iH«iQl^J^iWWlAS41^8All^liSAJl^^ found i;gG244Q public Hihvaxv This Volume is for REFERENCE USE ONLY From the collection of the ^ m o Prelinger V Jjibrary t P San Francisco, California 2008 THE BELL SYSTEM TECHNICAL JOURNAL A JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION EDITORS R. W. King J. O. Perrine EDITORIAL BOARD W. H. Harrison O. E. Buckley O. B. Blackwell M. J. Kelly H. S. Osborne A. B. Clark J. J. PiLLioD F. J. Feely TABLE OF CONTENTS AND INDEX VOLUME XXVI 1947 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK PRINTED IN U. S. A. '^-^H^^-'Vx. cS= THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXVI 1347 Table of Contents January, 1947 Development of Silicon Crystal Rectifiers for Microwave Radar Re- ceivers — /. H. Scaff and R. S. Ohl 1 End Plate and Side Wall Currents in Circular Cylinder Cavity Reso- nator — J. P. Kinzer and 1 . G. Wilson 31 First and Second Order Equations for Piezoelectric Crystals Expressed in Tensor Form — W. P. Mason 80 The Biased Ideal Rectifier — W. R. Bennett 139 Properties and Uses of Thermistors — Thermally Sensitive Resistors — /. A. Becker, C. B. Green and G. L. Pearson 170 April, 1947 Radar Antennas — H. T. Friis and W. D. Lewis 219 Probability Functions for the Modulus and Angle of the Normal Com- plex Variate — Ray S. Hoyt 318 Spectrum Analysis of Pulse Modulated Waves — /. C. Lozier 360 July, 1947 Telephony by Pulse Code Modulation— If . M. Goodall 395 Some Results on Cylindrical Cavity Resonators — /. P. Kinzer and l.G. Wilson 410 Precision Measurement of Impedance Mismatches in Waveguide — Allen F. Pomeroy 446 Reflex Oscillators — J. R. Pierce and W. G. Shepherd 460 iii 126^^40 MI\R 9 1348 iv bell system technical journal October, 1947 The Radar Receiver — L. W . Morrison, Jr 693 High-\'acuum Oxide-Catliode Pulse Modulator Tubes — C. E. Fay . . . . 818 Polyrod Antennas — G. E. Mueller and W . A. Tyrrell 837 Targets for Microwave Radar Navigation — Sloan D. Robertson 852 Tables of Phase Associated with a Semi-Inhnite Unit Slope of Atten- uation — D. E. Thomas 870 Index to Volume XXVI Analysis, Spectrum, of Pulse Modulated Waves, /. C. Lozier, page 360. Antennas, Polyrod, G. E. Mueller and W . A . Tyrrell, page 837. Antennas, Radar, E. T. Frits and W. D. Lewis, page 219. Attenuation, Tables of Phase Associated with a Semi-Infinite Unit Slope of, D. E. Thomas, page 870. B Becker, J. A., C. B.Green and G. Z.Pear^ow, Properties and Uses of Thermistors — Therm- ally Sensitive Resistors, page 170. Bennett, W. R., The Biased Ideal Rectifier, page 139. Cavity Resonator, Circular Cylinder, End Plate and Side Wall Currents in, /. P. Kinzer and I. G. Wilson, page 31 . Cavity Resonators, Cylindrical, Some Results on, /. P. Kinzer and I. G. Wilson, page 410. Code Modulation, Pulse, Telephony by, W. M. Goodall, page 395. Crystal, Silicon, Rectifiers for Microwave Radar Receivers, Development of, /. H. Scaff and R. S. Ohl, page 1. Crystals, Piezoelectric, Expressed in Tensor Form, First and Second Order Ecjuations for, W. P. Mason, page 80. Fay, C. E., High-Vacuum Oxide-Cathode Pulse Modulator Tubes, page 818. Friis, H. T. and W. D. Lewis, Radar Antennas, page 219. Goodall, W. M., Telephony by Pulse Code Modulation, page 395. Green, C. B.,G. L.PearsonandJ . A. Seeder, Properties and Uses of Thermistors- — Therm- ally Sensitive Resistors, page 170. H Hoyt, Ray S., Probability Functions for the Modulus and Angle of the Normal Complex Variate, page 318. Impedance Mismatches in Waveguide, Precision Measurement of, Allen F.Pofneroy, page 446. K Kinzer, J. P. and /. G. Wilson, End Plate and Side Wall Currents in Circular Cylinder Cavity Resonator, page 31. Kinzer, J. P. and I. G. Wilson, Some Results on Cylindrical Cavity Resonators, page 410 Lewis, W . D. and H. T. Friis, Radar Antennas, page 219. Lozier, J . €., Spectrum Analysis of Pulse Modulated Waves, page 360. M Mason, W . P., First and Second Order Equations for Piezoelectric Crystals Expressed in Tensor Form, page 80. vi BELL SYSTEM TECHNICAL JOURNAL Microwave Radar Navigation, Targets for, Sloan D. Robertson, page 852. Microwave Radar Receivers, Development of Silicon Crystal Rectifiers for, /. H. Scajf and R. S. Ohl, page 1 . Mismatches, Impedance, in Waveguide, Precision Measurement of, Allen F. Pomeroy, page 446. Modulated Waves, Pulse, Spectrum Analysis of, /. C. Lozier, page 360. Modulation, Pulse Code, Telephony by, W. M . Goodall, page 395. Modulator Tubes, High-Vacuum Oxide-Cathode Pulse, C. E. Fay, page 818. Morrison, Jr., L. W ., The Radar Receiver, page 693. Mueller, G. E. and W . A. Tyrrell, Polyrod Antennas, page 837. N Navigation, Microwave Radar, Targets for, Sloan D. Robertson, page 852. O Ohl, R. S. and J. H. Scaf, Development of Silicon Crystal Rectifiers for Microwave Radar Receivers, page 1 . Oscillators, Reflex,/. R.Pierce and W . G. Shepherd, page 460. P Pearson, G. L., J. A . Becker and C. B. Green, Properties and Uses of Thermistors — Therm- ally Sensitive Resistors, page 170. Phase, Tables of, Associated with a Semi-Infinite Unit Slope of Attenuation, D. E. Thomas, page 870. Pierce, J. R. and W. G. Shepherd, Reflex Oscillators, page 460. Piezoelectric Crystals Expressed in Tensor Form, First and Second Order Equations for, W. P. Mason, page 80. Polyrod Antennas, G. E. Mueller and W. A . Tyrrell, page 837. Pomeroy, Allen F., Precision Measurement of Impedance Mismatches in Waveguide, page 446. Probability Functions for the Modulus and Angle of the Normal Complex Variate, Ray S. Hoyt, page 318. Pulse Code Modulation, Telephony by, W. M. Goodall, page 395. Pulse Modulated Waves, Spectrum Analysis of,/. C. Lozier, page 360. Pulse Modulator Tubes, High-Vacuum Oxide-Cathode, C. E. Fay, page 818. R Radar: High -Vacuum Oxide-Cathode Pulse Modulator Tubes, C. E. Fay, page 818. Radar: End Plate and Side Wall Currents in Circular Cylinder Cavity Resonator, /. P. Kinzer and I. G. Wilson, page 31. Radar: Some Results on Cylindrical Cavity Resonators, /. P. Kinzer and I. G. Wilson, page 410. Radar: Polyrod Antennas, G. E. Mueller and W. A . Tyrrell, page 837. Radar: Reflex Oscillators, /. R. Pierce and W. G. Shepherd, page 460. Radar Antennas, H. T. Friis and W. D. Lewis, page 219. Radar Navigation, Microwave, Targets for, Sloan D. Robertson, page 852. Radar Receiver, The, L. W. Morrison, Jr., page 693. Radar Receivers, Microwave, Development of Silicon Crystal Rectifiers for, /. H. Scaff and R. S. Ohl, page 1. Receiver, Radar, The, L. W. Morrison, Jr., page 693. Receivers, Microwave Radar, Development of Silicon Crystal Rectifiers for,/. H. Scaff and R. S. Ohl, page 1. Rectifier, Biased Ideal, The, W. R. Bennett, page 139. Rectifiers, Silicon Crystal, for Microwave Radar Receivers, Development of,/. E. Scaff and R.S. Ohl, page 1 . Reflex Oscillators, /. R. Pierce and W. G. Shepherd, page 460. Resistors, Thermally Sensitive — Properties and Uses of Thermistors, /. A . Becker, C. B. Green and G. L. Pearson, page 170. Resonator, Circular Cylinder Cavity, End Plate and Side Wall Currents in, /. P. Kinzer and I. G. IFz/50M,page31. Resonators, Cylindrical Cavity, Some Results on, /.P. Kinzer and LG. H^i/^OM, page 410. Robertson, Sloan D., Targets for Microwave Radar Navigation, page 852. INDEX Scajf, J. B. and R. S. Ohl, Development of Silicon Crystal Rectifiers for Microwave Radar Receivers, page 1 . Shepherd, W . G. and J. R. Pierce, Reflex Oscillators, page 460. Silicon Crystal Rectifiers for Microwave Radar Receivers, Development of, J. E. Sea ff and R. S. Ohl, page 1. Spectrum Analysis of Pulse Modulated Waves, /. C. Lozier, page 360. Tensor Form, First and Second Order Equations for Piezoelectric Crystals Expressed in» W. P. Mason, page 80. Thermistors, Properties and Uses of — Thermally Sensitive Resistors, /. A . Becker, C. B. Green and G. L. Pearson, page 170. Thomas, D. E., Tables of Phase Associated with a Semi-Infinite Unit Slope of Attenuation, page 870. Tyrrell, W. A . and G. E. Mueller, Polyrod Antennas, page 837. V Vacuum, High-, Oxide-Cathode Pulse Modulator Tubes, C. E. Fay, page 818. W ' Waveguide, Precision Measurement of Impedance Mismatches in, Allen F. Pomeroy, page 446. Wilson, I. G. and J. P. Kinzer, End Plate and Side Wall Currents in Circular Cylinder Cavity Resonator, page 31. Wilson, I. G. and J. P. Kinzer, Some Results on Cylindrical Cavity Resonators, page 410. VOLUME XXVI JANUARY, 1947 no. i THE BELL SYSTEM TECHNICAL JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION Development of Silicon Crystal Rectifiers for Microwave Radar Receivers J. H. Scaff and R. S. Ohl 1 End Plate and Side Wall Currents in Circular Cylinder Cavity Resonator J. P. Kinzer and I. G. Wilson 31 First and Second Order Equations for Piezoelectric Crys- tals Expressed in Tensor Form W. P. Mason 80 The Biased Ideal Rectifier W. R, Bennett 139 Properties and Uses of Thermistors — Thermally Sensitive Resistors . .J.A. Becker, C. B. Green and G. L. Pearson 170 Abstracts of Technical Articles by Bell System Authors. . 213 Contributors to This Issue 217 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 50^ per copy $1.50 per Year THE BELL SYSTEM TECHNICAL JOURNAL Published quarterly by the American Telephone and Telegraph Company 195 Broadway J New York, N. Y- EDITORS R. W. King J. O. Perrine EDITORIAL BOARD W. H. Harrison O. B Blackwell H. S. Osborne J. J. PiUiod O. E. Buckley M. J. KeUy A. B. Clark S. Bracken «»i«i« »■■«■■■ 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. •i«i«>«i^i«a«i«>«>«i« Copyright, 1947 American Telephone and Telegraph Company PRINTED IN U. S A CORRECTION FOR ISSUE OF OCTOBER, 1946 In the article SPARK GAP SWITCHES FOR RADAR, lines 2-14 inclusive on page 593 should have appeared be- tween lines 10 and 11 on page 588. The Bell System Technical Journal Vol. XXVI Ja72uary, 1947 No. i Development of Silicon Crystal Rectifiers for Microwave Radar Receivers By J. H. SCAFF and R. S. OHL Introduction TO THOSE not familiar with the design of microwave radars the exten- sive war use of recently developed crystal rectifiers^ in radar receiver frequency converters may be surprising. In the renaissance of this once familiar component of early radio receiving sets there have been develop- ments in materials, processes, and structural design leading to vastly improved converters through greater sensitivity, stability, and ruggedness of the rectifier unit. As a result of these developments a series of crystal rectifiers was engineered for production in large quantities to the exacting electrical specifications demanded by advanced microwave techniques and to the mechanical requirements demanded of combat equipment. The work on crystal rectifiers at Bell Telephone Laboratories during the war was a part of an extensive cooperative research and development program on microwave weapons. The Office of Scientific Research and Development, through the Radiation Laboratory at the Massachusetts Institute of Technology, served as the coordinating agency for work con- ducted at various university, government, and industrial laboratories in this country and as a liaison agency with British and other Allied organiza- tions. However, prior to the inception of this cooperative program, basic studies on the use of crystal rectifiers had been conducted in Bell Telephone Laboratories. The series of crystal rectifiers now available may thus be considered to be the outgrowth of work conducted in three distinct periods. First, in the interval from 1934 to the end of 1940, devices incorporating point contact rectifiers came into general use in the researches in ultra- high-frequency and microwave communications techniques then under way at the Holmdel Radio Laboratories of Bell Telephone Laboratories. ' A crystal rectifier is an assymmetrical, non-linear circuit element in which the seat of rectification is immediately underneath a point contact applied to the surface of a semi- conductor. This element is frequently called "point contact rectifier" and "crystal de tector" also. In this paper these terms are considered to be S3'nonymous. 1 2 BELL SYSTEM TECHNICAL JOURNAL At that time the improvement in sensitivity of microwave receivers employ- ing crystal rectiliers in the frequency converters was clearly recognized, as were the advantages of rectifiers using silicon rather than certain well known minerals as the semi-conductor. In the second period, from 1941 to 1942, the advent of important war uses for microwave devices stimulated increased activity in both research and development. During these years the pattern for the interchange of technical information on microwave devices through government sponsored channels was established and was continued through the entire period of the war. With the extensive inter- change of information, considerable international standardization was achieved. In view of the urgent equipment needs of the Armed Services emphasis was placed on an early standardization of designs for production. This resulted in the first of the modern series of rectifiers, namely, the ceramic cartridge design later coded through the Radio Manufacturers Association as type 1N21. In the third period, from 1942 to the present time, process and design advances accruing from intensive research and development made possible the coding and manufacture of an extensive series of rectifiers all markedly superior to the original 1N21 unit. It is the purpose of this paper to review the work done in Bell Telephone Laboratories on sihcon point contact rectifiers during the three periods mentioned above, and to discuss briefly typical properties of the rectifiers, several of the more important applications and the production history. Crystal Rectifiers in the Early Microwave Research The technical need for the modern crystal rectifier arose in research on ultra-high frequency communications techniques. Here as the frontier of the technically useful portion of the radio spectrum was steadily advanced into the microwave region, certain limitations in conventional vacuum tube detectors assumed increasing importance. Fundamentally, these limitations resulted from the large interelectrode capacitance and the finite time of transit of electrons between cathode and anode within the tubes. At the microwave frequencies (3000 megacycles and higher), they became of first importance. As transit time effects are virtually absent in point contact rectifiers, and since the capacitance is minute, it was logical that the utility of these devices should again be explored for laboratory use. The design of the point contact rectifiers used in these researches was dictated largely, of course, by the needs of the laboratory. Frequently the rectifier housing formed an integral part of the electrical circuit design while other structures took the form of a replaceable resistor-like cartridge. A variety of structures, including the modern types, arranged in chrono- logical sequence, are shown in the photograph, Fig. 1. In general, the SILICON CRYSTAL RECTIFIERS 3 principal requirements of the rectifiers for laboratory use were that the units be sensitive, stable chemically, mechanically, and electrically, and ^v-^ 1934 Ti 1937 i^ -«~ Jt* Fig. 1— Point contact rectifier structures. 1934-1943. Approximately f actual size. that they be easily adjusted. Considering the known vagaries of the device's historical counterpart, it was considered prudent to provide in the structures means by which the unit could be readjusted as frequently as might prove necessary or desirable. 4 BELL SYSTEM TECHNICAL JOURNAL As the properties of various semi-conductors were known to vary widely, an essential part of the early work was a survey of the properties of a number of minerals and metalloids potentially useful as rectifier materials. There were examined and tested approximately 100 materials, including zincite, molybdenite, galena, iron pyrites, silicon carbide, and silicon. Of the materials investigated most were found to be unsuitable for one reason or another, and iron pyrites and silicon were selected as having the best overall characteristics. The subsequent studies were then directed toward improving the rectifying material, the rectifying surface, the j^oint contact and the mounting structure. Fig. 2 — Rectilicr inserts untl contact jxjints lor use in early 3(K)t) megacycle converters. Overall length of insert ^-inch approximately. i'"()r use at freciuencies in the region of .-^OOO megac}-cles standard demount- able elements, consisting of rectitier "inserts" and contact points, were develojied for use in various housings or mounting blocks, depending upon the j)articular circuit requirements. The rectitier "inserts" consisted of small wafers of iron pyrite or silicon, soldered to hexagonal brass studs as shown in Fig. 2a. In these devices the surface of the semi-conductor was prei)ared by grinding, polishing, and etching to develop good rectification characteristics. Our knowledge of the metallurgy of silicon had acKanced by this time to the stage where a uniformly acti\e rcctilier surface could be j)roduccd and searching for active spots was not nccessar\'. l'\irther- more, it was jiossible to ])repare inserts of a jiositive or negative \ariety, signifying that the easy direction of current llow was obtained with the silicon i)ositive with respect to the point or \ice \ersa. Owing to a greater noiilincarity of the current \-oUage characteristic, the n-t)"pe or negative SILICON CRYSTAL RECTIFIERS 5 insert tended to give better performance as microwave converters while the p-type, or positive insert, because of greater sensitivity at low voltages, proved to be more useful in test equipment such as resonance indicators in frequency meters. In certain instances also, it was advantageous for the designer to be able to choose the polarity best suited to his circuit design. In contrast, however, to the striking uniformity obtained with the silicon processed in the laboratory, the pyrite inserts were very non-uniform. Active rectification spots on these natural mineral specimens could be found only by tediously searching the surface of the specimen. More- over, rectifiers employing the pyrite inserts showed a greater variation in properties with frequency than those in which silicon was used. In addition to providing a satisfactory semi-conductor, it was necessary also to develop suitable materials for use as point contacts. For this use metals were required which had satisfactory rectification characteristics with respect to silicon or pyrites and sutBcient hardness so that excessive contact areas were not obtained at the contact pressures employed in the rectifier assembly. The metals finally chosen were a platinum-iridium alloy and tungsten, which in some cases was coated with a gold alloy. These were employed in the form of a fine wire spot welded to a suitable spring member. The spring members themselves were usually of a wedge shaped cantilever design and were made from coin silver to facilitate elec- trical connection to the spring. Several contact springs of two typical designs are shown in the photograph, Figs. 2b and 2c. A typical mounting block arranged for use with the inserts and points }: is shown in Fig. 1 (1940) and in Fig. 3. This block was so constructed that I it could be inserted in a 70 ohm coaxial line without introducing serious |l discontinuities in the line. The contact point of the rectifier was assembled 1 in the block to be electrically connected to the central conductor of the I coaxial radio frequency input fitting, while the crystal insert screwed into I a tapered brass pin electrically connected to the central conductor of the } coaxial intermediate frequency and d-c output fitting. The tapered pin I fitted tightly into a tapered hole in a supporting brass cylinder, but was : insulated from the cylinder by a few turns of polystyrene tape several ; thousandths of an inch thick. This central pin was thus one terminal of a i coaxial high-frequency by-pass condenser. The capacitance of this con- ' denser depended upon the general nature of the circuits in which the block was to be used, and was generally about 15 mmfs. The arrangement of the point, the crystal insert and their respective supporting members was I such that the point contact could be made to engage the surface of the silicon at any spot and at the contact pressure desired and thereafter be clamped firmly in a fixed position by set screws. Typical direct current characteristics of the positive and negative silicon inserts and of pyrite inserts assembled and adjusted in this mounting block are shown in Fig. 4. BELL' SYSTEM TECHNICAL JOURNAL INSULATING BEAD BRASS BLOCK - TAPERED BRASS PIN POLYSTYRENE TAPE INTERMEDIATE FREQUENCY ^ AND DIRECT CURRENT OUTPUT DETAIL OF COAXIAL CONDENSER ASSEMBLY , I Fig. 3 — Schematic diagram of one of the early crystal converter blocks. The inserts and points in appropriate mounting blocks were widely used in centimeter wave investigations prior to 1940.- The principal laboratory uses were in frequency converter circuits in receivers, and as radio fre- 2 G. C. Southworth and A. P. King, "Metal Horns as Directive Receivers of Ultra- short Waves," Proc. L R. E. v. 27, pp. 95-102, 1939; Carl R. Englund, "Dielectric Con- stants and Power Factors at Centimeter Wave Lengths," Bell Sys. Tech. Jour., v. 23, pp. 114-129, 1944; lirainerd, Koehler, Reich, and WoodrulT, "Ultra High Frequency Tech- niques," D. Van Nostrand Co., Inc., 250-4th Avenue, New York, 1942. SILICON CRYSTAL RECTIFIERS 7 quency instrument rectifiers. They were also used to a relatively minor extent in some of the early radar test equipment. Moreover, the avail- ability of these devices and the knowledge of their properties as microwave converters tended to focus attention on the potentialities of radar designs employing crystal rectifiers in the receiver's frequency converter. Similarly, the techniques established for preparation of the inserts tended to orient subsequent manufacturing process developments. For example, the methods now generally used for preparing silicon ingots, for cutting the rectifying element from the ingot with diamond saws, and for forming the lO-i 10-2 10-3 ^ .' y <- ^ / R^- -'' ,^' ^^ y^ NEGATIVE SILICON GOLD ALLOY POINT / • • '^^' F.- — ■ " z i ^' ./' f X /^A / • / POSITIVE SILICON > ' PLATINUM --*yr ALLOY point/] r F = FORWARD CURRENT R= REVERSE CURRENT V /iron PYRITES /gold ALLOY POINT 10-8 10-6 10-5 lO"'^ CURRENT IN AMPERES 10-3 10-2 10-1 Fig. 4 — Direct-current characteristics of silicon and iron pyrite rectifiers fabricated as inserts, 1939. back contact to the rectifying element by electroplating procedures, are still essentially similar to the techniques used for preparing the inserts in 1939. As a contribution to the defense research effort, this basic informa- tion, with various samples and experimental assemblies, was made available to governmental agencies for dissemination to authorized domestic and foreign research establishments. Development of the Ceramic Type Cartridge Structure The block rectifier structure previously described was well adapted to various laboratory needs because of its flexibility, but for large scale utiliza- tion certain Umitations are evident. Not only was it necessan^- that the parts be accurately machined, but also the adjustment of the rectifier in 8 BELL SYSTEM TECHNICAL JOURNAL the block structure required considerable skill. With recognition of the military importance of silicon crystal rectifiers, effort was intensified in the development of standardized structures suitable for commercial pro- duction. In the 1940-1941 period, contributions to the design of silicon crystal rectifiers were made by British workers as a part of their development of new military implements. For these projected military' uses, the problem of replacement and interchangeability assumed added importance. The design trend was, therefore, towards the development of a cartridge type structure with the electrical adjustment fixed during manufacture, so that the unit could be replaced easily in the same manner as vacuum tubes. In the latter part of 1941 preliminary information was received in this country through National Defense Research Committee channels on a rectifier design originating in the laboratories of the British Thomson- Houston Co., Ltd. A parallel development of a similar device was begun in various American laboratories, including the Radiation Laboratory at the Massachusetts Institute of Technology, and Bell Telephone Labora- tories. In the work at Bell Laboratories, emphasis was placed both on development of a structure similar to the British design and on explora- tion and test of various new structures which retained the features of socket interchangeability but which were improved mechanicalh- and electrically. In the work on the ceramic cartridge, the external features of the British design were retained for reasons of mechanical standardization but a number of changes in process and design were made both to improve performance and to simplify manufacture. To mention a few, the position of the silicon wafer and the contact point were interchanged because measurements indicated that an improvement in performance could thereby be obtained. To obviate the necessity for searching for active spots on the surface of the silicon and to improve performance, fused high purity silicon was substituted for the "commercial" silicon then employed by the British. The rectifying element was cut from the ingots by diamond saws, and carefully polished and etched to develop optimum rectification character- istics. Similar improvements were made in the prej^aration of the point or "cats whisker", replacing hand operations l:)y machine techniques. To protect the unit from mechanical shock and the ingress of moisture, a sjiecial imjjregnating comjjound was de\'eloped which was completely satisfactory even under conditions of rapid changes in temperature from —40° to 4-70°C. All such improvements were directed towards ini]iro\ing quality and establishing techniques for mass production. In this early work time was at a jircmium because of the need for prompt standardization of the design in order that radar system designs might in SILICON CRYSTAL RECTIFIERS 9 turn be standardized, and that manufacturing facilities might be estabhshed to supply adequate quantities of the device. The development and initial production of the device was accomplished in a short period of time. This was possible because process experience had been acquired in the insert development, and centimeter wave measurements techniques and faciUties were then available to measure the characteristics of experimental units at the operating frequency. By December 1941, a pattern of manufacturing techniques had been established so that production by the Western Electric Company began shortly thereafter. This is believed to have been the first commercial production of the device in this country. As a result of the basic information on centimeter wave measurements techniques which was available from earlier microwave research at the Holmdel Radio Laboratory, it was possible also, at this early date, to propose to the Armed Services that each unit be required to pass an ac- ceptance test consisting of measurement of the operating characteristics at the intended operating frequency. This plan was adopted and standard test methods devised for production testing. Considering the complexity of centimeter wave measurements, this was an accomplishment of some magnitude and was of first importance to the Armed Services because it assured by direct measurement that each unit would be satisfactory for field use. The cartridge structure resulting from these developments and meeting the international dimensional standards is shown in Fig. 5. It consists of two metal terminals separated by an internally threaded ceramic insu- lator. The rectifying element itself consists of a small piece of silicon (p- type) soldered to the lower metal terminal or base. The contact spring or "cats whisker" is soldered into a cylindrical brass pin which slides freely into an axial hole in the upper terminal and may be locked in any desired position by set screws. The spring itself is made from tungsten wire of an appropriate size, formed into an S shape. The free end of the wire, which in a finished unit engages the surface of the silicon and establishes rectifica- tion, is formed to a cone-shaped configuration in order that the area of contact may be held at the desired low value. The silicon elements used in the rectifiers are prepared from ingots of fused high purity silicon. Alloying additions are made to the melt when required to adjust the electrical resistivity of the silicon to the value desired. The ingots are then cut and the silicon surfaces prepared and cut into small Dieces approximately 0.05 inch square and 0.02 inch thick suitable for use n the rectifiers. The contact springs are made from tungsten wire, gold Dlated to facilitate soldering. Depending upon the application, the wires 10 BELL SYSTEM TECHNICAL JOURNAL may be 0.005 inch, 0.0085 inch, or 0.010 inch in diameter. After forming the spring to the desired shape, the tip is formed electrolytically. In assembUng the rectifier cartridge, the two end terminals, consisting of the base with the silicon element soldered to it, and the top detail con- taining the contact spring, are threaded into the ceramic tube so that the free end of the spring does not engage the silicon surface. An adhesive wfifflSBtfSS^SSJ^ ■ i A ■ . I . ^M CERAMIC TUBE POINT ASSEMBLY— I TERMINAL Fig. 5 — Ceramic cartridge rectifier structure and parts. Overall length of assembled rectifier is approximately finch. is employed to secure the parts firmly to the ceramic. The rectifier is then "adjusted" by bringing the point into engagement with the silicon surface and establishing optimum electrical characteristics. Finally the unit is impregnated with a special compound to protect it from moisture and from damage by mechanical shock. Units so prepared are then ready for the final electrical tests. The adjustment of the rectifier is an interesting operation for at this SILICON CRYSTAL RECTIFIERS 11 stage in the process the rectification action is developed, and to a considerable degree, controlled. If the point is brought into contact with the silicon surface and a small compressional deflection applied to the spring, direct- current measurements will show a moderate rectification represented by the passage of more current at a given voltage in the forward direction than in the reverse. If the side of the unit is now tapped sharply by means of a small hammer, the forward current will be increased, and, at the same time, the reverse current decreased.^ With successive blows the reverse current is reduced rapidly to a constant low value while the forward current increases, but at a diminishing rate, until it also becomes relatively constant. The magnitude of the changes produced by this simple operation is rather surprising. The reverse current at one volt seldom decreases by less than a factor of 10 and frequently decreases by as much as a factor of 100, while the forward current at one volt increases by a factor of 10. Paralleling these changes are improvements in the high-frequency properties, the conversion loss and noise both being reduced. The tapping operation is not a haphazard searching for better rectifying spots, for with a given silicon material and mechanical assembly the reaction of each unit to tapping is regular, systematic and reproducible. The condition of the sihcon surface also has a pronounced bearing on "tappability" for by modifications of the surface it is possible to produce, at will, materials sensitive or insensitive in their reaction to the tapping blows. In the development of the compounds for filling the rectifier, special problems were met. For example, storage of the units for long periods of time under either arctic or tropical conditions was to be expected. Also, for use in air-borne radars operating at high altitudes, where equipment might be operated after a long idle period, it was necessary that the units be capable of withstanding rapid heating from very low temperatures. The temperature range specified was from —40° to -|-70°C. Most organic materials normally solid at room temperature, as the hydrocarbon waxes, are completely unsuitable, as the excessive contraction which occurs at i low temperatures is sufficient to shift the contact point and upset the precise adjustment of the spring. Nor are liquids satisfactory because of their tendency to seep from the unit. However, special gel fillers, consisting of a wax dispersed in a hydrocarbon oil, were devised in Bell Telephone Laboratories to meet the requirements, and were successfully applied by the leading manufacturers of crj^stal rectifiers in this country-. Materials of a similar nature, though somewhat different in composition, were also used subsequently in Britain. Further improvements in these compounds have been made recently, extending the temperature range 10°C at low ' Southworth and Kin^; loc. cit. 12 BELL SYSTEM TECHNICAL JOURNAL temperatures and about 30°C at high temperatures in response to the design trend towards operation of the units at higher temperatures. The units employing this compound may, if desired, be repeatedly heated and cooled rapidly between — 50°C and +100°C without damage. Use of the impregnating compound not only improves mechanical stability but prevents ingress or absorption of moisture. Increase of humidity would subject the unit not only to changes in electrical properties such as variation in the radio frequency impedance, but also to serious corrosion, for the galvanic couple at the junction would support rapid corrosion of the metal point. In fact, with condensed moisture present in unfilled units corrosion can be observed in 48 hours. For this reason alone, the develop- ment of a satisfactory filling compound was an important step in the suc- cessful utilization of the units by the Armed Services under diverse and drastic field conditions. Table I Shelf Aging Data on Silicon Crystal Rectifiers of the Ceramic Cartridge Design Initial Values Values After Storage for 7 Months Storage Conditions Conversion Loss (Median; (L) Noise Ratio (Median) (Nr) Conversion Loss (median) (L) Noise Ratio (median) (Nr) 75°F. 65% Relative Humidity 110°F. 95% Relative Humidity - 40°C . ... db 6.8 6.9 7.0 dh 3.9 3.9 3.9 dh 6.7 6.9 6.8 db 4.3 4.3 3.9 The large improvement in stability achieved in the present device as compared with the older crystal detectors may be attributed to the design of the contact spring, correct alignment of parts in manufacture and to the practice of filling the cavity in the unit with the gel developed for this purpose. Considering the apparently delicate construction of the device, the stability to mechanical or thermal shock achieved by these means is little short of spectacular. Standard tests consist of drojiping the unit three feet to a wood surface, immersing in water, and of ra])idly lieating from —40 to 7()°C None of these tests im])airs the quality of the unit. Similarly the unit will withstand storage for long periods of time under adverse conditions. Table I summarizes the results of tests on units which were stored for approximately one year under arctic ( — 40°), tropical (114°F — 95% relative humidity), and temi)erate conditions. Though minor changes in the electrical characteristics were noted in the accelerated tropical test, none of the units was inoperative after this drastic treatment. SILICON CRYSTAL RECTIFIERS 13 Development or the Shielded Rectifier Structure Rectifiers of the ceramic cartridge design, though manufactured in very large quantities and widely and successfully used in military apparatus, have certain well recognized limitations. For example, they may be ac- cidentally damaged by discharge of static electricity through the small point contact in the course of routine handhng. If one terminal of the unit is held in the hand and the other terminal grounded, any charge which may have accumulated will be discharged through the small contact. Since such static charges result in potential differences of several thousand volts it is understandable that the unit might suffer damage from the dis- charge. Although damage from this cause may be avoided by following a few simple precautions in handling, the fact that such precautions are needed constitutes a disadvantage of the design. Certain manufacturing difficulties are also associated with the use of the threaded insulator. The problem of thread fit requires constant attention. Lack of squareness at the end of the ceramic cyhnder or lack of concentricity in the threaded hole tends to cause an undesirable eccen- tricity or angularity in the assembled unit which can be minimized only by rigid inspection of parts and of final assemblies. At the higher frequencies (10,000 megacycles), uniformity in electrical properties, notably the radio frequency impedance, requires exceedingly close control of the internal mechanical dimensions. In the cartridge structure where the terminal connections are separated by a ceramic insulating member, the additive variations of the component parts make close dimensional control inherently difficult. To eliminate these difficulties the shielded structure, shown in Fig. 6, was developed. In this design the rectifier terminates a small coaxial line. The central conductor of the line, forming one terminal of the rec- tifier, is molded into an insulating cylinder of silica-filled bakelite, and has spot welded to it a 0.002-inch diameter tungsten wire spring of an offset C design. The free end of the spring is cone shaped. The rectifying element is soldered to a small brass disk. Both the disk, holding the rectifying element, and the bakelite cylinder, holding the point, are force- fits in the sleeve which forms the outer conductor of the rectifier. By locating the bakelite cylinder within the sleeve so that the free end of the central conductor is recessed in the sleeve, the unit is effectively protected from accidental static damage as long as the holder or socket into which the unit fits is so designed that the sleeve establishes electrical contact with the equipment at ground potential before the central conductor. The sleeve also shields the rectifying contact from effects of stray radiation. The radio frequency impedance of the shielded unit can be varied within certain limits by modifying the diameter of the central conductor. For 14 BELL SYSTEM TECHNICAL JOURNAL example, in the 1N26 unit, which was designed for use at frequencies in the region of 24,000 megacycles, a small metal slug fitting over the central conductor makes it possible to match a coaxial line having a 65-ohm surge impedance. For certain circumstances this modification in design is advantageous, while in others it is a disadvantage because the matching slug is effective only over a narrow range of frequencies. IS POINT ASSEMBLY OUTER CONDUCTORn METAL DISC Fig. 6 — Shielded rectifier structure and parts. Overall length of assembled rectifier is approximately | inch. The shielded structure was developed in 1942 and since it was of a sim- plified design with reduced hazard of static damage, it was proposed to the Armed Services for standardization in June of that year. However, because of the urgency of freezing the design of various radars and because the British had aheady standardized on the outhne dimensions of the ceramic type cartridge, Fig. 5, the Services did not consider it advantageous to standardize the new structure when first proposed. In deference to this international standardization program, plans for the manufacture of this 'i SILICON CRYSTAL RECTIFIERS 15 structure were held in abeyance during 1942 and 1943. However, an opportunity for realizing the advantages inherent in the shielded design was afforded later in the war and a sufficient quantity of the units was pro- duced to demonstrate its soundness. As anticipated from the construc- tional features, marked uniformity of electrical properties was obtained. Types, Applicatioks, akd Operating Characteristics Various rectifier codes, engineered for specific military uses, were manu- factured by Western Electric Company during the war. These are listed in Table II. The units are designated by RMA type numbers, as 1N21, 1N23, etc., depending upon their properties and the intended use. Letter suflixes, as 1N23A, 1N23B, indicate successively more stringent perform- ance requirements as reflected in lower allowable maxima in loss and noise ratio, and, usually, more stringent power proof-tests. In general, different codes are provided for operation in the various operating frequency ranges. For example, the 1N23 series is tested at 10,000 megacycles while the 1N21 series is tested at 3,000 megacycles and the 1N25 at 1000 megacycles, approximately. Since higher transmitter powers are frequently employed at the lower frequencies, somewhat greater power handling ability is provided in units for operation in this range. One of the more important uses of sihcon crystal rectifiers in military equipment was in the frequency converter or first detector in superheter- odyne radar receivers. This utilization was universal in microwave re- ceivers. In this application the crystal rectifier serves as the non-linear circuit element required to generate the difference (intermediate) frequency between the radio frequency signal and the local oscillator. The inter- mediate frequency thus obtained is then amplified and detected in conven- tional circuits. As the crystal rectifier is normally used at that point in the receiving circuit where the signal level is at its lowest value, its perform- ance in the converter has a direct bearing on the overall system performance. It was for this reason that continued improvements in the performance of crystal rectifiers were of such importance to the war effort. For the converter application, the signal-to-noise properties of the unit at the operating frequency, the power handling ability, and the uniformity of impedance are important factors. Tlie signal-to-noise properties are measured as conversion loss and noise ratio. The loss, L, is the ratio of the available radio frequency signal input power to the available inter- mediate frequency output power, usually expressed in decibels. The noise ratio, Nr, is the ratio of crystal output noise power to thermal (KTB) noise power. The loss and noise ratio are fundamental properties of the 16 BELL SYSTEM TECHNICAL JOURNAL "^ ^ ^^ Co ^ ^ OJ »-i t3 1-1 <u o )-"S o 1> > ^ > "c5 0. 1- > OJ > 1 OJ o 2; c o U 1 1-3 Q o o Q tn "o bj) bu bO be M bc be Ui U »-. U Lh ^ OJ OJ OJ 1) OJ " ro O "^ O <~0 "0 3^ OiO "3 u oc<io^o<M^d lO 'H lO H " CN — "^".1 o-S gS l^ll Lo ir:) lO »o Lo O irj "0 o »o 1-H 4J — 1> rt O -i 1 >< 8 rt rt rO SIQ kf. o E r^ ^« > S 1 O >i O C r-4 O o -* o . 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S ^ rt "" 3 S ■.Sec O PL, pLiC/J^Z SILICON CRYSTAL RECTIFIERS 17 » * CRYSTAL PARTS I I RECTIFIER (ENLARGED) RETAINING PLUG Fig. 7 — Converter for wave guide circuits as installed in the radio frequency unit of the AN/APQ13 radar system. This was standard equipment in B-29 bombers for radar bombing and navigation. 18 BELL SYSTEM TECHNICAL JOURNAL converter. From these data and other circuit constants, the designer may calculate* expected receiver performance. For operation as converters,^ crystal rectifiers are employed in suitable holders. These may be arranged for use with either coaxial line or wave guide circuits, depending upon the application. Figure 7 shows a converter for wave guide circuits installed in the radio frequency unit of an air-borne \ radar system. A typical converter designed for use with coaxial lines is [ shown in the photograph Fig. 8. A schematic circuit of this converter '. is shown in Fig. 9. In such circuits the best signal-to-noise ratio is realized when an optimum amount of beating oscillator power is supplied. The optimum power depends, in part, on the properties of the rectifier itself, and, in part, on other circuit factors as the noise figure of the intermediate Fig. 8 — Converter for use at 3000 megacycles. The crystal rectifier is located adjacent to its socket in the converter. frequency amplifier. For a well designed intermediate frequency amplifier with a noise figure of about 5 decibels, the optimum beating oscillator power is such that between 0.5 and 2.0 milliamperes of rectified current flows through the rectifier unit. Under these conditions and with the unit matched to the radio frequency line, the beating oscillator power absorbed by the unit is about one milliwatt. For intermediate frequency amplifiers ■" The quantities L and Ni? are related to receiver performance bj' the relationship F^ = Z.(N/? - 1 + FiF) where Fr is the receiver noise figure and Fip is the noise figure of the intermediate fre- quency amplifier. All terms are expressed as power ratios. A rigorous definition of receiver noise figure has been given l)v H. T. Friis "Noise Figures of Radio Receivers," Proc. L R. E., vol. 32, pp. 419-422; July, 1944. * C. F. Edwards, "Microwave Converters," presented orally at the Winter Technical Meeting of the /. R. E., January 1946 and submitted to the /. R. E. for publication. SILICON CRYSTAL RECTIFIERS 19 with poorer noise figures, the drive for optimum performance is higher than the figures cited above. Conversely, for intermediate frequency amphfiers with exceptionally low noise figures, optimum [performance is obtained with lower values of beating oscillator drive. If desired, somewhat higher currents than 2.0 milliamperes may be employed without damage to the crystal. The impedance at the terminals of a converter using crystal rectifiers, both at radio and intermediate frequencies, is a function not only of the rectifier unit, but also of the circuit in which the unit is used and of the SILICON RECTIFIER BY PASS CONDENSER y^^ SIGNAL INPUT Fig. 9 — Schematic diagram of crystal converter. power level at which it is operated. Consequently the specification of an impedance for a crystal rectifier is of significance only in terms of the circuit in which it is measured. Since the converters used in the production testing of crystal rectifiers are not necessarily the same as those used in the field, and since in addition there are frequently several converter designs for the same type of unit, a specification of cr>'stal rectifier impedance in pro- duction testing can do little more than select units which have the same impedance characteristic in the production test converter. The impedances at the terminals of two converters of different design but using the same crystal rectifier may vary by a factor of 3 or even more, with the inter- mediate frequency impedance generally varying more drastically than the radio frequency impedance. The variation is also a function of the con- 20 BELL SYSTEM TECHNICAL JOURNAL version loss. Crystals with large conversion losses are less susceptible to impedance changes from reactions in the radio frequency circuit than are low conversion loss units. The level of power to which the rectifiers can be subjected depends upon the way in which the power is applied. The application of an excessive amount of power or energy results in the electrical destruction of the unit by ru{)ture of the rectifying material. Experimental evidence indicates that the electrical failure may be in one of three categories. The total energ}^ of an applied pulse is responsible for the impairment when the pulse length is shorter than 10~' seconds, the approximate thermal time constant of the crystal rectifier as given by both measurement and calcula- tion. For pulse lengths of the order of 10~^ seconds the peak power in the pulse is the determining factor, and for continuous wave operation the limitation is in the average power. In performance tests in manufacture all units for which burnout tolerances are specified are subjected to proof-tests at levels generally comparable with those which the unit may occasionally be expected to withstand in actual use, but greater than those to be employed as a design maximum. The power or energy is applied to the unit in one of two types of proof-test equipment. The multiple, long time constant (of the order of 10" seconds) pulse test is applied to simulate the plateau part of a radar pulse reaching the crystal through the gas discharge transmit-receive switch.^ This test uses an artificial line of appropriate impedance triggered at a selected repetition rate for a determined length of time. The power available to the unit is computed from the usual formula, 4Z' where P is the power in watts, V is the potential in volts to which the pulse generator is charged, and Z is the impedance in ohms of the pulse generator. In general, where this test is employed, a line is used which matches the impedance of the unit under test at the specified voltage. The second type of test is the single discharge of a coaxial line through the unit to simulate a radar pulse spike reaching the crystal before the transmit-receive switch fires. The pulse length is of the order of 10~^ second. The energy in the test si)ike mav be computed from the relation where E is the energy in ergs, C the capacity of the coaxial line in farads, and r the potential in volts to whicli the line is charged. "A. L. Samuel, J. W. Clark, and W. W. Mumford, "The Gas Discharge Transmit- Receive Switch," Bell Sys. Tech. Jour., v. 25 No. 1, pp. 48-101. Jan. 1946. SILICON CRYSTAL RECTIFIERS 21 Specification proof-test levels are, of course, not design criteria. Since the units are generally used in combination with protective devices, such as the transmit-receive switch, it is necessary to conduct tests in the circuits I of interest to establish satisfactory operating levels. I In general, however, the units may be expected to carry, without deteriora- tion, energy of the order of a third of that used in the single d-c spike proof- ; test or peak powers of a magnitude comparable with that used in the multiple I flat-top d-c pulse proof-test. The upper Hmit for applied continuous wave i signals has not been determined accurately, but, in general, rectified currents i below 10 milliamperes are not harmful when the self bias is less than a few tenths of a volt. ' The service life of a crystal rectifier will depend completely upon the ; conditions under which it is operated and should be quite long when its ! ratings are not exceeded. During the war, careful engineering tests con- ! ducted on units operating as first detectors in certain radar systems revealed j no impairment in the signal-to-noise performance after operation for several [ hundred hours. A small group of 1N21B units showed only minor impair- I ments when operated in laboratory tests for 100 hours with pulse powers I (3000 megacycles) up to 4 watts peak available to the unit under test. Another important military application of silicon crystal rectifiers was as low-power radio frequency rectifiers for use in wave meters or other items of radar test equipment. Here the rectification properties of the unit at the operating frequency are of primary interest. Since units which are satisfactory as converters also function satisfactorily as high-frequency rectifiers special types were not required for this application. Units were also used in military equipment as detectors to derive directly the envelope of a radio frequency signal received at low power levels. These signals were modulated usually in the video range. The low-level performance is a function of the resistance at low voltages and the direct- current output for a given low-power radio frequency input. These may be combined to derive a figure of merit which is a measure of receiver performance.^ Typical direct-current characteristics of the silicon rectifiers at tempera- tures of —40°, 25° and 70°C are given in Fig. 10. It will be noted in these curves that both the forward and reverse currents are decreased by reducing the temperature and increased by raising the temperature. The reverse current changes more rapidly with temperature than the forward current, however, so that the rectification ratio is improved by reducing the tempera- ture, and impaired by raising the temperature. The data shown are for typical units of the converter type. It should be emphasized, however, '' R. Beringer, Radiation Laboratory Report No. 61-15, March 16, 1943. 22 BELL SYSTEM TECHNICAL JOURNAL that by changes in processing routines the direct-current characteristics shown in Fig. 10 may be modified in a predictable manner, particularly with respect to absolute values of forward current at a particular voltage. Modern Rectifier Processes When the development of the type 1N21 unit was undertaken, the scien- tific and engineering information at hand was insufficient to permit inten- tional alteration or improvement in electrical properties of the rectifier. In these early units, the control of the radio frequency impedance, power handling ability and signal-to-noise ratio left much to be desired. Within a short time, some improvements in performance were realized by process improvements such as the elimination of burrs and irregularities from the point contact to reduce noise. Substantial improvements were not obtained, I0-' 1 1 1 1 REVERSE CURRENT FORWARD CURRENT ^^ '"' / ^ 'r' ^^ • l ■^ ^ ^ -^ / / ^^■7 ^ ■^^"^ / / '^A 0-9 Fig. 1( K 3— Direct- 1 cu rrent c? 1 an var 3-6 CURREr icterist lous tei 1 JT 1 ics np 0-5 N AMPE of P-ty ;rature 1 RE pe< s. 0-4 silicon ( 1 :ryE 0-3 tal rec 10-2 titier at 10-1 J however, until certain improved materials, processes, and techniques were developed. In the engineering development of improved cr>'stal rectifier materials and jjrocesses, basic data have been acquired which make it possible to alter the properties of the rectifier in a predictable manner so that tlie units may now be engineered to the specific electrical requirements desired by the circuit designer in much the same manner as are modern electron tubes. This has led not only to improvements in performance but also to a diver- sification in types and applications. The simplified equivalent circuit for the point contact rectifier, shown in Fig. 11, provides a basis for consideration of the various process features. In Fig. 11, Cb represents the electrical capacitance at the boundary between the point contact and the semi-conductor, Rn the non-linear resistance at this boundary, and /^s is the spreading resistance of the semi-conductor SILICON CRYSTAL RECTIFIERS 23 proper, that is the total ohmic resistance of the siHcon to current through the point. The capacitance Cb being shunted across the rectifying bound- ary, decreases the efficiency of the device by its by-pass action because the current through it would be dissipated as heat in the resistance Rs. Losses from this source increase rapidly with increased frequency because of the enhanced by-pass action. It would appear, therefore, that to improve effi- ciency it would be important to minimize both Rs and Cb by some method such as reducing the area of the rectifying contact and lowering the body resistance of the silicon employed. For a given silicon material, the imped- ances desired for reasons of circuitry and considerations of mechanical stabiUty place a limit on the extent to which performance may be improved by reducing the contact area. Rs may be reduced by using silicon of lower resistivity, but this generally results in poorer rectification. This impair- ment is due apparently to some subtle change in the properties of the rectifying junction resulting from decreasing the specific resistance of the silicon material. Rg (NON-LINEAR BARRIER RESISTANCE) Rs I WV (SPREADING RESISTANCE) vw Cb (barrier capacity) Fig. 11 — Simplified equivalent circuit of crystal rectifier. The answer to this apparent dilemma lies in the application of an oxidizing heat treatment to the surface of the semi-conductor. This process derives from researches conducted independently in this country and in Britain, though there was considerable interchange of information between various interested laboratories. In the oxidizing treatment, apparently the im- purities in the silicon which contribute to its conductivity diffuse into the adhering silica film, thereby depleting impurities from the surface of the silicon. When the oxide layer is then removed by solution in dilute hydro- fluoric acid, the underlying silicon layer is exposed and remains intact as the acid does not readily attack the silicon itself. Since decreasing the impurity content of a semi-conductor increases its resistivity, the silicon surface has higher resistivity after the oxidizing treatment than before. Thus by oxidation of the surface of low resistance silicon it is possible to secure the enhanced rectification associated with the high resistance surface layer, while by virtue of the lower resistivity of the underlying material the PR losses through Rs are reduced. 24 BELL SYSTEM TECHNICAL JOVRXAL In actual practice the i)roperties of the rectifier are governed by the resistivity of the silicon material, the contact area, and the degree of oxida- tion of the surface. By the controlled alteration of these factors units may be engineered for specific applications. The body resistance of the silicon is controlled by the kind and quantity of the impurities present. Aluminum, beryllium or boron may be added to purified silicon to reduce its resistivity to the desired level. Boron is especially effective for this purpose, the quantity added usually being less than 0.01 per cent. As little as 0.001 per cent has a very pronounced effect upon the electrical properties. The contact area is determined by the design of contact spring employed and the deflection applied to it in the adjustment of the rectifier. The degree of oxidation is controlled by the time and temperature of the treat- ment and the atmosphere employed. In the development of the present rectifier processes, certain experimental relationships were obtained between the performance and the contact area on the one hand, and the power handling ability and contact area on the other. These show the manner in which the processes should be changed to produce a desired change in properties. For example. Fig. 12 shows the relationship between the spring deflection applied to a unit and the conver- sion loss at a given frequency. The apparent contact area, (i.e., the area of the flattened tip of the spring in contact with the silicon surface, as measured microscopically) also increases with increasing spring deflection. It will be seen in Fig. 12 that for a given silicon material, the conversion loss at 10,000 megacycles increases rapidly with the contact area. The curves tend to reach constant loss values at the higher spring deflections. It is believed that this may be ascribed to the fact that for a given spring size and form, the increment in contact area obtained by successive increments in spring deflection would diminish and finally become zero after the elastic limit of the spring is exceeded. The losses plotted in Fig. 12 were measured on a tuned basis, that is, the converter was adjusted for maximum intermediate frequency output at a fixed beating oscillator drive for each measurement. Were these measure- ments made on a fixed tuned basis, that is, with the converter initially ad- justed for maximum intermediate frequency output for a unit to which the minimum spring deflection is applied, and the units with larger deflections then measured without modification of the converter adjustment, even greater degradation in conversion loss than that shown in Fig. 12 would be observed. This results from the dependence of the radio frequency imped- ance upon the contact area. In loss measurements made on the tuned basis, changes in the radio frequency impedance occasioned by the changes in the contact area do not affect the values of mismatch loss obtained, while on the SILICON CRYSTAL RECTIFIERS 25 fixed tuned basis they would result in an increase in the apparent loss be- cause of the mismatch of the radio frequency circuits. While the conversion loss is degraded by increasing the contact area, the power handling ability^ of the rectifiers is improved, as shown in Fig. 13. FREQUENCY = 10,000 MEGACYCLES Q ^^^ A y ""'^ unitC ( ) /^ y — - J /A Y -^ i I \y n 1 2 3 4 5 6 7 8 SPRING DEFLECTION IN THOUSANDTHS OF AN INCH Fig. 12 — Relationshi]) between sjjring deflection and conversion loss in silicon crystal rectifiers. This is not surprising because the larger area contact gives a wider current distribution and thus minimizes the localized heating effects near the con- tact. Generally, therefore, in the development of units for operation at a *The measurement of power handling ability of crystal rectifiers by application of radio freciuency jwwcr is comi)licated by the fact that the impedance of the unit under test varies with power level. If a unit is matched in a converter at a low-power level and ]iower at a higher level is then applied, not all of the j^ower available is absorbed by the unit but a portion of it is reflected (due to the change in impedance). This factor has been called the self protection of the unit and it necessitates the distinction between the powei absorbed hy and the power available to the unit under test. The data for Fig. 13 were acquired by first matching the unit in converters at low powers (about 0.3 milliwatts CW 30C0 mc's) and then exposing it for a short period to successively higher levels of pulse power cf sc[uare wave form of 0.5 microseconds width at a rei:)etition rate of 20CO pulses per seccnd, measuring the loss and noise ratio after each power application. The power handling ability is then expressed as the available peak power required to cause a 3 db impairment in the conveision loss or the receiver noise figure. This method was employed because in ladar receivers the units are matched for low-power levels. In this lespect the method simulates field operating conditions, but the "spike" of radar pulses is absent. The increase in power handling abilit\' with increasing area shown in Fig. 13 is confirmed by similar measure ments with radio frequenc>- pulse power with the unit matched at high-level powers, b\- direct-current tests, and by simple 60-cycle continuous wave tests. The magnitude of the increase depends, however, upon the particular method employed for measurement. 26 BELL SYSTEM TECHNICAL JOURNAL given frequency, a compromise must be effected between these two impor- tant performance factors. Because of increased condenser by-pass action a smaller area must be used to obtain a given conversion loss at a higher fre- quency. For this reason the power handling ability of units designed for use at the higher frequencies is somewhat less than that of the lower-fre- II) — - > uj O Q. t -I UJ 2i <l UJ Z << 100 80 60 - FREQUENCY= 3000 MEGACYCLES • • » • • - • • a ( - « •• - • • • • 1 • - 1 • ■•>•• • • > •• • a • • - - 4 > t - • • • - 1 1 1 1 1 1 1 0.02 0.04 0.06 0.1 0.2 0.4 0.6 0.8 1.0 APPARENT CONTACT AREA IN SQUARE INCHES XIO" Fig. 13- -Correlation between power handling ability measured with microsecond radio frequency pulses and contact area in silicon crystal rectifiers. quency units because emphasis has been placed upon achieving a given sig-- nal-to-noise performance in each frequency band . Use of the improved materials and processes produced rather large im- provements in the d-c rectification ratio, conversion loss, noise, power handling ability, and uniformity. Typical direct-current rectification char- acteristics of units produced by both the old and the new processes are shown in Fig. 14. These curves show that reverse currents at one volt were de- creased by a factor of about 20 while the forward currents were increased by SILICON CRYSTAL RECTIFIERS 27 a factor of approximately 2.5 giving a net improvement in rectification ratio of 50 to 1. The parallel improvement in receiver performance resulting from process improvements is shown in Fig. 15. A comparison in power handling ui a. UJ u. a. r D \iS il< Q UJ UJ o tr ZUJ cnz UJ (J cr UJ 10-2 REVERSE CURRENT FORWARD CURRENT s/ ^"^ ' ,J "^ y ' y ■x"* ■^ ^ '■^ / ■^' \0-^ lO"'* 10" CURRENT IN AMPERES 10-2 Fig. 14 — Improvement in the direct-current rectification characteristics of sihcon crystal rectifiers in a four-year period. 10,000 MEGACYCLES 3000 MEGACYCLES 16 15 ■ (/5 .■ ■ UJ •■ • -1 -'. ,■ CL ■ :- </) ■■ 14 - ■ 2 ;-■ •; -1 • ;.cn-. . UJ ■• • •Q ■• ■• o ■ •■ 2 •. .■ <ri :. ■' -J .• ■ ' tu ■' • ; Q ■ \i " ■.a. ■• ; ■ o . • ■.■••-:• ■ 1- '■'■ •. O ■ '■ •" ■'. ;•<•; ■.-. z . • UJ • ■ ' t- ■ .' ■ D •■ •■' CO ■. ■ 12 '. tr •■ ■.o■.•. ;-(D •. .< ■• ; ; -J .-_ :■' Q-". : z ■': •■. Z '■ ■■o-: ;q: ;.. ■-!- .■• •z .-.• ■<J ■■■ 2 .' Q. •■o ■.■ ;•_!■. •. UJ •■ -■■ > ;■ .•■UJ .• ;■. Z ■■ • UJ ■, ■-. 2 ■ • .■• Q. ." :.0 ■■ .. _i • •• UJ ■, '. Z '■ •' 2 ■ ■,' 1- ■'■ :• o .■ :■ t ■: :• z:- ■■- ^.■■ .'z'.-. ; o •• .'z ■■■ . O . ■' ■ ■ to- II ,■.■■ 01 : >t ctz 'OZ> h- ■ 10 ■ UJ • . . H ■ :• u ;• • UJ •: '.-!■. •; Q • • o •; irz. . ■ Q •. ■. _J ■. ■• cr. ■■ ■. > • •. UJ ■ ;■ Q ;. ■ : >- ■ •• O . ■ a.- ■: Q- .'• . u ■• • 1- ■•. . o ■■■ m ao : < ■■ . _j ■ : <-!■ • Q--/ ■ Q ■•■.; orr '.■ ^ . •.■ cc '. •: t^ '■'. ' a. ■ • CEO ■•' o ." :o v (CK- :o(r .cr • ; <z .. UJ ■ , o ■ •o ■■ .cDi-: y - -lO' • o . ■■o •: <^ ■X- • o .. vo ::• .-. o- ■•fO .', ■_]0. 8 • •- • 1 ■ ■ '.'.■.■•-.• ". • ■ . , OCT 1942 DEC 1942 MAR APR 1945 SEPT JAN JULY SEPT NOV APR 1941 1942 1943 1944 1945 DATE * note: — 6 DECIBELS IS THE MINIMUM RECEIVER-NOISE FIGURE ATTAINABLE WITH A DOUBLE DETECTION RE- CEIVER EMPLOYING A CRYSTAL CONVERTER AND A 5-DB INTERMEDIATE-FREQUENCY AMPLIFIER. Fig. 15 — Effect of continued improvement in the crystal rectifier on the microwave receiver performance. The noise figures plotted are average values. ability of the 3000-megacycle converter types made by the improved pro- cedures and the older procedures is shown in Fig. 16. The flexibility of the processes may be illustrated by comparison of two 28 BELL SYSTEM TECHNICAL JOURNAL very different units, tlie 1X26 and the 1N25. Though direct comparison of power handling ability is complicated by the fact that the burnout test methods employed in the de^•elopment of the two codes were widely different, it may be stated conservatively that while the 1X26 would be damaged after absorbing something less than one watt peak pulse power, the 1X25 unit will withstand 25 watts peak or more. The 1X26 unit is, however, capable of satisfactory operation as a converter at a frequency of some 20 times that of the 1X25. These two units have been made by essentially the same pro- cedures, the difference in properties being principally due to modification of alloy composition, heat treatment, and contact area. u 6.0 a. 7.0 (j 8.0 6.2 7.2 8.2 9.2 notes: I. TEST FREQUENCY = 3000 MEGACYCLES 2. NOISE RATIO IS THE RATIO OF THE AVAILABLE OUTPUT NOISE POWER OF THE CRYSTAL RECTIFIER TO KTB IMPROVED PROCESS ""■-^ ~_ ■^ V 1 1 1 1 1 I— 1 1 1 1 1 1 3.1 ^ - . , III > 1 V 1 1 1 ^ ^ 1 111 ■^ 6.3 > 7.3 O 8.3 ^ 9.3 INITIAL PROCESS 1 ^^ T^ = :v 1 1 i 1 1 , 1, v^ 1 1 1 1 2.3 3.3 0.1 0.2 0.4 0.6 1.0 2 4 6 8 10 20 40 60 100 200 AVAILABLE PEAK PULSE POWER IN WATTS Fig. 16 — Comparison of Uie radio fre([uency power handling aljilit_\- of silicon crystal rectifiers prepared by different processes. Prior to the process developments described above, in the interests of simplifying the field supply problem one general purpose unit, the type 1X21 , had been made available for field use. However, it became obvious that the advantages of having but a single unit for field use could be retained only at a sacrifice in either power handling ability or high-frequency conversion loss. Since the higher power radar sets operated at the lower microwave frequencies, it seemed quite logical to employ the new processes to improve power handling ability at the lower microwave frequencies and to impro\e the loss and noise at the higher frequencies. A recommendation accordingly w^as made to the Services that different units be coded for operation at v^OOO megacycles and at 10, ()()() megacycles. The decision in the matter was SILICOX CRYSTAL RECTIFIERS 29 INCREASING POWER-HANDLING ABILITY IN25 (14.7) IN2IB (12.2) IN28 (13.2) IN26 (15.2) IN23B (12.7) IN23A (14.2) NEW PROCESS INTRODUCED IN2IA (14.6) IN23 (17.1) SELECTION IN21 (16.4) — NOTE— I NUMBERS IN PARENTHESES ARE RECEIVER NOISE FIGURES IN DECIBELS CALCULATED FOR THE POOREST UNIT ACCEPTABLE UNDER EACH SPECIFICATION AND BASED ON AN INTERMEDIATE-FREOUENCy AMPLIFIER NOISE FIGURE OF 5 DECIBELS ' 24,000 FREQUENCY IN MEGACYCLES PER SECOND Fig. 17 — Evolution of coded silicon crystal rectifiers. 24 - 22 ■M-^:-- 20 - 18 - 16 - 14 - 12 - 10 - 8 6 - 4 - 2 - * EXTRAPOLATED _ [ ■:\ 1942 1943 1944 1945 * YEAR Fig. 18 — Relative annual production of silicon crystal rectifiers at the Western Electric Company 1942-1945. affirmative. The importance of this decision may be appreciated from the fact that it permitted the coding and manufacture of units such as the 1X2 IB and 1N28, high burnout units with improved performance at 3000 30 BELL SYSTEM TECHNICAL JOURNAL megacycles, and the 1N23B unit which was of such great importance in 10,000 megac3xle radars because of its exceptionally good performance. From this stage in the development the diversification in types was quite rapid. The evolution of the coded units, of increasing power handling ability for a given performance level at a given frequency, and of better per- formance at a given frequency is graphically illustrated in Fig. 17. The large improvements in calculated receiver performance are again evident, especially when it is considered that the receiver performances given are for the poorest units which would pass the production test limits. Extent of Manufacture and Utilization An historical resume of the development of crystal rectifiers would be incomplete if some description were not given of the extent of their manu- facture and utilization. Commercial production of the rectifiers by Western Electric Company started in the early part of 1942 and through the war years increased very rapidly. Figure 18 shows the increase in annual production over that of the first year. By the latter part of 1944 the production rate was in excess of 50,000 units monthly. Production figures, however, reveal only a small part of the overall story of the development. The increase in production rate was achieved simultaneously with marked improvements in sensitivity, the improvements in process techniques being reflected in manu- facture by the ability to deliver the higher performance units in increasing numbers. The recent experience with the silicon rectifiers has demonstrated their utility as non-linear circuit elements at the microwave frequencies, that they may be engineered to exacting requirements of both a mechanical and elec- trical nature, and that they can be produced in large quantities. The defi- ciencies of the detector of World War I, which limited its utility and contribu- ted to its retrogression, have now been largely eliminated. It is a reasonable expectation that the device will now find an extensive application in commu- nications and other electrical equipment of a non-military character, at microwave as well as lower frequencies, where its sensitivity, low capacitance, freedom from aging effects, and its small size and low-power consumption may be employed advantageously. Acknowledgements Tlie development of crystal rectifiers described in this paper required the cooperative effort of a number of the members of the staff of Bell Telephone Laboratories. The authors wish to acknowledge these contributions and in j)articular the contributions made by members of the Metallurgical group and the Holmdel Radio Laboratory with wliom they were associated in the development. End Plate and Side Wall Currents in Circular Cylinder Cavity Resonator By J. p. KINZER and I. G. WILSON Formulas are given for the calculation of the current streamlines and in- tensity in the walls of a circular cylindrical cavity resonator. Tables are given which permit the calculation to he carried out for many of the lower order modes. The integration of / '.,,,'" dx is discussed; the integration is carried out for Jo -^^'-*'"» C = \,2 and 3 and tables of the function are given. The current distribution for a number of modes is shown by plates and figures. Introduction In waveguides or in cavity resonators, a knowledge of the electromagnetic field distribution is of prime importance to the designer. Representations of these fields for the lower modes in rectangular, circular and elliptical waveguide, as well as coaxial transmission line, have frequently been de- scribed. For the most i)art, however, these representations have been diagram- matic or schematic, intended only to give a general physical picture of the fields. In actual designs, such as high Q cavities for use as echo boxes,^ accurately made plates of the distributions were found necessary to handle adequately problems of excitation of the various modes and of mode sup- pression. One use of the charts is to determine where an exciting loop or orifice should be located and how the held should be oriented for maximum coup- ling to a particular mode. Optimum locations for both launchers and ab- sorbers can be found. Naturally, when attention is concentrated on a single mode these will be located at the maximum current density points. ! If, however, two or more modes can coexist, and only one is desired, com- I promise locations can sometimes be found which minimize the unwanted phenomena. Also, in a cylindrical cavity resonator of high Q with diameter large com- pared with the operating wavelength, there are many high order modes of j oscillation whose resonances fall within the design frequency band. Some I of these are undesired and one of the objectives of a practical design is to ! reduce their responses to a tolerable amount. This process is termed ! ' "High Q Resonant Cavities for Microwave Testing," Wilson, Schramm, Kinzer, I B.S.T.J., July 1946. I 31 32 BELL SYSTEM TECIIMCAL JOURNAL "suppression of the extraneous modes". In this process, an exact knowledge of the distribution of the currents in the cavity walls has been found highly useful. For example, it has been found experimentally that annular cuts in the end pliUes of the cylinder give a considerable amount of suppression to many types of extraneous modes with very little effect on the performance of the desired TE Oln mode. These cuts are narrow slits concentric with the axis of the cylinder and going all the way through the metallic end plates into a dielectric beyond.- The physical explanation is that an annular slit cuts through the lines of current fiow of the extraneous modes, and thereby interrupts the radial component of current and introduces an impedance which damps, or suppresses, the mode. For the TE Oln mode, the slits TE Modes TM Modes Ph C W II, = \'j'({k,p) COs(d K 1 kl kip kip He = J'fikip) cos (d 1 v. „ r .hJfikiD/2)'' ^^'-l^krkVDrr _ [sin (Q cos ^3 2I IL ^ Jf(ki DID cos iQ sin ^3 s He = J'f(ki D/2) cos (6 cos ^3 2 //. = k = ^ ^ kl-^ kl A ^1 2r , _ nv D ' ~ L r = ;;;"' root of J f{x) = for TM Modes. = m"' root of /;.(.v) = for TE Modes. D = cavity diameter L = cavity length Fig. 1 — Components of H vector at walls of circular c_\ Under cavity resonator. are parallel to the current streamlines and there is no such interruption; presumably there is a slight increase in current density alongside the slit, 2 Similar cuts through the side wall of tlie cylinder in planes i)erpendicular to the cylinder axis are also henctkial, hut are more troublesome mechanically. CIRCULAR CYLINDER CAVITY RESONATOR 33 as the current formerly on the surface of the removed metal crowds over onto the adjacent metal, but this is a second-order effect. To determine the best location of such cuts, therefore, it is necessary to know the vector distributions of the wall currents for the various modes. This current vector, /, is proportional to and perpendicular to the mag- netic vector, //, of the field at the surface. Expressions for the components of the //-vector at the surfaces of the end plates and side walls are given in Fig. 1. End Plate: Contour Lines At the end plates, the magnitude of the //-vector at any point is given by: IP = H,' + lie'. (1) Xow substitute values of Hp and He from Fig. 1 into (1); drop any constant factors common to Hp and He as these can be swallowed in a final propor- tionality constant; introduce the new variable x: X = kip = r ^. (2) where R = D/2 = cavity radius. Thus is obtained; //' = [J fix) cos (df + - J fix) sin (6 X (3) Now Jf and Jf, are expressed in terms of Jf^i and Jf^i and a further re- duction leads to. //"' = (//_ cos (d)' + iJf+ sin Cey (4) where Jf. = Jf.,ix) - Jf^.ix) (5) and Jf+ = Jf.r(x) + Jf,:ix) (6) The formulas (4) to (6) apply to both TE and TM modes. The values obtained depend on r, which is different for each mode. When ^ = 0, / is proportional to Jf. and when 6 — ir/lf, I is proportional to Jf+ . Relative values of / are thus easily calculated for these cases, once tables of // are available. Such tables have been prepared and are attached. For TE modes, when d = 0, He — 0, and the currents are all in the 6 direction. For TM modes, when 6 = 0, Hp = 0, and the currents are all in the p-direction. When d = tt/K, the converse holds. Figures 3 to 18 are a set of curves showing the relative magnitude of H (or /) for several of the lower order TE and TM modes. The abscissae 34 BF.Ll. SYSTEM TKCHNICAI. JOURNAL are relative radius, i.e., p/R; the ordinates are relative magnitude referred to the maximum value. The drawings also give r = ttD/Xc for each mode, where Xc is the cutoff wavelength in a circular guide of diameter D. Values for any point of the surface of the end plate can be calculated by using these curves in Conjunction with equation (4). In general, for each mode there are certain radii at which the current flow is entirely radial, (/« =0). At these radii, which correspond to zeros of Jt(x) or Jf(x), the annular cuts mentioned in the introduction are quite effective. However, the maxima of Ip do not coincide with the zeros of fe; and a more sophisticated treatment gives the best radius as that which maximizes pip-. X'alues of the relative radius for this last condition are given in Table IV. Contour lines of equal relative current intensity are obtained by setting H^ constant in (4), which then expresses a relation between x and 6. The easiest and quickest way to solve (4) is graphically, by plotting H vs. x for different values of 6. End Plate: Current Streamlines It is easy to show that the equations of the current streamlines are given by the solutions of the differential equation Ie^~'Hp- ^^^ In the case of the TE modes, (7) is easily solved by separation of the vari- ables, leading to the final result: J((x) cos fd = C (8) in which C is a i)arameter whose value depends on the streamline under consideration. In the TE modes, the £-lines in the interior of the cavity also satisfy (8), hence a {)lot of the current streamlines in the end plate serves also as a plot of the E lines. In the case of the TM modes, (7) is not so easily solved. Separation of the variables leads to: f f-J({x) -logsm^^ = j ^j'^^dx. (9) The right-hand side of (9) can be reduced somewiiat, yielding -log sin te = log [xJt{x)\ + \ i/, dx (10) J Jf(x) but no further reduction is possible. The remaining integral represents a new function which must be tabulated. Its ev^aluation is discussed at CIRCULAR CYLINDER CAVITY RESONATOR 35 length in the Appendix, where it is denoted by Fi{x). Table II of the Ap- pendix gives its values (for ( — \, 2 and 3) and also those of G({x) where Fi{x) = -\ogG({x) (11) Thus (10) becomes -log sin (d = log [x Jt{x)/G({x)] + C (12) and the final equation for the current streamlines is [xJt{x)/Gl{x)] sin (d ^ C (13) where C is a parameter as before. It is not difficult to show that G({x)/Jc{x) has zeros at the zeros of J((x). For these values of x, sin €6=0 whatever the value of C, and all stream- lines converge on (or diverge from) 2(m points on the end plate. The flow lines of (13) are orthogonal to the family (8) and could readily be drawn in this manner. However, better accuracy is obtained by plotting (13). End Plate: Distributions The 32 attached plates show the distribution of current in the end plates of a circular cylinder cavity resonator for a number of modes. In the first set of 21, the scaling is such that the diameters of the figures are proportional to those of circular waveguides which would have the same cutoff frequency. This group is of particular interest to the wave- guide engineer. In a second group of 11, the scaling is such as to make the outside diam- eters of the cylinders uniform. This group is of particular interest to a cavity designer. This distribution is a vector function of position; that is, at each point in the end plate the surface current has a different direction of flow and a dif- ferent magnitude or intensity. The variation in current intensity is repre- sented by ten degrees of background shading. The lightest indicates re- gions of least current intensity and the darkest greatest intensity. The direction of current flow is shown by streamlines. Streamlines are lines such that a tangent at any point indicates the direction of current flow at that point. The modes represented are the r£ 01, 02, 03 TM 01,02, 03 r£ 11, 12, 13 TM U, 12, 13 TE 21, 22, 23 TM 21, 22 TE3l,32 TM3l,32 36 BELL SYSTEM TECHNICAL JOURNAL in the nomenclature which has become virtually standard. In this system, TE denotes transverse electric modes, or modes whose electric Lines lie in planes perpendicular to the cylinder axis; TM denotes transverse mag- netic modes, or modes whose magnetic lines lie in transverse planes. The first numerical index refers to the number of nodal diameters, or to the order of the Bessel function associated with the mode. The second numerical index refers to the number of nodal circles (counting the resonator boundary as one such) or to the ordinal number of a root of the Bessel function asso- ciated with the mode. On the end plates, the distribution does not depend upon the third index (number of half wavelengths along the axis of the cylin- der) used in the identiiication of resonant modes in a cylinder. This con- siderably simplifies the problem of presentation. The orientation of the field inside the cavity and hence the currents in the end plate depend on other things; thus the orientation of the figures is to be considered arbitrary. The plates also apply to the corresponding modes of propagation in a cir- cular waveguide as follows: The background shading represents the in- stantaneous relative distribution of energy across a cross section of guide. For TE modes, the current streamlines depict the E lines; for the TM modes, they depict the projection of the E lines on a plane perpendicular to the cylinder axis. Side Wall: The current distribution in the side walls is easily obtained from the field equations of Fig. 1. For TM modes, the currents are entirely longi- tudinal; their magnitudes vary as cos (6 cos nirz/ L. This distribution is so simple as not to require plotting. For TE modes, the situation is more complicated, since both Hz and He exist along the side wall. The current streamlines are given by the solu- tions of the differential equation dz DHe ,.,. de-~2H/ ^^^^ By .separation of the variables, the solution is found to be Contour lines of constant magnitude of the current are given by \k\D In the above, C and A' are j)arameters, different values of which correspond to difTerent streamlines or contour lines, respectively. log (C cos (6) = log cos ksZ. (15) 2 ^ sin (d cos ksZj -\- (cos fd sin k^z)' = K\ (16) CIRC ULA R C I UNDER CA VIIY RESONA TOR 37 Since both streamlines and contours are periodic in z and 6, it is not essential to represent more than is covered in a rectangular piece of the side wall corresponding to quarter periods in :: and d. These are covered in a L . ttD length T~ along the cavity and in a distance ~t~ around the cavitv. If 2h 4' such a piece of the surface be rolled out onto a plane it forms a rectangle irnD of proportions ~. . The ditliculty in depicting the side wall currents of TE modes, as com- pared with the end plate currents, is now apparent. For the end plate, the "proportions" are fixed as being a circle. Furthermore, for a given f, as m increases the effect is merely to add on additional rings to the previous streamline and contour plots. Here, however, the proportions of the rec- tangle are variable, in the first place. And for a given rectangle the stream- lines and contours both change as ( and )n are varied. Another way of ex- pressing the same idea is that for end plates the current distribution does not depend upon the mode index n, and varies only in an additive way with the index m, whereas for the side walls the distribution depends in nearly equal strength on f, m and ;/. Some simplification of the situation is accomplished by introducing two new parameters, the "shape" and the "mode" parameters, defined by: irnD ( S = — M=^ (17) and two new variables Z = hz <f> = (d. (18) Substitution of the above, and also the expressions for k\ and ^3 (see Fig. 1) into (15) and (16) yields cos Z = C(cos (/)) (streamlines) (19) T-2 2 . ni/2 cos Z {S^M^ sin2 4> - cos- <^). (contours). (20) For given proportions S, one can calculate the streamlines and contours for various values of M. Thus a "square array" of side wall currents can be prepared, such as shown on Fig. 2. The mode parameter, if, in the physical case takes on discrete values which depend on the mode. Some of its values are given in the following table. They all lie between and 1 and there are an infinite number of them. 38 BELL SYSTEM TECHNICAL JOURNAL Valxjz ot a/ = l/r FOR TE Modes t 1 2 3 4 5 6 10 15 ' 20 m= 1 2 3 4 .5432 .1875 .1172 .0854 .6549 .2982 .2006 .1519 .7141 .3743 .2644 .2057 .7522 .4309 .3154 .2506 .7793 .4753 .3575 .2888 .8000 .5113 .3930 .3219 .8495 .6080 .4945 4209 .8813 .6774 .5730 .9001 For any given mode in any given cavity, the values of S and M can be calculated from (17). In general, these values will not coincide with those which have been plotted, but by the same token, they will lie among a group of four combinations which have been plotted. Since the changes in dis- tribution are smooth, mental two-way interpolation will present no difficulty. Acknowledgment The final plates depicting the current distributions are the result of the efforts of many individuals in plotting, spray tinting of the background, inking of the streamlines on celluloid overlay, and photographing. Special mention must be made, however, of the contribution of Miss Florence C. Larkey, who carried out all the lengthy calculations of the tables hereto attached and of the necessary data for the plotting. 1 d. 1 \j / 1 -4\ j 4 '"^ ^^ \ / / 1 "'"7/ yi \ or TE modes (/ > 0) CIRCULAR CYLINDER CAVITY RESONATOR 39 U. UJ oir ZqC r.o 0.8 ^^ ::^ ~— ^^ .^...^ "^v^^ ^V Sw He OR I^"*^"-. ^^^ 0.6 \ AT e = 90" ^^^^ Hp OR leN. 0.4 0.2 ATe=0" > V ^^=1.841 ^C N \ \ V N^ 0.4 0.5 0.6 RELATIVE RADIUS Fig. 3 — End plate currents in TE 11 mode. ^y '"^ He OR I,^ ■ (Hp 0Rle=0) ^ y y X / ^° -2.405 Ac / 0.4 0.5 0.6 RELATIVE RADIUS Fig. 4 — End plate currents in TM 01 mode. Oa ^ "'^ He OR Ip AT e = 45» ^-- ^ y / Hp OR le AT e = 0» X ^ = 3.054 ^ / \ 0.3 0.7 Fig. 5- 0.4 0.5 0.6 RELATIVE RADIUS -End plate currents in TE 21 mode 40 BELL SYSTEM TECHNICAL JOURNAL "^^ "n 1 \ X \ N \ N N N \ \ V HpOR le ^\aT e = 90° \ ^ \^ s V Hq or Ip \ AT e = 0" \ s. ID -- 3.832 K \ v^ •J 0.4 0.4 0.5 O.a RELATIVE RADIUS Fig. 6 — End plate currents in TM 11 mode. / HpOR I^^V^ (He OR lp=0) X J / N \ / N \ A / Ac \ \ / \ 4 0.4 0.5 0.6 RELATIVE RADIUS Fig. 7 — End plate currents in TE 01 mode. CIRCULAR CYLINDER CAVITY RESONATOR 41 ^ ^'^HeOR Ip ATG-30» ^ im. 4.201 Xc / / / /^ •"y^po^. le ATe^O" V y\ ^ \ V ^^ y K 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 RELATIVE RADIUS Fig. 8 — End plate currents in T£ 31 mode. / ^ / /^ HpOR le ^ AT e=45« \f \ \ \ / f N V He OR Ip \ ATe:^ \ s. / \ \ N \ ^ = 5.,36 Ac \ \ \ s. V -. . 