alone when the mix is good, but still not comparable to TFS. The basic principle here of simple ciphers separated by a mixing trans- formation can of course be extended. For example one could use TkFiSjFiRi with two mixes and three simple ciphers. One can also simplify by using the same ciphers, and even the same keys as well as the same mixing transforma- tions. This might well simplify the mechanization of such systems. The mixing transformation which separates the two (or more) appear- ances of the key acts as a kind of barrier for the enemy — it is easy to carry a known element over this barrier but an unknown (the key) does not go easily. By supplying two sets of unknowns, the key for S and the key for T, and separating them by the mixing transformation F we have "entangled" the unknowns together in a way that makes solution very difficult. Although systems constructed on this principle would be extremely safe they possess one grave disadvantage. If the mix is good then the propaga- tion of errors is bad. A transmission error of one letter will affect several letters on deciphering. 714 BELL SYSTEM TECHNICAL JOURNAL 27. Incompatibility of the Criteria for Good Systems 'J'he five criteria for good secrecy systems given in section 5 appear to have a certain incompatibility when appHed to a natural language with its complicated statistical structure. With artificial languages having a simple statistical structure it is possible to satisfy all requirements simultaneously, by means of the ideal type ciphers. In natural languages a compromise must be made and the valuations balanced against one another with a view toward the particular application. If any one of the five criteria is dropped, the other four can be satisfied fairly well, as the following examples show: 1. If we omit the first requirement (amount of secrecy) any simple cipher such as simple substitution will do. In the extreme case of omitting this condition completely, no cipher at all is required and one sends the clear! 2. If the size of the key is not limited the Vernam system can be used. 3. If complexity of operation is not limited, various extremely compli- cated types of enciphering process can be used. 4. If we omit the propagation of error condition, systems of the type TFS would be very good, although somewhat complicated. 5. If we allow large expansion of message, various systems are easily devised where the "correct" message is mixed with many "incorrect" ones (misinformation). The key determines which of these is correct. A very rough argument for the incompatibility of the five conditions may be given as follows: From condition 5, secrecy systems essentially as studied in this paper must be used; i.e., no great use of nulls, etc. Perfect and ideal systems are excluded by condition 2 and by 3 and 4, respectively. The high secrecy required by 1 must then come from a high work characteristic, not from a high equivocation characteristic. If the key is small, the system simple, and the errors do not propagate, probable word methods will gen- erally solve the system fairly easily, since we then have a fairly simple sys- tem of equations for the key. This reasoning is too vague to be conclusive, but the general idea seems quite reasonable. Perhaps if the various criteria could be given quantitative significance, some sort of an exchange equation could be found involving them and giving the best physically compatible sets of values. The two most difficult to measure numerically are the complexity of operations, and the complexity of statistical structure of the language. APPENDIX Proof of Theorem 3 Select any message Mi and group together all cryptograms that can be obtained from Mi by any enciphering operation T, . Let this class of crypto- CO.\[.\fl'N/CAriO.\ THEORY OF S/'ICRF.CV SYSTEMS 715 grams be Ci . Group willi Mi all messages that can l)e (;btainecl from i/| by T~i^TjM\ , and call this class C\ . The same C\ would be obtained if we started witii any other M in C\ since TsT^TiMi = TiMi. Similarly the same G would be obtained. Choosing an M not in C'l (if any such exist) we construct C> and C'^ in the same way. Continuing in this manner we obtain the residue classes with properties (1) and (2). Let Mi and M2 be in Ci and suppose M2 = TiTYMi. If El is in Ci and can be obtained from Mi by El = TaMi = TpMi =■••== T„Mi, then El = TaT'^TiMi = TpT~2'TiM. = • • • - TxMo = T^Mi ■ ■ ■ Thus each Af , in d transforms into £1 by the same number of keys. Simi- larly each Ei in Ci is obtained from any M in Ci by the same number of keys. It follows that this number of keys is a divisor of the total number of keys and hence we have properties (3) and (4). The Design of Reactive Equalizers* By A. P. BROGLE, Jr. This paper describes a systematic method of approximating with a finite number of network elements a transfer characteristic which is a prescribed func- tion of frequency, rather than a constant, over the useful frequency band. Al- though applied here only to input and output couphng networks as reactive equalizers and where loss equalization to an extremely high degree of precision over a wide frequency band is desired, the mathematical expressions which form the basis for the design are applicable to any 4-terminal network whose transfer characteristic is specified in a similar manner over the real frequency range. The selection of the appropriate form of the transfer function for equalization purposes is the fundamental consideration. A squared Tchebycheff polynomial is found to be particularly suitable to produce a desired cut-off characteristic with- out impairing the precision of equalization in the useful band. A method of polynomial approximation based on the transformation co = tan (p/i is used to obtain the coefficients of the in-band approximating function. Predistorting the transfer specification and minimizing the mean-square error, the coefficients become the Fourier cosine coefficients for an infinite frequency range; and are the solutions of a linear set for a finite range, o <

OUTPUT COUPLING CIRCUIT LINE AND LINE EQUIPMENT INPUT COUPLING CIRCUIT --B V 1 / Co Q 4 ^ Fig. 2 — Simplified section of a broad-band transmission system. systems, and to outline the external requirements and limitations imposed by the system itself on these networks. The characteristics of input and output coupling networks which are of engineering interest are: (1) The contribution of the coupling circuits to the transmission per- formance of the system as a whole. (2) The impedance matching requirements between the coupling net- works and the transmission line. (3) The limitation on the maximum performance of a coupling network imposed by the parasitic capacitance usually present in the termination. These characteristics are perhaps best illustrated by a somewhat idealized section of a broad-band transmission system. Figure 2 represents the output 2Ref. 1, pp. 383-392. 718 BELL SYSTEM TECHNICAL JOURNAL stage of a repeater, a section of the associated transmission line, and the first stage of the succeeding repeater of a simplified system. The specification of a flat transmission characteristic over the useful freciuency band l)et\veen A and B in the figure indicates that equaHzation for the Une loss of the section must occur in either or both coupling circuits, in the line equipment, or in all three of these circuits. For feedback amplifiers, the most desirable type, a flat characteristic between A and B can be specified only if the feedback circuits, or /3 circuits, of the amplifiers are designed to have no transmission variation with frequency. In general, it is possible to suppose the feedback factor, /3, of the amplifiers to be the appropriately varying function of frequency to equalize a part of the line loss, thus altering the transmission specification from A to B. However, the /3 circuits must include regulation of other types in most cases. Hence, it is impractical to include much loss equalization in these circuits. Since satisfactory performance of the section is dependent also on the maintenance of a large signal-to-noise ratio, it is important that the line contain no sources of additional loss. It is clear, then, that the best trans- mission performance is obtained (1) without the use of equalization in the line^ and (2) when the reactive input and output coupling circuits equalize as large a percentage as possible of the total line loss. Physically, the coupling circuits will be transformers, plus any number of tuning and shaping elements. In addition to the primary function of metal- lically separating the line from the repeater amplifiers, it will be seen later that the transformers provide the means of adjusting, independent of the value of the prescribed line impedance, the final impedance level of the net- work to conform with the value of the parasitic capacitance present. Besides the contribution of the various networks in the system to the overall transmission performance, there is the problem of matching the coupling circuits to the line. For constant-resistance equalization, this problem is immediately solved by the relationship Z1Z2 = R-. Well-estab- lished techniques make it a relatively simple matter to design for a specified attenuation variation with frequency at the same time that the impedance of the equalizer is matched to the line. This same procedure, with certain modifications, can be carried over to the design of reactive equalizers. In Fig. 2, the transformers of the input and output coupling circuits are un- terminated. That is, the input of the output circuit and the output of the input circuit are terminated in substantially open circuits. In order to pre- vent the reflection of power at the junctions of the coupling circuits and the line, the impedances of the input and output circuits as viewed from the line must be made equal to the impedance of the line. This impedance re- ' In practice, the /3 circuits and constant resistance networks associated with the line actually equalize a certain percentage of the total line loss characteristic. DESIGN OF REACTIVE EQUALIZEKS 719 quiremenl is fulfilled by providing both coupling circuits with a balancing network, connected as shown in Fig. ^. By accepting a small constant trans- mission loss,' the relationship ZiZo = R- is satisfied if the im[)edance Zo of the balancing network is made the inverse of the transmission circuit inii)edance Zi. Because of the relative ease of designing an inverse impedance /■:, once Zi is known in the final stages of a particular design, it is appropriate to omit from further discussion the presence of the balancing networks. The fundamental theoretical limitation in the maximum transmission performance of these coupling networks is due directly to the presence of the parasitic tube capacitances Co and d . If the parasitic capacitances were not present, the turns ratios of the transformers in the coupling circuits could quite evidently be made extremely high in order to produce over any specified frequency band as large a transmission response as desired. How- ever, even though these capacitances are usually small, they always tend to short circuit the coupling networks whenever the impedance ratios of the R i:a LINE *R — vw > R pi z, COUPLING CIRCUIT BALANCING NETWORK Rl - 2R ZiZ2= a2R2 — *■ Z2 Fig. 3 — Balancing network arrangement. transformers are made too high. The determination of the maximum re- sponse of these networks over a prescribed frequency range is thus a basic problem in the design of reactive equalizers. The fundamental limitation on the response of these networks is expressed in terms of the total area available under the transfer characteristic.^ When this characteristic is a desired function over a finite frequency band, the maximum utilization of the area available is obviously attained when all the area is included in the useful band. This condition is described as a resistance efficiency of 100 per cent. A smaller resistance eflficiency, 75 per cent for example, means that three-fourths of the total area under the characteristic is available in the useful frequency region, while the remainder of the area may be utilized to decrease the rate at which the characteristic is cul-ojf. Hence, the realization of a prescribed resistance efiiciency in the * The effective impedance of the hne as viewed from the coupling circuit is equal to twice the actual line impedance. Thus, a penalty of 10 log -^ = 3db is imposed by the K presence of the balancing network. ^ See eq. (4) and discussion in the following section. 720 BELL SYSTEM TECHNICAL JOURNAL design of a reactive equalizer places a definite requirement on the behavior of the transfer characteristic outside the useful frequency band. Although the precision of equalization as a design requirement actually is inclusive in the term transmission performance as used previously, it is included here as a separate requirement to emphasize its importance in this problem. The specification of a flat transmission from A to B in Fig. 2 provides the means of assigning to the tolerance of equalization a quantita- tive meaning. Hence, the tolerance per repeater section of the system may be expressed as the maximum allowable db deviation from the flat trans- mission characteristic, A to B, over the useful frequency band. For extremely broad-band systems, such as a coaxial system for simultaneous long-distance telephone and television transmission, many repeater sections appear in tandem between terminals. Thus, the deviations in each of these sections contribute to the system as a whole. In addition to the distances usually involved, repeater spacing becomes closer as the effective transmission band of these systems is increased. In order to design new systems with increas- ingly better overall tolerances, at the same time that the broad-banding requirements call for a greatly increased number of repeater sections per system, the tolerances imposed on the mdividual sections become exceed- ingly small. As a consequence, the maximum tolerance for an individual section must be specified as perhaps less than ±0.05 db deviation. 