0.4 0.5 0.6 RELATIVE RADIUS Fig. 9 — End plate currents in TM 21 mode. 42 BELL SYSTEM TECHNICAL JOURNAL 2z "■a A ^ AT e-22'/2° ^ = 5.3,e Ac / / / /^ Hp OR le^ AT e=o° V .^' ^ y^ K \ . ^ ^ \ 0.4 0.5 0.6 RELATIVE RADIUS Fig. 10 — End plate currents in TE 41 mode. 2 -0.2 »- -0.4 ^ \^ \ \ \ \ "" He OR Ip Sw AT 0-90" \ ^P OR le e=o» \ ^~ -/ N \ / / ''0=5.332 Ac \ / \^ y / 0.4 0.5 0.6 RELATIVE RADIUS 1.0 Fig. 11 — End plate currents in TE 12 mode. * ^ \ / r \ \ / \ ^ \ \ > \ HeORlp TTD ^r — -5.520 Ac \ V K (HpOB Ie=0) 0.2 0.3 0.7 0.4 0.5 0.6 RELATIVE RADIUS Fig. 12 — End plate currents in TM 02 mode. / X HpOR le ATe = 30« / \ A / ^ "^ He OR Ip V AT 6=0° \ y r \ \ \ y \ \ \ ^ = 6.3eo Ac \ \ k \ \ ^^ 0.3 0.7 Fig. 13- 0.4 . 0.5 0.« RELATIVE RADIUS -End plate currents in TM 31 mode. 43 0.8 0.9 1.0 44 BELL SYSTEM TECHNICAL JOURNAL t 0.4 / /^eOR Ip AT 6 = 18* / / / / 1 /^ ^ •"Hp OR le AT 6=0" N \, . ^-^ ^ \ 0.4 0.5 0.6 RELATIVE RADIUS Fig. 14 — End plate currents in TE 51 mode. 0.3 0.4 0.5 0.6 RELATIVE RADIUS 0.7 0.1 0.2 Fig. 15 — End plate currents in TE 22 mode. Z 0.8 I 0.4 ^ ^1 \ \ He OR \p AT e = o» . ^ \ N \ > \^ \ V ^P OR le e=9o« / \. ^ / \ "^ / ^^ / / / ^ = 7.016 Ac \^ ^y / 0.2 0.3 0.4 0.5 0.6 0.7 O.a 0.9 1.0 RELATIVE RADIUS Fig. 16 — End plate currents in TM 12 mode. 5 -0.2 5 -0.6 / ^ ^ \ / \ \ \ (He OR lp = 0) Hp OR Ie\ \ \ / \ \ J / P= 7.016 \ ^^ y 04 0.5 0.6 RELATIVE RADIUS Fig. 17 — End plate currents in TE 02 mode. 45 4(3 BELL SYSTEM TECHNICAL JOURNAL < to.e Oct ^3 0.4 0.2 y >^eOR Ip ''^ AT 9 = 15' il^= 7.501 J / / k^ y^p OR le AT 9=0° "N \ .^ k" ^ \ 0.3 0.7 Fig.. 18- 0.4 0.5 0.6 RELATIVE RADIUS -End plate currents in TE 61 mode. CIRCULAR CYLINDER CAVITY RESONATOR 47 Fig. 19— TE 01 mode. Fig. 20— TE 02 mode. 48 BELL SYSTEM TECHNICAL JOURNAL Fig. 21— TE 03 mode. Fig. 22— TK 11 mode. Fig. 23— TE 12 mode. Fig. 24— TE 13 mode. 49 50 BELL SYSTEM TECHNICAL JOURNAL Fig. 25^TE 21 mode. Fig. 26 — TE 22 mode. Fig. 27— TE 23 mode. Fig. 28— TE 31 mode. 51 52 BELL SYSTEM TECHNICAL JOURNAL Fig. 29 — TE il mode. Fig. 30— TM 01 mode. Fig. 31— TM 02 mode. Fig. 32— TM 03 mode. 53 54 BELL SYSTEM TECHNICAL JOURNAL Fig. 33— TM 11 mode. Fig. 34— TM 12 mode. CIRCULAR CYLINDER CAVITY RESONATOR 55 Fig. 35— TM 13 mode. a__ Fig. 37 TM 11 mode. 56 CIRCULAR CYLINDER CAVITY RESONATOR 57 Fig. 38— TM 31 mode. 58 BELL SYSTEM TECHNICAL JOURNAL Fig. 39— TM il mode. CIRCULAR CYLIXDER CAVITY RESONATOR 59 Fig. 40— TE 11 mode. 60 BELL SYSTEM TECHNICAL JOURNAL Fig. 41— TE 12 mode. CIRCULAR CYLINDER CAVITY RESONATOR 61 Fig. 42— TE 13 mode. 62 BELL SYSTEM TECHNICAL JOURNAL Fig. 43— TE 21 mode. CIRCULAR CYLINDER CAVITY RESONATOR 63 Fig. 44— TE 22 mode. 64 BELL SYSTEM TECHNICAL JOURNAL Fig. 45— TE 31 mode. CIRCULAR CYLINDER CAVITY RESONATOR 65 Fig. 46— TE 32 mode 66 BELL SYSTEM TECHNICAL JOURNAL Fig. 47— TM 11 mode. CIRCULAR CYLINDER CAVITY RESONATOR 67 Fig. 48— TM 12 mode. 68 BELL SYSTEM TECHNICAL JOURNAL Fig. 49— TM 21 mode CIRCULAR CYLINDER CAVITY RESONATOR 69 Fig. 50— TM 22 mode. 70 BELL SYSTEM TECHNICAL JOURNAL APPENDIX /•■'" J fix) INTEGRATION OF / 777- dx The discussion here is concerned only with integral values of ^ > 0. The integral is not simply expressible in terms of known (i.e., tabulated) func- tions, hence what amounts to a series expansion is used. The method follows Ludinegg^ who gives the details for ^ = 1. The value of the integrand at :r = is first discussed. For ^ = 1 , /i(0) = and /i(0) = 0.5, hence the integrand has the value zero. For I > \, both numerator and denominator are zero, hence the value is indeterminate. Evaluation by (f — 1) differentiations of numerator and denominator separately leads to the result that the integrand (and the integral also) is zero at X = for all C. We now introduce a constant p(. and a function 4>({x) which are such that the following equation is satisfied, at least for a certain range of values of x: Ji= -pcij'i-^^^^^^ + <i>tJl (1) Denote the desired integral by F(.{x), i.e.: Then substitution of (1) into (2) yields: F( = -pC log For X = 0, J (/ x^ ^ is indeterminate, but evaluation by difTerentiating numerator and denominator separately (/' — 1) times gives the value iM^-l)! If we can now arrange matters so that 4>c remains finite in the range (0, x), its integration can be carried out, a) by expansion into a power series and integration term-by-term, or, b) by numerical integration. Solving (1) for (j)C one obtains «= ^, ^-^. (4): Jf Equation (4) becomes indeterminate at .v = 0, when (■ > \. Evaluation by differentiating numerator and denominator separately € times shows </)^(0) = 0. > Uoclifrcqiicnztech. u. Elckhoak., V. 62, j)]). .VS-44, .Auk- 1943. CIRCULAR CYLINDER CAVITY RESONATOR 71 At the first zero of Je (the value of x at a zero of j'i will be denoted by r), 4>l is held finite by choice of the value of p( . It is clear that (4) becomes indeterminate at x = r, if Since // satisfies the differential equation j7 + -j(-h {1 - fyx')j( = (6) X and J(ir) = 0, one has by substitution Values of p for several cases are: ^=1234 1 1 n = 1.841 3.054 4.201 5.318 r-z = 5.331 r, = 8.536 pf = 1.418 1.751 2.040 2.303 1.036 1.014 4>iir)=-0.n6 -0.286 -0.446 -0.604 -0.180 -0.115 Evaluation of 4>f{r) by the usual process- gives: Mr^ ^ -S^l^ (S) Values of (f)({r) are given in the preceding table. Since <p( is finite at the origin and at the first zero of Jf , it may be ex- panded into a Maclaurin series whose radius of convergence does not, however, exceed the value of x at the second zero of J( . Alternatively, by choosing p{ to keep <^f finite at the second (or'^"") zero of J( it may be expanded into a Taylor series about some point in the interval between the first (or (k — 1)"') and third (or (k + l)"") zeros. Expansions about the origin are given in Table I. Unfortunately, the convergence of these power series is so slow that they are not very useful. Instead, equation (4) is used to calculate (l>( and / 4>( dx is obtained by numerical integration. With pt fixed to hold 4>( finite at the first root, f i , of J( , it is soon found that 4>f becomes infinite at the higher roots. This is because different values -Substitute (6) into (4) to eliminate JJ; dilTerentiate numerator and denominator separately; use (6) to eliminate J^; allow x — > r, using J'Ar) = and value of p^ from (7). 72 BELL SYSTEM TECHNICAL JOURNAL of p are required at the difl"erent roots, as shown for ( — 1 in the table above. A logical extension would therefore be to make p a function of .v such that it takes on the required values at ri , r-j , rs , • • • . When this is done and p({x) is introduced into (1) and (2), one has to integrate / K.v)/"(..-) ,,^ and this is intractable. Hence p{x) is made a discontinuous function, such that p has the value pi corresponding to ;'i for values of .v from zero to a point bi between ri and r-i ; the value p2 corresponding to r^ for values of .v from bi to a point bi be- tween r-i and rs; and so forth. This introduces discontinuities in </>. No discontinuities exist, however, in the function G( = e~'( (9) which is given in Table II. The calculations were made by Miss F. C. Larkey; numerical integration was according to Weddle's rule. Within the limits of this tabulation, then, G( and F( are now considered to be known functions. Table I Power Series Expansions of 4>t{x) / ^p\ /I 17A /7 19p\ ,,,,, , (■ , _ _f j , + (^^ _ -^ j ,. + (^- - _ j ...+ .,. = -0.063813.V -0.001 178x3 -0.0000358.v5 _ ... *,W - - ^) .V + (i - '^^ .V. + [^ - ^^ -V + . . . = +0.15451.V +0.01648.r' - O.OO.SSO.v^ - ••■ /! Sp\ ( \ 41/. \ / 13 103/> \ '^^^■^' = (i - 2ij -^ + Vn - 5760 j -^"^ + (,17280 " 276480 j "^ + = +0.12210.V +0.00667.V' +0.00375.vS - ••• . ^ Unless p = b + cJ' {b and c constants), which is not of any use. CIRCULAR CYLINDER. CAVITY RESONATOR 73 Table II r Ji ix) Values OF FiU) = / --— dx;G,{x) = e^^i Jo '^i(^"'' F,{x) y .1 .2 .3 .4 .5 1291 .6 .7 .8 .9 0050 0201 0455 0816 1887 2616 3493 4539 1 5782 7261 9036 1.1192 1.3874 1.7336 2.2103 2.9577 4.6961 4.1846 2 2.7727 2.0801 1.6199 1.2775 1.0073 7864 6018 4454 3117 1970 3 0987 0147 -0564 -1157 -1640 -2018 -2296 -2475 -2556 -2537 4 -2416 -2188 -1845 -1377 -0769 +0960 2153 3646 5549 5 8060 1.1595 1.7307 3.2014 2.3851 1.4478 9635 6373 3939 2024 6 0470 -0812 -1879 -2768 -3506 -4111 -4594 -4966 -5233 -5398 7 -5463 -5429 -5292 -5049 -4693 -4214 -3598 -2826 - 1868 -0685 8 +0789 2657 5107 8530 1.3992 2.7313 2.1565 1 . 1974 7154 3942 9 1562 -0300 -1802 -3034 -4053 -4897 -5590 -6150 -6591 -6921 G,{x) .1 •2 .3 .4 .5 .6 .7 .8 1.0000 9950 9801 9555 9216 8789 8280 7698 7052 5609 4838 4051 3265 2497 1766 1097 0519 0091 0625 1249 1979 2787 3652 4555 5478 6406 7322 9060 9854 1.0580 1.1226 1.1781 1.2236 1.2581 1.2808 1.2912 1.2733 1.2445 1.2026 1.1476 1.0799 1.0000 9085 8063 6945 4467 3136 1772 0407 0921 2351 3816 5287 6744 9541 1.0846 1.2067 1.3190 1.4200 1.5084 1.5831 1.6432 1.6877 1.7269 1.7209 1.6976 1.6568 1.5989 1.5241 1.4331 1.3265 1.2054 9241 7667 6001 4261 2468 0813 1157 3020 4890 8554 1.0304 1.1974 1.3545 1.4998 1.6318 1..7489 1.8497 1.9330 6351 0152 8212 1.2888 5741 8168 1.7157 1.0709 6742 1.9978 74 BELL SYSTEM TECHNICAL JOURS A L Valuks ok Fi{x) rMx) Jo J^ix] dx; (l,{x) = e-"': F,{x) X .1 .2 .3 .4 .5 .6 .7 .8 .9 0025 0100 0226 0403 0632 0914 1251 1645 2097 1 2612 3192 3840 4563 5365 6253 7236 8323 9528 1.0866 2 1.2357 1.4008 1.5913 1.80C1 2.0541 2.3456 2.0972 3.1380 •3.7263 4.6110 3 6.4527 6.7644 4.7528 3.8572 3.2808 2.8597 2.5316 2.2658 2.0451 1.8590 4 1.7002 1.5641 1.4470 1.3466 1.2607 1.1881 1 . 1275 1.0783 1.0396 1.0112 5 9928 9843 9858 9974 1.0190 1.0530 1.0985 1.1573 1.2311 1.3223 6 1.4345 1.5726 1.7447 1.9040 2.2555 2.6743 3.3910 6.5119 3.5122 2.7144 7 2.2595; 1.9432 1.7034 1.5131 1.3579 1.2294 1 . 1223 1.0328 .9586 .8977 S .84901 .8115 .7846 .7679 .7612 .7615 .7779 .8020 .8372 .8845 y .9452; 1.0212 1 1.1149 1.2301 1.3725 1.5512 1.7817 2.0950 2.5660 3.4864 .V 1.0000 .1 9975 .2 .3 .4 .5 .6 .7 .8 8483 .y 9900 9777 9605 9388 9127 8824 8108 1 7701 7267 6811 6336 5848 5351 4850 4350 3856 3373 2 2906 2459 2036 1643 1282 0958 0674 0434 0241 0099 3 0017 0012 0086 0211 0376 0573 0795 1037 1294 1558 4 1826 2093 2353 2601 2834 3048 3238 3402 3536 3638 5 3705 3737 3731 3688 3607 3489 3334 3143 2920 2665 6 2383 2075 1747 1403 1048 0690 0337 0015 0298 0662 7 1044 1432 1821 2202 2572 2925 3255 3560 3834 4075 8 4278 4442 4563 4640 4671 4656 4593 4484 4329 4129 9 8886 3602 3280 2923 2535 2120 1683 1231 0768 0306 CIRCULAR CYLINDER CAVITY RESONATOR 75 Values ok Fs(x) Gi{x) = e'^i X .1 .2 .3 0152 .4 .5 .6 0604 .7 .K M 0017 0067 0268 0420 0826 1081 1373 1 1703 2070 2476 2922; 3410 3942 4518 5141 5814 6539 2 7319 8158 9060 1.00281 1.1070 1.2192 1.3401 1.4706 1.6118 1.7650 3 1.9321 2.1150 2.3165 2.5402 2.7908 3.0752 3.4034 3.7905 4.2624 4.8669 4 5.7117 7.1373 16.2303 7.2383 5.8409 5.0409 4.4852 4.0843 3.7292 3.4543 5 3.2239 3.0282 2.8605 2.7160 2.5913 2.4838 2.3914 2.3128 2.2467 2.1922 6 2.1487 2.1156 2.0927 2.0798 2.0768 2.0838 2.1012 2.1293 2.1685 2.2208 7 2.2864 2.3674 2.4664 2.5868 2.7340 2.9159 3.1460 3.4491 3.8790 4.5950 8 6.9408 4.9414 4.0348 3.5348 3.1912 2.9324 2.7276 2.5608 2.4227 2.3074 9 2.2108 2.1302 2.0637 2.0097 1.9676 1.9361 1.9147 1.9036 1.9025 1.9115 G,{x) X .1 9983 .2 .3 .4 9734 ..s 9589 .6 .7 .8 .9 1.0000 9933 9849 9413 9208 8975 8717 1 8434 8130 7806 7466 7110 6742 6365 5980 5591 5200 2 4810 4423 4041 3668 3305 2955 2618 2298 1995 1712 3 1448 1206 0986 0789 0614 0462 0333 0226 0141 0077 4 0033 0008 0000 0007 0029 0065 0113 0172 0240 0316 5 0398 0484 0572 0661 0749 0834 0915 0990 1057 1117 6 1166 1206 1233 1250 1253 1244 1223 1189 1143 1085 7 1016 0937 0849 0753 0650 0542 0430 0318 0207 0101 8 0010 0071 0177 0292 0411 0533 0654 0772 0887 0995 9 1096 1188 1270 1340 1398 1443 1474 1490 1492 1479 Table III Bkssel FiNCTioN.s OF The First Kind /o{x) X .0 .1 .2 .3 .4 9604 .5 .6 .7 .8 8463 .9 + 1.0 9975 9900 9776 +9385 9120 8812 8075 1 +7652 7196 6711 6201 5669 +5118 4554 3980 3400 2818 ?. +2239 1666 1104 0555 0025 -0484 0968 1424 1850 2243 3 -2601 2921 3202 3443 3643 -3801 3918 3992 4026 4018 4 -3971 3887 3766 3610 3423 -3205 2961 2693 2404 2097 5 -1776 1443 1103 0758 0412 -0068 +0270 +0599 +0917 + 1220 fi + 1506 1773 2017 2238 2433 +2601 2740 2851 2931 2981 7 +3001 2991 2951 2882 2786 +2663 2516 2346 2154 1944 8 + 1717 1475 1222 0960 0692 +0419 0146 -0125 -0392 -0653 9 -0903 1142 1367 1577 1768 -1939 2090 2218 2323 2403 Jdx) +0 +4401 +5767 +3391 -0660 -3276' 6 I -2767 7 -0047 8 +2346 9 +2453 .1 .2 .3 .4 1960 .5 .6 2867 .7 .8 0499 0995 1483 +2423 3290 3688 4709 4983 5220 5419 +5579 5699 5778 5815 5683 5560 5399 5202 +4971 4708 4416 4097 30()i) 2613 2207 1792 + 1374 0955 0538 0128 1033 1386 1719 2028 -2311 2566 2791 2985 3371 3432 3460 3453 -3414 3343 3241 3110 2559 2329 2081 1816 -1538 1250 0953 0652 +0252 +0543 +0826 + 1096 + 1352 1592 1813 2014 2476 2580 2657 2708 +2731 2728 2697 2641 2324 2174 2004 1816 + 1613 1395 1166 0928 1 4059 5812 3754 -0272 3147 2951 0349 2192 2559 0684 J-iix) X .0 .1 .2 .3 .4 .5 .6 .7 0588 .8 .9 +0 0012 0050 0112 0197 0306 0437 0758 0946 1 + 1149 1366 1593 1830 2074 2321 2570 2817 3061 3299 ?, +3528 3746 3951 4139 4310 4461 4590 4696 4777 4832 3 +4861 4862 4835 4780 4697 4586 4448 4283 4093 3879 4 +3641 3383 3105 2811 2501 2178 18-16 1506 1161 0813 5 +0466 0121 -0217 -0547 -0867 -1173 1464 1737 1990 2221 6 -2429 2612 2769 2899 3001 3074 3119 3135 3123 3082 7 -3014 2920 28(X) 2656 2490 2303 2097 1875 1638 1389 8 -1130 0864 0593 0320 0047 +0223 0488 0745 0993 1228 9 + 1448 1653 1840 2008 2154 2279 2380 2458 2512 2542 Jz{x) X .0 .1 .2 .3 0006 .4 .5 .6 0044 .7 0069 .8 0102 .9 +0 0002 0013 0026 0144 1 +0196 0257 0329 0411 0505 0610 0725 0851 0988 1134 2 + 1289 1453 1623 1800 1981 2166 2353 2540 2727 2911 3 +3091 3264 3431 3588 3734 3868 3988 4092 4180 4250 4 +4302 4333 4344 4333 4301 4247 4171 4072 3952 3811 5 +3648 3466 3265 3046 2811 2561 2298 2023 1738 1446 6 + 1148 0846 0543 0240 -0059 -0353 0641 0918 1185 1438 7 -1676 1896 2099 2281 2442 2581 2696 2787 2853 2895 8 -2911 2903 2869 2811 2730 2626 2501 2355 2190 2007 9 -1809 1598 1374 1141 0900 0653 0403 0153 +0097 +0343 76 Ja{x) X .0 .1 .2 .3 .4 .5 .6 .7 0006 .8 .9 +0 0001 0002 0003 0010 0016 1 +0025 0036 0050 0068 0091 0118 0150 0188 0232 0283 2 +0340 0405 0476 0556 0643 0738 0840 0950 1037 1190 3 + 1320 1456 1597 1743 1892 2044 2198 2353 2507 2661 4 +2811 2958 3100 3236 3365 3484 3594 3693 3780 3853 5 +3912 3956 3985 3996 3991 3967 3926 3866 378S 3691 6 +3576 3444 3294 3128 2945 2748 2537 2313 2077 1832 7 + 1578 1317 1051 0781 0510 0238 -0031 -0297 -0557 -0810 8 1054 1286 1507 1713 1903 2077 2233 2369 2485 2581 9 -2655 2707 2736 2743 2728 2691 2633 2553 2453 2334 J,(x) X .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 +0 0001 0001 1 +0002 0004 0006 0009 0013 0018 0025 0033 0043 0055 2 +0070 008S 0109 0134 0162 0195 0232 0274 0321 0373 3 +0430 0493 0562 0637 0718 0804 0897 0995 1098 1207 4 + 1321 1439 1561 1687 1816 1947 2080 2214 2347 2480 5 +2611 2740 2865 2986 3101 3209 3310 3403 3486 3559 6 +3621 3671 3708 3731 3741 3736 3716 3680 3629 3562 7 +3479 3380 3266 3137 2993 2835 2663 2478 2282 2075 8 + 1858 1632 1399 1161 0918 0671 0424 0176 -0070 -0313 9 -0550 0782 1005 1219 1422 1613 1790 1953 2099 2229 /6(X) X .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0001 0001 0002 0002 0003 0005 0007 0009 2 0012 0016 0021 0027 0034 0042 0052 0065 0079 0095 3 0114 0136 0160 0188 0219 0254 0293 0336 0383 0435 4 0491 0552 0617 0688 0763 0843 0927 1017 1111 1209 5 1310 1416 1525 1637 1751 1868 1986 2104 2223 2341 6 2458 2574 2686 2795 2900 2999 3093 3180 3259 3330 7 3392 3444 3486 3516 3535 3541 3535 3516 3483 3436 8 3376 3301 3213 3111 2996 2867 2725 2571 2406 2230 9 2043 1847 1644 1432 1215 0993 0768 0540 0311 0082 J7{X) X .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0001 0001 0001 2 0002 0002 0003 0004 0006 0008 0010 0013 0016 0020 3 0025 0031 0038 0047 0056 0087 0080 0095 0112 0130 4 0152 0176 0202 0232 0264 0300 0340 0382 0429 0479 5 0534 0592 0654 0721 0791 0866 0945 1027 1113 1203 6 1296 1392 1491 1592 1696 1801 1908 2015 2122 2230 7 2336 2441 2543 2643 2739 2832 2919 3001 3076 3145 8 3206 3259 3303 3337 3362 3376 3379 3371 3351 3319 9 3275 3218 3149 3068 2974 2868 2750 2620 2480 2328 77 V'i(x) J .0 .1 .2 4925 .3 .4 .5 .6 .7 .8 .9 +5000 4981 4832 4703 4539 4342 4112 3852 3565 1 +3251 2915 2559 2185 1798 1399 0992 0581 0169 -0241 2 -0645 1040 1423 1792 2142 2472 2779 3060 3314 3538 3 -3731 3891 4019 4112 4170 4194 4183 4138 4059 3948 4 -3806 3635 3435 3210 2962 2692 2404 2100 1782 1455 5 -1121 0782 0443 0105 +0227 +0552 0867 1168 1453 1721 6 + 1968 2192 2393 2568 2717 2838 2930 2993 3027 3032 7 +3007 2955 2875 2769 2638 2483 2307 2110 1896 1666 8 + 1423 1169 0908 0640 0369 0098 -0171 -0435 -0692 -0940 9 -1176 1398 1604 1792 1961 2109 2235 2338 2417 2472 J'2ix} X .0 .1 .2 0497 .3 .4 .5 .6 .7 1610 .8 .9 +0 0250 0739 0974 1199' 1412 1793 1958 1 +2102 2226 2327 2404 2457 2485 2487 2463 2414 2339 2 +2239 2115 1968 1799 1610 1402 1178 0938 0685 0422 3 +0150 -0128 -0409 -0691 -0971 -1247 1516 1777 2026 2261 4 -2481 2683 2865 3026 3165 3279 3368 3432 3469 3479 5 -3462 3419 3349 3253 3132 2988 2821 2632 2424 2199 6 -1957 1702 1436 1161 0879 0592 0305 0018 +0266 +0544 7 +0814 1074 1321 1553 1769 1967 2144 2300 2434 2543 8 +2629 2689 2725 2734 2719 2679 2614 2526 2415 2283 9 +2131 1961 1774 1572 1358 1133 0899 0659 0416 0170 A{x) X .0 .1 .2 .3 0056 .4 0098 .5 .6 .7 .8 0374 .9 +0 0006 0025 0152 0217 0291 0465 1 +0562 0665 0772 0881 0991 1102 1210 1315 1415 1508 2 + 1594 1671 1737 1792 1833 1861 1875 1873 1855 1821 3 + 1770 1703 1619 1519 1403 1271 1125 0965 0793 0609 4 +0415 0212 0003 -0213 -0432 -0653 0874 1094 1310 1520 5 -1723 1918 2101 2272 2429 2570 2695 2801 2889 2956 6 -3003 3028 3031 3013 2973 2911 2828 2724 2600 2457 7 -2296 2118 1925 1719 1500 1270 1033 0789 0540 0289 8 -0038 +0211 +0457 +0696 +0928 + 1150 1360 1557 1739 1904 9 +2052 2180 2288 2376 2441 2485 2507 2506 2483 2438 J[{x) X .0 .1 .2 0001 0003 .4 - 0007 .5 .6 0022 .7 .8 .9 +0 0013 0034 0051 0071 1 +0097 0126 0161 0201 0246 0296 0350 0409 0473 0539 2 +0610 0GS2 0757 0833 0909 0985 1060 1133 1203 1269 3 + 1330 1385 1434 1475 1508 1532 1545 1549 1541 1522 4 + 1490 1447 1391 1323 12431 1150 1045 0929 0802 0665 5 +0518 0363 0200 0030 -0145 -0324 0506 0690 0874 1057 6 -1237 1412 1582 1745 1900 2045 2178 2299 2407 2500 7 -2577 2638 2683 2709 2718! 2708 2679 2633 2568 2485 8 -2385 2267 2134 1986 1824' 1649 1462 1265 1060 0847 9 -0629 0408 0184 +0039 +0261 +0480 0694 0900 1098 1286 78 CIRCULAR CYLINDER CAVITY RESONATOR 79 /5(X) X .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 +0 0001 0002 0003 0005 0008 1 +0012 0018 0025 0034 0045 0058 0073 0092 0113 0137 2 +0164 0194 0228 0265 0305 0348 0394 0443 0494 0548 3 +0603 0660 0718 0777 0836 0895 0952 1008 1062 1113 4 +1160 1203 1242 1274 1301 1321 1333 1338 1335 1322 5 +1301 1270 1230 1180 1120 1050 0970 0881 0782 0675 6 +0559 0435 0304 0166 0023 -0126 0278 0433 0591 0749 7 -0907 1064 1217 1368 1513 1652 1783 1906 2020 2123 8 -2215 2294 2360 2412 2449 2472 2479 2470 2446 2405 9 -2349 2277 2190 2088 1972 1842 1700 1546 1382 1208 A{x) X .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 +0 0001 1 +0001 0002 0003 0004 0006 0009 0012 0016 0021 0027 2 +0034 0043 0053 0065 0078 0094 0111 0130 0152 0176 3 +0202 0231 0262 0295 0331 0368 0408 0450 0493 0538 4 +0585 0632 0680 0728 0776 0823 0870 0916 0959 1000 5 +1039 1074 1105 1132 1155 1172 1183 1188 1187 1178 6 + 1163 1139 1108 1069 1022 0967 0904 0833 0753 0666 7 +0572 0470 0362 0247 0127 0002 -0128 -0261 -0397 -0535 8 -0674 0813 0952 1088 1222 1352 1478 1597 1710 1816 9 -1912 2000 2077 2143 2198 2240 2270 2287 2290 2279 Table IV Relative Radius for Maximum of pll Mode TE 11 .737 12 .982 .254 13 .993 .613 .159 21 .894 22 .988 .407 23 .995 .664 .274 31 .937 32 .991 .491 41 .956 42 .993 .548 51 .967 CI .974 TM 01 .901 02 .983 .393 03 .993 .627 .250 11 .961 12 .989 .525 13 .995 .682 .362 21 .977 22 .992 .596 31 .984 32 .994 .643 41 .988 51 .990 61 .992 1 First and Second Order Equations for Piezoelectric Crystals Expressed in Tensor Form By W. P. MASON Introduction AEOLOTROPIC substances have been used for a wide variety of elastic piezoelectric, dielectric, pyroelectric, temperature expansive, piezo- optic and electro-optic effects. While most of these effects may be found treated in various publications there does not appear to be any integrated treatment of them by the tensor method which greatly simplifies the method of writing and manipulating the relations between fundamental quantities. Other short hand methods such as the matrix method can also be used for all the linear effects, but for second order effects involving tensors higher than rank four, tensor methods are essential. Accordingly, it is the purpose of this paper to present such a derivation. The notation used is that agreed upon by a committee of piezoelectric experts under the auspices of the Insti- tute of Radio Engineers. In the first part the definition of stress and strain are given and their inter- relation, the generalized Hookes law is discussed. The modifications caused by adiabatic conditions are considered. When electric fields, stresses, and temperature changes are applied, there are nine first order effects each of which requires a tensor to express the resulting constants. The effects are the elastic effect, the direct and inverse piezoelectric effects, the temperature expansion effect, the dielectric effect, the pyroelectric effect, the heat of deformation, the electrocaloric effect, and the specific heat. There are three relations between these nine effects. Making use of the tensor trans- formation of axes, the results of the symmetries existing for the 32 types of crystals are investigated and the possible constants are derived for these nine effects. Methods are discussed for measuring these properties for all 32 crystal classes. By measuring the constants of a specified number of oriented cuts for each crystal class, vibrating in longitudinal and shear modes, all of the elastic, dielectric and piezoelectric constants can be obtained. Methods for calculating the properties of the oriented cuts are given and for deriving the fur.damental constants from these measurements. 1 For example Voigt, "Lehrl)uch der Kiistall Physik," B. Tcul)ner, 1910; Wooster, "Crystal Physics," Cainl)ridge Press, 1938; Cady "Piezoelectricity" McGraw Hill, 1946. * The matrix method is well described 1)V W. L. Bond "The Mathematics of the Ph\sical Properties of Crystals," B. S. T. J., Vol. 22, pp. 1-72, 1943. 80 PIEZOELECTRIC CR YSTA LS IN TENSOR FORM 81 Second order effects are also considered. These eflfects (neglecting second order temperature eflfects) are elastic constants whose values depend on the applied stress and the electric displacement, the electrostrictive eflfect, piezoelectric constants that depend on the applied stress, the piezo-optical effect and the electro-optical effect. These second order equations can also be used to discuss the changes that occur in ferroelectric type crystals such as Rochelle SaU, for which between the temperature of — 18°C. and -f24°C.,a spontaneous polarization occurs along one direction in the crystal. This spontaneous polarization gives rise to a first order piezoelectric deforma- tion and to second order electrostrictive effects. It produces changes in the elastic constants, the piezoelectric constants and the dielectric constants. Some measurements have been made for Rochelle Salt evaluating these second order constants. Mueller in his theory of Rochelle Salt considers that the crystal changes from an orthorhombic crystal to a monoclinic crystal when it becomes spontaneously polarized. An alternate view developed here is that all of the new constants created by the spontaneous polarization are the result of second order eflfects in the orthorhombic crystal. As shown in section 7 these produce new constants proportional to the square of the spontaneous polarization which are the ones existing in a monoclinic crystal. 0.i this view "morphic" eflfects are second order eflfects produced by the spontaneous polarization. 1. Stress and Strain Relations in Aeolotropic Crystals I.I. Specification of Stress The stresses e.xerted on any elementary cube of material with its edges along the three rectangular axes X, Y and Z can be specified by considering the stresses on each face of the cube illustrated by Fig. 1. The total stress acting on the face ABCD normal to the X axis can be represented by a resultant force R, with its center of application at the center of the face, plus a couple which takes account of the variation of the stress across the face. The force R is directed outward, since a stress is considered posi- tive if it exerts a tension. As the face is shrunk in size, the force R will be proportional to the area of the face, while the couple will vary as the cube of the dimension. Hence in the limit the couple can be neglected with respect to the force R. The stress (force per unit area) due to R can be resolved into three components along the three axes to which we give the designation Here the first letter designates the direction of the stress component and the second letter x^ denotes the second face of the cube normal to the X axis. Similarly for the first X face OEFG, the stress resultant can be resolved 82 BELL SYSTEM TECHNICAL JOU R^AL into the compo7ients 7„, , Ty,, , T,., , which are oppositely directed to those of the second face. The remaining stress components on the other four faces have the designation Face OABE CFGD OADG bcfp: r. n n (2) Fig. 1. — Cube showing method for specifying stresses. The resultant force in the X direction is obtained by summing all the forces with components in the X direction or F\ = (n., - r„J dydz + {T^y, - T.y,) dxdz + (n„ - T^,) dxdy. (3) But Tzxt — ~~Txx^ 4" —^ — dx; iiyj — J xyj + 'I^'.,r. r„.= -r„,+^v. (4) and equation (3) can be written in the form /dTxx , dTxv 1 dT; J'' l-^' + --j^^ + ''-±^^dxdydz. \ dx dy dz ) (5) Similarlv the resultant forces in the other directions are (6) PIEZOELECTRIC CRYSTALS IN TENSOR FORM 83 We call the components r T. T T21, 7^22, T,, T31 , Tz2 , T33 (7) the stress components exerted on the elementary cube which tend to deform it. The rate of change of these stresses determines the resultant force on the cube. The second form of (7) is commonly used when the stresses are considered as a second rank tensor. Fiff. 2. — Shearing stresses exerted on a cube. It can be shown that there is a relation between 3 pairs of these compo- nents, namely T = T 1 TV 1 ■ T = T T = T (8) To show this consider P'ig. 2 which shows the stresses tending to rotate the elementary cube about the Zaxis the cube about the Z axis by producing the couple The stresses Ty^^'dnd Ty^^ tend to rotate Tyx dx dy dz (9) The stresses Tjy^ and T^y.^ produce a couple tending to cause a rotation in the opposite direction so that ^ {Tyj, — T:ry) dx dy dz = couple I (hi (10) is the total couj^ie ter.ding to produce a rotation around the Z axis. But from dynamics, it is known that tliis cou])le is equal to the product of the moment of inertia of the section times the angular acceleration. This moment of inertia of the section is proportional to the fourth power of the cube edge and the angular acceleration is fmite. Hence as the cube edge M 84 BELL SYSTEM TECHNICAL JOURNAL approaches zero, the right hand side of (10) is one order smaller than the left hand side and hence T = T (11) The same argument applies to the other terms. Hence the stress com- ponents of (7) can be written in the symmetrical form r. T. T. n. Tn, Tn, Tu T., n, n Ty^ = Tn, T22 , Tiz =. Te, T2, T, r„ Tn , T,,, Tiz T,, T,, Tz (12) The last form is a short hand method for reducing the number of indices in the stress tensor. The reduced indices 1 to 6, correspond to the tensor indices if we replace llbyl; 22 by 2; 33 by 3; 23 by 4; 13 by 5; 12 by 6. This last methcd is the mcst common way for writing the stresses. 1.2 Strain Component, The types of strain present in a body can be specified by considering two points P. and ^ of a medium, and calculating their separation in the strained condition. Let us consider the point P at the origin of coordinates and the point Q having the coordinates x, y and z as shown by Fig. 3. Upon strain- Fig. 3. — Change in length and position of a hne due to strain in a solid body. ing the body, the points change to the positions P', Q'. In order to specify the strains, we have to calculate the difTerence in length after straining, or have to evaluate the distance P'Q'-P Q. After the material has stretched the point P' will have the coordinates ^i , 7?i , f 1 , while Q' will have the coordinates -v + I2 ; v + 772 ; 2 + ^> . But the displacement is a continuous function of the coordinates .r, y and z so that we have ^2 = ^1 + ^ X + / >' + ^ 3- dx dy dz PIEZOELEC TRIG CR YS TA LS IN TENSOR FORM 85 Similarly . dr} , drj drj ox oy dz (13) i ^' = ^'^dx'^dyy^dz'- ' Hence subtracting the two lengths, we iind that the increases in separation \ in the three directions are 5x = .T ^ + V / + S -^ I dx dy dz I ' dr] d-q drj ,... 5v = ^^+>'t-+2^ (14) ox dy dz dx dy dz d^ The net elongation of the line in the x direction is x — and the elongation dx . d^ . . . per unit length is —^ which is detined as the linear strain in the x direction. dx We have therefore that the linear strains in the x. y and s directions are 5, = |f; S.^p; 53 = ^f. (15) dx dy dz The remaining strain coefficients are usually defined as oy dz dz dx dx dy and the rotation coefficients by the equations _ d^ dtf _ d^ d^ _ drj d^ dy dz dz dx dx dy Hence the relative displacement of any two j.oints can be expressed as h = xS, + y [-~^) + z [-^) (17) (18) 86 BELL SYSTEM TECIINICA L JOI 'UNA L which represents the most general type of disj^lacement that the Hne P Q can undergo. As discussed in section 4 the definition of the shearing strains given by equation (16) does not allow them to be represented as part of a tensor. If however we defined the shearing strains as 25,3 = S, = \dy dzj 25|3 — Si, = i^ + ^i • dz dx ' 25. = S. = p + 'J dx ay (19) they can be expressed in the form of a symmetrical tensor S(, 65 ^11 S\2 012 'S13 S22 '-*>'23 = s,. ^^33 Si Se S2 s, 2 2 s. s, S; 2 2 (20) For an element suffering a shearing strain S^ — 2Si2 only, the displace- ment along X is proportional to y, while the displacement along y is propor- tional to the X dimiension. A cubic element of volum.e will be strained into a rhombic form, as shown by Fig. 4, and the cosine of the resulting angle 6 Fig. 4. — Distortion due to ;i shear! iig strain. measures the shearing deformation. For an element suffering a rotation ccz only, the dis])lacement along x is proj;ortional to y and in the negative y direction, while the dis])laccmcnt along y is in the ]>ositive .v direction. Hence a rectangle has the displacement shown by lig. 5, which is a pure rotation of the body without change of form, about the z axis. For any PIEZOELECTRIC CRYSTALS IN TENSOR FORM 87 body in equilibrium or in nonrotational vibration, the co's can be set equal to zero. The total potential energy stored in a general distortion can be calculated as the sum of the energies due to the distortion of the various modes. For fih example in expanding the cube in the x direction by an amount — dx = ox Si dx, the work done is the force times the displacement. The force wil Fig. 5. — A rotation of a solid body. be the force Ti and will be Ti dy dz. Hence the potential energy stored in this distortion is T\ dSi dx dy dz For a shearing stress T^ of the type shown by Fig. 4 the displacement dS(,dx 7r» T times the force T^ dy dz and the displacement — ^-^ times the force T(, dx dz equals the stored energy or AP^e = \ (dS^Te + dSeT^) dx dy dz = dS^T^ dx dy dz. Hence for all modes of motion the stored potential ener gy is equal to APE = [Ti dSi +■ Ti dS2 + Ti dSi + Ti dSi + T^, dSs (21) + Tt dSe] dx dy dz. 1 .3 Generalized Hookers Law Having specified stresses and strains, we next consider the relationship ; between them. For small displacements, it is a consequence of Hooke's I Law that the stresses are proportional to the strains. For the most un- I symmetrical medium, this proportionality can be written in the form (22) 88 BELL S YSTEAf TECH NIC A L JOURNA L T\ = CnSi + C12S2 -f- C13S3 -\- CuSi -\- Ci^Si -\- CioSe T2 = C21S1 + C22S2 + C23S3 + C24S4 -\- C2bSs + ^26^ 6 7^3 = ^31'5*1 + CS2S2 + ^33^3 + €3484 + ^35^6 + ^36-^6 Ti = C41S1 + €4282 + r43'5'3 + CiiSi -\- €4^3 f, -\- ^46^6 Tt = Cr,iSi + f52^2 + ^53^3 + C^Si + Ci^S;, + ^56.5 6 7^6 = CeiSl -\- f 62'?2 + f e3'S'3 + C64Si + f 65^5 + ^66^6 where Cn for example is an elastic constant expressing the proportionality between the Si strain and the Ti stress in the absence of any other strains. It can be shown that the law of conservation of energy, it is a necessary consequence that C12 = C21 and in general c,,- = Cji. (23) This reduces the number of independent elastic constants for the most unsymmetrical medium to 21. As shown in a later section, any symmetry existing in the crystal will reduce the possible number of elastic constants and simplify the stress strain relationship of equation (22). Introducing the values of the stresses from (22) in the expression for the potential energy (21), this can be written in the form 2PE = cnSl + 2C12S1S2 + IcnSiSs + 2fi4^i54 + 2cuSiS\ + Ici&SiS^ + ^22^2 + 2r23^2'S'3 + 2C24'S'25'4 + 2f25'S'2^5 + 2C2oS'26'6 + C33S3 -{- IcsiSsSi -\- IczffSzSi, -f- IcsgS^S^ + f44'^4 + 2r45^4^'5 + ICi^'iSfi (24) The relations (22) thus can be obtained by differentiating the potential energy according to the relation c)PF c)PF It is sometimes ad\antageous to exi)ress the strains in terms of the stresses. This can be done by solving the equations (22) simultaneously for the strains resulting in the equations PIEZOELECTRIC CRYSTALS IN TENSOR FORM S9 Si = 511^1 + 512^2 + SuTz + SuTi + 51575 + Sy^Ti, Si = S21T1 -\- S22T2 + 523^3 + S^iTi + 5257^6 + 526^6 53 = S31T1 + 532^2 + 533^3 + 53474 + 53575 + 53676 54 = 54i7i + 54272 + 54373 + 5447i + 54575 + 54676 '^'5 = S^iTi -\- Sf,iTl -\- 55373 + 55474 + 55575 + ^6676 Si = 56l7i + 56272 + 56373 + 56474 + 56575 + 56676 (26) Inhere i+i Sii = _(-i)'"^A:y (27) for which A*^ is the determinant of the dj terms of (28) and'A^y the minor obtained by suppressing the ith andjth columo A'^ = <"ll Ci2 Ci3 Cu '"15 <^16 ^12 ^22 <r23 Cu C25 ^26 Cl3 C23 ^33 C34 <"36 ^36 ri4 C24 C34 f44 C45 C46 ^15 ^25 <"35 Cib Cbb ^56 ^16 <^26 ''36 ^46 C{,( Ce6 (28) Since c.-y = cy, it follows that 5,y = 5y,. The potential energy can be expressed in the form. 27£ = 5ii7? + 2S12T1T2 + 25i37\73 + IsuTiTi + 25i57i76 + 25i67i76 + 52272 + 2S23T2T3 + 25247274 + 2S26T2T5 + 2S2iT2T ^ + •^3373 + 253^X3X4 -\- 2S3bT3Tb + 25367376 + 54474 + 25457475 + 25467476 (29) + 55575 + 2sb%Ti,Ti -\- SbbTe- The relations (26) can then be derived from expressions of the type 5i = dPE S, = dPE (30) dTi ' ' "" 576 1.4 Isothermal and Adiabatic Elastic Constants We have so far considered only the elastic relations that can be measured statically at a constant temperature. The elastic constants are then the isothermal constants. For a rapidly vibrating body, however, there is no 90 BELL SYSTEM TECHNICAL JOURNAL chance for heat to equalize and consequently the elastic constants operative are the adiabatic constants determined by the fact that no heat is added or subtracted from any elemental volume. For gases there is a marked difference between the adiabatic and the isothermal constants, but for piezoelectric cr^'stals the difference is small and can usually be neglected. To investigate the relation existing we can write from the first and second laws of thermodynamics, the relations dV = [Ti dSi 4- T2 dS2 + T3 dSs (31) + T, dSi + Ts dS, + 7^6 dS,] -\-ed(r which expresses the fact that the change in the total energy U is equal to the change in the potential energy plus the added heat energy dQ = Q da where is the temperature and cr the entropy. Developing the strains and entropy in terms of the partial differentials of the stresses and temperature, we have dS, = ^^ dT, + ?i^ dT, + ^' dTs dTi dT2 ST. oli 01^ die oQ dS, = '^Ut. -h ^^' dT. + §' dn oil 01 2 alz (32) do = l^ AT. + If AT, + If dT^ all 01 2 01 i ^^dT, + ^dT, + ^ dT, + ^dQ. dTi an - dTe ae The partial derivatives of the strains with regard to the stresses are readily seen to be the isothermal elastic compliances. The partial derivatives of the strains by the temperatures are the six temperature coefficients of ex- pansion, or dSi dSi ... ae ' ae To evaluate the partial derivatives of the entropy with re.^pect to the stresses we make use of the fact that U is a perfect difTerential so that dS\ da dS^ da ,,.-. PIEZOELECTRIC CR YSTA LS IN TENSOR FORM 91 Finally multiplying through the last of equation (32) by 9 we can write them as Si = snTi + 512^2 + suTz + SuTi + Si^T^ + suT^ + oci dQ Si = SieTi + -^267^2 + ■^36^3 + SisT4 + 5667^6 + •^662^6 + OC^ dO dQ = Q d(T = 6[aiTi + q:27'2 + otsTs + 0474 + ai,T^ + a^Te] + pCpdQ since ©t^ is the total heat capacity of the unit volume at constant stress, which is equal to pCp, where p is the density and Cp the heat capacity at constant stress per gram of the material. To get the adiabatic elastic constants which correspond to no heat loss from the element, or dQ = 0, dQ can be eliminated from (35) giving ^1 = s'nTi + 5127^2 + SnTs + 3^X4 + s[f,Tf, + s'^Tf, + (ai/pCp) dQ (36) Se = s'uTi + sIbT^ + SuTs + s'teTi + sl^T^ + s^Te + (as/pCp) dQ where ,-, = s% - «-i^. (37) pLp For example for quartz, the expansion coeffxients are ai = 14.3 X 10"V°C; 02 = 14.3 X 10"V°C; a, = 7.8 X 10"V°C; The density and specific heat at constant pressure are p — 2.65 grams/cm ; Cp= 7.37 X 10^ergs/cm^ Hence the only constants that differ for adiabatic and isothermal values are •^11 = 522 ; .^12 ; -^13 ; -^33 • Taking these values as sn = 127.9 X 10~'* cmVdyne; Su = -15.35 X 10"''; su = 11.0 X 10"'*; 533 = 95.6 X 10"''. We find that the corresponding isothermal values are sfi = 128.2 X 10"'*; 5?2 = -15.04 X 10"'*; 5?3 = 10.83 X 10"'*; s% = 95.7 X 10"'* cmVdyne ^See "Quartz Crystal Applications" Bell System Technical Journal, Vol. XXII> No. 2, July 1943, W. P. Mason. 92 BELL SYSTEM TECHNICAL JOURNAL at 25°C. or 298° absolute. These differences are probably smaller than the accuracy of the measured constants. If we express the stresses in terms of the strains by solving equation (35) simultaneously, we find for the stresses (38) 7^6 = Ci^Si -\- c^^S'i -\~ Cz^Si + Cif,SA + CjfrSs + Ces'S'e — Xe dQ where The X's represent the temperature coefficients of stress when all the strains are zero. The negative sign indicates that a negative stress (a compression) has to be applied to keep the strains zero. If we substitute equations (38) in the last of equations (35), the relation between increments of heat and temperature, we have dO = Qda = e[\iSi + MSi + XsSs + XiS^ + X56-5 + Xe^e] (39) + [pCp — 0(aiXi + 012X2 + 0:3X3 + 0:4X4 + 0:5X5 + a^X6)]dQ. If we set the strains equal to zero, the size of the element does not change, and hence the ratio between dQ and dB should equal p times the specific heat at constant volume C„. We have therefore the relation p[Cp — Cv] = B[a:iXi + 02X2 + 0:3X3 + 04X4 + 0:5X5 + osXe]. (40) The relation between the adiabatic and isothermal elastic constants Cij thus becomes c'j = cl + ^'. (41) Since the difference between the adiabatic and isothermal constants is so small, no differentiation will be made between them in the following sections. 2. Expression for The Elastic, Piezoelectric, Pyroelectric and Dielectric Relations of a Piezoelectric Crystal When a crystal is piezoelectric, a potential energy is stored in the crystal when a voltage is applied to the crystal. Hence the energy expressions of (31) requires additional terms to represent the increment of energy dl'. If we employ C(iS units which have so far been most widely used, as applied PIEZOELECTRIC CR VST A LS IN" TENSOR FORM 93 to piezoelectric cn^stals, the energy stored in any unit volume of the crystal is dU = Ti dSi + T2 dS2 + T3 dS^ + Ti dS, + Ts dS, + Te dSe , J, dD, , ^ dD, , _, dDi ,^, (42) ■iir 47r ■iir where Ei , E2 and £3 are the components of the field existing in the crystal and Di , A and D3 the components of the electric displacement. In order to avoid using the factor l/4ir we make the substitution The normal component of 5 at any bounding surface is fo the surface charge. On the other hand if we employ the MKS systems of units the energy of any component is given by Zn^/^^n directly and in the following formulation 5 can be replaced by D. There are two logical methods of writing the elastic, piezoelectric, pyro- electric and dielectric relations. One considers the independent variables as the stresses, fields, and temperature, and the dependent variables as the strains, displacements and entropy. The other system considers the strains, displacements and entropy as the fundamental independent variables and the stresses, fields, and temperature as the independent variables. The first system appears to be more fundamental for ferroelectric types of crystals. If we develop the stresses, fields, and temperature in terms of their partial derivatives, we can write i/d.<t 0^2/ D.a OCis/D.tr OOi/ D,a \/ s,a da /a Obz/S.a Off Js.D T, = ^-^\ dS,^^-^") dS2-\-^-^^ ^-^3 + ^^^^ dS, (44 A) O'Jl/D.a 002/ D.a OOs/D.a 004/0,0 a^)5/D,o O0(,/D,a O0\ / s.a 002/ S,a 003 /s.a dcr /S,D 94 BELL S YSTEM TECH NIC A L JOURNA L £x = £i = ^^ ) dS, + )>b/D,<r O06/D,o OOi/s.a OOi / S.a + f) ,,, + fl) ,. Oh/a.a OCT /S.D £. = £, = ^A ''51 + ^sl) ''5, + ^') ''•Ss + lf) i& OOi/Cff U02/ D,a OOZ/ D,a OOi/D.a + ^^) ,5. + f) .6^, + f) .a, + f) .a, OOf,/ D,a OO^/D.a OOl/s.a OO2 / S,a dds/s.a OCT /a,D ,e=|f) .5. + 11) .5, + 11) .53 + 11) </5, OOl/D.a OOi/D.a OOs/D.a OO4/ D.a J D,a 00%/ D,a OOl/s.o O02/S,a ) d5z+f) da. :/S,a Off/s.D 883/8,0 The subscripts under the partial derivatives indicate the quantities kept constant. A subscript D indicates that the electric induction is held constant, a subscript a indicates that the entropy is held constant, while a subscript 5 indicates that the strains are held constant. Examining the first equation, we see that the partial derivatives of the stress Ti by the strains are the elastic constants c,-, which determine the ratios between the stress Ti and the appropriate strain with all other strains equal to zero. To indicate the conditions for the partial derivatives, the superscripts D and a are given to the elastic constants and they are written c^j'. The partial derivatives of the stresses by 5 = D/^t are the piezo- electric constants //,/ which measure the increases in stress necessary to hold the crystal free from strain in the presence of a displacement. Since if the crystal tends to expand on the application of a displacement, the stress to keep it from exi)anding has to be a compression or negative stress, the negative sign is given to the /{"a constants. As the only meaning of the // constants is obtained by measuring the ratio of the stress to 5 = D/iir at constant strains, no superscript S is added. However there is a difference I.etween isothermal and adiabatic piezoelectric constants in general, so PIEZOELECTRIC CR VST A LS IN TENSOR FORM 95 that these piezoelectric constants are written Z/"^^. Finally the last partial derivatives of the stresses by the entropy a can be written dT 'da ") ^' = 1,^-P) Q^'^^ST^") 'iQ = -yrdQ (45) • /s,D 6 da /s,D 6 oa /s.d where dQ is the added heat. We designate 1/6 times the partial derivative as — Yn and note that it determines the negative stress (compression) necessary' to put on the cr>'stal to keep it from expanding when an increment of heat dQ is added to the crystal. The electric displacement is held constant and hence the superscripts S, and D are used. The first six equa- tions then can be written in the form (46) — h'nxhi — /U'Jo — h'na^s — y^f dQ. To evaluate the next three equation? involving the fields, we make use of the fact that the expression for dU in equation (42) is a perfect differential. As a consequence there are relations between the partial derivatives, namely (47) ar„. _ a£„. dT^ ae . dEn _ dQ dbn dSj da dSm da dhn We note also that dEA d8n / S.a = 47r/3f;; (4.S) where /3 is the so called "impermeability" matrix obtained fiom '.he dielectric matrix e„m by means of the equation &r.n = ^-^ (40) where A is the determinant fll , fl2 , CIS €12 , fno COS fKi , ^s , e.s3 (5(!) and a"''" the minor obtained by suppressing the wth row and ;/th column. The partial derivatives of the fields by the entropy can he written dE^ da A . 1 dE„\ 1 dE„,\ .s,z. ,,, .... /S.D U da /S.D 6 da /s n where q'n is a pyroelectric constant measuring the increa:£e in field required to produce a zero charge on the surface when a heat /() is added to the 96 BELL SYSTEM TECH NIC A L JOURNA L crystal. Since the voltage will be of opposite sign to the charge generated on the surface of the crystal in the absence of this counter voltage a nega- • • • , S,D tive sign is given to g „ . Finally the last partial derivative 6e\ , 1 ae\ _ , i ae\ ._ dQ aa/s.D U OCT /s.D U da /s,d pC„ represents the ratio of the increase in temperature due to the added amount of heat dQ when the strains and electric displacements are held constant. It is therefore the inverse of the specific heat at constant volume and constant electric displacement per gram of material times the density p. Hence the ten equations of equation (44) can be written in the generalized forms — h'nlh — llnlh — il'nzh " In dQ Em, = —h\mSl — him^l ~ I'Sm'^S ~ ^UmSi — ll^mS^ — IlimSt ~\~ -iTrfSml^l + ^Tr&^ + -iw^^ - qlf dQ (53) Je=— e[7i ^1 + 72 02 + 73 03 + 74 04 + 75 OS + 76 OeJ —Q[qi 5i + 92 ^2 + 93 53] + -ttd • 11= 1 to 6; m = 1 to 3 If, as is usually the case with vibrating crystals the vibration occurs with no interchange of heat between adjacent elements dQ — and the ten equations reduce to the usual nine given by the general forms Tn = CnlSl + Cn^Si + CnsSi + CniSi + CnbSf, + C ntS e — hni5i — hnih — hnzh Em = —JllmSl — IhviSi — IhmSz — /74m'S'4 — IhmSb — /'Cm'S'e + 47r/3mi5i + 4T/3I262 + 47ri3'l3 53. (54) In these equations the superscript a has been dropj^ed since the ordinary constants are adiabatic. The tenth equation of {S3>) determines the increase in temperature caused by the strains and displacements in the absence of any flow of heat. If we introduce the e.xpression of equations (53) into equation (42) the total energy of the crystal is per unit volume. PIEZOELECTRIC CR YSTA LS IN TENSOR FORM 97 21 = rii 61 + 2fio ^1^2 + 2^13 oi-Js + ^i^H 'Ji'J4 + 2ri5 ^165 + 2ci6 oiOe + r.?i'5l + 2c^fS,S, + 2r?4'^^2>S4 + 2c^_,''SoS, + 2f?6%56 ^33 -J3 "T" ^^^34 03^4 "T" ^^'35 0305 -f- Z('36 03O6 (■44 O4 i- Zf45 O4O5 -j- Zr46 O4O6 + D,(T ^2 I rj Z).ff o O ' + f66''^'6 (55) -(2//Ii5,5'i + 2/;I,5i52 + 2//l35i.93 + 2/;l45i54 + 2li%5,S, + 2//l65i5-6) -(2//2l5,5l + 2J1U2S2 + 2//235253 + 2//24^2^^4 + 2111-^^3^ + 2//26526'6) -(2//3l53.Si + 2hl.MS2 + 2//33^3^3 + 2/;345354 + 2//35636'5 + 2//3653^6) -(27i'%^/() + 272'%f/<3 + 2yl'^SsdQ + 274'''6'4fi?(? + 275'°55rf() + 2y'l''S,dQ) +iirWiUl + 2/3^;r6if2 + 2(Sf,'d,bs + /3^;r62 + 2f32zdod, + /Sf^^i] -(29f%r/C' + 2qt-''5,dQ + 2gt''''W0 + ~§r". Equations (53) can be derived from this expression by employing the partial 1 derivatives i The other form for writing the elastic, f)iezoelectric, pyroelectric and di- j electric relations is to take the strains, displacements, and entropy as the ! fundamental variables and the stresses, fields and temperature increments ■ as the dependent variables. If we develop them in terms of their partial j derivatives as was done in (44), use the relations between the partial deriva- t tives shown in equation (57). (57) and substitute for the partial derivatives their equivalent elastic, piezo- electric, pyroelectric, temperature expansions, dielectric and specific heat constants, there are 10 equations of the form ddm _ dSn . dSr, _ da d5^ da dTn dEm ' dQ ~ dT„ ' 60 dE„ 98 BELL SYSTEM TECH NIC A L JOURNA L + ^2^2 + (tzEz + a^Je 5m = (iimTl + dirnT-i. + d^mTz + dimTi + d^^Th + d^^Te + |l£, + ^|l £, + !pi £3 + /'Ic/e (58) 47r 47r 47r </^ = 9 (/o- = 6[ai Ti + Q!2 7^2 + af Ts + af 7^4 + af Ts + af rej + eiplE, + Pa'^Es + plE,] + />C^(/e. w = 1 to 6, m = 1 to 3 The superscripts E, 0, and T indicate respectively constant field, constant temperature and constant stress for the measurements of the respective constants. It will be noted that the elastic compliance and the piezo- electric constants d^n are for isothermal conditions. The a^ constants are the temperature expansion constants measured at constant field, while the p^ constants are the pyroelectric constants relating the ratio of 5 == D/47r to increase in temperature ^6, measured at constant stress. Since there is constant stress, these constants take into account not only the "true" pyro- electric effect which is the ratio of 5 = Z>/47r to the temperature at constant volume, but also the so called "false" pyroelectric effect of the first kind which is the polarization caused by the temperature expansion of the crystal. This appears to be a misnomer. A better designation for the two effects is the pyroelectric effect at constant strain and the pyroelectric effect at constant stress. Cp is the specific heat at constant pressure and constant field. If we substitute these equations into equation (42), the total free energy becomes !^ = E Z s^nTmTn + 2 ^^ Xl d'toT^Eo 4- 2 i; a'„Tje n = l 0=1 3 T,e + Z E ^ £o£, + 2 E PoEpde + ^^ ^e. 0=1 p=i 47r 0=1 t) Equation (58) can then be obtained by partial derivatives of the sort at/ _ d£ _dQ dU (59) dTn' dEp' e d(de)' By tensor transformations the expression for U in (59) can be shown to be equal to the expression for U in (55). The adiabatic equations holding for a rapidly vibrating crystal can be PIEZOELECTRIC CR VST A LS IN TENSOR FORM 99 obtained by setting dQ equal to zero in the last of equations (58) and elim- inating dQ from the other nine equations. The resulting equations are Bm = dim Ti + d^m T2 + dzm Ti + dim Ti (60) + d,m n ^ d^T,+ '^ El + ^' £2 + '-^^ £3 47r 4t 47r where the symbol a for adiabatic is understood and where the relations between the isothermal and adiabatic constants are given by E E (^ B .T f^ T,a T,Q l.T .T r\ Hence the piezoelectric and dielectric constants are identical for isothermal and adiabatic conditions provided the crystal is not pyroelectric, but differ if the crystal is pyroelectric. The difference between the adiabatic and isothermal elastic compliances was discussed in section (1.4) and was shown to be small. Hence the equations in the form (60) are generally used in discussing piezoelectric crystals. Two other mixed forms are also used but a discussion of them will be delayed until a tensor notation for piezoelectric crystals has been discussed. This simplifies the writing of such equations. 3. General Properties of Tensors The expressions for the piezoelectric relations discussed in section 2 can be considerably abbreviated by expressing them in tensor form. Further- more, the calculation of elastic constants for rotated crystals is considerably simplified by the geometrical transformation laws established for tensors. Hence it has seemed worthwhile to express the elastic, electric, and piezo- electric relations of a piezoelectric crystal in tensor form. It is the purpose of this section to discuss the general properties of tensors applicable to Cartesian coordinates. If we have two sets of rectangular axes (Ox, Oy, Oz) and (Ox', Oy' , Oz) having the same origin, the coordinates of any point P with respect to the second set are given in terms of the first set by the equations x' — (iX -\- miy -\- Jhz y' = lix -\- m^y + «22 (61) z' = I3X + m^y -\- HiZ. 100 BELL SYSTEM TECH NIC A L JOURNA L The quantities (^i , • • • , ;/3) are the cosines of the angles between the various axes; thus A is the cosine of the angle between the axes Ox', and Ox; n^ the cosine of the angle between Oz' and Oz, and so on. By solving the equations (61) simultaneously, the coordinates .v, y, z can be expressed in terms of .t', y' , z' by the equations. X = l,x' + t^' + t,z' y = mix' + Woy' + nviz' (62) z = nix' + n<iy' + r^z' . We can shorten the writing of equations (61) and (62) considerably by changing the notation. Instead of x, y, z let us write .Ti , x? , Xz and in place of x' , y' , z' we write X\ , X2 , Xs. We can now say that the coordinates with respect to the first system are .Ti , where i may be 1, 2, 3 while those with respect of the second system are Xj , where / = 1, 2 or 3. Then in (61) each coordinate Xj is expressed as the sum of three terms depending on the three x, . Each x, is associated with the cosine of the angle between the direction of x, increasing and that of x, increasing. Let us denote this cosine by c , y . Then we have for all values of j, 3 x'j = aijXi + a2jX2 + asjXs = ^ aijXi. (63) Conversely equation (62) can be written 3 Xi = XI ^•■y-'^y (64) y=i where the a ,; have the same value as in (63), for the same values of i and 7, since in both cases the cosine of the angle is between the values of x; and x; increasing. Such a set of three quantities involving a relation between two coordinate systems is called a tensor of the first rank or a vector. We note that each of the equations (63), (64) is really a set of three equa- tions. Where the suffix i or j appears on the left it is to be given in turn all the values 1, 2, 3 and the resulting equation is one of the set. In each such equation the right side is the sum of three terms obtained by giving j or / the values 1, 2, 3 in turn and adding. Whenever such a summation occurs a suffix is repeated in the expression for the general term as dijXj . We make it a regular convention that whenever a suffiix is repeated it is to be given all possible values and that the terms arc to be added for all. Then (63) can be written simply as x^ = a,;X,- the summation being automatically understood by the convention. PIEZOELECTRIC CRYSTALS IN TENSOR FORM 101 There are single quantities such as mass and distance, that are the same for all systems of coordinates. These are called tensors of the zero rank or scalars. Consider now two tensors of the first rank «, and Vk ■ Suppose that each component of one is to be multiplied by each component of the other, then we obtain a set of nine quantities expressed by Ui Vk , where i and k are independently given all the values 1. 2, 3. The components of «; Vk with respect to the Xj set of axes are Uj V( , and tijVi = (aijtii) (aicfk) = anQkiUiVk (65) The suffixes / and k are repeated on the right. Hence (65) represents nine equations, each with nine terms. Each term on the right is the product of two factors, one of the. form a ijOki, depending only on the orientation of the axes, and the other of the form UiVk , representing the products of the components referred to the original axes. In this way the various Uj Vf can be obtained in terms of the original UiVk . But products of vectors are not the only quantities satisfying the rule. In general a set of nine quantities IV ik referred to a set of axes, and transformed to another set by the rule ^';Y = OijQki u>ik (66) is called a tensor of (he second rank. Higher orders tensors can be formed by taking the products of more vectors. Thus a set of n quantities that transforms like the vector product XiXj • • • Xp is called a tensor of rank /?, where n is the number of factors. On the right hand side of (66) the / and k are dummy suffices; that is, they are given the numbers 1 to 3 and summed. It, therefore, makes no difference which we call i and which k so that ^^'j7 jakfiCik — OkjaifCkf (67) Hence Wk( transforms by the same rule as u' ik and hence is a tensor of the second rank. The importance of this is that if we have a set of quantities li'n U'i2 U'i3 W21 K'22 'iC'23 ■Z^'31 li-'SO IC-i^ fthe second ra Wn K'21 ^C'31 «'12 1^22 W'32 "d'n K'23 "^£'33 which we know to be a tensor of the second rank, the set of quantities (68) (69) is another tensor of the second rank. Hence the sum (idk + i^'ki) and the difference (^c',k — iVk,) are also tensors of the second rank. The first of 102 BELL SYSTEM TECH NIC A L JOURNA L these has the property that it is unaltered by interchanging i and k and therefore it is called a symmetrical tensor. The second has its components reversed in sign when i and k are interchanged. It is therefore an antisym- metrical tensor. Clearly in an antisymmetric tensor the leading diagonal components will all be zero, i.e., those with i = k will be zero. Now since Wik= \ {wik + Wki) + h (u'ik — Wki) (70) we can consider any tensor of the second rank as the sum of a symmetrical and an antisymmetrical tensor. Most tensors in the theory of elasticity are symmetrical tensors. The operation of putting two suffixes in a tensor equal and adding the terms is known as contraction of the tensor. It gives a tensor two ranks lower than the original one. If for instance we contract the tensor ut Vk we obtain UiVi = UiVi + U2V2 + U3V3 (71) which is the scalar product of u i and Vk and hence is a tensor of zero rank. We wish now to derive the formulae for tensor transformation to a new set of axes. For a tensor of the first rank (a vector) this has been given by equation (61). But the direction consines A to «3 can be expressed in the form (72) _ dx' _ axi dx dxi ' dx' dxi Wi = —- = -— ; dy dxt dx' dz dxs _ dy' _ dX2 dx dxi ' dy' dX2 W2 = ^ = r— ; dy 6x2 dy' dz _ dX2 8x3 _ dz' _ dx'z dx dxi ' dz' dx'i dy dX2 dz' dz _ dx's dxs Hence equation {61) can be expressed in the tensor form X / dXj dXi (73) Similarly since a tensor of the second rank can be regarded as the product of two vectors, it can be transformed according to the equation / / /dXj \ /dXf \ dXj dXf .»,v \dXi / \dxis / dXi dXk which can also be expressed in the generalized form / dXj dXf /-rv PIEZOELECTRIC CRYSTALS IN TENSOR FORM 103 In general the transformation equation of a tensor of the ;zth rank can be written xi OXj^ OXj., a.V/„ (76) 4. Application of Tensor Notation to the Elastic, Piezoelectric AND Dielectric Equations of a Crystal Let us consider the stress components of equation (7) T T T ^ XX ^ xy •* J2 T T T ^ yx ^ yy •'2/2 T,x T,y r,, from which equation (8) is derived ■i xy I yx ] ^ xz -i zi , ^ yz •* zj/ and designate them in the manner shown by equation (77) to correspond with tensor notations (77) by virtue of the relations of (8). We wish to show now that the set of 9 elements of the equation constitutes a tensor, and by virtue of the relations of (8) a symmetrical tensor. The transformation of the stress components to a new set of axes x', y', z' has been shown bv Love to take the form Tn Tn Tn T21 T22 Toa = Tn T,2 7^33 Tn Tn Tn Tn T22 T2, Tn ^23 7^33 T^x = fl T^j, + rn\Tyy ~\- nlT,, + lliMiT^y + 2(iUiTj,z + ImiUiTy, (78) Txy = (ifiTjcx + fnitnoTyy-'r nin2T,,-{- (Awo + limi)T^y + (A«2 + hnifT^^ + {mini + niniiiTy^ where A to 113, are the direction cosines between the axes as specified by equation (61). Noting that from (72) «3 = dXj dx3 the first of these equations can be put in the form ^ See "Theory of Elasticity," Love, Page 80. 104 BELL SYSTEM TECHNICAL JOURNAL , /dx'i^\ dx[ dx'i \ dxi I d.Ti 0x2 + ''P '-^ Tn + (g) 8x2 dxi dxi dxi dXi dxs dXi dxi _ dxi dxi i 22 -r T— - ^— i 23 — r — - — 1 k( 0X2 0X3 OXk dX( (79) 5xi dxi dxi dxi ( dx ~r -X — -z — i 31 "T -r — -7 — -/ 32 "rl -r- d.T3 dxi dX3 dX2 \0iC3 :)■ while the last equation takes the form / _ dxi 8x2 . dxi dx2 „ , dxi 8x2 „ ■t 12 — -^ — -z — i 11 ~r -7, — -;:— i 12 -r r — r — i 13 dxi 0X1 dxi 6x2 oxi 0x3 dxi dxo ™ , dxi 6x2 rp , dxi dxo ^ _„„,„... 1 -7. — -z — i 21 "t" -r — - — -1 22 ~r r — - — i 23 — r — 'Z — i kf 0X2 oXi 0X2 0X2 d.Vo 0X3 ax/c oXf , dxi 6x2 „ , dxi 6x2 „ , ~r ~ — -;; — i 31 "T T— - -r — i 32 "T* 0X3 d.V] 0x3 0x2 The general expression for any component then is r' . = ^^ f '' dXk dxf dxi 8x2 dxf dxi dx'2 dxs 6x3 (80) (81) which is the transformation equation of a tensor of the second rank. Hence the stress components satisfy the conditions for a second rank tensor. The strain components •J XX '^xy "Jxz •^yx '^yy '^yz >J zx ^ zy ^ zz do not however satisfy the conditions for a second rank tensor. This is shown by the transformation of strain components to a new set of axes, which have been shown by Love to satisfy the equations Sxy — 2A^2'5'ii + 2viim2Syy + luirioSzz + (Aw2 + ^2Wi)5'j (82) + (A"2 + fl\(2)S^z + (Wl"2. + m2lh)S:c PIEZOELECTRIC CRYSTALS IN TENSOR FORM If, however, we take the strain components as 105 c _ c _ ^^ ■'11 — 'In — TT 1 ax S,2 dr) By ' c _ c _ ^f O33 — 'Jjz — :r- dz 2 \dx dx/ ' (83) Si-i — Siy-> — 1 (dj dy + dr,\ dzj the nine components ^n .SV2 A'l3 .V21 .Vo, .V23 .V31 .S'32 A'33 (83) will form a tensor of the second rank, as can be sh(jwn by the transformation equations of (82). The generaUzed Hooke's hiw given by equation {22) becomes '/'.-.= CijkfSkt (84) CijkC is a fourth rank tensor. The right hand side of the equation being the product of a fourth rank tensor by a second rank tensor is a sixth rank tensor, but since it has been contracted twice by having k and ^ in both terms the resultant of the right hand side is a second rank tensor. Since dm is a tensor of the fourth rank it will, in general, have 81 terms, but on account of the symmetry of the T , j and Sic( tensors, there are many equiva- lences between the resulting elastic constants. These equivalences can be determined by expanding the terms of (84) and comparing with the equiva- lent expressions of (22). For example + ^1121621 -f- ril22'S'22 + ("1123»^23 + <"n3 Al + <"1132-S'32 + CU33'S33 • (85) Comparing this equation with the tirst of (22) noting that Su — S21 = — ', etc., we have t'UU — C\\ ; ('1112 — ("1121 — '"in ; <"1133 — '"iS ', f-'llU ^^1122 = fl2 ; f'll23 = t'll32 = 6"l4 • t-1131 (86) 106 BELL SYSTEM TECH NIC A L JOURNA L In a similar manner it can be shown that the elastic constants of (22) correspond to the tensor elastic constants djui according to the relations C\\ = fun ; Cl2 = <'1122 = C22II ; Cl3 — Ca33 = f33n ', ^14 = ^1123 = ^132 = Cnn = C32U ; Cib = diw = ^1131 = ^'isu = Cun ', Cu = fiii2 = Cn2i = <^i2ii = ^2111 ', C22 — <^2222 ', C2Z — <^2233 — ^3322 ', ^24 = ^2223 = ^2232 = ^2322 = ^3222 ', C2b — ^2213 = <"2231 = '"1322 = ^3122 ', <^26 = <^2212 = <^2221 = <'l222 = ^2122 ', C33 = C3333 ; C34 = ^3323 = ^3332 = ^2333 = ^3233 ', ^36 = 3313 = ^3331 = '^1333 — ^3133 J (87) ^36 = ^3312 — C3321 — C1233 — ^2133 ', ^44 — ^2323 — ^2332 — ^3223 — f3232 y ^46 — ^2313 — ^2331 = ^3213 = <^3231 = 1323 = 1332 = ^3132 = ^3123 ', ''46 = ^2312 = £"2321 — C32I2 = C322I — ^1223 = C1232 = C2I23 — C2132 ; C55 = C1313 = C1331 = f3U3 = ^^3131 ; Cb6 — fl312 = 0321 = ^3112 = C3121 = fl213 = ^1231 = ^2113 = £"2131 ) f 66 = f 1212 = <"1221 = ^2112 — <^2121 • Hence there are only 21 independent constants of the 81 djkf constants which are determined from the ordinarily elastic constants c,/ by replacing 1 by 11 ; 2 by 22; 3 by 33; 4 by 23; 5 by 13; 6 by 12 (88) and taking all possible permutations of these constants by interchanging them in pairs. The inverse elastic equations (26) can be written in the simplified form Sij = SijkfTk(. (89x By expanding these equations and comparing with equations (26) we can establish the relationships _ _ Su _ _ _ _ Sn = ^1111 ; -^12 = 51122 — -^2211 ; -^13 — 51133 — -^3311 ; "y — -^1123 — -51132 — -52311 — ■^16 _ _ ■^16 _ _ _ _ •^3211 ; -W — -51113 — -51131 — -51311 — -53111 ', -y — -5lll2 — ■51121 — -51211 — -52111 ; •522 — •52222 ; 523 = -52: 233 = -53322 ; 2 1 _ Sr. !23 — •52232 =^ -52322 = 53222 ; 526 _ 2 ^2213 = = -52231 = -51322 = = -53122 5? 6 '■' 2 -5221 2 = : 5222 1 = ^51222 = -5212-2 ; 533 = = 53333 (90 A) ^34 _ 2 -53323 '■ = -53332 = -52333 = -5.^233 ; 2 = -53313 = -53331 - = -51333 = = 5;tl33 . 536 ' 2 -- PIEZOELECTRIC CRYSTALS IN TENSOR FORM 107 -^44 ^45 ^3312 — •^3321 — -^1233 — -52133 ', -J — ■^2323 — -^2332 — -^3223 — -^3232 ', — — ^2313 = _ _ _ _ _ _ -^46 _ ■^2331 — •^3213 — -^3231 — -^1323 — -^1332 — .^3123 — -^3132 ', -J — -^2312 — -^2321 = (90 B) _ _ _„_ _ ■>55_ _ ^3212 — ■^3221 — -^1223 — J1232 — -^2123 — -^2132 ; ~J — ■^1313 — -^1331 — •^3113 = •^56 _ _ . •^3131 ; -J" ~ "^^^12 ~ "^13-1 ~ "^3112 — •^3121 — -^1213 — -^1231 — ■^2113 — •^2131 ', •^66 _ _ _ _ -; '■ •^1212 — •^1221 — -^2112 — 52121 • 4 Here again the SijkC elastic constants are determined from the ordinary elastic constants 5,y by replacing 1 by 11, 2 by 22, 3 by 33, 4 by 23, 5 by 13, 6 by 12. However for any number 4, 5, or 6 the elastic compliance Sij has to be di- vided by two to equal the corresponding SijkC compliance, and if 4, 5 or 6 occurs twice, the divisor has to be 4. The isothermal elastic compliance of equations (39) can be expressed in tensor form Si,^slk(T,c + a,,dQ (91) 1 where as before a,; is a tensor of the second rank having the relations to the ordinary coefficients of expansion Oil = «ii ; 02 = "22 ; "3 = «33 ,* y = ^23 i oib ae The heat temperature equation of (35) is written in the simple form I dQ = + akt Tut e + pCp de. (92) ' . . . ii By eliminating dO from (92) and substituting in (91) the adiabatic constants !i are given in the simple form SijkC = SijkC - —^ — . (93) The combination elastic and piezoelectric equations (60) can be written in the tensor form T Sii = S^jkCTkC + d^ijEm ; hr, = ~ Eyn + dnkCTkC- (94) 4ir 108 BELL SYSTEM TECH NIC A L JOVRNA L Here d^ij is a tensor of third rank and €,„„ one of second rank. The dmi) constants are related to the eighteen ordinar}"- constants (/,/ by the equations du = d\n ; dn — d\oo ; dy di6 2 "133 ; — "123 — "132 , — — "113 — "131 ; '^222 ; ^^23 — '^233 ', -Z '/223 </o32 -T- — dnu = ^231 ; ~r = 'A>i2 = fi'221 ; '/31 2 (})h) ^34 _ , _ , ^35 — "323 — "332 ; -^ </313 — dz ' 2 — "311 ; "32 — "322 ; "33 — "333 = "312 — "321 • The tensor equations (94) give a simple method of expressing the piezo- electric equations in an alternate form which is useful for some purposes. This involves relating the stress, strain, and displacement, rather than the applied field strength as in (94;. To do this let us multii')ly through the right hand equation of (94) by the tensor 47r,S,L, , obtaining A-K'Sl „ 5 „ = e J, ntimnEm + 47r(/ „ kt l^m n T k( (96) where /il,, is a icn:or of the "free" dielectric impermeability obtained from the determinant. ^L = (-1)' ,.yJ. *r . whe e A is the determinant fu €12 fl3 T fl2 T C22 r €23 r ei3 T €23 T €33 (97) (98) and Am,, the minor obtained from this by suppressing the wth row and nth column. If we take the i)roduct el„ /i„.„ for the three values of w, we have as multiijliers of E\ , Eo , E^ , respectively €11 Pn + €12 Pl2 + €13 Pl3 = 1 €21 P2I + €22 P22 + €23 P23 == 1 €31 P31 -r €32 P32 "T" €33 P33 — 1- (99): Bui by virtue of equations (97) and (98) it is obvious that the value of each term of (99) is unity. Hence we have E„ — Aw0mn 5„ — (47r dnkt iSmn) 'i\t (100) i PIEZOELECTRIC CRYSTALS IN TENSOR FORM 109 Since the dummy index n is summed for the values 1, 2, and 3, we can set the value of the terms in brackets equal to and equation (100) becomes Em = 47r (3mn 5„ — gmkC Tkl . Substituting this equation in the first equations of (94) we have where Si,k( = Sijkf. — d„ni gmkl = Sijkt — 4:X[j8„„ d nkt dmij\. (101) (102) (103) By substituting in the various values of i,j, k and ^ corresponding to the 21 elastic constants, the difference between the constant displacement and constant potential elastic constants can be calculated. If equations (102) and (103) are expressed in terms of the Si,- ■ -, S^ strains and Ti,- ■ •, T^ stresses, the gnij constants are related to the gij constants as are the corre- sponding dij constants to the (/„,/ constants of equation (95). Another variation of the piezoelectric equations which is sometimes em- ployed is one for which the stresses are expressed in terms of the strains and field strength. This form can be derived directly from equations (9-i) by multiplying both sides of the first equation by the tensor c^jkC for the elastic constants, where these are defined in terms of the corresponding s^j elastic compliances by the equation 4 = (-i)^'"^^a:;/a (104) where A is the determinant A^ = ^11 5l2 SlZ 5i4 ^15 5l6 .f. 5^2 E •^23 E 524 E 525 E 526 E E •^23 E 533 •^34 sl. 536 E E S2i E Sz\ E 544 E 545 54% 515 525 535 545 555 556 516 526 536 546 566 566 and A*y in the minor obtained by suppressing the /th row and^'th column. Carrying out the tensor multiplication we have Cijkt Sij = djkt Sijkt Tkf + dmij c-jkC E„ (105) no BELL SYSTEM TECHNICAL JOURNAL As before \vc find that the tensor product of cijk( Si,k( is unity for all values of k and (. Hence equation (105) can be written in the form Tu(= clu(Si,- e„.uE„, (106) where Cmk( is the sum CmkC = d,„ij cljkl (107) surrn ed for all values of the dummy indices / and 7. If we substitute the equation (106) in the last equation of (94) we lind s bn=^-PEm + er^^Sij (108) where e"™,, the clamped dielectric constant is related to the free dielectric constant emn by the equation ein ^ tin- MdnUtemkt]. (109) Expressed in two index piezoelectric constants involving the strains ^u- • -Svi and stresses Tw • • T12 the relation between the two and three index piezo- electric constants is given by the equation en = ^ni ; ^12 = ^122 ; ^13 = ^133 ; ^14 — ^123 = ^132 ; ^15 = ^U3 = ^131 e\e = «U2 = em ; ^21 = ^211; ^22 = ^222 ; ^23 = ^233 ; ^24 = ^223 = ^232 e25 = ^213 — ^231 ; ^26 = ^212 = ^221 ", ^31 = ^3U ; <'32 = <'322 ', «33 = ^333 ^34 = ^323 = ^332 ', ^35 = ^313 = ^331 i ^36 = €312 = ^321 • (110) Finally, the fourth form for expressing the piezoelectric relation is the one given by equation (53). Expressed in tensor form, these equations become TkC = c'^]k(S,j — h„ktb„ ; Em = 47r^'l„ bn — hmijSij (111) In this equation the three index piezoelectric constants of equation (HI) are related to the two index constants of equation (53) as the e constants of (110). These equations can also be derived directly from (106) and (108) by eliminating Em. from the two equations. This substitution yields the additional relations h„k( = -^T^e-rnkf (imn \ ^ikf = cfjkf + C„,k( I'mrj = C^ijkl (112) + 47r emk( Cnij 0mn where i3L = (-i)^"'*"'a:;Va'' PIEZOELECTRIC CRYSTALS IN TENSOR FORM 111 in which s en S €12 S €13 s ei2 S €22 .S €23 s €13 6' €2.f .S €33 The four forms of the piezoelectric equations, and the relation between them are given in Table I. Table I Four Forms of the Elastic, Dielectric, and Piezo Electric Equations AND their Interrelations Form Elastic Relation 5,,- = Si,k(Tu( + d„,,E,, Electric Relation bn = -~ En+ dnkfTkf 47r 2 Sii = Si,kfTk( + gn^jSn E„, = 4x^„n5„ - gmk(Tk( 3 Tk( = Cij(kSi, - emk(E„ s -iTT 4 Tk( = CijkfSu - h„kfbn Em = iTT^ijn - hmiiSii Form Relation Between Elastic Conjlaii.j Relation Between Piezoelectric Constants Relation Between Dielectric Constants 1 <*^= ^O^Z-'^-W^mAf g^,(= 47r^l,d,,f ^L = (-i)""+">A^yA*^ 2 cf^ = (-1)(' + ^-'a^^^/a«^ e,nkt = d„,,cf^^^ 'tn = e^ - ■i-^idnkfe„kf) 3 'iikf. = 'f,kf+''n>'f/^'"'i k„k( = 47r^'L.'',„i/' ^L - ^T ^ Rnkthn^kt mn 4^ 4 cO,= (-1) (' + '■' A'^^/A^"" hnk( = SniiC'^,,( ^t. = (-1)('"+")A^V^'^ I 5. Effect of Symmetry and Orientation on the Dielectric Piezo- j electric and Elastic Constants of Crystals j All crystals can be divided into 32 classes depending on the type of sym- 1 metry. These groups can be divided into seven general classifications il depending on how the axes are related and furthermore all il classes can ^ be built out of symmetries based on twofold (binary) axes, threefold (irig- 1 onal) axes, fourfold axes of symmetry, sixfold axes of symmetry, planes of j' reflection symmetry and combinations of axis reflection symmetry besides a simple symmetry through the center. Each of these types of symmetry 1 1 2 BELL SYSTEM TECH NIC A L JOURNA L result in a reduction of the number of dielectric, piezoelectric, and elastic constants. Since the tensor equation is easily transformed to a new set of axes by the transformaion equations (76) this form is particularly advantageous for determining the reduction in elastic, piezoelectric and dielectric con- stants. For example consider the second rank tensors, c^^ and ak( for the dielectric constant and the expansion coefficients. Ordinarily for the most general symmetry each tensor, since it is symmetrical, requires six inde- pendent coefficients. Suppose however that the X axis is an axis of twofold or binary symmetry, i.e., the properties along the positive Z axis are the same as those along the negative Z axis. If we rotate the axes 180° about the A' axis so that -f Z is changed into — Z, the direction cosines are (113) / - ^^1 - 1 . ,dxi bxx ^ Wi = -— = ; dx2 dxi „ dX3 dX2 . 9X2 n „2 = = dxs ^3 = f^-0; dx\ dx's "•' - a., ~ " ■ dx's «3 = ^- = -1- dxs transformation equations for a second / dx'i dxj dxk dxt rank tensor are (114) Applying (113) to (114) summing for all values of k and / for each value of i, and J we have the six components ' ' _ ' _ ' _ ' _ ' _ ('1 1 -\ €11 — CU ; «12 — ~ €12 ; tl3 — — ei3 ; €22 — €22 ; ^23 — ^23 ', ^33 — ^33 • \ll^) Since a crystal having the A' axis a binary axis of symmetry must have the same constants for a -\-Z direction as for a — Z direction, this condition can only be satisfied by €12 = €13 = 0. (116) The same condition is true for the expansion coefficients since they form a second rank tensor and hence «12 = «13 = 0. (117) In a third rank tensor such as dijk , enk , gnh , I' nk , we similarly find that of the eighteen independent constants hm = //le ; //ii3 = //i5 ; /?2ii = /'2i ; //222 = /'22 ; //223 = hi ; (118) //233 = /'23 ; /'311 = //31 ', /'322 = /'32 ', Ihi^i — ll'M ', //333 — "33 • are all zero. The same terms in the dijk , ^nk , gnk tensors are also zero. PIEZOELECTRIC CR VST A LS IN TENSOR FORM 113 In a fourth rank tensor such as Cijk(, Sijkt, applying the tensor trans- formation equation _ dXi dXj dXk dxe . . '^*^tn ^'^n v'V'o ""vp and the condition (113) we similarly find Cl6 = Cl6 = ^25 = C26 = C35 = C36 = C45 = Ca = 0. (120) If the binary axis had been the Y axis the corresponding missing terms can be obtained by cyclically rotating the tensor indices. The missing terms are for the second, third and fourth rank tensors, transformed to two index symbols. Cu , Cl6 , C24 , C26 , C34 , C36 , C45 , C55 . Similarly if the Z axis is the binary axis, the missing constants are ei3 , fi2 ; hn , hn , Ihz , hn , hi , h^ , ha , A26 , hzi , hzf, ; (121) (122) Cu , CiB , C2A , C25 , Czi , C35 , C46 , Cb6 • Hence a cr>'stal of the orthorhombic bisphenoidal class or class 6, which has three binary axes, the X, Y and Z directions, will have the remaining terms, Cu , ^22 , ^33 ; hu , ^25 , ^'36 ', Cn , Cn , Cl3 , C21 , C23 , C33 , C44 , C55 , Cee (123) with similar terms for other tensors of the same rank. Rochelle salt is a crystal of this class. If Z is a threefold axis of symmetry, the direction cosines for a set of axes rotated 120° clockwise about Z are, f I = --- = - .5 ; wi = -— = - .866 ; «i = t— = oxi 0X2 dXz ^3 = ^^ = .866; m2=^=-.5; «2 = ^^ = (124) 0x1 0x2 0x3 , dx'z dx'z ^ dx'z 4 = — -=0; m3=-— = 0; riz = ^— = 1. dxi 0X2 0X3 Applying these relations to equations (114) for a second rank tensor, we find for the components €11 = .25eii+ .433ei2+ •75e22 ; ei2 = —. 433 cu + .25 €12 + .433^22 ei3 = — -Seis — .866e23 ; €22 = .75€u — .433ei2 + .25c22 (125) €23 = .866 en — .5e2j ; €33 = €33 • 114 BELL SYSTEM TECHNICAL JOURNAL For the third and tifth equations, since we must have ei3 = cis ; €23 = f2;> in order to satisfy the symmetry relation, the equations can only be satis- fied if e.3 = eo3 = 0. (126) Similarly solving the lirst three equations simultaneously, we find fl2=0;6u= 622. (127) Hence the remaining constants are en = 622 ; 633 • (128) Similarly for third and fourth rank tensors, for a crystal having Z a trigonal axis, the remaining terms are hn , hu = —lh\ , hn = 0; hu , //15 , /'le = — /'22 /?21 = — /'22, //22 , /'23 = 0, //24 = /'l5 ] hb = — hi , hi = " /?I1 (129) //31 ; ^32 = //31 ; //33 ; /'34 = 0; /;35 = 0; //36 = cn ; ^12 ; ^13 ; cu ; fis = ~<^25 ; ^le = c\2 ; C21. — c\\ ; C23 = c\i ; C24 = — '"14 ; C25 ; ^26 = Cn ; C20 = C\3 ; f33 ; ("34 = 0; czh — ^\ C36 = (130) Cu ; ^24 — ~Cu ;czi — ^; cu ; f45 — 0; C46 — c\^ C\i = —^25; ^25 ; <'35 = 0; f45 = 0; Css = C44 ; C56 = Cu C16 = 0; ^26 — 0; r36 = 0; C46 — C21, ; ("56 "^ Cu ; fee = 2 vn~Ci2)- If the Z axis is a trigonal axis and the X a binary axis, as it is in quartz, the resulting constants are obtained by combining the conditions (116), (118), (120) with conditions (128), (129), (130) respectively. The resulting second, third and fourth rank tensors have the following terms 611 ; 612 = 0; 613 = 612 = 0; 622 = 6U ; 623 = (131) 613 = 0; €23 = 0; €33 flu ; fin = — //ii ; //13 = 0; //i4 ; //I5 = 0; //i6 = //21 = 0; //22 = 0; //23 = 0; //24 = 0; h, = -hu ; //26 = -hn (132) //3l = 0; //32 = 0; //33 = 0; //34 = 0; hy, - 0; //36 - PIEZOELECTRIC CRYSTALS IN TENSOR FORM 115 (133) Cn ; Ci2 ; C\3 ; Cu ; cis = 0; cie = Cn 5 ^22 = ^11 ; ^23 = C\3 ; C24 = Ci4 ; C25 =0; C26 = Ci3 ; ^23 = Ciz ; (^33 ; C34 = ; C35 = ; Cae = fi4 ; C24 = — Ci4 ; r34 = 0; <:44 ; C45 = 0; r46 = C15 = 0; ^26 = 0; f35 = 0; C45 = 0; C55 = ("44 ; C56 = Cu <^i6 = 0; C26 = 0; f36 = 0; Css = 0; C55 = ru ; Cee = 2 (<^ii~fi2)- vS.l Second Rank Tensors for Crystal Classes The symmetry relations have been calculated for all classes of crystals. For a second rank tensor such as e,/, the following forms are required Triclinic Classes 1 and 2 eu , €12 , €13 ei2 , ^22 , C23 «13 , «23 , ^33 fU , , €13 , €22 , ei3 , , €33 €11,0 ,0 , 622 , (134) ,0 , €33 €11,0 ,0 , €„ , ,0 , €33 €11,0 ,0 , €„ , 0,0, €„ 5.2 Third Rank Tensors of the Piezoelectric Type for the Crystal Classes hn , hu , his , /'i4 , /'15 , /'le Monoclinic sphenoidal, 1' a binary axis, Class 3 MonocHnic domatic, Y a plane of symmetry. Class 4 Monoclinic prismatic, Center of symmetry, Class 5 Orthorhombic Classes 6, 7, 8 Tetragonal, Trigonal Hexagonal Classes 9 to 27 Cubic Classes 28 to 32 Triclinic Assymetric (Class 1) No Symmetry //21 , ^/22 , //23 , //24 , //26 , ^'26 /'31 , hsi , /?33 , //34 , //35 , hzr, 116 BELL SYSTEM TECHNICAL JOV RNAL Triclinic pinacoidal, (center of symmetry) h = (Class 2) ,0 ,0 , //14 , , /?16 hii , lin , fhz ,0 , //26 , ,0 ,0 , //34 , , /;,6 hn , Ih2 , hn ,0 , /7i6 , ,0 ,0 , /724 , ,ht hi , /'32 , /'33 , , hsB ,0 Monoclinic prismatic (center of symmetr>0 h = (Class 5) ,0 ,0 , /7i4 , ,0 ,0 ,0 ,0 ,//26,0 ,0 ,0 ,0 ,0 ,//36 ,0 ,0 ,0 ,/;i6,0 ,0 ,0 , //24 , ,0 /?31 , //32 , //33 , ,0 ,0 Orthorhombic bipyramidal (center of s}mmetr>-) // = (Class 8) , 0,0, liu , liib , Monoclinic Sphenoidal (Class 3) Y is binary axis Monoclinic domatic (Class 4) Y plane is plane of symmetry Orthorhombic bisphenoidal (Class 6) X, Y, Z binary axes Orthorhombic pyramidal (Class 7) Z binary-, X, Y, planes of s\Tnmetry Tetragonal bisphenoidal (Class 9) Z is quaternar}^ alternating Tetragonal pyramidal (Class 10) Z is quaternar}' , 0,0, -//15, /7l4,0 //31 , -/'31 , , 0,0, //36 ,0 ,0 , Ihi , //15 , ,0 ,0 ,//l5, -//i4,0 //31 , //31 , //33 , , ,0 Tetragonal scalenohedral (Class 11) / I ,0 ,0 , liu ,0 ,0 quaternar\'. A' and I' binary ,^ ,^ ,, ,, , n ^ ' , , , , //i4 , ,0 ,0 ,0 ,0 , //36 Tetragonal trapezohedral (Class 12) jO ,0 ,0 , Im , ,0 Z quaternar^^ A' and F binar^^ 0.0,0,0, -/;. , I , , , , 0,0 (135) PI EZOELECTKTC CRYSTALS TN TENSOR FORM 117 Ditetragonal pyramidal (Class 14) Z quaternary, X and 1' planes of sy mmet ry Tetragonal bipyramidal (center of symmtery) h — Q (Class 13) , () ,0 ,0 , /;,5 , .0 .0 ,//i5,0 ,0 /-■■U , /?.l , //33 , ,0 ,0 Ditetragonal bipyramidal (center of symmetry) // = (Class 15) Trigonal pyramidal (Class //u , — //u , , hu , /?i5 , —fi'n 16) Z trigonal axis / a / / / — //22, /^22 , , /;i5 —hu,—fin hn , //;u , //3.3 , , , Trigonal rhombohedral (Class 17) center of symmetry, // = Trigonal trapezohedral (Class 18), Z trigonal, .Y binary Trigonal bipyramidal (Class 19), / trigonal, plane of symmetry Ditrigonal pyramidal (Class 20) Z trigonal, Y plane of symmetry Ditrigonal bipyramidal (Class 22) Z trigonal, Z plane of sym- metry and 1' plane of symmetry Hexagonal pyramidal (Class 2i) Z hexagonal Hexagonal trapezohedral (Class 24) Z hexagonal, .Y binary //u, -//u , /?14 , , () , ,0 , -Ihi , -hn , f) , , //ll. -//11,() , , — A22 -//22, //22 , , , -hn , , , , , (» , , //15 -//22 -//22, //?2 , //15, hu , Ihl , //33 , . 1) center of symmetry. // = hn, -//u ,0 , , , , ,0 ,0 , , -hn , , ,0 , , , , , Ihi , //15 , , , flu , — hu , hi , //31 , //33 , , , , , , //14 , , (• , ,0 ,0 , -//14 , , ,0 ,0 , , 1 18 BELL SYSTEM TECH NIC A L JOURNA L Hexagonal bipyramidal (Class 25) center of symmetry, /? = Dihexagonal pyramidal (Class 26) .Y hexagonal Y plane of symmetry ,0 ,0 ,0 ,/7i5,0 ,0 ,0 ,/7i5,0 ,0 h\ , //31 , /'33 , ,0 ,0 Dihexagonal bipyramidal (Class 27) center of symmetry, h = Cubic tetrahedral-pentagonal-dedo- cahedral (Class 28) A', V, Z binary ,0 ,0 ,hu,0 ,0 ,0 ,0 ,0 , //i4 , ,0 ,0 ,0 ,0 ,/;,4 Cubic pentagonal-icositetetrahedral (Class 29) ^ = Cubic, dyakisdodecahedral (Class 30) center of symmetry, // = Cubic, hexakisletrahedral (Class 31) X, I', / quaternary alternating ,0 ,0 , /;i4 , ,0 ,0 ,0 ,0 , //i4 , ,0 ,0 ,0 ,0 ,/7i4 Cubic, hexakis-octahedral (Class 32) center of symmetry, // = This third rank tensor has been expressed in terms of two index symbols rather than the three index tensor symbols, since the two index symbols are commonly used in expressing the piezoelectric effect. The relations for the // and e constants are // 14 , /' i5 , // lb are equivalent to // ,23 , // 113 , /' 112 (136) in three index symbols, whereas for the d ij and gij constants we have the relations </,4 fl,5 1 ' T' dit are equivalent to r/,23 , d,n, ^,12 (137) Hence the </, relations for classes 16, 18, 19, and 22 will be somewhat dif- ferent than the // symbols given above. These classes will be PIEZOELECTRIC CRYSTALS IN TENSOR FORM 119 Class 16 Class 18 Class 19 Class 22 dn —dn du dn —Id^i — dvt d^i </i5 —du —2dn dn dsi d33 ^u -dn du -du -2dn dn -dn -2^22 -da (/22 -2dn ^11 -dn -2dn (138) 5.3 Fourth Rank Tensors of the Elastic Type for the Crystal Classes Triclinic System cn C\2 ^13 Cu Cl5 ^6 The 5 tensor is (Classes 1 and 2) 21 moduli Cn Coo Cos Coi ^25 C06 entirely analo- gous Cl3 Cos C33 C34 <"35 C36 Cli C2i C34 ^44 a 5 f46 fl5 <:26 C3& C45 f55 Cb6 ("16 ^20 C36 C46 ^56 ^66 (139) Monoclinic System Cn C\o Cn fl5 The s tensor is (Classes 3, 4 and 5) 12 moduli Cl2 Co.i C03 C2b entirely analo- gous C\3 Co.3 C33 Csb Cii C4f, Cl5 f"25 <"36 (^55 C46 ^66 120 BEl Rhombic System (Classes 6, 7 and 8) 9 moduli Tetragonal system, Z a fourfold axis (Classes 9, 10, 13) 7 moduli Tetragonal system, Z a fourfold axis, X a two- fold axis (Classes 11, 12, 14, 15) 6 moduli Trigonal system, Z a twofold axis, (Classes 16, 17) 7 moduli L SYSTEM 7 ^ECH .V/Cl / JOIRNAL '11 Cu (-'i.i The s tensor is en C22 C23 {) entirely analo- gous Cn C23 C33 C44 (-"55 CbG Cn C\2 Cn {) Cu The s tensor is C\2 Cn Cn — Cu entirely analo- gous Cn Cn C33 C44 Cu ^16 -C16 Cf,e, Cn C\2 Cn The i- tensor is Cu Cn Cn entirely analo- gous ("13 Cn C33 () Cii (44 (-"6fi C\\ Cl2 Cn ("14 - -t-25 The 5 tensor is cu Cn Cn — ("14 ("26 (^ analogous ex- cept that 546 = Cn Cn C33 n 2^25 , ■^56 = 2^14 , Cu -(14 -(■44 '25 ^66 = 2 (511 — ^12) — f26 (-25 ("44 ("14 ("25 ("14 "11 — ^12 • PIEZOELECTRIC CRYSTALS IN TENSOR FORM 121 Trigonal system, Z a trigonal axis, X a binary axis (Classes 18, 20, 21) 6 moduli Hexagonal system, Z a sixfold axis, X a two- fold axis (Classes 19, 22, 23, 24, 25, 26, 27) 5 moduli Cubic system (Classes 28, 29, 30, 31, 32) 3 moduli Isotropic bodies, moduli Cl3 Cu Cn Cu — Ci4 Cn C\1 C\3 Cxi Cn Cn Cn Cn Cn Cn C\z Cn Cn Cn Cn Cn Cn Cn Cu Cn — Cu C33 Cn Cn C3Z Cn Cn Cn Cn Cn Cn C44 C44 C44 Cu Cu Cn- Cn 2 Cii Cn Cn Cn C44 ^11 ~ Cn 2 Cn Cn ~ Cn Cn — Cn The 5 tensor is analogous ex- cept that 556 = 2^14 , Stt — 2(511— 512) The 5 tensor is analogous ex- cept 566 = 2 (511 — 512) The 5 tensor is entirely analo- gous The . 5 tensor analogous ex- cept last three diagonal terms are 2 (511 — 512) 122 BELL SYSTEM TECH NIC A L JOURNA L 5.4 Piezoelectric Equations for Rotated Axes Another application of the tensor equations for rotated axes is in deter- mining the piezoelectric equations of crystals whose length, width, and thick- ness do not coincide with the crystallographic axes of the crystal. Such oriented cuts are useful for they sometimes give properties that cannot be obtained with crystals h'ing along the crystallographic axes. Such proper- ties may be higher electromechanical coupling, freedom from coupling to undesired modes of motion, or low temperature coefficients of frequency. Hence in order to obtain the performance of such crystals it is necessary to be able to express the piezoelectric equations in a form suitable for these orientations. In fact in first measuring the properties of these crystals a series of oriented cuts is commonly used since by employing such cuts the resulting frequencies, and impedances can be used to calculate all the pri- mary constants of the crystal. The piezoelectric equations (111) are Tkl = CijkfSij — hnkC^n ; Em. = ^TTPmn^ n ~ hmijSij . (HI) The first equation is a tensor of the second rank, while the second equation is a tensor of the first rank. If we wish to transform these equations to another set of axes x', y', z', we can employ the tensor transformation equations , ^ dx[dx^ ^ dxldxf dxk dX( dxk dx( [CukfSn -\~ 2Ci2k(Sl2 -\- 2Cl3t^5'l5 + C22k(S22 + 2c23ktS-a + C33ktS3z] - '- —-[hikth + h2k(b2 + hklh] (140) axk oxf EL = 47r p^ [/3li5i + ^':2 62 + ^isd^] - ^' OXm dx„, [hmllSl] + lllmuS 12 + 2llml3Sli + Am22«S'22 + 2(1^23^23 "f" hmSiSzs]. These equations express the new stresses and fields in terms of the old strains and displacements. To complete the transformation we need to express all quantities in terms of the new axes. For this purpose we employ the tensor equations dXi dXj , dXn , where ~r~i are the direction cosines between the old and new axes. It is OXi dx ■ 3x ■ obvious that -— ' = -— ^ and the relations can be written OXi dx i PIEZOELECTRIC CR YSTA LS IN TENSOR FORM 123 A = Wi = ;^ ; ^"2 = —/ ; ^3 = ^ (142) Hence substituting equations (141) in equations (140) the transformation equations between the new and old axes become dxi dxi dxi dxi (2 — ^ ' \ dXi dXi dXi dX2 dx[ ' dX2 dX2 W3 = - / dxz dxz dxs dX2 dX3 dxz rp' _ D dXk dXf dXi dXj , _ dx^ dxf dx^ / dXk ax I dXi axj dXk dXf dxn (143) These equations then provide means for determining the transformation of constants from one set of axes to another. As an example let us consider the case of an ADP crystal, vibrating longi- tudinally with its length along the xi axis, its width along the X2 axis and its thickness along the X3 axis, which is also the X3 axis, and determine the elastic, piezoelectric and dielectric constants that apply for this cut when Xi is 9 = 45° from xi . Under these conditions dx'i dxi A = z— = 3-/ = cos 9; dxi 0x1 dxi dXi . mi = —- = —-, = sm 8; W2 0x2 OXl dxi dxs Ml =—- = —-,= 0; Hi = 6x3 dxi SX2 _ dxi . dxi , bill 17, bX2 dxi dx'z _ 6x2 _ dX2 . dx2 , cost;, 0X2 dxs dx2 dX2 0x3 6X2 - ^""^ - 0- dXs dx^ "' dx'z dxz dXz dx3 (144) Since ADP belongs to the orthorhombic bisphenoidal (Class 6), it will have the dielectric, piezoelectric and elastic tensors shown by equations (134), (135), (139). Applying equations (143) and (144) to these tensors it is 124 BELL SYSTEM TECH NIC A L JOURNA L readily shown that the stresses for 6 = 45° are given by the equations ex- pressed in two index symbols ^38 5 a r = (cfl + 2 + 2c?