2. The Problem of Reactive Equalization In this section the problem of reactive equalization will be formulated in terms of the special problems of input and output coupling circuit design. Broadly speaking, the general characteristics of input and output coupling networks, as outlined in the introduction to establish the practical basis for reactive equalization, will be further developed in order to give them a quantitative meaning. Because of the complexity of some derivations and their extensive treatment elsewhere, detailed proofs in general will be merely outlined. The method of analysis follows Bode's treatment of the problem while the principal results taken from network theory are Guillemin's. As previously stated, the untermmated case for input and output coupling circuits arises whenever the terminating resistance is infinite in comparison with the other impedances of the network.® Figures 4 and 5 represent, re- spectively, an output and an input couplmg network of the type illustrated in Fig. 2 with infinite terminations. In each figure, Rl represents the line, N is the lossless coupling network, and C„ is the parasitic shunt capacitance « The so-called terminated case exists when the parasitic capacitance Co or C,- in Fig. 2 is shunted by a finite resistance. Since no essential differences exist between the two cases with respect to the approximation problem, an analysis for the unterminated case alone is sufficient to clarify the more important design considerations. DESIGN OF REACTIVE EQUALIZERS 721 which Hmits ihe response over any specified frequency band. For purj)oses of analysis and design, it is convenient to represent the coupling transformers in the manner indicated. By adopting this equivalent representation of a physical transformer, the so-called high-side equivalent circuit of the trans- former, which includes the leakage reactance, the magnetizing inductance, and the input and output winding capacitances, is incorporated as part of the coupling network itself. By excluding the ideal transformer portion of the equivalent represen- tation of the physical transformer from the network itself, a simpliftcation is possible. As shown in Figs. 6 and 7, the combination of the resistance Rl IDEAL Fig. 4 — Output coupling circuit. Fig. 5 — Input coupling circuit. and the ideal transformer may, in each case, be replaced by a resistance Ro = o^Rl , where "a" is the step-up turns ratio of the ideal transformer. Rl is the specified resistance, and Rq and "a" are determined in the design procedure from the maximum response obtainable with the prescribed capacitance Cn in the termination. The starting point for the study of these circuits is a consideration of the limitation on the amplitude response of these networks with frequency due to the presence of C„ in the terminations. Since the current ratio — in Fig. 6 and the voltage ratio ~ in Fig. 7 might be as large as desired if it were not for the presence of C„, the immediate problem is that of relating the magni- tude of these ratios, as functions of the real frequency, to the capacitance C„. This relationship is dependent on a necessary condition for the physical 722 BELL SYSTEM TECHNICAL JOURNAL realizability of a driving-point impedance function. If this function is chosen as the Z = R-\- jX in the figures, the necessary condition of interest is that Z, as an analytic function, have no poles in the right half of the complex fre- quency plane and that Z approach ~^ as co approaches infinity. By inte- grating this function over the appropriate path in the right half of the X (complex frequency) plane and setting the result equal to zero, the desired expression becomes *'0 *<'" = 2C.- To show that the resistance R is related to the ratios J, E and — I hi. (1) it is Z = RtJx ^ Ro 0 for CO > 0. This condi- network of the tvpe described is that tion will be insured if the polynomial, 1 -|- Bicj + • • • + /i„w ", has no negative real X- roots of odd multiplicity.^^ In addition to the sufficiency of eq. (7), if the -^ — = fV4 derived from "" in the usual manner Ao //(A) 1 Ro is to be the transfer impedance of a lossless network terminated in resistance, it is necessary that g{\) be either even or odd and that li(\) be a Hurwitz polynomial.'