«) ^, Ol + ((:fi + Cl2 2 ~ 2C66) c' 02 + C\zSz (rfl + 2 ~ 2(;66) e' Ol 4- (cfi + Cl2 + 2C?6) ^' + D c,' Ci9 Oa (145) Tz — CizSl 4" Cl3 02 4" C33O3 r; = Cf4 5l + //14 62 ; £1 = -/?145b + 47rLSuai'] Te = cf4 ^5 - /?i4 5i ; £2 = h^'x + 47r[)Su52] J,, ^ icn - c^2 ) _^^ . £^ ^ _^^^f^| „ 5^j ^ 4x1/333 53]. For a long thin longitudinally vibrating crystal all the stresses are zero except the stress Ti along the length of the crystal. Hence it is more ad- vantageous to use equations which express the strains in terms of the stresses since all the stresses can be set equal to zero except Ti . All the strains are then dependent functions of the strain Si and this only has to be solved for. Furthermore, since plated cjystals are usually used to determine the properties of crystals, and the field perpendicular to a plated surface is zero, the only field existing in a thin crystal will be £3 if the thick- ness is taken along the ^3 or Z axis. Plence the equations that express the strains in terms of the stresses and fields are more advantageous for calcu- lating the properties of longitudinally vibrating crj^stals. By orienting such crystals with respect to the crystallographic axis, all of the elastic constants except the shear elastic constants can.be determined. All of the piezoelectric and dielectric constants can be determined from measure- ments on oriented longitudinally vibrating crystals. For such measurements it is necessary to determine the appropriate elastic, piezoelectric, and dielectric constants for a crystal oriented in any direction with respect to the crystallographic axes. We assume that the length lies along the Xi axis, the width along the .T2 axis and the thickness along the Xz axis. Starting with equations of the form O t; ^^ Sxjlc(llcC ~l d i jmt-'m T (146) 47r k PIEZOELECTRIC CR YSTA LS IX TENSOR FORM 125 and transforming to a rotated system of axes whose direction cosines are given by (142), the resulting equation becomes (147) , _ £ dx'i dx'j dXk dxt rp' , J dXi dx, dXm j^i »'■ ~ ^''''^ ^. ^ f)r[ f)r'. ^^ '"* ax- ax- 'ax' ' OXi OXj OXk OXf UXt VXj UXm ./ _ emn dXn dXm 77' i j , ^^n dXk dx( f 47r daPn OX,n OXn OXk dX( All the stresses except Tn can be set equal to zero and all the fields except Ez vanish. Furthermore, all the strains are dependently related to ^n . Hence for a thin longitudinal crystal the equation of motion becomes , _ £ dx[ dx'i dXk dxt rp' , . dx[ dxi dx„ / "^*'^'^ dx- dx- dx'y dx^ '"" dx-dX-dx{ ' . , , (148) ./ _ c^ ^ 5^ p' a: J dxzdxkdx( , 47r 5x;, ^jcs " 5x„ 5a;i dx'i In terms of the two index symbols for the most general type of crystal, we have E' E' £ /)4 I /^ E I E \ i)2 2 1 /T £ I E \ el 2 51111 = ^11 = SiiW + (2^12 + 566)^1^^1 + (2^13 + 55b)4Wi + 2{Sii + 5f6)^iWl«l + Isf^Vh + 25f6AWi + 5^2^! + /0 £ I jB \ 2 2 I r. E 3 , r,/ E , E\ Iff (isiz + summi + isufmni + 2(^25 + 546)wi^i?h 2s26fniCi + 533W1 + IsziHinii + 2536^1^1 + 2(5^6 + 5f5)«iAwi (149) ! din = dn = dn^sd + du^ml + ^is^^i + dutzmiiti + dif^t^itii I + dwtilinii + diinizli + doomm + d^sntsfii + dumsmifh I + </25«3A"i + di^niztinii 4- c?3i"3^i + dsiUzml + dzs,mn\ i + dummini + dziUzkni + dz&nz^inii \ €33 = «ii4 + leiitzmz + 2i.iz(znz + €22^3 + 2€23W3W3 + €33^3 I Hence by cutting 18 crystals with independent direction cosines 9 elastic constants and 6 relations between the remaining twelve constants can be I determined. All of the piezoelectric constants and all of the dielectric constants can be determined from these measurements. These constants can be measured by measuring the resonant and antiresonant frequencies \ and the capacity at low frequencies. The resonant frequency Jr is deter- I mined by the formula h = Yi V^ ^^^^ ^^ y psii 126 BELL SYSTEM TECHNICAL JOURNAL for any long thin crystal vibrating longitudinally. Hence when the density is known, Sn can be calculated from the resonant frequency and the length of the crystal. Using the values of Sn obtained for 15 independent orienta- tions enough data is available to solve for the constants of the first of equations (149). The capacities of the different crystal orientations meas- ured at low frequencies determine the dielectric constant 633 and si.x orienta- tions are sufficient to determine the six independent dielectric constants tmn ■ The separation between resonance and antiresonance Af = /a — Jr determines the piezoelectric constant dn according to the formula d\i = ;1/ £33 4^ ^11 (151) The \-alues of dn measured for 18 independent orientations are sufficient to determine the eighteen independent piezoelectric constants. The remaining six elastic constants can be determined by measuring long thin crystals in a face shear mode of motion. Since this is a contour mode of motion, the equations are considerably more complicated than for a longitudinal mode and involve elastic constants that are not constant field or constant displacement constants. It can be shown that the fundamental frequency of a crystal with its length along x\ , width (frequency determining direction) along .Vo and thickness (direction of applied field) along xs , will be 1 / c.E I c,E , a// c.E c,E\2 1 . c. { = — i/ ^ 22 -\- C66 ± V (C22 — ^66 ) + 4C26 ^ 2C y 2p (152) where the contour elastic constants are given in terms of the fundamental elastic constants by E E £2 c.E -^ll •^66 ■^16 C21 = ; E E E E c,E -^12 ■^16 •^11 -^26 C26 = 1 E E £2 c.B _ SnS22 ^12 C66 — : (153) where A is the determinant A = Su , SV2 , Sl6 E S\2 , B S22 } E 526 E Sl6 , E -^26 , E ■^66 (154) Since all of the constants except svi and ^ee can be determined by measure- ments on longitudinal crystals and the value of (25f2 + ^ee) has been de- ' This is proved in a recent paper "Properties of Dipotassium Tartrate (DKT)- Crys- tals," Phys. Rev., Nov., 1946. PIEZOELECTRIC CR VST A LS IN TENSOR FORM 127 termined, the measurement of the lowest mode of the face shear crystal gives one more relation and hence the values of 5i2 and S6& can be uniquely determined. Similar measurements with crystals cut normal to Xi and width along Xs and with crystals cut normal to X2 and width along Xi determine the constants SAi , 523 and 555 , Siz respectively. The equivalent constants are obtained by adding 1 to each subscript 1, 2, 3 or 4, 5, 6 for the iirst crystal with the understanding that 3+1 = 1 and 6+1 =4. For the second crystal 2 is added to each subscript. Finally the remaining three constants can be determined by measuring the face shear mode of three crystals that have their lengths along one of the crystallographic axes and their width (frequency determining. axis) 45° from the other two axes. Any symmetry existing in the crystal will cut down on the number of constants and hence on the number of orientations to determine the funda- mental constants. 6. Temperature Effects in Crystals In section 2 a general expression was developed for the effects of tempera" ture and entropy on the constants of a crystal. Two methods were given, one which considers the stresses, field, and temperature differentials as the independent variables, and the second which considers the strains, displace- ments and entropy as the independent variables. In tensor form the 10 equations for the first method take the form Em= — hm i jS i J + ■iir^m'n 5 n " qll dQ (155) The piezoelectric relations have already been discussed for adiabatic condi- tions assuming that no increments of heat dQ have been added to the crystal. If now an increment of heat dQ is suddenly added to any element of the crystal, the first equation shows that a sudden expansive stress is generated S.D proportional to the constant X;t^ which has to be balanced by a negative stress (a compression) in order that no strain or electric displacement shall be generated. This effect can be called the stress caloric effect. The second equation of (155) shows that if an increment of heat dQ is added to the crystal an inverse field Em has to be added if the strain and surface charge are to remain unchanged. This effect may be called the field caloric 128 BELL SYSTEM TECH NIC A L JOURNA L effect. Finally the third equation of (155) shows that there is a reciprocal efifect in which a stress or a displacement generates a change in temperature even in the absence of added heat dQ. These effects can be called the strain temperature and charge temperature effects. The second form of the piezoelectric equations given by (58) are more familiar. In tensor form these can be written Sij = sfjktT.cl + dZijEm + afy do 8n = dlk( Tk( + '4^E^ + pi dQ (156) 47r dQ = eda ^ QatcTut + QplErr, + pCl dS The afy are the temperature expansion coefficients measured at constant field. In general these are a tensor of the secjnd rank having six com- ponents. The constants pn are the pyroelectric constants measured at displacements which relate the increase in polarization or surface charge due to an increase in temperature. They are equal to the so-called "true" pyroelectric constants which are the polarizations at constant volume caused by an increase in tempeiature plus the "false" pyioelectric effect of the first kind which represents the polarization caused by a uniform temperature expansion of the crystal as its temperature increases by dQ. As mentioned previously it is more logical to call the two effects the pyroelectric effects at constant stress and constant strain. By eliminating the stresses from the first of equations (156) and substituting in the second equation it is readily shown that Pn = Pn — OC^,enij (157) Hence the difference between the pyroelectric effect at constant stress and the pyroelectric effect at constant strain is the so-called "false" pyroelectric effect of the first kind a^je^a . The first term on the right side of the last equation is called the heat of deformation, for it represents the heat generated by the application of the stresses TkC ■ The second term is called the electrocaloric effect and it represents the heat generated by the application of a field. The last term is p times the specific heat at constant pressure and constant field. The temperature expansion coefficients a.-y form a tensor of the second rank and hence have the same components for the various crystal classes as do the dielectric constants shown by equation (134). The pyroelectric tensor pn and /?'„ are tensors of the first rank and in general will have three components pi , p2 , and Ps . In a similar manner to that used for second, third and fourth rank tensors it can be shown that the various crystal classes have the following comi)onents for the first rank tensor />,. . FIEZOELECTKIC CRYSTALS IN TENSOR FORM 129 Class 1 : components pi , p-i , ps ■ Class 3 : I' axis of binary symmetry, components 0, p2 ,0 (158) Class 4: components pi , 0, ps . Classes 7, 10, 14, 16, 20, 23, and 26: components 0, 0, pz ; and Classes 2, 5, 6, 8, 9, 11, 12, 13, 15, 17, 18, 19, 21, 22, 24, 25, 27, 28, 29, 30, 31, and 32: components 0, 0, 0, i.e., /> = 0. For a hydrostatic pressure, the stress tensor has the components Tn = T22= Tss^ —p = pressure; T12 = Tn = ^23 = (159) Hence the displacement equations of (156) can be written in the form K = '4^ Em- <^np + pldQ (160) where <^np = dnlJn + d n22T22 + <^n33?'3.3 that is the contracted tensor d nkkTkk ■ This is a tensor of the tirst rank which has the same components as the pyroelectric tensor pn for the various cPv'stal classes. 7. Second Order Effects in Piezoelectric Crystals We have so far considered only the conditions for which the stresses and tields are linear functions of the strains and electric displacements. A number of second order effects exist when we consider that the relations are not linear. Such relations are of some interest in ferroelectric crystals such as Rochelle salt. A ferroelectric crystal is one in which a spontaneous polarization exists over certain temperature ranges due to a cooperative effect in the crystal which lines up all of the elementary dipoles in a given "domain" all in one direction. Since a spontaneous polarization occurs in such crj'stals it is more advantageous to develop the equations in terms of the electric displacement rather than the external field. Also heat effects are not prominent in second order effects so that we develop the strains and potentials in terms of the stresses and electric displacements D. By means of McLaurin's theorem the first and second order terms are in tensoi form _ dSij dSij 1 r d'^Sij ^'' ~ dTkf ^'^ ^ a5„ ^" + 21 IdTkCdT^r ^'^^'^ + 2 „„ „j TkCdn + rr^r 6„5o + ■ • • higher terms d'E„ (161) dTktdTn TklT^r d^Em d^E„, 1 + 2 ^^T^T ^i<^^" + ^777 5„5o + • • • higher terms dTktdSn d5„d5o whereas before 8 = D/4ir 130 BELL S YSTEM TECH NIC A L JOURNA L In this equation the linear partial differentials have already been discussed and are given by the equations dSij y dSij dEn dEm T where s^nkt are the elastic compliances of the crystal at constant displace- ment, gijn the piezoelectric constants relating strain to electric displacement /At, and /3l„ the dielectric "impermeability" tensor measured at constant stress. We designate the partial derivatives dTddT,r ^'''^''■' dT,m„, dn^dT^r y'^" d Sij _ d'En _ ^D . d'Em __(^D ddnddo dTijdSo dSndSo (163) The tensors N, M, Q, and are respectively tensors of rank 6, 5, 4 and 3 whose interpretation is . discussed below. Introducing these definitions equations (161) can be written in the form Em = Tkflgmkf.+ h^^ikfnTqr + Qkfmn^,] + SnlATT^mn + ^Omno^ Written in this form the interpretation of the second order terms is obvious. N'ijkfgr represents the nonlinear changes in the elastic compliances s^jj caused by the application of stress Tgr . Since the product of N nklqrTqr represents a contracted fourth rank tensor, there is a correction term for each elastic compliance. The tensor M'^jkfn can represent either the non- linear correction to the elastic compliances due to an applied electric dis- placement Dn or it can represent the correction to the piezoelectric constant gijn due to the stresses Tk( . By virtue of the second equation of (162), the second equivalence of (163) results. The fourth rank tensor ^Qnno represents the electrostrictive effect in a crystal" for it determines the strains existing in a crystal which are proportional to the square of the electric displacement. Twice the value of the electrostrictive tensor ^Q^j„o , which appears in the second equation of (164) can be interpreted as the change in the inverse dielectric constant or "impermeability" constant. Since a change in dielectric constant with applied stress causes a double refraction of light through the crystal, this term is the source of the piezo-optical effect in crystals. The third rank tensor Omno represents the change in the "im- permeability" constant due to an electric field and hence is the source of the electro-optical effect in crystals. These equations can also be used to discuss the changes that occur in ferroelectric type crystals such as Rochelle Salt when a spontaneous polariza- PIEZOELECTRIC CR YSTA LS IN TENSOR FORM 131 tion occurs in the crystal. When spontaneous polarization occurs, the dipoles of the crystal are Uned up in one direction in a given domain. For Rochelle salt this direction is the ±X axis of the crystal. Now the electric displacement Dz is equal to 47r 47r 47r = f:^^ = f:f + P,„ + P,^ = ^o p,^ (165) where Px^ is the electronic and atomic polarization, and Px^ the dipole polarization The electronic and atomic polarization is determined by the field and hence can be combined with the field through the dielectric constant eo , which is the temperature independent part of the dielectric constant. When the crystal becomes spontaneously polarized, a field E^ will result, but this soon is neutralized by the flow of electrons through the surface and volume conductance of the crystal and in a short time Ez = 0. Hence for any permanent changes occurring in the crystal we can set 8x = — =PxD = dipole polarization (166) 47r which we will write hereafter as Pi . In the absence of external stresses the direct effects of spontaneous polari- zation are a spontaneous set of strains introduced by the product of the spontaneous polarization by the piezoelectric constant, and another set produced by the square of the polarization times the appropriate electro- strictive components. For example, Rochelle salt has a spontaneous polarization Pi along the Xi axis between the temperatures — 18°C to + 24°C. The curve for the spontaneous polarization as a function of temperature is shown by Fig. 6. The only piezoelectric constant causing a spontaneous strain will be ^14/2 = gnz • Hence the spontaneous polariza- tion causes a spontaneous shearing strain S, = guPz = 120 X 10"' X 760 = 9.1 X 10~* (167^ if we introduce the experimentally determined values. Since .5'4 is the cosine of 90° plus the angle of distortion, this would indicate that the right angled axes of a rhombic system would be distorted 3.1 minutes of arc. This is the value that should hold for any domain. For a crystal with several domains, the resulting distortion will be partly annulled by the different signs of the polarization and should be smaller. Mueller measured an angle of 3'45" at 0°C for one crystal. This question has also been * This has been measured by measuring the remanent polarization, when ail the domains are lined up. See "The Dielectric Anomalies of Rochelle Salt," H. Mueller, Annals of the N. Y. Acad. Science, Vol. XL, Art. 5, page 338, Dec. 31, 1940. ^ "Properties of Rochelle Salt," H. Mueller, Phys. Rev., Vol. 57, No. 9, May 1, 1940. 132 BELL SYSTEM TECH NIC A L JOURNAL investigated by the writer and Miss E. J. Armstrong by measuring the temperature expansion coefficients of the Y and Z axes and comparing their average with the expansion coefficient at 45° from these two axes. The difference between these two expansion coefficients measures the change in angle between the Y and Z axes caused by the spontaneous shearing strains. The results are shown by Fig. 7. Above and below the ferro- electric region, the expansion of the 45° crystal coincides with the average expansion of the Y and Z axes measured from 25°C as a reference tempera- ture. Between the Curie temperatures a difference occui^ indicating thai the Y and Z crystallographic axes are no longer at right angles. The dif- ference in expansion per unit length at 0°C (ihe maximum point) corresponds to 1.6 X 10"* cm per cm. This represents an axis d istortion of 1 .1 minutes 700 600 t3 No <5 500 -II- Q.O a. m 400 DO ^ai 300 0<J 200 O- 7 100 -20 -16 -12 -8-4 4 8 12 16 20 24 28 TEMPERATURE IN DEGREES CENTIGRADE Fig. 6.- — Spontaneous polarization in Rochelie Salt along the X axis. of arc. Correspondingly smaller values are found at other temperatures in agreement with the smaller spontaneous polarization at other tempera- tures. It was also found that practically the same curve resulted for either 45° axis, indicating that the mechanical bias put on by the optometer used for measuring expansions introduced a bias determining the direction of the largest number of domains. The second order terms caused by the square of the spontaneous polariza- tion is given by the expression S,i = QlnP\ (168) Since Q is a fourth rank tensor the possible terms for an orthorhombic bisphenoidal crystal (the class for Rochelie salt) are 5u = QinxPl ; ^22 = Q2inP\ ; ^33 = QunPl (169) PIEZOELECTRIC CRYSTALS IN TENSOR FORM 133 In an effort to measure these effects, careful measurements have been made of the temperature expansions of the three axes X, Y and Z. The results are shown by Table II. On account of the small change in dimension from (lO-* O -16 at -18 .' ' / / / / f J i V > y / V J f 1 > V ^ } • M.n' OF ARC J • r / ^ I • • ^ t A r / f X / y Y' r -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 TEMPERATURE IN DEGREES CENTIGRADE Fig. 7. — Temperature expansion curve along an axis 45° between Y and Z as a function of temperature. the Standard curve it is difiScult to pick out the spontaneous components by plotting a cur\-e. By expressing the expansion in the form of the equation AL ^ = ai(r-25) + 02(^-25)' + ^3(^-25)' (170) 134 BELL SYSTEM TECHNICAL JOURNAL Table II Measured Temperature Expansions for the Three Crystalographic .\xes Temperature Expansion Temperature in °C. Expansion X lO-i Y Axis Temperature in °C. Expansion in °C. X io-« a: Axis X 10-« Z Axis 39.6 38.7 35.2 10.2 9.46 6.96 + 35.0 30.3 25.25 4.45 2.5 0.2 -1,34.9 29.9 25.05 +4.9 2.5 + .05 30.2 27.2 26.2 3.63 1.41 0.71 23.9 22.9 19.35 -0.42 -0.88 -2.4 24.0 19.95 14.95 -.5 -2.62 -5.11 25.15 24.0 23.0 0.06 -0.71 -1.39 14.9 10.0 5.4 -4.25 -6.25 -8.18 +9.75 +4.9 -7.55 -9.9 -12.31 21.8 16.0 15.2 -2.37 -6.5 -7.05 +0.3 -9.7 -16.3 -10.15 -13.98 -16.41 -6.35 -10.5 -15.0 -15.3 -17.29 -19.42 4.9 +0.3 -4.7 -14.12 -17.28 -20.3 -20.85 -25.1 -30.3 -17.94 -19.22 —20.8 -18.0 -23.2 -25.1 -20.86 -23.08 -23.96 -10.7 -15.3 -20.7 -24.0 -26.6 -30.2 -35.0 -39.7 -53.2 -22.23 -23.54 -27.60 -31.1 -35.0 -40.0 -26.59 -28.28 -30.4 -25.7 -30.1 -34.7 -32.7 -35.2 -37.85 -40.7 -45.0 -50.5 -41.25 -44.0 -47.0 and evaluating the constants by employing temperatures outside of the ferroelectric range, a normal curve was established. For the X, Y, and Z axes these relations are AL lO/T, -i-\3 AL = 69.6 X 10""'(r-25) + 7.4 X 10""(T-25)' - 3.13 X 10 "'(T-25) {X direction) = 43.7 X 10~*(T-25) + 8.16 X 10''(T-25)' - 3.60 X 10~'''(T-25)' (I' direction) = 49.4 X 10~'(r-25) + 1.555 X 10"'(r-25)' - 2.34 X nr'\T-25) {Z direction) (i7i; PIEZOELECTRIC CRYSTALS IN TENSOR FORM 135 The difference between the normal curves and the measured values in the Curie region is shown plotted by the points of Fig. 8. The solid and dashed curves represent curves proportional to the square of the spontaneous polarization and with multiplying constants adjusted to give the best fits for the measured points. These give values of Qim , Qizn , Qasn equal to Qnu = -86.5 X 10-^^'; Q^,u = +17.3 X 10~''; Q^^n=-2A.2xm-'' (172) ^^^^^ Another effect noted for Rochelle salt is that some of the elastic constants suddenly change by small amounts at the Curie temperatures. This is a consequence of the tensor Mfy^^,,, for if a spontaneous polarization P 5 -15 o -40 A ---- '\ r A --^^ ^S22 ^ ^^ >--'' A ^-r^ -^ '\ ^ .n"'''^ ^ "^-^ D D %>' '''' 1 \ "*~^ 1 1 —''■ n / \ / o \ / \ / \| / \o / \ / °X / -4 4 8 12 16 TEMPERATURE IN DEGREES CENTIGRADE Fig. 8. — Spontaneous electrostrictive strain in Rochelle Salt along the three crystallographic axes. occurs, a sudden change occurs in some of the elastic constants as can be seen from the first of equations (164). The second equation of (164) shows that this same tensor causes a nonlinear response in the piezoelectric con- stant. Since a change in the elastic constant is much more easily deter- mined than a nonlinear change in the piezoelectric constant, the first effect is the only one definitely determined experimentally. Since all three crys- tallographic axes are binary axes in Rochelle salt, it is easily shown that the only terms that can exist for a fifth rank tensor are terms of the types Mxxm ; Mf2223 ; iWf2333 (173) with permutations and combinations of the indices. Hence when a spon- taneous polarization l\ occurs, the elastic constants become s%kt - MtikdPx (174) 136 BELL SYSTEM TECHNICAL JOURNAL Comparing these with the relation of (90) we see that the spontaneous polarization has added the elastic constants D {Minn + Minii + Mnni + Mz2ni)Pi (175) 014 - 2 <r" - (Af 22231 + M2232I + M2322I + 3/32221) -fl J24 - 2 V (M^3331 + Mf2331 + ^33321 + MiZ2ii)Pi Sb6 — (iWf21bl + -M'f32U + M3II2I + M312II + Mf2i3i + Mf23ii + Mnni + A/^i3ii)/^i between the two Curie points. Hence while the spontaneous polarization Pi exists, the resulting elastic constants are ^11 , 5l2, 5l3 , Sh , , •^12 , 522, 523, 524 , , •^13 , ■^23 , 533 , 534 , , •^14 , ■^24 , Sm , 544 , , , , , , 555 , 5&6 , , , , 556 , 566 (176) Comparing this to equation (139) which shows the possible elastic constants for the various crystal classes, we see that between the two Curie points, the crystal is equivalent to a monoclinic sphenoidal crystal (Class 3) with the X axis the binary axis. Outside the Curie region the crystal becomes orthorhombic bisphenoidal. This interpretation agrees with the tempera- ture expansion curves of Fig. 7. The sudden appearance of the polarization 1\ will affect the frequency of a 45° ,Y-cut crystal, for with a crystal cut normal to the .Y axis and with the length of the crystal at an angle B with the Y axis, the value of the elastic compliance 522 along the length is 522' = 5^2 cos* G -f 25^4 cos^ B sin B + (25^3 + 54*4) sin B cos B (177) + 2^34 sin B cos B + Sn sin B Hence for a crystal with its length 45° between the Y and Z axes, elastic compliance becomes '« _ 522 ~1~ 2(524 ~1~ 523 + 534) + ^44 + 533 S21 — (178) PIEZOELECTRIC CRYSTALS IN TENSOR FORM 137 For a 45*^ X-cut crystal we would expect a sudden change in the value of 522 as the crystal becomes spontaneously polarized between the two Curie points due to the addition of the s^i and s^^ elastic compliances. Such a change has been observed for Rochelle salt* as shown by Fig. 9 which shows the frequency constant of a nonplated crystal for which the elastic com- pliances s^j should hold. uj 217 5 Z 216 UJ u QC 215 O 209 2 208 q 207 \^ \ ^ *.^ ^ X ■-^ /^l _Q_ 1 ^ \ "x" *x^ 1 \ \ ^ >> N^ ^^^ 1 1 \ V ^ V_ 1 1 \ 1 1 ^*> ^ \ \ \ \ ^_^ , , ">s^ N \ \ \ ^/' ^-' N \ , V -n\ V ,^_y V \\ \ u. o 6 Hi 3 4 3 2 1 -20 -16 -12 -8-4 4 8 12 16 20 24 28 32 TEMPERATURE IN DEGREES CENTIGRADE 36 40 44 48 Fig. 9. — Frequency constant and Q of an unplated 45°X cut Rochelle Salt ctystal plotted as a function of temperature. Hence the sudden change in the elastic constant is a result of the two second order terms s^ -f s^i , which are caused by the spontaneous polariza- tion. The value of the sum of these two terms at the mean temperature of the Curie range, 3°C is •^24 + ^34 = 4.1 X 10 cm"/ dyne (179) Crystals cut normal to the Y and Z axes should not show a spontaneous change in their frequency characteristic since the spontaneous terms Su , 524 , 534 and 5b6 do not affect the value of Young's modulii in planes normal to Y and Z. Experiments on a 45° F-cut Rochelle salt crystal do not show a spontaneous change in frequency at the Curie temperature, although there is a large change in the temperature coefficient of the elastic compliance between the two Curie points. This is the result of third order term and is ' "The Location of Hysteresis Phenomena in Rochelle Salt Crystals," W. P. Mason, Phys. Rev., Vol. 50, p. 744-750, October 15, 1940. gn ^12 gl3 gl4 138 BELL S YSTEM TECH NIC A L JOURNA L not considered here. The spontaneous ^ae constant affects the shear con- stant ^66 for crystals rotated about the A' axis and could be detected experi- mentally. No experimental values have been obtained. The effects of spontaneous polarization in the second equation of (164) are of two sorts. For an unplated crystal, a spontaneous voltage is gen- erated on the surface, which, however, quickly leaks off due to the surface and volume leakage of the crystal. The other effects are that the spon- taneous polarization introduces new piezoelectric constants through the tensor Qkfmn , changes the dielectric constants through the tensor Omno and introduces a stress effect on the piezoelectric constants through the tensor Mkfmqr ■ Siuce piezoelectric constants are not as accurately measured as elastic constants, the first effect has not been observed. The additional piezoelectric constants introduced by the tensor Qkfmn are shown by equa- tion (180) g2, g26 (180) Since the only constants for the Rochelle salt class, the orthorhombic bisphenoidal, are gu , g2b , gse , this shows that between the two Curie points the crystal becomes monoclinic sphenoidal, with the A' axis being the binary axis. The added constants are, however, so small that the accuracy of measurement is not sufficient to evaluate them. From the expansion measurements of equation (172) and the spontaneous polarization values, three of them should have maximum values of gn = -6.6 X 10"^ gu = +1.3 X 10"'; gn = -1.8 X 10"' (181) These amount to only 6 per cent of the constant gu , and hence they are not easily evaluated from piezoelectric measurements. The effect of the tensor Omuo is to introduce a spontaneous dielectric constant €23 between the Curie temperatures so that the dielectric tensor becomes en, , 0, e,,, 623 (182) , €23 , f33 As discussed at length by Mueller'"* this introduces a spontaneous bire- fringence for light passing through the crystal along the A', 1' and Z axes which adds to the birefringence already present. « "Proi)crtics of Rochcilc Salt I and IV," Phvs. Rev. 47, 175 (1935); 58, 805 November 1, 1940. i The Biased Ideal Rectifier By W. R. BENNETT Methods of solution and specific results are given for the spectrum of the response of devices which have sharply defined transitions between conducting and non-conducting regions in their characteristics. The input wave consists of one or more sinusoidal components and the operating point is adjusted by bias, which may either be independently applied or produced bv the rectified output itself. Introduction THE concept of an ideal rectifier gives a useful approximation for the analysis of many kinds of communication circuits. An ideal rectifier conducts in only one direction, and by use of a suitable bias may have the critical value of input separating non-conduction from, conduction shifted to any arbitrary value, as illustrated in Fig. 1. A curve similar to Fig. 1 might represent for example the current versus voltage relation of a biased diode. By superposing appropriate rectifying and linear characteristics with different conducting directions and values of bias, we may approximate the characteristic of an ideal limiter. Fig. 2, which gives constant response when the input voltage falls outside a given range. Such a curve might approximate the relationship between flux and magnetizing force in certain ferromagnetic materials, or the output current versus Signal voltage in a negative-feedback amplifier. The abrupt transitions from non-conducting to conducting regions shown are not realizable in physical circuits, but the actual characteristics obtained in many devices are much sharper than can be represented adequately by a small number of terms in a power series or in fact by any very simple analytic function expressible in a reasonably small number of terms valid for both the non-conducting and conducting regions. In the typical communication problem the input is a signal which may be expressed in terms of one or more sinusoidal components. The output of the rectifier consists of modified segments of the original resultant of the individual components separated by regions in which the wave is zero or constant. We are not so much interested in the actual wave form of these chopped-up portions, which would be very easy to compute, as in the fre- quency spectrum. The reason for this is that the rectifier or limiter is usually followed by a frequency-selective circuit, which delivers a smoothly varying function of time. Knowing the spectrum of the chopped input to the selective network and the steady-state response as a function of 139 140 BELL SYSTEM TECHNICAL JOURNAL BIAS APPLIED VOLTAGE Fig. 1. — Ideal biased linear rectifier characteristic. (1) LINEAR CHARACTERISTIC (2) BIASED POSITIVE RECTIFIER (3) BIASED NEGATIVE RECTIFIER bi ^ (4) BIASED IDEAL LIMITER I'ig. 2. — Synthesis of liniiter characteristic. THE BIASED IDEAL RECTIFIER 141 frequency of the network, we can calculate the output wave, which is the one having most practical importance. The frequency selectivity may in many cases be an inherent part of the rectifying or limiting action so that discrete separation of the non-linear and linear features may not actually be possible, but even then independent treatment of the two processes often yields valuable information. The formulation of the analytical problem is very simple. The standard theory of Fourier series may be used to obtain expressions for the amplitudes of the harmonics in the rectifier output in the case of a single applied fre- quency, or for the amplitudes of combination tones in the output when two or more frequencies are applied. These expressions are definite integrals involving nothing more compUcated than trigonometric functions and the functions defining the conducting law of the rectifier. If we were content to make calculations from these integrals directly by numerical or mechanical methods, the complete solutions could readily be written down for a variety of cases covering most communication needs, and straightforward though often laborious computations could then be based on these to accumulate eventually a suflficient volume of data to make further calculations un- necessary. Such a procedure however falls short of being satisfactory to those who would like to know more about the functions defined by these integrals without making extensive numerical calculations. A question of consider- able interest is that of determining under what conditions the integrals may be evaluated in terms of tabulated functions or in terms of any other func- tions about which something is already known. Information of this sort would at least save numerical computing and could be a valuable aid in studying the more general aspects of the communication system of which the rectifier may be only one part. It is the purpose of this paper to present some of these relationships that have been worked out over a considerable period of time. These results have been found useful in a variety of prob- lems, such as distortion and cross-modulation in overloaded ampUfiers, the performance of modulators and detectors, and efifects of saturation in magnetic materials. It is hoped that their publication will not only make them available to more people, but also stimulate further investigations of the functions encountered in biased rectifier problems. The general forms of the integrals defining the amplitudes of harmonics and side frequencies when one or two frequencies are applied to a biased rectifier are written down in Section I. These results are based on the standard theory of Fourier series in one or more variables. Some general relationships between positive and negative bias, and between limiters and biased rectifiers are also set down for further reference. Some discussion is given of the modifications necessary when reactive elements are used in the circuit. 142 BELL SYSTEM TECHNICAL JOURNAL Section 11 summarizes specific results on the single-frequency biased rectitier case. The general expression for the amplitude of the -typical harmonic is evaluated in terms of a hypergeometric function for the power law case with arbitrary exponent. Section HI takes up the evaluation of the two-frequency modulation products. It is found that the integer-power-law case Tan be expressed in finite form in terms of complete elliptic integrals of the first, second, and third kind for almost all products. Of these the first two are available in tables, directly, and the third can be expressed in terms of incomplete integrals of the first and second kinds, of which tables also exist. No direct tabulation of the complete elliptic integrals of the third kind encountered here is known to the author. They are of the hyperbolic type in contrast to the circular ones more usual in dynamical problems. Imaginary values of the angle /3 would be required in the recently published table by Heuman . A few of the product amplitudes depend on an integral which has not been reduced to elliptic form, and which is a transcendental function of two variables about which little is known. Graphs calculated by numerical integration are included. The expressions in terms of elliptic integrals, while finite for any product, show a rather disturbing complexity when compared with the original integrals from which they are derived. It appears that elliptic functions are not the most natural ones in which the solution to our problem can be expressed. If we did not have the elliptic tables available, we would prefer to define new functions from our integrals directly, and the study of such functions might be an interesting' and fruitful mathematical exercise. Solutions for more than two frequencies are theoretically possible by the same methods, although an increase of complexity occurs as the first few components are added. When the number of components becomes very large, however, limiting conditions may be evaluated which reduce the problem to a manageable simplicity again. The case of an infinite number of components uniformly spaced along an appropriate frequency range has been used successfully as a representation of a noise wave, and the detected output from signal and noise inputs thus evaluated . The noise problerri will not be treated in the present paper. I. The General Problem Let the biased rectifier characteristic, Fig. 1, be expressed by / 0, E < b\ I = (1.1) \f{E -b), b < eJ 1 Carl Heuman, Tables of Comi)letc Ellii)tic Integrals, Jour. Math, and Phvsics, Vol. XX, No. 2, pp. 127-206, April, 1941. . ^ W. R. Hcnnctt, Response of a Linear Rectifier to Signal and Noise, Jour. Acous. Soc. Amer., Vol. 15, pp. 164-172, Jan. 1944. THE BIASED IDEA L RECTIFIER 143 Then if a single frequency wave defined by E = P cos pt, - P < b < P, (1.2) is applied as input, the output contains only the tips of the wave, as shown in Fig. 3. It is convenient to place the restrictions on P and b given in Eq. (1.2). The sign of P is taken as positive since a change of phase may be introduced merely by shifting the origin of time and is of trivial interest. If the bias b were less than —P, the complete wave would fall in the con- ducting region and there would be no rectification. If b were greater than ,-«-Pcos pt Fig. 3. — Response of biased rectifier to single-frequencj' wave. P, the output would be completely suppressed. Applying the theory of Fourier series to (1.1) and (1.2), we have the results Oo 2 r a„ = - If Jo 2 n=l arc cos h/P -\- Zli (^n COS n pt f(P COS X — b) cos nx dx (1.3) (1.4) When two frequencies are applied, the output may be represented by a double Fourier series. The typical coefficient may be found by the method explained in an earlier paper by the author^. The problem is to obtain the double Fourier series expansion in x and y of the function g{x,y) defined by: /O, P cos x -\- Q cos y < b \ Six, V) = (1.5) \f{P cos -T + () COS y — b), b < P cos .v + Q cos v/ ' W. R. Bennett, New Results in the Calculation of Modulation Products, B. 5. T./., Vol. XII, pp. 228-243, April, 1933. 144 BELL SYSTEM TECHNICAL JOURNAL We substitute the special values x = pt,y = qt after obtaining the expansion. Let ^1 = Q/P, h = -b/P (1.6) The most general conditions of interest are comprised in the ranges: 0<y^i<l, -2<^o<2'* (1.7) To P J /; \ \ CASE 1 1 n -TT "2 / ^ 1 ° 2 \ CASE n TT \ V X— >. /case hi / \: / Fig. 4. — Regions in x3'-plane bounded by ^o + cos x ■\- k\ cos )» = 0. The regions in the x^-plane in which g{x,y) does not vanish are bounded by the various branches of the curve : ^0 + cos :v + ^1 COS T = (1 .8) We need to consider only one period rectangle bounded by x = ±x, y = zLir, since the function repeats itself at intervals of lir in both x and y. The shape of the curve (1.8) within this rectangle may have three forms, which are depicted in Fig. 4. In Case I, ^o -\- ki > k, ko — ki < 1, the curve divides into four branches which are open at both ends of the x- and y-axes. In Case (2), )^o + ^i <1, ^o — ^i > —1, the curve has two branches open THE BIASED IDEAL RECTIFIER 145 at the ends of the y-axis. In Case (3), —1 < ^o + ^i < 1, ^o — ^i < —1, a single closed curve is obtained. The limits of integration must be chosen to fit the proper case. The Fourier series expansion of g{x,y) may be written : 00 00 g(^) y) = zL ^ O'mn COS mx cos ny (1.9) where amn is found from integrals of the form: A = -^^ / dy I j{P cos X -\- Q cos y — b) cos mx cos ny dx (1.10) Here, as usual, «„ is Neumann's discontinuous factor equal to two when m is not zero and unity when m is zero. The values of the limits for the dif- ferent cases are : Case I, flmn = Ai-\- A2 ({xi = 0, X2 = arc cos (—^0 — ki cos y) 1-/^0 I (^-^^^ yi = arc cos — , y2 — tt (ari = 0, :i:2 = X 1 _ ^j, I (1.12) yi = 0, ^2 == arc cos — — Xo = arc cos ( — ^0 ~ ^1 cos y y2 = TT X2 = arc cos (—^0 — ^1 cos y) (1.13) y2 = arc cos {-'^) (1.14) For a considerable variety of rectifier functions/, the inner integration may be performed at once leaving the final calculation in terms of a single definite integral. A somewhat different point of view is furnished by evaluating the integral (1.4) for the biased single-frequency harmonic amplitude, and then replacing the bias by a constant plus a sine wave having the second frequency. When each harmonic of the first frequency is in turn expanded in a Fourier series 146 BELL 5VSTEM TECHNICAL JOURNAL in the second frequency, the two-frequency modulation coefficients are ob- tained. Some early calculations carried out graphically in this way are the source of the curves plotted in Figs. 18 to 21 inclusive, for which I am indebted to Dr. E. Peterson. If reactive elements are used in the rectifier circuit, the voltage across the rectifying element may depart from the input wave shape applied to the complete network. The solution then loses its explicit nature since the rectifier current is expressed in terms of input voltage components which in turn depend on voltage drops produced in the remainder of the network by the rectifier currents. Practical solutions can be worked out when relatively few components are important. n In+ Ii BIASED RECTIFItR UNIT E-InR effective: bias on Fig. 5. — Biased rectifier in series with RC network. As an example consider the familiar case of a parallel combination of resistance R and capacitance C in series with the biased rectifier, Fig. 5. If C has negligible impedance at all frequencies of importance in the rectifier circuit except zero, we may assume that the voltage across R is constant and equal to loR, where /o is the d-c. component of the rectifier current. The voltage across the rectifier unit is then E — loR- The effect is a change in the value of bias from b io b -\- IqR. If the d-c component in the output is calculated for bias b + IqR, we obtain the value of /o in terms of 6 -f- IqR, an implicit equation defining Io- If this equation can be solved for /n, the bias b + !oR can then be determined and the remaining modulation products calculated. A more imj)ortant case is that of the so-called envelope detector, in which the imjjcdance of the condenser is very small at all frequencies contained in the input signal, but is very large at frequencies comparable with the band width of the s[)cctrum of the input signal. These are the usual conditions prevailing in the detection oi audio or video signals from modulated r-f or i-f waves. The sf)lution dei)en(ls on writing the input signal in the form of a slowly varying positive valued envelope function multiplying a rapidly THE BIASED IDEAL RECTIFIER 147 oscillating cosine function. That is, if the input signal can be repre- sented as E= A (0 COS0 (/), (1.15) where .1 (/) is never negative and has a spectrum confined to the frequency range in which lirfC is negligibly small compared with 1/7?, while cos 0(/) has a spectrum confined to the frequency range in which \/R is negligibly small compared with 2irfC, we divide the components in the detector output into two groups, viz.: 1. A low-frequency group /;/ containing all the frequencies comparable with those in the spectrum of .1 (/). The components of this group flow through R. 2. A high-frequency group Ihf containing all the frequencies comparable to and greater than those in the spectrum of cos (f) (/). The components of this group flow through C and produce no voltage across R. The instantaneous voltage drop across R is therefore equal to Ii/R, and hence the bias on the rectifier is 6 + Ii/R. If .1 and </> were constants, we could make use of (1.3) and (1.4) to write: .arc cos [(b+Ii/R)/A] I If + hf = :^" + 2 <^n cos nd (1.16) rt pare COS 1(0-1-1 If a )i A i On = - I f{A cos X — b — IifR) cos nx dx (1.17) TV Jo If .4 and (f) are variable, the equation still holds provided Ii/R < .1 at all times. Assuming the latter to be true (keeping in mind the necessity of checking the assumption when /;/ is found), we note that terms of the form fln cos n d consist of high frequencies modulated by low frequencies and hence ; the main portion of their spectra must be in the high-frequency range. I Hence we must have as a good approximation when the envelope frequencies ii are well separated from the intermediate frequencies, \ ■> /«arc cos [(6+/;/K)/4] 1 hf = ^ = - \ f{A cosx - b - IifR)dx (1.18) 1 I TT Jq jl This equation defines /;/ as a function of A, and if it is found that the ! condition b -\- IifR<\ is satisfied by the resulting value of Ii/, the problem j is solved. If the condition is not satisfied, a more complicated situation , exists requiring separate consideration of the regions in which b + Ii/R < A ' and 6 -f IifR > A . I To be specific, consider the case of a linear rectifier wnth forward con- ductance a = l/R, and write V — Ij/R. Then '^'V ^ Va - {b a- Vy- - (b A- V) arc cos ^-^tZ (1.19) XV A 148 BELL SYSTEM TECHNICAL JOURNAL When 6 = (the case of no added bias), this equation may be satisfied by setting V = cA,() <c < 1, (1.20) which leads to R yd' 1 — arc cos c, ^ (1-21) defining c as a function of Ro/R- The value of c approaches unity when the ratio of rectifier resistance to load resistance approaches zero and falls off to zero as Ro/R becomes large. The curve may be found plotted else- where . This result justifies the designation of this circuit as an envelope detector since with the proper choice of circuit parameters the output voltage is proportional to the envelope of the input signal. The equations have been given here in terms of the actual voltage applied to the circuit. The results may also be used when the signal generator contains an internal impedance. For example, a nonreactive source inde- pendent of frequency may be combined with the rectifying element to give a new resultant characteristic. If the source impedance is a constant pure resistance tq throughout the frequency range of the signal input but is negligibly small at the frequencies of other components of appreciable size flowing in the detector, we assume the voltage drop in ro is roCi cos (/). We then set n — 1 in (1.17) and replace ai by {Aq — A)/rQ, where Aq is the voltage of the source. The value of lu in terms of A from (1.18) is then substituted, giving an implicit relation between A and Ao . A further noteworthy fact that may be deduced is the relationship be- tween the envelope and the linearly rectified output. By straightforward Fourier series expansion, the positive lobes of the wave (1.15), may be written as: (E, £>0\ p £r - =-4(/) - + ' cos 4>{t) \ 0, E <0 / TT 2 2 Y^ ( — )"* cos 2m 0(/) (1.22) 7rm=i 4w2 — 1 Hence if we represent the low-frequency components of Er by Ei/, we have: £,/ = ^ (1.23) IT or A (/) = wE,f (1.24) * See, for example, the top curve of Fig. 9-25, p. 311, H. J. Reich, Theory and AppHca- tions of Electron Tubes, McGraw-Hill, 1944. THE BIASED IDEAL RECTIFIER 149 Equation (1.23) expresses the fact that we may calculate the signal com- ponent in the output of a half-wave linear rectifier by taking I/tt times the envelope. Equation (1.24) shows that we may calculate the response of an envelope detector by taking t times the low-frequency part of the Fourier series expansion of the linearly rectified input. Thus two procedures are in general available for either the envelope detector or linear rectifier solution, and in specific cases a saving of labor is possible by a proper choice between the two methods. The final result is of course the same, although there may be some difficulty in recognizing the equivalence. For example, the solution for linear rectification of a two-frequency wave P cos pt -^ Q cos qt was given by the author in 1933', while the solution for the envelope was given by Butterworth in 1929^ Comparing the two expressions for the direct-current component, we have: - 2P o Elf = -y[2E — (1 — k") K], where K and £are complete elliptic integrals of the first and second kinds with modulus k = Q/P — 2P A {t) = — (1 + k) El, where Ei is a complete elliptic intregal of the TT second kind with modulus ki = 2 \/k/{\ + k). Equation (1.24) implies the existence of the identity (1 +k)Ei^ 2E- {\ - k') K (1.25) The identity can be demonstrated by making use of Landen's transforma- tion in the theory of elliptic integrals. 2. Single-Frequency Signal The expression for the harmonic amplitudes in the output of the rectifier can be expressed in a particularly compact form when the conducting part of the characteristic can be described by a power law with arbitrary ex- ponent. Thus in (1.4) if /(c) = az\ we set X = b/P and get •arc cos X 2 73" /<arc cos a aP i , ^ y , I fln = / (cos X — A) COS nx ax TT Jo ! 2^T{p + DaPW - X)"^^ I * S. Butterworth, Apparent Demodulation of a Weak Station by a Stronger One Experimental Wireless, Vol. 6, pp. 619-621, Nov. 1929. 150 BELL SYSTEM TECHNICAL JOURNAL The equation holds for all real values of v greater than —1. The symbol F represents the Gaussian hypergeometric function*: f (a, 6; .; .) = , + "*. + °(" + D ^^^ + D ,.+ ... (2.2) c 1! c(c + 1) 2! The derivation of (2.1) requires a rather long succession of substitutions, expansions, and rearrangements, which will be omitted here. When V is an integer, the hypergeometric function may be expressed in finite algebraic form, either by performing the integration directly, or by making use of the formulas: F{yi/2, — n/2; 1/2; z) — cos (^i arc sin z), (2.3) sin (fi arc sin z) .(i±-M^-i-.0 HZ together with recurrence formulas for the f'-f unction. When p is an odd multiple of one half, the /-'-function may be expressed in terms of complete elliptic integrals of the first and second kind with modulus [(1 — X)/2] " by means of the relations, F(hh;i;k') =-K, IT F{-h^;^;k') =-E, (2.4) and the recurrence formulas for the /''-function. For the case of zero bias, we set X = 0, and applv the formula F{a. X-a-c; 1/2) = ^J^T]^^M+Zzj\ ^ obtaining the result: We point out that the above results may be applied not only when the api)lied signal is of the form P cos pt with P and p constants, but to signals " For an account of the ])roi)crties of the hypergeometric function, see Ch. XIV of Whittaker and Watson, Modern Analysis, Cambridge, 1940. A discussion of elliptic integrals is given in ("h. XXII of the same hook. THE BIASED IDEAL RECTIFIER 151 in which F and p are variable, provided that P is always positive. We thus can apply the results to detection of an ordinary amplitude-modulated wave or to the detection of a frequency-modulated wave after it has passed through a slope circuit. A case of considerable practical interest is that of an amplitude-modulated wave detected by a diode in series with a parallel combination of resistance R and capacitance C. The value of C is assumed to be sufficiently large so that the voltage across R is equal to the ao/2 component of the current through the diode multiplied by the resistance. This is the condition for envelope detection mentioned in Part 1. The diode is assumed to follow Child's law, which gives v = 3/2. We write V _ r(5/2)(l -X^aP^'' (, ...l-A .2-. where X = V/P. Note that K is a constant equal to the direct-voltage output if P is constant. If P varies slowly with time compared with the high-frequency term cos pt, V represents the slowly varying component of the output and hence is the recovered signal. But Hh, h 3; k') = i|^ [2(2^=^ - 1)£ + (2 - 3k'){l - k')K] (2.8) where A' and E are complete elliptic integrals of the first and second kind with modulus k. Hence 37r (1 -t- 3X)(1 -f X) ^:vp = ^ = — I — ^-^^ <"' where the modulus of A' and E is \/(l — X)/2. This equation defines p as a function of X, and hence by inversion gives X as a function of p. The resulting curve of X vs. p is plotted in Fig. 6 and may be designated as the function X = g (p). If we substitute X = V/P we then have V = P g{3Tr/Ra V2P) (2.10) This enables us to plot V as a function of P, for various values of Ra, Fig. 7. Since P may represent the envelope of an amplitude-modulated (or diflf- erentiated FM) wave, and V the corresponding recovered signal output voltage, the curves of Fig. 7 give the complete performance of the circuit as an envelope detector. In general the envelope would be of form P — Po[l -f c s{l)\, where s{t) is the signal. We may substitute this value of P directly in (2.10) provided the absolute value of c s{t) never exceeds unity. 152 BELL SYSTEM TECHNICAL JOURNAL Fig. 6. — The Function X = g{f>) defined by Eq. (2.9). Fig. 7. — Performance of 3/2 — power-law rectifier as an envelope detector with low-imped- ance signal generator. To express the output in terms of a source voltage f o in series with an impedance equal to the real constant value ro at t,he signal frequency and zero at all other frequencies, we write ra ai 3C,P3/2(1 _ X)2 :^7|— /^(^f, -I;3r^-j (2.11) THE BIASED IDEAL RECTIFIER 153 or Po = (i+|^)p. (2.12) where E = 3i?a(l - X)2/'i 4\/2 = ?^ / ^ 1 -*X \ P 1 1> "ij ^' 2 / (2.13) V2P[2(1 - ife' + )fe')£ - (2 - yfe')(l - k')K]. 1.4 1.2 15 20 25 Pq in volts Fig. 8. — Performance of 3/2 — power-law rectifier as an envelope detector with impedance of signal generator low except in signal band. By combining the curves of Fig. 7 giving V in terms of P with the above equations giving the relation between P and Pq, we obtain the curves of Figs. 8, 9, 10, giving F as a function of Pq. The curves approach linearity as Ra is made large. On the assumption that the curves are actually linear, we define the conversion loss D of the detector in db in terms of the ratio of maximum power available from the source to the power delivered to the load: D = 10 log! Po/8ro vyR = 10 logi m R^ Sro (2.14) Curves of D vs r^/R are given in Figs. 11 and 12. The optimum relation between r^ and R when the forward resistance of the rectifier vanishes has long been known to be r^/R. — .5. The curves show a minimum in this 154 BELL SYSTEM TECHNICAL JOURNAL region when Ra is large. In the limit as Ra approaches infinity, we may show that the relation between f o and V approaches: (2.15) 15 20 25 P„ IN VOLTS Fig. 9. — Performance of 3/2 — power-law rectifier as an envelope detector with impedance of signal generator low except in signal band. Fig. 10. — Performance of 3/2 — power-law rectifier as an envelope detector with impedance of signal generator low except in signal l)and. The corresponding limiting formula for D is (2.16) THE BIASED IDEAL RECTIFIER 155 The minimum value of D is then found to occur at tq = R/2 and is zero db. We note from the curves that the minimum loss is 1.2 db when Ra = 10 and 0.4 when Ra = 100. This example is intended mainly as illustrative rather than as a complete tabulation of possible detector solutions. The methods employed are sufficiently general to solve a wide variety of problems, and the specific evaluation 'process included should be sufficiently indicative of the proce- dures required. Cases in which various other selective networks are asso- ciated with the detector have been treated by Wheeler^. Fig. 11. — Conversion loss of 3/2 — power-law rectifier as envelope detector with impedance of signal generator low except in signal band. m 14 o Z 12 to 10 <n 3 8 -Ra = 10 1 X — xRa=100 1 \. 27= VOLTS OUTPUT — = \ ! 1 ^ ssss* V^=i2_5_^^^ y — ^^S^^iSr^^^HO . 3 Fig. 12. — Conversion loss of 3/2 — power-law rectifier as envelope detector with impedance of signal generator low except in signal band. 3. Two-Frequency Inputs The general formula for the coefficients in the two-frequency case depends on a double integral as indicated by (1.10). In many cases one integration may be performed immediately, thereby reducing the problem to a single definite integral which may readily be evaluated by numerical or mechanical ' H. A. \Mieeler, Design Formulas for Diode Detectors, Proc. I. R. E., Vol. 26, pp. 745-780, June 1938. 156 BELL SYSTEM TECHNICAL JOURNAL means. It appears likely in most cases that the expression of these results in terms of a single integral is the most advantageous form for practical purposes, since the integrands are relatively simple, while evaluations in terms of tabulated functions, where possible, often lead to complicated terms. Numerical evaluation of the double integral is also a possible method in cases where neither integration can be performed in terms of functions suitable for calculation. One integration can always be accomplished for the integer power-law case, since the function / (P cos x -\- Q cos y — J) in (1.12) then becomes a polynomial in cos x and cos y. Cases of most practical interest are the zero-power, linear, and square-law detectors, in which /(z) is proportional to z", z , and z" respectively. The zero-power-law rectifier is also called a total limiter, since it limits on infinitesimally small amplitudes. We shall tabulate here the definite integrals for a few of the more important low-order OS pt+Qcos qt RESPONSE OF LIMITER _A_ I m\mm ■kw////////A m "^ TIME ». ^ f Fig. 13. — Response of biased total limiter to two-frequency wave. coefficients. To make the listing uniform with that of our earlier work, we express results in terms of the coefhcient Amn, which is the amplitude of the component of frequency mp ± nq. The coefl&cient Amn is half of «„„ when neither m nor n is zero. When w or » is zero,, we take Amn = a^n and drop the component with the lower value of the i sign. When both m and n are zero, we use the designation Aqq/I for ooo, the d-c term. In the tabula- tions which follow we have set/(z) = otz' with v taking the values of zero and unity. We first consider the biased zero-power-law rectifier or biased total limiter. This is the case in which the current switches from zero to a constant value under control of two frequencies and a bias as illustrated by Fig. 13. The results are applicable to saturating devices when the driving forces swing through a large range compared with the width of the linear region. It is also to be noted that the response of a zero-power-law rectifier may be regarded as the Fourier series expansion of the conductance THE BIASED IDEAL RECTIFIER 157 of a linear rectifier under control of two carrier frequencies and a bias. The results may therefore be applied to general modulator problems based on the method described by Peterson and Hussey**. We may also combine the Fourier series with proper multiplying functions to analyze switching between any arbitrary forms of characteristics. We give the results for positive values of ^o- The corresponding coefficients for —ko can be ob- tained from the relations: (3.1) -^00 ^00 Here we have used plus and minus signs as superscripts to designate co- efficients with bias +^o and — ^o respectively. We thus obtain a reduction in the number of different cases to consider, since Case III consists of nega- tive bias values only, and these can now be e'xpressed in terms of positive bias values falling in Cases I and II. It is convenient to define an angle 6 by the relations: ^ T^ ^^^-^ k,> \,h- h<\ . (Case I) \ ,h + h<\,h- k,> -\ (Case 11)/ arc cos - Zero-Power Rectifier or Total-Limiter Coefficients Setting y(2) = a in (1.10), —^ = 1 — — / arc cos (^o + ki cos y) dy. 2a r Je — = 4 f Vl - {ko + kr cos yy dy An ^ 2h r sin^ y dy a TT^ ie \/l — (^0 + ^1 cos yY — = — / cos Vl — (^0 + ^1 cos y)- dy \ {2>3) a TT^ Je — = — -^ / (^0 + ki COS y) Vl - (^0 + ki cos yy dy a TT^ J e Aw. _ 2^1 r sin^ y cos y dy a TT^ h Vl — (^0 + ^1 cos yy — - — — / (^0 + ^1 COS y) COS y Vl — (^o + ki cos y)' dy a TT^ Jft J ' E. Peterson and L. W. Hussey, Equivalent Modulator Circuits, B. S. T. J., Vol. 18, pp. 32-48, Jan. 1939. 158 BELL SYSTEM TECHNICAL JOURNAL Similarly for a linear rectifier: 1 + 2 2 Au — aP — AiQ ^01 = aQ — Aq\ Amn ^^ \ ) A mn J W + « > 1 (.3.4) We have shown in Fig. 2 how an ideal limiting characteristic, which trans- mits linearly between the upper and lower limits, may be synthesized from two biased linear rectification characteristics. Equation (3.4) shows how to calculate the corresponding modulation coefficients, when the coefficients for bias of one sign are known. The limiter characteristic is equal to az— h (2) - h (2), where /i (2) = oc z - bi, 0, z > —bi z < —bi z > bi 1 /2 (2) = a I 0, z < bxj \z + 62 The expression for/2 (zj may also be written: 'z — ( — 62), 2 > —bi 0, Z < -^2 ji (z) = a (z + 62) — a ) (3.5) (3.6) Hence the modulation coefficient A^n for the limiter may be expressed in terms of y4„,„ (61) and A^n ( — 62) as follows: (61) + {-T^^'Amn (62), m ^ n 7^ \ (3.7) A -mn — A 1 If the limiter is symmetrical {b\ = 62), the even -order products vanish and the odd orders are doubled. The terms aP, aQ are to be added to the dexter of (3.7) for .4 10, ^01 respectively. The odd Hnear-rectifier coefficients, when multiplied by two, thus give the modulation products in the output of a symmetrical limiter with maximum amplitude ^0, as may be seen by substituting fti = 62 = —^0 in (3.7). For the fundamental components aP and aQ respectively must be subtracted from twice the Aio and Aoi co- efficients for ^n- Linear Rectifier Coefficients D.C. ^00 2 /aP = ko-\- \ f [Vl - (*o + ki cos 3-)^ (3.8) — (^0 + ki cos y) arc (cos ^0 + ^1 cos y)] dy THE BIASED IDEAL RECTIFIER FXJNDAMENTALS 159 (3.9) (3.10) (3.11) (3.12) AWaP = 1 + -^ f f(^o + ^1 COS y) Vl - {h + ^1 cos yY — arc cos (^o + ^i cos y)] dy Aoi/aP = ki-^-f [Vl - (ko + ki COS yy •K^ J e — {ko + ^1 COS y) arc cos (^o + ^i cos y)] cos y dy Sum and Difference Products — Second Order ^11 = ^ / [(^0 + ki cos y) Vl - (^0 + ki cos yy — arc cos (^o "1" ki cos y)\ cos y dy Sum and Difference Products — Third Order A21 = ^ I [1 — (^0 + ki cos yYf~ cos y dy 6t~ Je The above products are the ones usually of most interest. Others can readily be obtained either by direct integration or by use of recurrence formulas. The following set of recurrence formulas were originally derived by Mr. S. O. Rice for the biased linear rectifier: 2n Amn + ^1 (« — m — 3) Am+l,n-l -{- ki (m -\- n -{- 3)Am+i,n~i + 2kon .4„,+i,„ = 2» Amn + kl (n -j- m — 3) Am-l.n+l + ^1 (w — w + 3) A „,-!,„+! + 2kon Am-\.n = 2m ki Amn -\- {m — n — 3) Am-l,n^l + (m -f n + 3)A„.+i,„+i + 2^ow ^m,„+i = 2 m h Amn + {m -]r n — 3) Am-i.n-\ -\- {m — n -\- 3)Am+l,n-l + 2^oW A^.n-l = By means of these relations, all products can be expressed in terms of .4 00, ^10, Aoi, and An. The following specific results are tabulated: .^20 = 3(^00 ~ 2kiAn ~ 2^0^10) _ 1 \ (3.14) A02 — -TT- (^1^1 no ~ 2^4 11 — 2^0 -4 01) ' 3ki ) (3.13) 160 BELL SYSTEM TECHNICAL JOURNAL 1 [ (3.15) An = jr {kiAio — yloi — ^0^11) ^30 = —^0^20 — ^1^21 1 1 (3.16) ^03 = — r (^0^02 + ^112) ki J The third-order product A21 is of considerable importance in the design of carrier ampHfiers and radio transmitters, since the (2/> — 9)-product is the cross-product of lowest order falling back in the fundamental band when overload occurs. Figure 14 shows curves of .I21 calculated by Mr. J. O. Edson from Eq. (3.12) by mechanical integration. We point out also that the Unear-rectifier coefficients give the Fourier series expansion of the admittance of a biased square-law rectifier when two frequencies are applied. We shall next discuss the problem of reduction of the integrals appearing above to a closed form in terms of tabulated elliptic integrals^. This can be done for all the coefficients above except the d-c for the zero-power law and for the d-c and two fundamentals for the linear rectifier. These contain the integral H(i^o , ^1) = / arc cos (^0 + ^1 cos y) dy (3.17) which has been calculated separately and plotted in Fig. 22. When the arc cos term is accompanied by cos wy as a multiplier with m ?^ 0, an integra- tion by parts is sufficient to reduce the integrand to a rational function of cos y and the radical \^\ — {ko + ki cos yY, which may be reduced at once to a recognizable elliptic integral by the substitution z = cos y. It is found that all the integrals except that of (3.17) appearing in the results can be expressed as the sum of a finite number of integrals of the form: • cos 9 gm ^2 By differentiating the expression z""" V'(l — z)^[l — {ko -f- kiz"-] with respect to 2, we may derive the recurrence formula: ^rn = —7 7Tr2 K2W — 3)^0^12™-! [m — l)ki + (w - 2){kl - k\- 1)Z^_2 (3.19) - (2W - S)hkiZra-3 + (W - 3)(1 - kl)Z^-i\ ' Power series expansions of coefficients such as treated here have been given by A. G. Tynan, Modulation Products in a Power Law Modulator, Proc. L K. E., Vol. 21, pp. 1203-1209, Aug. 1933. /cc 1 I THE BIASED IDEAL RECTIFIER 161 o o V A ■= UJ u / ' O 9 II o T r— A^A- 3 ~ II • a|Q- ^ \. 1 — ^^ >* ^ / -^ ^/^ <^ PCOS QCOS />- ^ ^ ^ ^^/ / i y ^ y 1 Zi v 1 /" ( \ % \ \ ^ o<x ^^^s^ ^ \^ \ \ ^*!a ^ ^ onla ~ -H d/'^V 1+ 1+ 1+ 1+ ionaoad( b+cl2) jo sanindwv 162 BELL SYSTEM TECHNICAL JOURNAL It thus is found that the value of Zm for all values of m greater than 2 can be expressed in terms of Zq, Z\, and Z2. Eq. (3.18) may be written in the form: z'^-dz Z3 = Zi The substitution V(Z - 2i) (S - Z2)(Z3 - z)(Zi - 2) Zl = — (1 + ^o)Al , Zo = — 1 / (1 - /to)Ai , Case I) \ \ 1, Case II / (1, Case I \ (1 - /feo)//fei, Casell, / Z2CZ3 — Zl) — Zi(Z3 — Z2)U^ Z = reduces the integral to ^m — Z3 — 2i — (Z3 — Z2)m2 — Zl) *'o / du h V(24 - Z2)(23 - zO h ^7 (73 -,2)(1 _ ^^2 -) where: tl = Z3 — Z2 23 — Zl 2 (Z4 — Zi)(23 — Z2) X = (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) (24 - Z2)(23 - 2i) Hence if A', E and 11 represent respectively complete elliptic integrals of the first, second, and third kinds with modulus «, and in the case of third kind with parameter —t], we have immediately: 2K Zo = Zl = z,= kl-\/(Zi — Z2)(23 — 2i) 2 [2i K -\- {Z2- 2i) n] ki\/{Zi - 22) (Z3 - 2i^ ^lV(24-l)(23-2i) [^^ ^' + '^'(^^ - ^^^" (3.26) (3.27) (3.28) THE BIASED IDEAL RECTIFIER 163 To complete the evaluation of Z2, assume a relation of the following type with undetermined constants Ci, C2, C3, C4: I (1 - dii h (1 - T/w')' V(l - «') (1 - k' w2) c 'io V(l - m2)(1 -k'u^ du u^ ^«^ + C3 y^ (J _ ^^2) ^^j _ ^^^ ^^ _ ^3^,^ + C4 z y/Cl - z^) (1 - K^ 2^) 1 - 1722 (3.29) 2 1.2 UJ < o Z 1.0 5 0.2 < -0.2 O H-0.4 < N. -1 — ^\s 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.5 IS 2.0 RATIO OF BIAS TO LARGER FUND^MENTAL Fig. 15. — Fundamentals and (Ip ± q) — product from full-wave biased zero-power-law rectifier with ratio of applied fundamental amplitudes equal to 0.5. Fi = larger funda- mental, F2 = smaller fundamental, F3 = (2/> ± q) — product. Differentiate both sides with respect to z, set z = 1, and clear fractions. Equating coefificients of like powers of z separately then gives four simul- taneous equations in G, C2, C3, C4. Solving for C], C2, C3 and setting z = 1 in (3.29) gives r^ du 1 r i (1 - vuf \/(l - W) (1 -^?^) " 2(r, - 1) [^ "^ _j_ (2,? - 3) k" - 7,(77 - 2) jjl 77^ .2 (3.30) 164 BELL SYSTEM TECHNICAL JOURNAL u Q _) i\ O o q: a. uj Q. o u a. 0.8 0.6 0.4 0.2 -0.2 ^2 Q2 0.4 0.6 08 1.0 1.2 1.4 1.6 1.8 2 ^0 RATIO OF BIAS TO LARC^TD FiJMD.tMENTAL Fig. 16. — Fundamentals and (2/> ± q) — product from full-wave biased zero-power-law rectifier with equal applied fundamental amplitudes. < t<0 1 1 i "^ ^ ^0 «_^ A V 3-^ ^ ^ ^.^i^___^ 0.2 0.4 0.8 Fig. 17. — The integral Zm with ^i = 0.5. Since the necessary tables of FI are not available, we make use of Legendre's Transformation, which in this case gives: '" Legendre, Traites dcs Fonctions EUiptiques, Paris, 1825-28, Vol. I, Ch. XXIII. THE BIASED IDEAL RECTIFIER 165 2.0 Aqo ^ •^Or^.O ^ ^ — 0, ■^ ^ ^ -1 ^ ^ 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Fig. 18. — D-c. term in linear rectifier output with two applied frequencies. 1.0 0.8 0.6 Aqi 0.4 02 K,= 1 -^ 0.8 0.6 0.4 ■ 0.2 0.2 0.4 0.6 0.8 1.0 12 1.4 1.6 1.8 2.0 «0 Fig. 19. — Smaller fundamental in biased linear rectifier output. n = ir + tan <^ V 1 — K sin^ ^ 1/2 = arc sin Jo £(0) = f vn^ K dd Vl - K^ sin2 e 2 sin2 e dd (3.31) (3.32) (3.33) (3.34) The functions F(0) and E(0) are incomplete elliptic integrals of the first and second kinds. They are tabulated in a number of places. Fairly good tables, e.g. the original ones of Legendre, are needed here since the difference between KE(«^) and EF(0) is relatively small. 166 BELL SYSTEM TECHNICAL JOURNAL 1 1 M M ■^ / / / M //. ^ / \ /// /// // i\ /, /V / / 7 1 o /Ol /CO / f d/ 6/ 6/ o •O / yt \ (0/ o/ d/ d/ o 1 d 1 / 1 \ / f THE BIASED IDEAL RECTIFIER 167 V / ^ / / /oy i^ / / / ^ ^ V. \. l. ^ V X > ^^^ ^ ^ \^ ^^^ s_ \ ^ \ O C d C i l > ^ c > c i c 1 r ) ■■*. ) C ■>!■ 3 3 d -•^ •?; PlH 168 BELL SYSTEM TECHNICAL JOURNAL _l 4 < O 2 UJ 3 _l ^ 1 ^° = °-'o^2 0.5 0.8 , _JJ— _ ^ ■-' , ,.-- K4_^ J-i— ■^ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 •^1 Summarizing: Fig. 22. — Graph of the integral E (^o ^i). Case I, ^0 + ^1 > 1, ^0 — ^1 < 1 K Zo = V^i Zi = - [KE{4>) - EFm - Z,, Z2 ^ib — ko '^-^A K = v^ -n <P - - arc sin fv 2*1 + *o + *1 Case II, ^0 + ^1 < 1, ^0 — ^1 > — 1 Zo = 2K V(i + hY - kl Zx = I [ir£(<^) - EFm - Zo ^2 = ^2 (1 + ^I - i^o)Zo - 2^o)^iZ, - 2£ V(l + ^i)' - ^5 a/ 4^1 = arc sin /' — ^0 + ^1 (3.35) (3.36) THE BIASED IDEAL RECTIFIER 169 The values of the fundamentals and third-order sum and difference products for the biased zero-power-law rectifier have been calculated by the formulas above for the cases ki — .5 and ^i = 1. The resulting curves are shown in Fig. (15) and (16). The values of the auxiliary integrals Zo, ^i , and Zo are shown for ^i = .5 in Fig. (17). These integrals become infinite at kit = I — ki so that the formulas for the modulation coefficients become indeterminate at this point. The limiting \-alues can be evaluated from the integrals {^.3), etc., directly in terms of elementary functions when the relation ^o = 1 — ^i is substituted, except for the H-function. Limiting forms of the coefficients when k„ is small are of value in calcu- lating the effect of a small signal superim[)osed on the two sinusoidal com- ponents in an unbiased rectifier. By straightforward power-series expan- sion in ^oi we find : Zero-Power- Law Rectifier, ko Small: Aro = -„£ - 2£ 7r2(l - k'') kl + Aoi = ~ [£ - (1 - kl)K\ + -^^^ r^-^)^» + [ (3.37: A21 = - -,r, [(1 + TT'^l K - 2k\)E 1 - 2k\ 1 - k\ (1 - k\)K\ '^kl + In the above expressions, the modulus of K and E is ki. When k^ = 0, these coefficients reduce to half the values of the full-wave unbiased zero- power-law coefficients, which have been tabulated in a previous publication. Acknowledgment In addition to the j)ersons already mentioned, the writer wishes to thank Miss M. C. Packer, Miss J. Lever and Mrs. A. J. Shanklin for their assistance in the calculations of this paper. " R. M. Kalb and W. R. Bennett, Ferromagnetic Distortion of a Two-Frequency Wave, B. S. T. J., Vol. XIV, .\pril 1935, Eq. (21), p. 336. Properties and Uses of Thermistors — Thermally Sensitive Resistors ' By J. A. BECKER, C. B. GREEN and G. L. PEARSON A new circuit element and control device, the thermistor or thermally sensitive resistor, is made of solid semiconducting materials whose resistance decreases about four per cent per centigrade degree. The thermistor presents interesting opportunities to the designer and engineer in many fields of technology for ac- complishing tasks more simply, economically and better than with available devices. Part I discusses the conduction mechanism in semiconductors and the criteria for usefulness of circuit elements made from them. The fundamental physical properties of thermistors, their construction, their static and dynamic characteristics and general principles of operation are treated. Part II of this paper deals with the applications of thermistors. These include : sensitive thermometers and temperature control elements, simple temperature compensators, ultrahigh frequency' power meters, automatic gain controls for transmission systems such as the Types K2 and LI carrier telephone systems, voltage regulators, speech volume limiters, compressors and expandors, gas pres- sure gauges and flowmeters, meters for thermal conductivity determination of liquids, and contactless time delay devices. Thermistors with short time con- stants have been used as sensitive bolometers and show promise as simple com- pact audio-frequency oscillators, modulators and amplifiers. PART I— PROPERTIES OF THERMISTORS Introduction THERMISTORS, or thermsMy sensitive resistors, are devices made of solids whose electrical resistance varies rapidly with temperature. Even though they are only about 15 years old they have already found im- portant and large scale uses in the telephone plant and in military equip- ments. Some of these uses are as time delay devices, protective devices, voltage regulators, regulators in carrier systems, speech volume limiters, test equipment for ultra-high-frequency power, and detecting elements for very small radiant power. In all these applications thermistors were chosen because they are simple, small, rugged, liave a long life, and require little maintenance. Because of these and other desirable properties, ther- mistors promise to become new circuit elements which will be used exten- sively in the fields of communications, radio, electrical and thermal instrumentation, research in physics, chemistry and biology, and war tech- nology. Specific types of uses which will be discussed in the second part of this paper include: 1) simple, sensitive and fast responding thermometers, * Published in Elec. Engg., November 1946. The authors acknowledge their indebtedness to Messrs. J. H. Scaff and H. C. Theuercr for furnishing samples for most of the curves in Fig. 4, and to Mr. G. K. Teal for the data for the lowest curve in that figure. 170 PROPERTIES AND USES OF THERMISTORS 171 temperature compensators and temperature control devices; 2) special switching devices witiiout moving contacts; 3) regulators or volume limiters; 4) pressure gauges, flowmeters, and simple meters for measuring thermal conductivity in liquids and gases; 5) time delay and surge suppressors; 6) special oscillators, modulators and amplifiers for relatively low frequencies. Before these uses are discussed in detail it is desirable to present the physical principles which determine the properties of thermistors. The question naturally arises "why have devices of this kind come into use only recently?" The answer is that thermistors are made of semi- conductors and that the resistance of these can vary by factors up to a thousand or a million with surprisingly small amounts of certain impurities, with heat treatment, methods of making contact and with the treatment during life or use. Consequently the potential application of semiconduc- tors was discouraged by experiences such as the following: two or more units made by what appeared to be the same process would show large variations in their properties. Even the same unit might change its re- sistance by factors of two to ten by exposure to moderate temperatures or to the passage of current. Before semiconductors could seriously be con- sidered in industrial applications, it was necessary to devote a large amount of research and development efifort to a study of the nature of the conduc- tivity in semiconductors, and of the effect of impurities and heat treatment on this conductivity, and to methods of making reliable and permanent contacts to semiconductors. Even though Faraday discovered that the resistance of silver sulphide changed rapidly with temperature, and even i though thousands of other semiconductors have been found to have large \ negative temperature coefficients of resistance, it has taken about a century i of effort in physics and chemistry to give the engineering profession this j new tool which may have an influence similar to that of the vacuum tube I and may replace vacuum tubes in many instances. If thermistors are to be generally useful in industry: ! 1) it should be possible to reproduce units having the same character- istics; I 2) it should be possible to maintain constant characteristics during use; the contact should be permanent and the unit should be chemically inert ; 3) the units should be mechanically rugged; 4) the technique should be such that the material can be formed into various shapes and sizes; 5) it should be possible to cover a wide range of resistance, temperature coefficient and power dissipation. Thermistors might be made by any method by which a semiconductor 172 BELL SYSTEM TECHNICAL JOURNAL could be shaped to definite dimensions and contacts applied. These meth- ods include: 1) melting the semiconductor, cooling and solidifying, cutting to size and shape; 2) evaporation; 3) heating compressed powders of semi- conductors to a temperature at which they sinter into a strong compact mass and firing on metal powder contacts. While all three processes have been used, the third method has been found to be most generally useful for mass production. This method is similar to that employed in ceramics or in powder metallurgy. At the sintering temperatures the powders recrystallize and the dimensions shrink by controlled amounts. The powder process makes it possible to mix two or more semiconducting oxides in varjnng proportions and obtain a homogeneous and uniform solid. It is thus possible to cover a considerable range of specific resistance and tem- Fig. 1. — Thermistors made in the form of a bead, rod, disc, washer and flakes. perature coefficient of resistance with the same system of oxides. By '' means of the powder process it is possible to make thermistors of a great variety of shapes and sizes to cover a large range of resistances and power handling capacities. Figure 1 is a photograph of thermistors made in the form of beads, rods, discs, washers and flakes. Beads are made by stringing two platinum alloy wires parallel to each other with a spacing of five to ten times the wire diam- eter. A mass of a slurry of mixed oxides is applied to the wires. Surface tension draws this mass into the form of a bead. From 10 to 20 such beads are evenly spaced along the wires. The beads are allowed to dry and are heated slightly until they have sufficient strength so that the string can be handled. They then are passed through the sintering furnace. The oxides I shrink onto the i)latinum alloy wires and make an intimate and permanent, I electrical contact. The wires then are cut to separate the individual beads. i( PROPERTIES AND USES OF THERMISTORS 173 The diameters of the beads range from 0.015 to 0.15 centimeters with wire diameters ranging from 0.0025 to 0.015 centimeters. Rod thermistors are made by mixing the oxides with an organic binder and solvent, extruding the mixture through a die, drying, cutting to length, heating to drive out the binder, and sintering at a high temperature. Con- tacts are applied by coating the ends with silver, gold, or platinum paste as used in the ceramic art, and heating or curing the paste at a suitable temperature. The diameter of the rods can ordinarily be varied from 0.080 to 0.64 centimeter. The length can vary from 0.15 to 5 centimeters. Discs and washers are made in a similar way by pressing the bonded I powders in a die. Possible disc diameters are 0.15 to ,^ or 5 centimeters; |l thicknesses from 0.080 to 0.64 centimeter. Flakes are made by mixing the oxides with a suitable binder and solvent to a creamy consistency, spreading a film on a smooth glass surface, allowing ! the film to dry, removing the film, cutting it into flakes of the desired size and shape, and firing the flakes at the sintering temperatures on smooth \ ceramic surfaces. Contacts are applied as described above. Possible dimensions are: thickness, 0.001 to 0.004 centimeter; length, 0.1 to 1.0 ! centimeter; width, 0.02 to 0.1 centimeter. ! In any of these forms lead wires can be attached to the contacts by solder- ' ing or by firing heavy metal pastes. The dimensional limits given above , are those which have been found to be readily attainable. In the design of a thermistor for a specific application, the following characteristics should be considered: 1) Mechanical dimensions including ^ those of the supports. 2) The material from which it is made and its prop- ; erties. These include the specific resistance and how it varies with tem- I perature, the specific heat, density, and expansion coefficient. ^) The i dissipation constant and power sensitivity. The dissipation constant is I the watts that are dissipated in the thermistor divided by its temperature [ rise in centigrade degrees above its surroundings.. The power sensitivity is I the watts dissipated to reduce the resistance by one per cent. These con- stants are determined by the area and nature of the surface, the surrounding 'medium, and the thermal conductivity of the supports. 4) The heat ca- j.pacity which is determined by specific heat, dimensions, and density. 5) :The time constant. This determines how rapidly the thermistor will heat [or cool. If a thermistor is heated above its surroundings and then allowed to cool, its temperature will decrease rapidly at first and then more slowly until it finally reaches ambient temperature. The time constant is the time ! required for the temperature to fall 63 per cent of the way toward ambient i temperature. The time constant in seconds is equal to the heat capacity tin joules per centigrade degree divided by the dissipation constant in watts 174 BELL SYSTEM TECHNICAL JOURNAL per centigrade degree. 6) The maximum permissible power that can be dissipated consistent with good stability and long life, for continuous opera- tion, and for surges. This can be computed from the dissipation constant and the maximum permissible temperature rise. This and the resistance- temperature relation determine the maximum decrease in resistance. Properties of Semiconductors As most thermistors are made of semiconductors it is important to discuss the properties of the latter. A semiconductor may be defined as a substance io« 10* 2 I O . I 10' ill o z ,< y,n-2 KT' \ v\ \ \ \ \i > ^ \ 1 1- c^ -v^ ^ Cr ^ ^ ■^, PL .ATir guM - -100 100 200 TEMPERATURE °C 300 400 Fig. 2. — Logarithm of specific resistance versus temperature for three thermistor ma- terials as compared with platinum. whose electrical conductivity at or near room temperature is much less than that of typical metals but much greater than that of typical insulators. While no sharp boundaries exist between these classes of conductors, one might say that semiconductors have specific resistances at room tempera- ture from 0.1 to 10* ohm centimeters. Semiconductors usually have high h negative temperature coefKicients of resistance. As the temperature is increased from O^C. to 300°C., the resistance may decrease by a factor of a thousand. Over this same temperature range the resistance of a typical metal such as platinum will increase by a factor of two. Figure 2 shows how the logarithm of the specific resistance, p, varies with temperature, T, in degrees centigrade for three typical semiconductors and for platinum. PROPERTIES AND USES OF THERMISTORS 175 Curves 1 and 2 are for Materials No. 1 and No. 2 which have been ex'ten- sively used to date. Material No. 1 is composed of manganese and nickel oxides. Material No. 2 is composed of oxides of manganese, nickel and cobalt. The dashed part of Curve 2 covers a region in which the resistance- temperature relation is not known as accurately as it is at lower tempera- tures. Curve 3 is an experimental curve for a mixture of iron and zinc 2 U 10- 2 5 y 10- / r y / / — / / ) / / -^ -- -y— — ^ /^ — f v^ / / / / / / / J. — ~A~ - -- t 7. 'szr / / " / / .r / / / / J / / / / / / / /' ' / / / 3.0 xiO'' temperature: °k Fig. 3. — Logarithm of the si)ecific resistance of two thermistor materials as a function of inverse absolute temperature. See equation (1). oxides in the proportions to form zinc ferrite. From Fig. 2 it is obvious that neither the resistance R nor log R varies linearly with T. Figure 3 shows plots of log p versus l/T, for Materials No. 1 and No. 2. These do form approximate straight lines. Hence BlT Pooe or p = poe (,bIt)-{bitq) (1) where T = temperature in degrees Kelvin; p„ — p when T = oo or \/T = 0; P{i = p when T = To ; e = Naperian base = 2.718 and 5 is a constant equal to 2.303 times the slope of the straight lines in Fig. 3. The dimensions of B 176 BELL SYSTEM TECHNICAL JOURNAL are Kelvin degrees or centigrade degrees; it plays the same role in equation (1) as does the work function in Richardson's equation for thermionic emission. For Material No. \, B — 392()C°. This corresponds to an elec- tron energy equivalent to 3920 11600 or 0.34 volt. While the curves in Fig. 3 are approximately straight, a more careful investigation shows that the slope increases linearly as the temperature increases. From this it follows that a more precise expression for p is: , T — c PIT p = A 1 6 or log p = log .1 - r log T + D/2.303r (2) The constant c is a small positive or negative number or zero. For Ma- terial No. 1, log A = 5.563, < = 2.73 and D = 3100. For a particular form of Material No. 2 log .1 = 11.514, c = 4.83 and D = 2064. If we define temperature coefilicient of resistance, a, by the equation a = {\/R) {(IR/dT) (3) it follows from equation (1) that a = -B/r. (4) For Material No. 1 and T - 300°K, a - -3920/90,000 = -0.044. For platinum, a — +0.0037 or roughly ten times smaller than for semiconduc- tors and of the opposite sign. From equation (2) it follows that «= -{D/D- (c/T). (5) From equation (3) it follows that a = (1 2.303) {(flogR'dT). (6) For a discussion of the nature of the conductivit}^ in semiconductors, it is simpler and more convenient to consider the conductivity, a, rather than the resistivity, p. a = \/p and logo- = —log p. (7) The characteristics of semiconductors are brought out more clearly if the conductivity or its logarithm are plotted as a function of \/T over a wide temj:;erature range. Figure 4 is such a j)lot for a number of silicon sam- ples containing increasing amounts of impurity. At high temperatures all the samples have nearly the same conductivity. This is called the intrinsic conductivity since it seems to be an intrinsic properly of silicon. At low temperatures the conductivity of different sami:)les varies by large factors. Tn this region silicon is said to be an impurity semiconductor. For extremely i)ure silicon only intrinsic conductivity is present and the PROPERTIES AND USES OF THERMISTORS 177 resistivity obeys equation (1). As the concentration of a particular im- purity increases, the conductivity increases and the impurity conductivity predominates to higher temperatures. Some impurities are much more effective in increasing the conductivity than others. One hundred parts per million of some impurities may increase the conductivity of pure silicon at room temperature by a factor of 10^ Other impurities may be present 7 '0 O 310 I o bio- o §'0- o —I \ \ 2 \ 1 V /^^"^ \ 1 ^^. ..^ \ "-~-^ "^ ^ s *""*** ^^ • \, ^ -1 \. .. S =s— ^ X \ — -^ \ s. S, A N ^ s. ■? \ \ ■\ >^ \ \ \ \ \ ^ X -3 \ \ ""^^ — \— -Vi— \ \, \ V V 4 \ \ s. ^ ^^^-^ -5 \_ >-^ -\ — "^*^«< \ g i-i- \ o s -6 7 V 1 l' xlO"' TEMPERATURE °K Fig. 4. — Logarithm of the conductivity of various specimens of silicon as a function of inverse absolute temperature. The conductivity increases with the amount of im- purity. in 10,000 parts per million and have a small effect on the conductivity. Two samples may contain the same concentration of an impurity and still differ greatly in their low temperature conductivity; if the impurity is in solid solution, i.e., atomically dispersed, the effect is great; if the impurity is segregated in atomically large particles, the effect is small. Since heat treatments affect the dispersion of impurities in solids, the conductivity of semiconductors may frequently be altered radically by heat treatment. Some other semiconductors are not greatly affected by heat treatment. 