- In this problem g(X) = 1 is surely even since all zeros of -^ — occur at inhnity; and the method of forming — ^ — always insures that h{\) = m -\- n, wliere m is the even part and ;/. is the odd part of /;(X), is a Hurwitz polynomial. Thus, the fulfillment of the sufficient condition that there be no negative real X^ roots of odd multiplicity of B(co') is the assurance that the JB's of eq. (7) will always produce a physical network of the con- figuration of Fig. 9. Once the conditions for physical realizability have been fulfilled, and a ZM R, .... . , / m elements are easily calculated from a partial traction expansion of Z22 = — has been found in the final stages of a particular design, the network according to the following relation: Zi2(X) _ ZuO^)/Ro _ g(X) _ g(X)/n Ro 1 + S22(X)/i?c ni+n I + m/n' (9) where zU\) = ^^ and zUX) n n The previous discussion of the special problems of input and output coupling circuit design has been based, broadly, on (1) a consideration of the terminating or load impedance, (2) a consideration of the shape of the transfer characteristic, and (3) a consideration of the conditions for physical realizability. A major problem in the design is the choice of an appro.ximat- ing function which satisfactorily matches the stated transfer characteristic over the useful frequency band and, at the same time, sharply changes slope near the cut-off frequency so that it approximates zero outside the useful band in a prescribed manner. When the transfer characteristic is a constant over the useful frequency band, e.g., the impedance matching and low-pass filter cases, techniques which employ TchebychefT [iolynomials as the ap- 11 Ref . 4. '2 A Hurwitz polynomial is defined as a polynomial in X which has the property that the quotient of its even and odd parts, 0 for 0 < a; < ^, and referring to Fig. 16, J\X ) ~r C V n\^) a negative value of f{x~) in the immediate out-band region might be of sufficient magnitude to cancel the positive effect of e-F„(.v) and, hence, produce a negative value oif{x^) + €-Vn{x). However, at higher frequencies, the squared Tchebycheff polynomial takes on very large positive values. Thus, negative values and variations in f{x^) are effectively reduced in the magnitude of their effect on K Znijx) /(/) + e Vi(x) in direct relation to the increase in the magnitude of €-F„(.t). In order that an accurate prediction of the resistance efficiency may be made, it is necessary that the slope oif(x-) + eWl{x) increase in a uniform manner in the immediate out-band region. Since variations in the slope of f{x^) have their largest effect ui the region just outside the useful band, it is, of course, best to prevent rapid variations in this region. The remaining condition on the form oifix"^) is that Ao should be adjusted so that Ao < e"". By providing the transfer specification with a less steep slope requirement at low frequencies it is possible to obtain over the valuable DESIGN OF REACTIVE EQUALIZERS 733 IHjrtiou of the useful band an increased precision of equalization.'^ This adjustment represents an increased transmission at low frequencies. Thus. it is sometimes necessary to employ an ecjualizer of the constant resistance t_\-pe when additional equalization is desired at low frequencies. Figures 16 and 1 7 have been drawn to reflect this condition on Aq . After an/(.v-) which conforms with the requirements outlined above has been found, it is necessary to find a eW\{x) = A'J + .I2V + • • • + A'J" (15) which, when added to/(.v'-), produces the desired B{x~). This procedure is greatly facilitated by the known properties of Tchebycheff polynomials: A Tchebycheff polynomial of order n is defined by Vn{x) = cos (« cos~%). (16) This function oscillates between plus one and minus one for | .v | < 1 and approaches ± » for | .t | > 1 . Tabulated below are the expanded analytical expressions for the polynomials for n = 1 through ;; = 8. Fi(.v) = X F6(-v) F2(.v) - 2.V- - 1 V,{x) = 32/ Vz{x) = 4x - 3x V7{x) \\{x) = 8.V* - Sx-' + 1 T^8(.v) = 128x' - 256.v' + 160.v' - 32x' + 1 With the help of the recursion formula, xVn(x) = ^[Vn+l{x) + Vn-l(x)], (17) the corresponding expressions for w > 8 may be systematically calculated. Figure 18 shows a plot of the Tchebycheff polynomial for n = 5. In the case of low-pass filters'^ and impedance matching networks,'^ Tchebycheff polynomials are often used for the solution of the ai)proxima- tion problem. The function | Zi2(jx) |- in these cases has an oscillatory be- havior which approximates unity in the useful band, and has all its zeros at infinity so that the network consists of n elements of an unbalanced ladder structure of alternating series inductances and shunt capacitances. The appropriate function for | Zi2{jx) |- is I ^'='^'^' I' = 1 + } V'M ' ''«^ " There is a practical limit to the reduction of A 0 below e^"o. Referring to Figs. 13 and 14, it is apparent that A' = — . Thus, .4o is a direct measure of the impedance level over the useful band, and must not be made too small if the highest practical level of response is to be attained. '8 Ref. 2, pp. 53-79. " Ref. 3, pp. 26-34. 