178 BELL SYSTEM TECHNICAL JOURNAL The impurity need not even be a foreign element; in the case of oxides or sulphides, it can be an excess or a deficiency of oxygen or sulphur from the exact stoichiometric relation. This excess or deficiency can be brought about by heat treatment. Figure 5 shows how the conductivity depends on temperature for a number of samples of cuprous oxide, CU2O, heat ID' 1.0- o ^io-« I- y 8io- tCT^°i — \i s^- — X, N v^ ^k. - ■ N X. \^ ^^ V^ V. ^ * t^ ^ ^, K \\\. ^ >v \ 1: X- V =»^ \v ^ ^ — *% \ ^ \l v ^S \ X s. \ \^ k \ \, \ N ^i^^ \ -^^^ \ \ ^ t V — \ \ \ — ^ \ > \ -^ \ — OJ \ 1^ ^ 1 1 xlQ- temperature: °k Fig. 5. — Logarithm of the conductivity of various specimens of cuprous oxide as a function of inverse absolute temperature. The conductivity increases with the amount of excess oxygen above the stoichiometric value in CuoO. Data from reference 1. treated in such a way as to result in varying amounts of excess oxygen from zero to about one per cent.' The greater the amount of excess o.xygen the greater is the conductivity in the low temperature range. At high tem- peratures, all samples have about the same conductivity. Semiconductors can be classified on the basis of the carriers of the current into ionic, electronic, and mixed conductors. Chlorides such as NaCl and some sulphides are ionic semiconductors; other sulphides and a few oxides PROPERTIES AND USES OF THERMISTORS 179 such as uranium oiide are mixed semiconductors; electronic semiconductors include most oxides such as MnsOs, FejOs, NiO, carbides such as silicon carbide, and elements such as boron, silicon, germanium and tellurium. In ionic and mixed conductors, ions are transported through the solid. This changes the density of carriers in various regions, and thus changes the conductivity. Because this is undesirable, they are rarely used in mak- ing thermistors, and hence we will concentrate our interest on electronic semiconductors. The theoretical and experimental physicists have established that there are two types of electronic semiconductors which can be called N and P type, depending upon whether the carriers are negative electrons or are equivalent to positive "holes" in the filled energy band. In N type, the ACCEPTOR M PURITIES INTRINSIC Fig. 6. — Schematic energy level diagrams illustrating intrinsic, N and P types of semi- conductors. carriers are deflected by a magnetic field as negatively charged particles would be and conversely for P type. The direction of deflections is ascer- tained by measurement of the sign of the Hall effect. The direction of the thermoelectric effect also fixes the sign of the carriers. By determining the resistivity, Hall coefficient and therm.oelectric power of a particular specimen at a particular temperature it is possible to determine the density of carriers, whether they are negative or positive, and their mobility or mean free path. The mobility is the mean drift velocity in a field of one volt per centimeter. The existence of these classifications is explained by the theoretical physi- cist^ . 3 , 4 j^ terms of the diagrams in Fig. 6. In an intrinsic semiconductor at low temperatures the valence electrons completely fill all the allowable energy states. According to the exclusion principle only one electron can occupy a particular energy state in any system. In semiconductors and 180 BELL SYSTEM TECHNICAL JOURNAL insulators there exists a region of energy values, just above the allowed band, which are not allowed. The height of this unallowed band is expressed in equivalent electron volts, A£. Above this unallowed band there exists an allowed band; but at low temperatures there are no electrons in this band. When a iield is applied across such a semiconductor, no electron can be accelerated, because if it were accelerated its energy would be increased to an energy state w^hich is either tilled or unallowed. As the temperature is raised some electrons acquire sufficient energy to be raised across the un- allowed band into the upper allowed band. These electrons can be ac- celerated into a slightly higher energy state by the applied field and thus can carry current. For every electron that is put into an "activated" state there is left behind a "hole" in the normally filled band. Other electrons having slightly lower energies can be accelerated into these holes by the applied field. The physicist has shown that these holes act toward the applied field as if they were particles having a charge equal to that of an electron but of opposite sign and a mass equal to or somewhat larger than the electronic mass. In an intrinsic semiconductor about half the con- ductivity is due to electrons and half due to holes. The quantity A£ is related to B in equation (1) by: 2B = (A£) e/k (8) in which B is in centigrade degrees, A£ is in volts, e is the electronic charge in coulombs, k is Boltzmann's constant in joules per centigrade degree. The value of e/k is 11,600 so that A£ = Z^/5800. (8a) The difference between metals, semiconductors, and insulators results from the value of A£. For metals A£ is zero or very small. For semicon- ductors A£ is greater than about 0.1 volt but less than about 1.5 volts. For insulators A£ is greater than about 1.5 volts. Some impurities with positive valencies which may be present in the semi- conductor may have energy states such that A£i volts equivalent energy can raise the valence electron of the impurity atom into the allowed con- duction band. See Figure 6. The electron now can take part in conduc- tion; the donator impurity is a positive ion which is usually bound to a par- ticular location and can take no part in the conductivity. These are excess or A^ type conductors. The conductivity de[)ends on the density of dono- tors, A£i , and T. Similarly some other impurity with negative valencies may have an energy state A/S2 volts above the top of the lilled band. At room temi)era- ture or higher, an electron in the filled band may be raised in energy and PROPERTIES AND USES OF THERMISTORS 181 accepted by the impurity which then becomes a negative ion and usually is immobile. However, the resulting hole can take part in the conductivity. In all cases represented in Fig. 6 an electron occupying a higher energy level than a positive ion or a hole has a certain probability that in any short interval of time it will drop into a lower energy state. However, dur- ing this same time interval there will be electrons which will be raised to a higher energy level by thermal agitation. When the number of electrons per second which are being elevated is equal to the number which are de- scending in energy, equilibrium prevails. The conductivity, a, is then a = N evi-i- P ev2 (9) where N and P are the concentrations of electrons and holes respectively, e is the charge on the electron, z'l and V2 are the mobilities of electrons and holes respectively. The above picture explains the following experimental facts which other- wise are difficult to interpret. 1) A^ type oxides, such as ZnO, when heated in a neutral or slightly reducing atmosphere become good conductors, presumably because they contain excess zinc which can donate electrons. If they then are heated in atmospheres which are increasingly more oxidiz- ing their conductivity decreases until eventually they are intrinsic semi- conductors or insulators. 2) P type oxides, such as NiO, when heat treated in strongly oxidizing atmospheres are good conductors. Very likely they contain oxygen in excess of the stoichiometric relation and this oxygen accepts additional electrons. When these are heated in less oxidizing or neutral atmospheres they become poorer conductors, semiconductors, or insulators. 3) When a P type oxide is sintered with another P type oxide, the conductivity increases. Similarly for two N type oxides. But when a P type is added to an N type the conductivity decreases. 4) If a metal forms several oxides the one in which the metal exerts its highest valence is N type, while the one in which it exerts its lowest valence will be P type.^ For several reasons it is desirable to survey the whole field of semicon- ductors for resistivity and temperature coefficient. One way in which this might be done is to draw a line in Figure 3 for each specimen. Before long such a figure would consist of such a maze of intersecting lines that it would be difficult to single out and follow any one line. The information can be condensed by plotting log po versus B in equation (1) for each specimen.^ The most important characteristics of a specimen thus are represented by a single point and many more specimens can be surveyed in a single diagram. Figure 7 shows such a plot for a large number of semiconductors investi- gated at these Laboratories or reported in the literature. Values for po and B are given for T = 25 degrees centigrade. The points form a sort of 182 BELL SYSTEM TECHNICAL JOURNAL milky way. Semiconductors having a high po are Ukely to have a high value of B and vice versa. If a series of semiconductors have points in Fig. 7 which fall on a straight line with a slope of 1/2.37^0 , they have a common intercept in Fig. 3 for (l/T) = 0. 10" - - - ~ - — - - - r- - — - - — - 10^ , 10* J • - 10* o ?n f\J ijlO* 5 . |l03 z "~ W • w ' y id' o s 10° , I0-' irrZ 6 xlO'^ '"0 I 2 3 4 B IN X AT 25t Fig. 7.— Logarithm of the resistivity of various semiconducting materials as a func- tion of B in equation (I). The quantity, B, is proportional to the temperature coefiicient of resistance as given in equation (4). Physical Properties of Thermistors One of the most interesting and useful properties of a thermistor is the way in which the voltage, F, across it changes as the current, /, through it increases. Figure 8 shows this relationship for a 0.061 centimeter diam- eter bead of Material No. 1 suspended in air. Each time the current is PROPERTIES AND USES OF THERMISTORS 183 changed, sufficient time is allowed for the voltage to attain a new steady value. Hence this curve is called the steady state curve. For sufficiently small currents, the power dissipated is too small to heat the thermistor appreciably, and Ohm's law is followed. However, as the current assumes larger values, the power dissipated increases, the temperature rises above ambient temperature, the resistance decreases, and hence the voltage is less than it would have been had the resistance remained constant. At some current, !„ , the voltage attains a maximum or peak value, Vm • Beyond /^^ Xso h \ \ 6(fS. ^^s^ I 100 ^^^-- ' .. """"^55 2 0.5 5 10 MILLIAMPERES Fig. 8. — Static voltage-current curve for a typical thermistor. The numbers on the curve are the centigrade degrees rise in temperature above ambient. i this point as the current increases the voltage decreases and the thermistor is said to have a negative resistance whose value is dV/dl. The numbers on the curve give the rise in temperature above ambient temperature in centi- grade degrees. Because currents and voltages for different thermistors cover such a large range of values it has been found convenient to plot log V versus log /. Figure 9 shows such a plot for the same data as in Fig. 8. For various points on the curve, the temperature rise above ambient temperature is given. In a log plot, a line with a slope of 45 degrees represents a constant resist- ance; a line with a slope of —45 degrees represents constant power. 184 BELL SYSTEM TECHNICAL JOURNAL For a particular thermistor, the position of the log V versus log I plot is shifted, as shown in Fig. 10, by changing the dissipation constant C. This IjO MILLIAMPERES Fig. 9. — Logarithmic plot of static voltage-current curve for the same data as in Figure 8. The diagonal hnes give the values of resistance and power. B=3900 R= 50,000 OHMS T=300°K 100 V\/ / K / X ■!o4 \ \ /X y < \ X /^ X A y X X k / 10"' 10"" 10' ■" 10'' 10"' I 10 CURRENT IN AMPERES Fig. lO.^Logarithmic plots of voltage versus current for three values of the dissipa- tion constant C. These curves are calculated for the constants given in the upper jiart of tlje figure. can be done by changing the air pressure surrounding the bead, changing the medium, or changing the degree of thermal coupling between the thermi§- PROPERTIES AND USES OF THERMISTORS 185 tor and its surroundings. The value of C for a particular thermistor in given surroundings can readily be determined from the V versus / curve in either Figs. 8 or 9. For each point, V/I is the resistance while V times / is IF, the watts dissipated. The resistance data are converted to tempera- ture from R versus T given by equation (2). A plot is then made of W versus T. For thermistors in which most of the heat is conducted away, W will increase linearly with T, so that C is constant. For thermistors suspended by fine wires in a vacuum, W will increase more rapidly than pro- portional to T, and C will increase with T. For thermistors of ordinary size and shape, in still air, C/Area = 1 to 40 milliwatts per centigrade degree per square centimeter depending upon the size and shape factor. B=3900 C=5X10 WATTS/DEG. aoo'K 100 / rfp/\ \* Ay \ \^^ -5p /y \ / \4/ X 10"* 10-** 10"' 10-^ 10"' I K) CURRENT IN AMPERES Fig. 11.— Logarithmic plots of voltage versus current for three values of the resistance, Ro , at ambient temperature. These curves are calculated for the constants given in the upper part of the figure. The user of a thermistor may want to know how many watts can be dis- sipated before the resistance decreases by one per cent. This may be called the power sensitivity. It is equal to C/{a X 100), and amounts to about one to ten milliwatts per square centimeter of area in still air. Both C and the power sensitivity increase with air velocity. The dependence of C on gas pressure and velocity is the basis of the use of thermistors as manom- eters and as anemometers or flowmeters. Note that in Fig. 10 one curve can be superposed on any other by a shift along a constant resistance line. Figure 1 1 shows a family of log V versus log / curves for various values on Ro while B, C, and To are kept constant. This can be brought about by changing the length, width and thickness to vary Ro while the surface area is kept constant. If the resistance had been changed by changing the am- bient temperature. To , the resulting curves would not appear very different 186 BELL SYSTEM TECHNICAL JOURNAL from those shown. Note that one curve can be superposed on any other curve by a shift along a constant power Hne. Figure 12 shows a family of log V versus log / curves for eight different values of B while C, Ra , and To are kept constant. In contrast to the curves in Figs. 10 and 11 in which any curve could be obtained from any other curve by a shift along an appropriate axis, the curves in Fig. 12 are each distinct. For each curve there exists a limiting ohmic resistance for low C=5X10"'^WATTS/DEG. Ro-SQPOO OHMS T = 300 K 1000 o-* 10- 10- 10- 10 10-2 CURRENT IN AMPERES Fig. 12. — Logarithmic plots of voltage versus current for eight values of B in equation (1). These carves are calculated for the constants given in the upper part of the figure. currents and another for high currents. For B = these two are identical. As B becomes larger the log of the ratio of the two limiting resistances in- creases proportional to B. Note also that for B > 1200 A'°, the curves have a maximum. For large B values this maximum occurs at low powers and hence at low values of T — To . This follows since W = C{T — To). As B decreases, Vm occurs at increasingly higher powers or temperatures. For B < 1200 K°, no maximum exists. The curves in Figs. 10 to 12 have been drawn for the ideal case in which the resistance in series with the thermistor is zero and in which no tempera- ture limitations have been considered. In any actual case there is always PROPERTIES AND USES OF THERMISTORS 187 some unavoidable small resistance, such as that of the leads, in series with the thermistor and hence the parts of the curves corresponding to low re- sistances may not be observable. Also at high powers the temperature may attain such values that something in the thermistor structure will go to pieces thus limiting the range of observation. These unobservable ranges have been indicated by dashed lines in Fig. 12. The exact location of the dashed portions will of course depend on how a completed thermistor is con- structed. In setting these limits consideration is given to temperature limi- tations beyond which aging efifects might become too great. The curves in Figs. 9 to 12 have been computed on the basis of the follow- ing equations: W = C(T - To) = VI (11) For these curves the constants Rq , To , B, and C are specified. The values of temperature, T^ , power, W^ , resistance, R^. , voltage, F„ , and current, Im , that prevail at the maximum in the voltage current curve are given by the following equations in which T^ is chosen as the independent param- eter. By differentiating equations (10) and (11) with respect to /, putting the derivatives equal to zero, one obtains Tl = B{Tm - To) (12) whose solution is r„ = {B/2) (1 T Vl - 4To/B). (13) The minus sign pertains to the maximum in Figs. 10 to 12 while the plus sign pertains to the minimum. Note that Tm depends only on B and To , and not on R, Ro or C. From equations (4), (10) and (11) it follows that: - a^ {T^ - To) = 1 (14) \V„. = C{T„, - To) (15) i?,„ = Ro r^""'^" ^ Ro t-'iX - (r„ - To)/To + (1/2) {(n.- To)/ToV ] (16) F„ = [C Ro {Tm - To) {e-'-'^')]'" = \\C Ro (r„. - To) €-' [1 - {Tm - To)/ To 4- (1/2) \{Tm- To)/ToV- WV" (17) Jr. = [{C/Ro) {Tm - To) e'-'^^r- = {{{C/Ro) {Tm - To) e[\ + {Tm - To)/To + (1/2)1 (r.- To)/To}'+ ■■■ ]}V'' (18) 188 BELL SYSTEM TECHNICAL JOURNAL Thus far the presentation has been limited to steady state conditions, in which the power supplied to the thermistor is equal to the power dissipated by it, and the temperature remains constant. In many cases, however, it is important to consider transient conditions when the temperature, and any quantities which are functions of temperature, var}^ with time. A simple case which will illustrate the concepts and constants involved in such problems is as follows: A massive thermistor is heated to about 150 to 200 degrees centigrade by operating it well beyond the peak of its voltage 200 100 ' \ V 80 N^ \^ 60 \ ' V o N. N. Z ^ "^20 H \ 10 \, \ 8 \ 4 2 k 150 200 TIME IN SECONDS Fig. 13.— Cooling characteristic of a massive thermistor: log of temperature above ambient versus time. current characteristic. At time / = 0, the circuit is switched over to a con- stant current having a value so small that PR is always negligibly small. The voltage across the thermistor is then followed as a function of time. From this, the resistance and temperature are computed. Figure 13 shows i a plot of log (r - Ta) versus / for a rod thermistor of Material No. 1 about 1.2 centimeters long, 0.30 centimeter in diameter and weighing 0.380 gram. In any time interval Al, there are C(T - To) A/ joules being dissipated. | .As a result the temperature will decrease by A7" given by -HAT = ar - Ta) A/ or (7' - 7'„) - -{H/C) (A7'/A/) iV)) PROPERTIES AND USES OF THERMISTORS 189 where H = heat capacity m joules per centigrade degree. The solution of this equation is (r - r„) = (r„ - r„) in which 2\ — T when / = and r = H/C, (20) (21) where r is in seconds. It is commonly called the time constant. From equation (20) it follows that a plot of log {T — T a) versus t should yield a straight line whose slope = — 1/2.303t. If // and C vary slightly with temperature then t will vary slightly with T and /. The line will not be perfectly straight but its slope at any t or (T — To) will yield the appro- Table I. — Values of C, t, H as Functions of T for a Thermistor of Material No. 1 ABOUT 1.2 Centimeters Long, 0.30 Centimeters in Diameter and Weighing 0.380 Gram Ta = 24 degrees centigrade T Degrees Centigrade C Watts per C. degree T Seconds // Joules per C. degree h Joules per gram per C. degree 44 64 0.0037 0.0037 76 74 0.28 0.27 0.75 0.72 84 104 0.0038 0.0037 71 69 0.27 0.26 0.71 0.68 124 144 0.0038 0.0038 68 67 0.26 0.26 0.67 0.67 164 ; 184 0.0039 0.0041 67 66 0.26 0.27 0.69 0.71 204 0.0042 66 0.28 0.73 priate t or H/C for that T. As previously described, C can be determined from a plot of watts dissipated versus T. For this thermistor this curve became steeper at the higher temperatures so that C increased for higher temperatures. Table I summarizes the values of C, r, and // at various T for the unit in air. When a thermistor is heated by passing current through it, conditions are somewhat more involved since the PR power will be a function of time. At any time in the lieating cycle the heat power liberated will be equal to the watts dissipated or C{T — Ta) plus watts required to raise the tem- perature or HdT/dl. The heat power liberated will de})end on the circuit conditions. In a circuit like that shown in the upper corner of Figure 14, the current varies with time as shown by the six curves for six values of the battery voltage E. If a relay in the circuit operates when the current reaches a definite value, a considerable range of time delays can be achieved. 190 BELL SYSTEM TECHNICAL JOURNAL This family of curves will be modified by changes in ambient temperature and where rather precise time delays are required, the ambient temperature must be controlled or compensated. Since thermistors cover a wide range in size, shape, and heat conductivity of surrounding media, large variations in //, C, and t can be produced. The time constant can be varied from about one millisecond to about ten minutes or a millionfold. One very important property of a thermistor is its aging characteristic or how constant the resistance at a given temperature stays with use. To obtain a stable thermistor it is necessary to: 1) select only semiconductors which are pure electronic conductors; 2) select those which do not change chemically when exposed to the atmosphere at elevated temperatures; 3U 40 KEY 1 THERMISTOR^ E=£ JOVCLTS ■■^ n 1^ 1 (/I ^ 70 20 |1 II tf 30 66- OSCIi^ GRAPH ENT // /" ^ W < ^0 5 20 ■7 I V /^ ^ 40 ^ ,n " 30 Ld 10 h /^ . ^ • ^ D P i ^ \ 3 i 3 1 f 3 9 TIME IN SECONDS Fig. 14. — Current versus time curves for six values of the battery voltage in the circuit shown in the insert. 3) select one which is not sensitive to impurities likely to be encountered in manufacture or in use; 4) treat it so that the degree of dispersion of the critical impurities is in equilibrium or else that the approach to equilibrium is very slow at operating temperatures; 5) make a contact which is intimate, sticks tenaciously, has an expansion coefficient compatible with the semi- conductor, and is durable in the atmospheres to which it will be exposed; 6) in some cases, enclose the thermistor in a thin coat of glass or material impervious to gases and liquids, the coat having a suitable expansion coeffi- cient; 7) preage the unit for several days or weeks at a temperature some- what higher than that to which it will be subjected. By taking these pre- cautions remarkably good stabilities can be attained. Figure 15 shows aging data taken on three-quarter inch diameter discs of Materials No. 1 and No. 2 with silver contacts and soldered leads. These discs were measured soon after production, were aged in an oven at 105 degrees centigrade and were periodically tested at 24 degrees centigrade. PROPERTIES AND USES OF THERMISTORS 101 The percentage change in resistance over its initial value is plotted versus the logarithm of the time in the aging oven. It is to be noted that most of the aging takes place in the first day or week. If these discs were preaged for a week or a month and the subsequent change in resistance referred to the resistance after preaging, they would age only about 0.2 per cent in one year. In a thermistor thermometer, this change in resistance would cor- respond to a temperature change of 0.05 centigrade degree. Thermistors mounted in an evacuated tube or coated with a thin layer of glass age even less than those shown in the figure. For some applications such high stability is not essential and it is not necessary to give the thermistors special treatment. " .-rMM *\=. _ MATEe\£i=^-^ ■ .0 ^^■^^ ' i^' y wiATrRlAL'**^2 - 5 '^ ■ — — '' i[ AY IV EEK 1 MOt ITH 6 MONTI- S 1 YEAF ^ 2YRS 5YRS KD' 10^ lO'' 10^ TIME IN HOURS AT 105° C. Fig. 15. — Aging characteristics of thermistors made of Materials No. 1 and No. 2 aged in an oven at rG5°C. Per cent increase in resistance over its initial value versus time on a logarithmic scale. Thermistors have been used at higher temperatures with satisfactory aging characteristics. Extruded rods of Material No. 1 have been tested for stab- ility by treating them for two months at a temperature of 300 degrees centigrade. Typical units aged from 0.5 to 1.5 per cent of their initial resistance. Similar thermistors have been exposed alternately to tempera- tures of 300 degrees centigrade and —75 degrees centigrade for a total of 700 temperature cycles, each lasting one-half hour. The resistance of typ- ical units changed by less than one per cent. In some applications of thermistors very small changes in temperature produce small changes in potential across the thermistor which then are amplified in high gain amplifiers. If at the same time the resistance is fluctuating randomly by as little as one part in a million, the potential across the thermistor will also fluctuate by a magnitude which will be 192 BELL SYSTEM TECHNICAL JOURNAL directly proportional to the current. This fluctuating potential is called noise and since it depends on the current it is called current noise. In order to obtain the best signal to noise ratio, it is necessary that the current noise at operating conditions be less than Johnson or thermal noise.'^ ■* To make noise-free units it is necessary to pay particular attention to the raw mate- rials, the degree of sintering, the grain size, the method of making contact and any steps in the process which might result in minute surface cracks or fissures. POWER IN WATTS 0.1 I 10 THERMISTOR ELEMENT CURRENT IN MILLIAMPERES 100 Fig. 16. — Logarithmic plots of voltage versus current for six values of heater curren in an indirecth' heated thermistor. Resistance and power scales are given on the diag onal lines. All the thermistors discussed thus far were either directly heated by the current passing through them or by changes in ambient temperature. In indirectly heated thermistors, the temperature and resistance of the thermis- tor are controlled primarily by the power fed into a heater thermally coupled to it. One such form might consist of a 0.038 centimeter diameter bead of Material No. 2 embedded in a small cylinder of glass about 0.38 centimeter long and 0.076 centimeter in diameter. A small nichrome heater coil hav- ing a resistance of 100 ohms is wound on the glass and is fused onto it with more glass. Figure 16 shows a plot of log V versus log / for the bead ele- ment at various currents through the heater. In this way the bead resist- ance can be changed from 3000 ohms to about 10 ohms. Indirectly heated thermistors are ordinarily used where the controlled circuit must be iso- lated electrically from the actuating circuit, and where the power from the latter must be fed into a constant resistance heater. PROPERTIES AND USES OF THERMISTORS 193 PART II— USES OF THERMISTORS The thermistor, or thermally sensitive resistor, has probably excited more interest as a major electric circuit element than any other except the vacuum tube in the last decade. Its extreme versatility, small size and ruggedness were responsible for its introduction in great numbers into communications circuits within five years after its first appUcation in this field. The next five year period spanned the war, and saw thermistors widely used in addi- tional important applications. The more important of these uses ranged from time delays and temperature controls to feed-back amplifier automatic gain controls, speech volume limit ers and superhigh frequency power meters. It is surprising that such versatility can result from a temperature dependent resistance characteristic alone. However, this effect produces a very useful nonlinear volt-ampere relationship. This, together with the ability to pro- duce the sensitive element in a wide variety of shapes and sizes results in applications in diverse fields. (The variables of design are many and inter- related, including electrical, thermal and mechanical dimensions. The more important uses of thermistors as indication, control and cir- cuit elements will be discussed, grouping the uses as they fall under the primary characteristics: resistance-temperature, volt-ampere, and current- time or d^mamic relations. Resistance-Temperature Relations It has been pointed out in Part I that the temperature coefficient of elec- trical resistance of thermistors is negative and several times that of the or- dinary metals at room temperature. In Thermistor Material No. 1, which is commonly used, the coefficient at 25 degrees centigrade is —4.4 per cent per centigrade degree, or over ten times that of copper, which is +0.39 per cent per centigrade degree at the same temperature. A circuit element made of this thermistor material has a resistance at zero degrees centigrade which is nine times the resistance of the same element at 50 degrees centigrade. For comparison, the resistance of a copper wire at 50 degrees centigrade is 1.21 times its value at zero degrees centigrade. The resistance-temperature characteristics of thermistors suggest their use as sensitive thermometers, as temperature actuated controls and as compensators for the effects of varying ambient temperature on other ele- ments in electric circuits. Thermometry The application of thermistors to temperature measurement follows the usual principles of resistance thermometry. However, the large value of temperature coefficient of thermistors permits a new order of sensitivity to be obtained. This and the small size, simplicity and ruggedness of thermis- 194 BELL SYSTEM TECHNICAL JOURNAL tors adapt them to a wide variety of temperature measuring applications. VV^hen designed for this service, thermistor thermometers have long-time stability which is good for temperatures up to 300 degrees centigrade and excellent for more moderate temperatures. A well aged thermistor used in precision temperature measurements was found to be within 0.01 centi- grade degree of its calibration after two months use at various temperatures up to 100 degrees centigrade. As development proceeds, the stability of thermistor thermometers may be expected to approach that of precision platinum thermometers. Conventional bridge or other resistance measuring circuits are commonly employed with thermistors. As with any resistance thermometer, consideration must be given to keeping the measuring current sufficiently small so that it produces no appreciable heating in order that the Table II. — Temperature-Resistance Characteristic of a Typical Thermistor -Thermometer Temperature CoefBcients Temperature Resistance B a -25°C. 580,000 ohms 3780 C. deg. -6.1%/ C. deg. 145,000 3850 -5.2 25 46,000 3920 -4.4 50 16,400 3980 -3.8 75 6,700 4050 -i.i 100 3,200 4120 -3.0 150 830 4260 -2.4 200 305 4410 -2.0 275 100 4600 -1.5 Dissipation constant in still air, approx 4 milliwatts/C. deg. Thermal time constant in still air, approx 70 seconds Dimensions of thermistor, diameter approx 0.11 inch length approx 0. 54 inch thermistor resistance shall be dependent upon the ambient temperature alone. Since thermistors are readily designed for higher resistance values than metallic resistance thermometers or thermocouples, lead resistances are not ordinarily bothersome. Hence the temperature sensitive element can be located remotely from its associated measuring circuit. This permits great flexibility in application, such as for instance wire line transmission of temperature indications to control points. Table II gives the characteristics of a typical thermistor thermometer. The dissipation constant is the ratio of the power input in watts dissipated in the thermistor to the resultant temperature rise in centigrade degrees. The time constant is the time required for the temperature of the thermistor to change 63 per cent of the difference between its initial value and that of the surroundings. As a sensitive thermometer, this thermistor with a simple Wheatstone bridge and a galvanometer whose sensitivity is 2 X PROPERTIES AND USES OF THERMISTORS 195 U zi^ f^ 196 BELL SYSTEM TECHNICAL JOURNAL 10"^° amperes per millimeter per meter will readily indicate a temperature change of 0.0005 centigrade degree. For comparison a precision platinum resistance thermometer and the required special bridge such as the Mueller will indicate a minimum change of 0.003 centigrade degree with a similar galvanometer. Several thermistors which have been used for thermometry are shown in Fig. 17. Included in the group are types which are suited to such diverse applications as intravenous blood thermometry and supercharger rotor temperature measurement. In Fig. 17, A is a tiny bead with a response time of less than a second in air. B is a probe type unit for use in air streams or liquids. C is a meteorological thermometer used in automatic radio transmission of weather data from free balloons. D is a rod shaped imit. E is a disc or pellet, adapted for use in a metal thermometer bulb. Discs like the one shown have been sweated to metal plates to give a low thermal impedance connection to the object whose temperature is to be determined. F is a large disc with an enveloping paint finish for use in humid surroimd- ings. The characteristics of these types are given in Table III. The temperature of objects which are inaccessible, in motion, or too hot for contact thermometry can be determined by permitting radiation from the object to be focussed on a suitable thermistor by means of an elliptical mirror. Such a thermistor may take the form of a thin flake attached to a solid support. Its advantages compared with the thermopile and resistance bolometer are its more favorable resistance value, its ruggedness, and its high temperature coefficient of resistance. It can be made small to reduce its heat capacity so as rapidly to follow changing temperatures. Flake thermistors have been made with time constants from one millisecond to one second. Since the amount of radiant power falling on the thermistor may be quite small, sensitive meters or vacuum tube amplifiers are required to measure the small changes in the flake resistance. Where rapidly vary- ing temperatures are not involved, thermistors with longer time constants and simpler circuit equipments can be utilized. Temperature Control The use of thermistors for temperature control purposes is related closely to their application as temperature measuring devices. In the ideal tem- perature sensitive control element, sensitivity to temperature change should be high and the resistance value at the control temperature should be the proper value for the control circuit used. Also the temperature rise of the control element due to circuit heating should be low, and the stability of calibration should be good. The size and shape of the sensitive element are dictated by several factors such as the space available, the required speed of response to temperature changes and the amount of power which must PROPERTIES AND USES OF THERMISTORS 197 be dissipated in the element by the control circuit to permit the arrange- ment to operate relays, motors or valves. Because of their high temperature sensitivity, thermistors have shown much promise as control elements. Their adaptability and their stability at relatively high temperatures led, for instance, to an aircraft engine con- trol system using a rod-shaped thermistor as the control element.^ The Table III. — Thermistor Thermometers A B C D E F Nominal Resistance, Ohms at -25°C 5,000 2,000 900 460 250 95 -3.4 150 0.1 1 Bead 0.015 0.02 325,000 100,000 33,000 13,000 6,000 1,600 500 80 -4.4 300 1 7 30 4 Probe 0.1 0.6 87,500 37,500 18,000 9,700 5,500 3,700 -2.8 100 7 25 Rod 0.05 1.2 610,000 153,000 48,500 17,300 7,100 3,400 870 -4.4 150 7 60 Rod 0.15 0.7 490 175 71 32 16 4.5 1.6 -3.8 200 Disc 0.2 0.1 13,000 25 50 3,200 950 340 75 145 100 150 70 200 300 Temp. Coeff. «, %/C. deg. at 25°C -4.4 Max. Permissible Temp., °C. . Dissipation Constant, C, mw/C deg. Still air 100 20 Still water — Thermal Time Constant, Seconds Still air Still water — Shape Disc Dimensions, Inches Diameter or Width Length or Thickness (less leads) 0.56 0.31 thermistor, mounted in a standard one-quarter inch diameter temperature bulb assembly, operated at approximately 275 degrees centigrade. It was associated with a differential relay and control motor on the aircraft 28 volt d-c system. The power dissipation in the thermistor was two watts. The resistance of a typical thermistor under these high temperature con- ditions remained within ±1.5 per cent over a period of months. This corresponds to about ± one centigrade degree variation in calibration. Several other related designs were developed using the same control system 198 BELL SYSTEM TECHNICAL JOURNAL with other thermistors designed for both higher and lower temperature operation. In the lower temperature applications, typical thermistors maintained their calibrations within a few tenths of a centigrade degree. In general, electron tube control circuits dissipate less power in the ther- mistor than relay circuits do. This results in less temperature rise in the thermistor and leads to a more accurate control. While the average value of this temperature rise can be allowed for in the design, the variations in different installations require individual calibration to correct the errors if they are large. The corrections may be different as a result of variations of the thermal conductivity of the surrounding media from time to time or from one installation to another. The greater the power dissipated in the thermistor the greater the absolute error in the control temperature for a given change in thermal conductivity. This follows from the relation ^T = W/C (22) where AT is the temperature rise, W is the power dissipated and C is the dis- sipation constant which depends on thermal coupling to the surroundings. For the same reason, the temperature indicated by a resistance thermometer immersed in an agitated medium will depend on the rate of flow if the tem- perature sensitive element is operated several degrees hotter than its sur- roundings. The design of a thermistor for a ventilating duct thermostat might pro- ceed as follows as far as temperature rise is concerned : 1 . Determine the power dissipation. This depends upon the circuit selected and the required overall sensitivity. 2. Estimate the permissible temperature rise of the thermistor, set by the expected variation in air speed and the required temperature control accur- acy. 3. Solve Equation (22) for the dissipation constant and select a thermistor of appropriate design and size for this constant in the nominal air speed. Where more than one style of thermistor is available, the required time constant will determine the choice. Compensators It is a natural and obvious application of thermistors to use them to com- pensate for changes in resistance of electrical circuits caused by ambient temperature variations. A simple example is the compensation of a copper wire line, the resistance of which increases approximately 0.4 per cent per centigrade degree. A thermistor having approximately one-tenth the resistance of the copper, with a temperature coefficient of —4 per cent per centigrade degree placed in series with the line and subjected to the same ambient temperature, would serve to compensate it over a narrow tempera- PROPERTIES AND USES OF THERMISTORS 199 ture range. In practice however, the compensating thermistor is associated with parallel and sometimes series resistance, so that the com.bination gives a change in resistance closely equal and opposite to that of the circuit to be compensated over a wide range of temperatures. See Fig. 18. 2000 -40 -20 20 40 60 TEMPERATURE IN DEGREES CENTIGRADE 80 Fig. 18. — Temperature compensation of a copper conductor by means of a thermistor network. A copper winding having a resistance of 1000 ohms at 25 degrees centi- grade can be compensated by means of a thermistor of 566 ohms at 25 degrees centigrade in parallel with an ohmic resistance of 445 ohms as shown in Fig. 18. The winding with compensator has a resistance of 1250 ohms constant to ± 1.6 per cent over the temperature range —25 degrees centi- grade to -t-75 degrees centigrade. Over this range the copper alone varies from 807.5 ohms to 1192.5 ohms, or ± 19 per cent about the mean. The 200 BELL SYSTEM TECHMCAL JOURI^AL total resistance of the circuit has been increased only 6.1 per cent at the upper temperature limit by the addition of a compensator. This increase is small because of the high temperature coefficient of the compensating thermistor. The characteristics of such a thermistor are so stable that the resistance would remain constant within less than one per cent for ten years if maintained at any temperature up to about 100 degrees centigrade. Figure 15 shows aging characteristics for typical thermistors suitable for use in compensators. These curves include the change which occurs during the seasoning period of several days at the factory, so that the aging in use is a fraction of the total shown. In many circuits which need to function to close tolerances under wide ambient temperature variation, the values of one or more circuit elements may var>' undesirably with temperature. Frequently the resultant overall variation with temperature can be reduced by the insertion of a simple ther- mistor placed at an appropriate point in the circuit. This is particularly true if the circuit contains vacuum tube amplifiers. In this manner fre- quency and gain shifts in communications circuits have been cancelled and temperature errors prevented in the operation of devices such as electric meters. The change in inductance of a coil due to the variation of magnetic characteristics of the core material with temperature has been prevented by partially saturating the coil with direct current, the magnitude of which is directly controlled by the resistance of a thermistor imbedded in the core. In this way the amount of d-c magnetic flux is adjusted by the thermistor so that the inductance of the coil is independent of temperature. In designing a compensator, care must be taken to ensure exposure of the thermistor to the temperature affecting the element to be compensated. Power dissipation in the thermistor must be considered and either limited to a value which will not produce a significant rise in temperature above am- bient, or offset in the design. Volt-Ampere Characteristics The nonlinear shape of the static characteristic relating voltage, current, resistance and power for a typical thermistor was illustrated by Fig. 9. The part of the curve to the right of the voltage maximum has a negative slope, applicable in a large number of ways in electric circuits. The par- ticular characteristic showTi begins with a resistance of approximately 50,000 ohms at low power. Additional power dissipation raises the temperature of the thermistor element and decreases its resistance. At the voltage maximum the resistance is reduced to about one-third its cold value, or 17,000 ohms, and the dissipation is 13 milliwatts. The resistance becomes approximately 300 ohms when the dissipation is 100 milliwatts. Such resistance-power characteristics have resulted in the use of thermistors as sensitive power measuring devices, and as automatically variable resistances PROPERTIES AND USES OF THERMISTORS 201 for such applications as output amplitude controls for oscillators and am- plifiers. Their nonlinear characteristics also fit thermistors for use as volt- age regulators, volume controls, expandors, contactless switches and remote control devices. To permit their use in these applications for d-c as well as a-c circuits, nonpolarizing semiconductors alone are employed in thermistors with the exception of two early types. Power Meter Thermistors have been used very extensively in the ultra and superhigh frequency ranges in test sets as power measuring elements. The particular advantages of thermistors for this use are that they can be made small in size, have a small electrical capacity, can be severely overloaded without 0.5 ONE INCH Fig. 19. — Power measuring thermistors with different sized beads. change in calibration, and can easily be calibrated with direct-current or low-frequency power. For this application the thermistor is used as a power absorbing terminating resistance in the transmission line, which may be of Lecher, coaxial or wave-guide form. Methods of mounting have been worked out which reduce the reflection of high frequency energy from the termination to negligible values and assure accurate measurement of the power over broad bands in the frequency spectrum. Conventionally, the thermistor is operated as one arm of a Wheatstone bridge, and is biased with low frequency or d-c energy to a selected operating resistance value, for instance 125 or 250 ohms in the absence of the power to be measured. The application of the power to be measured further decreases the thermistor resistance, the bridge becomes unbalanced and a deflection is obtained on the bridge meter. A full scale power indication of one miUiwatt is customary for the test set described, although values from 0.1 milliwatt to 200 milli- watts have been employed using thermistors with different sized beads as shown in Fig. 19. 202 BELL SYSTEM TECH MCA L JOURNAL Continuous operation tests of these tliermistors indicate very satisfactory stability with an indelinitcly long life. A grouj) of eight power meter ther- mistors, normally operated at 10 milliwatts and having a maximum rating of 20 milliwatts, were o])erated for over 3000 hours at a power input of 30 milliwatts. During this lime the room temperature resistance remained within 1.5 per cent of its initial value, and the power sensitivity, which is the significant characteristic, changed by less than 0.5 per cent. When power measuring test sets are intended for use with wide ambient temjierature variations, it is necessary to temperature compensate the ther- mistor. This is accomplished conventionally by the introduction of two other thermistors into the bridge circuit. These units are designed to be insensitive to bridge currents but responsive to ambient temperature. One of the compensators maintains the zero point and the other holds the meter scale calibration independent of the effect of temperature change on the measuring thermistor characteristics. Automatic Oscillator Amplitude Control Meacham, and Shepherd and Wise" have described the use of thermis- tors to provide an effective method of amplitude stabilization of both low and high frequency oscillators. These circuits oscillate because of positive feedback around the vacuum tube. The feedback circuit is a bridge with at least one arm containing a thermistor which is heated by the oscillator output. Through this arrangement, the feedback depends in phase and magnitude upon the output, and there is one value of thermistor resistance which if attained would balance the bridge and cause the oscillation ampli- tude to vanish. Obviously this condition can never be exactly attained, and the operating point is just enough different to keep the bridge slightly unbalanced and produce a predetermined steady value of oscillation output. Such oscillators in which the amplitude is determined by thermistor non- linearity have manifold advantages over those whose amplitude is limited by vacuum tube nonlinearity. The harmonic content in the output is smaller, and the performance is much less dependent upon the individual vacuum tube and upon variations of the supply voltages. It is necessary that the thermal inertia of the thermistor be sufficient to prevent it from varying in resistance at the oscillation frequency. This is easily satisfied for all frequencies down to a small fraction of a cycle per second. Figure 20 shows a thermistor frequently used for oscillator control together with its static electrical characteristic. This thermistor is satisfactory in oscillators for frequencies above approximately 100 cycles per second. Similar types have been developed with response characteristics suited to lower frequencies and for other resistance values and powers. PROPERTIES AND USES OF THERMISTORS 203 WTiere the ambient temperature sensitivity of the thermistor is dis- advantageous in oscillator controls, the thermistor can be compensated by Fig. 20A. — An amplitude control thermistor. The glass bulb is 1.5 inches in length. 102 nO \ / X X K/ X X XX X ^x x:x XX X K y( 6 V \ V \ ><X/i v^^^ XK XV V \ < / \ / /, xV^x-A^^ \X\ / \ y// / <l X ^."^ \ \ / /s \ Vi\\sD XS ^' z X / \ 2 ^ •■ y> X ^?^X>rX^,>^ K 4 S ^ }" n ^ ^ / \ A? § ^^ ^ >^ / X/ X^^^ X ^ X X >< ^d'. z'C^^C^O"*^ x" y 6 Af/ \ K V \ K V \ ''H t^ X v<S>- y^ /. / < /\\x y ^ / < / \ ^< XXx / \ ^OJ >i if) a. ^4 t// ^ ^ 2 1 K r X / N ^ ^ X / \ ^ SxM <^ X \ ^ / f. Ox ^ / X^ ^ / :m^ t? X 8 6 X K^ X X Y/ X u y V \^ x- X V < XV N \ < s K X^^^ V X X XV / \ / < X<: X / K / / \ X/ / <^ 2 yy\ X / \ X >^ /> X / N ^1 O^ -c- X ^ X \ ^ / X ^ ^ / "m / S V ^ X I0-* 2 4 6 810"^ 2 4 6 810'^ 2 4 6 8I0'' 2 4 6 8 | CURRENT IN MILLIAMPERES Fig. 20B. — Steady state characteristics of amplitude control thermistor shown in Figure 20A. thermostating it with a heater and compensating thermistor network, as shown in Fig. 21. Amplifier Automatic Gain Control Since the resistance of a thermistor of suitable design varies markedly with the power dissipated in it or in a closely associated heater, such ther- 204 BELL SYSTEM TECIIMCAL JOlh'XAL mistors have proven to be very valuable as automatic gain controls, es- pecially for use with negative feedback ampliliers. This arrangement has seen extensive use in wire communication circuits for transmission level regulation, and has been described in some detail elsewhere.^-- ^^' ^^ In one form, a directly heated thermistor is connected into the feedback circuit of the amplifier in such a way that the amount of feedback voltage is varied to compensate for any change in the output signal. By this arrangement, the gain of each amplifier in the transmission system is continually adjusted to correct for variations in overall loss due to weather conditions and other factors, so that constant transmission is obtained over the channel at all times. In the Type K2 carrier sj^stem now in extensive use, the system gain is regulated principally in this way. In this system the transmission loss variations due to temperature are not the same in all parts of the pass band. The loss is corrected at certain repeater points along the transmission line by two additional thermistor gain controls: slope, proportional to fre- H EATER T^'PE /T HERMISTOR constantI /;t\ ipRi I^CCt^ to" CURRENTS (^) ^f^2 (Nif) CONTROLLED SOURCE T Vp^t I rV^W CIRCUIT DISC THERMISTOR HEATER THERMISTOR Fig. 21. — Circuit employing an auxiliary disc thermistor to compensate for effect of varying ambient temperature on a control thermistor. quency, and bulge, with a maximum at one frequency. These thermistors are indirectly heated, with their heaters actuated by energy dependent upon the amplitude of the separate pilot carriers which are introduced at the send- ing end for the purpose. In this type of application, the thermistor will react to the ambient tem- perature to which it is exposed, as well as to the current passing through it. Where this is important, the reaction to ambient temperature can be elimi- nated by the use of a heater type thermistor as shown in Fig. 21. The heater is connected to an auxiliary circuit containing a temperature com- pensating thermistor. This circuit is so arranged that the power fed into the heater of the gain control thermistor is just sufficient at any ambient temperature to give a controlled and constant value of tejnjjerature in the vicinity of the gain control thermistor element. Another interesting form of thermistor gain control utilizes a heater type thermistor, with the heater driven by the output of the amplifier and with the thermistor element in the input circuit, as shown in Fig. 22. In this arrangement the feedback is accomplished by thermal, rather tiian electrical coupling. A broad-band carrier system, Type LI, is regulated PROPERTIES AND USES OF THERMISTORS 205 with this type of thermistor. In this system a pilot frequency is suppHed, and current of this frequency, selected by a network in the regulator, actu- ates the heater of the thermistor to give smooth, continuous gain control. By utilizing a heater thermistor of diflferent characteristics, the circuit and load of Fig. 22 may be given protection against overloads. In this application the sensitivity and element resistance of the thermistor are chosen so that the thermistor element forms a shunt of high resistance value so as to have negligible effect on the amplifier for any normal value of output. However, if the output power rises to an abnormal level, the thermistor element becomes heated and reduced in resistance. This shunts the input to the amplifier and thus limits the output. Choice of a thermistor having a suitable time constant permits the onset of the limiting eflfect to be delayed for any period from about a second to a few minutes. LOAD THERMISTORS"^ ^HEATER HEATER nPE THERMISTOR Fig. 22. — Thermal feedback circuit for gain control purposes. This arrangement has also been used as a protective circuit for overloads. Regulators and Limiters A group of related applications for thermistors depends on their steady state nonlinear volt-ampere characteristic. These are the voltage regulator, the speech volume limiter, the compressor and the expandor. The com- pressor and expandor are devices for altering the range of signal amplitudes. The compressor functions to reduce the range, while the expandor increases it. In Fig. 23, Curve 1 is a typical thermistor static characteristic having negative slope to the right of the voltage maximum. Curve 2 is the charac- teristic of an ohmic resistance R having an equal but positive slope. Curve 3 is the characteristic obtained if the thermistor and resistor are placed in series. It has an extensive segment where the voltage is almost inde- pendent of the current. This is the condition for a voltage regulator or limiter. If a larger value of resistance is used, as in Curve 4, its combination with the thermistor in series results in Curve 5, the compressor. In these uses the thermistor regulator is in shunt with the load resistance, so that in the circuit diagram of Fig. 23, E = Eo = Ei- IRs. (23) Here E is the voltage across the thermistor and resistor R, Eo is the output 206 BELL SYSTEM TECHNICAL JOURNAL voltage, and Er , I and Rs are respectively the input voltage, current and resistance. If the thermistor and associated resistor are placed in series between the generator and load resistance, an expandor is obtained, and Eg = Ej — E. (24) As the resistance R in series with the thermistor is increased, the degree of expansion is decreased and vice versa. 4 8 12 16 CURRENT IN MILLIAMPERES 20 Fig. 23. — Characteristics of a simple thermistor voltage regulator, limiter or com- pressor circuit. The treatment thus far in this section assumes that change of operating point occurs slowly enough to follow along the static curves. For a suffi- ciently rapid change of the operating point, the latter departs from the static curve and tends to progress along an ohmic resistance line intersecting the static curve. For sufficiently rapid fluctuations,- control action may then be derived from the resistance changes resulting from the r.m.s. power dis- sipated in the thermistor unit. In speech volume limiters, the thermistor is designed for a speed of response that will produce limiting action for the changes in volume which are syllabic in frequency or slower, and that will not follow the more rapid speech fluctuations with resulting change in wave PROPERTIES AND USES OF THERMISTORS 207 shape or nonlinear distortion. Speech volume limiters of this type can ac- commodate large volume changes without producing wave form distor- tion. i^.i^ Remote Control Swiches The contactless switch and rheostat are natural extensions of the uses just discussed. The thermistor is used as an element in the circuit which is to be controlled, while the thermistor resistance value is in turn dependent upon the energy dissipated directly or indirectly in it by the controlling cir- cuit. By taking advantage of the nonlinearity of the static volt-ampere characteristic, it is possible to provide snap and lock-in action in some applications. Manometer Several interesting and useful applications such as vacuum gauges, gas analyzers, flowmeters, thermal conductivity meters and liquid level gauges of high sensitivity and low operating temperature are based upon the physical principle that the dissipation constant of the thermistor depends on the thermal conductivity of the medium in which it is immersed. As shown in Fig. 10, a change in this constant shifts the position of the static characteristic with respect to the axes. In these applications, the unde- sired response of the thermistor to the ambient temperature of the medium can in many cases be eliminated or reduced by introducing a second thermis- tor of similar characteristics into the measuring circuit. The compensating thermistor is subjected to the same ambient temperature, but is shielded from theeflfect being measured, such as gas pressure or flow. Thetwo therm- istors can be connected into adjacent arms of a Wheatstone bridge which is balanced when the test effect is zero and becomes unbalanced when the effective thermal conductivity of the medium is increased. In gas flow measurements, the minimum measurable velocity is limited, as in all '*hot wire" devices, by the convection currents produced by the heated thermistor. The vacuum gauge or manometer which is typical of these appHcations will be described somewhat in detail. The sensitive element of the thermis- tor manometer is a small glass coated bead 0.02 inch in diameter, suspended by two fine wire leads in a tubular bulb for attachment to the chamber whose gas pressure is to be measured. The volt-ampere characteristics of a typical laboratory model manometer are shown in Fig. 24 for air at several absolute pressures from 10~® millimeters of mercury to atmospheric. The operating point is in general to the right of the peak of these curves. Electrically this element is connected into a unity ratio arm Wheatstone bridge with a similar but evacuated thermistor in an adjacent arm as shown in the circuit 208 BELL SYSTEM TECHNICAL JOURNAL schematic of Fig. 25. The air pressure caHbration for such a manometer is also shown. The characteristic will be shifted when a gas is used having a thermal conductivity different from that of air. Such a manometer has been found to be best suited for the measurement of pressures from 10~^ to 10 millimeters of mercury. The lower pressure limit is set by practical considerations such as meter sensitivity and the ability to maintain the zero setting for reasonable periods of time in the presence of the variations of supply voltage and ambient temperature. The upper pressure measure- ment limit is caused by the onset of saturation in the bridge unbalance 4~> ^ 10-2 4 6 8I0-' 2 4-68! 2 46 8|0' CURRENT IN MILLIAMPERES 4 6 810^ Fig. 24. — Characteristics of a typical thermistor manometer tube, showing the effect of gas pressure on the volt-ampere and resistance-power relations. voltage versus pressure characteristic at high pressures. This is basically because the mean free path of the gas molecules becomes short compared with the distance between the thermistor bead and the inner surface of the manometer bulb, so that the cooling effect becomes nearly independent of the pressure. The thermistor manometer is specially advantageous for use in gases which may be decomposed thermally. For this type of use, the thermistor element temperature can be limited to a rise of 30 centigrade degrees or less above ambient temperature. For ordinary applications, however, a temperature rise up to approximately 200 centigrade degrees in vacuum PROPERTIES AND USES OF THERMISTORS 209 permits measurement over wider ranges of pressure. Special models have also been made for use in corrosive gases. These expose only glass and plati- num alloy to the gas under test. Timing Devices The numerically greatest application for thermistors in the communication field has been for time delay purposes. The physical basis for this use has 4 6 6|0 2 4 6 810-2 2 4 6 B|0- PRESSURE IN MM OF MERCURY 6 8 I Fig. 25. — Operating circuit and calibration for a vacuum gauge utilizing the thermistor of Figure 24. been discussed in Part I for the case of a directly heated thermistor placed in series with a voltage source and a load to delay the current rise after circuit closure. This type of operation will be termed the power driven time delay. By the use of a thermistor suited to the circuit and operating conditions, power driven time delays can be produced from a few milliseconds to the order of a few minutes. Thermistors of this sort have the advantage of small size, light weight, ruggedness, indefinitely long life and absence of contacts, moving parts, or pneumatic orifices which require maintenance 210 BELL SYSTEM TECHNICAL fOURNAL care. Power driven time delay thermistors tre best fitted for applications where close limits on the time interval arc not required. In some com- munications uses it is satisfactory to permit a six to one ratio between maxi- mum and minimum times as a result of the simultaneous variation from nominal values of all the following factors which affect the delay : operating voltage ± 5 per cent; ambient temperature 20 degrees centigrade to 40 degrees centigrade; operating current of the relay ± 25 per cent; relay resistance zt 5 per cent; and thermistor variations such as occur from unit to unit of the same type. After a timing operation a power driven time delay thermistor should bs allowed time to cool before a second operation. If this is not done, the second timing interval will be shorter than the first. The cooling period depends on particular circuit conditions and details of thermistor design, but generally is several times the working time delay. In telephone relay circuits requiring a timing operation soon after previous use, the thermistor usually is connected so that it is short circuited by the relay contacts at the close of the working time delay interval. This pe: nits the thermistor to cool during the period when the relay is locked up. If this period is suffi- ciently long, the thermistor is available for use as soon as the relay drops out. Time delay thermistors have been operated more than half a million times on life test with no significant change in their timing action. To avoid the limitations of wide timing interval limits and extended cool- ing period between operations usually associated with the power driven time delay thermistor, a cooling time delay method of operation has been used. In this arrangement, two relays or the equivalent are employed and the thermistor is heated to a low resistance-value by passing a relatively large current through it for an interval short compared with the desired time interval. The current then is reduced automatically to a lower value and the thermistor cools until its resistance increases enough to reduce the cur- rent further and trip the working relay. This part of the operating cycle accounts for the greater part of the desired time interval. With this ar- rangement, the thermistor is available for re-use immediately after a com- pleted timing interval, or, as a matter of fact, after any part of it. By proper choice of operating currents and circuit values, wide variations of voltage and ambient temperature may occur with relatively little effect upon the time interval. The principal variable left is the cooling time of the thermis- tor itself. This is fixed in a given thermistor unit, but may vary from unit to unit, depending upon dissipation constant and thermal capacity, as pointed out above. In addition to their use as definite time delay devices, thermistors have been used in several related applications. Surges can be prevented from PROPERTIES AND USES OF THERMISTORS 211 operating relays or disturbing sensitive apparatus by introducing a ther- mistor in series with the circuit component which is to be protected. In case of a surge, the high initial resistance of the thermistor holds the surge current to a low value provided that the surge does not persist long enough to overcome the thermal inertia of the thermistor. The normal operating voltage, on the other hand, is applied long enough to lower the thermistor resistance to a negligible value, so that a normal operating current will flow after a short interval. In this way, the thermistor enables the circuit to distinguish between an undesired signal of short duration and a desired signal of longer duration even though the undesired impulse is several timss higher in voltage than the signal. Oscillators, Modulators and Amplifiers A group of applications already explored in the laboratory but not put into engineering use includes oscillators, modulators and amplifiers for the low and audio-frequercy range. If a thermistor is biased at a point on the negative slope portion of the steady-state volt-ampere characteristic, and if a small alternating voltage is then superposed on the direct voltage, a small alternating current will flow. If the thermistor has a small time con- stant, T, and if the applied frequency is low enough, the alternating volt- ampere characteristic will follow the steady-state curve and dV/dl will be negative. As the frequency of the applied a-c voltage is increased, the value of the negative resistance decreases. At some critical frequency, /c , the resistance is zero and the current is 90 degrees out of phase with the voltage. In the neighborhood of /c , the thermistor acts like an induc- tance whose value is of the order of a henry. As the frequency is increased beyor.d/c , the resistance is positive and increases steadily until it approach- es the d-c value when the current and voltage are in phase. The critical frequency is given approximately by /c = l/2r. If T can be made as small as 5 X 10~ seconds, fc is equal to 10,000 cycles per second and the thermistor would have an approximately constant negative resistance up to half this frequency. Point contact thermistors having such critical frequencies or even higher have been made in a number of laboratories. However, none of them have been made with sufficient reproducibility and constancy to be useful to the engineer. It has been shown both theoretically and experimentally that any negative resistance device can be used as an oscillator, a modulator, or an amplifier. With further development, it seems probable that thermistors will be used in these fields. 212 BELL SYSTEM TECHNICAL JOURNAL Summary The general principles of thermistor operation and examples of specific uses have been given to facilitate a better understanding of them, with the feeling that such an understanding will be the basis for increased use of this new circuit and control element in technology. References 1. Zur Elektrischen Leitfahigkeit von Kupferoxydul, W. P. Juse and B. VV. K5rtschatow. Physikalische Zeitschrift Der Sovvjetunion, Volume 2, 1932, pages 453-67. 2. Semi-conductors and Metals (book), A. H. Wilson. The University Press, Cam- bridge, England, 1939. 3. The Modern Theory of Solids (book), Frederick Seitz. McGraw-Hill Book Company, New York, N. Y., 1940. 4. Electronic Processes in Ionic Crystals (book), N. F. Mott and R. W. Gurney. The Clarendon Press, Oxford, England, 1940. 5. Die Elektronenleitfahigkeit von Festen Oxyden Verschiedener Valenzstufen, M. Le- Blanc and H. Sachse. Physikalische Zeitschrift, Volume 32, 1931, pages 887-9. 6. Uber die Elektrizitatsleitung Anorganischer Stofle mit Elektronenleitfahigkeit, Wil- fried Meyer. Zeitschrift Fur Physik, Volume 85, 1933, pages 278-93. 7. Thermal Agitation of Electricity in Conductors, J. B. Johnson. Physical Review, Volume 32, July 1928, pages 97-113. 8. Spontaneous Resistance Fluctuations in Carbon Microphones and Other Granular Resistances, C. J. Christensen and G. L. Pearson. The Bell System Technical Journal, Volume 15, April 1936, pages 197-223. 9. Automatic Temperature Control for Aircraft, R. A. Gund. AIEE Transactions, Volume 64, 1945, October section, pages 730-34. 10. The Bridge Stabilized Oscillator, L. A. Meacham. Proc. IRE, Volume 26, October 1938, pages 1278-94. 11. Frequency Stabilized Oscillator, R. L. Shepherd and R. O. Wise. Proc. IRE, Vol- ume 31, June 1943, pages 256-68. 12. A Pilot-Channel Regulator for the K-1 Carrier System, J. H. Bollman. Bell Labora- tories Record, Volume 20, No. 10, June 1942, pages 258-62. 13. Thermistors, J. E. Tweeddale. Western Electric Oscillator, December 1945, pages 3-5, 34-7. 14. Thermistor Technics, J. C. Johnson. Electronic Industries, Volume 4, August 1945, pages 74-7. 15. Volume Limiter for Leased-Line Service, J. A. Weiler. Bell Laboratories Record, Volume 23, No. 3, March 1945, pages 72-5. Abstracts of Technical Articles by Bell System Authors Capacitors — Their Use in Electronic Circuits} M. Brotherton. This book tells how to choose and use capacitors for electronic circuits. It ex- plains the basic factors which control the characteristics of capacitors and determine their proper operation. It helps to provide that broad under- std.nding of the capacitor problem which is indispensable to the efficient design of circuits. It tells the circuit designer what he must vmderstand and consider in transforming capacitance from a circuit symbol into a practi- cal item of apparatus capable of meeting the growing severity of today's operation requirements. Mica Capacitors for Carrier Telephone Systems.^ A. J. Christopher AND J. A. Kater. Silvered mica capacitors, because of their inherently high capacitance stability with temperature changes and with age, now are used widely in oscillators, networks, and other frequency determining circuits in the Bell Telephone System. Their use in place of the previous dry stack type, consisting of alternate layers of mica and foil clamped under high pressures, has made possible considerable manufacturing econ- omies in addition to improving the transmission performance of carrier telephone circuits. These economies are the result of their relatively simple unit construction and the ease of adjustment to the very close capacitance tolerance required. Visible Speech Translators with External Phosphors.^ Homer Dudley AND Otto 0. Gruenz, Jr. This paper describes some experimental ap- paratus built to give a passing display of visible speech patterns. These patterns show the analysis of speech on an intensity-frequency-time basis and move past the reader like a printed line. The apparatus has been called a translator as it converts speech intended for aural perception into a form suitable for visual prception. The phosphor employed is not in a cathode-ray tube but in the open on a belt or drum. The Pitch, Loudness and Quality of Musical Tones {A demonstration- lecture introducing the new Tone Synthesizer)} Harvey Fletcher. Re- lations are given in this paper which show how the pitch of a musical tone » Published by D. Van Nostrand Company, Inc., New York, N. Y., 1946. ' Elec. Engg., Transactions Section, October 1946. ^Jour. Acous. Soc. Anier., July 1946. * Amer. Jour, of Physics, July- August 1946. 213 214 BELL SYSTEM TECHNICAL JOURNAL depends upon the frequency, the intensity and the overtone structure of the sound wave transmitting the tone. Similar relations are also given which show how the loudness and the quality depend upon these same three physical characteristics of the sound wave. These relationships were de- monstrated by using the new Tone Synthesizer. By means of this in- strument one is able to imitate the quality, pitch and intensity of any musi- cal tone and also to produce many combinations which are not now used in music. The Sound Spectrograph.^ W. Koenig, H. K. Dunx, and L. Y. Lacy. The sound spectrograph is a wave analyzer which produces a permanent visual record showing the distribution of energy in both frequency and time. This paper describes the operation of this device, and shows the mechanical arrangements and the electrical circuits in a particular model. Some of the problems encountered in this type of analysis are discussed, particularly those arising from the necessity for handling and portraying a wide range of component levels in a complex wave such as speech. Spectrograms are shown for a wide variety of sounds, including voice sounds, animal and bird sounds, music, frequency modulations, and miscellaneous familiar sounds. Geometrical Characterizations of Some Families of Dynamical Trajectories} L. A. MacColl. a broad problem in differential geometry is that of characterizing, by a set of geometrical properties, the family of curves which is defined by a given system of differential equations, of a more or less special form. The problem has been studied especially by Kasner and his students, and characterizations have been obtained for various families of curves which are of geometrical or physical importance. However, the interesting problem of characterizing the family of trajectories of an electri- fied particle moving in a static magnetic field does not seem to have been considered heretofore. The present paper gives the principal results of a study of this problem. Visible Speech Cathode-Ray Translator."^ R. R. Riesz and L. Schott. A system has been developed whereby speech analysis patterns are made continuously visible on the moving luminescent screen of a special cathode- ray tube. The screen is a cylindrical band that rotates with the tube about a vertical axis. The electron beam always excites the screen in the same vertical plane. Because of the persistence of the screen phosphor and the rotation of the tube, the impressed patterns are spread out along a horizon- ^ Jour. Acous. Soc. Amer., July 1946. ^ Amer. Math. Soc. Transactions, July 1946. ' Jour. Acous. Soc. Amer., July 1946. ABSTRACTS OF TECHNICAL ARTICLES 215 tal time axis so that speech over an interval of a second or more is always visible. The upper portion of the screen portrays a spectrum analysis and the lower portion a pitch analysis of the speech sounds. The frequency band up to 3500 cycles is divided into 12 contiguous sub-bands by filters. The average speech energy in the sub-bands is scanned and made to control the excitation of the screen by the electron beam which is swept synchro- nously across the screen in the vertical direction. A pitch analyzer pro- duces a d-c. voltage proportional to the instantaneous fundamental fre- quency of the speech and this controls the width of a band of luminescence that the electron beam produces in the lower part of the screen. The translator had been used in a training program to study the readability of visible speech patterns. Derivatives of Composite Functions.^ John Riordan. The object of this note is to show the relation of the Y polynomials of E. T. Bell, first to the formula of DiBruno for the wth derivative of a function of a function, then to the more general case of a function of many functions. The sub- ject belongs to the algebra of analysis in the sense of Menger; all that is asked is the relation of the derivative of the composite function to the derivatives of its component functions when they exist and no questions of analysis are examined. The Portrayal of Visible Speech.^ J. C. Steinberg and N. R. French. This paper discusses the objectives and requirements in the protrayal of visible patterns of speech from the viewpoint of their effects on the legibility of the patterns. The portrayal involves an intensity-frequency-time analy- sis of speech and the display of the results of the analysis to the eye. Procedures for accomplishing this are discussed in relation to information on the reading of print and on the characteristics of speech and its inter- pretation by the ear. Also methods of evaluating the legibility of the visible patterns are described. Short Survey of Japanese Radar — 1}° Roger I. Wilkinson. The result of a study made immediately following the fall of Japan and recently made available for public information, this two-part report is designed to present a quick over-all evaluation of Japanese radar, its history and de- velopment. As the Japanese army and navy developed their radar equip- ment independently of each other, Part I of this article concentrates on the army's contributions. *Amer. Math. Soc. Bulletin, August 1946. ^ Jour. Aeons. Soc. Amer., July 1946. ^"Elec. Engg., Aug.-Sept. 1946. 216 BELL SYSTEM TECHNICAL JOURNAL A Variation on the Gain Fcrmula for Feedback Amlifters for a Certain Driving-Impedance Configuration.^^ T. W. Winternitz. An expression for the gain of a feedback amplifier, in which the source impedance is the only significant impedance across which the feedback voltage is developed, is derived. As examples of the use of this expression, it is then applied to three common circuits in order to obtain their response to a Heaviside unit step-voltage input. " Proc. LR.E., September 1946. Contributors to This Issue Joseph A. Becker,. A. B., Cornell University 1918; PhD., Cornell Univer- sity, 1922. National Research Fellow, California Institute of Technology, 1922-24; Asst. Prof, of Physics, Stanford University, 1924. Engineering Dept., Western Electric Company, 1924-1925; Bell Telephone Laboratores, 1925-. Mr. Becker has worked in the fields of X-Rays, magnetism, thermio- nic emission and adsorption, particularly in oxide coated filaments, the properties of semiconductors, as applied in varistors and thermistors. W. R. Bennett, B. S., Oregon State College, 1925; A.M., Columbia University, 1928. Bell Telephone Laboratories, 1925-. Mr. Bennett has been active in the design and testing of multichannel communication systems, particularly with regard to modulation processes and the effects of nonlinear distortion. As a member of the Transmission Research De- partment, he is now engaged in the study of pulse modulation techniques for sending telephone channels by microwave radio relay. C. B. Green, Ohio State University, B.A. 1927; M.A. in Physics, 1928. Additional graduate work at Columbia University. Bell Telephone Lab- oratories, 1928-. For ten years Mr. Green was concerned with trans- mission development for telephotography and television systems and with the design of vacuum tubes. Since 1938 he has been engaged in the developl- ment and application of thermistors. J. P. Kinzer, M. E., Stevens Institute of Technology, 1925. B.C.E., Brooklyn Polytechnic Institute, 1933. Bell Telephone Laboratories, 1925-. Mr. Kinzer's work has been in the development of carrier telephone repeat- ers; during the war his attention was directed to investigation of the mathe- matical problems involved in cavity resonators. W. P. Mason, B.S. in E.E., Univ. of Kansas, 1921; M.A., Ph.D., Co- lumbia, 1928. Bell Telephone Laboratories, 1921-. Dr. Mason has been engaged principally in investigating the properties and applications of piezoelectric crystals and in the study of ultrasonics. R. S. Ohl, B. S. in Electro-Chemical Engineering, Pennsylvania State College, 1918; U. S. Army, 1918 (2nd Lieutenant, Signal Corps); Vacuum tube development, Westinghouse Lamp Company, 1919-21; Instructor in 217 218 BELL SYSTEM TECHNICAL JOURNAL Physics, University of Colorado, 1921-1922. Department of Development and Research, American Telephone and Telegraph Company, 1922-27; Bell Telephone Laboratories, 192 7-. Mr. Ohl has been engaged in various exploratory phases of radio research, the results of which have led to nu- merous patents. For the past ten or more years he has been working on some of the problems encountered in the use of millimeter radio waves. G. L. Pearson, A. B., Willamette University, 1926; M. A. in Physics, Stanford University, 1929. Bell Telephone Laboratories, 1929-. Mr. Pearson 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. J. H. ScAFF, B.S.E. in Chemical Engineering, University of Michigan, 1929. Bell Telephone Laboratories, 1929-. Mr. Scaff's early work in the Laboratories was concerned with metallurgical investigations of impurities in metals with particular reference to soft magnetic materials. During the war he was project engineer for the development of silicon and germanium crystal rectifiers for radar applications. At the present time, he is re- sponsible for metallurgical work on varistor and magnetic materials. I. G. Wilson, B.S. and M.E., University of Kentucky, 1921. Western Electric Co., EngineeringDepartment, 1921-25. Bell Telephone Labora- tories, 1925-. Mr. Wilson has been engaged in the development of am- plifiers for broad-band systems. During the war he was project engineer in charge of the design of resonant cavities for radar testing. VOLUME XXVI APRIL, 1947 no. 2 THE BELL SYSTEM TECHNICAL JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION Publk Ubnn Radar Antennas H. T. Friis and W. D. Lewis 219 Probability Functions for the Modulus and Angle of the Normal Complex Variate Ray S. Hoyt 318 Spectrum Analysis of Pulse Modulated Waves /. C. Lozier 360 Abstracts of Technical Articles by Bell System Authors. . 388 Contributors to This Issue 394 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 50i 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 W. H. Harrison O. E. Buckley O. B. Blackwell M. J. KeUy H. S. Osborne A. B. Clark J. J. Pilliod S. Bracken «■■■■»■■ 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. 1947 American Telephone and Telegraph Company PRINTED IN U. S. />. BOOKS AND PERIODICALS FOR WORLD RECOVERY The desperate and continued need for American publica- tions to serve as tools of physical and intellectual recon- struction abroad has been made vividly apparent by appeals from scholars in many lands. The American Book Center for War Devastated Libraries has been urged to continue meeting this need at least through 1947. The Book Center is therefore making a renewed appeal for American books and periodicals — for technical and scholarly books and periodicals in cdl fields and particularly for puhlicalions of the past ten years. We shall particularly welcome complete or incomplete files of the Bell System Technical Journal. The generous support ^^■hich has been given to the Book Center has made it possible to ship more than 700,000 volumes abroad in the past year. It is hoped to double this amount before the Book Center closes. The books and periodicals which individuals as well as institutional libraries can spare are urgently needed and will help in the reconstruction A\hich must preface world understanding and peace. Ship your contributions to the American Book Center, % The Library of Congress, Washington 25, D. C, freight prepaid, or write to the Center for fiu-therinfoiTiiation. The Bell System Technical Journal Vol. XXVI April, 1947 No. 2 Radar Antennas By H. T. FRIIS and W. D. LEWIS Table of Contents Introduction 220 Part I — Electrical Principles 224 1 . General 224 2. Transmission Principles 226 2 . 1 Gain and Effective Area of an Antenna 226 Definition of Gain 226 Definition of Effective Area 226 2.2 Relationship between Gain and Effective Area 227 2.3 The Ratio G/A for a Small Current Element 227 2.4 The General Transmission Formula 230 2.5 The Reradiation Formula 230 2.6 The Plane, Linearly Polarized Electromagnetic Wave 231 3. Wave Front Analysis 232 3 . 1 The Huygens Source 233 3.2 Gain and Effective Area of an Ideal Antenna 235 i.i Gain and Effective Area of an Antenna with Aperture in a Plane and with Arbitrary Phase and Amplitude 236 3.4 The Significance of the Pattern of a Radar Antenna 237 3.5 Pattern in Terms of Antenna Wave Front 238 3.6 Pattern of an Ideal Rectangular Antenna 239 3.7 Effect on Pattern of Amplitude Taper 240 3.8 Effect on Pattern of Linear Phase Variation 241 3 . 9 Effect on Pattern of Scjuare Law Phase Variation 242 3. 10 Effect on Pattern of Cubic Phase Variation 244 3.11 Two General Methods 245 3. 12 Arrays 246 3. 13 Limitations to Wave Front Theory 246 4. Application of General Principles 247 Part II — Methods of Antenna Construction 247 5. General 247 6. Classification of Methods 248 7. Basic Design Formulation 250 7 . 1 Dimensions of the Aperture 250 7 . 2 Amplitude Distribution 251 7 . 3 Phase Control 251 8. Parabolic Antennas 251 8.1 Control of Phase 251 8.2 Control of Amplitude 253 8 . 3 Choice of Configuration 254 8.4 Feeds for Paraboloids 258 8.5 Parabolic CyUnders between Parallel Plates 260 8.6 Line Sources for Parabolic Cylinders 262 8.7 Tolerances in Parabolic Antennas 264 9. Metal Plate Lenses 266 9. 1 Lens Antenna Configurations 269 9.2 Tolerances in Metal Plate Lenses 269 9.3 Advantages of Metal Plate Lenses 270 10. Cosecant Antennas 270 10. 1 Cosecant Antennas based on the Paraboloid 271 10.2 Cylindrical Cosecant Antennas 274 219 220 BELL S YS TEM TECH NIC A L JO URN A L 1 1 . Lobing 274 11.1 Lobe Switching 275 11.2 Conical Lolling 276 12. Rapid Scanning 276 12.1 Mechanical Scanning 277 12.2 .\rray Scanning 278 12.3 Optical Scanning 282 Part III — Military Radar Antennas Developed by the Bell Laboratories 284 13. General .' 284 14. Naval Shipborne Radar Antennas 286 14. 1 The SE Antenna 286 14.2 The SL .\ntenna 286 14.3 The SJ Submarine Radar Antenna 291 14.4 The Modified SJ/Mark 27 Radar Antenna 294 14.5 The SH and Mark 16 Radar Antennas 294 14.6 Antennas for Early Fire Control Radars 297 14. 7 \ Shipborne .\nti-Aircraft Eire Control Antenna 298 14.8 The Polyrod Eire Control Antenna '. . 300 14.9 The Rocking Horse Eire Control Antenna 301 14. 10 The Mark 19 Radar Antenna 302 14. 1 1 The Mark 28 Radar Antenna 305 14. 12 A 3 cm Anti-.\ircraft Radar Antenna 307 15. Land Based Radar Antennas 307 15. 1 The SCR-545 Radar " Search" and "Track" Antennas 307 15.2 The AN/TPS-IA Portable Search Antenna 309 16. Airborne Radar Antennas 312 16. 