16.v^ - 20.v' + 5.V 32.v^ - 48.1-' + 18.V- - 1 64v' - 112.v' -f 56.t' - 7x 734 BELL SYSTEM TECHNICAL JOURNAL where e is an arbitrary constant. Figure 19 shows the plot of the squared Tchebycheff polynomial, eWn{x), for the values of » = 5, and e = 0.5 and 6 = 0.1, while Fig. 20 shows a j)lot of the transfer function expressed in eq. (18). It is to be noted that the oscillatory behavior with equal maxima and minima of squared Tchebycheff polynomials for values oi x < 1 and the rapid approach to + °c for values of x > 1 make their use particularly suitable as the solution of the approximation problem for low-pass filters and impedance matching networks. It is now apparent that these same Fig. 18 — Tchebycheff polynomial, F„(.t), for w = 5. properties validate their use as the out-band approximating function for reactive equalizers.-" Another useful property of squared Tchebycheff polynomials as ap- proximating functions for low-pass filters and impedance matching net- works is the inclusion of the specification of the tolerance as a factor in the transfer function. The allowable db deviation over the useful band is related to e by €' = e - 1, where ap is the maximum pass-band loss in nepers. Thus, the appropriate choice of e always realizes the specified tolerance over the useful band. 2" When better tolerances are required and when the network configuration is not rigidly specified, Jacobian elliptic functions, rather than TchebychefT polynomials, might be employed. DESIGN OF REACTIVE EQUALIZERS 735 However, it is imi)ortaiit to observe that a given value of e automatically determines l)()th the pass-band tolerance and the rate of cut-oflF in the out- band region. Hence, if a specified tolerance is to be realized in the useful band, no control exists over the determination of the resistance elTiciency. Also, it is ap[)arent from Figs. 19 and 20 that small in-band deviations are always obtained at the expense of lower resistance efficiencies, and vice versa. Fig. 19— Squared Tchebycheff polynomials, i-Vl{x), for n = 5, and e = 0.5 and « = 0.1. Fig. 20— Transfer function expressed in eq. (18) for the values of n and e shown in Fig. 19. Returning to the problem of reactive equalization, for // odd, e-F^(.r) may be expressed as eWl{x) = e2(Ci.v2 + C.J + • • • + C„.r'"). (19) Thus, any A, of eq. (15) is given by aI = e-C . By using the expressions for Fi(.v) through V&{x) tabulated previously, or eq. (17), it is a very simple task to fmd the C, for any desired //. Thus, V\{x) = C\x'- + C2.V'* + • • • -f Cn.v"" is readily ascertained, and the only real problem is the choice of e'-. If /(.V-) has already been chosen, this is accomplished by an addition of /(.V-) and i-Vn{x) for several values of €-. When a c- is found such that the combination, when reciprocated, very closely apjiroximates the specified resistance efficiency, B{x-) is completely defmed. 736 BELL SYSTEM TECHNICAL JOURNAL The final expression for i^(.v-) may now be written as i^(.x:2) = fix') + eWlix) = (/!„ + AyX' + • • • + Anx'") + (A[x~+ ••• -i-Anx'"). (20) In terms of eq. (20), the corresponding expression for the special transfer function for equalization purposes becomes Zn(jx) 2 K Ro 1 (21) Ao + (^1 + A[)x' + (^2 + A2)x' + ... + (^„ + A'Jx' When all the Ay and A;, are known in a particular design, the coefiticients Bi • • ■ Bn of eq. (7) may be readily determined. Hence, the elements of the network may be found by using the appropriate equations of Section 2. 