1 The AN/APS-4 Antenna 312 16.2 The SCR-520, SCR-717 and SCR-720 .Antennas 313 16.3 The AN/APQ-7 Radar Bombsight Antenna 315 Introduction "O ADAR proved to be one of the most important technical achieve- -'-^ ments of World War II. It has many sources, some as far back as the nineteenth century, yet its rapid wartime growth was the result of military necessity. This development will continue, for radar has increasing applications in a peacetime world. In this paper we will discuss an indispensable part of radar — the antenna. In a radar system the antenna function is two-fold. It both projects into space each transmitted radar pulse, and collects from space each received reflected signal. Usually but not always a single antenna performs both functions. The effectiveness of a radar is influenced decisively by the nature and quality of its antenna. The greatest range at which the radar can de- tect a target, the accuracy with which the direction to the target can be determined and the degree with which the target can be discriminated from its background or other targets all depend to a large e.xtent on electrical properties of the antenna. The angular sector which the antenna can mechanically or electrically scan is the sector from which the radar can provide information. The scanning rate determines the frequency with which a tactical or navigational situation can be ex- amined. RADAR A NTENNA S 221 Radar antennas are as numerous in kind as radars. The unique character and particular functions of a radar are often most clearly evident in the design of its antenna. Antennas must be designed for viewing planes from the ground, the ground from planes and planes from other planes. They must see ships from the shore, from the air, from other ships, and from submarines. In modern warfare any tactical situation may require one or several radars and each radar must have one or more antennas. Radar waves are almost exclusively in the centimeter or microwave region, yet even the basic microwave techniques are relatively new to the radio art. Radar demanded antenna gains and directivities far greater than those previously employed. Special military situations required antennas with beam shapes and scanning characteristics never imagined by communication engineers. It is natural that war should have turned our efforts so strongly in the direction of radar. But that these efforts were so richly and quickly rewarded was due in large part to the firm technical foundations that had been laid in the period immediately preceeding the war. When, for the common good, all privately held technical information was poured into one pool, all ingredients of radar, and of radar antennas in particular, were found to be present. A significant contribution of the Bell System to this fund of technical knowledge was its familiarity with microwave techniques. Though Hertz himself had performed radio experiments in the present micro- wave region, continuous wave techniques remained for decades at longer wavelengths. However, because of its interest in new communication channels and broader bands the Bell System has throughout the past thirty years vigorously pushed continuous wave techniques toward the direction of shorter waves. By the middle nineteen-thirties members of the Radio Research Department of the Bell Laboratories were work- ing within the centimeter region. Several aspects of this research and development appear now as particularly important. In the first place it is obvious that knowledge of how to generate and transmit microwaves is an essential factor in radar. Many lower frequency oscillator and transmission line tech- niques are inapplicable in the microwave region. The Bell Laboratories has been constantly concerned with the development of generators which would work at higher and higher frequencies. Its broad famil- iarity with coaxial cable problems and in particular its pioneering work with waveguides provided the answers to many radar antenna problems. Another telling factor was the emphasis placed upon measurement. Only through measurements can the planners and designers of equip- 222 BELL SYSTEM TECHNICAL JOURNAL ment hope to evaluate performance, to chose between alternatives or to see the directions of improvement. Measuring technicjues employing double detection receivers and intermediate frequency amplifiers had long been in use at the Holmdel Radio Laboratory. By employing these techniques radar engineers were able to make more sensitive and accurate measurements than would have been possible with single de- tection. Antennas are as old as radio. Radar antennas though different in form are identical in principle with those used by Hertz and Marconi. Consequently experience with communication antennas provided a valuable background for radar antenna design. As an example of the importance of this background it can be recalled that a series of experi- Fig. 1 — An Electromagnetic Horn. ments with short wave antennas for Transatlantic radio telephone service had culminated in 1936 in a scanning array of rhombic antennas. The essential principles of this array were later applied to shipborne fire control antenna which was remarkable and valuable because of the early date at which it incorporated modern rapid scanning features. In addition to the antenna arts which arose directly out of communi- cation problems at lower frequencies some research specifically on micro- wave antennas was under way before the war. Earl\- workers in wave- guides noticed that an open ended waveguide will radiate directly into space. It is not suri)rising therefore that these workers developed the electromagnetic horn, which is essentially a waveguide tapered out to an aperture (Fig. 1). One of the first used and simplest radio antennas is the dipole (Fig. MDAR ANTENNAS 111 2). Current oscillating in the dipole generates electromagnetic waves which travel out with the velocity of light. A single dipole is fairly non-directive and consecjuently produces a relatively weak, field at a distance. When the wave-length is short the field of a dipole in a i^ o o Fig. 2 — A Microwave Dipole. Fig. 3 — x\ Dipole Fed Paraboloid. chosen direction can be increased many times by introducing a re- flector which directs or 'focusses' the energy. In communication antennas the focussing reflector is most com- monly a reflecting wire array. Even at an early date in radar the wave- length was so short that 'optical' reflectors could be used. These were 224 BELL S YSTEM TECH NIC A L JOURNA L sometimes paraboloids similar to those used in searchlights (Fig. 3). Sometimes they were parabolic cylinders as in the Mark III, an early shipl)orne fire control radar developed at the Whippany Radio Labora- tory. From these relatively simple roots, the communication antenna, the electromagnetic horn and the optical reflector, radar antennas were developed tremendously during the war. That this development in the Bell Laboratories was so well able to meet demands placed on it was due in large part to the solid foundation of experience possessed by the Research and Development groups of the Laboratories. Free inter- change of individuals and information between the Laboratories and other groups, both in the United States and Great Britain, also con- tributed greatly to the success of radar antenna development. Because of its accelerated wartime expansion the present radar an- tenna field is immense. It is still growing. It would be impossible for any single individual or group to master all details of this field, yet its broad outline can be grasped without "difficulty. The purpose of this paper is two-fold, both to provide a general dis- cussion of radar antennas and to summarize the results of radar antenna research and development at the Bell Laboratories. Part I is a dis- cussion of the basic electrical principles which concern radar antennas. In Part II we will outline the most common methods of radar antenna construction. Practical military antennas developed by the Bell Laboratories will be described in Part III. The reader who is interested in general familiarity with the over all re- sult rather than with technical features of design may proceed directly from this part to Part III. PART I ELECTRICAL PRINCIPLES 1. General Radar antenna design depends basically on the same broad principles which underlie any other engineering design. The radar antenna designer can afford to neglect no aspect of his problem which has a bearing on the final product. Mechanical, chemical, and manufacturing considerations are among those which must be taken into account. It is the electrical character of the antenna, however, which is connected most directly with the radar performance. In addition it is through atten- tion to the electrical design problems that the greatest number of novel antennas have been introduced and it is from the electrical viewpoint that the new techniques can best be understood. An antenna is an electromagnetic device and as such can be understood RADAR ANTENNAS 225 through the appUcation of electromagnetic theory. Maxwell's equations provide a general and accurate foundation for antenna theory. They are the governing authority to which the antenna designer may refer directly when problems of a fundamental or bafHing nature must be solved. It is usually impracticable to obtain theoretically exact and simple solu- tions to useful antenna problems by applying Maxwell's Equations directly. We can, however, use them to derive simpler useful theories. These theories provide us with powerful analytical tools. Lumped circuit theory is a tool of this sort which is of immense practical importance to electrical and radio engineers. As the frequency becomes higher the approximations on which lumped circuit theory is based become inaccurate and engineers find that they must consider distributed in- ductances and capacitances. The realm of transmission line theory has been invaded. Transmission line theory is of the utmost importance in radar antenna design. In the first place the microwave energy must be brought to the antenna terminals over a transmission line. This feed line is usually a coaxial or a wave-guide. It must not break down under the voltage which accompanies a transmitted pulse. It must be as nearly lossless and reflec- tionless as possible and it must be matched properly to the antenna terminals. The importance of a good understanding of transmission line theory does not end at the antenna terminals. In any antenna the energy to be trans- mitted must be distributed in the antenna structure in such a way that the desired radiation characteristics will be obtained. This may be done with transmission lines, in which case the importance of transmission line theory is obvious. It may be done by 'optical' methods. If so, certain trans- mission line concepts and methods will still be useful. While it is true that transmission line theory is important it is not nec- essary to give a treatment of it in this paper. Adequate theoretical dis- cussions can be found elsewhere in several sources.^ It is enough at this point to indicate the need for a practical understanding of transmission line principles, a need which will be particularly evident in Part II, Methods of Antenna Construction. We may, if we like, think of the whole radar transmission problem in terms of transmission line theory. The antenna then appears as a trans- former between the feed line and transmission modes in free space. We cannot, however, apply this picture to details with much effectiveness unless we have some understanding of radiation. In the sections to follow we shall deal with some theoretical aspects of radiation. We shall begin with a discussion of fundamental transmission 1 See, for example, S. A. Schelkunoff, Electromagnetic Waves, D. Van Nostrand Co., Inc., 1943, in particular. Chapters VII and VIII, or F. E. Terman, Radio Engineer's Hand- book:, McGraw-Hill Book Co., Inc., 1943, Section 3. 226 BELL SYSTEM TECHNICAL JOURNAL principles. This discussion is applicable to all antennas regardless of how they are made or used. When applied to radar antennas it deals chiefly with those properties of the antenna which affect the radar range. Almost all microwave radar antennas are large when measured in wave- lengths. When used as transmitting antennas they produce desired radia- tion characteristics by distributing the transmitted energy over an area or Svave front'. The relationships between the phase and amplitude of elec- trical intensity in this wave front and the radiation characteristics of the antenna are predicted by 'ivave front analysis. Wave front analysis is essentially the optical theory of diffraction. Although approximate it applies excellently to the majority of radar antenna radiation problems. We shall discuss wave front analysis in Section 3. 2. Transmission Principles 2.1 Gain and Effective Area of an Antenna An extremely important property of any radar antenna is its ability to project a signal to a distant target. The gain of the antenna is a number which provides a quantitative measure of this ability. Another important property of a radar antenna is its ability to collect reflected power which is returning from a distant target. The efectiie area of the antenna is a quantitative measure of this ability. In this section these two quantities will be defined, and a simple relation between them will be derived. Their importance to radar range will be established. Definition of Gain. When power is fed into the terminals of an antenna some of it will be lost in heat and some will be radiated. The gain G of the antenna can be defined as the ratio G = P/Po (1) where P is the power flow per unit area in the plane linearly polarized elec- tromagnetic wave which the antenna causes in a distant region usually in the direction of maximum radiation and Po is the power flow per unit area which would have been produced if all the power fed into the terminals had been radiated equally in all directions in space. Definition of Effective Area. When a plane linearly polarized electromag- netic wave is incident on the receiving antenna, received power Pr will be available at the terminals of the antenna. The effective area of the antenna is defined, by the equation A = Pn/P' (2) where P' is the j^ower per unit area in the incident wave. In other words the received power is equal lo ihc j)ower flow through an area that is equal to the effective area of the antenna. RA DA RAN TENNA S 227 2.2 Relationship behveen Gain and Efeclive Area Figure 4 shows a radio circuit in free space made up of a transmitting antenna T and a receiving antenna R. If the transmitted power 7^r had TRANSMITTING ANTENNA Fig. 4 — Radio Circuit in Free Space. been radiated equally in all directions, the power flow per unit area at the receiving antenna would be 47r(/2 Definition (1) gives, therefore, for the power flow per unit area at the receiving antenna P = p,Gr = ^" (4) and definition (2) gives for the received power ^« = ''■'' = '-^ (') From the law of reciprocity it follows that the same power is transferred if the transmitting and receiving roles are reversed. By (5) it is thus evident that KJT-Aji = QtrAt or Gt/At = Gr/Ar (6) Equation (6) shows that the ratio of the gain and effective area has the same constant value for all antennas at a given frequency. It is necessary, therefore, to calculate this ratio only for a simple and well known antenna such as a small dipole or uniform current element. 2.3 The Ratio G/A for a Small Current Element In Fig. 5 are given formulas' in M.K.S. units for the free space radiation from a small current element with no heat loss. We have assumed that 2 See S. A. Schelkunoff, Electromagnetic Waves, D. Van Nostrand Co., Inc., 1943, p. 133 228 BELL SYSTEM TECHNICAL JOURNAL X CURRENT ELEMENT (LENGTH i METERS) (i<< A) MAGNETIC INTENSITY^H, le -^~T^ — -e ^ 5IN9 L KV AMPERES METER ELECTRIC INTENSITY = Eg = 120TrH<(, ^ ^^^ I I fr^l'^ p WATTS POWER FLOW =P = |H4,Ee| = 30^^— J SIN'^e y^^^^^ Z o ■ r. .^ fr^l^ WATTS P 15 MAXIMUM FOR 6=90. ce., P^^ =30Tr|^— J -jj^^r^z U) (i) POWER FLOW ACROSS SPHERE OF RADIUS r OR r^ n -61 ^ TOTAL RADIATION = W =/ P2TTr SINS rde = 80Tt2 I yJ WATTS (s) ,2 (6) (7) RADIATION RESISTANCE = R r^q " T? " ^°'" TJ ^^^^ BY (4) AND (5) : P MAX anr^ W WATTS METERS Fig. 5 — Free Space Radiation from a Small Current Element with Uniform Current I Amperes over its Entire Length. this element is centered at the origin of a rectangular coordinate system and that it lies along the Z axis. At a large distance r from the element RADAR ANTENNAS 229 the maximum power flow per unit area occurs in a direction normal to it and is given by _ 3W w^atts ,,_. SttH meter^ where T'F is the total radiated power. If W had been radiated equally in all directions the power flow per unit area would be p ^ W_ watts .gv 47rr2 meters^ It follows that the gain of the small current element is p Gdiople = -^— = 1-5 (9) The effective area of the dipole will now be calculated. When it is used to receive a plane linearly polarized electromagnetic wave, the available output power is equal to the induced voltage squared divided by four times the radiation resistance. Thus Pn = ^ Watts (10) 4i?rad where E is the effective value of the electric field of the wave, i is the length of the current element and i?rad is the radiation resistance of the current element. From Fig. 5 we see that i?rad — , ohms. Since the power A" flow per unit area is equal to the electric field squared divided by the im- pedance of free space, in other words Po — tt— we have u. 1 ZOir P ^X^ ^dipoie = ^ = ^- meter" (11) We combine formulas (9) and (11) to find that 6^dipole _ 4t ■^dipole A Since, as proved in 2.2 this ratio is the same for all antennas, it follows that for any antenna ^=^ (12) 230 BELL S YSTEM TECH NIC A L JOURNA L 2.4 The General Transmission Formula Transmission loss between transmitter and receiver through the radio circuit shown in Fig. 4 was given by ecjuation (5). By substituting the relation (12) into (5) w-e can obtain the simple free space transmission formula: Ph = Pt 4^" watts (13) Although this formula applies to free space only it is believed to be as useful in radio engineering as Ohm's law is in circuit engineering. 2.5 The Reradialion Formula One further relation, the radar reflection formula is of particular interest. Consider the situation illustrated in Fig. 6. Let Pt be the power radiated REFLECTING OBJECT (As= PROJECTED AREA IN DIRECTION OF RADAR) RADAR ' At.Gt TRANSMITTER h- Ar,Gr Fig. 6 — Radar with Separate Receiving and Transmitting Antennas. from an antenna with effective area A t, As the area of a reflecting object at distance d from the antenna and Ph the power received by an antenna of effective area ^k . By equation (13) the power striking As is — — — — . If this power were reradiated equally in all directions the reflected power flow at the receiving antenna would be — — 3—— but since the average reradiation is larger toward the receiving antenna, the power flow per unit area there is usually K J ,J^f where A' > 1. It follows from (2) that 4Trd*\^ r> T^r PtAtArAs (..s. Formula (14) shows clearly why the use of large and efflcient antennas will greatly increase the radar range. Formula (14) applies to free space only. Application to other conditions RADAR A NTENNAS 231 may require corrections for the effect of the "ground", and for the effect of the transmission medium, which are beyond the scope of this paper. 2.6 The Plane, Linearly Polarized Electromagnetic Wave In the foregoing sections we have referred several times to 'plane, linearly polarized electromagnetic waves'. These waves occur so commonly in antenna theory and practice that it is worth while to discuss them further here. Some properties of linearly polarized, plane electromagnetic waves are illustrated in Fig. 7. At any point in the wave there is an electric field and a magnetic field. These fields are vectorial in nature and are at right angles to each other and to the direction of propagation. It is customary to give the magnitude of the electric field only. If we use the M.K.S. system of units the magnitudes of the fields are e.xpressed in familiar units. Electric intensity appears as volts per meter and magnetic intensity as amperes per meter. The ratio of electric to magnetic intensity has a value of 1207r or about 377 ohms. This is the 'impedance' of free space. The power flow per unit area is e.xpressed in watts per square meter. We see, therefore, that the electromagnetic wave is a means for carrying energy not entirely unlike a familiar two wire line or a coaxial cable. Electromagnetic waves are generated when oscillating currents flow in conductors. We could generate a plane linearly polarized electromagnetic wave with a uniphase current sheet consisting of a network of fine wires backed up with a conducting reflector as shown in Fig. 7. This wave could be absorbed by a plane resistance sheet with a resistivity of 377 ohms, also backed up by a conducting sheet. The perfectly conducting reflecting sheets put infinite impedances in parallel with the current sheet and the resistance sheet, since each of these reflecting sheets has a zero impedance at a spacing of a quarter wavelength. A perfectly plane electromagnetic wave can exist only under certain ideal conditions. It must be either infinite in extent or bounded appropriately by perfect electric and magnetic conductors. Nevertheless thinking in terms of plane electromagnetic waves is common and extremely useful. In the first place the waves produced over a small region at a great distance from any radiator are essentially plane. Arguments concerning receiving antennas therefore generally assume that the incident waves are plane. In the second place an antenna which has dimensions of many wavelengths can be analyzed with considerable profit on the basis of the assumption that it transmits by producing a nearly plane electromagnetic wave across its aperture. This method of analysis can be applied to the majority of micro- wave radar antennas, and will be discussed in the following sections. 232 BELL SYSTEM TECHNICAL JOURNAL 3. Wave Front Analysis The fundamental design question is "How to get what we want?" In a radar antenna we want specified radiation characteristics; gain, pattern and polarization. Electromagnetic theory tells us that if all electric and magnetic currents in an antenna are known its radiation characteristics may be derived with the help of Maxwell's Equations. However, the es- sence of electromagnetic theory insofar as it is of use to the radar antenna WAVE GENERATOR REFLECTING SHEET A ^ CURRENT SHEET WAVE RECEIVER REFLECTING SHEET RESISTANCE SHEET i 2rr ^ MAGNETIC INTENSITY = H = Ie"~?r AMPERES METER ELECTRIC INTENSITY= E = 120nH ^OLTS METER POWER FLOW = P = EH ^^'^ ^^ s METERS CURRENT DENSITY^I At^^^^J^^^ METER RESISTIVITY =R = 120Tr 0.HM5 Fig. 7 — Linearly Polarized Plane Electromagnetic Waves. designer can usually be expressed in a simpler, more easily visualized and thus more useful form. This simpler method we call wave front analysis. In a transmitting microwave antenna the power to be radiated is used to produce currents in antenna elements which are distributed in space. This distribution is usually over an area, it may be discrete as with a dipole array or it may be continuous as in an electromagnetic horn or paraboloid. These currents generate an advancing electromagnetic wave over the aperture of RADAR ANTENNAS 233 the antenna. The amplitude, phase and polarization of the electric intensity in portions of the wave are determined by the currents in the antenna and thus by the details of the antenna structure. This advancing wave can be called the 'wave front' of the antenna. When the wave front of an antenna is known its radiation characteristics may be calculated. Each portion of the wave front can be regarded as a secondary or 'Huygens' source of known electric intensity, phase and polari- zation. At any other point in space the electric intensity, phase and polari- zation due to a Huygens source can be obtained through a simple expression given in the next section. The radiation characteristics of the antenna can be found by adding or integrating the effects due to all Huygens sources of the wave front. This procedure is based on the assumption that the antenna is transmit- ting. A basic law of reciprocity assures us that the receiving gain and radia- tion characteristics of the antenna will be identical with the transmitting ones when only linear elements are involved. This resolution of an antenna wave front into an array of secondary sources can be justified within certain limitations on the basis of the induc- tion theorem of electromagnetic theory. These limitations are discussed in a qualitative way in section 3.13. 3.1 The Huygens Source Consider an elementary Huygens source of electric intensity £opolarized parallel to the X axis with area dS in the XY plane (Fig. 8). This can be thought of as an element of area dS of a wave front of a linearly polarized plane electromagnetic wave which is advancing in the positive z direction.^ From Maxwell's Equations we can determine the field at any point of space due to this Huygens Source. The components of electric field, are found to be Ee = t — — e (1 + cos 6) cos <^ , , Tkr (l5) Ea, = —I — - — e (1 -1- cos 6) sm </> 2Kr where X is the wavelength. We see at once that this represents a vector whose absolute magnitude at all points of space is given by \E\ =^(l-^cose). (16) ^ S. A. Schelkunoff, Loc. Cit., Chap. 9. 234 BELL SYSTEM TECHNICAL JOURNAL Here Ef^dS is an amplilude factor which depends on the wavelength, intensity and area of the elementar}' source and \/r is an amplitude factor which specilies the \ariation of field with distance. (1 + cos 6) is an amjilitude factor which shows that the directional pattern of the elementary source is a cardioid with maximum radiation in the direction of propagation and no radiation in the reverse direction. When we use the properties of the Huygens source in analyzing a micro- Fig. 8 — The Huygens Source. wave antenna we are usually concerned principally with radiation in or near the direction of propagation. For such radiation Equation 16 takes a par- ticularly simple form in Cartesian Coordinates E, .£^^_,(,WX)r.^^^Q.^^^Q_ (17) This represents an electric vector nearly parallel to the electric vector of the source. The amplitude is given by the factor ^ and the phase by the RADAR ANTENNAS 235 factor i e *''^''' ^'^. With this equation as a basis we will now proceed to study some relevant matters concerning radar antennas. 3.2 Gain and EJJective Area of an Ideal Anlenna On the basis of (17) we can now determine the gain of an ideal antenna of area S {S ^ X^). This antenna is assumed to be free of heat loss and to transmit by generating an advancing wave which is uniform in phase and amplitude in the XY plane. Let the electric intensity in the wave front of Fig. 9 — An Ideal Antenna. the ideal antenna be E^ polarized parallel to the X axis (Fig. 9). The trans- mitted power Pr is equal to the power flow through S and is given by (18) At a point Q on the Z axis the electric intensity is obtained by adding the effects of all the Huygens sources in S. If the distance of Q from is so great that r = d + ^ 236 BELL S YSTEM TECH NIC A L JOURNA L where A is a negligibly small fraction of a wavelength for every point on .9 then we see from (17) that the electric vector at Q is gi\en by Js \r Xd The power flow per unit area at Q is therefore 1 £^5' PtS P = UOir \W \H' Po the power flow per unit area at Q when power is radiated isotropically from is found by assuming that Pt is spread evenly over the surface of a sphere of radius d. The gain of a lossless, uniphase, uniamplitude, linearly polarized antenna is, by the definition of equation 1, the ratio of 19 and 20. It follows from 12 that the effective area of the ideal antenna is A ^ S (22) In other words in this ideal antenna the effective area is equal to the actual area. This is a result which might have been obtained by more direct arguments. 3.3 Gain and Efeclive Area of an A ntenna with Aperture in a Plane and with Arbitrary Phase and Amplitude Let us consider an antenna with a wave front in the XY plane which has a known phase and amplitude variation. Let the electric intensity in the wave front be E{x, y) = Eoaix, y)e'*^''''^ (23) polarized parallel to the x axis. The radiated power is equal to the power flow through 5 and is given by _ E'o I a'{x, y) dS P... = " J " " (24) 1207r The input power to the antenna is Pt = PradA (25) RADAR AN TENNA S liT where Z is a loss factor (< 1). At a point Q on the Z axis the electric inten- sity is obtained by adding the effects of all the Huygens sources in S. If OQ is as great as in the above derivation for the gain of an ideal antenna then we see from 17 that the electric intensity at Q is £x = i ^^^- £o I a{x, y)e"^^'-'US; Ey = 0; E, = 0. (26) Ad J The power flow per unit area at Q is given by ^^T^rl^-I' (27) and Po the power flow per unit area at Q when Pt is radiated isotropically is given by equation (3). The power gain of the antenna, by definition 1 is therefore Po 1207r / 47rrf2 x2 f a{x, y)6'*^- dS / a(x, y) Js (28) dS The gain expressed in db is given by Gdb = 10 log.o G (29) We combine 12 and 28 to obtain A = L I a{x,y)e'*^'''''dS (30) / a^{x, y) dS a formula for the effective area of the antenna. 3.4 The Significance of the Pattern of a Radar A ntenna The accuracy with which a radar can determine the directions to a target depends upon the beam widths of the radar antenna. The ability of the radar to separate a target from its background or distinguish it from other targets depends upon the beam widths and the minor lobes of the radar antenna. The efficiency with which the radar uses the available power to view a given region of space depends on the beam shape of the antenna. These quantities characterize the antenna pattern. In the following sec- tions means for the calculation of antenna patterns in terms of wave front theory will be developed, and some illustrations will be given. 238 BELL SYSTEM TECHNICAL JOURNAL 3.5 Pattern in Terms of Antenna Wave Front If the relative phase and amplitude in a wave front are given by E{x, y) = a(x, y)e"''^''' (31) the relative phase and amplitude at a distant point Q not necessarily on the Z axis (Fig. 10) in the important case where the angle QOZ between the direction of propagation and the direction to the point is small, is given from (17) by adding the contributions at Q due to all parts of the wave front. This gives Xa Js dS. (32) Fig. 10 — Geometry of Pattern Analysis. The quantity r in (32) is the distance from any point P with coordinates .r, y, 0, in the XY, plane to the point Q (Fig. 10). Simple trigonometry shows that when OQ is very large r = d — X sin a — y sin ^ (33) where d is the distance OQ, a is the angle ZOQ' between OZ and OQ' the projection of OQ on the XZ plane and /3 is similarly the angle ZOQ". The substitution of 33 into 32 gives Eo = • -i(2WX)d ^] *^ i ^ t(2ir/X)(T8ino+i/sin/3) + i \d f *(!,!/) a{x, y) dS. (34) RADAR ANTENNAS 239 In most practical cases this equation can be simplified by the assumptions cf>(x,y) = <t>'{x) + ct>"iy) a{x,y) = a'ix)a''{y) from which it follows that I £q I = Fid)Fia)F(fi) (35) where F{d) is an amplitude factor which does not depend on angle, F{a) = j e*'^-'^''^"'""+'*'^^^a'(x)^x- (36) is a directional factor which depends only on the angle a and not on the angle (8 or d, and F(/3) similarly depends on /3 but not on a or d. The pattern of an antenna can be calculated with the help of the simple integrals as in 36, and illustrations of such calculations will be given in the following sections. 3.6 Pattern of an Ideal Rectangular Antenna Let the wave front be that of an ideal rectangular antenna of dimensions a, b ; with linear polarization and uniform phase and amplitude. The dimen- sions a and b can be placed parallel to the .Y and F axes respectively as sketched in Fig. 9. Equation 36 then gives F{a) = r'\'''-'^''^'''" dx = a'-^ (37) J-al2 W , , X a sin a where ^ = . Similarly F^0)=b'^ (38) where i/' = , _ TT 6 sin /3 The pattern of the ideal rectangular aperture, in other words the distribution of electrical field in angle is thus given approximately by F(a)F(ff) = ai'^'^ . (39) The function is plotted in Fig. 11. It is perhaps the most useful function of antenna theory, not because ideal antennas as defined above are particularly desirable in practice but because they provide a simple stand- 240 BELL SYSTEM TECHNICAL JOURNAL ard with which more useful but more complex antennas can profitably be compared. 3.7 Efect OH Pattern oj Amplitude Taper The — — pattern which results from an ideal wave front has undesirably high minor lobes for most radar applications. These minor lobes will be reduced if the wave front of constant amplitude is replaced by one which retains a constant phase but has a rounded or 'tapered' amplitude dis- tribution. OFF AXIS \l /APERTURE UNIFORM PHASE AND AMPLITUDE ACROSS APERTURE -5n -AV\ -3n -2TT -no n 2n 3n 4tt 5n ^ TTO SIN a Fig. 1 1 — Pattern of Ideal Rectangular Antenna. If such an amplitude taper is represented analytically by the function a'{x) = Ci + C'i cos ttx (40) then equation (36) is readily integrable. To integrate it we utilize the identity cos — = a 2 upon which the integral becomes the sum of three simple integrals of the form . ka ,an sm e""dx = a all ka y (41) RADAR ANTENNAS 241 We therefore obtain , . sin ^ C2 F{a) = aCx — ^ + ^ y sin (.+i)^ sin(,--y U^-^d (*-^) (42) The patterns resulting from two possible tapers are given by substi- tuting Ci == 0, C2 = 1 and Ci = 1/3, C2 = 2/3 in (42). These patterns are sin a evidently calculable in terms of the known function . They are plotted a in Figs. 12 and 13. 0.8 < 0.2 - 5n -4TT ■3n -2n -non ... no sma 3TT 4-n Fig. 12 — Pattern of Tapered Rectangular Antenna. It will be observed that minor lobe suppression through tapering is ob- tained at the expense of beam broadening. In addition to this the gain is reduced by tapering, as could have been calculated from 28. These unde- sirable effects must be contended with in any practical antenna design. The choice of taper must be made on the basis of the most desirable com- promise between the conflicting factors. 3.8 Efect on Pattern of Linear Phase Variation If we assume a constant amplitude and a linear phase variation 4>'{x) = —k\x 242 BELL SYSTEM TECHNICAL JOURNAL over an aperture —a/2 < x < a/2 then 36 becomes a simple integral of the form (41) and we obtain sin xp" „ ira . kia . . /' (a) = a —777— where \f/ — — sm a — --- (43) yp A 2 The physical interpretation of^(43) is simply that the pattern is identical to the pattern of an antenna with constant amplitude and uniform phase but rotated through an angle 6 where sm 6 = — — 27r - 2"n -non u,_ no SIN a 2rr Fig. 13 — Pattern of Tapered Rectangular Antenna. Simple examination shows that the new direction of the radiation maximum is at right angles to a uniphase surface, as we would intuitively expect. This phenomenon has particular relevance to the design of scanning antennas. 3.9 Effect on Pattern of Square Law Phase Variation If we assume a constant amplitude and a square law phase variation (t)'{x) = —kix over the aperture a/2 < x < a/2 then the substitution 27r . X = 1 i2 L sm a X + 2k 2 _ (44) RADAR ANTENNAS reduces (36) to the form /- (a) = - e V '^ ^ - e k-2 J Equation (45) can be evaluated with the help of Fresnel's Integrals [ cos X' dX, j sin X' dX dX 243 (45) ANGLE f A 1 OFF AXIS / • * \ \ N t — \ < 4 Uq 1 / i p J\ Ab) /v n 2n -2n -n y ^ _ j]_g__5iN_a -2n -no T[ 2T\ -2n -n o tt 2n Fig. 14 — Patterns of Rectangular Apertures with Square Law Phase Variation. which are tabulated^, or from Cornu's Spiral which is a convenient graphical representation of the Fresnel Integrals. Typical computed patterns for apertures with square law phase variations are plotted in Fig. 14. These theoretical curves can be applied to the fol- lowing important practical problems. (1) The pattern of an electromagnetic horn. ■• For numerical values of Fresnel's Integrals and a plot of Cornu's Spiral see Jahnke and Emde, Tables of Functions B, G, Teubner, Leipzig, 1933, or Dover Publications, New York Citv, 1943, 244 BELL S YS TEM TECH NIC A L JOURNA L (2) The defocussing of a reflector or lens due to improper placing of the primary feed. (3) The defocussing of a zoned reflector or lens due to operation at a fre- quency off mid-band. In addition to providing distant patterns of apertures with curved wave fronts (44) provides theoretical 'close in' patterns of antennas with plane wave fronts. This arises from the simple fact that a plane aperture appears as a curved aperture to close in points. The degree of curvature depends on the distance and can be evaluated by extremely simple geometrical con- siderations. When this has been done we find that Fig. 14 represents the so-called Fresnel diffraction field. With this interpretation of square law variation of the aperture we can examine several additional useful problems. We can for instance justify the commonly used relation for the minimum permissible distance of the field source from an experi- mental antenna test site. This distance produces an effective phase curva- ture of X/16. We can examine optical antenna systems employing large primary feeds, in particular those employing parabolic cylinders illuminated by line sources. 3.10 Ejffed on Pattern of Cubic Phase Variation If we assume a constant amplitude and a cubic phase variation <l>'{x) = — kzx over the aperture from — a/2 < x < a/2 then equation (36) becomes F{a) = f"'e-"^'.e''^''''>"'°".(ix (46) J- a/2 If ksx < ~ then it is a fairly good approximation to write e-^'l^' = I - ikW - ^Af -^ ... (47) from which it follows that (46) can be integrated since it reduces to a sum of three terms each of which can be integrated. Typical computed patterns for apertures with cubic phase variation are plotted in Figs. 15 and 16. Cubic phase distortions are found in practice when reflectors or lenses are illuminated by primary feeds which are off axis either because of inaccurate alignment or because beam lobing or scanning through feed motion is desired. The beam distortion due to cubic phase variation is known in optics as 'coma' and the increased unsymmetrical lobe which is particularly evident in Fig. 16 is commonly called a 'coma lobe'. RADAR ANTENNAS 245 u 0.6 Q- 0.4 5 > 0.2 r \ t V^'^^^ ANGLE V 1 OEF AXIS \ 1 \j >; APERTURE 1 \ 1 -45"/- - 1 < I CL * 1 cuBic phase: variation TO ± 45° AT EXTREMES OF APERTURE ^, / ^ i \ / -^ / ^ \ V \ J \ y ^ ^ \J -srr -4n -srr - 2n -no n 2n sn 4n 5n _ no SIN a Fig. 15 — Pattern of Rectangular Antenna with Cubic Phase Variation. O- 0.4 < / APERT °o T UREn \ ^ ANGLE OFF AXIS \ ^c H J y °o ^ O) \ I CL t [ \ UNIFORM AMPLITUDE. CUBIC PHASE VARIATION TO + 90° AT EXT REMES OF APERTURE ^ / \ / \ r \ ,^ \ \ J \ \ 1 V / V \ 1/ ■2n Fig. 16 — Pattern of Rectangular Antenna with Cubic Phase Variation. 3.11 Two General Methods In sections 3.7 and 3.8 we integrated (36) by expressing a'(x)e'*'^''^ asa sum of terms of the form e* "". Since c'(x)e' "" for finite amplitudes in a finite 246 BELL S YSTEM TECH NIC A L JOURNA L aperture can always be expressed as a Fourier sum of this form this solution can in princij)le always be found. Alternatively in section 3.10 the integral was evaluated as a sum of inte- grals of the general type / x"g''"^</.v. Since d'(.v)e'* ^'^' for finite amplitudes in a finite aperture can always be expressed in terms of a power series, this solution can also in principle always be found. 3.12 xirrays When the aperture consists of an array of component or unit apertures the evaluation of (36) must be made in part through a summation. When all of the elementary apertures are ulike this summation can be reduced to the determination of an 'Array Factor'. The pattern of the array is given by multiplying the array factor by the pattern of a single unit. The pattern of an array of identical units spaced equally at distances some- what less than a wavelength can be proved to be usually almost equivalent to the pattern of a continuous wave front with the same average energy density and phase in each region. 3.13 Limitations to Antenna Wave Front Analysis Through the analysis of antenna characteristics by means of wave front theory as based on equation (17) we have been able to demonstrate some of the fundamental theoretical principles of antenna design. The use of this simple approach is justified fully by its relative simplicity and by its applica- bility to the majority of radar antennas. Nevertheless it cannot always be used. It will certainly be inaccurate or inapplicable in the following cases: (1) When any dimension of the aperture is of the order of a wavelength or smaller (as in many primary feeds). (2) Where large variations in the amplitude or phase in the aperture occur in distances which are of the order of a wavelength or smaller (as in dipole arrays). (3) Where the antenna to be considered does not act essentially through the generation of a plane wave front (as in an end lire antenna or a cosecant antenna). When the wave front analysis breaks down alternative satisfactory ap- proaches based on Maxwell's equation are sometimes but not always fruit- ful. Literature on more classical antenna theory is available in a variety of sources. For much fundamental and relevant theoretical work the reader is referred to Schelkunoff.'' " S. A. Schelkunoff, Loc. Cit. RADAR ANTENNAS 247 4. Application of General Principles In the foregoing sections we have provided some discussion of what hap- pens to a radar signal from the time that the pulse enters the antenna on transmission until the time that the reflected signal leaves the* antenna on reception. We have for convenience divided the principles which chiefly concern us into three groups, transmission line theory, transmission prin- ciples and wave front theory. With the aid of transmission line theory we can examine problems con- cerning locally guided or controlled energy. The details of the problems of antenna construction, such as those to be discussed in Part II frequently demand a grasp of transmission line theory. With it we can study local losses, due to resistance or leakage, which affect the gain of the antenna. We can examine reflection problems and their effect on the match of the antenna. Special antennas, such as those employing phase shifters or trans- mission between parallel conducting plates, introduce many special prob- lems which lie wholly or partly in the transmission line field. An understanding of the principles which govern transmission through free space aids us in comprehending the radar antenna field as a whole. Through a general understanding of antenna gains and effective areas we are better equipped to judge their significance in particular cases, and to evaluate and control the effects of particular methods of construction on them. Wave front theory provides us with a powerful method of analysis through which w^e can connect the radiation characteristics produced by a given antenna with the radiating currents in the antenna. Through it we can examine theoretical questions concerning beam widths and shape, unwanted radiation and gain. An understanding of theory is necessary to the radar antenna designer, but it is by no means sufficient. It is easy to attach too much importance to theoretical examination and speculation while neglecting physical facts which can 'make or break' an antenna design. Theory alone provides no substitute for the practical 'know- how' of antenna construction. It cannot do away with the necessity for careful experiment and measurement. Least of all can it replace the inventiveness and aggressive originality through which new problems are solved and new techniques are developed. PART II METHODS OF ANTENNA CONSTRUCTION 5. General Techniques are essential to technical accomplishment. An understanding of general principles alone is not enough. The designing engineer must have 248 BELL S YSTEM TECH NIC A L JOURNA L at his disposal or develop practical methods which can produce the results he requires. The effectiveness and simplicity of these methods are fair measures of the degree of technical development. The study of methods of radar antenna construction is the study of the means by which radar antenna requirements are met. In a broader sense this includes an examination of mechanical structures, of the metals and plastics from which antennas are made, of the processes by which they are assembled, and of the finishes by which they are protected from their envi- ronment. It might include a study of practical installation and maintenance procedures. But these matters, which like the rest of Radar have unfolded widely during the war, are beyond the scope of this paper. An adequate discussion of them would have to be based on hundreds of technical reports and instruction manuals and on thousands of manufacturing drawings. The account of methods which is to follow will therefore be restricted to a dis- cussion, usually from the electrical point of view, of the more useful and common radar antenna configurations. 6. Classification of Methods During the history of radar, short as it is, many methods of antenna con- struction have been devised. To understand the details of all of these methods and the diverse applications of each is a task that lies beyond the ability of any single individual. Nevertheless most of the methods fall into one or another of a limited collection of groups or classifications. We can grasp most of what is generally important through a study of these groups. In order to provide a basis for classification we will review briefly, from a transmitting standpoint, the action of an antenna. Any antenna is in a sense a transformer between a transmission line and free space. More explicitly, it is a device which accepts energy incident at its terminals, and converts it into an advancing electromagnetic wave with prescribed amph- tude, phase and polarization over an area. In order to do this the antenna must have some kind of energy distributing system, some means of amplitude control and some means of phase control. The distributed energy must be suitably controlled in phase, amplitude and polarization. All antennas perform these functions, but different antennas perform them by different means. Through an examination of the means by which they are performed and the differences between them we are enabled to classify methods of antenna construction. To distribute energy over its aperture an antenna can use a branching system of transmission lines. When this is done the antenna is an array. Arrays are particularly common in the short wave communication bands, but somewhat less common in the microwave radar bands. In a somewhat simpler method the antenna distributes energy over an area by radiating it RADAR ANTENNAS 249 from an initial source or 'primary feed'. This distribution can occur in both dimensions at once, as from a point source. Alternatively the energy can be radiated from a primary source but be constrained to lie between parallel conducting plates so that it is at first distributed only over a long narrow aperture or 'line source'. Distribution over the other dimension occurs only after radiation from the line source. In order to control the amplitude across the aperture of an array antenna we must design the branching junctions so that the desired power division occurs in each one. When the energy is distributed by radiation from a primary source we must control the amplitude by selecting the proper pri- mary feed directivity. We can control the phase of an array antenna by choosing properly the lengths of the branching lines. Alternatively we can insert appropriate phase changers in the lines. When the energy is distributed by primary feeds, methods resembling those of optics can be used to control phase. The radiation from a point source is spherical in character. It can be 'focussed' into a plane wave by means of a paraboloidal reflector or by a spherical lens. The radiation from a point source between parallel plates or from a uniphase line source is cylindrical in character. It can be focussed by a parabolic cylinder or a cylindrical lens. In Table A we have indicated a possible classification of methods of radar antenna construction. This classification is based on the differences dis- cussed in the foregoing paragraphs. Table A Classification of Methods of Radar Antenna Construction - Dipoles r Arrays of Methods of Radar Antenna Construe tion Optical Methods Polyrods Optical Elements r Point sources Spherical < and Optics Dipole Arrays Wave Guide Apertures I Spherical r^"^"^^"''^ -- Elements-! Lenses - Arrays r Line Reflectors] Cylindrical ^0"^^^^ [ j^^^^^^ Optics < and Cylindrical J deflectors - Elements "1 t Lenses 250 BELL SYSTEM TECH NIC A L JOURNA L 7. Basic Design Formulation Certain design factors are common to almost all radar antennas. Because of their importance it would be well to consider these factors in a general way before proceeding with a study of particular antenna techniques. Almost every radar antenna, regardless of how it is made, has a well de- fined aperture or wave front. Through wave front analysis we can often examine the connections between the Huygens sources in the antenna aper- ture and the radiation characteristics of the antenna. We can, in other words, use wave front analysis to study the fundamental antenna design factors, provided the analysis does not violate one of the conditions of section 3.13. 7.1 Dimensions oj the Aperture The dimensions of the aperture of a properly designed antenna are related to its gain by simple and general approximate relations. If the aperture is Uniphase and has an amplitude distribution that is not too far from constant the relation ^ 47ryl is useful in connecting the gain of an antenna with the area of its aperture. The effective area is related to the area of the aperture by the equation A = rjS where ij is an efficiency factor. In principle 77 could have any value but in practice for microwave antennas 77 has always been less than one. Its value for most Uniphase and tapered amplitude antennas is between 0.4 and 0.7. In special cases, e.g. for cosecant antennas or for some scanners its value may be less than 0.4. The necessary dimensions for the aperture may be determined from the required beam widths in two perpendicular directions. Beam widths are usually specified as half power widths, that is by the number of degrees between directions for which the one way response is 3 db below the maxi- mum response. Figure 11 shows that for an ideal rectangular antenna with uniform phase, polarization and amplitude ap/2= 51 - degrees where a^/o == a half power width in degrees, a = aperture dimension and X = wavelength. The relation ap/2 = 65 - degrees is more nearly correct for the majoritv of a practical antennas with round or elliptical apertures and with uniform phase and reasonably tapered amplitudes. RADAR ANTENNAS 251 7.2 Amplitude Distribution Except where special, in particular cosecant, patterns are desired the principle factors affecting amplitude distribution are efficiency and required minor lobe level. The amplitude distribution or taper of an ideal uniphase rectangular wave front affects the minor lobe level as indicated by Figures 1 1 , 12 and 13. Practical antennas tend to fall somewhat below this ideal picture because of non-uniform phase and because of variations from the ideal amplitude distribution due to discontinuities in the aperture and undesired leakage or spillover of energy. Nevertheless a commonly used rule of thumb is that minor lobes 20 db or more below the peak radiation level are tolerable and will not be exceeded with a rounded amplitude taper of 10 or 12 db. 7.3 Phase Control Uniphase wave fronts are used whenever a simple pattern with prescribed gain, beam widths and minor lobes is to be obtained with minimum aperture dimensions. When special results are desired such as cosecant patteri^s or scanning beams the phase must be varied in special ways. Mechanical tolerances in the antenna structure make it impossible to hold phases precisely to the desired values. The accuracy with which the phases can be held constant in practice varies with the technique, the antenna size and the wave length. Undesired phase variations increase the minor lobes and reduce the gain of an antenna. The extent to which phase variations can be expected to reduce the gain is indicated in Fig. 17. 8. Parabolic Antennas The headlights of a car or the searchlights of an antiaircraft battery use reflectors to produce beams of light. Similarly the majority of radar anten- nas employ reflectors to focus beams of microwave energy. These reflectors may be exactly or approximately parabolic or they may have special shapes to produce special patterns. If they are parabolic they may be paraboloids which are illuminated by point sources and focus in both directions, or they may be parabolic cylinders which focus in only one direction. If they are parabolic cylinders they may be illuminated by line sources or they may be confined between parallel conducting plates and illuminated by point sources to produce line sources. 8.1 Control of Phase A simple and natural way to distribute energy smoothly in space is to radiate it from a relatively nondirectional 'primary' source such as a dipole array or an open ended wave guide. This energy will be formed into a direc- tive beam if a reflector is introduced to bring it to a plane area or wave front with constant phase. If the primary source is effectively a point as far as 252 BELL SYSTEM TECHNICAL JOURNAL phase is concerned, that is if the radiated energy has the same phase for all points which are the same distance from a given point, then the reflector should be parabolic. This can be proved by simple geometrical means. In Fig. 18 let the point source .V coincide with the point .v = /", y = of a coordinate system and let the uniphase wave front coincide with the line X — f. Let us assume that one point of the reflector is at the origin. Then it can be shown that any other point of the reflector must lie on the curve A'2 = Afx A square: phase variations / 1 <1> / y / 3 SAW TOOTH PHASE VARIATIONS 01 B i /\ /\ _l ID J — \/ \y \/ > Q 2 / / Z / in If) / o -J y / ^ ^ \y B^__^ ^^ n ^ -^ 20 40 60 80 4>= MAXIMUM PHASE VARIATION IN DEGREES Fig. 17 — Loss due to IMiase Variation in Antenna Wave Front. This is a parabola with focus at/, o and focal length/. The derivation based on Fig. 18 is two dimensional and therefore in principle applies as it stands only to line source antennas employing para- bolic cylinders bounded by parallel conducting planes (Fig. 24 and 25). If Fig. 18 is rotated about the X axis the parabola generates a paraboloid of revolution (Fig. 3). This paraboloid focusses energy spreading spherically from the point source at .5 in such a way that a uniphase wave front over a plane area is produced. Alternatively Fig. 18 can be translated in the Z direction perjiendicular to the XY plane. The parabola then generates a RADAR ANTENNAS 253 parabolic cylinder and the point source S generates a line source at the focal line of the parabolic cylinder (Fig. 19). The energy spreading cylindrically from the line source is focussed by the parabolic cylinder in such a way that a Uniphase wave front over a plane area is again produced. Parabolic cylinders and paraboloids are both used commonly in radar antenna practice. In the discussion so far it has been assumed that the primary source is effectively a point source and that the reflector is exactly parabolic. If the primary source is not effectively a point source, in other words if it produces waves which are not purely spherical, then the reflector must be distorted from the parabolic shape if it is to produce perfect phase correction. When Fig. 18 — -Parabola. this occurs the correct reflector shape is sometimes specified on the basis of an experimental determination of phase. 8.2 Control of Amplitude When a primary source is used to illuminate a parabolic reflector there are two factors which affect the amplitude of the resulting wave front. One of these is of course the amplitude pattern of the primary source. The other is the geometrical or space attenuation factor which is different for different parts of the wave front. In most practical antennas each of these factors tends to taper the amplitude so that it is less at the edges of the antenna than it is in the central region. The effective area of the antenna is reduced by this taper. In any finite parabolic antenna some of the energy radiated by the primary 254 BELL SYSTEM TECHNICAL JOURNAL source will fail to strike the reflector. The effective area must also be re- duced by the loss of this 'spill-over' energy. The maximum effective area for a parabolic antenna is obtained by design- ing the primary feed to obtain the best compromise between loss due to taper and loss due to spill-over. It has been shown theoretically that this best compromise generally occurs when the amplitude taper across the aperture is about 10 or 12 db and that in the neighborhood of the optimum the efficiency is not too critically dependent on the taper. This theoretical result is well justified by experience and has been applied to the majority of practical parabolic antennas. It applies both when the reflector is paraboloidal so that taper in both directions must be considered : — PARABOLIC CYLINDER LINE SOURCE ANTENNA Fig. 19 — A Parabolic Cylinder with Line Feed. and when the reflector is a parabolic cylinder with only a single direction of taper. It is a fortunate by-product of a 10 or 12 db taper that it is gen- erally sufficient to produce satisfactory minor lobe suppression. 8.3 Choice of Configuration We have shown how a simple beam can be obtained through the use of a paraboloidal reflector with a point source or alternatively through the use of a reflecting parabolic cylinder and a line source. The line source itself can be ])roduced with the help of a parabolic cylinder bounded by parallel conducting plates. We will now outline certain practical considerations. These considerations may determine which of the two reflector types will be ' C. C. Cutler, Parabolic Antenna Design for Microwaves, paper to be [published in Proc. of the I. R. E. RADAR ANTENNAS 255 used for a particular job. They may help in choosing a focal length and in determining which tinite portion of a theoretically infinite parabolic curve should be used. Finally they may assist in determining whether reflector technique is really the best for the purpose at hand or whether we could do better with a lens or an array. In designing a parabolic antenna it must obviously be decided at an early stage whether a paraboloid or one or more parabolic cylinders are to be employed. This choice must be based on a number of mechanical and elec- trical considerations. Paraboloids are more common in the radar art than parabolic cylinders and are probably to be preferred, yet a categorical a priori judgment is dangerous. It will perhaps be helpful to compare the two alternatives by the simple procedure of enumerating some features in which each is usually preferable to the other. Paraboloidal antennas (a) are simpler electrically, since point sources are simpler than line sources. (b) are usually lighter. (c) are more efficient. (d) have better patterns in the desired polarization. (e) are more appropriate for conical lobing or spiral scanning. Antennas employing parabolic cylinders (a) are simpler mechanically since only singly curved surfaces are required. (b) have separate electrical control in two perpendicular directions. This last advantage of parabolic cylinders is important in special antennas, many of which will be described in later sections. It is useful where an- tennas with very large aspect ratios (ratio of dimensions of the aperture in two perpendicular directions) are desired. It is highly desirable where con- trol in one direction is to be achieved through some special means, as in cosecant antennas, or in antennas which scan in one direction only. Let us suppose that we have selected the aperture dimensions and have decided whether the reflector is to be paraboloidal or cylindrical. The reflector is not yet completely determined for we are still free in principle to use any portion of a parabolic surface of any focal length. In order to obtain economy in physical size the focal length is generally made between 0.6a and 0.25a where 'a' is the aperture. For the same reason a section of the reflecting surface which is located symmetrically about the vertex is often chosen (Figures 3 and 19). When a symmetrically located section of the reflector is used certain diffi- culties are introduced. These difficulties, if serious enough so that their removal justifies some increase in size can be bypassed through the use of an 256 BELL SYSTEM TECHNICA L JOURNAL ofifset section as shown in Fig. 20. We can comment on these difficulties as follows : 1. The presence of the feed in the {)ath of the reflected energy causes a region of low intensity or 'shadow' in the wave front. The effect of this shadow on the antenna pattern depends on the size and shape of the feed and on the characteristics of the portion of the wave front where it is located. Its effect is to subtract from the undisturbed pattern a 'shadow pattern' component which is broad in angle. This decreases the gain and increases the minor lobes as indicated in Fig. 21.^ \ V-FEED Fig. 20 — Offset Parabolic Section. 2. Return of reflected energy into the feed introduces a standing wave of impedance mismatch in the feed line which is constant in amplitude but varies rapidly in phase as the frequency is varied. A mismatch at the feed which cancels the standing wave at one frequency will add to it at another frequency. A mismatch which will compensate over a band can be introduced by placing a raised plate of proper dimensions at the vertex of the reflector as indicated in Fig. 22, but such a jilate produces a harmful effect on the pattern. In an antenna which must operate over a broad band it is consequently usually better to match ' Figures 21, 22, and 23 arc taken from V. C. Cutler, loc. cit. J RADAR ANTENNAS 257 Fig. 21- -5 5 DEGREES OFF AXIS -Effect of Shadow on Paraboloid Radiation Pattern. Fig. 22 — Apex Matching Plate for Improving the Impedance Properties of a Parabola. 258 BELL S YSTEM TECH NIC A L JOURNA L the feed to space and accept the residual standing wave, or if this is too great to use an offset section of the parabolic surface. 8.4 Feeds for Paraboloids We have seen that an antenna with good wave front characteristics and consequently with a good beam and pattern can be constructed by illu- minating a reflecting paraboloid with a properly designed feed placed at its focus. In this section we will examine the characteristics which the feed should have and some of the ways in which feeds are made in practice. A feed for a paraboloid should a. be appropriate to the transmission line with which it is fed. This is sometimes a coaxial line but more commonly a waveguide. b. Provide an impedance match to this feed line. This match should usually be obtained in the absence of the reflector but sometimes, for narrow band antennas, with the reflector present. c. have a satisfactory phase characteristic. For a paraboloid the feed should be, as far as phase is concerned, a true point source radiating spherical waves. As discussed at the end of 8.1, if the wave front is not accurately spherical, a compensating correction in the reflector can be made. d. have a satisfactory amplitude characteristic. According to 8.2 this means that the feed should have a major radiation lobe with its maxi- mum striking the center of the reflector, its intensity decreasing smoothly to a value about 8 to 10 db below the maximum in the direc- tion of the reflector boundaries and remaining small for all directions which do not strike the reflector. e. have a polarization characteristic which is such that the electric vec- tors in the reflected wave front will all be polarized in the same di- rection. f. not disturb seriously the radiation characteristics of the antenna as a whole. The shadow efl'ect of the feed, the feed line and the necessary mechanical supports must be small or absent . Primary radiation from the feed which does not strike the reflector or reflected energy which strikes the feed or associated structure and is then reradiated must be far enough down or so controlled that the antenna pattern is as required. In addition to the electrical requirements for a paraboloid feed it must of course be so designed that all other engineering requirements are met, it must be firmly suj^ported in the required position, must be connected to the antenna feed line in a satisfactory manner, must sometimes be furnished with an air tight or water tight seal, and so forth. RA DA R A NTENNA S 259 From the foregoing it is evident that a feed for a paraboloid is in itself a small relatively non-directive antenna. Its directivity is somewhat less than that obtained with an ordinary short wave array. It is therefore not surprising that dipole arrays are sometimes used in practice to feed paraboloids. A simple dipole or half wave doublet can in itself be used to feed a parabo- loid, but it is inefficient because of its inadequate directivity. It is prefer- able and more common to use an array in which only one doublet is excited directly and which contains a reflector system consisting of another doublet ov a reflecting surface which is excited parasitically. Dipole feeds although useful in practice have poor polarization charac- teristics and although natural when a coaxial antenna feed line is used are less convenient when the feed line is a waveguide. Since waveguides are more common in the microwave radar bands it is to be expected that wave- guide feeds would be preferred in the majority of paraboloidal antennas. The most easily constructed waveguide feed is simply an open ended waveguide. It is easy to permit a standard round or rectangular waveguide transmitting the dominant mode to radiate out into space toward the parabo- loid. It will do this naturally with desirable phase, polarization and ampli- tude characteristics. It is purely coincidental, however, when this results in optimum amplitude characteristics. It is usually necessary to obtain these by tapering the feed line to form a waveguide aperture of the required size and shape. The aperture required may be smaller than a standard waveguide cross section so that its directivity will be less. In this case it may be necessary to 'load' it with dielectric material so that the power can be transmitted. It may be greater, in which case it is sometimes called an 'electromagnetic horn'. It may be greater in one dimension and less in the other, as when a paraboloidal section of large aspect ratio is to be illuminated. If a single open ended waveguide or electromagnetic horn is used to feed a section of the paraboloid which includes the vertex, the waveguide feed line must partially block the reflected wave in order to be connected to the feed. To avoid this difficulty several rear waveguide feeds have been used. In this type of feed the waveguide passes through the vertex of the parabo- loid and serves to support the feed at the focus. The energy can be caused to radiate back towards the reflector in any one of several ways, some of which involve reflecting rings or plates or parasitically excited doublets. The 'Cutler' feed is perhaps the most successful and common rear feed. It operates by radiating the energy back towards the paraboloid through two apertures located and excited as shown in Fig. 23. * C. C. Cutler, Loc. Cit. 260 BELL S YSTEM TECH NIC A L JOURNA L 8.5 Parabolic Cylinders beticceii Parallel Plates In «S.O we saw thai parabolic cylinders may be illuminated by line sources or that they may be confined between parallel plates and illuminated by point sources to produce line sources. In either of these two cases the char- acteristics which the feed should have are specilled accurately by the con- ditions stated at the beginning of 8.4 for paraboloidal feeds with the excep- tions that condition c must be reworded so that it applies to cylindrical rather than to spherical optics. We will first consider parabolic cylinders bounded by parallel plates because in doing so we describe in passing one form of feed for unbounded parabolic cylinders. Two forms of transmission between parallel plates are used in practice. r!" Fig. 23 — Dual Aperture Rear Feed Horn. a. The transverse electromagnetic (TEM) mode in which the electric vector is perpendicular to the plates. This is simply a slice of the familiar free space wave and can be propagated regardless of the spacing between the plates. It is the only mode that can travel between the plates if they are separated less than half a wavelength. Its velocity of propagation is independent of plate spacing. b. The TEoi mode in which the electric vector is parallel to the plates. This mode is similar to the dominant mode in a rectangular waveguide and differs from it only in that it is not bounded by planes perpen- dicular to the electric vector. It can be transmitted only if the plate spacing is greater than half a wavelength, is the only parallel mode that can exist if the spacing is under a wavelength and is the only sym- metrical parallel mode that can exist if the plate spacing is under three RADAR ANTENNAS 261 halves of a wavelength. Its phase velocity is determined by the plate spacing in a manner given By the familiar waveguide formula Va = where 'c' is the velocity of light, e is the dielectric constant relative to free space of the medium between the plates, X is the wavelength in air and 'a' is the plate spacing. The TEM mode between parallel plates can be generated by extending the central conductor of a coaxial perpendicularly into or through the wave space and backing it up with a reflecting cylinder as indicated in Fig. 24. PARALLEL PLATES REFLECTING CYLINDER PARABOLIC CYLINDER Fig. 24 — Parabolic Cylinder Bounded by Parallel Plates. Probe Feed. Alternatively this mode can be generated as indicated by Fig. 25 by a wave- guide aperture with the proper polarization. The TEni mode, when used, is usually generated by a rectangular wave- guide aperture set between the plates with proper polarization as indicated in Fig. 25. Care must be taken that only the desired mode is produced. The TEM mode will be unexcited if only the desired polarization is present in the feed. The next parallel mode is unsymmetrical and therefore even if it can be transmitted will be unexcited if the feed is placed symmetrically with respect to the two plates. Parallel plate antennas as shown in figures 24 and 25 are useful where particularly large aspect ratios are required. The aperture dimension per- pendicular to the plates is equal to the plate spacing and therefore small. 262 BELL SYSTEM TECHNICAL JOURNAL It can be increased somewhat by the addition of flares. The other dimen- sion can easily be made large. Fig. 25 — Parabolic Cylinder Bounded b>' Parallel Plates. Wave Guide Feed. Fig. 26. — Fxperimental 7' x 32' Antenna. 8.6 Line Sources for Parabolic ( 'yliiulcrs A line source for a parabolic cylinder is physically an antenna with a long narrow aperture. Any means for obtaining such an aperture can be used in {)ro(lucing a line source. Parallel plate systems as described in 8.5 have been used as line sources in several radar antennas. The large (7' x 32') RADAR ANTENNAS 263 experimental antenna shown in Fig. 26 was one of the first to illustrate the practicality of this design. The horizontal pattern of the 7' x 32' antenna is plotted in Fig. 27. The horizontal beam width is seen to be of the order of 0.7 degrees. The antenna illustrated in Fig. 26 is interesting in another way for it is a good example of a type of experimental construction which was extremely useful in wartime antenna development. Research and development engi- 2 2 4 6 DEGREES Fig. 27 — 7' X 32' Antenna, Horizontal Pattern. neers found that they could often save months by constructing initial models of wood. Upon completion of a wooden model electrically im- portant surfaces were covered with metal foil or were sprayed or painted with metal. Thus, where tolerances permitted, the carpenter shop could replace the relatively slow machine shop. Another form of parallel plate line feed results when a plastic lens is placed between parallel plates and used as the focussing element. A linear array 264 BELL SYSTEM TECHNICAL JOURNAL of elements excited with the proper phase and amplitude can also be used. Some discussion of alternative approaches will appear in the section on scanning techniques. 8.7 Tolerances in Parabolic Antcinias The question of tolerances will always arise in practice. Ideal dimensions are only approximated, never reached. The ease of obtaining the required accuracy is an important engineering factor. The tolerances in paraboloidal antennas or in parabolic cylinders illu- minated by line sources can be divided into three general classes: (a) Tolerances on reflecting surfaces. (b) Tolerances on spacial relationships of feed and reflector. (c) Tolerances on the feed. When the actual reflector departs from the ideal parabolic curve deviations in the phase will result. These will tend to reduce the gain and increase the minor lobes. The effects of such deviations on the gain can be estimated with the help of Fig. 17. We should recall that an error of a in the reflector surface will produce an error of about 2<j in the phase front. Based on this kind of argument and on experience reflector tolerances are generally set in X X practice to about ± 77 or ± ~ dependmg on the amount of beam deteriora- tion that can be permitted. In Fig. 28 are compared some electrical characteristics of two paraboloidal antennas, one employing a precisely constructed paraboloidal searchlight mirror and the other a carefully constructed wooden paraboloidal reflector with the same nominal contour. This comparison is revealing for it shows the harm that can be done even by small defects in the reflector surface. Although the two patterns are almost identical in the vicinity of the main beam, the general minor lobe level of the wooden reflector remains higher at large angles and its gain is less. It must not, however, be assumed that a solid reflecting surface is neces- sary to insure excellent results. Any reflecting surface which reflects all or most of the power is satisfactory provided that it is properly located. Per- forated reflectors, reflectors of woven material and reflectors consisting of gratings with less than half wavelength spacing are commonly used in radar antenna practice. These reflectors tend to reduce weight, wind or water resistance and visibility. Many of them will be described in Part III of this paper. The feed of a parabolic reflector should be located so thai its i)hase center coincides with the focus of the reflector. If it is located at an incorrect dis- RADAR ANTENNAS 265 tance from the vertex a circular curvature of phase results and the system is said to be 'defocussed' (Sec. 3.9). As the feed is moved off the axis of the reflector the first effect is a shifting of the beam due to a linear variation of the phase (Sec. 3.8). For greater distances off axis a cubic component of phase error becomes effective (Sec. 3.10). Phase error, whether circular, cubic or more complex, results in a reduction in gain and usually in an in- crease of minor lobes. Although the effects of given amounts of phase curva- ture on the radiation characteristics of an antenna can be estimated by theo- retical means, it is usually easier and quicker to find them experimentally. 5 S 25 UJ in 10 30 1 1 1 1 n \ ENVELOPES OF MINOR LOBE PEAKS ]j \ A -^•^ • A J \ \^ A T 7 \ B ^/^ \^ B 45 50 55 30 25 20 15 10 5 5 10 15 20 25 30 HORIZONTAL ANGLE IN DEGREES Fig. 28 — Effect of Small Inaccuracies in Reflector. The tolerances on the feed itself appear in various forms, many of which can be examined with the aid of transmission line theory and most of which are too detailed for discussion in this paper. It is generally true here also that experiment is a more effective guide than theory. Experience has shown that when parallel plate systems are used, either as complete antennas or as line feeds for other elements, tolerances on the parallel conducting plates must be considered carefully. It is obvious that when the TEoi mode is used the plate spacing must be held closely, since the phase velocity is related to the spacing. This spacing can be controlled through the use of metallic spacers perpendicular to the plates. These 266 BELL S YSTEM TECH NIC A L JOVRNA L spacers, if small enough in cross section, do not disturb things unduly. The velocity of the TEM mode is, on the other hand, almost independent of the plate spacing. This mode is, however, more likely to cause trouble by leaks through joints and cracks in the plates. 9. Metal Plate Lenses At visible wavelengths lenses have, in the past, been far more common than in the microwave region, due chiefly to the absence of satisfactory lens materials. A solid lens of glass or plastic with a diameter of several feet is a massive and unwieldy object. By zoning, which will be discussed below, these difticulties can be reduced but they still remain. A new lens technique, particularly effective in the microwave region was developed by the Bell Laboratories during the war.^ It is evident that any material in which the phase velocity is different from that of free space can be used to make a phase correcting lens. The material which is used in this new technique is essentially a stack of equally spaced metal plates parallel to the electric vector of the wave front and to the direction of propagation. Lenses made from this material are called 'Metal Plate Lenses'. When the spacing between neighboring plates is between X/2 and X only one mode with electric vector parallel to the plates can be transmitted. This is the TEoi mode for which the phase velocity is given in Sec. 8.5. When the medium between the plates is air this equation can be converted into the expression N= i/l \2a[ for the effective index of refraction. Here X is the wavelength in air and a is the plate spacing. As a varies between X/2 and X, A' varies as indicated in Fig. 29. In the neighborhood of a = X, N is not far from 1 and as a approaches X/2, N ap- proaches 0. Since A^ is always less than 1 we see that there is an essential difference between metal plate lenses and glass or plastic lenses for which N is always greater than 1. This difference is seen in the fact that a glass lens corrects phases by slowing down a travelling wave front, while a metal lens operates in the reverse direction by speeding it up. This means that a convergent lens with a real focus must be thinner in the center than the edge, the opposite of a convergent optical lens (Fig. 30). Unless the value of A^ is considerably different from 1 it is evident that very thick lens sections must be used to produce useful phase corrections. For this reason values of 'a' not far from X/2 should be chosen. On the other hand values of *a' too close to X/2 would cause undesirably large reflections 9 W. E. Kock, "Metal Lens Antennas", Proc. I. R. E., Nov., 1946. RADAR ANTENNAS 267 from the lens surfaces and impose severe restrictions on the accuracy of plate spacings. The compromises that have been used in practice are N = 0.5 for which a = 0.577X and N = 0.6 for which a — 0.625X. Even with N' = 0.5 or 0.6 lenses become thick unless inconveniently lon<7 focal distances are used. Thick lenses are undesirable not only because they occupy more space and are heavier but also because the plate spacing must be held to a higher degree of accuracy if the phase correction is to be as ± 0.4 0.2 ^ ^ -^/KS7 0.75X PLATE SPACING l.OOx Fig. 29 — Variation of Effective Index of Refraction with Plate Spacing in a Metal Plate Lens. required. To get around these difficulties the technique of zoning is used. Zoning makes use of the fact that if the phase of an electromagnetic vector is increased or decreased by any number of complete cycles the effect of the vector is unchanged. When applied to a metal plate lens antenna this means simply that wherever the phase correction due to a portion of the lens is greater than a wavelength this correction can be reduced by some integral number of wavelengths such that the residual phase correction is under one wavelength. If this is done it is evident that no portion of the 268 BELL SYSTEM TECHNICAL JOURNAL lens needs to correct the phase by more than one wavelength. It follows that no portion of the lens need to be thicker than X/(l — A^). (0) FEED FEED Fig. 30 — Comparison of Dielectric and Metal Plate Lenses. (b) [{l|||i|||M||||l ""i>i|{|||||l||||l (0) (b) FEED FEED Fig. 31 — Comparison of Unzoncd and Zoned Metal Plate Lenses. A cross section of a ty])ical metal j)hite lens before and after zoning is illustrated in Fig. 31. This figure shows clearly why zoning reduces con- siderably the size and mass of a lens. RADAR ANTENNAS 269 Zoning is not without disadvantages. One disadvantage is obviously that a zoned lens which is designed for one frequency will not necessarily work well at other frequencies. It is in principle possible to design a broad band zoned metal plate lens corresponding to the color compensated lenses used in good cameras. So far, however, this has not been necessary since band characteristics of simple lenses have been adequate. Another difficulty that zoning introduces is due to the boundary regions between the zones. The wave front in this region is influenced partly by one zone and partly by the other and may as a result have undesirable phase and amplitude characteristics. This becomes serious only if especially short focal distances are used. 9.1 Lens Antenna Configuratio7is Any of the configurations which are possible with parabolic reflectors have their analogues when metal plate lenses are used. Circular lenses illumi- nated by point sources and cylindrical lenses illuminated by line sources are not only theoretically possible but have been built and used. Since a lens has two surfaces there is actually somewhat more freedom in lens design than in reflector design. Metal Plate Lenses have usually been designed with one surface flat, but the possibility of controlling both surfaces is emerging as a useful design factor where special requirements must be met. Feeds for lenses should fulfill most of the same requirements as feeds for reflectors. We find a difference in size in lens feeds in that they must gen- erally be more directive because of greater ratios of focal length to aperture. A difference in kind occurs because the feed is located behind the lens where none of the focussed energy can enter the feed or be disturbed by it. As a result some matching and pattern problems which arise in parabolic antennas are automatically absent when lenses are used. In choosing a design for a lens antenna system with a given aperture one must compromise between the large size which is necessary when a long focal length is used and the more zones which result if the focal length is made short. Most metal plate lenses so far constructed have had focal lengths somewhere between 0.5 and 1.0 times the greatest aperture dimension. 9.2 Tolerances in Metal Plate Lenses It is not difficult to see that phase errors resulting from small displace- ments or distortions of a metal plate lens are much less serious than those due to comparable distortions of a reflector surface. This follows from the fact that the lens operates on a wave which passes through it. If a portion of the lens is displaced slightly in the direction of propagation it is still operating on roughly the same portion of the wave front and gives it the same phase correction. If a portion of a reflector were displaced in the same way the error in the wave front would be of the order of twice the 270 BELL SYSTEM TECHNICAL JOURNAL displacement. Quantitative arguments show that less severe tolerances apply to all major structural dimensions of a metal lens antenna. It is true of course that the dimensions of individual portions of the metal lens must be held with some accuracy. The metal plate spacing determines the eflfective index of refraction of the lens material. Where A^ = 0.5 it is customary to require that this be held to ±X/75, and where A' = 0.6 to ±X/50. The thickness of the lens in a given region is less critical, and must be held to ± ttt., T7\ where it is desired to hold the phase front to ±X/16. 10 (1 — A') Fig. 32 illustrates clearly the drastic way in which the location of a lens can be altered without seriously afifecting the pattern. It shows, inci- dentally, how a lens may behave well when used as the focussing element in a moving feed scanning antenna. ■- TV (b) l-^ " 5 10 \ \ ' ^ 1 15 20 \—f^ i\ 25 Uk, ^nJ 1 vV U Fig. 32 — Effect on Pattern of Lens Tilting. 9.3 Advantages of Metal Plate Lenses On the basis of the above discussion we can see that metal plate lenses have certain considerable advantages. The most important of these is perhaps found in the practical matter of tolerances. It is a comparatively simple matter to hold dimensions of small objects to close tolerances but quite another thing to hold dimensions of large objects closely under the conditions of modern warfare. This advantage emerges with increasing importance as the wavelength is reduced. Metal plate lenses have contributed a great degree of flexibility to radar antenna art. When they are used two surfaces rather than one may be controlled, and the dielectric constant can be varied within wide limits. Independent control in the two polarizations may be applied. We can con- fidently expect that they will become increasingly popular in the radar field. 10. Cosecant Antennas One of the earliest uses of radar was for early warning against aircraft. i RADAR ANTENNAS 271 The skies were searched for possible attackers with antennas which rotated continuously in azimuth. An equally important but later use appeared with the advent of great bombing attacks. Bombing radars 'painted' maps of the ground which permitted navigation and bombing during night and under even the worst weather conditions. In these radars also the antennas were rotated in azimuth, either continuously through 360° or back and forth through sectors. The majority of radars designed to perform these functions provided verti- cal coverage by means of a special vertical pattern rather than a vertical scan. It can easily be seen that such a pattern would have to be 'special.' If we assume, for example, that a bombing plane is flying at an altitude of 10.000 feet, then the radar range must be 10,000 esc 60° = 11,500 feet if a target on the ground at a bomb release angle of 60° from the horizontal is to be seen. Such a range would by no means be enough to pick up the target at say 10° in time to prepare for bombing, for then a range of 10,000 esc 10° = 57,600 feet would be required. This range is far more than is neces- sary for the 60° angle. It appears then that in the most efficient design the radar range and therefore the radar antenna gain, must be different in dif- ferent directions. The required variation of gain with vertical direction could be specified in any one of several ways. It seems natural to specify that a given ground target should produce a constant signal as the plane flies towards it at a con- stant altitude. Neglecting the directivity of the target this will occur if the amplitude response of the antenna is given by £ = E^cscd. This same con- dition will apply by reciprocity to an early warning radar antenna on the ground which is required to obtain the same response at all ranges from a plane which is flying in at a constant altitude. This condition is not alone sufficient to specify completely the vertical pat- tern of an antenna. For one thing it can obviously not be followed when ^ = 0, for this would require infinite gain in this direction. Therefore a lower limit to the value of 6 for which the condition is valid must be set. In addition an upper limit less than 90° is specified whenever requirements per- mit, since control at high angles is especially difficult. When the limits have been set it still remains to specify the magnitude of the constant £o- This can be done by specifying the range in one particular direction. This speci- fication must of course be consistent with all the factors that determine gain, including the reduction due to the required vertical spread of the pattern. 10.1 Cosecant Antennas based on the Paraboloid It is evident that the standard paraboloidal antennas so far discussed will not produce cosecant patterns. These patterns being unsymmetrical will result only if the wave front phase and amplitude are especially controlled. 