4. Approximation Method This section will consider the second of the two main tasks in the formu- lation of the design method. Broadly speaking, the special transfer function derived in the previous section, eq. (13), provides the approximating func- tions to be used in this problem while this section develops the systematic method of determining the coefficients of these functions for a finite number of network elements. The function of most interest in the approximation problem is the in-band approximating function f{x-). Thus, the develop- ment of the approximation method for reactive equalizers is concerned specifically with the determination, consistent with the previous limitations and requirements, of the coeflScients, Ao • • • ^„ , of the polynomial /(x-) . The Fourier method of polynomial approximation, first introduced by Wiener, ^1 is characterized by a transformation of the independent variable to make the approximating function in the new frequency domain a periodic function. Thus, the well-known method of Fourier analysis is available as a general polynomial approximation method. This method has not been ap- plied extensively in practical applications. However, the uniform nature of Bix"^) over the useful frequency range makes its application to the design of reactive equalizers of the type described here seem feasible. By the transformation x = tan

Fig. 21 — Graphical representation of the transformation x = tan ^. the Ak in eq. (14) to the au in eq. (22). This is accomplished by means of the following relationships: X = tan /; — cos (f + cos (p 1 2 1 -~ X COS Hip = Vn (cos (p). Thus, the corresponding expression for eq. (22) in the frequency domain becomes fi{() + ^1 COS if + ^2 COS- ip -\- bz CO?, ip -\- ■■■+ bn cos" (p -'^^•' ^ (1 + x'Y + = /(.v^)/.(.v^), where /2(.v ) = 1 (1 + .t2)»' Therefore, it is necessary to predistort the approximatefl function B{x'-) by redefining the /( L k=0 J (23) is minimized. The set of linear equations which relates the a^ of the approximating function /ly to the approximated function a(ip) is derived for a range 0 to 5 in the ip domain with s < t hy minimizing eq. (23).- The minimum con- dition is specified when the derivative with respect to each coefficient aj is zero. Thus, da-^ i ^ I ^(^) - IJ ak cos kip [-cos 7V] dip = 0 (24) is the analytical expression for this condition. Collecting terms, — = -2 / [a(ip) cos jip] dip + 2 / X) a.f, cos kip [cos />] dip "";■ ''0 ^0 L'^^^o J -"-■' T- = —2 [a(ip) cos jip] dip + 2a j / cos 7V cos kip dip = 0 •'0 JQ ' cos jip cos kip dip and C^ = / [a(ip) cos Mdip, the set of '^ This derivation is similar to one given by R. M. RedhelTer in Ref. 6, pp. 8-10. 740 BELL SYSTEM TECHNICAL JOURNAL linear equations becomes 2^ Pjk aj = Ck . 3=0 (i = 0,1,2, ■■■,n) (25) Therefore, the procedure for determining the optimum coefficients for the range 0 to 5 in the ip domain is as follows: First, compute the Ck which depend on the approximated function a{ip). Ck = I [aicp) cos k(p] dip. Jo (26) Next, compute the elements of Pjk given by Pik — _ sin ij - k)s sin (j -j- k)s ,^ .. 2(7 - k) ^" 2' 2(i + k) Poo = -y. (27) These elements depend only on the range 5 and termmate with the desired n in any design. For convenience, these numbers may be arranged in the form of a symmetrical matrix [Pjk]. Hence, the optimum coefficients are found by solving the matrix equation, [Pjk] X k] = [Ck]. (j,k = 0,l,2,'--, n) (28) In this problem of approximating B(x^) to a high degree of precision over the useful frequency range, the range in the cp domain of most interest is 0 TT to - . However, before the approximation over only part of the frequency range is considered, it is helpful to set down the relations which apply when a{(p) is approximated over the whole frequency range, 5 = tt. In this case, the matrix [Pjk] takes on a form in which all non-diagonal entries are zero. Thus, "'tt 0 0 0 • • [P,A.] = Poo Pw Poi Pn PnO Pon Pm ■t^nn 0^00 0 0^0 0 0 0 J DESIG.X OF REACTIVE EQUALIZERS 741 The solution in this case is particularly simple, and gives the well-known Fourier coeilicients, 0 = - / a(:2) ¥> BW) BW cos«| /i(^) /(*») 0 3.00 2.98 0° 3.00 3.00 2.98 2.98 0.1 2.87 2.91 10° 2.88 2.80 2.77 2.87 0.2 2.69 2.74 20° 2.74 2.49 2.48 2.73 0.3 2.49 2.48 30° 2.56 2.09 2.09 2.57 0.4 2.09 2.17 40° 2.21 1.54 1.58 2.28 0.5 1.80 1.85 50° 1.87 1.05 1.07 1.95 0.6 1.57 1.57 60° 1.60 0.68 0.70 1.65 0.7 1.37 1.39 70° 1.37 0.42 0.43 1.39 0.8 1.22 1.23 80° 1.17 0.24 0.24 1.17 0.9 1.11 1.13 90° 1.00 0.13 0.13 1.00 1.0 1.00 1.00 1.1 — 0.56 1.2 — -0.32 1.3 — -2.12 1.5 — -11.4 2.0 — -115.0 0.3 < a; < 1, becomes B{x) = e"^ e -2k/{x) _ 2(.1-Vi) In addition, the specification for B(x-) may be extended to zero frequency by reciprocating the dashed portion of the curve of Fig. 23 in the range 0< X < 0.3. 746 BELL SYSTEM TECHNICAL JOURNAL In this illustration a simplified f{x-) = .4o + AiX + A2X + A3X of order (n — 2) may be chosen such that the transfer characteristic is matched within the specified tolerance over the useful band.^^ The specification a{