272 BELL SYSTEM TECHNICAL JOURNAL On the other hand, because paraboloidal antennas are simple and common it is natural that many cosecant designs should be based on them. These designs can be classified into two grouj)s, those in which the reflector is modified and those in which the feed is modified. Some early cosecant antennas were made by introducing discontinuities in paraboloidal reflectors as illustrated in Fig. i3. These controlled the radiation more or less as desired over the desired cosecant pattern but pro- NORMAL PARABOLOID / SURFACE / PARABOLOID SURFACE Fig. 33 — Some Cosecant Antennas Based on the Paraboloid (Cosecant Energy Down- ward). duced rather serious minor lobes elsewhere. This difficulty can be overcome through the use of a continuously distorted surface as illustrated in Fig. 34. This reflector, flrst used at the Radiation Laboratories, is a normal parabo- loid in the lower part whereas the upper part is the surface that would be obtained by rotating the parabola through the vertex of the upper part about its focal j)oint. Several types of feed have been used in combination with paraboloids to produce cosecant patterns. These are usually arrays which operate on the princij)lc that each element is a feed which contributes principally to one RADAR ANTENNAS 273 region of the vertical pattern. The elements may be dipoles or waveguide apertures fed directly through the antenna feed line or they may be reflectors which reradiate reflected energy originating from a single primary source. No matter how excited they must be properly controlled in phase, amplitude and directivity. Cosecant antennas based on the paraboloid are common and can some- times fulfill all requirements with complete satisfaction. Nevertheless they Fig. 34- — Barrel Cosecant Antenna (Cosecant Energy Downward). suffer from certain disadvantages. The most serious of these is that they lack resolution at high vertical angles, that is the beam is wider horizontally at high angles. This is to be expected for reasons of phase alone, for a paraboloidal reflector is, after all, designed to focus in only one direction. If phase difliculties were completely absent however, azimuthal resolution at high angles would still be destroyed because of cross polarized components of radiation. These components arise naturally from doubly curved reflectors, even simple paraboloids. They are sometimes overlooked when antennas are measured in a one way circuit with a linearly polarized test field, but must obviously be considered in radar antennas, 274 BELL S YSTEM TECH NIC A L JOURNA L 10.2 Cylindrical Cosecant Antennas Harmful cross polarized radiation is produced by doubly curved reflectors. This radiation is dillicult to control and therefore undesirable where a closely controlled cosecant characteristic at high angles is required. Although not at first evident, it seems natural now to bypass polarization difficulties through the use of singly curved cylindrical reflectors. These reflectors if illuminated with a line source of closely controlled linear polarization provide a beam which is linearly polarized. This beam has also in azimuth approxi- mately the directivity of the line source at all vertical angles. It is thus superior in two significant respects to cosecant beams produced by doubly curved reflecting surfaces. A cylindrical cosecant antenna consists of a cylindrical reflector illumi- nated by a line source. Part of the cylinder is almost parabolic and con- tributes chiefly to the strong part of the beam which lies closest to the hori- zontal. This part is merged continuously into a region which departs considerably from the parabolic and contributes chiefly to the radiation at higher angles. Although wave front principles can be used and certainly must not be violated, the principles of geometrical optics have been particularly effective in the determination of cosecant reflector shapes. The detailed application of these principles will not be discussed here. While applying the geo- metrical principles the designer must be sure that the over-all size and con- figuration of the antenna can produce the results he wants. He must design a line source with the desired polarization and horizontal pattern and a vertical pattern which fits in with the cosecant design. In addition he must take particular care to reduce sources of pattern distortion to a level at which they cannot interfere significantly with the lowest level of the cose- cant 'tail'. 11. LOBING In many of the tactical situations of modern war radar can be used to provide fire control information. Radar by its nature determines range and microwave radar with its narrow well defined beams is a natural instrument for finding directions to a target, whether the missile to be sent to that target is a shell, a torpedo or a bomb. In fire control radar, as opposed to search or navigational radar, two properties of the antenna deserve par- ticular attention. These are the accuracy and the rate with which direc- tion to a target can be measured. Lobing is a means which utilizes to the fullest extent the accuracy avail- able from a given antenna aperture and which increases, usually as far as is desired, the rate at which this information is provided and corrected. RADAR ANTENNAS 275 A lobing antenna which is to provide information concerning one angle only, azimuth for example, is capable of producing two beams, one at a time, and of switching rapidly from one to the other. This process is called Lobe Stvitclniig. The two beams are nearly coincident, differing in direction by about one beam width. When the signals from the two beams are com- pared, they will be equal only if the target lies on the bisector between the beams (Fig. 35). The two signals can be compared visually on an indicator screen of the radar or they can be compared electrically and fed directly into circuits which control the direction of fire. ANTENNA DIRECTED TO LEFT OF TARGET ANTENNA DIRECTED AT TARGET ANTENMA DIRECTED TO RIGHT OF TARGET RELATIVE SIGNALS FROM TWO BEAMS Fig. 35 — Lobe Switching. When two perpendicular directions are to be determined, such as the elevation and azimuth required by an anti-aircraft battery, four or in prin- ciple three discrete beams can be used. Radar antennas designed for solid angle coverage more commonly, however, produce a single beam which ro- tates rapidly and continuously around a small cone. This rotation is known as conical lobing. A comparison of amplitudes in a vertical plane can then be used to give the elevation of the target and a similar comparison in a horizontal plane to give its azimuth. Here too the electrical signals can be compared visually on an indicator screen, but an electrical comparison will provide continuous data which can be used to aim the guns and at the same time to cause the radar antenna to follow the target automatically. 11.1 Lobe Sivitching Two methods of lobe switching are common. In one of these the lobing antenna is an array of two equally excited elements. Each of these ele- 276 BELL SYSTEM TECHNICAL JOURNAL nients occujnes one half of the final antenna aperture, and provides a Uni- phase front across this half. If the two elements were excited with the same phase the radiation maximum of the resulting antenna beam would occur in a direction at right angles to the combined phase front. If the phase of one element is made to lag behind that of the other by a small amount, 60° say, the phase of the combined aperture will of course be discontinuous with a step in the middle. This discontinuous phase front will approximate with a small error, a uniphase wave front which is tilted somewhat with respect to the wave fronts of the elements. The phase shift will there- fore result in a slight shift of the beam away from the normal direction. When the phase shift is reversed the beam shift will be reversed. Two properly designed elementary antennas in combination with a means for rapidly changing the phase will therefore constitute a lobe switching an- tenna. Such an antenna is described more in detail in Sec. 14.6. Another method of lobe switching is more natural for antennas based on optical principles. In this method two identical feeds are placed side by side in the focal region of the reflector. When one of these feeds illu- minates the reflector a beam is produced which is slightly ofif the normal axial direction. Illumination by the other feed produces a second beam which is equally displaced in the opposite direction. The lobe of the an- tenna switches rapidly when the two feeds are activated alternately in rapid succession. The antenna must use some form of rapid switching appropri- ate to the antenna feed line. In several applications switches are used which depend on the rapid tuning and detuning of resonant cavities or irises. 11.2 Conical Lobiiig A conically lobing antenna j)roduces a beam which nutates rapidly about a fixed axial direction. This is usually accomplished by rotating or nutat- ing an antenna feed in a small circle about the focus in the focal plane of a paraboloid or lens. This antenna feed can be a spinning asymmetrical dipole or a rotating or nutating waveguide aperture. It can result in a beam with linear polarization which rotates as the feed rotates, or prefer- ably in a beam for which the polarization remains parallel to a tixed direction. The beam itself must be nearly circularly symmetric so that the radar re- sponse from a target in the axial direction will not vary with the lobing. The reflector or lens aj)erture is consequently usually circular. When the antenna is small it is sometimes easier to leave the feed fixed and to produce the lobing by moving the reflector. 12. RAPID SCANNING A lobing radar can j)rovide range and angular information concerning a single target rapidi)' and accurately but these things arc not always enough. RADAR ANTENNAS 277 It is sometimes necessary to obtain accurate and rapid information from all regions within an agular sector. It may be necessary to watch a certain region of space almost continuously in order to be sure of picking up fast moving targets such as planes. To accomplish any of these ends we must use a rapid scanning radar. A rapid scanning radar antenna produces a beam which scans continuously through an angular sector. The beam may sweep in azimuth or elevation alone or it may sweep in both directions to cover a solid angle. An azimuth or elevation scan may be sinusoidal or it may occur linearly and repeat in a sawtooth fashion. Solid angle scanning may follow a spiral or flower leaf pattern or it might be a combination of two one way scans. A combination of scanning in one direction and lobing in the other is sometimes used. Scanning antennas must, unfortunately, be constructed in obedience to the same principles which regulate ordinary antennas. The same attention to phase, amplitude, polarization and losses is necessary if comparable results are to be obtained. When scanning requirements are added to these ordinary ones new problems are created and old ones made more difficult. An antenna in order to scan in any specified manner must act to produce a wave front which has a constant phase in a plane which is always normal to the required beam direction. This can be done in several different ways. The simplest of these, electrically, is to rotate a fixed beam antenna as a whole in the required fashion. This can be called mechanical scanning. Alternatively an antenna array can be scanned if it is made up of suitable elements and the relative phases of these elements can be varied appro- priately. This can be called array scanning. Thirdly, optical scanning can be produced by moving either the feed or the focussing element of a suitably designed optical antenna. 12.1 MecJianical Scanning Electrical complexities of other types of rapid scanners are such that it is probably not going too far to say that the required scan should be accom- plished by mechanical means wherever it is at all practical. This applies to radar antenna scans which occur at a slow or medium rate. Search antennas, whether they rotate continuously through 360° or back and forth over a sector are scanners in a sense but the scan is usually slow enough to be performed by rotating the antenna structure as a whole. As the scan becomes more rapid, mechanical problems become more severe and elec- trically scanning antennas appear more attractive. Mechanical ingenuity has during the war extended the range in which mechanical scanners are used. One important and eminently practical mechanical rapid scanner, the 'rocking horse' is now in common use (Fig. 36). This antenna is electrically a paraboloid of elliptical aperture illu- 278 BELL SYSTEM TECHNICAL JOURNAL minated by a liorn feed, a combination which produces excellent electrical characteristics. The paraboloid and feed combination is made structurally strong and is pivoted to permit rotational oscillation in a horizontal plane. It is forced to oscillate by a rigid crank rod which is in turn driven by an eccentric crank on a shaft. The shaft is belt driven by an electric motor and its rotational rate is held nearly constant by a flywheel. The mechanical arrangement described so far would oscillate rotationally in an approxi- mately sinusoidal fashion. Since every action has an equal and opposite reaction it would, however, react by producing an oscillatory torque on its Fig. 36 — Experimental Rocking Horse Antenna. mounting. Since the antenna is large and the oscillation rapid this would J reduce a ssvere and undesirable vibration. To get around this difficulty an opposite and balancing rotating moment is introduced into the mechan- ical system. This appears in the form of a pivoted and weighted rod which is driven from the same eccentric crank by another and almost parallel crank arm. Although not theoretically perfect the rotational 'dynamic' balancing described permits the antenna to scan without serious vibration. One form of this antenna will be described in a later section. 12.2 Array Scanning During our discussion of general principles in Part II, we saw that an antenna wave front can be synthesized by assembling an array of radiating RADAR ANTENNAS 279 elements and distributing power to it through an appropriate transmission line network. If the radiation characteristics of the array are to be as de- sired the electrical drive of each element must have a specified phase and ampHtude. In addition each element must in itself have a satisfactory characteristic and the elements must have a proper spacial relationship to each other. Such array antennas have been extremely useful in the 'short wave' bands where wavelengths and antenna sizes are many times larger than at most radar wavelengths but for fixed beam radar antennas they have been largely superceded by the simpler optical antennas. Where a rapidly scanning beam is desired, however, they possess certain advantages which were put to excellent use in the war. These advantages spring from the possibility of scanning the beam of an array through the introduction of rapidly vary- ing phase changes in its transmission line distributing system. Let us first examine certain basic conditions that must be fulfilled if an array antenna is to provide a satisfactory scan. The pattern of any array is merely the sum of the patterns of its elements taking due account of phase, amplitude and spacial relationships. If all elements are alike and are spaced equally along a straight line it is not difficult to show that a mathematical expression for the pattern can be obtained in the form of a product of a factor which gives the pattern of a single element and an array factor. The array factor is an expression for the pattern of an array of elements each of which radiates equally in all directions. Since each of the elements is fixed in direction it is only through control of the array factor that the scan can be obtained. If we excite all points of a continuous aperture with equal phase and a smoothly tapered amplitude the aperture produces a beam with desirable characteristics at right angles to itself and no comparable radiation else- where. Similarly if we excite all elements of an array of identical equally spaced circularly radiating elements with equal phase and a smoothly tapered amplitude the array will produce a beam with desirable charac- teristics at right angles to itself. It will also produce a beam in any other direction for which waves from the elements can add up to produce a wave front. Such other directions will exist whenever the array spacing is greater than one wavelength. In order to see this more clearly let us examine Fig. 37, where line XX' represents an array of elements. From each element to the line AA' is a constant distance, so A A' is obviously parallel to a wave front when the elements are excited with equal phase. If we can find a line BB' to which the distance from each element is exactly one wavelength more or less than from its immediate neighbors then it too is parallel to a wavefront, for energy reaching it from any element of the array will have the same phase 280 BELL SYSTEM TECIIXICAL JOURNAL except for an integral number of cycles. The same will apply to a line CC , to which the distance from each element is exactly two wave lengths more or less than from its immediate neighbors, or to any other line where this dif- ference is any integral number of wavelengths. Now in no radar antenna do we desire two or more beams for they will result in loss of gain and probably in target confusion. The array must therefore be designed so that for all positions of scan all beams except one will be suppressed. This will automatically occur if the array spacing is somewhat less than one wavelength. If the array spacing is greater than one wavelength these extra beams will appear in the array factor; they Fig. 37 — Some Possible Wave Fronts of an Array of Elements Spaced 2.75 X. must therefore be suppressed by the pattern of a single element. The pat- tern of an element must in other words, have no significant components in any direction where an extra beam can occur. Where elements with only side fire directivity are spaced more than a wavelength apart in a scanning array it is almost impossible to obtain adequate extra lobe sui)pression. If these elements are spaced by the minimum amount, that is by exactly the dimensions of their apertures and all radiate in phase the}- may indeed just manage to produce a desirable beam. A little analysis shows however that an appreciable phase variation from element to element, e\'en though linear, will introduce a serious ex- tra lobe. To get around this difiKulty elements with some end lircdirec- livity must be used. RADAR ANTENNAS 281 A simple end fire element, and one that has been used in practice, is the 'polyrod' (Fig. 38). A polyrod, is as its name implies, a rod of polystyrene. This rod, if inserted into the open end of a waveguide, and if properly pro- portioned and tapered, will radiate energy entering from the waveguide from points which are distributed continuously along its length. If the Fig. 38— A Polyrod. l'>x[)erimental Polyrod Array. wave in the polyrod travels approximately with free space velocity it will produce a radiation maximum in the direction of its axis. The radia- tion pattern of the polyrod will have a shape which is characteristic of end fire arrays, narrower and flatter topped than the pattern of a side fire array which occupies the same lateral dimension. This elementary pattern can be fitted in well with the array factor of a scanning array. Such a scanning array is shown in Fig. 39 and will be described in 282 BELL SYSTEM TECH NIC A L JOURNA L greater detail in section 14.8. Each element of this array consists of a fixed vertical array of three polyrods. This elementary array provides the re- quired vertical pattern and has appropriate horizontal characteristics. Fourteen of these elements are arranged in a horizontal array with a spacing between neighbors of about two wavelengths. Energy is distributed among the elements with a system of branching waveguides. Thirteen rotary phase changers are inserted strategically in the distributing system. Each phase change is rotating continuously and shifts the phase linearly from 0° to 360° twice for each revolution. As the phase changers rotate the array produces a beam which sweeps repeatedly linearly and continuously across the scanning sector. When elements of a scanning array are spaced considerably less than one wavelength it is a very simple matter to obtain a suitable elementary pattern, for the array factor itself has only a single beam. This advantage is offset by the greater number of elements and the consequent greater com- plexity of distributing and phase shifting equipment. In one useful type of scanning antenna however distributing and phase shifting is accomplished in a particularly simple manner. Here the distributing system is merely a waveguide which can transmit only the dominant mode. The wide dimen- sion of the guide is varied to produce the phase shifts required for scanning. The elements are dipoles. The center conductor of each dipole protrudes just enough into the guide to pick up the required amount of energy. It is evident from the above discussion that such a waveguide fed dipole array will produce a single beam in the normal direction only if the dipoles are all fed in phase and are spaced less than a wavelength. It is therefore not satisfactory to obtain constant phase excitation by tapping the dipoles into the guide at successive guide wavelengths for these are greater than free space wavelengths. Consequently the dipoles are tapped in at suc- cessive half wavelengths in the guide and reversed successively in polarity to compensate for the successive phase reversals due to their spacing. This type of array provides a line source which can be scanned by moving the guide walls. In order to leave these mechanically free suitable wave trapping slots are provided along the length of the array. A practical antenna of this type will be described in Sec. 16.3. 12.3 Optical Scanning With a camera or telescope all parts of an angular sector or field are viewed simultaneously. We would like to do the same thing by radar means, but since this so far appears impossible we do the next best thing by looking at the parts of the field in rapid succession. Nevertheless certain points of similarity appear. These points are emphasized by a survey of the fixed RA DA R A NT EN N A S 2 83 beam antenna field for there we find optical instruments in abundance, parabolic reflectors and even lenses. It is not a very big step to proceed from an examination of optical systems to the suggestion that a scanning antenna can be provided by moving a feed over the focal plane of a reflector. Nevertheless experience shows that this will not be especially profitable unless done with due caution. The first efl'ect of moving the feed away from the focus in the focal plane of a paraboloid is indeed a beam shift but before this process has gone far a third order curvature of the phase front is produced and is accompanied by a serious deterioration in the pattern and reduction in gain. This difficulty or aberration is well known in classical optical theory and is called coma. Coma is typified by patterns such as the one shown in Fig. 16. It is the first obstacle in the path of the engineer who wishes to design a good moving feed scanning antenna. Coma is not an insuperable obstacle however. Its removal can be accomplished by the application of a very simple geometrical principle. This principle can be stated as follows: "The condition for the absence of coma is that each part of the focussing reflector or lens should be located on a circle with center at the focus." This condition can be regarded as a statement of the spacial relation- ship required between the feed and all parts of the focussing element. It is a condition which insures that the phase front will remain nearly linear when the feed is moved in the focal plane. It can be applied approximately whether the focussing element is a reflector or a lens and to optical systems which scan in both directions as well as those which scan in one direction. Coma is usually the most serious aberration to be reckoned with in a scanning optical system, but it is by no means the only one. Any defect in the phase and amplitude characteristic which arises when the feed is moved can cause trouble and must be eliminated or reduced until it is toler- able. Another defect in phase which arises is 'defocussing'. Defocussing is a square law curvature of phase and arises when the feed is placed at an improper distance from the reflector or lens. Its effect may be as shown in Fig. 14. It can in principle always be corrected by moving the feed in a correctly chosen arc, but this is not always consistent with other require- ments on the system. In addition to troubles in phase an improper ampli- tude across the aperture of the antenna will arise when the feed is trans- lated unless proper rotation accompanies this motion. To combat the imperfections in an optical scanning system we can choose over-all dimensions in such a way that they will be lessened. Thus it is generally true that an increase in focal length or a decrease in aperture will increase the scanning capabilities of an optical system. This alone is usually not enough, however, we must also employ the degrees of free- 284 BELL S YSTEM TECH NIC A L JOURNA L dom available to us in the designing of the focussing element and the feed motion to improve the performance. If the degrees of freedom are not enough \vc must, if we insist on an optical solution introduce more. This could in principle result in microwave lenses similar to the four and five element glass lenses found in good cameras, but such complication has not as yet been necessary in the radar antenna art. Since military release has not been obtained as this article goes to press we must omit any detailed discussion of optically scanning radar antenna techniques. PART III MILITARY RADAR ANTENNAS DEVELOPED BY THE BELL LABORATORIES i3. General In the fuial jxirt of this paper we will describe in a brief fashion the end products of radar antenna technology, manufactured radar antennas. Without these final practical exhibits the foregoing discussion of principles and methods might appear academic. By including them we hope to illustrate in a concrete fashion the rather general discussion of Parts I and II. The list of manufactured antennas will be limited in several ways. Severe but obviously essential are the limitations of military security. In addition we will restrict the list to antennas developed by the Bell Laboratories. In cases where invention or fundamental research was accomplished elsewhere due credit will be given. Finally the list will include only antennas manu- factured by contract. This last limitation excludes many experimental antennas, some initiated by the Laboratories and some by the armed forces. It is worthwhile to begin with an account of the processes by which these antennas were brought into production. The initiating force was of course military necessity. The initial human steps were taken sometimes by members of the armed forces who had definite needs in mind and sometimes by members of the Laboratories who had solutions to what they believed to be military needs. With a definite job in mind conferences between military and Laboratories personnel were necessary. Some of these dealt with legal or financial matters, others were princi})ally technical. In the technical conferences it was necessary at an early date to bring military requirements and tech- nical {)ossibilities in line. As a result of the conferences a program of research and development was oflen undertaken by the Laboratories. An initial contract was signed which RA DA R A NT EN N A S 285 called for the delivery of technical information, and sometimes for manu- facturing drawings and one or more completed models. Usually the antenna was designed and manufactured as part of a complete radar sys- tem, sometimes the contract called for an antenna alone. After prehminary work had been undertaken the status of the job was reviewed from time to time. If preliminary results and current mihtary requirements warranted a manufacturing contract was eventually drawn up and signed by Western Electric and the contracting government agency. This contract called for delivery of manufactured radars or antennas ac- cording to a predetermined schedule. Research and development groups of the Laboratories cooperated in war as in peace to solve technical problems and accomplish technical tasks. Under the pressure of war the two functions often overlapped and seemed to merge, yet the basic differences usually remained. Members of the Research Department, working in New York and at the Deal and Holmdel Radio Laboratories in New Jersey were concerned chiefly with electrical design. It was their duty to understand fully electrical principles and to invent and develop improved methods of meeting mili- tary requirements. During the war it was usually their responsibility to prescribe on the basis of theory and experiment the electrical dimensions of each new radar antenna. A new and diificult requirement presented to the Research Department was sometimes the cause of an almost personal competition between alter- native schemes for meeting it. Some of these schemes were soon eliminated by their own weight, others were carried side by side far along the road to production. Even those that lost one race might reappear in another as a natural winner. In the Development Groups working in New York and in the greatly expanded Whippany Radio Laboratory activity was directed towards coor- dination of all radar components, towards the establishment of a sound, well integrated mechanical and electrical design for each component and towards the tremendous task of preparing all information necessary for manufacture. It was the job of these groups also to help the manufacturer past the unavoidable snarls and bottlenecks which appeared in the hrst stages of production. In addition development personnel frequently tested early production models, sometimes in cooperation with the armed forces. As we have intimated, research and development were indistinguishable at times during the war. Members of the research department often found themselves in factories and sometimes in aircraft and warships. Develop- ment personnel faced and solved research problems, and worked closely with research groups. 286 BELL S YS TEM TECH NIC A L JO URN A L For several years when pressure was high the effort was intense; at times feverish. Judging by miUtary results it was highly effective. Some of the material results of this effort are described in the following pages. 14. Naval Shipborne Radar Antennas 14.1 The SE Auleiiiia'° Very early in the war, the Navy requested the design of a simple search radar s3-stem for small vessels, to be manufactured as quickly as possible in order to till the gap between design and production of the more complex- search systems then in {process of develo])ment. The proposed system was to be small and simple, to permit its use on vessels which otherwise would be unable to carry radar equipment because of size or power supply capabil- ity. This class of vessel included PT boats and landing craft. The antenna designed for the SE system is housed as shown in Fig. 40. It was adapted for mounting on the top or side of a small ship's mast, and is rotated in azimuth by a mechanical drive, hand operated. The para- boloid reflector is 42 inches wide, 20 inches high, and is illuminated by a circular aperture 2.9 inches in diameter. In the interests of simplicity, the polarization of the radiated beam was permitted to vary with rotation of the antenna. The SE antenna was operated at 9.8 cm, and fed by 1^x3 rectangular waveguide. At the antenna base, a taper section converted from the rectangular waveguide to 3" round guide, through a rotating joint directly to the feed opening. Characteristics of the SE antenna are given below: Wavelength 9.7 to 10.3 cm Reflector 42" W x 20" H Gain 25 db Horizontal Beam Width 6° Vertical Beam Width 12°, varj'ing somewhat with polarization Standing Wave 9.7tolO.Ocm 4.0 db 10.0 to 10.3 cm 6.0 db 14.2 The SL Radar Antenna^' The SL radar is a simple marine search radar developed by Bell Tele- phone Laboratories for the Bureau of Ships. During the war, over 1000 of these radars were produced by the Western Electric Company and installed on Navy vessels of various categories. The principal tield for installation was destroyer escort craft ("DE"s). Figure 41 shows an SL antenna installation al)oard a DE. 'J'he antenna is covered, for wind and " Written by R. J. Phillips. " Written by H. T. Budenbom. RADAR ANTENNAS 287 weather protection, in a housing which can transmit 10 cm radiation. \'isible also is the waveguide run down the mast to the r.f. unit. The SL radar provides a simple non-stabiUzed PPI (Plan Position Indicator) display. The antenna is driven by a synchronous motor at 18 rpm. Horizontal polarization is used to minimize sea clutter. The 'f^T"^^ Fitr. 4U — SE Antenna. radiating structure, shown in Figure 42, consists of a 20" sector of a 42" paraboloid. The resulting larger beam width in the vertical plane is pro- vided in order to improve the stability of the pattern under conditions of ship roll. Figure 43 illustrates the path of the transmitted wave from the SL r.f. unit to the antenna. It also illustrates the manner in which horizon- tally polarized radiation is obtained. The diagram shows the position of 288 BELL SYSTEM TECHNICAL JOURNAL "ft: RADAR ANTENNAS 289 ^ %j ■^ X, ■^i ^^P'^""" ^; Fig. 42 — SL Antenna. / 290 BELL SYSTEM TECHNICAL JOURNAL the electric force vector in traversing the waveguide run. The path from the r.f. unit is in rectangular guide (TEi, o mode) through the right angle bend, to the base of the rotary joint. A transducer which forms the base portion of the joint converts to the TMoi mode in circular pipe. For this mode, the electric held has radial symmetry, much as though the wave- guide were a coaxial line of vanishingly small inner conductor diameter. PIPE CONTAINING SPIRAL SEPTUM TE. INDICATES DIRECTION OF ELECTRIC VECTOR. INDICATES VECTOR LIES X TO PLANE OF PAPER. REFLECTOR ROTARY JOINT I AND CHOKE TEio Fig. 43 — SL Radar Antenna — Wave Guide Path. The energy passes the rotary joint in this mode; choke labyrinths are pro- vided at the joint to minimize radio frequency leakage. The energy then flows through another transducer, from TMoi mode back to TEio mode. The lower horizontal portion of the feed pipe immediately tapers to round guide, the mode being now TEn. Ne.xt the energy transverses a 90° elbow, which is a standard 9i)° pipe casting, and enters the vertical section im- RADAR ANTENNAS 291 mediately below the feed aperture. The E vector is in the plane of the paper at this point. However, the ensuing vertical section is fitted with a spiral septum. This gradually rotates the plane of polarization until at the top of this pipe the E vector is perpendicular to the plane of the paper. Thus, after transversing another 90° pipe bend, the energy emerges horizon- tally polarized, to feed the main reflector. Specific electrical characteristics of the SL antenna are: Polarization — Horizontal Horizontal Half Power Beamwidth — 6° Vertical Half Power Beamwidth — 12° Gain — about 22 db. 14.3 The SJ Submarine Radar Antenna It had long been expected that one of the early offensive weapons of the war would be the submarine. It was therefore natural that early in the history of radar the need for practical submarine radars was felt. The principal components of this need were twofold, to provide warning of ap- proaching enemies and to obtain torpedo fire control data. The SJ Sub- marine Radar was the first to be designed principally for the torpedo fire control function. Work on the SJ system was under way considerably before Pearl Harbor. When this work was initiated the advantages of lobing fire control systems were clearly recognized, but no lobing antennas appropriate for submarine use had been developed. Requirements on such an antenna were ob- viously severe, for in addition to fulfiUing fairly stringent electrical con- ditions, it would have to withstand very large forces due to water resistance and pressure. The difficulties evident at the outset of the work were overcome by an ingenious adaptation of the simple waveguide feed. It was recognized that a shift of the feed in the focal plane of a reflector would cause a beam shift. Why not, then, use two waveguide feeds side by side to produce the two nearly coincident beams required in a lobing antenna? When this was tried it was found to work as expected. It remained to devise a means of switching from one waveguide feed to the other with the desired rapidity. This in itself was no simple problem, but was solved by applying principles learned through work on waveguide filters. The switch at first employed was essentially a branching filter at the junction of the single antenna feed line and the line to each feed aper- ture. Both branches of this filter were carefully tuned to the same fre- quency, that of the radar. The switching was performed by the insertion of small rapidly rotating pins successively into the resonant cavities of the 292 BELL SYSTEM TECHNICAL JOURNAL two filters (Fig. 44). Presence of the ])ins in one of the filters detuned it and therefore prevented ])o\vcr from Uowing through it. Rotation of the pins accordingly produced switching as desired. In a later modification of this switch the same general princi})les were used but resonant irises rather than resonant cavities were employed. The SJ Submarine Radar was in use at a comi)aralively early date in the war and saw much ser\-ice with the Pacific submarine lleet. Despite some early doubts, submarine commanders were soon convinced of its powers. .*<C SWITCH UNIT CHAMBERS OFTUIMING Pi MS Fig. 44— The SJ Tuned Cavity Switch. It is believed that in the majority of cases it replaced the periscope as the principle fire control instrument. In addition it served as a valuable and unprecedented aid to navigation. It is interesting and relevant to quote from two letters to Laboratories engineers concerning the SJ. One dated October 3, 1943, from the radar officer of a submarine stated that there were twenty "setting sun" fiags painted on the conning tower and asked the engineer to "let your mind dwell on the fact that you helped to put more than 50% of those flags there". RADAR ANTENNAS 293 The commander of another submarine wrote in a similar vein, "You can rest assured that we don't regard your gear as a bushy-brain space taker, but a very essential part of our armament". I'ig. 45 -Tlie SJ Submarine Radar Antenna. Figure 45 is a photograph of an SJ antenna, characteristics are as follows: Its principal electrical Gain > 19 db Horizontal Half Power Beamwidth 8° Vertical Half Power Beamwidth 18° Vertical Beam Character — Some upward radiation Lobe Switching Beam Separation — approximately 5° Gain reduction at beam cross-over < 1 db Polarization — Horizontal 294 BELL S YSTEM TECH NIC A L JOURNA L 14.4 The %rodified S J/ Mark 27 Radar Antenna The SJ antenna described above performed a remarkable and timely fire control job as a lobing antenna but was found to be unsatisfactory when rotated continuously to produce a Plan Position Indicator (PPI) presenta- tion. In the PPI method of presentation range and angle are presented as radius and angle on the oscilloscope screen. Consequently a realistic map of the strategic situation is produced. This map is easily spoiled by false signals due to large minor lobes of the antenna. Since it had been established that the PPI picture was valuable for navigation and warning as well as for target selection it was decided to modify the antenna in a way that would reduce these undesirably high minor lobes. These were evidently due principally to the shadowing effect of the massively built double primary feed. Accordingly a new reflector was de- signed which in combination with a slightly modified feed provided a much improved pattern. The new reflector was different in configuration principally in that it was a partially offset section of a paraboloid. The reflector surface was also markedly different in character since it was built as a grating rather than a solid surface. This reduced water drag on the antenna. In addition the grating was less visible at a distance, an advantage that is obviously appreciable when the antenna is the only object above the water. This modified antenna was used not only on submarines as part of the SJ-1 radar but also on surface vessels as the Mark 27 Radar Antenna. Figure 46 shows one of these antennas. Its electrical characteristics are as follows: Gain > 20 db Horizontal Half Power Beamwidth = 8° Vertical Half Power Beamwidth = 17° Vertical Beam Character — Some upward radiation Lobe Switching Beam Separation — approximately 5° Gain reduction at beam cross-over < 1 db Polarization — -Horizontal 14.5 The SH and Mark 16 Autenna^^ The antennas designed for the SH and Mark 16 Radar Equipments are practically identical. The SH system was a shipborne combined fire con- trol and search system, and the Mark 16 its land based counterpart was used by the Marine Corps for directing shore batteries. These systems operated at 9.8 cm. The requirement that the system, operate as a fire control as well as a search system imposed some rather stringent mechanical requirements on the antenna. For search purposes, the antenna was rotated at 180 rpm, and indications were presented on a plan position indicator. For fire control data, slow, accurately controlled motion was recjuired. Bearing accuracy is attained by lobe switching in '^Written by R. J. Philipps. RADAR ANTENNAS 295 much the same manner as in the SJ and SJ-1 antennas previously described. The antenna is illustrated in Fig. 47, With the SH system, the unit is mast mounted; for the Mark 16, the unit is mounted atop a 50 foot steel Fig. 46 — The SJ-1 /Mark 27 Submarine Radar Antenna. tower which can be erected in a few hours with a minimum of personnel. The electrical characteristics are as follows: Gain— 21. db Reflector Dimensions 30" W x 20" H Horizontal beam width — 7.5° Vertical beam width — 12° Lobe separation — 5° approximately Loss in gain at lobe crossover — 1 db approximately Scan — (1) 360°, at 180 rpm for PPI operation (2) 360°, at approximately 1 rpm for accurate azimuth readings, with lobe switching 296 BELL SYSTEM TECHNICAU JOURNAL SH systems were most successfully used in invasion operations in the Aleutians. They were installed on landing craft, and the use of the high A Fig. 47— SH Antenna. speed scan enabled the craft to check constantly their relative positions in the dense fogs encountered during the landing operations. RADAR ANTENNAS 297 14.6 Allien lias for Early Fire Control Radars^^ The first radars to be produced in quantity for fire coiitrol on naval ves- sels were the Mark 1, Mark 3 and Mark 4 (originally designated FA, FC and FD). These radars were used to obtain the position of the target with sufficient accuracy to permit computation of the firing data required by the guns. The first two (Mark 1 and Mark 3) were used against enemy surface targets while the Mark 4 Radar was a dual purpose system for use against both surface and aircraft targets. These radars were described in detail in an earlier issue. ^'^ However, photographs of the antennas and per- tinent information on the antenna characteristics are repeated herein for the sake of completeness. (See Table B and Figures 48, 49 and 50.) Table B Radar Mark 1 Mark 3 Mark 4 Dimensions 6'x6' 3'xl2' 1 6'x6' 6'x7' Operating Frequency 500 or 700 MC 680-720 MC 680-720 MC Beam Width in Degrees (Between half power points one way.) Azimuth 12° 6° 12° 12° Elevation 14° 30° 14° 12° Antenna Gain 22 db 22 db 22 db 22.5 db. Beam Shift in Degrees Azimuth 0° ±1.5° ±3° ±3° Elevation 0° 0° 0° ±3° An antenna quite similar to the Mark 3, 6 ft. x 6 ft. antenna, was also used on Radio Set SCR-296 for the Army. This equipment was similar to the Mark 3 in operating characteristics but was designed mechanically for fixed installations at shore points for the direction of coast artillery gun fire. For these installations the antenna was mounted on an amplidyne controlled turntable located on a high steel tower. The entire antenna and turntable was housed within a cylindrical wooden structure resembling a water tower. Equipments of this type were used as a part of the coastal defense system of the United States, Hawaiian Islands, Aleutian Islands and Panama. " Written by W. H. C. Higgins. ""Early Fire Control Radars for Naval Vessels," W. C. Tinus and W. H. C. Higgins, B. S. T. J. 298 BELL SYSTEM TECHNICAL JOURNAL 14.7 A Shipborne A nti- Aircraft Fire Control Antenna}^ A Shipborne Anti- Aircraft Fire Control Antenna is shown in Fig. 51. This antenna consists of two main horizontal cylindrical parabolas in each "^t3l ■•#*» h-: L' Vl^ '33®' ^j^SsF ^m jE'^^^^CI?)^^ ^^ ^^^^^ ^^^^^ ^_^ ^j^ Fig. 48— Mark 1 Antenna. of which two groups of four half-wave dipoles are mounted with their axes in a horizontal line at the focus of the parabolic reflectors. The four groups of dipoles are connected by coaxial lines on the back of the antenna to a lobe 16 Written by C. A. Warren. RADAR ANTENNAS 299 switcher, which is a motor driven capacitor that has a single rotor plate and four stator plates, one for each group of dipoles. The phase shift intro- duced into the four feed lines by the lobe switching mechanism causes the antenna beam to be "lobed" or successively shifted to the right, up, left and down as the rotor of the capacitor turns through 360 degrees. Mounted centrally on the front of the antenna at the junction of the two parabolic antennas is a smaller auxiliary antenna consisting of two dipole elements and a parabolic reflector, the purpose of which is to reduce the minor lobes that are present in the main antenna beam. The auxiliary Fig. 49 — Mark 3 Radar Antenna on Battleship New Jersey. antenna beam is not lobe switched and is sufficiently broad in both the horizontal and vertical planes to overlap both the main antenna beam and the first minor lobes. The auxiliary antenna feed is so designed that its field is in phase with the field of the main beam of the main antenna. This causes the feed of the auxiliary antenna to "add" to the field of the main antenna in the region of its main beam, but to subtract from the field in the region of its first minor lobes. This occurs because the phase of the first minor lobes differs by 180 degrees from that of the main beam. As a result, the field of the main beam is increased and the first minor lobes are greatly 300 BELL SYSTEM TECHNICAL JOURNAL reduced. By re(lucin<f these minor lobes to a low value, the region around the main beam is free of lobes, thus greatly reducing the possibility of false tracking due to "cross overs" between the main beam and the minor lobes. 14.8 The Polyrod Fire Control Antenna The Polyrod Fire Control antenna is an arra}' scanner emplo}ing essen- tially the same principles as those used in the multii)le unitsteerable antenna Fig. 50 — Mark 4 Radar Antenna on Ikittleship Tennessee. system (MUSA) developed before the war for short-wave transatlantic telephony. Some of these principles have been discussed in Sec. 12.2. That they could be applied with such success in the microwave region was due to a firm grounding in waveguide techniques, to the invention of the polyrod antenna and the rotary phase changer, and especially to excellent technical work on the part of research, development and production person- nel. It is perhaps one of the most remarkable achievements of wartimq RADAR ANTENNAS 301 radar that the polyrod antenna emerged to fill the rapid scanning need a early and as well developed as it did. The Polyrod Fire Control antenna is a horizontal array of fourteen identi- cal fixed elements, each element being a vertical array of three polyrods. Energy is distributed to the elements through a waveguide manifold. The phase of each element is controlled and changed to produce the desired scan by means of thirteen rotary phase changers. These phase shifters are J 1 ly ""f|~ Tf?ANSMISS10N MINOR LOBE SUPPRESSOR ANTENNA '—MMN ANTENNA LINE Fig. 51. — Shi[)borne Anti-Aircraft Fire Control Antenna geared together and driven synchronously. Figure 52 is a schematic diagram of the waveguide and phase changer circuits. Figure 39 shows an experimental polyrod antenna under test at Holmdel. Figure 53 is another view of the Polyrod antenna. 14.9 The Rocking Horse Fire Control Antenna It was long recognized that an important direction of Radar develop- ment lay towards shorter waves. This is particularly true for fire control antennas where narrow, easily controlled beams rather than great ranges are needed. The Polyrod antenna had pretty thoroughly demon- 302 BELL SYSTEM TECHNICAL JOURNAL strated the value of rapid scanning, yet the problem of producing a rapid scanning higher frequency antenna of nearly equal -dimensions was a new and different one. Several possible solutions to this problem were known. The array technique applied so effectively to the polyrod antenna could have been applied here also, but only at the expense of many more elements and greater complexity. After much preliminary work it was finally concluded that a mechanically scanning antenna, the "rocking horse," provided the best solution to the higher frequency scanning problem. This solution is practical and relatively simple. - DELAY EQUALIZING WAVE GUIDE LENGTHS UNIT ANTENNAS (VERTICAL POLYROD TRIDENTS) WAVE GUIDE DISTRIBUTING MANIFOLD WITH ROTARY PHASE CHANGERS (720° PHASE CHANGE PER REV.) INPUT Fig. 52. — Schematic Diagram of Poljrod Fire Control Antenna. The operation of the rocking horse is described in Sec. 12.1. It is essen- tially a carefully designed and firmly built paraboloidal antenna which oscillates rapidly through the scanning sector. Its oscillation is dynamically balanced to eliminate undesirable vibration. Figure 54 is a photograph of a production model of the rocking horse antenna. 14.10 The Mark 19 Radar Aiilcmia^'^ In Anti-aircraft Fire Control Radar Systems for Heavy Machine Guns it is necessary to em])loy a highly directive antenna and to obtain continu- ous rapid comparison of the received signals on a number of beam positions "Sections 14.10, 14.11 and 14.12 were written by F. E. Nimmcke. RADAR ANTENNAS 303 304 BELL SYSTEM TECHNICAL JOURNAL as discussed in Section 11.2. Such an antenna is also required to obtain the high angular precision for anti-aircraft fire control. These require- ments are achieved by the use of a conical scanning system. The beam from the antenna describes a narrow cone and the deviation of the axis of the cone from the line of sight to the target can be determined and meas- ured by the phase difference between the amplitude modulated received signal and the frequency of the reference generator associated with the Fig. 54. — Rocking Horse Fire Control Antenna. antenna. This information is presented to the pointer-trainer at the direc- tor in the form of a wandering dot on an oscilloscope. The antennas described in sections 14.10, 14.11 and 14.12 were all designed by the Bell Laboratories as anti-aircraft fire control radar systems, particu- larly for directing heavy machine guns. They were designed for use on all types of Naval surface warships. In Radar Kquii)ment Mark 19, the first system to be associated with the control of 1.1 inch and 40 mm anli-aircraft machine guns, the antenna was designed for operation in the 10 cm region. This antenna consisted of a spinning half dipole with a coaxial transmission line feed. The antenna RADAR ANTENNAS 305 was driven by 115-volt, 60 cycle, single phase motor to which was coupled a two-phase reference voltage generator. The motor rotated at approxi- mately 1800 rpm which resulted in a scanning rate of 30 cycles per second. This antenna was used with a 24-inch spun steel parabolic reflector which provided, at the 3 db point, a beam width of approximately 11° and a beam shift of 8.5° making a total beam width of approximately 20° when scan- ning. The minor lobes were down more than 17 db (one way) from the maximum; and the gain of this antenna was 21 db. This antenna assembly JUNCTION BOX 'M PARABOLOIDAL REFLECTOR Fig. 55— Mark 19 Ant^ was integral with a transmitter-receiver (Fig. 55) which was mounted on the associated gun director. Consequently, the size of the reflector was limited by requirements for unobstructed vision for the operators in the director. As a matter of fact, for this type of radar system serious con- sideration must be given to the size and weight of the antenna and asso- ciated components. 14.11 The Mark 28 Radar Antenna The beam from the antenna used in Radar Equipment Mark 19 was relatively broad and to improve target resolution, the diameter of the 306 BELL SYSTEM TECBNICAL JOURNAL reflector for the antenna in Mark 28 was approximately doubled. The Mark 28 is a 10 cm system and employs a conical scanning antenna similar to that described for Mark 19. The essential difference is that the spun steel parabolic reflector is 45 inches in diameter which provides a beam width of ai)pr<).\imately 6.5° and a beam shift of 4.5° making a total of 11°. Fig. 56 — Mark 28 Antenna Mounted on 40 MM Gun. The minor lobes are down more than 17 db (one way) from the maximum; and the gain of this antenna is 26 db. It was found necessary to perforate the reflector of this dimension in order to reduce deflection caused by gun blast and by wind drag on the antenna assembly. The antenna assembly for Radar Equipment Mark 28 is shown in Fig. 56. This assembly i§ shown mounted on a 40 mm Gun. i?^ DARAN TENNA S 307 14.12 .1 3 CM Anti-Aircraft Radar Antenna. To obtain greater discrimination between a given target and other targets, or between a target and its surroundings, the wavelength was reduced to the 3 cm region. An antenna for this wavelength was designed to employ the conical scan principle. In this case the parabolic reflector was 30 inches in diameter and transmitted a beam approximately 3° wide at the 3db point with a beam shift of 1.5° making a total of 4.5° with the antenna scanning. The minor lobes are down more than 22 db (one way) from the maximum; and the gain of this antenna is ?)5 db. In the 3 cm system in which a Cutler feed was used, the axis of the beam was rotated in an orbit by "nutation" about the mechanical axis of the antenna. This was accomplished by passing circular waveguide through the hollow shaft of the driving motor. The rear end of the feed (choke coupling end) was fixed in a ball pivot while the center (near the reflector) was off set the proper amount to develop the required beam shift. This off set was produced by a rotating eccentric driven by the motor. The latter was a 440 volt, 60 cycle, 3 phase motor rotating at approximately 1800 rpm which resulted in a scanning rate of 30 cycles per second. The two- phase reference voltage generator was integral with the driving motor. It was found necessary at these radio frequencies to use a cast aluminum reflector and to machine the reflecting surface to close tolerances in order to attain the consistency in beam width and beam direction required for accurate pointing. An antenna assembly for the 3 cm anti-aircraft radar is shown in Fig. 57. 15. Land Based Radar Antennas 15.1 The SCR-545 Radar ''Search'' and "Track" Antennas''' The SCR-545 Radar Set was developed at the Army's request to meet the urgent need for a radar set to detect aircraft and provide accurate tar- get tracking data for the direction of anti-aircraft guns. This use required that a narrow beam tracking antenna be employed to achieve the necessary tracking accuracy, furthermore, a narrow^ beam antenna suitable for accurate tracking has a very limited field of view and requires additional facilities for target acquisition. This was provided by the search antenna which has a relatively large field of view and is provided with facilities for centering the target in its field of view. These two an- tennas are integrated into a single mechanical structure and both radar axes coincide. The "Search" antenna operates in the 200 mc band and is com- " Section 15.1 was written by A. L. Robinson. 308 BELL SYSTEM TECHNICAL JOURNAL posed of an array of 16 quarter wave dipoles spaced 0.1 wave-length in front of a flat metal refletlor. All feed system lines and impedance matchinj,' (Icxiccs arc made uj) of coaxial transmission line sections. The array is divided into four quarters, each being fed from the lobe switching mechanism. This division is required to i)ermit lobe switching in both horizontal and vertical planes. The function of the lobe switching mecha- 3C'M Anti-.\irciaft Radar Antenna. nism is to introduce a particular phase shift in the excitation of the elements of one half of the antenna with respect to the other half. The theory of this tyjjc of lobe switching is discussed in section 11.1. The antenna beam spends a])j)roximately one quarter of a lobing cycle in each one of the four lobe positions. Each of the four lobe positions has the same radiated field intensity along the antemia axis and therefore when a target is on axis equal signals will be received from all four lobe positions. RADAR ANTENNAS 309 The "Track" antenna operates in the 10 cm. region and consists of a reflec- tor which is a parabola or revolution, 57 inches in diameter, illuminated by a source of energy emerging from a round waveguide in the lobing mechanism. Conical lobing is achieved by rotating the source of energy around the parabola axis in the focal plane of the parabola. Conical lobing is discussed in section 11.2. The round waveguide forming the source is filled with a specially shaped polystyrene core to control the illumination of the para iola and to seal the feed system against the weather. The radio frequency power is fed through coaxial transmission line to a coaxial-waveguide transition which is attached to the lobing mechanism. The "Search" and "Track" antenna lobing mechanisms are synchronized and driven by a common motor. The radio frequency power for both antennas is transmitted through a single specially constructed coaxial transmission line to the common antenna structure, where a coaxial transmission line filter separates the power for each antenna. Figure 58 is a photograph of a production model of the SCR-545 Radar Set. The principal electrical characteristics of the antennas are tabulated below: Antennas Search Track Gain 14.5 db 30 db Horizontal Beamwidth 23.5° 5° Vertical Beamwidth 25.5° 5° Polarization Horizontal Vertical Type of Lobing Lobe switching Conical lobing Angle between lobe positions 10° 3° Lobing rate 60 cycles/sec. 60 cycles/sec. The SCR-545 played an important part in the Italian campaign, particu- larly in helping to secure the Anzio Beach Head area, as well as combating the "V" bombs in Belgium. However the majority of SCR-545 equip- ments were sent to the Pacific Theater of Operations and played an im- portant part in operations on Leyte, Saipan, Iwo Jima, and Okinawa. 15.2 The AN/TPS-IA Portable Search Antenna^ In order to provide early warning information for advanced units, a light weight, readily transportable radar was designed under Signal Corps contract. i« Written by R. E. Crane. 310 BELL SYSTEM TECHNICAL JOURNAL «rV RADAR ANTENNAS 311 The objective was to obtain as long range early warning as possible with moderate accurracy of location. Emphasis was placed on detection of low flying planes. The objectives for the set indicated that the antenna should be built as large as reasonable and placed as high as reasonable for a portable set. Some latitude in choice of frequency was permitted at first. For rugged- ness and reliability reasons which seemed controlling at the time, the fre- quency was pushed as high as possible with vacuum tube detectors and R.F. amplifiers. This was finally set at 1080 mc. Fig. 59— AN/TPS-IA Antenna. The antenna as finally produced was 15 ft. in width and 4 ft. in height" The reflecting surface was paraboloidal. The mouth of the feed horn was approximately at the focus of the generating parabola. The feedhorn was excited by a probe consisting of the inner conductor of the coaxial transmission line extended through the side of the horn and suitably shaped. To reduce side lobes and back radiation the feedhorn was dimensioned to taper the illumination so that it was reduced about 10 db in the horizontal and vertical planes at the edges of the reflector. Dimensions of probe and exact location of feed, etc. were determined empirically to secure acceptable impedance over the frequency band needed. This band, covered by spot frequency magnetrons, was approximately ±2.5% from mid frequency. Figure 59 shows the antenna in place on top of the set. 312 BELL SYSTEM TECHNICAL JOURNAL The characteristics of this antenna are summarized below: Gain 27.3 db. Horizontal Half Power Bcamwidth 4.4° Vertical Half Power Beamwidth 12.6° Vertical Beam Characteristic Symmetrical Polarization Horizontal Impedance (SWR over ±2.5% <4.0db band) 16. Airborne Rad.vr Antennas 16.1 The AX APS-4 Anten)ia^ AN/APS-4 was designed to provide the Navy's carrier-based planes with a high performance high resolution radar for search against surface and airborne targets, navigation and intercej^tion of enemy planes under conditions of fog and darkness. For this service, weight was an all im- portant consideration and throughout a production schedule that by \"-J day was approaching 15,000 units, changes to reduce weight were con- stantly being introduced. In late production the antenna was responsible for 19 lbs. out of a total equipment weight of 164 lbs. The military require- ments called for a scan covering 150° in azimuth ahead of the plane and 30° above and below the horizontal plane in elevation. To meet this require- ment a Cutler feed and a parabolic reflector of 6.3" focal length and 14|" diameter was selected. Scanning in azimuth was performed by oscillating reflector and feed through the required 150° while elevation scan was per- formed by tilting the reflector. Beam pattern was good for all tilt angles. In early flight tests the altitude line on the B scope due to reflection from the sea beneath was found to be a serious detriment to the performance of the set. To reduce this, a feed with elongated slots designed for an elliptical reflector was tried and found to give an improvement even when used with the approximately round reflector. The elliptical reflector was also tried, but did not improve the performance sufficiently to justify the increased size. As will be noted in Fig. 60, the course of the mechanical development brought the horizontal pivot of the reflector to the form of small ears pro- jecting through the ])arabola. No appreciable deterioration of the beam {)attern due to this unorthodox expedient was noted. The equipment as a whole was built into a bomb-shaped container hung in the bomb rack on the underside of the wing. Various accidents resulted in this container being torn ofT the wing in a crash landing in water or dropped on the deck of the carrier. After these mishaps, the equipment was frequently found to be in good working order with little or no repair required. » Written by F. C. Willis. RADAR ANTENNAS 313 Gain 28 db Beamwidth 6° approx. circular Polarization Horizontal Scan Mechanical Scanning Sector- -Azimuth 150° Scanning Sector- -Elevation 60° Scanning Rate one per sec. Total weight 19 lbs. Fig. 60— AX/.\PS-4 Antenna. 16.2 The SCR-520, SCR-717 and SCR-720 Antennas-' The antenna shown in Fig. ol is typical of the type used with the SCR-520 and SCR-720 aircraft interception (night fighter) airborne radar equip- ment, as well as the SCR-717 sea search and anti-submarine airborne radar equipment. The parabolic reflector is 29 inches in diameter and produces a radiation beam about 10° wide. The absolute gain is approximately 25 db. RF energy is supplied to a pressurized emitter through a pressurized transmission line system which includes a rotary joint located on the ver- so Written by J. F. Morrison, 314 BELL SYSTEM TECHNICAL JOURNAL tical axis and a tilt joint on the horizontal axis. Either vertical or hori- zontal polarization can be used by rotating the mounting position of the emitter. Vertical polarization is preferred for aircraft interception work and horizontal polarization is i)referred for sea search work. Fig. 61— SCR-520 Antenna. For aircraft interception the military services desired to scan rapidly a large solid angle forward of the pursuing airplane, i.e. 90° right and left, 15° below and 50° above the line of flight. The data is presented to the opera- tor in the form of both "B" and ''C" })resentations and for this purpose potentiometer data take-offs are provided on the antenna. The reflector is spun on a vertical axis at a rate of 360 rpm and at the same time it is RADAR ANTENNAS 315 made to nod up and down about its horizontal axis by controllable amounts up to a total of 65° and at a rate of 30° per second. In the sea search SCR-717 equipment, selsyn azimuth position data take- offs are provided which drive a PPI type of indicator presentation. The rotational speed about the vertical axis in this case is either 8 or 20 rpm as selected by the operator. The reflector can also be tilted about its horizontal axis above or below the line of flight as desired by the operator. It wUl be noted that the emitter moves with the reflector and accordingly it is always located at the focal point throughout all orientations of the antenna. 16.3 T/ie AN/APQ-7 Radar Bombsight Antenna^^ Early experience in the use of bombing-through-overcast radar equip- ment indicated that a severe limitation in performance was to be expected as the result of the inadequate resolution offered by the then available air- borne radar equipments. This lack of resolution accounted for gross errors in bombing where the target area was not ideal from a radar standpoint. To meet this increased resolution requirement in range, the transmitted pulse width was shortened considerably. In attempting to increase the azimuthal resolution, higher frequencies of transmission were employed. This enabled an improvement in azimuthal resolution without resorting to larger radiating structures, a most important consideration on modern high speed military aircraft. To extend the size of the radiating structure without penalizing the air- craft performance, the use of a linear scanning array which would exhibit high azimuthal resolution was considered. This array was originally con- ceived in a form suitable to mount within the existing aircraft wing and transmit through the leading edge. As development proceeded, the restric- tions imposed on the antenna structure as well as the aircraft wing design resulted in the linear array scanner being housed in an appropriate separate air foil and attached to the aircraft fuselage (Fig. 62). The above study resulted in the development of the AN/APQ-7 radar equipment, operating at the X-band of frequencies. This equipment provided facilities for radar navigation and bombing. The AN/APQ-7 antenna consisted of an array of 250 dipole structures spaced at | wavelength intervals and energized by means of coupling probes extending into a variable width waveguide. The vertical pattern was arranged to exhibit a modified esc distribution by means of accurately shaped "flaps" attached to the assembly. " Written by L. W. Morrison. *' A large part of the antenna development was carried out at the M. I. T. Radiation Laboratory. 316 BELL SYSTEM TECHNICAL JOURNAL ANTENNA AIRFOIL ASSEMBLY Fig. 62— AN/APQ-7 AntennaMounted on B24 ;Bomber. CHOKE JOINT COUPLING SLIDING SURFACES Fig. 63— AN/APQ-7 Antenna. Left- Expanded Wave Guide Assembly. -Contracted Wave Guide Assembly. Right — The scanning of the beam is accomj)lished by varying the width of the feed waveguide. This is accomplished l)y means of a motor driven actuated cam which drives a push rod extending along the waveguide assembly back RADAR ANTENNAS 317 and forth. Toggle arms are attached to this push rod at frequently spaced intervals which provides the motion for varying the width of the waveguide while assuring precise parallelism of the side walls throughout its length (Fig. 63). The normal range of horizontal scanning exhibited by this linear array, extends from a line perpendicular to the array to 30° in the direction of the feed. By alternately feeding each end, a total scanning range of ±30° from the perpendicular is achieved. Appropriate circuits to synchronize the indicator for this range are included. The use of alternate end feed on the AN/APQ-7 antenna requires that the amount of energy fed to the individual dipoles is somewhat less than if a single end feed is employed. The AN/APQ-7 antenna is 16| feet in length and weighs 180 pounds exclusive of air foil housing. The following data applies; Gain = 32.5 db Horizontal beamwidth = 0.4° Vertical beam characteristic = modified csc^ Scan — Array scanning Scanning Sector — ± 30° Horizontal Scanning Rate = 45°/second Acknowledgments Contributors to the research and development of the radar antennas described in this paper included not only the great number of people directly concerned with these antennas but also the many people engaged in general research and development of microwave components and measuring tech- niques. A complete list of credits, therefore, will not be attempted. In addition to the few individuals mentioned in footnotes throughout the paper, the authors would like to pay special tribute to the following co-workers in the Radio Research Department: C. B. H. Feldman who with the assistance of D. H. Ring made an outstanding contribution in the development of the polyrcd array antenna; W. A. Tyrrell for his work on lobe switches; A. G. Fox, waveguide phase changers; A. P. King, paraboloids and horn antennas; A. C. Beck, submarine antennas; G. E. Mueller, polyrods. Probability Functions for the Modulus and Angle of the Normal Complex Variate By RAY S. HOYT This paper deals mainly with various 'distribution functions' and 'cumulative distribution functions' pertaining to the modulus and to the angle of the 'normal' comy)lex variate, for the case where the mean value of this variate is zero. Also, for auxiliary uses chiefly, the distribution function pertaining to the recijirocal of the modulus is included. For all of these various probability functions the paper derives convenient general formulas, and for four of the functions it supplies comprehensive sets of curves; furthur, it gives a table of computed values of the cumulative distribution function for the modulus, serving to verify the values computed by a difTerent method in an earlier paper by the same author.^ Introduction IN THE solution of problems relating to alternating current networks and transmission systems by means of the usual complex quantity method, any deviation of any quantity from its reference value is naturally a complex quantity, in general. If, further, the deviation is of a random nature and hence is variable in a random sense, then it constitutes a 'complex random variable,' or a 'complex variate,' the word 'variate' here meaning the same as 'random variable' (or 'chance variable' — though, on the whole, 'random variable' seems preferable to 'chance variable' and is more widely used). Although a complex variate may be regarded formally as a single ana- lytical entity, denotable by a single letter (as Z), nevertheless it has two analytical constituents, or components: for instance, its real and imaginary constituents (X and F); also, its modulus and amplitude (|Z| and 6). Correspondingly, a complex variate can be represented geometrically by a single geometrical entity, namely a plane vector, but this, in turn, has two geometrical components, or constituents: for instance, its two rec- tangular components (X and F); also, its two polar components, radius vector and vectorial angle (R = \ Z \ and 6). This paper deals mainly with the modulus and the angle of the complex variate,^ which are often of greater theoretical interest and practical im- '"Probabihty Theory and Telephone Transmission Engineering," Bell System Tech- nical Journal, January 1933, which will hereafter be referred to merely as the "1933 paper". ' Throughout the paper, I have used the term 'complex variate' for any 2-dimensional variate, because of the nature of the contemplated applications indicated in the first 318 PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 319 portance than the real and imaginary' constituents. The modulus variate and the angle variate, individually and jointly, are of considerable the- oretical interest; while the modulus variate is also of very considerable practical importance, and the angle variate may conceivably become of some practical importance. The paper is concerned chiefly with the 'distribution functions'^ and the 'cumulative distribution functions' pertaining to the modulus (Sections 3 and 5) and to the angle (Sections 6 and 7) of the 'normal' complex variate, for the case where the mean value of this variate is zero. The distribution function for the reciprocal of the modulus is also included (Section 4). The term 'probability function' is used in this paper generically to include 'distribution function' and 'cumulative distribution function.' To avoid all except short digressions, some of the derivation work has been placed in appendices, of which there are four. These may be found of some intrinsic interest, besides faciUtating the understanding of the paper. 1. Distribution Function and Cumulative Distribution Function IN General: Deeinitions, Terminology, Notation, Relations, AND Formulas The present section constitutes a generic basis for the rest of the paper. Let T denote any complex variate, and let p and a denote any pair of real quantities determining r and determined by t. (For instance, p and (7 might be the real and imaginary components of r, or they might be the modulus and angle of t.) Geometrically, p and a may be pictured as gen- eral curvilinear coordinates in a plane, as indicated by Fig. 1.1. Let T denote the unknown value of a random sample consisting of a single r-variate, and p' and a' the corresponding unknown values of the constituents of r'. Further, let G(p, a) denote the 'areal probability density' at any point p,a- in the p,(7-plane, so that G(p,a)dA gives the probability that t falls in a differential area dA containing the point r; and so that the integral of paragraph of the Introduction, and also because the present paper is a sort of sequel to my 1933 paper, where the term 'complex variate' (or rather, 'complex chance-variable') was used throughout since there it seemed clearly to be the best term, on account of the field of applications contemplated and the specific applications given as illustrations. However, for wider usage the term 'bivariate' might be preferred because of its prevalence in the field of Mathematical Statistics; and therefore the paper should be read with this alternative in view. ^The term 'distribution function' is used with the same meaning in this paper as in my 1933 paper, although there the term ' probability law' was used much more frequently than 'distribution function,' but with the same meaning. 320 BELL SYSTEM TECHNICA L JOURNA L G(p,(T)dA over the entire p,o--plane is equal to unity, corresponding to certainty. For the sake of subsequent needs of a formal nature, it will now be as- sumed that G{p,(t) = at all points p,o outside of the pi , P2 , ci , a^ quad- rilateral region in the p,o--plane, Fig. 1.1, bounded by arcs of the four heavy curv'es, for which p has the values pi and p2 and a the values ai and ao , with pi and en regarded, for convenience, as being less than p2 and a^ respec- tively. Further, G(p,a) will be assumed to be continuous inside of this p+dp P^ Pa Pi Fig. 1.1 — Diagram of general curvilinear coordinates. quadrilateral region, and to be non-infinite on its boundary. Hence, for probability purposes, it will suffice to deal with the open inequalities Pi < P < P2, (1.1) ai < a < (T2, (1.2) which pertain to this quadrilateral region excluding its boundary; and thus it will not be necessary to deal with the closed inequalities pi ^ p ^ P2 and (Ti ^ 0- ^ ao , which include the boundary."* ' The matters dealt with generically in this paragraph may he illustrated b>- the fol- lowing two important particular cases, which occur further on, namely: POLAR COORDINATES: p=|r| = 7?, <r=0 = angle of r. Then p, = A', = 0, P2 = Ri = 'X' , <Ti = di = 0, ffi = $2 = 2ir, whence (1.1) and (1.2) become < R < oc and Q < 6 < lir, respectively. RECTANGULAR COORDIN.^TES : p = Re r = .v, <r = Im t = y. Then p, = .v, = — x ^ P2 = X2 = 00, o"! = yi = — =0, 0-2 = vs = «= , whcucc (1.1) and (1.2) become — oo < X < <» and — =»_< y < <«, respectively. PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 321 A generic quadrilateral region contained within the quadrilateral region Pi , P2 , 0^1 , o'2 in Fig. 1.1 is the one bounded by arcs of the dashed curves P3 , Pi , (T3 , (Ti , where ps < p4 and as < <j\ . Here, as in the preceding paragraph, it will evidently suffice to deal with open inequalities. Referring to Fig. 1.1, the probability functions with which this paper will chiefly deal are certain particular cases of the probability functions P{p, a), P{p I 0-34) and Q{pz\ , C734) occurring on the right sides of the follow- ing three equations respectively: p{p < p' < p ^ dp, (J < a' < a + d<r) = P(p,a)dpda, (1.3) p(p < p < p -^ dp, az < a' < (Ji) = P{p I (T3i)dp, (1.4) p{pz < p < Pi , (T3 < a' < (Ti) = Q{p3i , 0-34). (1.5) These equations serve to define the above-mentioned probability functions occurring on the right sides in terms of the probabilities denoted by the left sides, each expression p( ) on the left side denoting the probability of the pair of inequalities within the parentheses. Inspection of these equations shows that: P(p,(r) is the 'distribution function' for p and a jointly; P{p \ 0-34) is a 'distribution function' for p individually, with the understanding that a' is restricted to the range a^-to-ai ; Qipsi ,o'34) is a 'cumulative distribution function' for p and a jointly. Since the left sides of (1.3), (1.4) and (1.5) are necessarily positive, the right sides must be also. Hence, as all of the probability functions occur- ring in the right sides are of course desired to be positive, the differentials dp and da must be taken as positive, if we are to avoid writing | dp \ and I (/(T I in place of dp and da respectively. Returning to (1.3), it is seen that, stated in words, P{p,a) is such that P{p.a)dpda gives the probability that the unknown values p' and a' of the constituents of the unknown value r' of a random sample consisting of a single r-variate lie respectively in the differential intervals dp and da containing the constituent values p and a respectively. Thus, unless dpda is the differential element of area, Pip,a) is not equal to the 'areal probability density,' G{p,a), defined in the fourth paragraph of this section. In general, if £ is such that Edpda is the differential element of area, then P(p, a) = EG{p, a). (An illustration is afforded incidentally by Appendix A.) P{p,a), defined by (1.3), is the basic 'probabiUty function,' in the sense that the others can be expressed in terms of it, by integration. Thus ^ Thus p in p( ) may be read 'probability that' or 'probabiHty of.' 322 BELL SYSTEM TECHNICAL JOURNAL P{p I 0-34) and P{(T I p3i), defined respectively by (1.4) and by the correlative of (1.4), can be expressed as 'single integrals,' as follows*: P(p I as,) = f * P(p,a) da, (1.6) P{a \ ps,) = H P{p,a) dp. (1.7) (?(P34 , (T34), defined by (1.5), can be expressed as a 'double integral,' funda- mentally; but, for purposes of analysis and of evaluation, this will be replaced by its two equivalent 'repeated integrals': Q(p3i , Cr 3i) f P{p,a) da dp = X^ I j ^(P.<^) dp\da, (1.8) the set of integration limits being the same in both repeated integrals because these limits are constants, as indicated by Fig. 1.1. On account of (1.6) and (1.7) respectively, (1.8) can evidently be written formally as two single integrals: Q(P34, ^34) = / P(p 1 a34) dp = / P{a\ P34) da, (1.9) but implicitly these are repeated integrals unless the single integrations in (1.6) and (1.7) can be executed, in which case the integrals in (1.9) will actually be single integrals, and these will be quite unlike each other in form, being integrals with respect to p and a respectively — though of course yielding a com.m.on expression in case the indicated integrations can be executed. The particular cases of (1.4) and (1.5) with which this paper will chiefly deal are the following three: p{p < p' <p + dp, a, <a' < a^) = P{p | a^:) dp = P (p) dp, (1.10) Pipi <p' <p,a,<a' < a.) = Q{< p,a,o) ^ Q{p), (1.11) p{p <p' <p2,ai<a' < 0-,) - Q{> p,an) = (?*(p). (1.12) ^ The single-integral formulation in (1.6) can be written down directly by mere inspec- tion of the left side of (1.4). Alternatively, (1.6) can be obtained by representing the left side of (1.4) by a repeated integral, as follows: Pip I (^34.) dp = pp-\dp P r'Ci •' P L"'''3 Pip, a)da dp = f Pip, <T)da dp, whence (1.6); the last equality in the above chain equation in this footnote evidently results from the fact that, in general fix)dx = f(x)dx, since each side of this equa- tion represents dA, the differential element of area under the graph of /(.v) from x to X -f dx. PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 323 In each of these thice equations the very abbreviated notation at the ex- treme right will be used wherever the function is being dealt with exten- sively, as in the various succeeding sections. Such notation will not seem unduly abbreviated nor arbitrary if the following considerations are noted: In (1.10), «T]2 corresponds to the entire effective range of a, so that P(p \ o-]2) is the 'principal' distribution function for p. Similarly, in (1.11), Q(< p,on) is the 'principal' cumultive distribution function for p. In (1.12), the star indicates that Q*ip) is the 'complementary' cumulative distribution func- tion, since Q(p) + Q*(p) = Q(pi2 , 0-12) = 1, unity being taken as the measure of certainty, of course. For occasional use in succeeding sections, the defining equations for the probabiUty functions pertaining to four other particular cases will be set down here: p{p<p' <P + dp, (Tx<a' <a) = P(p I < (t) dp, • (1.13) p(p< p' < p-{- dp, a < a' < (X2) ^ P(p \ > a) dp, (1.14) Pip, <p' <p,a,<a' <ct) = Q{< p, < a), (1.15) Pip <p' < p2 ,ai<a' <a) = Qi> p, < a). (1.16) It may be noted that (1.13) and (1.14) are mutually supplementary, in the sense that their sum is (1.10). Similarly, (1.15) and (1.16) are mutually supplementary, in the sense that their sum is ()(p]?,< a) = Qi< (r,pi2), which is the correlative of (1.11). This section will be concluded with the following three simple trans- formation relations (1.17), (1.18) and (1.19), which will be needed further on. They pertain to the probability functions on the right sides of equa- tions (1.3), (1.4) and (1.5) respectively, h and k denote any positive real constants, the restriction to positive values serving to simplify matters without being too restrictive for the needs of this paper. P{hp,ka) = ^^P{p,<t), (1.17) P{hp\k<rz,) =\Pip\ <^34), (1-18) Q{hpu,kazi) = Q{pzi, (T34). (1.19) Each of the three formulas (1.17), (1.18), (1.19) can be rather easily derived in at least two ways that are very different from each other. One way depends on probability inequality relations of the sort p{t<t'<t'Vdt) = p{gt<gt'<gt-^d[gt]), (1.20) p{h<t'<U) = p{gh<gl'<gh), (1.21) 324 BELL SYSTEM TECHNICAL JOURNAL where / stands generically for p and for a, and g is any positive real constant, standing generically for h and for k; (1.20) and (1.21) are easily seen to be true by imagining every variate in the universe of the /-variates to be multiplied by g, thereby obtaining a universe of (g/)-variates. A second way of deriving each of the three formulas (1.17), (1.18), (1.19) depends on general integral relations of the sort ( f{t) di = ^^ r fit) d{gt) ^u" f (-) d\. (1.22) •'« g ^ga g Jga \g/ A third way, which is distantly related to the second way, depends on the use of the Jacobian for changing the variables in any double integral; thus, P(p,<r) dXdn dpdcr = d{p,(T) = 1 -^ a(p,cr) d(X,M) (1.23) the first equality in (1.23) depending on the fact that the two sets of vari- ables and of differentials have corresponding values and hence are so re- lated that p(p<p'<p-\-dp, a<y<(T-\-da) = p(\<y<X-\-d\ m<m'<M+^/)u), (1-24) whence P(p,a) 1 dpd<j I = Pi\,fi) I dXdfjL |. 2. The Normal Complex Variate and Its Chief Probability Functions The 'normal' complex variate may be defined in various equivalent ways- Here, a given complex variate z = x -\- iy will be defined as being 'normal' if it is possible to choose in the plane of the scatter diagram of s a pair of rectangular axes, u and r, such that the distribution function P{u,v) for the given complex variate with respect to these axes can be written in the form^ P{u,v) 1 ZTTOuOv exp 2Sl 41 2Sl\ P(u)Piv). (2.1) We shall call w = u -\- iv the 'modified' complex variate, as it represents the value of the given complex variate g — .t -f iy when the latter is referred to the w,r-axes; P(u) and P{v) are respectively the individual distribution 1 functions for the u and r components of the modified complex variate ; and ■^ Defined by equation (L3) on setting p = it and a = v. "This ecjuation is (12) of my 1933 paper. It can he easily verified tliat the (double) integral of (2.1) taken over the entire n, ii-plane is equal to unity. PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 325 Su and Sv are distribution parameters called the 'standard deviations' of w and V respectively. If / stands for u and for v generically, then P(t) = -7^ vfe,^-^;]' <'•'' ^' = /_j'^«'"- P.3) From the viewpoint of the scatter diagram, the distribution function Pin,v) is, in general, equal to the 'areal probability density' at the point u,v in the plane of the scatter diagram, so that the probabihty of falling in a differential element of area dA containing the point ti,v is equal to P{u,v)dA ; similarly, P{;u) and P{v) are equal to the component probability densities. In particular, the probability density is 'normal' when P{u,v) is given by (2.1). Geometrically, equation (2.1) evidently represents a surface, the normal 'probability surface,' situated above the u, r-plane; and P{u, v) is the ordinate from any point u,v in the u,v-p\a.ne to the probability surface. The M,T'-axes described above will be recognized as being the 'principal central axes,' namely that pair of rectangular axs which have their origin at the 'center' of the scatter diagram of s = x + iy and hence at the center of the scatter diagram of u> — u -\- iv, so that w = 0, and are so oriented in the scatter diagram that m; = (whereas 2^0 and xy 9^ 0, in general). In equation (2.1), which has been adopted above as the analytical basis for defining the 'normal' complex variate, the distribution parameters are Su and Sv ; and they occur symmetrically there, which is evidently natural and is desirable for purposes of definition. Henceforth, however, it will be preferable to adopt as the distribution parameters the quantities S and b defined by the pair of equations S' = Sl + Sl , (2.4) bS' = Sl - S; , (2.5) whence , __ Su Sy _ 1 [Sy/Su) ,r. ,,. »Jm "r Sy 1 -\- {Sy/SuJ From (2.4), S is seen to be a sort of 'resultant standard deviation.' The last form of (2.6) shows clearly that the total possible range of b is — l^b^l, corresponding to '^^Sy/Su^O. The pair of simultaneous equations (2.4) and (2.5) give 2Sl = {\ + b)S-, (2.7) 2^; = (1-^.)^-, (2.8) which will be used below in deriving (2.11). 'Equations (2.4) and (2.6) are respectivelj- (14) and (13) of my 1933 paper. 326 BELL S YSTEM TECH NIC A L JOURNA L With the purpose of reducing the number of parameters by 1 and of dealing with variables that are dimensionless, we shall henceforth deal with the 'reduced' modified variate W = U ■\- iV defined by the equation W ^ w/S = u/S + iv/S = U + iV. (2.9) Thus we shall be directly concerned with the scatter diagram of W = U + iV instead of with that oi w = u -\- iv. The distribution function P(L'*,T') for the rectangular components U and 1' of any complex variate W — U -\- iV is defined by (1.3) on setting p = i' and cr = T; thus, p{u,v)dudv = p{U<u'<u-\-du,v<r'<vi-dV). (2.10) When the given variate z — x -\- iy is normal, so that the modified variate 11) — u -{■ iv is normal, as represented by (2.1), then, since S is a mere con- stant, the reduced modified variate W — U -{- i]' defined by (2.9) will evidently be normal also, though of course with a different distribution parameter. Its distribution function P(t',l ) is found to have the formula 1 r t/2 F2 ■ where P{1) and P{V) are the component distribution functions: t/2 = F{U)P{V), (2.11) ^(^) = vOT)^-r P(V) = ./..; . ^exp[-^4 (2.12) (2.13) \/ir(l - b) These three distribution functions each contain only one distribution parameter, namely b; moreover, the variables U = u/S and 1' = v/S are dimensionless. ' The distribution function P{R,6) for the polar components R and 6 of any complex variate W = R{cos 6 -\- i sin 6) is defined by (1.3) on setting p = R and a — 6; thus P{R,e)dRd9 = p{R<R'<R^dR. d<d' <d-\-de). (2.14) For the case where 11' is 'normal,' it is shown in Appendix A that R [ -R' VT ^'(^'^) = -Wr^-T. exp -^-fi:2 (1 - & cos 2d) (2.15) exp[-L(l - 6 cos 20)], (2.16) "This formula can be obtained from (2.1) by means of (2.7), (2.8), (2.9) and (1.17) after specializing (1.17) by the substitutions p = u,a = v and h = k = 1/5. It is (16) of my 1933 paper, but was given there without proof. PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 327 where L= Ry{\-b-'). (2.17) In P{R,d) it will evidently suffice to deal with values of 6 in the first quadrant, because of symmetry of the scatter diagram. The fact that P(R,6) depends on 6 as a parameter when W is 'norma]' may be indicated explicitly by employing the fuller symbol P{R,d;b) when desired; thus the former symbol is here an abbreviation for the latter. In P{R,d) = P(R, 6; b) it will suffice to deal with only positive values of b, that is, with O^b^l (whereas the total possible range of b is — l^^^l). For (2.15) shows that changing b to —b has the same effect as changing 2d to 7r±2e, or d to T/2±d; that is, P{R,d; -b) ^ P(R, ir/2±d; b). Seven formulas which will find considerable use subsequently are obtain- able from the integrals corresponding to equations (1.13) to (1.16), by setting p = R and a = 6 or else p = 6 and c = R, whichever is appropriate, and thereafter substituting for P{R,6) the expression given by (2.16), and lastly executing the indicated integrations wherever they appear possible." The resulting formulas are as follows: P(R \ < d) = y^ exp(-Z) / expibL cos 26) dd, (2.18) T Jo (2.19) P{e \ < R) = ^^ ~ ^' 1 - exp[-i:(l - b cos 2d)] 2ir I — b cos 20 P(e \> R) = ^^ ~ ^' exp[-£(l - b cos 29)] 2t 1 — b cos 26 (2.20) dR (2.21) Q{< R, < 6) = - [ \ \/l exp(-L) [ exp{bL cos 26) dd TT Jo L "^O Vnili r" 1 - exp[-I(l - b cos 26)] ~27~ io 1 - b cos 26 ^^' ^^-^^^ Q(> R, < 6) =- I VL exp(-L) j exp {bL cos 26) dd dR (2.23) Ztt Jo 1 — b cos 26 Formulas (2.21) to (2.24) are obtainable also by substituting (2.18) to (2.20) into the appropriate particular forms of (1.9). When a ^-range of integration is 0-to-5(7r/2), where q = 1, 2, 3 or 4, this " Except that in (2.22) the part 1/(1 — b cos 26) is integrable, as found in Sec. 7, equations (7.6) and (7.7). 328 BELL S YSTEM TECH NIC A L JOURNA L range can be reduced to 0-to-7r/2 provided the resulting integral is mul- tiplied by q; that is, /«5(7r/2) ^jr/2 / F{e)(W = q / F{e)dd, (2.25) Jo •'0 because of symmetry of the scatter diagram. 3. The Distribution Function for the Modulus The distribution function P{R | dv2) = F{R) for the modulus R of any complex variate IT = R(cos 6 + / sin 0) is defined by equation (1.10) on setting p = R, a = 9, ffi = 6] — and (r2 — 62 — 2ir; thus P{R)dR = p(R<R'<R+dR, (xe'KlTv). (3.1) An integral formula for F(R) is immediately obtainable from (1.6) by setting p = R, o — 6, (Ti = ai = 61 = and 04 = a^ ~ S2 = 2x; thus F{R) = [ F{R,d) do. (3.2) Jo The rest of this section deals with the case where \V = R(cos 6 + / sin 6) is 'normal.' Since this case depends on i as a parameter, F(R) is here an abbreviation for F{R;h). A formula for F{R;b) can be obtained by sub- stituting F{R, 6) from (2.15) into (3.2) and executing the indicated integra- tion by means of the known Bessel function formula i: exp(r} cos \f/) dip = 7r/o(r/), (3.3) /o( ) being the so-called 'modified Bessel function of the first kind,' of order zero.^'- The resulting formula is found to be^^ 2R .1 - d^Ti bR^ - b' (3.4) This can also be obtained as a particular case of the more general formula (2.18) by setting 6 — 2t in the upper limit of integration and then apply- ing (3.3). In F(R;b) it will suffice to deal with positive values of b, that is, with U^6^1, as (3.4) shows that F(R; -b) = F{R;b). 12 It may be recalled that /o(c) = /o(/-), and in general that /„(;) = i-"Jn{i~). In the list of references on Bessel functions, on the last page of this paper, the 'modified Bessel function' is treated in Ref. 2, p. 20; Ref. 3, p. 102; Ref. 4, p. 41; Ref. 1, p. 77. Regarding formula {3.3), see Ref. 1, p. 181, Eq. (4), i. = 0; Ref. 1, p. 19, Eq. (9), fourth expression, p = 0; Ref. 2, p. 46, Eq. (10), n = 0; Ref. 3, p. 164, Eq. 103, n = 0. ^' This formula was given in its cumulative forms, / P{R; b)dR, as fornuilas (Sl-.A) and (53-A) of the unpublished .\ppendix A to my 1933 paper. PROBABILITY FUNCTIONS FOR COMPLEX VARIATi: 32^ It will often be advantageous to express P^; 6 in terms of b and one or the other of the auxiliary variables L and T defined by the equations ^ = r^2' (3-5) ^ = ^^ = 1^2- (^-6) Formula (3.4) thereby becomes, respectively, P{R;b) = 2VLexp{-L)h{bL), (3.7) P(R;b) = 2 y^l exp[^j h{T). (3.8) Formula (3.8) will often be preferable to (3.7) because the argument of the Bessel function in (3.8) is a single quantity, T. Because tables of /o(-V) are much less easily interpolated than tables of Mo(X) defined by the equation Mo(X) = exp(-X)h(X), (3.9) extensive tables of wiiich have beeo published," it is natural, at least for computational purposes, to write (3.4) in the form 2R r -R' 1 Vl - b^ Mo • bR' 1 -b' (3.10) For use in equation (3.16), it is convenient to define here a function Mi(X) by the equation M,(X) = exp(-A')/i(X), (3.11) corresponding to (3.9) defining Mo{X). Mi(X) has the similar property that it is much more easily interpolated than is Ii(X); and extensive tables of Ml (A') are constituent parts of the tables in Ref. 1 and Ref. 6. The quantity bR-/{l — b') = T, which occurs in (3.4) and (3.8) as the argument of /o( ), and in (3.10) as the argument of Mo{ ), evidently ranges from to co when R ranges from to co and also when b ranges from to 1. Formula (3.10) is suitable for computational purposes for all values of the above-mentioned argument bR~/(l — b'^) = T not exceeding the largest values of X in the above-cited tables in Ref. 1 and Ref. 6. For larger values of the argument, and partiularly for dealing with the limiting i-* Ref. 1, Table II (p. 698-713), for X = to 16 by .02. Ref. 6, Table VIII (p. 272- 283), for A^ = 5 to 10 by .01, and 10 to 20 by 0.1. Each of these references conveniently includes a table of exp(A^) whereby values of /o(A') can be readily and accurately evalu- ated if desired. Values of /o(A') so obtained would enable formulas (3.4), (3.7) and (3.8) of the present paper to be used with high accuracy without any difficult interpolations, since the table of exp(A'') is easily interpolated by utilizing the identity exp(A'i -)- A'2) = exp(Ai) exp(A^2). 330 BELL SYSTEM TECHNICAL JOURNAL case where the argument becomes infinite, formula (310)- — and hence (3.4) — may be advantageously written m the form where No{X) = V2^exp(-X)/o(X) = \/2^Mo{X), (3.13) an extensive table of which has been published.'^ The natural suitabiUty of the function A^o(^) for dealing with large values of A' is evident from the structure of the asymptotic series for No{X), for sufficiently large values of X, which runs as follows:^® iVo(X) ~ 1 + jl^ + jl^, + jl^, + . . . , (3.14) whence it is evident that No{oo) = 1. (3.15) For use in Appendix C, it is convenient to define here a function A^i(A") by the equation" Ni{X) = V'2^exp(-X)/i(X) - V2^M,{X), (3.16) corresponding to (3.13) defining No(X), with Mi(X) defined by (3.11). The asymptotic series for Ni{X), which will be needed in Appendix C, is^^ NiiX) -- 1 - 3 whence it is evident that 1 . 0-5) (l -5)(3-7) 1 .1!8X 2I(8X)2^ 31(8X)» ^ J' ^^ ^ Ni{oo) = 1. (3.18) When b is very nearly but not exactly equal to unity, so that bR" R" R" (3.19) 1-^2 1-62 2(1 - 6) ' it is seen from (3.4) that P{R;b) is, to a very close approximation, a function 15 Ref. 7, pp. 45-72, for X = 10 to 50 by 0.1, 50 to 200 by 1, 200 to 1000 by 10, and for various larger values of X. 16 Ref. 1, p. 203, with (u, m) defined on p. 198; Ref. 5, p. 366; Ref. 2, p. 58; Ref. 3, p. 163, Eq. 84; Ref. 4, pp. 48, 84. 1^ N i{X) is tabulated along with N^iX) in Ref. 7 already cited in connection with equa- tion (3.13). " Ref. 1, p. 203, with {v, m) defined on d. 198; Ref. 5, p. 366; Ref. 2, p. 58; Ref. 3, p. 163, Eq. 84. PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 331 of only a single quantity, which may be any one of the three very nearly equal expressions in (3.19) — but the last of them is evidently the simplest. Fig. 3.1 gives curves of P(,R;b), with the variable R ranging continuously Fig. 3.1 — Distribution function for the modulus {R = to 2.8). from to 2.8 and the parameter b ranging by steps from to 1 inclusive, which is the complete range of positive b. Fig. 3.2 gives an enlargement (along the i?-axis) of the portion of Fig. 3.1 between R — and R = 0.4, 332 BELL SYSTEM TECHNICAL JOURNAL l/\ V\ \ 0) 6 II X) A 1 \\ 1 m \\ l\ A \ ^ I \ \\ \= - w I \V\\ o 6 y 1 \ \ \V \ M \ L d\ V \ /) eo\ d\ \\ \\ \// \ \ \ > \\ A/ x \ \V d A q \ \ V \\\ ' ' / \ ^ \ V \^ \ d / \ \ \ \ \V \ y \ \ V \ V ^ d w \ y. \ \ \, \ /^ k^ \ N, \ \ \\ % 1 \ >«^ X \ s\ 1 ^ -Q g: d"- d -*«= >i --^ ^ ^ 1 ^ ' ■ is ^ ^ DISTRIBUTION FUNCTION, P(R;b) Fig. 3.2— Distribution function for the modulus (/^ = to 0.4). PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 333 and includes therein curves for a considerable number of additional values of b between 0.9 and 1 so chosen as to show clearly how, with b increasing toward 1, the curves approach the curve for 5 = 1 as a limiting particular curve; or, conversely, how the curve iov b — 1 constitutes a limiting par- ticular curve — which, incidentally, will be found to be a natural and con- venient reference curve. This curve, iov b = 1, will be considered more fully a little further on, because it is a limiting particular curve and be- cause of its resulting peculiarity at i? = 0, the curve iov b = 1 having at R = a. projection, or spur, situated in the P{R;b) axis and extending from 0.7979 to 0.9376 therein (as shown a little further on). The formulas and curves iov b = and b = 1, being of especial interest and importance, will be considered before the remaining curves of the set. For the case b = 0, formula (3.4) evidently reduces immediately to F{R;0) = 2Rexp(-R^). (3.20) This case, 6 = 0, is that degenerate particular case in which the equiprob- ability curves in the scatter diagram of the complex variate, instead of being ellipses (concentric), are merely circles, as noted in my 1933 paper, near the bottom of p. 44 thereof (p. 10 of reprint). For the case b = 1, the formula for the entire curve of P{R; b) = P(R;1), except only the part at R = 0, can be obtained by merely setting b = I in^^ (3.12) as this, on account of (3.15), thereby reduces immediately to 2_ V2^ P'iR;\) denoting the value of P{R;b) when b = 1 but i? 5^ 0, the restriction i? 5^ being necessary because the quantity R~/(l — b^) in (3.12) — and in (3.4) — does not have a definite value when b — 1 if i? = 0. Thus, in Figs. 3.1 and 3.2, the curve of P'(R;\) is that part of the curve iov b = 1 which does not include any point in the P{R; b) axis (where R — 0) but extends rightward from that axis toward R = -f 00. The curve of P'{R;l) is the 'effective' part of the curve of P{R;l), in the sense that the area under the former is equal to that under the latter, since the part of the curve of P{R;l) at R = can have no area under it. P(0;1) denoting (by convention) the value, or values, of P{R;b) when R — and b — 1, that is, the value, or values, of P{R'S) when R = 0, it is seen, from consideration of the curves of P{R;b) in Figs. 3.1 and 3.2 when b approaches 1 and ultimately becomes equal to 1, that the curve of P(0;1) consists of all points in the vertical straight line segment extending upward in the PiR;b) axis, from the origin to a height 0.9376 [= Max P(i?;l)],20 '^ Use of (3.12) instead of (3.4), which is transformable into (3.12), avoids the indefinite expression « .0.^ which would result directly from setting 6 = 1 in (3.4). ^^ As shown near the end of Appendix B, MaxP(^;l) is situated at /? = and is equal to 0.9376. ^'(^; 1) = r7^exp|^-f]> (R ^ 0)> (3.21) 334 BELL S YSTEM TECH NIC A L JOURNA L together with all points in the straight line segment extending downward from the point at 0.9376 to the point at 0.7979 [= 2/ \/2^ = P'{R\\) for R = 0+]. The curve of P(0; 1), because it has no area under it, is the 'non-effective' part of the curve of P{R\\). Starting at the origin of coordinates, where i? = 0, the complete curve of P{R\\) consists of the curve of P(0;1), described in the preceding para- graph, in sequence with the curve of P'(R;\), given by (3.21). Thus the complete curve of P(R;\) is the locus of a tracing point moving as follows: Starting at the origin of coordinates, the tracing point first ascends in the P{R; b) axis to a height 0.9376 [= MaxP(i?;l)]; second, descends from 0.9376 to 0.7979 [= 2/ V2^ = P'iR'A) for R = 0-\-]; and, third, moves rightward along the graph of P'(R;\) [b = l] toward i? = -f co . The locus of all of the points thus traversed by the tracing point is the complete curve'' of P{R;l). In addition to being the principal part ('effective' part) of the curve of P{R;\), the curve of P'(R;\), whose formula is (3.21), has a further impor- tant significance. For the right side of (3.21), except for the factor 2, will be recognized as being the expression for the well-known 1 -dimensional 'normal' law; the presence of the factor 2 is accounted for by the fact that the variable i? = | i? | can have only posiive values and yet the area under the curve must be equal to unity. This case, b = 1, is that degenerate particular case in which the equiprobability curves, instead of being ellipses, are superposed straight line segments, so that the resulting 'probability density' is not constant but varies in accordance with the 1-dimensional 'normal' law (for real variates), as noted in my 1933 paper, at the top of p. 45 thereof (p. 11 of reprint). All of the curves of P{R;b), where O^b^l, pass through the origin, the curve of PiR;\.) [b = 1] being no exception, since the part P(0;1) passes through the origin. Formula (3.12), supplemented by (3.15), shows that P(R; b) = at i? = 00 ; and this is in accord with the consideration that the total area under the curve of P{R;b) must be finite (equal to unity). Since P{R;b) — slI R — and a.t R — co, every curve of P{R;b) must have a maximum value situated somewhere between R ~ Q and R — oo — as confirmed by Figs. 3.1 and 3.2. These figures show that when b increases from to 1 the maximum value increases throughout but the value of R where it is located decreases throughout. The maxima of the function P{R;b) and of its curves (Figs. 3.1 and 3.2) are of considerable theoretical interest and of some practical importance. ''^ The presence, in the curve of F{R; 1), of the vertical projection, or spur, situated in the P{K; b) axis and extending from 0.7979 to 0.9376 therein, is somewhat remindful (qualitatively) of the'Gibbs phenomenon' in the representation of discontinuous periodic functions by Fourier series. PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 335 The cases b — Q and b = \ will be dealt with first, and then the general case {b = b). For the case J = it is easily found by differentiating (3.20) that P{R;b) = P{R; 0) is a maximum Sit R — 1/ \^2 = 0.7071 and hence that its maximum value is \/2exp (—1/2) = 0.8578, agreeing with the curve for 6 = in Fig. 3.1. For the case b = I, which is a limiting particular case, the maximum value of P(R;b) — P(i?;l) apparently cannot be found driectly and simply, as will be realized from the preceding discussion of this case. Near the end of Appendix B, it is shown that the maximum value of P{R;\) occurs at 7? = (as would be expected) and is equal to 0.9376. This is the maximum value of the part P(0;1 of P(R;1). The remaining part of P(R;l), namely P'{R;1), whose formula is (3.21), is seen from direct inspection of that formula to have a right-hand maximum value a.t R = 0+, whence this m-aximum value is 2/v 2ir = 0.7979. For the general case when b has any fixed value within its possible positive range (O^i^ 1), it is apparently not possible to obtain an explicit expression (in closed form) either for the value of R at which P{R;b) has its maximum value or for the maximum value of P(R;b); and hence it is not possible to make explicit computations of these quantities for use in plotting curves of them, versus b, of which they will evidently be functions. However, as shown in Appendix B, these desired curves can be exactly computed, in an indirect manner, by temporarily taking b as the dependent variable and taking T, defined by (3.6), as an intermediate independent variable. For let Re denote the critical value of R, that is, the value of R at which PiR;h) has its maximum value; and let Tc denote the corresponding value of T, whence, by (3.6), Tc= bRl/il-b'). (3.22) uj 0.8 I UJ O 5 0.4 gO.2 »- o z 2 MAX P(R;b) ■;^ "^ Rr " Vi-b2 "~~~~ ■^ Pc "" \ \ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 PARAMETER, b 0.8 0.9 1.0 Fig. 3.3 — Functions relating to the maxima of the distribution function for the modulus. 336 BELL SYSTEM TECHNICAL JOURNAL Then, computed by means of the formulas derived in AppendLx B, Fig. 3.3 gives a curve of Re and a curve of Max P(R;b), each versus b. Since the curve of Re cannot be read accurately at 6 ?5r; 1, there is included also a curve of Rc/y/l — b-, from which Re can be accurately and easily com" puted for any value of b; incidentally, the curve of Re/y/l — 6' is simul- taneously a curve of -s/Telb, on account of (3.22). From Fig. ?i.7i it is seen that Re varies greatly with b but that Max Pji-;^ varies only a little, as also is seen from inspection of Figs. 3.1 and 3.2 giving curves of P{R\b) as function of R with b as parameter. In Fig. }).?), the curve of Re shows that for 6 = 1 the maximum of P{R;b) occurs ai R = 0; and the curve of Max P{R;b) shows that Max P{R;\) ^ 0.94, agreeing to two significant figures with the value 0.9376 found near the end of Appendix B. - 4. The Distribution Function for the Reciprocal of the Modulus At first, let R denote any real variate, and P{R) its distribution function. Also let r denote the reciprocal of R, so that r = \/R; and let P{r) denote the distribution function for r. Then -- P{r) = R'PiR) = P{R)/r\ (4.1) If P{R) depends on any parameters, P{r) will evidently depend on the same parameters. The rest of this section deals with the case where W = R(cos + i sin 6) is 'normal.' Since this case depends on 6 as a parameter, P(R) and P(r) are here abbreviations for P{R;b) and P{r;b) respectively. As PiR;b) has the distribution function given by (3.4), the distribution function for r will be P{r;b) = (Vl - b'-)r 3 exp -1 (1 - &VJ "L(i - b'yy (4.2) obtained from the right side of (3.4) by changing R to l/r and multiplying " For if r and R denote any two real variates that are functionally related, sa}- F{r, K) = 0, and if dr and dR are corresponding small increments, then evidently P{r) \dr\ == P{R) \ dR \ whence Pir) PiR) dR dr bF/br dF/dR In particular, if r = \/R, whence F = r — l/R, then (4.1) results immediately. For a somewhat ditYerent and more detailed treatment of change of the variable in distribution functions, see Thorton C. Fry, "Probability and its Engineering Uses," 1928, pp. 1.S3-155. (Cases of more than one variate are treated on pp. 155-174 of the same reference.) PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 337 the result by 1/r, in accordance with (4.1). Evidently P{r; — b) = P(r;b). By means of (4.1), formulas (3.7) and (3.8) give, respectively, P{r;b) = 2(l-b^')L"'exp{-L)Io{bL), (4.3) P(r;b) = 2(1 - b') l^lj exp|^^j /o(T), (4.4) wherein L and T are defined by (3.5) and (3.6) respectively, but will now be written in the equivalent forms i = (T^- (4.5) r = Si=_^_A_, (4.6) which are evidently more suitable for the present section. A few particular cases that are especially important will be dealt with in the following brief paragraph, ending with equation (4.8). For the two extreme values of r, namely and oc , P{r;b) is zero for all values of b in the b- range (0^6^ 1). When b = 0, When b = I, P{r-b) = P{r;0) = ^^expf-ij. (4.7) ^f ] Pir;b) = P{r;\) = ^^ ;;;, exp| ~, \. (4.8) Fig. 4.1 gives curves of P(r;b), with the variable r ranging continuously from to 1.4 and the parameter b ranging by steps from to 1; however, in the r-range where r is less than about 0.6, alternate curves had to be omitted to avoid undue crowding. Fig. 4.2 gives an enlargement of the section betwen r = 0.2 and r = 0.5, and includes therein the curves that had to be omitted from Fig. 4.1. In Fig. 4.1 it will be noted that with the scale there used for P(r;b) the values of P(r;b) are too small to be even detectable for values of r less than about 0.25. Even in the enlargement supplied by Fig. 4.2, the values of P{r;b) are not detectable for r less than about 0.2. The curves of P{r;b) in Figs. 4.1 and 4.2 would have had to be computed from the lengthy formula (4.2) — or its equivalents — except for the fact that curves of P{R;b) had already been computed in the preceding section of the paper. The last circumstance enabled the P{r;b) curves to be obtained from the P{R;b) curves by means of the very simple relation (4.1). It will be observed that each curve of P{r;b) [Fig. 4.1] has a maximum 338 BELL SYSTEM TECHNICAL JOURNAL ordinate, whose value and location depend on b. When b increases from to 1, the maximum ordinate decreases throughout but the value of r where it is located remains nearly constant, at about 0.82, until b becomes about 0.40 Z O 3 0.35 tr ^ 0.30 a 0.25 0.20 0.15 -^ <i ~v k / Yf ^ vs I /^ 0.6 x^ ^ k — .Ov ^ k m N> ^ ^ /! \ N \ \^ li ^ c^ \ 1 1 \ k^ \ X 1 . \, \ 1 j "^ /I b = t.a o.e-j< 0.6-^ Ijl 111 oaU- / h ////, 4 w 0.1 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 1.3 RECIPROCAL OF THE MODULUS, P Fig. 4.1 — Distribution function for the reciprocal of the modulus (r = to 1.4). 0.7, after which the location of the maximum value moves rather rapidly to about 0.71 for ft = 1. For the cases 6=0 and b = 1, it is easily found, by differentiating (4.7) and (4.8), that the maximum ordinates are located at r = \/2/3 = 0.8165 and at r = l/'\/2 = 0.7071 respectively; and hence, by (4.7) and (4.8). that the values of these maximum ordinates are (3\/3/2 exp (—3/2) = PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 339 0.8198 and (4/V27r) exp (-1) = 0.5871 respectively. These results for the cases 6 = and 6=1 agree with the corresponding curves in Fig. 4.1. 0.20 0.23 0.32 0.36 0.40 RECIPROCAL CF THE MODULUS, T Fig. 4.2 — Distribution function for the reciprocal of the modulus {r = 0.2 to 0.5). For the general case where b has any fixed value in the 6-range (0^6^ 1), it is apparently not possible to obtain an explicit expression (in closed form) either for the value of r at which P{r;b) has its maximum value or for the 340 BELL SYSTEM TECH NIC A L JOURNAL maximum value of P(r;b). However, as shown in Appendix C, curves of these quantities versus b can be computed, in an indirect manner, by temporarily taking b as the dependent variable and taking T, defined by (4.6), as an intermediate independent variable. For let Tc denote the critical value of r, that is, the value of r at which P(r;b) has its maximum value; and let Tc denote the corresponding value of T, whence, by (4.6), Tc= b/{\-b'-)r\ (4.9) Then, computed by means of the formulas derived in Appendix C, Fig. 4.3 gives a curve of Vc and a curve of Max P{r;b), each versus b. From these curves it is seen that re and Max P{r\b) do not vary greatly with b, as also is seen from inspection of Fig. 4.1 giving curves of P{r\b) as function of r with b as parameter. Tc MAX F ^(ribT -^ ^ — < g 0.4 to z 2 0.2 t- u z £ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 PARAMETER, b Fig. 4.3 — Functions relating to the maxima of the distribution function for the reciprocal of the modulus. 5. The Cumulative Distribution Function for the Modulus The cumulative distribution function Q{<R,di2) = Q{R) for the modulus R of any complex variate W = R{cos 6 + i sin 6) is defined by equation (1.11) on setting p = R, a = 6, pi = Ri ~ 0, ai = 6i — and (72 = 6-. = Itt; thus QiR) = p{{)<R'<RA)<d'<2Tr). (5.1) Similarly, from (1.12), the complementary cumulative distribution function Q{>R,di2) = Q*{R) is defined by the equation Q*{R) - p(R<R'<-^^,{)<e'<2Tr). (5.2) Q*iR) is usually more convenient than Q{R) for use in engineering ap- plications, because it is usually mor? convenient to deal with the relatively PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 341 small probability of exceeding a preassigned rather large value of R than to deal with the corresponding rather large probability (nearly equal to unity) of being less than the preassigned value of R. A 'double integral' for Q{R), in the form of two 'repeated integrals,' can be written down directly by inspection of the p{ ) expression in (5.1) or by specialization of (1.8); thus ' / P{R,d) de clR = / P{R,d) dR dd. (5.3) Evidently these can be written formally as two 'single integrals,' Q{R) = / P{R) dR = / P{e\ < R) dd, Jo Jn (5.4) by means of the distribution functions P(R) = P(R | ^i.) and P{e \ <R) given by the formulas P{R) = [ P{R,e) dd, (5.5) P{d\<R) = [ P{R,d)dR. (5.6) Jo Jo (5.5) is the same as (3.2). (5.6) is a special case of (1.6), and the left side of (5.6) is a special case of P{p \ <a) detined by (1.13). Similarly, from (5.2), we arrive at the following formulas corresponding to (5.3), (5.4), (5.5), and (5.6) respectively: dd, Q*{R) = ■ / PiR,d) dd dR = / P(R,d) dR J R \_Jo J •'oL'^'' ^00 /.27r Q*(R) = p{R) dR = P{d\ > R) dd, J R Jo P{d\ > R) = f P{R,d) dR. J R P{R) = [ P{R,d) dd, Jo (5.9) (5.7) (5.8) (5.10) The rest of this section deals with the case where W = i?(cos d -\- i sin 6) is 'normal.'-^ Since this case depends on 6 as a parameter, Q{R) and Q*(R) are here abbreviations for Q{R;b) and Q*{R;b) respectively. A natural and convenient way for deriving formulas for Q{R) is afforded by the general formula (5.4) together with the auxiliary general formulas (5.5) and (5.6), beginning with the two latter. For the 'normal' case, P{R,d) is given by (2.15). When this is sub- stituted into (5.5) and (5.6), it is found that each of the indicated integra- 23 For the 'normal' case, the cumulative distribution function was treated in a very different manner in my 1933 paper and its unpublished Appendix A. That paper included applications to two important practical problems, and its unpublished Appendix C treated a third such problem. (The unpublished appendices, A, B and C, are mentioned in foot- note 3 of the 1933 paper.) 342 BELL SYSTEM TECHNICAL JOURNAL tions can be executed, giving the two previously obtained formulas (3.4) and (2.19) for P(i?) = P(R;b) and P{d\ <R) respectively. When these are substituted into (5.4), there result two types of single-integral formulas for Q{R): A prirrary type, involving an indicated integration as to R; and a secondary tyj^e, involving an indicated integration as to 6. Formulas of these two types for Q{R) will now be derived. An integral formula of the primary type for Q{R) = Q{R;b) can be ob- tained by substituting P(R) = P(.R',b) from (3.4) into the first integral in (5.4), giving Q{R) = 2 [ Jo X Vl - b- exp r ~^' 1 r ^^' 1 Li - b'i h Li - h'\ d\. (5.11) This can also be obtained as a particular case of the more general formula (2.21) by setting d = 2ir in the upper limit of integration and then apply- ing {i.2,). In (5.11), X is used instead of R as the integration variable in order to avoid any possible confusion wdth R as an integration limit. Thus the integrand is a function of X with 6 as a parameter. Evidently Q{R;b) — Q(R;—b). Formula (5.11) is evidently suitable for evaluation of ()(i?) by numerical integration.-^ By suitably changing the variable in (5.11), we arrive at the following various additional formulas, which, though equivalent to (5.11), are very different as regards the integrand and the limits of integration. As previ- ously, L denotes R-/{\ — b-). Q{R) 1 Vl K2 Jo exp ■X 1 b' dX, Q{R) = Vl - b^ I exp(-X) h{b\) dX, Jq Q(R) = LVi - b'~ I exp(-LX) h{bLX) dX, Jo J PYn(—l (5.12) (5.13) (5.14) (5.15) Q{R) = Vl - ^'M h{b log X) r/X. Jexp{-L) These four additional formulas are of some theoretical interest, but ap- parently they are less suitable than (5.11) for numerical integration with respect to R. A formula differing slightly from (5.11) could evidently be obtained by taking X/-\/l — 6^ as a new variable, and hence R/y/l — b^ as the upper limit of integration. Corresponding formulas for Q*(R) = Q*{R;b) can of course be obtained from the preceding formulas (5.11) to (5.15) inclusive for Q{R) = Q{R;b) ^* In this connection, Appendix D may be of interest. PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 343 by merely changing the integration Hmits correspondingly — for instance, in (5.11), from 0, i? to i?, oo ; in (5.13), from 0, L to L, ^ \ and so on. How- ever, the first four formulas for Q*{K) so obtained would suffer .the disad- vantage of each having an infinite limit of integration, rendering those formulas unsatisfactory for numerical integration purposes. This difficulty can be avoided by making the substitution R = \/r in each of those formulas for Q*{R). The resulting formulas are the following five, corresponding to (5.11) to (5.15) respectively :24 ()*(i?) Vi Q*{R) = Q*(R) VT 2 rj_ _ /i2 Jo X^ b' Jo X2 exp ^-lAH .1 - b~_ h - exp r-i/A' Ll - b\ h b/}C 1 - F b/\ d\, (5.16) 1 ]jx, X' Vl - b^ [ exp exp 1 b /n xj lxJ L /"n ~bL X X dX, dX, expi—L) ' Io{b log X) dX a (5.17) (5.18) (5.19) (5.20) As a check on (5.16), it is obtainable from (4.2) by integrating the latter as to r. For purposes of evaluation by numerical integration, formula^ (5.11) to (5.15) inclusive may evidently differ greatly as regards the amount of labor involved and the nurrerical precision practically attainable. In each of these formulas except (5.14) the integrand contains only one param- eter, b, while the integration range involves either R or L = R-/{\ — b-). In (5.14) the integrand contains two independent parameters, b and L, while the integration range is a mere constant, 0-to-l. Similar statements apply to formulas (5.16) to (5.20) inclusive. A partial check on any formula for Q(R) can be applied by setting R = <x> ^ since Q(°o) should be equal to unity (representing certainty). If, for instance, this procedure is applied to formula (5.13), the right side is found to reduce to unity by aid of the known relation" exp (-^X) JoiBX) dX = } Jo 1 (5.21) together with Io{BX) — jQ(iBX). An integral formula of the secondary type for Q*(R) = Q*{R;b) can be obtained by substituting (2.20) into the last integral in (5.8), utilizing (2.25), » Ref. 1, p. 384, Eq. (1); Ref. 2, p. 65, Eq. (2); Ref. 4, p. 58, Eq. (4.5). 344 BELL S YSTEM TECH NIC A L JOURNA L changing the variable of integration by the substitution 6 = 0/2, and rearranging; thus it is found that Q*{R) = ylZ? r ^^P(^^ '^' ^) d<l>. (5.22) 7r exp L Jo 1 — b cos This formula can also be obtained as a particular case of the more general formula (2.24) by setting 6 = 27r in the upper limit of integration, utilizing (2.25), and changing the variable of integration by the substitution 6 = 0/2. Two partial checks on any general formula for Q{R) = Q{R;b) or for Q*{R) = Q*{R;b) can be applied by setting b — and b — 1, and comparing the resulting particular formulas with those obtained by integrating the formulas for P{R;0) and F'{R;\) obtained in Section 3, namely formulas (3.20) and (3.21) there. It is thus found that Q*(R;0) = exp(-R') = ( P{R;0)dR, (5.23) Q{R; 1) = 2 |-J= jf^xp -^ dR^=^ [ ^'^^'^ ^^ ^^- ^^'--^^ It will be recalled that the quantity between braces in (5.24) is extensively tabulated, and that ^t is sometimes called the 'normal probability integral.' Several of the above general formulas for QiR) = p{R'<R) and for Q*{R) = p{R'>R) are closely connected with my 1933 paper." Indeed, formulas (5.11), (5.14), (5.16), (5.19) and (5.22) above are the same as (53-A), (56-A), (52-A), (55-A) and (22-A), respectively, of the unpublished Appendix A to the 1933 paper; and (5.12), (5.13), (5.15), (5.17), (5.18) and (5.20) above were derived in the same connection, although they were not included in the Appendix A. Formula (5.22) was employed in the unpublished Appendix A of the 1933 paper, being (22-A) there, as a basis for deriving two very different kinds of series type formulas for computing the values of p{R'>R) = Q*{R) underlying the values of pb.t){R'>R) constituting Table I (facing Fig. 8) in that paper. -^ 2*^ This formula, (5.22), was derived by me in a somewhat different manner in the un- pubHshed Appendix A to my 1933 paper. Later I found that an efjuivalent formula, easily transformable into (5.22), had been given by Bravais as formula (51) in his classical paper ".Analyse mathcmatique sur les probabilites des erreurs de situation d'un point," published in Mcmoires de I'Academie Royale des Sciences do I'lnstitut de FVance, 2nd series, vol. IX, 1846, pp. 255-332. (This is available in the Public Library of New York City, for instance.) ^^ There the abbreviated symbols p(R' < R) and /)(/?' > R) were used with the same meanings as the complete symbols on the right sides of ecjuations (5.1) and (5.2), respec- tively, of the present paper. ^^ Each of the two kinds of series type formulas comprised a finite portion of a con- vergent series plus an exact remainder term consisting of a definite integral. In the PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 345 In the present paper, formulas (5.11) and (5.16) have been used for numer- ical evaluation of QiR) = p{R'<R) and of Q*(R) = p{R'>R) by numerical integration (employing 'Simpson's one-third rule'), aided by some of the considerations set forth in Appendix D. However, only a moderate number of values of these quantities have been thus evaluated — merely enough to afford a fairly comprehensive check on Table I of my 1933 paper, by means of a sample consisting of 60 values (about 26%) distributed in a somewhat representative manner over that table. These new values of Q*{R) = p{R'>R) = 1 — Q(R) are presented in Table 5.1 (at the end of this section) in such a way as to facilitate comparison with the old values, namely those in the 1933 paper. Thus, for any fixed value of R in Table 5.1, there are two horizontal rows of computed values of Q*{R), the first row (top row) coming from the 1933 paper, and the second row coming from the present paper. The third row of each set of four rows gives the deviations of the second row from the first row; and the fourth row expresses these deviations as percentages of the values in the first row. In the first row of any set of four rows, any value represents Q*{R) = pb{R'>R) obtained, in accordance with Eq. (22) of my 1933 paper, by adding exp (— i?-) to pb^o{R'>R) given in Table I there. In the second row of a set, any value represents Q*{R) = 1 — Q{R) as computed by for- mula (5.11) or (5.16) of the present paper: more specifically, the values for R = 0.2, 0.4, 0.6 and 0.8 were computed by (5.11); and the values for R = \.6 and i? = 2 by (5.16), taking r = 1/1.6 = 0.625 and r - 1/2 = 0.5 respectively." In the 1933 paper, the values of Pb{R'>R) = Q*{R;b) for J = and for b — I were omitted as being unnecessary there because their values could be easily obtained from the simple exact formulas to which the general formulas there reduced, ior b = and ^ = 1. Those reduced formulas were the same as (5.23) and (5.24) of the present paper, except that (5.24) gives Q(R;\) instead of giving Q*{R;\) = 1 - QiR;!). The values obtained from these two formulas, exact to the number of significant figures here retained, are given in Table 5.1 at the intersections of the first row of each set of four rows with the columns 6 = and b = I. Therefore in these two columns the deviations (in the third row of each set of four rows) are devia- tions from exact values; the values in the second row of each set are, as use of such a formula for numerical computations, the expansion producing the con- vergent series was carried far enough to insure that the remainder deiinite integral would be relatively small, though usually not negligible; and then this remainder definite integral was evaluated sufficiently accurately by numerical integration. 2s In the work of numerical integration, ' Simpson's one-third rule' was employed for R = 0.2, 0.4, 0.6, 0.8 and 2. For R = 1.6, so that r = 1/1.6 = 0.625, 'Simpson's one- third rule' was employed up to r = 0.620, and the ' trapezoidal rule' from r = 0.620 to r = 0.625. 346 BELL SYSTEM TECUNICAL JOURNAL already stated, those obtained by the methods of the present paper, employ- ing numerical integration. From detailed inspection of Table 5.1 it will presumably be considered that the agreement between the two sets of values of Q*{R\b) = pb(R'>R) is to be regarded as satisfactory, at least from the practical viewpoint, the largest deviation being less than one per cent (for R = 0.8, b — 0.9). Table 5.1 Valxjes of Q*{R) = p{R' > R) b R 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 1. 00 0.2 .9608 .9590 .9574 .9550 .9516 .9463 .9372 .9168 .8930 .84148 " .9623 .9605 .9590 .9567 .9528 .9473 .9387 .9206 .8925 .84124 " .0015 .0015 .0016 .0017 .0012 .0010 .0015 .0038 -.0005 -.00024 " .16 .16 .17 .18 .13 .11 .16 .41 -.06 -.03 0.4 .8521 .8462 .8410 .8335 .8228 .8071 .7830 .7420 .7127 .68916 " .8537 .8477 .8427 .8351 .8240 .8081 .7841 .7459 .7125 . 68897 " .0016 .0015 .0017 .0016 .0012 .ODIO .0011 .0039 -.0002 -.00019 K .19 .18 .20 .19 .15 .12 .14 .53 -.03 -.03 0.6 .6977 .6880 .6799 .6686 .6531 .6324 .6055 .5721 .5578 .54851 <( .6992 .6892 .6814 .6698 .6540 .6334 .6065 .5764 .5572 .54831 (( .0015 .0012 .0015 .0012 .0009 .0010 .0010 .0043 -.0006 -.00020 K .22 .17 .22 .18 .14 .16 .17 .75 -.11 -.04 0.8 .5273 .5167 .5081 .4969 . .4826 .4656 .4477 .4316 .4261 .42371 " .5290 .5183 .5099 .4982 .4840 .4672 .4488 .4357 .4266 .42355 " .0017 .0016 .0018 .0013 .0014 .0016 .0011 .0041 .0005 -.00016 II .32 .31 .35 .26 .29 .34 .25 .95 .12 -.04 1.6 .07730 .07986 .08207 .08522 .0891 .0938 .0990 .1042 .1070 . 10960 " .07727 .07988 .08210 .08536 .0892 .0938 .0989 .1042 .1069 . 10958 '< -.00003 .00002 .00003 .00014 .0001 .0000 -.0001 .0000 -.0001 -.00002 " -.04 .03 .04 .16 .11 .00 -.10 .00 -.09 -.02 2.0 .01832 .02153 .02394 .02681 .0301 .0337 .0375 .0414 .0435 .04550 " .01823 .02145 .02383 .02685 .0302 .0338 .0376 .0415 .0436 .04552 <( -.00009 - .00008 -.00011 .00004 .0001 .0001 .0001 .0001 .0001 .00002 (( -.49 -.37 -.46 .15 .a .30 .27 .24 .23 .04 6. The Distribution Function For The Angle The distribution function P{d \ Rn) = P{d) for the angle 9 of any complex variate W = R{cos 6 -\- i sin 9) is defined by equation (1.10) on setting p = 6, a = R, (Xi — Ri = and ao — R^ — 'x, -^ thus P{9)d9 = p{d<9'<d-\-d9,0<R'<x). (6.1) An integral formula for P(9) is immediately obtainable from (1.6) by setting p — 9, a = R, as = (Xi — Ri = and 0-4 = ao — R2 = °o ', thus p(e) = [ P{R, 9) (JR. (6.2) PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 347 The rest of this section deals with the case where W = R{cos d -\- i sin 6) is 'normal.' Since this case depends on 6 as a parameter, P{d) is here an abbreviation for P{B\b). A formula for P{d;b) = P{d) can be obtained by substituting P{R,d) from (2.15) into (6.2) and executing the indicated integration, which can be easily accomplished. The resulting formula is found to be 2x(l — bcosld) This formula can also be obtained as a particular case of either of the more general formulas (2.19) and (2.20) by setting R = co m (2.19) or 7? = in (2.20); also by adding (2.19) to (2.20) and then utilizing (1.10). In P{d) = P{d;b) it will evidently suffice to deal with values of 6 in the first quadrant, because of symmetry of the scatter diagram. In P{d;b) it will suffice to deal with only positive values of b, as (6.3) shows that changing b to —b has the same effect as changing 26 toir±26, or 6 to 7r/2±0; that is, P{e;-b) = P{j/2±d;b). Fig. 6.1 gives curves of P{6;b), computed from (6.3), as function of 6 with b as parameter, for the ranges 0^^^90° and Q^b^l. The curves in Fig. 6.1 indicate that P{6;b) is a maximum at = 0° and a minimum at 9 = 90°. These indications are verified by formula (6.3), as this formula shows that: Max P{d;b) = P{0°;b) = ^ \/ H^ , (6.4) Thence Min P{e;b) = P{90°;b) = i- ^ j-qj] • (6-5) MmP{d;b)/MsixP(6;b) ^ (l-6)/(l + 6), (6.6) P{e;b)/MiixP{e;b) = P{d;b)/PiO°;b) = {l-b)/{l-b cos2d). (6.7) The curves in Fig. 6.1 indicate also that P{d;b) is independent of d in the case b = 0. This is verified by formula (6.3), as this formula shows that P{6;0) = l/27r. (6.8) Thence (6.3) can be written P{d;b)/P{e;0) = (Vn^y2)/(l-6cos2^). (6.9) 3" Beginning here, 6 will usually be expressed in degrees instead of radians, for prac- tical convenience. 348 BELL SYSTEM TECHNICAL JOURNAL By setting cos 20 = in (6.3), so that d = 45°, it is found that (vT^^2)/27r - P(45°';6), (6.10) c d f\l rn ^ I/) (0 r- 00 c* o> ddddd dd d d 1 111 / i 1 1 7 1// / f// 7 1 '1 / i V / i / / r ^ / ^ -^ f/^ II 0) 0) d Ol -^ ^ / / / J ^^^ o ^ a o y ^ — 7 If)/ T o7 c i/ d c 3 d o r "^ / 1 ( 1 DISTRIBUTION FUNCTION, P (9 ; b) Fig. 6.1 — Distribution function for the angle. whence (6.3) can be written P{d;b)/Pi45°;b) = 1/(1-6 cos 20). (6.11) PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 349 ■ 1 \ ' o / 1 \ / 1 / / , C\j/ d/ / 1 / / m / 61 J / / / / 1 0.5 / / / \ / / / <0 / 6/ / / \ / / V / / ' / / / } / o / / // '/ / / / / '01 d / / / // // / / y / \ /// O / y / /en / o / 'I l/// // / ^ ^ X y F /y x^^ ^ ^ y t REDUCED DISTRIBUTION FUNCTION, P(e;b)/MAX P(9;b) Fig. 6.2 — Reduced distribution function for the angle. 350 BELL SYSTEM TECHNICAL JOURNAL In the case b — \, the curves in Fig. 6.1 suggest, by Hmiting considera- tions, that P(0;1) is zero for all 6 except d = 0°, and that P{d;\) is infinite for 6 = 0°. These conclusions are verified by formula (6.3), as this formula shows that: P{d;\) = for ()°<d<mr; P{d;\) = --c for 6 = 0°, 180°. The curves in Fig. 6.1, though having the advantage of directly rep- resenting P{d;b) as function of 6 with b as parameter, are somewhat trouble- some to use because of their numerous crossings of each other. This difficulty is not present in Fig. 6.2, which gives curves of P{d;b)/Ma,x P(6;b), obtained by dividing the ordinates P{6;b) of the curves in Fig. 6.1 by the respective maximum ordinates of those curves, as given by (6.4), so that the equation of the curves in Fig. 6.2 is formula (6.7). 7. The Cumulative Distribution Function for the Angle The cumulative distribution function Q{<6,R]2) = Q{6) for the angle 6 of any complex variate TF ^ R{cos 6 + / sin 6) is defined by equation (1.11) on setting p = d, a ^ R, pi = di =^ 0, ai = Ri = and 02 = R2 = »= ; thus Q{d) = p{0<d'<d, 0<R'<oo). (7.1) A 'double integral' for Q{d), in the form of two 'repeated integrals,' can be written down directly by inspection of the p( ) expression in (7.1) or by specialization of (1.8); thus Q(d) = f \ [ P(R, d)dR dd ^ I f P(R, e) dd dR. (7.2) Ja \_Jtii J Jo L*'o J Evidently these can be written formally as two 'single integrals,' Q{d) = f P(9) dd = \ P{R\< d) dR, (7.3) by means of the distribution functions P{d) = P(e\ R12) and P{R\ <d) given by the formulas P(d) - [ P{R, 6) dR, (7.4) P(R \ <d) = f P(R, 6) dd. (7.5) Jo Jo (7.4) is the same as (6.2). (7.5) is a special case of (1.6), and the left side of (7.5) is a special case of P{p \ <a) defined by (1.13). The rest of this section deals with the case where W = R{cos d -\- i sin 6) is 'normal.' Since this case depends on b as a parameter, Q{d) is here an abbreviation for Q{6;b). A natural and convenient way for deriving formulas for Q(d) is afforded PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 351 by the general formula (7.3) together with the auxiliary general formulas (7.4) and (7.5), beginning with the two latter. It will be convenient to dispose of (7.5) before dealing with (7.4), as (7.5) turns out to be the less useful. For when P{R,d) given by (2.16) is sub- stituted into (7.5), the indicated integration cannot be executed in general, as (7.5) becomes (2.18), wherin the indicated integration can be executed only for certain special values of the integration limit 6 — by means of the special Bessel function formula (3.i). When PiR,d) given by (2.15), which is equivalent to (2.16) used above, is substituted into (7.4), it is found that the indica^^ed integration can be executed, giving the previously obtained formula (6.3) for F{d) = P{&',b). A 0-integral formula for Q{d) = Q{Q\h) can be obtained by substituting P{e) = P{d;b) from (6.3) into the first integral in (7.3), giving Vi - 6- f' dd Vi - 62 r'" d<f> ^^ ' ' 27r h \ - b cos 28 47r h 1 b cos (7.6) This formula can also be obtained as a particular case of the more general formulas (2.22) and (2.24) by setting i? = ^ in (2.22) or i? = in (2.24); also by adding (2.22) to (2.24) and then utilizing (1.11). The integral in (7.6) is of well-known form, and the indicated integration can be executed, yielding the following two equivalent formulas for Q{d\h): 27r tan -1 1 cos 2^ - 6 n ''' L i-6cos2d r In Q{d;b) it will evidently suffice to deal with values of 6 in the first quad- rant, because of symmetry of the scatter diagram, and the resulting fact that Q{n 90°) = n/i, where n = 1, 2, 3 or 4. In Q{6;b) it will suffice to deal with positive values of b, as (7.7) shows that^i Q{e; -b) I-e i±M Fig. 7.1 gives curves of Q{d;b) = Q{6) computed from (7.7), as function of d with b as parameter, for the ranges 0^0^90° and 0^6^ 1. Consideration of the scatter diagram of IF or of its equiprobability curves, which are concentric similar ellipses, affords several partial checks on the curves in Fig. 7.1 and on formula (7.7) from which they were plotted. ^1 This relation can also be derived geometrically from the fact that the scatter dia- gram for —b is obtainable by merely rotating that for b through 90°, as shown by (2.6), or (2.7) and (2.8), or (2.11). 352 BELL SYSTEM TECHNICAL JOURNAL \ \ 1 k \ \ Uvi \v \\J\ \\ vv \ \\ N \ ^ \\ ^ \ :^i \ 1 \ V \\ \^ ^v\ \ \ \ N] V \ •* \ ■f\< ^' t$; V \ V :S^ ^ x\ \ <3\\ V V^ ^^ ^1 ^ v. 4 o' ^ \ \ ^\ ^\ ^ i \, q ij \ \ \ \\ ^ s^ ^ \ \, ^ sV ;:$ NNV ^ X ,^ :^ ^ ^ ^ — ^ ^ CUMULATIVE DISTRIBUTION FUNCTION, Q(e;b) Fig. 7.1 — Cumulative distribution function for the angle. Thus, the fact that the curve for ^ = is a straight Hne, whose equation is (3(0 ;0) = e/2-w = 07360°, {b = 0), corresponds to the fact that for 6 = the equiprobability curves are circles. PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 353 The fact that the curve for 6 = 1 is the straight Hne Qid;l) = 1/4 = 0.25 corresponds to the fact that for 6 = 1 the scatter diagram has degenerated to be merely a straight Hne coinciding with the real axis, so that no point outside of this line makes any contribution to Q{d;\). The fact that, at ^ = 90°, Qi9;b) = Q{90°;b) has for all b the value 1/4 = 0.25 corresponds to the fact that the area of a quadrant of the scatter diagram is one-fourth the area of the entire scatter diagram. Hence Q(360°;b) = 4Q{90°;b) = 1, which is evidently correct. Acknowledgment The computations and curve-plotting for this paper were done by Miss M. Darville; those for the 1933 paper, by Miss D. T. Angell. APPENDIX A Derivation of Formula (2.15) for P{R,d) (2.15) will here be derived from (2.11) by utiHzing the fact that the 'areal probabiUty density', G, at any fixed point in the scatter diagram must be independent of the system of coordinates; for G dA gives the probability of faUing in any differential element of area dA, and this probabiUty must evidently be independent of the shape of dA (assuming that all linear dimen- sions of dA are differential, of course). Thus, indicating the element of area by an underline, we have, in rectangular coordinates, G dUdV = P{U,V)dUdV, (Al) whence G = PiU,V). (A2) In polar coordinates, GRdddR - P(R,d)dRde, (A3) whence G = P{R,d)/R. (A4) Comparing these two expressions for G shows that P{R,e) = RP(U,V). (A5) Thus, a formula for P(R,6) can be obtained from (2.11) by merelv multiply- ing both sides of that formula by R. However, in the resulting formula it will remain to express U and F in terms of R and 6, by means of the relations U ^ R cos d, (A6) V = R sin d. (A7) The final result, after a simple reduction, is (2.15), which is thus proved. APPENDIX B Formulas of the Curves in Fig. 3.3 As in equation (3.22), Re will here denote the critical value of R, that is, the value of R at which P{R) = P{R',b) has its maximum value; and Tc '2 Formula (A5) can be easily verified by the entirely different method which utilizes (1.23). 354 BELL SYSTEM TECHNICAL JOURNAL will denote the corresponding value of T, whence Tc is given in terms of Re and b by (3.22). A formula for dP{R)/dR could of course be obtained directly from (3.4) but it will be found preferable to obtain it indirectly from the less cumber- some formula (3.8) containing the auxiliary variable T defined by (3.6). Evidently, since b does not depend on R, dP{R) ^ dPjR) dT_ ^ 2bR dP{R) dR dT dR 1 - b'- dT ' ^ ^ Thus, since the factor IbR/il — b") cannot vanish for any value of R (except R = 0), the only critical value of R must be that corresponding to the value of T at which dP{R)'/dT vanishes, namely Tc, since Tc has been defined to be the value of T corresponding to Re- (Incidentally, equation (Bl) shows that Tc is equal to the value of T at which P(R) is an extremum when P(R) is regarded as a function of T.) From (3.22), Rl Tc (32) 1 - b' b Evidently Tc and Re must ultimately be functions of only b. The next paragraph deals with Tc, which evidently has to be known before Re can be evaluated. From (3.8) it is found that, since dh{T)/dT = I\{T), = nm -^ + r_L , h{T) 1 (B3) '12T h{T) b_ Hence, since P(i?) does not vanish for any value of R (except R = Q and R = oo), Tc will be a root of the conditional equation obtained by equating to zero the expression in brackets in (B3). This conditional equation is transcendental in Te and apparently has no closed form of explicit solution for Tc ; and its solution by successive approximation, or otherwise, would likely be rather slow and laborious. However, the bracket expression in (B3) shows that b can be immediately expressed explicitly in terms of Te by the equation ^ ^ 1 + 2Teh{Tc)/h{Tc) ' ^^^^ For some purposes, the following two equations, each equivalent to (B4), will be found more convenient: T- 2 + ^^/727)' ^^^^ l£ = IZ? (B6) b 1 - bh{Te)/h{Te) ^ ^ PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 355 On account of (B2), the right sides of (B5) and (B6) are equal not only to Tc/b but also to i?c/a-6"). Since the utilization of formulas (B4), (B5) and (B6) for computing the curves in Fig. 2).di will involve taking Tc as the independent variable and assigning to it a set of chosen numerical values, the natural first step is to find approximately the range of Tc corresponding to the 6-range, O^^^l, in order to be able to choose only useful values of Tc. This step will be taken in the next paragraph. Equation (B6) shows that Tc/b = 1/2 when 6 = 0, and hence that Tc ~ when b = 0; and this last is verified by (B4). The other end-value of the Tc-range, namely the value of Tc iox b = 1, cannot be found explicitly and exactly. However, rough values of limits between which it must lie can be found fairly easily as follows: To begin with, each of the equations (B5) and (B6) shows that Tc^ b/2, for all values of b in O^b^l; in par- ticular, Tc > 1/2 when b = I. An upper limit for Tc for any value of b can be found from (B5) by utilizing the power series expressions for Ii{Tc) and lo(Tc), whereby it is found that ^ -H^, (B7) where H =^ I - %' < 1. (B8) Io{I c) ^ o On substituting (B7) into (B5) and then solving for Tc in terms of b and H, it is found that Tc = b/(l + Vl - Hb'). (B9) On account of (B8), (B9) shows that Tc < b/{l + Vn^2), (BIO) whence, in particular, Tc<l when b = 1. By successive approximation or otherwise, it can now be rather quickly found that, when b — 1, Tc = 0.79 (to two significant figures).^^ From the preceding paragraph, it is seen that, when b ranges from to 1, Tc ranges from to about 0.79; Tc/b ranges from 0.5 to about 0.79; and, on account of (B2), Re ranges from ^/O.S = 0.707 down to 0. The curves in Fig. 3.3 are constructed with the aid of the formulas and methods of this appendix as follows: First, a set of values of Tc is chosen, ranging from to 0.79 and slightly larger. Second, for each such chosen Tc the right side of (B5) is computed, thereby evaluating Tc/b and also Rc/{l — b^), these two quantities being equal by (B2). Third, the cor- responding value of b is found by dividing Tc by Tc/b; less easily, it could ^' Because of the special importance oi b = 1 in other connections, Tc for b = I was later evaluated to four significant figures and found to be Tc = 0.7900; thence, by sub- stituting this value of T into (3.8), along with b = 1, it was found that Max. P{R;l) = 0.9376, which occurs at R = Re = 0,hy (B2). 356 BELL SYSTEM TECHNICAL JOURNAL be foun d by substituting Tc into (B4). Fourth, from Tc/b the value of \/Tc/b is found, and thereby the value of Rc/y/l — b"^ and thence Re . Finally, Max. P{R;b) is computed by inserting the critical values into any of the various (equivalent) formulas for PiR;b), namely (3.4), (3.7), (3.8), (3.10) or (3.12). APPENDIX C FOMULAS OF THE CURVES IN FiG. 4.3 The first six equations of this appendix are given without derivation and almost without any comments because they correspond exactly and simply to the first six equations, respectively, of Appendix B. Beginning with the second paragraph of the present appendix, the close correspondence ceases. dP(r) _ dP{r) dT _ -2b dP(r) dr dT dr (1 - ^2);^ dT (1 - bVc ~ T • dP(r) dT (CI) (C2) = P(r) ^ + [l+b^- 1] (C3) 12T ^ h{T) b\ ' ^^^^ * 3 + 2r, h(T,)/Io(Tc) ' ^^^^ b 2 Io{Pc) Tl = 3/2 b 1 - bh{Tc)/Io(Tc) ' ^"-"^ The bracketed expression in (C3) is seen to be obtainable from that in (B3) by merely changing T to T/3 wherever T does not occur as the argument of a function; hence the three equations following (C3) are obtainable from the three equations following (B3) by correspondingly changing Tc to Tc/S. (In this appendix, as in Section 4, small c is purposely used as a subscript to indicate a 'critical' value, whereas in Section 3 and in Appendix B, capital C is used for that purpose.) For use below, it will here be noted that h{Tc)/h(Tc) = N,{Tc)/No{Tc), (C7) as will be seen by dividing (3.16) by (3.13). On account of (3.17) and (3.14), (C7) shows that for large values of Tc the right side of (C7) is equal to 1 as a first approximation, and to 1 — 1/2 Tc as a second approximation; thus, for large Tc, PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 357 h(T,)/hiT,) = 1 - l/2r, = 1. (C8) The first step toward computing the curves in Fig. 4.3 is to find approxi- mately the Tc-range corresponding to the 6-range, O^b^l. This is done in the course of the next four paragraphs. When b = 0, equation (C6) shows that Tc/b = 3/2 and hence that Tc = 0; or, what is equivalent, b/Tc = 2/3 and hence l/Tc = oo (since b^ 0). When 6 = 1, Tc = CO, as can be easily verified from equation (C4), (C5) or (C6) by utilizing (C8). Thus, from the two preceding paragraphs, it is seen that, when b ranges from to 1, b/Tc ranges from 2/3 to 0; Tc/b from 3/2 to cc ; and Tc from to 00. Since Tc = "^ when b = 1, the choosing of a set of finite values of Tc will necessitate an approximate formula for computing Tc for values of b nearly equal to 1 , which means for very large values of T. Such a formula is easily obtainable from (C5) by utiUzing the approximation 1 — 1/2 Tc in (C8), whereby it is found that, for large Tc, Tc = b/{l-b), (C9) b/Tc = l-^*. (CIO) As examples, these approximate formulas give: When b = 0.99, Tc ~ 99, b/Tc = 0.01; when b = 0.9, Tc = 9, b/Tc = 0.1. It will be found that even in the second example the results are pretty good approximations. The curves in Fig. 4.3 are constructed with the aid of the formulas and methods of this appendix as follows: First, a set of values of Tc is chosen, ranging from to about 100 (the latter figure corresponding approximately to b = 0.99). Second, for each such chosen Tc the right side of (C5) is computed, thereby evaluating Tc/b and also 1/(1 — 6-)^^, these two quan- tities being equal by (C2). Third, the corresponding value of b is found by dividing Tc by Tc/b; less easily, it could be found by substituting Tc into (C4). Fourth, from Tc/b the value of \/Tc/b is found, and thereby the value of I/tc s/l — b^ and thence Tc . Finally, Max P{r;b) is computed by inserting the critical values into any of the (equivalent) formulas for P{r;b), namely (4.2), (4.3) or (4.4). APPENDIX D Some Simple General Considerations Regarding the Evaluation of Cumulative Distribution Functions by Numerical Integration This appendix gives some simple general considerations and relations that may sometimes facilitate and render more accurate the evaluation of cumulative distribution functions by numerical integration. 358 BELL S YSTEM TECH NIC A L JOURNA L Some of these considerations and relations have found application in Section 5 in the evaluation of the cumulative distribution function for the modulus R = I ir |. For this reason, the variate in the present section will be denoted by R. though without thereby restricting R to denote the modulus; rather, R will here denote any positive real variate, though it should preferably be a 'reduced' variate, so as to be dimensionless, as in equation (2.9). The restriction of R to positive values is imposed because it is strongly conducive to simplicity and brevity of treatment, without constituting an ultimate limitation. The reciprocal of R will be denoted by r, as previously.^* We may wish to evaluate numerically the cumulative distribution func- tion p{R'<R) = Q{R) or p{R'>R) = Q*{R) or both. Since these are not independent, their sum being equal to unity, the evaluation of either one determines the other, theoretically. However, when the evaluated one is nearly equal to unity, the remaining one may perhaps not be evaluable with sufficient accuracy (percentagewise) by subtracting the evaluated one from unity. Then it would presumably be advantageous to introduce for auxiliary purposes the variable r — 1/R, since evidently p(R'>R) = p{\/R'<l/R) = p{r'<r), (Dl) p(R'<R) = p{r'>r) = 1 - p{r'<r). (D2) Thus, if p{R'>R), in (Dl), is small compared to unity, it is presumably evaluable with higher accuracy percentagewise by dealing with p{r'<r) than with 1 — p{R'<R). Incidentally, after p{r' <r) has been evaluated, it might be used in (D2) to arrive at a still more accurate value of p{R' <R) than had originally been obtained directly by numerical integration. Assuming that we have a plot (or a table) of the distribution function P{R), we can evidently evaluate P{R'<R') = / P{R)dR (D3) Jo directly by numerical integration, provided the plot is sufficiently extensive to include R ; if not, we can, by (D2), resort to P(R'<R') = 1 - p(r'<r') = 1 - / P{r)dr, (D4) Jo assuming that a sulficiently extensive i)lot (or table) of P{r) is available and applying numerical integration to it. Even if the plot of P{R) used in (D3) is sulficiently extensive to include '■• The restriction of R, and hence of r, to positive values is seen to be absent from equa- tions (Dl), (D2), (D5) and (D6) but present in (D3), (D4), (D7) and (D8). PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 359 R , so that (D3) could be evaluated, it might be that (D4) would result in greater accuracy; this would presumably be the case when p{R' <R ) is nearly equal to unity. Evidently an evaluation of P(R'>R') = P(R)dR (D5) directly by numerical integration would be less satisfactory than the evalua- tion of p{R' <R ) in the preceding paragraph. For, due to the presence of the infinite limit in the integral in (D5), the plot of P{R) would have to be carried to a large enough value of R so that the integral from there to «^ would be known to be negligible. This diflficulty can be avoided by start- ing with the relation piR'>R') = 1 - piR'KR") (D6) and substituting therein the value of p{R' <R ) given by (D3) or (D4), resulting respectively in the following two formulas: p(R'>R') = I - P(R)dR, (D7) P(R'>R') = p(r'<r') = / P{r)dr, (D8) the integrals in which are evidently suitable for evaluation by numerical integration, none of the integration limits being infinite. If p{R'>R'^) is small compared to unity, (D8) would presumably be more accurate (percentagewise) than (D7). If the plot of P(R) is not sufficiently exten- sive to include R , (D7) evidently could not be used; but, instead, (D8) could be used if the plot of P{r) were sufficiently extensive to include r . References on Bessel Functions 1. Watson, "Theory of Bessel Functions," 1st. Ed., 1922; or 2nd Ed., 1944. 2. Gray, Mathews and MacRobert, "Bessel Functions," 2nd Ed., 1922. 3. McLachlan, "Bessel Functions for Engineers," 1934. 4. Bowman, "Introduction to Bessel Functions," 1938. 5. Whittaker and Watson, "Modern Analysis," 2nd Ed., 1915. 6. "British Association Mathematical Tables," Vol. VI: Bessel Functions, Part I, 1937. 7. Anding, "Sechsstellige Tafeln der Bessel'schen Funktionen imaginaren Arguments," 1911 (mentioned on p. 657 of Ref. 1). Spectrum Analysis of Pulse Modulated Waves By J. C. LOZIER The problem here is to find the frequency spectrum produced by the simul- taneous application of a number of frequencies to various forms of amplitude limiters or switches. The method of solution presented here is to first resolve the output wave into a series of rectangular waves or pulses and then to combine the spectrum of the individual pulses by vectorial means to find the spectrum of the output. The rectangular wave shape was chosen here as the basic unit in order to make the method easy to apply to pulse modulators. Introduction The rapidly expanding use of pulse modulation^ in its various forms is bound to make the frequency spectrum of pulse modulated waves a subject of increasing practical importance. The purpose of this paper is to show how to determine the frequency spectrum of these waves by methods based as far as possible on physical rather than mathematical considerations. The physical approach is used in an attempt to maintain throughout the analysis a picture of the way in which the various factors contribute to a given result. To further this objective the fundamentals involved are reviewed from the same point of view. The method is used here to analyze two distinct types of pulse modulation, namely, pulse position and pulse width modulation.^ These two cases are especially important for illustrative purposes because their spectra can be tied back to more familiar methods of modulation. Thus it will be shown that, as the ratio of the pulse rate to the signal frequency becomes large, pulse position modulation becomes a phase modulation of the various carrier frequencies that form the frequency spectrum of the unmodulated pulse wave, and pulse width modulation becomes a form of amplitude modulation of its equivalent carriers. The analysis also shows certain interesting input- output relationships that may be obtained from such modulators, treating them as straight transmission elements at the signal frequency. These relationships are of more than theoretical interest. The pulse position modulator has already been used as phase or frequency modulator to good advantage.^ The use of a pulse width modulator as an amplifier is ' E. M. Deloraine and E. Labin, "Pulse Time Modulation", Electrical Communications , Vol. 22, No. 2, pp. 91-98, Dec. 1944; H. S. Black "AN-TRC-6 A Microwave Relay Sys- tem", Bell Labs. Record, V. 33, pp. 445-463, Dec. 1945. 2 By pulse position modulation is meant that form of pulse modulation in which the length of each pulse is kept fixed but its position in time is shifted by the modulation, and by pulse width modulation that form in which the length of each pulse varies with the modulation but the center of each pulse is not shifted in position. ' L. R. Wrathall, "Frequency Modulation by Non-linear Coils", Bell Labs. Record, Vol. 23, pp. 445-463, Dec. 1945. 360 SPECTRUM ANALYSIS OF WAVES 361 another practical application, of which the self oscillating or hunting servo- mechanism is an example. The quantitative analysis of such systems depends on the ratio of the pulse repetition rate to the signal frequency. When this ratio is low, the solution can be obtained by a method shown here for resolving the modulated waves into selected groups of effectively unmodulated components. This technique is powerful since it can be done by graphical means whenever the complexity of either the system or the signal warrants it. When the ratio of pulse rate to signal frequency becomes high enough, such methods are no longer practical. However, under these conditions other methods become available, especially in cases like those mentioned above where the spectrum of the modulation approaches one of the more familiar forms. An important example of this occurs in the case of the pulse position modulator where, as the spectrum approaches that of phase modulated waves, the solution can often be found by the conventional Bessel's function technique used in analyzing phase and frequency modulators. The method proposed here for obtaining the spectrum analysis of pulse modulated waves is based on the use of the magnitude-time characteristic of the single pulse and its frequency spectrum as a pair of interchangeable building blocks, so that the analysis will develop this relationship. Before doing this the elementary theory of spectrum analysis will be reviewed Review or the Elementary Theory of Spectrum Analysis A complex wave may be represented in two ways. One way is by its magnitude at each instant of time. The other way is by its frequency spectrum, that is, by the various sinusoidal components that go to make up the wave. The two representations are interchangeable. The transformation from a given frequency spectrum to the corresponding magnitude vs. time function is straight-forward, for it is apparent that the various components in the frequency spectrum must add up to the desired magnitude-time function. The necessary additions may be difficult to make in some cases but they are not hard to understand. The reverse process of finding the frequency spectrum when the magni- tude-time characteristic is given is more involved, though using Fourier anal- ysis, the problem can generally be formulated readily enough. Furthermore the mathematical procedures involved can be interpreted physically in broad terms by modulation theory. However, these procedures become more difficult to perform, and the physical relationships more obscure, as the wave form under analysis becomes more complex. This is particularly true when general or informative solutions rather than specific answers are required. Pulse modulated waves are sufficiently new and complex to give such difficulties. 362 BELL SYSTEM TECHNICAL JOURNAL The process of finding the frequency spectrum of a complex wave from its magnitude-time function has a simple mathematical basis. It depends on the fact that the square of a sinusoidal wave has a positive average value over any interval of time, whereas the product of two sinusoidal waves of different frequencies will average zero over a properly chosen interval of time."* In theory then, as the magnitude-time function of a complex wave is the sum of all the components of the frequency spectrum, we have only to mul- ti])ly this magnitude-time function by a sinusoidal wave of the desired frccjuency and then average the product over the proper time interval to find the component of the spectrum at this frequency.^ One physical interpretation of this procedure can be given in terms of modulation theory. The product of the magnitude-time function with a sinusoidal wave will produce the beat or sum and difference frequencies be- tween the frequency of the sinusoid and each component of the frequency spectrum. Thus, if the spectrum contains the same frequency, a zero beat or dc term is produced, and this term may be evaluated by averaging the product over an interval that is of the proper length to make all the ac components vanish. The application of this principle for spectrum analysis is simple when the magnitude of the wave in question is a periodic function of time. The very fact that the wave is periodic is sufficient proof that the only frequencies that can be present in the wave are those corresponding to the basic repeti- tion rate and its harmonics. Thus the frequency spectrum is confined to these specific frequencies and so it takes the form of a Fourier series. Know- ing that the possible frequencies are restricted in this way, the problem of finding the frequency spectrum of a complex periodic wave is reduced to one of performing the above averaging process at each possible frequency. The period of the envelope of the Complex Wave is the proper time interval for averaging, and the integral formulation for obtaining this average is that for determining the coefficients in a Fourier series. The principle holds equally well when the magnitude-time function is non- periodic, but the concept is complicated by the fact that the frequency spectrum in such cases is transformed from one having a discrete number of components of harmonically related frequencies to one having a continuous- band of frequencies.*' Such s]:)ectra contain infinite numbers of sinusoidal ■• The i)roper time interval is generally some integral multiple of the period correspond- ing to the difference in frequency of the two sinusoid waves. * In practice it is generally necessary to multiply by both sine and cosine functions because of i)ossible phase differences. 8 One exception to this statement is the fact that any wave made up of two or more incommensurate frequencies is nonperiodic. Yet such waves will have a discrete spectrum if the number of components is finite. This incommensurate case is neglected throughout the discussion. SPECTRUM ANALYSIS OF WAVES 363 components, each of infinitesimal amplitude and so close together in fre- quency as to cover the entire frequency range uniformly. ' The continuous band type of frequency spectrum is just as characteristic of non-periodic waves as the discrete spectrum is of periodic waves. This can be shown as a logical extension of the Fourier series representation of periodic waves. The transition from a frequency spectrum consisting of a series of discrete frequencies to one consisting of a continuous band of fre- quencies can be made by treating the non-periodic function as a periodic function in which the period is allowed to become very large. As the period approaches infinity the fundamental recurrence rate approaches zero, so that the harmonics merge into a continuous band of frequencies. This does not of course change the basic realtionship between the fre- quency spectrum of a wave and its magnitude-time function. The mag- nitude-time function is still the sum of the components of the frequency spectrum. Also the frequency spectrum can still be obtained frequency by frequency, by averaging the product of the magnitude-time function and a unit sinusoid at each frequency. However, the actual transformations in the case of the non-periodic functions require summations over infinite bands of frequencies and over infinite periods of time and so fall into the realm of the Fourier and similar integral transforms. However, in any case the problem of spectrum analysis reduces to an averaging process. The process can be performed by mathematical inte- gration in all cases where a satisfactory analytical expression for the mag- nitude-time function is available. Fourier analysis provides a very powerful technique for setting up the necessary integrals in such cases. This averaging process can also be done graphically. It is apparent from the theory that if the product of the magnitude-time function and the sinusoid is sampled at a sufficient number of points, spaced uniformly over the proper time interval, then the average of the samples gives the desired value. This technique is fully treated elsewhere" so that it will not be con- sidered in detail here. However, use will be made of it in a qualitative way to augment the physical picture. Non-Linear Aspects The use of the frequency spectrum in transmission studies is generally limited to cases where the system in question is linear; that is, where the transmission is independent of the amplitude of the signal. However, the same techniques can still be used on systems employing successive linear and non-linear components, in cases where the transmission through the non-linear elements is independent of frequency. Under these conditions, the magnitude-time representation of the wave can be used in computing 'Whittaker and Robinson, Calculus of Observations. 364 BELL SYSTEM TECHNICAL JOURNAL llie transmission over each non-linear section, where the transmission is dependent only on the amplitude, and the frequency spectrum used over each linear section, where the transmission is dependent only on the fre- quency. This a technique can be used on most pulse modulating systems because such non-linear elements as the modulators and limiters generally encountered are substantially independent of frequency. Frequency Spectrum of the Single Pulse The single pulse is a non-periodic function of time and so has a continuous frequency spectrum. In this case the Fourier transforms are simple. They are derived in Appendix A. Figure 1 gives a graphical representation of the magnitude-time function and the frequency spectrum of the pulse. The expressions are general and hold for pulses of any length or amplitude. It is instructive to note that the frequency spectrum in this case can be MAGNITUDE-TIME FUNCTION, e (t) 1.0 LU qO.6 D 1- E 3 0.4 a n TIME, FREQUENCY SPECTRUM, g (f) -6C -4C -2C 2C 4C 6C FREQUENCY,!, IN TERMS OF C (WHERE C = VaO Fig. 1 — Magnitude time and frequency spectrum representations of a single pulse. determined by using the graphical technique mentioned previously. For example, consider the product of the magnitude-time function of the single pulse with a sinusoidal wave of given frequency and unit amplitude, so arranged in phase that its peak coincides with the center of the pulse. Theoretically the average of this product taken over the infinite period will give the relative magnitude of the component in the frequency spectrum of the pulse having the same frequency as the sinusoidal wave. In this case however, the average need only be taken over the length of the pulse, since the product vanishes everywhere else. Thus at very low frequencies, where the period of the sinusoidal wave is very much greater than the length of the pulse, the average is proportional to 2EL where E is the amplitude and 2L the length of the pulse. Then as the frequency increases, the average of the product, and hence the relative amplitude of the component in the spectrum, will first decrease. For the particular frequency such that the length of the pulse is one half the period, the relative ami)litude will have SPECTRUM ANALYSIS OF WAVES 365 2/2 fallen to 2EL X " I " being the average value of a half wave of unit ampli tude ). Similarly when the frequency is such that the length of the pulse is a full wavelength, the average will vanish, and when the pulse length is one and a half times the wavelength, the average is negative, having two negative and one positive half waves over the length of the pulse, and the 2 relative magnitude is 2EL X ^. These products are shown graphically on Fig. 2. Since these amplitudes correspond to those given in Fig. 1, for the spectrum components at/ = /o = 1/4Z, 2/o , and 3/o , it is apparent that the spectrum could be determined in this way. WHERE f = WHERE f = Val AVERAGE =2EL 1 E 1 - L +L TIME,t WHERE f = I/2L «-> / \ AVERAGE = <JJ r' / \ / \ 1- "-' / 1 a. (\j - 1 r^ TIME, t — »■ o <u ; V <o u r AVERAGE HVrr EL TIME, t a 4 3 rr "^ _4 '3TT RESULTANT SPECTRUM ^s J ^c -^,- L ,^^ ^'4C 3C FREQUENCY, f, IN TERMS OF C (WHERE C= V^O Fig. 2 — Graphical derivation of spectrum of single pulse by averaging product of pulse with sinusoidal waves of various frequencies. Basic Technique In the analysis presented here, the single pulse and its spectrum will be used in such a way that the need for individual integral transforms for each complex wave form under study is avoided. The theory is simple. A complex wave form may be approximated to any desired accuracy by a series of pulses, varying with respect to time in length, in amplitude, and in position. Now the spectra of these individual pulses are already known. Therefore, to find the frequency spectrum of the complex wave in question, it is necessary only to combine properly the spectra of the various pulses representing the complex wave. Thus the process is theoretically complete. The procedure is first to 366 BELL SYSTEM TECHNICAL JOURNAL break down the given complex wave into a series of single pulses. Next the spectrum of each pulse is determined separately. Then the spectrum of the complex wave is obtained by combining the spectra of the various single pulses involved. One of the things to be demonstrated here is that it is perfectly feasible in many cases to perform these summations graphically, even tliough basically it does involve the handling of spectra each containing an infinite number of frequency components. There are other wave forms that could be used as the fundamental build- ing block instead of the single pulse. The unit step function is one possi- bility, since it is used in transient analysis for a similar purpose. However, the single pulse has obvious advantages when the complex wave to be ana- lyzed is itself a series of pulses, as in pulse modulation. Again it would be nice to be able to choose as the fundamental unit a wave that has a discrete rather than a continuous band frequency spectrum, but it seems that any wave flexible enough to make a satisfactory building unit is inherently non- periodic and so has a continuous frequency spectrum. However the fact that the fundamental units have continuous spectra does not of itself compli- cate the results. If for example, the wave to be analyzed is periodic, the sum of the spectra of the various pulses must reduce to a discrete frequency spectrum. In the cases of interest here, when the pulse train under analysis is repetitive, combinations of identical pulses will be found to occur with the same fundamental period, and generally the first step in the summation of such spectra is to group the series of pulses into periodic waves with discrete spectra. Manipulations of Single Pulses In its use, the single pulse may be varied in amplitude, in length, and in position with respect to time. These changes have independent efifects on the frequency spectrum. A variation in the amplitude of a pulse does not change its spectrum, except to increase proportionately the magnitudes of all components. A change in position of a pulse with time does not change the amplitude vs. frequency characteristic of the spectrum, but it does shift the phase of each component by an amount proportional to the product of the frequency and the time interval through which the pulse was shifted. A change in the length of a pulse will change the shape of the amplitude vs. frequency characteristic of the spectrum. Figure 3 shows this effect. How- ever, if the center point of the pulse is not shifted in time, the relative phases of the components are not afifected by such changes in length. The single pulse can also be modulated to aid in the resolution of more complicated wave forms. This process is based on the use of the pulse as a function having a value of unity over a chosen time interval and a value of zero at all other times. Thus, to show a part of a sinusoidal wave, we need SPECTRUM ANALYSIS OF WAVES 367 only multiply this wave by a pulse of the correct length and proper phase with respect to the sinusoid to show only the desired piece of the wave. In this simple case it is not difficult to derive the spectrum because what are produced are the sum and the difference products of the modulating fre- quency with the spectrum of the pulse. This gives two single pulse spectra shifted up and down in frequency by the frequency of the modulation. An example of this is shown in Fig. 4, where the spectrum of a single half c>cle is determined. Pulse Position Modulation For the first example, a simple form of pulse position modulation will be analyzed. The pulse train in this case is made up of pulses spaced T seconds U 0.2 a -0.4 \^ \ s r^> \ \ \ \ ^x.-; puLse 3_L 2 LENGTHS: L 4L 3 jr \ \ N S > =— ■ -~'-^ ""' I 2 3 4 5 , FREQUENCY, f, IN TERMS OF C (WHERE C = — ) Fig. 3 — Change in frequency spectrum with pulse length. apart and the width of each pulse is a very small part of the spacing T. Such a pulse train is shown on Fig. 5. The pulse train is modulated by ad- vancing or retarding the position (time of occurance) of the pulses by an amount proportional to the instantaneous amplitude of the signal at sampled instants T seconds apart. Figure 5 also shows the signal, in this case a sine wave of frequency 1/lOr, and the resulting modulated pulse train. The peak amplitude of the modulating sine wave is assumed to shift the position of a pulse by 1 /-iT. The length and the amplitude of the pulses are the same since neither is affected in this type of modulation. The first step in the analysis is to determine the spectrum of the pulse train before modulation. Each pulse contributes a spectrum of the form 368 BELL SYSTEM TECHNICAL JOURNAL shown on Fig 1. Now the phase of each component in such a spectrum is so arranged that the spectrum forms a series of cosine terms all of which have zero phase angle at the center of the pulse. From successive pulses T SPECTRUM OF SINGLE PULSE UJ Q H _l Q- 5 < \ -L 0^ L TIME.t-* \ \ — ^^ V^ ^• / X MODULATION . PRODUCTS / / \ \ \ \ \ \ DIFFERENCE \ TERMS \ SUM TERMS \ ^^ ^-— --- \^ v^ ^ ~~~ — RESULTANT SPECTRUM Q D Q- 5 r'-/^ r\ --T "^ \ '/ N \ N 1/2 SUM + 1/2 DIFFERENCE -L L TIME.t— ♦ \^ — ^^ 2C 3C 4C 5C FREQUENCY, f, IN TERMS OF C (WHERE C^^t) Fig. 4 — Determination of spectrum of single half sine wave by modulation of single pulse spectrum with cos licet. seconds apart, the component at any given frequency will have the same amplitudes, but the relative phases will be 1-kJT radians apart. It is appar- ent that frequencies for which lirjT is 2x or some multiple of 27r radians SPECTRUM ANALYSIS OF WAVES 369 apart, the contributions from all pulses add in phase. These are the fre- quencies nc, where n = 1,2,3 and c "^ Tj.- It is also apparent that at fre- quencies for which the phase differences between the components are not an exact multiple of 2ir radians apart, the contributions from enough pulses must be spread in phase over an effective range of to 2x radians in such a way as to cancel one another. For example, take the particular frequency for which the difference in phase between pulses is 361° instead of 360°. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <u TIME.t— »■ u I inT ' ' o D 1- p.— ^\ 62 '°^:v '"" " H ^_^^ a < ^^\ \ \ "^^ ^^\ 1 i :i :i 1 TIME,t— »• 1 i 1 1 1 1 i 1 1 ;AT| -►1 U- -J U-AT2 TIME.t — »• -^- -^■ UNMODULATED PULSE TRAIN (PERIOD T) MODULATING FUNCTIOM OR SIGNAL (PERI0D=10T) POSITION MODULATED PULSE TRAIN (AT, ~e|,ETC) (REFERENCE) (AT,orO) TIME.t — »■ Fig. 5 — Formation of pulse position modulated pulse train and its resolution into subsidiary unmodulated pulse trains. The contribution from each preceding pulse will be effectively advanced in phase 1° with respect to its successor, so that the contributions from pulses 180 periods apart will be exactly 180° out of phase. Therefore over a sufBcient number of pulses, the net contribution is zero. The spectrum of the unmodulated pulse train is thus made up of a do term plus harmonics of the frequency C = \/T. The dc term is the average, and therefore is equal to £ X 2L/T, where E is the magnitude of the pulse. All of theother components have the same relative magnitudes that they have 370 BELL SYSTEM TECHNICAL JOURNAL in the single pulse spectrum. This gives a spectrum like that shown on Fig. 6. Figure 6 also shows for comparative purposes the spectrum of the subsidiary pulse wave consisting of every 6th pulse. Thus in the unmodulated case, the pulses have a uniform recurrence rate and the resultant spectrum, found by adding those of the individual pulses, reduces to a train of discrete frequencies comprised only of the harmonics of the recurrence rate of the pulses. The fundamental frequency, correspond- WHERE PULSE LENGTf \ = 1/6 PERIOC ) LENGTH 1.0 . o D ~- -~ -I E i 0.8 ~^"^--- 21 4T 6T 8T lOT 12T TlME.t 0.6 UJ 0.4 ""^-^ FREQUENCY ^^^^ SPECTRUM O 1- ^^^ O0.2 Hi OC cc UJ 1.0 2 0.8 0.6 0.4 0.2 C 2C 3C 4C 50 FREQUENCY, f, IN TERMS OF C (WHERE C =!/j) WHERE PULSE LENGTH = 1/36 PERIOD LENGTH FREQUENCY SPECTRUM TITTTITfTTITrTTrrn-rTT-n-T-r.-r 2V 4V 6V 8V lOV 12V 18V 24V 30V FREQUENCY, f, IN TERMS OF V (WHERE V = l/gC = l/gT) 36V Fig. 6 — Frequency spectrum of pulse trains where the spacing between the pulses is 6 and 36 times the pulse length respectively. ing to the recurrence rate, and its harmonics will be called the carrier fre- quencies of the pulse train. The effect of modulating the pulse train is to modulate each of these carriers, producing sidebands of the signal about them. When the pulse train is position modulated, the pulses are shifted in posi- tion by an amount AT, corresponding to the instantaneous ami^litudes of the modulating function. The spectrum of each pulse is unchanged, since the pulse length remains constant. However, components of successive SPECTRUM ANALYSIS OF WAVES 371 pulses at the carrier frequency c and its harmonics will no longer add directly, because of the phase shifts that accompany the change in position. This phase shift is equal to AT, the shift in position, times the radian frequency of the component in question. However, when the signal function is periodic, each pulse will have the same shift in position as any other pulse that occurs at the same relative instant in a later modulating cycle. Furthermore, when the carrier fre- quency is an exact multiple of the signal frequency i.e., c = nv, there will be a pulse recurring at the same relative instant in each cycle of v. Under these conditions, the pulse position modulated wave can be broken down into a group of unmodulated waves, each being made up of that series of pulses that recur at a given part of each modulating cycle, as shown in Fig. 5. These subsidiary waves are eflfectively unmodulated because, as each pulse recurs at the same instant in the modulating cycle, they are shifted to the same extent and hence will be uniformly spaced. This uniform spacing between pulses in a given wave is equal by definition to the period of the modulating function, and there will be as many of these unmodulated pulse trains as there are pulses in a single cycle. Thus, if c = nv, there will be n such pulse trains. The reason for grouping the pulses into these unmodulated pulse tarns is that unmodulated periodic trains have spectra of discrete frequencies. Since the pulse widths are all equal, and since the spacing between pulses is the same for each wave, the spectra of these unmodulated waves will all be identical. Furthermore, these spectra will be the same as that of the original carrier wave of pulses before modulation, except for two factors. First, the fundamental frequency is now i', corresponding to the modulating period, so that there are n times as many components as before. Secondly the amplitudes are reduced by the factor - because there is only one pulse in these new waves to every n pulses in the original wave. Thus, instead of having a spectrum made up of the carrier frequency and its harmonics, we now have one made up of harmonics of v. Since c = nv, such frequencies as c, c, ± t, c ± 2v, etc., are included. An example of the spectra of both the subsidiary and original pulse waves is shown on Fig. 6, for the case where n = 6. Thus the problem of finding the spectrum of such a pulse position modu- lated wave is reduced by this procedure to adding up the ;/ equal components at each of the frequencies of interest, such as c and c dz v, allowing for the phase difference between components corresponding to the position of one pulse with respect to that of the other n-l pulses in one modulating cycle. As an example, suppose n = 10 and the frequency to be computed is c + ^• Now <- + I) is 10% higher in frequency than c. Thus in the unmodulated 372 BELL SYSTEM TECHNICAL JOURNAL case, when the n pulses are equally spaced, they are 360° apart at c and consequently 360° + 36 or 396° at c + v. Therefore in the unmodulated case, each component would be advanced in phase 36° with respect to the previous one, so that the diagram of the 10 components would form the FREQUENCY C+V (a) ZERO MODULATION 10 (b) 50 PER CENT MODULATION 10 FREQUENCY C-V (C) ZERO MODULATION (d) 50 PER CENT MODULATION 9 8 10 2 I Fig. 7 — Vector pattern of subsidiary pulse components. vector pattern shown on Fig. 7A. The successive components are numbered 1 to 10. The sum in this unmodulated case is of course zero. Now the effect of modulation is to shift the relative jjhascs of these compo- nents by an amount determined by the shift in position of the corresponding pulses. When these relative phase shifts are such as to spoil the can- SPECTRUM ANALYSIS OF WAVES 373 cellation of the 10 components, a net component of this frequency is pro- duced in the frequency spectrum of the pulse wave. Taking the example shown in Fig. 5, the 10 components in Fig. 7A would be shifted to the posi- tions shown in Fig. 7B. These shifts in relative phase are determined in the following way. Figure 5 shows that the number 1 pulse is retarded an amount AT^i equal to 15% of T, the normal spacing between pulses. Thus at the carrier frequency c, the phase shift between the component from tkis retarded pulse and the reference pulse is 15% more than 360° or 414°. Thus the component at the carrier frequency c from the first subsidiary pulse train is shifted 54° from its unmodulated position. . At c -f- V, since the frequency is 10% higher, the net shift is 10% more than at c or 59.5°. Thus the number 1 component on the vector diagram of Fig. 7B is rotated 59.5° clockwise from its unmodulated position shown on Fig. 7A. Similarly pulses 2 and 3 are each shifted in position by equal amounts, AT2 and AT3 . These shifts in position give 85° phase shift at the carrier frequency. Hence components 2 and 3 Sit c -\-v are each rotated 10% more or 93.5° from their respective unmodulated reference positions shown on Fig. 12 A. Component number 4 is shifted 59.5° clockwise just as number 1 . Component 6 and 9 are also shifted 59.5° each, but in this case the modulat- ing function has the reverse polarity so that the components are rotated counterclockwise. Similarly components 7 and 8 are rotated 93.5° counterclockwise. The sum of these components in the vector diagram of Fig. 7B gives a resultant that is negative with respect to the reference direction and the magnitude that is 58% of the reference magnitude, where the reference mag- nitude and direction are those for the carrier c with no modulation. This gives the relative magnitude and phase of the c-\-v term produced by pulse position modulation for the case where the modulating function is a sine wave of frequency v — c/10 with a peak amplitude just large enough to shift a pulse by 1/4 of T, where T is the spacing between unmodulated pulses. A shift of this magnitude will be defined here as 50% modulation on the basis that 100% modulation should be 1/2 T, the maximum displacement that can be used without possible interference between pulses. In the same way the other component frequencies in the spectrum such as c,c — v,c±2v,etc., have been computed for the above case of 50% modulation, and for other peak ampUtudes of the modulating sine wave giving 25%, 70% and 100% modulation. In all cases the frequency of the modulating function was held at z; = c/10. This information is plotted on Fig. 8, show- ing V, c and the various components of the frequency spectrum that represent the sidebands about the carrier frequency c, as a function of the peak % modulation. 374 BELL SYSTEM TECHNICAL JOURNAL The above solution assumed a special case where c was an exact multiple of V. The purpose of this assumption was to simplify the problem to the extent that the periodicity of the modulated wave would be the same as that of the modulating function. There are two other possible cases. For one, the ratio of c to v could be such that a pulse would occur at the same instant of the modulating period only once every so many periods. The actual periodicity of the modulated pulse wave would be reduced accordingly because it would make the same number of periods of the modulating func- tion before the modulated pulse train is repeated. This is a result of the fact that pulse modulation provides for a discrete sampling rather than a continuous measure of the modulating wave. The technique of spectrum analysis demonstrated above is just as applicable to this case as it was to the simpler one. However, there will be comparatively more terms to be handled. The other possible case is the one where c and v are incommen- surate.^ In this case, the resulting modulated wave is non-periodic. How- ever, on the basis that the spectrum is practically always a continuous function of the signal frequency, this case has received no special attention here. At frequencies for which c is very much greater than v, so that the number of component pulse trains becomes too numerous to handle conveniently in the above fashion, the sidebands about each carrier or harmonic of the switching frequency can be computed by the standard methods for phase modulation, as the next section will demonstrate. This result follows directly from the theorem that as the carrier frequency c becomes large with respect to v, pulse position modulation merges into a linear phase modulation of each of the carriers. Pulse Position Modulation vs Phase Modulation When a pulse, in a pulse position modulated wave, is shifted by 1/2 the spacing between pulses (100% modulation) it is apparent from the previous discussion that the component of the carrier in the frequency spectrum of the pulse is shifted by 180°. Therefore to compare the spectrum of a pulse position modulated wave like that on Fig. 8 with the equivalent spectrum of a phase modulated wave, what is needed is Fig. 9, showing the frequency spectrum of a phase modulated wave of the form Cos{ct — k sin vt) as a func- tion of k for values of ^ up to -zr radians or 180°. The computation of the frequency spectrum of such a phase modulated wave has been adequately covered elsewhere and all that is done here is to give the brief development shown in appendix B. * Mr. W. R. Bennett has pointed out that this incommensurate case is the general one. It requires a double Fourier series, which reduces to a single series when the signal and carrier frequencies are commensurate. This analysis is based on the single Fourier series. SPECTRUM ANALYSIS OF WAVES 375 A comparison of the spectra on Figs. 8 and 9 shows that the sidebands have the same general pattern. However comparative sidebands are not 40 50 60 70 MODULATION IN PER CENT Fig. 8 — Spectrum of pulse position modulated wave for case where the carrier frequency C is 10 times the signal frequency v. quite equal in the two cases. In fact comparable upper and lower side- bands in the case of the pulse modulated wave shown on Fig. 8 are not 376 BELL SYSTEM TECHNICAL JOURNAL equal in absolute magnitude to each other. This lack of symmetry is due to the fact that c is ()nl\' 10 limes v. 115 3 8 2 8 4 PEAK PHASE SHIFT IN RADIANS Fig. 9— Spectrum of phase modulated wave cos {ct + k sin vt) as function of peak phase shift k for values of ^ up to tt radians. One way of proving this is to go through the process of computing the c — V term in this pulse modulated wave just as ihc c-\-v term was computed SPECTRUM ANALYSIS OF WAVES 377 earlier. Since the frequency c— I'is 10% less thane, the unmodulated pattern of the 10 subsidiary components, as shown on Fig. 7C, is the mirror image of that for c + ^ in 7A, for the first component is now 360° less 10% or 324°, and subsequent components are each retarded 36° with respect to the pre- vious one. When the pulse train is modulated the effect is similar to the case for c -\- v and, for the same per cent modulation, the Vector pattern of Fig. 7D is formed. The resultant in this case differs from that of 7B in sign as well as in magnitude. The difference in sign comes from the fact that, since component 1 in 7A corresponds to component 9 in 7C and com- ponent 2 in 7A to component 8 etc., the modulation in the case of c — t; rotates these corresponding components in opposite directions. The difference in magnitude is due to the fact that since c — v is an appreciably lower fre- quency than c -\- v\\\ this case (approx. 20%), the phase shift corresponding to a given shift in pulse position is proportionately less. Thus the corre- sponding Vector components are not shifted the same number of degrees. Thus the absolute magnitudes of c -f i' and c — v are not equal in this case. It is apparent that this difference in magnitudes oi c -\r v and c — v be- comes smaller as the carrier frequency c becomes larger with respect to v. In the limiting case of c very much greater than v, c -\- v and c — v would each be shifted the same number of degrees as c itself. If this more or less compromise shift of c is used to compute the c ± i', c ± 2v, and c db 3i; terms, then the resulting frequency spectrum is that of the phase modulated carrier on Fig. 9. The higher harmonics of c in the pulse position wave are similarly phase modulated and the interesting point is that 2c is modulated through twice as many degrees phase shift and 3c 3 times as many degrees, etc. Thus a single pulse position modulator could be designed to produce a harmonic of c with almost any desired degree of phase modulation. This is a useful method for obtaining a phase modulated wave, or with a 6 db per octave predistortion of the signal, a frequency modulated wave. Figure 8 also shows a term in v itself, which has been neglected so far in the discussion. It is apparent that the components at v contributed by the 10 subsidiar}' unmodulated waves must form the same kind of vector pattern as those oi c -\- v in Fig. 7. However, in this case c -\- v\% eleven times v in frequency, so that the components of v are rotated only one eleventh as much for a given pulse diplacement. Thus the magnitude of v at 100% modulation is equal to that oi c -\- v at approximately 9% modulation. For different frequency ratios of c to v the relationship of the v term io c -\- v will vary, and it is apparent that for c very much greater than v, the v term will vanish. The relationship is such that the amplitude of the v component out of the modulator at a given per cent modulation is directly proportional to its own frequency v for all frequencies less than approximately one quarter 378 BELL SYSTEM TECHNICAL JOURNAL of c, and the phase is 90° with respect to the input. Thus the modulator puts out a signal component that is the derivative of the input signal. To summarize the case of pulse position modulation, the frequency spec- trum may be determined by the methods based on subdividing the modu- lated pulse train into a series of unmodulated ones when the ratio oi c ta v is small, and by treating each harmonic of the carrier as a phase modulated wave of the form Cos n (ct -\- 6), where 6 is the modulating function, when the ratio of c to D is large. In the case treated here, the modulating function was a simple sinusoidal wave. Of course the analysis holds for more complicated wave shapes having frequency spectra of their own. In this event however the restriction on the relative magnitudes of the frequencies v and c should be taken as one on c and the highest frequency in the modulating spectrum. The complexity of the modulating function does not affect the analysis when it is done by this technique of subdividing the pulse train, since all that need be known is how much each pulse is shifted, and this can be done graphically. The analysis given here has neglected the length of the individual pulses. This was done when it was assumed that the individual contributions from the various pulse trains had the same amplitude at all frequencies. For any finite pulse width, the relative magnitudes of the various components must silt X be modified by the factor of the single pulse, as shown on Fig. 6. As mentioned in the introduction, a complex wave could be analyzed by multiplying its magnitude-time characteristic by unit sinusoids at each frequency in question, sampling the product at a sufficient number of points uniformly spaced over a cycle of the envelope of the complex wave, and then averaging the values of the product thus obtained. This technique is par- ticularly applicable to the analysis of pulse position modulated waves since, by taking the centers of the pulses of the modulated wave as the sampling instants, it is possible, with a finite number of samples (same as the number of pulses) to get the same results as though a very much greater number of uniformly spaced samples were taken. The interesting thing to note here is that the actual computations that would be involved in applying this sampling method of analysis to a pulse position modulated wave are almost identically the same calculations as required by the technique of resolving the pulse train into unmodulated subsidiary pulse trains used here. Pulse Width Modulation Pulse Width Modulation as defined here could also be termed "pure" pulse length modulation. The pulse train in the reference or unmodulated condition is a recurrent square wave, and the lengths of the pulses will be varied by the modulation without changing the position of the centers of the pulses. The term "pure" pulse length modulation is appHcable to this SPECTRUM ANALYSIS OF WAVES 379 special case where the phase relationship between spectra of adjacent pulses does not change with modulation because the centers of the pulses are not shifted by the modulation. The conventional form of pulse length modula- tion, where one end of the pulse is fixed in position, combines both this pulse width modulation and the pulse position modulation previously ana- lyzed. The interest in this case of pulse width modulation arose in con- nection with the analysis of ''hunting" ser\^omechanisms, and the analysis provides a basis for a general solution of the response of a two-position switch or ideal limiter to various forms of applied voltages. Since the unmodulated wave is a square wave with pulses of length 2L recurring at intervals of T = 4L, it has the familiar square wave spectrum including a d-c term, a fundamental term or carrier of frequency c = l/T, a 3rd harmonic with a negative ampUtude 1/3 that of the fundamental, etc. Figure 10 shows clearly that this spectrum is the sum of single pulses of width 2L spaced T = AL seconds apart. In the summation, all frequencies cancel except harmonics of c and, since they all add directly in phase, the component frequencies in the resultant spectrum have the same relative amplitudes as they have in one single pulse. When this pulse train is modulated, the width of each pulse becomes 2{L-\- AL), where the magnitude of AL depends in some specified way on the magnitude of thhe modulating function at the instant corresponding to the center of the pulse. For simplicity, the case will be taken where AL is proportional to the magnitude of the modulating function. For 100% modulation, AL will be assumed to vary from — L to +L. Figure 3 shows how the relative amplitude of the components of the frequency spectrum of a pulse vary for 3 different values of AL , along with the equation that gov- erns these amplitudes. If the modulating function has a periodicity v such that c = lOz', then every lOth pulse, recurring at the same instant in each modulating cycle, will be widened to the same extent and so can be formed into a subsidiary unmodulated pulse train, as was done on Fig. 5 for the pulse position modulated wave. Again vector diagrams like those in Fig. 7 may be formed showing the contribution of each of these subsidiary pulse trains at various frequencies such as c, r + v and c — v. ^^1len the waves are unmodulated, the vector diagrams for the same frequencies will be the same as those for the pulse position modulated case, except for the absolute amplitudes of the com- ponents, as long as c = lOr in each case. When the pulse width system is modulated, however, the modulation does not rotate the individual vector components as in the pulse position case since the spacing between pulses is not changed. What the pulse width modulation does is to change the length of the individual component vectors exactly as it does in the case of 380 BELL SYSTEM TECHNICAL JOURNAL the single pulses shown on Fig. 3. This change of magnitude, of course, can spoil the cancellation of the ten unmodulated components at some frequency- like c -\- 2v just as effectively as rotating them did in the case of the pulse position modulated wave, thus ])r<)during a sj)ectrum component at that frequency. As an example, the case will be taken where the modulating function is a 0.4 3 h. 0.2 5 ^o.sr Q 3 5^0,4 < liJ 0.3 > o h- Q. < ^ E 1 \ -L L TIME,t — * ---^ "V y ^ UlJ Q Q- < \ \ \ \ \ \ \ -5L -3L -L L 3L 5L 7L 9L TIME,t —*- \ \ \ \ \ \ ,-'"" — -^^ \ \ ^^^~ — IC 2C 3C 4C 5C 6C FREQUENCY, f, IN TERMS OF C (WHERE C =^) Fig. 10 — Comparative sjiectra of square wave and single pulse. sinusoid of frequency v. Then the change in width with modulation is given bv the formula ^L — k sin vl. Since c = lOr, the successive subsidiary pulse trains will be modulated an amount! — 1^ = ^sin( 1-k — las ;;/ lakes on the values from 1 to 10. Thus the spectra of these subsidiary pulse trains with ])ulses of length 2(L + SPECTRUM ANALYSIS OF WAVES 381 AZ,,„) recurring every l/v seconds will be a Fourier series of harmonics of v. The amplitude of the nth term of this series will be J^n = 77. — sm TTll 1 + ^ sin 27rw lo" This expression may be found from appendix C, equation (5a). Combining ^ 0.6 ^^ ^ ^. ^ X .-'' ^^' ^ ^^ • ' ^s Y''' y y '^- ^^ • • • y ^ y y • ^ • • X 2C-V ^- • y.^ ^^^ 2C7^ y^^ '^^ -^ _,'' "- 'J^ ^ ^ '^ ^ ^i^^**^ y ^ ci3^ _, '" ^.^ • • • y -' ""^^^^ ^:^2^ • X • ^ 40 50 60 70 MODULATION IN PER CENT Fig. 11 — Spectrum of pulse width modulated wave for case where carrier frequency C is 10 times the signal frequency v. the 10 such components at each frecjuency, as shown on Fig. 7 for the case of the pulse position modulated wave, the spectrum for this case of Pulse Width Modulation on Fig. 11 is produced. This spectrum is comparable to that on Fig. 8 for the pulse position modulated case. Pulse Width vs Amplitude Modulation That pulse width modulation is a form of amplitude modulation of the carriers of the unmodulated pulse train is shown mathematically by Equa- 382 BELL SYSTEM TECHNICAL JOURNAL 0.9 1.0 Fig. 12 — Response of ideal limiter to simultaneously applied isosceles triangle wave and sine wave inputs, k is the ratio of the peak amplitudes of sinusoidal and triangular waves at the input. . tion (8) in Appendix C, where the spectrum is developed as a Fourier series in harmonics of the pulse rale c with the modulation affecting only the amplitude of the coefficients. This mathematical analysis is continued in Appendix D where the fre- SPECTRUM ANALYSIS OF WAVES 383 quency spectrum is determined for AL = k sin vl. The spectrum thus computed is shown in Fig. 12. L An example of this type of pulse modulator is given by a two position switch or ideal limiter when the signal to be modulated is applied simul- taneously to the limiter with an isosceles triangle wave as carrier. The carrier should have a higher peak amplitude than the signal and a recurrence rate based on the desired carrier frequency. Figure 12 is arranged to show the output spectrum for such a limiter in terms of k, when k is the ratio of the peak amplitudes of the sinusoidal signal and triangular carrier wave inputs. A comparison of this spectrum with that on Fig. 11 shows that the two spectra have almost the same form, c and v have the same amplitude characteristics in each case. The c ± 2v and 2c ± v terms have differences that are like those found before in comparing the pulse position modulated wave on Fig. 8 and the phase modulated carrier on Fig. 9. As in that case, when c becomes very much greater than v the differences vanish. Application of Pulse Width Modulator Practical interest in this case lies in the fact that the signal is present in the output spectrum with a linear characteristic that makes such a modulator a linear amplifier. The "on-off" or "hunting" servomechanism is based on a modified form of such an amplifier in which the carrier is sup- plied by the self oscillation of the system. The term modified form is used because the self oscillations in general are more nearly sinusoidal than triangular in form and so do not give a linear change in pulse length over as wide a range of input amplitudes as does a triangular carrier. No attempt will be made to analyze such a system here since it has been handled elsewhere.^ However the above method is applicable to such problems regardless of the shape of the carrier or the signal. Other Forms of Pulse Modulation Another form of pulse modulation of interest is that of pulse length modu- lation in which either the start or the end of each pulse is fixed, so that the centers of the pulses vary in position with the length. This is a combination of both the pulse position and the pulse width modulations described above and can be analyzed by a combination of the methods developed. These same methods are also applicable to the analysis of frequency and phase modulated waves after they have been put through a limiter, as they generally are before detection. 9 See L. A. Macall, "The Fundamental Theory of Servomechanisms" D. Van Nostrand Company, 1945. 384 BELL SYSTEM TECIIMCAL JOURNAL APPENDIX A Fourier Transforms For Single Pulse The amplitude g{f) of the component of frequency/ in the spectrum of the Complex Magnitude-time function e{t) is given by the d-c component of the Moduhition products of c{t) and cos IttJI, found by averaging the product over the period of the comi)lex wave. Thus, for non-periodic waves, where the period is from — x to + x , the ampHtude of the spectrum at / is g(f) ^ f e(l) cos 2x/7 dt. (1) For the single pulse, where e{l) = £ for — L < / < L and e{l) = for all other values of /, equation (1) reduces to gif) ~ f E cos lirft dt. (2) Integrating, g(/) ^ :—. sin lirfi IttJ g{f)^. -.sin Itt/L. (3) Equation (3) is the expression for g(f) plotted on Fig. 1. Similarly, in the case of the single pulse, each increment in frequency df contributes a factor proportional to g{f) cos 27r// df to the composition of e{t), so that e(l) = f g(f) cos 27r// df. (4) Substituting in (4) the expression for g{f) given by equation (3), this becomes /A ^. -E /""sin 27r/Z, ^ ,^ ,. ,_, e(/) ^ - / -^^ cos 27r// df. (5) 7r J-oo / APPENDIX B Frequency Spectrum Or Phase Modulated Wave The Pliase Modulated Wave in this case is given by cos ((■/ — k sin vl) = cos {ct) cos (k sin vt) -f sin (ct) sin (k sin vt) Now cos (ct) cos (k sin ct) = Jo (k) cos {ct) + Jo (k) cos (c - 2v) t SPECTRUM ANALYSIS OF WAVES 385 + Jo (k) COS {c -\- 2v) t + ■■■ and sin (ct) sin {k sin cl) — Ji (k) cos (c — v) t - Ji (k) cos {c -\- v) t + /s (k) COS {c - 3v) I - /s (^) COS (c + 3v) t + ••• .'. COS (f/ — k sin ?'/) = Jq (k) COS (c/) + 7] (^) COS (c — z;) / - /i (y^) cos (c + v) t + /z (/^) cos (c - 2tO / + J2 (k) cos (c + 2tO / + /s (k) cos (c - 3z') t - J3 (k) cos (c + 3zO / H APPENDIX C In this Appendix the spectrum of a train of rectangular pulses of length 2(L + AL) recurring every T seconds, will be found from the spectrum of a single pulse of this train. For the single pulse at any frequency/, gin ^ -.sin 2^f{L + AL). (1) x/ For a series of such pulses recurring with a spacing T — 1/c, then the sum of spectra of the individual pulses form a Fourier series of harmonics of c. Thus e(t) = ^0 + Z) ^n cos liritd, (2) n = l where An is the sum of an iniinite number (one from each pulse) of infinitesi- mal terms g(;/c) and g{ — nc), shown in (1). Thus ^„ ^ 22 — sin 2Trnc{L + AL) (3) Tvnc Now to put an absolute value to the amplitudes g(/) shown in equation (1), it is necessary to average them over the recurrence period of the single pulse, making them infinitesimals. However, in the train of pulses recurring every T — \/c seconds, the amplitude of An can be determined by averaging the terms in (1) over an interval T. Then An = ^^sin 2Tvnc{L + AZ). (4) irncT When T = 4L = l/c, (4) reduce to 2E . — sm _, wn 2 , 2E . n-K (. . aA ... y4„ = — sm — ( 1 + —- j (5) 386 BELL SYSTEM TEC/LMCAL JOIKNAL For the example taken in the text, when the pulse train was subdivided into 10 subsiding pulse trains, the period T = 1/v = 10/c = 40L. Thus in this case, the Fourier coefficients of the harmonics of v are 2E . TTii / AL\ (5a) The expression for .1,, in equation (5) can be put in simpler form by using the formula for the sin of tlie sum of t wo angles. In this way, we get An — IE irn TTll sm I — I cos /irn AL L\ , /7r//\ . /irn AL (6) Now, for // odd, sia — alternately assumes the value ± 1 and cos — vanishes. (?) and for ii even, cos ( — - ) alternatelv assumes the value ±1 and sin irn vanishes. The A o term, being the d-c average of the pulse train, is given by E/2{L + AL) ^E (. , AL T 2 V T (7) If the pulse train is transformed by shifting the zero so that it alternates between db£/2 instead of and E, the first term in equation (7) vanishes and (2) becomes, from (6) & (7), e(t) = Ao A- Ai cos 27rf/ + Ai cos 2x 2cl + • Where etc. A, = A. = m 2E /t 1 = — cos ( - TT \Z ¥) 2L; . ML ^^ = 2. "" " U A, = 2E Stt /AL\ 3. ^^^ T \-l) (8) APPENDIX D The purjjose of this section is to comi)ule the si)ectrum of the carrier given by e(|ualion (S) in A])pendix C as their amplitudes vary with - = k sin vl. SPECTRUM ANALYSIS OF WAVES 387 For the J-c term, , EAL £, . , 9 T' "^ ? sin vt. For the fundamental or c term, 2E /tt . \ Ai cos ItcI = — ■ cos KT^k sin z;^ ) cos Iwcl Using the Bessel's expansion of cos (2 sin 6), we get, \Jo{k) cos 27rc _g +/2(^) cos 27r(c — 2v)t Ai cos 27rc/ = -— -^ +/-2(/^) COS 27r(c + 2v)t [-] etc. In a similar fashion, the other terms can also be computed, giving the spectrum shown on Fig. 12, where Joik) becomes the amplitude of c, J2{k) the amplitude of either c -{- 2votc — 2v, etc. Abstracts of Technical Articles by Bell System Authors Commercial Broadcasting Pioneer. The WEAF Experiment: 1922-1926} William Peck Banning. WEAF, the radio call letters which for nearly a quarter of a century designated a broadcasting station famous for its pioneering achievements, ceased last November to have its old significance. WNBC are the new call letters. This book is an excellent record of the four years during which this station was the experimental radio broad- casting medium of the American Telephone and Telegraph Company. The author indicates that the WEAF experiment aided the development of radio broadcasting in three ways: First, in the scientific and technological field. Second, in the emphasis of a high standard for radio programs. Third, in determining the means whereby radio broadcasting could support itself. When TF£/1 F changed hands from the American Telephone and Telegraph Company to new ownership, public reaction to almost every type of broad- cast had been tested, network broadcasting had been established and the economic basis upon which nationwide broadcasting now rests had been founded. A trail had been blazed that thereafter could be followed without hesitation. In so far as radio broadcasting is concerned, this book is a significant chapter in communication history. A Multichannel Microwave Radio Relay System} H. S. Black, J. W. Beyer, T. J. Grieser, F. A. Polkinghorn. An 8-channel microwave relay system is described. Known to the Army and Navy as AN/TRC-6, the system uses radio frequencies approaching 5,000 megacycles. At these frequencies, there is a complete absence of static and most man-made interference. The waves are concentrated into a sharp beam and do not travel along the earth much beyond seeing distances. Other systems using the same frequencies can be operated in the near vicinity. The transmitter power is only one four-millionth as great as would be required with nondirectional antennas. The distance between sets is limited but by using intermediate repeaters communications are extended readily to longer distances. Short pulses of microwave power carry the intclHgence of the eight messages utilizing pulse position modulation to modulate the 1 Published by Harvard University Press, Cambridge, Massacliusetts, 1946. ^ Elec. Engg., Trans. Sec, December 1946. 388 ABSTRjiCTS OF TECHNICAL ARTICLES 389 pulses and time division to multiplex the channels. The eight message circuits which each AN/TRC-6 system provides are high-grade telephone circuits and can be used for signaling, dialing, facsimile, picture transmission, or multichannel voice frequency telegraph. Two-way voice transmission over radio links totaling 1,600 miles, and one-way over 3,200 miles have been accomplished successfully in demonstrations. Further Observations of the Angle of Arrival of Microivaves? A. B. Crawford and William M. Sharpless. Microwave propagation measure- ments made in the summer of 1945 are described. This work, a continua- tion of the 1944 work reported elsewhere in this issue of the Proceedings of the I.R.E. and Waves and Electrons, was characterized by the use of an antenna with a beam width of 0.12 degree for angle-of-arrival measurements and by observations of multiple -path transmission. The Ejffect of Non-Uniform Wall Distributions of Absorbing Material on the Acoustics of Rooms} Herman Feshbach and Cyril M. Harris. The acoustics of rectangular rooms, whose walls have been covered by the non- uniform application of absorbing materials, is treated theoretically. Using appropriate Green's functions a general integral equation for the pressure distribution on the walls is derived. These equations show immediately that it is necessary to know only the pressure distribution on the treated surfaces to predict completely the acoustical properties of the room, such as the resonant frequencies, the decay constants, and the spatial pressure distribution. The integral equation is solved approximately using (1) perturbation method, and (2) approximate reduction of the integral equation to an equivalent transmission line. Criteria giving the range of validity of these approximations are derived. It was found useful to introduce a new concept, that of ^^efective admittance,''^ to express the results for the resonant frequency and absorption for then the amount of computation is reduced and the accuracy of the results is increased. The absorption of a patch of material was found as a function of the position of the absorbing material and was checked experimentally for a convenient case, an absorbing strip mounted on the otherwise hard walls of a rectangular room. Particular attention is given to the case where the acoustic material is applied in the form of strips. The results may then be expressed in series which converge very rapidly and are, therefore, amenable to numerical calculation. Ap- proximate formulas are obtained which permit estimates of the diffusion of sound in a non-uniformly covered room. In agreement with experience, these equations show that diffusion increases with frequency and with the ^ Proc. I.R.E. and Waves and Electrons, November 1946. ^Joiir. Aeons. Soc. America, October 1946. 390 BELL SYSTEM TECIIMCA L JOl RXA L number of nodes on the treated walls. The "interaction effect" of one strip on another is shown to decrease with an increase of the number of nodes. The results are then applied to the case of ducts with non-uniform distribution of absorbing material on its walls. Results are given which permit the calculation of the attenuation per unit length of duct. The methods of this paper hold for any distribution of absorbing material and also if the admittance is a function of angle of incidence. High Current Electron Guns J' L. M. Field. This j)aper presents a survey of some of the problems and methods which arise in dealing with the design of high current and high current-density electron guns. A discussion of the general limitations on all electron gun designs is followed by discussion of single and multiple potential guns using electrostatic fields only. A further discussion of guns using combined electrostatic and mag- netic fields and their limitations, advantages, and some possible design procedures follows. Reflection of Sound Signals in the Troposphere^' G. W. Gilman, H. B. CoxHEAD, and F. H. Willis. Experiments directed toward the detection of non-homogeneities in the first few hundred feet of the atmosphere were carried out with a low power sonic "radar." The device has been named the sodar. Trains of audiofrequency sound waves were launched vertically upward from the ground, and echoes of sufficient magnitude to be displayed on an oscilloscope were found. Strong displays tended to accompany strong temperature inversions. During these periods, transmission on a microwave radio path along which the sodar was located tended to be disturbed by fading. In addition, relatively strong echoes were received when the atmosphere was in a state of considerable turbulence. There was a well-defined fine-weather diurnal characteristic. The strength of the echoes was such as to lead to the conclusion that a more complicated distribu- tion of boundaries than those measured by ordinary meteorological methods is required in the physical picture of the lower troposphere. A Cathode-Ray Tube for Vieiving Continuous Patterns? J. B. Johnsox. A cathode-ray tube is described in which the screen of persistent phosphor is laid on a cylindrical portion of the glass. A stationary magnetic field bends the electron beam on to the screen, while rotation of the tube produces the time axis. When the beam is deflected and modulated, a continuous pattern may be viewed on the screen. 6 Rev. Mod. Pliys., July 1946. ^ Jour. Acous. Soc. Amer., October 1946. ''Jour. Applied Physics, November 1946. ABSTRA CTS OF TECHNICA L A RTICLES 391 The Molecular Beam Magnetic Resonance Method. The Radiofrequency Spectra of Atoms and Molecules.^ J. B. M. Kellogg and S. Millman. A new method known as the "Magnetic Resonance Method" which makes possible accurate spectroscopy in the low frequency range ordinarily known as the "radiofrequency" range was announced in 1938 by Rabi, Zacharias, Millman, and Kusch (R6, R5). This method reverses the ordinary pro- cedures of spectroscopy and instead of analyzing the radiation emitted by atoms or molecules analyzes the energy changes produced by the radiation in the atomic system itself. Recognition of the energy changes is accom- plished by means of a molecular beam apparatus. The experiment was first announced as a new method for the determination of nuclear magnetic moments, but it was immediately apparent that its scope was not limited to the measurement of these quantities only. It is the purpose of this article to summarize the more important of those successes which the method has to date achieved. Metal-Lens Antennas.^ Winston E. Kock. A new type of antenna is described which utilizes the optical properties of radio waves. It consists of a number of conducting plates of proper shape and spacing and is, in effect, a lens, the focusing action of which is due to the high phase velocity of a wave passing between the plates. Its field of usefulness extends from the very short waves up to wavelengths of perhaps five meters or more. The paper discusses the properties of this antenna, methods of construction, and applications. Underwater Noise Due to Marine Life}^ Donald P. Loye. The wide- spread use of underwater acoustical devices during the recent war made it necessary to obtain precise information concerning ambient noise condi- tions in the sea. Investigations of this subject soon led to the discovery that fish and other marine life, hitherto generally classified with the voiceless giraffe in noisemaking ability, have long been given credit for a virtue they by no means always practice. Certain species, most notably the croaker and the snapping-shrimp, are capable of producing noise which, in air, would compare favorably with that of a moderately busy boiler factory. This paper describes some of the experiments which traced these noises to their source and presents acoustical data on the character and magnitude of the disturbances. Elastic, Piezoelectric, and Dielectric Properties of Sodium Chlorate and Sodium Promote}^ W. P. Mason. The elastic, piezoelectric, and di- 8 Rev. Mod. PItys., July 1946. ^ Proc. I.R.E. and Waves and Electrons, November 1946. ^^ Jour. Aeons. Soc. America, October 1946. iip/m. Rev., October 1 and 15, 1946. 392 BELL SYSTEM TECH NIC A L JOURNA L electric constants of sodium chlorate (NaClOs) and sodium bromate (NaBrOs) have been measured over a wide temperature range. The value of the piezoelectric constant at room temperature is somewhat larger than that found by Pockels. The value of the Poisson's ratio was found to be positive and equal to 0.23 in contrast to Voigt's measured value of —0.51. At high temperatures the dielectric and piezoelectric constants increase and indicate the presence of a transformation point which occurs at a temperature slightly larger than the melting point. A large dipole piezo- electric constant (ratio of lattice distortion to dipole polarization) results for these crystals but the electromechanical coupling factor is small because the dipole polarization is small compared to the electronic and ionic polariza- tion and little of the applied electrical energy goes into orienting the dipoles. Paper Capacitors Containing Chlorinated Impregnants. Effects of Sulfur.^' D. A. McLean, L. Egerton, and C. C. Houtz. Sulfur is an effective stabilizer for paper capacitors containing chlorinated aromatics, in the presence of both tin foil and aluminum foil electrodes. Sulfur has unique beneficial effects on power factor which are especially marked when tin foil electrodes are used. The value of R (Equation 4) can be used as an index of ionic conductivity in the impregnating compound. Diagnostic power factor measurements on impregnated paper are best made at low voltages. Electron diffraction studies give results in line with the previously published theory of stabilization. Several previous findings are reaffirmed: (a) the importance of all components of the capacitor in determining its initial properties and aging characteristics, (b) the superiority of kraft paper over linen, and (c) widely different behavior of capacitors employing different electrode metals. A New Bridge Photo-Cell Employing a PJwio-Conductive Effect in Silicon. Some Properties of High Purity SiliconP G. K. Teal, J. R. Fisher, and A. W. Treptow. a pure photo-conductive effect was found in pyrolytically deposited and vaporized silicon films. An apparatus is described for making bridge type photo-cells by reaction of silicon tetrachloride and hydrogen gases at ceramic or quartz surfaces at high temperatures. The maximum photo-sensitivity occurs at 8400-8600A with considerable re- sponse in the visible region of the spectrum. The sensitivity of the cell appears about equivalent to that of the selenium bridge and its stability and speed of response are far better. For pyrolytic films on porcelain there are three distinct regions in the conductivity as a function of temperature. At low temperatures the electronic conductivity is given by the expression '^ Indus. & Eugg. Cliemislry, Noveni1)er 1946. ^^ Jour. Applied Pliysics, Novcmljcr 1946. ABSTRACTS OF TECHNICAL ARTICLES 393 <r = Af(T)exp-(E/2kT). At temperatures between 227°C and a higher temperature of 4(10 500°C a = Aexp—{E/2kT), where £ lies between 0.3 and 0.8 ev; and at high temperatures a = Aexp—(E/2kT), where E = 1.12 ev. The value 1.12 ev represents the separation of the conducting and non-conducting bands in silicon. The long wave limit of the optical absorp- tion of silicon was found to lie at approximately 10,500 A (1.18 ev). The data lead to the conclusion that the same electron bands are concerned in the photoelectric, optical, and thermal processes and that the low values of specific conductances found (1.8X10~* ohm~^ cm~^) are caused by the high purity of the silicon rather than by its polycrystalline structure. Non-Uniform Transmission Lines and Reflection Coefficients}^ L. R. Walker and N. Wax. A first-order differential equation for the voltage reflection coefficient of a non-uniform line is obtained and it is shown how this equation may be used to calculate the resonant wave-lengths of tapered lines. ^*Jour. Applied Physics, December 1946. Contributors to this Issue Harald T. Friis, E.E., Royal Technical College, Copenhagen, 1916; Sc.D., 1938; Assistant to Professor P. D. Pedersen, 1916; Technical Advisor at the Royal Gun Factory, Copenhagen, 1917-18; Fellow of the American Scandinavian Foundation, 1919; Columbia University, 1919. Western Electric Company, 1920-25; Bell Telephone Laboratories, 1925-. Formerly as Radio Research Engineer and since January 1946 as Director of Radio Research, Dr. Friis has long been engaged in work concerned with funda- mental radio problems. He is a Fellow of the Institute of Radio Engineers. Ray S. Hoyt, B.S. in Electrical Engineering, University of Wisconsin, 1905; Massachusetts Institute of Technology, 1906; M.S., Princeton, 1910. American Telephone and Telegraph Company, Engineering Department, 1906-07. Western Electric Company, Engineering Department, 1907-11. American Telephone and Telegraph Company, Engineering Department, 1911-19; Department of Development and Research, 1919 34. Bell Telephone Laboratories, 1934-. Mr. Hoyt has made contributions to the theory of loaded and non-loaded transmission lines and associated apparatus, theory of crosstalk and other interference, and probability theory with particular regard to applications in telephone transmission engineering. W. D. Lewis, A.B. in Communication Engineering, Harvard College, 1935; Rhodes Scholar, Wadham College, Oxford; B.A. in Mathematics, Oxford, 1938; Ph.D. in Physics, Harvard, 1941. Bell Telephone Labora- tories, 1941-. Dr. Lewis was engaged in radar antenna work in the Radio Research Department during the war; he is now engaged in microwave repeater systems research. J. C. LoziER, A.B. in Physics, Columbia College, 1934; graduate physics student, Princeton University, 1934-35. R.C.A. \'ictor Manufacturing Company, 1935-36; Bell Telephone Laboratories, Inc., 1936-. Mr. Lozier has been engaged in transmission development work, chiefly on radio telephone terminals. During the war he was concerned primarily with the theory and design of servomechanisms. 394 VOLUME XXVI JULY, 1947 NO. 3 THE BELL SYSTEM TECHNICAL JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION Telephony by Pulse Code Modulation W. M. Goodall 395 Some Results on Cylindrical Cavity Resonators J. P. Kinzer and I. G. Wilson 410 Precision Measurement of Impedance Mismatches in Waveguide Allen F. Pomeroy 446 Reflex Oscillators J. R. Pierce and W. G. Shepherd 460 Abstracts of Technical Articles by Bell System Authors. . 682 Contributors to This Issue 691 ■*y- AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 50^ per copy $1.30 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 W. H. Harrison O. E. Buckley O. B. Blackwell 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. Copyright, 1947 American Telephone and Telegraph Company PRINTED IN U. S. A. The Bell System Technical Journal Vol. XXVI July, 1947 No. 3 Telephony By Pulse Code Modulation* By W. M. Goodall An experiment in transmitting speech by Pulse Code Modulation, or PCM, is described in this paper. Each sample amplitude of a pulse amplitude modula- tion or PAM signal is transmitted ])y a code group of OX-OFF pulses. 2" amplitude values can be represented by an n digit binary number code. For a nominal 4 kc. speech band these n OX-OFF pulses are transmitted 8000 times a second. Experimental ef|uipment for coding the PAM pulses at the transmitter and decoding the PCM pulses at the receiver is described. Experiments with this equipment indicate that a three-unit code appears to be necessary for a minimum grade of circuit, while a six- or seven-unit code will provide good quality. Introduction THIS paper describes an experiment in transmitting speech by PCM, or pulse code modulation. The writer is indebted to his colleagues in the Research Department, C. E. Shannon, J. R. Pierce and B. M. Oliver, for several interesting suggestions in connection with the basic principles of PCM given in this paper. Work on a dififerent PCM system was carried on simultaneously in the Systems Development Department of the Bell Laboratories by H. S. Black. This in turn led to the development of an 8-channel portable system for a particular application. This system is being described in a forthcoming paper by H. S. Black and J. O. Edson.^ A method for pulse code modulation is proposed in a U. S. Patent issued to A. H. Reeves.2 The material now presented is competed of three parts. The first deals with basic principles, the second describes the experimental PCM system, while the last discusses the results obtained. Basic Principles PCM involves the application of two basic concepts. These concepts are namely, the time-division principle and the amplitude quantization * Paper presented in part at joint meeting of International Scientific Radio Union and Inst. Radio Engineers on May 5, 1947 at Washington, D. C. ^ Paper presented on June 11, 1947 at A. I. E. E. Summer General Meeting, Mont- real, Canada. Accepted for publication in forthcoming issue of A. I. E. E. Trans- actions. 2 A. H. Reeves. V . .S. Patent Hl.lllfilQ, Feb. 3, 1942, assigned to International Stand- ard Electric Corp.; also, French patent * 852, 183, October 23, 1939. 395 396 BELL SYSTEM TECHNICAL JOURNAL principle. The essence of the time-division principle is that any input wave can be represented by a series of regularly occurring instantaneous samples, provided that the sampling rate is at least twice the highest frequency in the input wave.^ For present purposes the amplitude quantization principle states that a complex wave can be approximated by a wave having a finite number of amplitude levels, each differing by one quantum, the size of the quantum jumps being determined by the degree of approximation desired. Although other arrangements are possible, in this paper we will consider the application of these two basic principles in the following order. First the input wave is sampled on a time-division basis. Then each of the samples so obtained is represented by a quantized amplitude or integer number. Each of these integer numbers is represented as a binary number of n digits, the binary number system being chosen because it can readily be ENVELOPE OF AUDIO SIGNAL NO AUDIO SIGNAL Fig. 1 — Pulses in a PAM System. represented by ON-OFF or two-position pulses. 2" discrete levels can be represented by a binary number of n digits.* Thus, PCM represents each quantized amplitude of a time-division sampling process by a group of ON-OFF pulses, where these pulses represent the quantized amplitude in a binary number system. The discussion so far has been in general terms. The principles just discussed will now be illustrated by examples. Multiplex transmission of speech channels by sending short pulses selected sequentially from the respective speech channels, is now well known in the telephone art and is called time-division multiplex. When the pulses consist simply of short samples of the speech waves, their varying amplitudes directly represent the speech waves and the system is called pulse amplitude modulation or PAM. In PAM the instantaneous amplitude of the speech wave is sampled at regular intervals. The amplitude so obtained is trans- ' This is because the DC, fundamental and harmonics of the wave at the left in Fig. 1 all become modulated in the wave at the right, and if the highest modulating frequency exceeds half the sampling rate, the lower sideband of the fundamental will fall in the range of the modulating frequency and will not be excluded l)y the low-pass filter. The result is distortion. ■• In a decimal system the digits can have any one of 10 values, to 9 inclusive. In a binary system, the digits can have only two values, either or 1. TELEPHONY BY PULSE CODE MODULATION 397 mitted as a pulse of corresponding amplitude. In order to transmit both positive and negative values a constant or d-c value of pulse amplitude can be added. (See Fig. 1.) When this is done positive values of the informa- tion wave correspond to pulse amplitudes greater than the constant value while negative values correspond to pulse amplitudes less than the constant value. At the receiver a reproduction of the original speech wave will be obtained at the output of a low-pass filter. The PCM system considered in this paper starts with a PAM system and adds equipment at the terminals to enable the transmission of a group of ON-OFF pulses or binary digits to represent each instantaneous pulse amplitude of the PAM system. Representation of the amplitude of a single PAM pulse by a finite group of ON-OFF pulses or binary digits requires quantization of the audio wave. In other words, we cannot represent the actual amplitude closer than ^ "quantum". The number of amplitude levels required depends upon the grade of circuit desired. The disturbance which results from the quantization process has been termed quantizing noise. For this type of noise a signal-to-noise ratio of 33 db would be ob- tained for 32 amplitude levels and this grade of circuit was deemed suffi- ciently good for a preliminary study. These 32 amplitude levels can be obtained with 5 binary digits, since 32 = 2^. Figure 2 shows how several values of PAM pulse amplitude can be represented by this binary code. The first column gives the digit pulses which are sent between the transmitter and receiver while the second column shows the same pulse pattern with each pulse weighted according to its assigned value, and the final column shows the sum of the weighted values. The sum, of course, represents the PAM pulse to the nearest lower amplitude unit. The top row where all the digits are present shows, in the middle wave form, the weighted equivalent of each digit pulse. By taking different combinations of the five digits all integer amphtudes between 31 and can be represented. The examples shown are for 31, 18, 3, and 0. Referring to Fig. 3 sampling of the audio wave (a) yields the PAM wave (b). The PAM pulses are coded to produce the code groups or PCM signal (c) . The PCM pulses are the ones sent over the transmission medium . For a sampling rate of 8000 per second, there would be 8000 PAM pulses per second for a single channel. The digit pulse rate would be 40,000 pps for a five-digit code. For a time-division multiplex of N channels both of these pulse rates would be multiplied by N. Wave form (d) shows the decoded PAM pulses where the amplitudes are shown under the pulses. The original audio wave is repeated as wave form (e). It will be noted that the received signal is delayed by one PAM pulse interval. It is also seen that the decoded pulses do not fit exactly on this curve. This is the result of quantization and the output of the low-pass 398 BELL SYSTEM TECHNICAL JOVRXAL filter will contain a quantizinjj; disturbance not shown in (e) which was not present in the input signal. A signal that uses regularly occurring ON-OFF pulses can be "regener- ated" and repeated indefinitely without degradation. A pulse can be "regenerated" by equipment which transmits an undistorted pulse provided a somewhat distorted pulse is received, and transmits nothing otherwise. BINARY NUMBER I I I i I I I I I i DECODED NUMBER WEIGHTED EQUIVALENT 16 i 8 ; 4 ; 2 1 1 I i I ; 1 I I J — L 16 ; I 1 2 ; n =^ Fis;. 2 — Binar\' and decimal equivalents. Thus, the received signal at the output of the final decoder is of the same quality as one produced by a local monitoring decoder. To accomplish this result, it is necessary, of course, to regenerate the digit pulses before they have been too badly mutilated by noise or distortion in tlie transmission medium. The regenerative ])roperty of a quantized signal can be of great importance in a long repeated system. I'"or example, with a con\cntional system each repeater link of a lOO-link system must huNc a signal-to-noise ratio 20 db 1> 1 o o a o o 3 @ 399 400 BELL SYSTEM TECHNICAL JOURNAL better than the complete system. For PCM, however, with regenerative repeaters the required signal-to-noise ratio in the radio part of the system is independent of the number of links. Hence, we have a method of trans- mission that is ideally suited to long repeated systems. At this point we might consider the bandwidth required to send this type of signal. For a 5-digit code the required band is somewhat less than 5 times that required for a PAM system. It is somewhat less than 5 times be- cause in a multiplex system crosstalk becomes a serious problem. In a PAM system this crosstalk would add up on a long system in somewhat the same manner as noise. In order to reduce the crosstalk it would probably be necessary to use a wider band for the PAM repeater system than would be required for a single-link system. For PCM, on the other hand, by using regeneration the whole system requirement for crosstalk can be used for each link. In addition, a relatively greater amount of crosstalk can be tolerated since only the presence or absence of a pulse needs to be determined. Both of these factors favor PCM. This is a big subject and for the present we need only conclude that from considerations of the type just given the bandwidth penalty of PCM is not nearly as great as might first be expected. . The same two factors that were mentioned in connection with crosstalk also apply to noise, and a PCM signal can be transmitted over a circuit which has a much lower signal-to-noise ratio than would be required to transmit a PAM signal, for example. Hence, we conclude that PCM for a long repeated system has some powerful arguments on its side because of its superior performance even though it may require somewhat greater bandwidth. There are other fac- tors where PCM differs from more conventional systems but a discussion of these factors is beyond the scope of this paper. The previous discussion may be summarized as follows: One begins with a pulse amplitude modulation system in which the pulse amplitude is modulated above and below a mean or d-c value as indicated in Fig. 1. It is assumed that it will be satisfactory to limit the amplitude range to be transmitted to a definite number of amplitude levels. This enables each PAM pulse to be represented by a code group of ON-OFF pulses, where the number of ampUtude levels is given by 2^, n being the number of elements in each code group. With this system the digit pulses can be "regenerated" and the quality of the overall transmission system can be made to depend upon the terminal equipment alone. Experimental PCM Equipment The experimental coder used in these studies might be designated as one of the "feedback subtraction type". It functions as follows: Each PAM pulse is stored as a charge on a condenser in a storage circuit. (See Fig. 4.) TELEPHONY BY PULSE CODE MODULATION 401 5H 3 < o CLO © © o a: UJ OQ HO p tr t-ui 2b H _l DO. 05 < (^ (^ ® 0, © (TtO < t-_i(r Z 3 u OQ. Z O LJ O ' > o o •^ Q. UJ 1- z UJ O ^§ QQ ui O °-2 > < _) UJ Q © >- < -I UJ o q: o (0 z UJ o o ZH Str 8^ St < D oa < o ^? m o D t/) © ©' I UJ _l Q. 2 < (0 © QZ ■D - < 402 BELL SYSTEM TECIIMCAL JOLRXAL The voltage across this condenser is compared with a reference voltage. The magnitude of this reference voltage corresponds to the d^c i)ulse amplitude of Fig. 1. The voltage has a magnitude of 16 units. If the magnitude of the condenser voltage exceeds the magnitude of the 16-unit voltage, a positive pedestal voltage is obtained in the output of the comparing circuit. This pedestal voltage is amplified, limited and applied to the pedestal modulator. The pedestal modulator serves as a gate for timing pulses from the timing pip generator. If the pedestal voltage and timing pulse are applied simultaneously to the pedestal modulator, a pulse is obtained in the output. In the jjresent case this pulse corresponds to the presence of the 16-unit digit in the code group which represents this PAM pulse. This digit pulse after amplification and limiting is (1) sent out over the line (PCM out) and (2) fed back through a suitable delay circuit to a subtraction circuit. The function of the subtraction circuit is to subtract a charge from the con- denser corresponding to the 16-unit digit. The charge remaining on the condenser is now compared with a new reference voltage which is h the magnitude of the first reference voltage or 8 units. If the magnitude of the voltage across the condenser exceeds this new reference voltage the above process is rei)eated and the second digit pulse is transmitted and another charge, this time corresponding to the 8-unit digit, is subtracted from the remaining charge upon the condenser. If the magnitude of the voltage across the condenser is less than the reference voltage, in either case above, then no pedestal will be produced and no digit pulse be transmitted. Since no pulse is transmitted, no charge will be subtracted from the condenser. Thus the charge remaining upon the condenser after each operation represents the part of the orig- inal PAM pulse remaining to be coded. The reference voltage wave consists of a series of voltages each of which is ^ of the preceeding one. There is one step on the reference voltage function for each digit to be coded. A better understanding of the coding process can be had by reference to the various wave forms involved. For completeness, wave forms from audio input to the coded pulse signal are shown for the transmitter in Figs. ^ and 5 and from the coded pulse signal to audio output for the receiver in Figs. 7 and 3. In the diagram the abscissas are time and the ordinates are amplitudes. Some of these wave forms have already been discussed in connection with Fig. .^. Since the coder functions in the same manner for each PAM pulse the detailed wave forms of the coding and decoding proc- esses are shown for only two amplitudes. The block schematic for the transmitter is given on Fig. 4, while that for the receiver is given in Fig. 6. The letters on Figs. 4 and 6 refer to the wave forms on Fig. 3, while the numbers refer to the wave forms in Figs. 5 and 7. 7. CODE ELEMENT TIMING PIPS Jl il i i 11 il 8. CODE GROUPS Fig. 5 — Detailed wave forms for PCM Transmitter (amplitude vs. time). 403 404 BELL SYSTEM TECHNICAL JOURNAL ©[ a. ai UJ J^ L. ® q: iij -J © 5 < (D , q: © a.v)< UJ . OQ-Z o 8- ® < _l UJ Q a. 1- UI z o CD O o V) i , © ® IT »"> UJ 1 > _i 5 < ®l© I 8i 0. :1_ .!_ Ul A 1. DELAYED CONTROL PULSE 2. STEP TIMING PIPS [l_ ^J R 3. REFERENCE STEP VOLTAGE ILJ 11 1 8. CODE GROUPS 11 JLJ 11 J J"L 9. OUTPUT OF RECEIVING SUBTRACTION CIRCUIT 10. RECEIVING STORAGE CONDENSER VOLTAGE '-J\ n II. UNDELAYED CONTROL PULSE n Fig. 7 — Detailed wave forms for PCM Receiver (amplitude vs. time). 405 406 BELL SYSTEM TECHNICAL JOURNAL Referring to Figs. 4 and 5, tlie "delayed control pulse" Curve 1 is the principal timing pulse for the transmitting coder. It is used to sample the audio wave and to start the step and timing-pip generators. Two sets of timing-pips are produced; one, ("urvc 2, is used to generate the reference step voltage while the other, Curve 7, is used for timing the digit pulses. The reference step voltage, Curve 3, is used in the comparing circuit and in the subtraction circuit. Curve 4 gives the output of the subtraction circuit, while Curve 5 is the voltage on the storage condenser. The next plot gives Curves 3 and 5 superimposed; the shaded area on this plot corresponds to the time during which a pedestal voltage is generated. The pedestal voltage is given by Curve 6, and the output of the pedestal modulator is given by Curve 8. This last curve is a plot of the two code groups corresponding to the two PAM pulses being coded . In studying these wave forms it will be noted that the delayed control pulse, the two sets of timing-pips and the reference step voltage curves are the same for each code group. On the other hand the storage condenser voltage, the pedestal voltage, the group of code pulses, and the group of pulses from the subtraction circuit are different for each code group. It will be recalled that a pedestal voltage is produced during the time that the condenser voltage exceeds the reference step voltage. The leading edge of each pedestal pulse is generated by the falling part of the reference step voltage. The trailing edge of each pedestal pulse is produced by the falling part of the condenser voltage. This drop in condenser voltage is the result of the operation of the subtraction circuit. The output of the subtraction circuit depends upon the delayed digit pulse which has just been passed by the pedestal pulse. Its magnitude depends upon the reference voltage step that applies to the particular digit being transmitted. The function of the delay in the feedback path is to allow the outgoing digit pulse to be com- pleted before the pedestal is terminated. It is seen that the pedestal voltage contains the same information as the transmitted code groups. Under ideal conditions the use of auxiliary timing pulses would not be required. However, in a practical circuit the leading edge of the pedestal varies, both as to relative timing and as to rate of rise. Under these conditions the auxiliary timing-pips permit accurate timing of the outgoing PCM pulses, as well as constant pulse shape for the input to the subtraction circuit. Summarizing the foregoing it is seen that in the coder under discussion a comparison is made for each digit between a reference voltage and the voltage across a storage condenser. Initially the voltage across this con- denser represents the magnitude of the PAM pulse being coded. After each digit I he voltage remaining on the condenser represents the magnitude of the f)riginal PAM j)ulse remaining to l)e coded. A pedestal voltage is TELEPHONY BY PULSE CODE MODULATION 407 obtained in the output of the comparing circuit whenever the storage con- denser voltage exceeds the reference step voltage. This pedestal, if present, allows a timing pulse to be sent out as a digit of the code group. This digit pulse is also delayed and fed back to a sub- traction circuit which reduces the charge on the condenser by a magnitude corresponding to the digit pulse just transmitted. This process is repeated step by step until the code is completed. Synchronizing the two control pulse generators, one at the transmitter and one at the receiver, is essential to the proper operation of the equipm.ent. This may be accomplished in a variety of ways. The best method of syn- chronizing to use would depend upon the application. Although the control could easily be obtained by transmitting a synchronizing pulse over the line, the equipment would have been somewhat more complicated and for these tests a separate channel was used to synchronize the control pulse generators at the terminals. Having thus established the timing of the receiving control pulse generator shown in Fig. 6 relative to the received code groups, the receiver generates a new set of waves as shown in Fig. 7. Except for delay in the transmission medium, the first three curves are the same as those shown in Fig. 5 for the transmitter. (1) is the delayed control pulse, (2) is the step timing wave, and (3) is the reference step voltage. Curve 8 is the received code group and (9) is the output current of the subtraction circuit. (10) gives the wave form of the voltage across the receiving storage circuit, and (11) gives the curve for the undelayed control pulse. The receiver functions as follows: The storage condenser is charged to a fixed voltage by each delayed control pulse. The charge on the condenser is reduced by the output of the subtraction circuit. The amount of charge that is subtracted depends upon which digit of the group produces the sub- traction pulse. This amount is measured by the reference step voltage. At the end of the code group the voltage remaining on the condenser is sampled by the undelayed control pulse. It is seen that the storage subtraction circuits in the transmitter and receiver function in similar ways. In the transmitter the original voltage on the condenser depends upon the audio signal, and after the coding process this voltage is substantially zero. The receiver starts with a fixed maximum voltage and after the decoding process the sample that is delivered to the output low-pass filter is given by the voltage reduction of the condenser during the decoding process. Except that the conditions at beginning and end of the coding and decoding periods are dififerent as discussed above, the subtraction process is the same for both units. The monitoring decoder in the transmitter operates in the same manner described above, except that it employs the various waves already generated for other uses in the transmitter (see Fig. 4). 408 BELL SYSTEM TECHNICAL JOURNAL Experimental Results An experimental system was set up as shown in Fig. 8. The pulse code modulator, radio transmitter, and antenna comprised the transmitting terminal; while an antenna, radio receiver and pulse code demodulator were used for the receiving terminal. A short air-path separated the terminals. The transmitter used a pulsed magnetron oscillator and the receiver em- ployed a broad-band superheterodyne circuit. The results obtained with this system were similar to those obtained by connecting the pulse code RADIO TRANSMITTER RADIO RECEIVER PULSE CODE MODULATOR PULSE CODE DEMODULATOR AUDIO INPUT A -O O— AUDIO OUTPUT Fig. 8— Block diagram of PCM system. modulator and demodulator together without the radio equipment. In fact, unless a large amount of attenuation was inserted in the path the presence of the radio circuit could not be detected. It was possible to adjust the PCM transmitter so that different numbers of digits could be produced. A brief study was made of the number of digits required. It was found that, with regulated volume, a minimum of three or four digits was necessary for good intelligibility for speech though, surprisingly enough, a degree of intelligibility was obtained with a single one. With six digits both speech and music were of good quality when regulated volume was used. Even with six digits, however, it was possible to detect the difference between PCM and direct transmission in A-B tests. This could be done most easily by a comparison of the noise in the two systems. If unregulated volume were used several more digits would proba- bly be desirable for high quality transmission. In listening to the speech transmitted over the PCM system one obtained the impression that the particular sound patterns of a syllable or a word TELEPHONY BY PULSE CODE MODULATION 409 could be transmitted with three or four digits. If the volume range of the talker varied it would be necessary to add more digits to allow for this variation. Over and above these effects, however, the background noise which is present to a greater or lesser extent in all communication circuits, is quantized by the PCM system. If the size of the quanta or amplitude step is too large the circuit will have a characteristic sound, which can easily be identified. Since the size of the quanta is determined by the number of digits, it is seen that the number of digits required depends not alone upon the speech but also upon the background noise present in the input signal. Summarizing, experimental results obtained indicate that at least 3 digits are desirable for a minimum grade of circuit and that as many as 6 or more will provide for a good quality circuit. If we wish to transmit a nominal speech band of 4000 cycles, PCM requires the 8000 pulses per second needed by any time-division system, multiplied by the number of digits transmitted. The extra bandwidth required for PCM however, buys some real advantages including freedom from noise, crosstalk and signal mutilation, and ability to extend the circuit through the use of the regenerative principle. The writer wishes to acknowledge the assistance of Mr. A. F. Dietrich in the construction and testing of the PCM equipment discussed in this paper. Some Results on Cylindrical Cavity Resonators By J. P. KINZER and I. G. WILSON Certain hitherto unpublished theoretical results on cylindrical cavity reson- ators are derived. These are: an approximation formula for the total number of resonances in a circular cylinder; conditions to yield the minimum volume cir- cular cylinder for an assigned (^; limitation of the frequency range of a tunable circular cylinder as set by ambiguity; resonant frequencies of'the elliptic cylinder; resonant frequencies and ^ of a coaxial resonator in its higher modes; and a brief discussion of fins in a circular cylinder. The essential results are condensed in a number of new tables and graphs. Introduction THE subject of wave guides and the closely allied cavity resonators was of considerable interest even prior to 1942, as shown in the bibliography. It is believed that this bibliography includes virtually everything published up to the end of 1942. During the war, many applications of cavity reso- nators were made. Among these was the use of a tunable circular cylinder cavity in the TE 01« mode as a radar test set; this has been treated in pre^ vious papers. ^'^ During this development, a num.ber of new theoretical results were obtained; some of these have been published.^ Here we give the derivation of these results together with a number of others not previ- ously disclosed. In the interests of brevity, an effort has been made to eliminate all material already published. For this reason, the topics are rather discon- nected, and it is also assumed that the reader has an adequate background in the subject, such as may be obtained from a study of references 3 to 7 of the bibliography, or a text such as Sarbacher and Edson.** A convenient reference and starting point is afforded by Fig. 1, taken from the Wilson, Schramm, Kinzer paper.- This figure also explains most of the notation used herein. Acknowledgement In this work, as in any cooperative scientific development, assistance and advice were received from many individuals and appropriate appreciation therefor is herewith extended. In some cases, explicit credit for special contributions has been given. Contents 1. Approximation formula for number of resonances in a circular cylin- drical cavity resonator. 2. Conditions for minimum volume for an assigned (). 410 NORMAL WAVELENGTHS "WWWf SAME AS TM MODES C- Vui = VELOCITY OF ELECTROMAGNETIC WAVES IN DIELECTRIC f = FREQUENCY SAME FORM AS FOR CYLINDER Tom HAS DIFFERENT VALUES Ic shape factors for recta ? 411 )\n mode xial reso- N A y are ob- (1) The dis- t, can be quency /o e the true es being a iped curve 1 approxi- mces of a Bolt^ and ? to apply e from the ionant fre- represent- to find the R = '^ C ators of re culai circu resor discu Th THI of It is beli up to th nators w cavity ir vious pa results v^ the dem ously dis In the material nected, a in the su of the bi A conv the Wils of the nc In this advice w( therefor i contributi 1. Appi drica 2. Cone SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 411 3. Limitation of frequency range of a tunable cavity in the TE Oln mode as set by ambiguity. 4. Resonant frequencies of an elliptic cylinder. 5. Resonant frequencies and Q of higher order modes of a coaxial reso- nator. 6. Fins in a circular cylinder. Approximation Formula for Number of Resonances in a Circular Cylinder From Fig. 1, the resonant frequencies of the cylindrical cavity are ob- tained from the equation: In which r is written in place of f/m , to simplify the equations. The dis- tribution of the resonant frequencies, starting with the lowest, can be approximated by a continuous function where N represents the total nunter of resonances up to a frequency /o or a wavelength Xo . This is bcur.d lo be en approxirraticn, since the true function F is discontinuous (or stepped) by virtue of the resonances being a series of discrete values. For practical purposes, if /*' fits the stepped curve so that the steps fluctuate above and below F, it will be a useful approxi- mation. Derivation of such a formula as applied to the acoustic resonances of a rectangular box has recently been a subject of investigation by Bolt^ and Maa.'" Only slight modifications of their method need be made to apply to the {^resent situation. MuUiply (1) thru by (- TTflA" 2 , /wan .7) -•■ +[-2L Hence, if a point ( r, — — J is plotted on the A'l' plane the distance from the origin to this point will be — - and hence a measure of the resonant fre- c quency. If all such points are plotted, they will form a lattice represent- ing all the possible modes of resonance. The problem, then, is to find the number of lattice ]X)ints in a quadrant of a circle with radius, R = — — . 412 BELL SYSTEM TECHNICAL JOURNAL The values of the Bessel zero, r, are not evenly spaced along the X axis; indeed the density, or number per unit distance, increases as r increases. Let the density be p{x). Then the problem becomes one of finding the weight of a quadrant of material whose density varies as p{x). Suppose the expression for M, the number of zeros r, less than some value X, is of the form M = Ax"^-]- Bx whence, by dififerentiation, p{x) = 2Ax-^B. The weight, IF, of the quadrant of a circle of radius R is then, by integra- tion, W =\aR^ + ^ BR^. 3 4 2L . . 2LW Since there are — lattice points per unit distance along the Y axis, ira iro. is apparently the total number of points in the quadrant. However, there are two small corrections to consider. First is that in this procedure a lattice point is represented by an area and for the points along the X axis Tra . . . . half the area, i.e., a strip — wide lying in the adjacent quadrant, has been omitted. Second is that the restriction w > for TE modes eliminates half the points along the X axis. As it happens, these corrections just cancel each other. Thus we have ^ - 3 xr 2 X^ in which 7 = ^^ S = ^aL Xo = ^ 4 /o From a tabulations^ of the first 180 values of r, the empirical values A = 0.262, B = Q were obtained. This gives V N = 4.39 -z . Ao Subsequently, from an analysis of over a thousand modes in a "square cylinder" (a = L), Dr. Alfredo Baiios, formerly of M.I.T. Radiation Lab- oratory, has calculated the empirical formula N = 4.38 -3 + 0.089 ;-2 (2) Aq Aq SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 413 from which A — 0.262,, B = 0.057. These values give better agreement with the 180 tabulated values of r. There is a two-fold degeneracy in a circular cylinder for modes with ■^ > 0, which is removed, for example, when the cylinder is made elliptical. The total number of modes, then, counting degeneracies twice, is about 2N, which brings (2) in line with the general result that, in any cavity resonator, Stt V the total number of modes is of the order -— r^ . 3 Ao Minimum Volume of Circular Cylinder for Assigned Q In practical applications of resonant cavities, the conditions of operation may require high values of Q which can be attained only by the use of high order modes. The total number of modes, most of which are undesired, can then be reduced only by making the cavity volume as small as possible, consistent with meeting the requirement on Q. It will be shown that, for a cylinder, operation in the TE 01m mode very probably gives the smallest volume for an assigned Q. Statement of Problem When the relative proportions (the shape) of a cavity and the mode of oscillation are fixed, both the Q and the volume, V, of the cavity are func- tions of the operating wavelength, X. Since we are primarily interested in the relationship between Q and V, with X fixed, some simplification can be made by eliminating X as a parameter. This may be done by a change of 8 V variables to () - and — , respectively; to simplify the typography, these A A quantities will be denoted by single symbols: We are, consequently, interested in the following specific problem: In a circular cylindrical resonator, which is the optimum mode family and what is the corresponding shape to obtain the smallest value of W for a preassigned value of P? A rigorous solution cannot be obtained by the methods of elementary calculus, since P is not a continuous function of the mode of oscillation. However, a possible procedure is to assume continuity, and examine the relation between P and W under this assumption. If sufficiently positive results are obtained, the conclusions may then be carried over to the dis- continuous (i.e., the physical) case with reasonable assurance that, except 414 BELL SYSTEM TECHXICAL JOURNAL perhaps for special \'alues, the correct answer is obtained. W'e proceed on this basis. Solntion To permit a more coherent presentation of the arguments, only their general outline follows. More mathematical details are given later. We start with the formulas for (^ - (= i^) as given in Fig. 1. A The lirst operation is to show that, under comparable conditions, i.e., X, r, n tixed, the TE Oniii modes give the highest values of P. That this is j)lausib!e can be seen in a general manner from the equations as they stand. For the TE modes, if ( — 0, the numerator of the fraction is largest. Also, P simplities, and the denominator roughly reduces the e.xpression in square brackets to the 1 2 power. Now compare this expression with those for the TM modes. That for the TM modes (// > 0) is smaller because of the factor (1 + R) in the denominator. Finally, that for the TAf modes (;/ = 0) is still smaller, because 1 < (1 + p-R-Y'-. This leaves only the TE Omii modes to be considered, and the next step is to show that ;;/ = 1 is the most favorable value. Since the relation be- tween P and ir is com{)licated, a j)arameter cp is introduced, with (p dehned by tan (^ = pR. (3) The resulting parametric equations are: r 1 P = ^ ^^— (4) ^TT .•? ,1.3 COS v? + - sm (f p pr^ 1 47r cos ip sm ip For each of the discrete values of r and n (;/ is related to p) then, plots of P vs W can be prepared as shown in Fig. 2 for the TE 01 » modes. Inspection of Fig. 2 shows that the best value of Q does not correspond to a minimum of W or a maximum of P for a given value of ;/, but rather to a point on the "envelope" of the curves. To get the envelope, we assume p to be continuous and proceed in the standard manner. It turns out that, by solving (4) f(^r p in terms of 7^ /- and v?, substituting the resulting e.x- (9 IF pression in TF, and setting --- = an equation is obtained which, when ^^p Sf)lvcd for <p, gi\'es the \'alucs of ^p which lie on the en\-clo]u\ SOME RESULTS OX CYLINDRICAL CAVITY RESONATORS 415 / 1 1 1 1 1 1 1 1 1 1 ( 1 / 1 1 1 r^ 10 1 1 no 1 c N \ 1 ( 1 1 1 1 1 1 J / / 1^1 \ ^ L 1 1 1 1 / / / ,'(\j ^^^^ 1*"--^ / _ ^ — \ ^< \y ^-'1 ij c V "\-. / / / / \- ■"r"" / 1 't V, / 1 1 / 1 / » / " 1 / '' 1 \_ ^ '' 1 I / /'/ / / 1 1 1 / / J 1 / V.- / O 416 BELL SYSTEM TECHNICAL JOURNAL We next substitute this expression for <p in W and calculate — assuming dr now that r is continuous, and find that W has no minimum. Practically, this means that the smallest value of r should be used, i.e., the TEOln mode. Finally, since from Fig. 2 it is seen that the envelope is reasonably smooth 8 for values of ^ - > 1, the expression for <p derived on the assumption of continuous p is used to obtain a simple relation of great utility in practical cavity design. Details of solution In (3), since R must be finite for a physical cylinder, < tan (p < oo , < sin v? < 1, and < cos v? < 1. Hence we may always divide by sin (p or cos <p. Note that (p ranges between 0° and 90°, From Fig. 1, 2d2\1/2 whence ^ ^ 2r(l + p'R') , . 2prR k sin (p = — — a (6) ^ cos ^ = — . (7) We define W by: a 3 ,3 X3 4R 87r3 ^^^ Substituting (6) and (7) in (8), pr^ 1 W = ^-2 —2 r— . (5') 47r cos cp sin <p ^ Substitution of (3) into the expression for Q- (= P) for the TE modes as A given in Fig. 1 yields, after some manipulation 2x 3 COS (p -\- - sin^ ^ + ( COS ^ — - sin ^ ) (^/r)^sin^ <p P \ P / SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 417 To show that any value of ^ > reduces P below its value when ^ = 0, let a = cos^ (p -{• - sin^ <p P b = { cos (f — - sin ^ 1 sin^ ip c = {l/r)\ It suffices to show that a a -\- he where the question is in doubt because h may take on negative values. If the inequality is to be valid, it is necessary only that (i + a) > 0, that is, cos «^ > 0. Hence, for the TE modes, only I — ^ needs be considered. For this case, the expression for P simplifies to r 1 P = For the TM modes, there is similarly obtained 27r 3 , 1 . 3 ' (4') cos ^ + - sm (^ P = P = r 1 2-K , 1 . w > (9) cos V? + - sm ip r cos (p 2-K , 1 . « = 0. 10) cos v? + ;r- sm <p Ip It is easy to show, since cos ^ < 1 and sin ^ < 1, that both (9) and (10) are less than (4'). Hence we have shown that, under comparable conditions, i.e., r and p constant, the TE Omn modes have higher values of P than any others. There is one flaw in the argument, viz., r takes on discrete values and cannot be made the same for all modes. It is conceivable, therefore, that for some specific values of P, a mode other than the TE Omn can be found which gives a smaller W than either of the two "adjacent" TE Omn modes, one having a value of r higher, the other lower, than the supposed high-P mode. This situation requires further refinement, and hence complication, in the analysis; we pass over this point. Having so far indicated that the TE Omn modes are the best, our next objective is find the best value of m, if possible. 418 BELL SYSTEM TECHNICAL JOURNAL By use of the parametric equatiuns (4) and (5), Fig. 2 has been ])lotted for r = ^.S^ (TE 01» modes) and values of n from 1 to 9. This drawing shows that, for each discrete value of r, minimum IT P is given by points on the "envelope" of the family of curves. The standard method of obtaining the envelope is to express If as a function of /' with )i as parameter (r is assumed fixed, for the moment), ■J 7,' i.e., ir = F(P, //),an(l then set — = 0. However, in this case it is easier dn to express IT = G(P, <p) and (p = H{ti), whence dF ^dG d^ dn dtp dn fir" fi /» and the envelope is obtained by setting r- = provided t- 5^ 0. We d<p on proceed, therefore, as follows. Assume p is continuous, and solve (4) for p, obtaining; sin^ tfi 2^ - cos ^ Now substitute (11) in (5). This gives TT' as a function of P and (p'. 3 47r- sm- (p cos ^ \ j-~p - cos^ ip (12) rJll To solve — = 0, we dilTerentiate and simplifv. This yields dip 5 cos (^ — 3 cos"* tp = — - . (13) irP Substituting (13) back into (11) yields 2 sin <p P = ^ 3 cos^ ip (14) The situation so far is that, with P and r assigned, W lies on the en- velope and is a minimum when v? satisfies (13); p is then given by (14). Obviously, for (13) to hold, it is necessary that 2-^<> '•'() obtain the best value of ;-, the ])rocedure is to differentiate ir„n„ with respect to r, assuming now that r is continuous, and examine for a mini- SOME RESULTS 0\ CYLINDRICAL CAVITY RESONATORS 419 mum. W'c can, however, first differentiate (12) by setting dW _ dW dW dip dr dr d(p dr dW and then substitute from (13). However, when (13) is satisfied, -— = 0. o<p This process yields dW ^ r (2 - 3 cos^ <p) dr IT- 9 sin- (p cos^ (p This shows -r- to be positive, when cosV < I • Hence -— = corresponds dr dr to a maximum, rather than a minimum.* If cos-(p < f, that is,^ > 35°16', then r should be as small as possible. The smallest r is 3.83, for the TE 01;/ modes. For r = 3.83, and (p > 35°, from (13) there is obtained P > 0.75. s The analysis thus indicates that, for values of P = ()- greater than 0.75, A the TE 01;/ mode yields the smallest ratio W/P or V/Q. An interesting and simple relation between /a and R for minimum W/P can easily be derived from the foregoing equations. Substitute (14) back into (6), thereby obtaining ■ *^^ (15) 3 a cos^ p Now use (7) with (15) to eliminate cos p, replace k by 27r/X, and r by 3.83, its numerical value for the TE 01;; modes. This gives ^] R = 2.23 or by substituting X = - , c = 3 X 10 , (fa)- R - 20.1 X 10-0. This useful relation was first discovered by W. A. Edson. Some further discussion is of interest. It is realized that a number of points have not been taken care of in a manner entirely satisfactory mathe- matically, but nevertheless important practical results have been obtained. As an example, since p and r can assume only discrete values, there are * It is for this reason that the determination of the stationary values of ]V{r, [>, f), subject to the constraint P(r, p, ^) = constant, by La Grange multipliers fails to yield the desired least value of W/P. 420 BELL SYSTEM TECHNICAL JOURNAL specific situations where some mode other than the TE Oln gives a smaller W/P. For example, it may be shown that for P between 0.97 and 1.14 the TE 021 mode yields a smaller W than the TE 013 or TE 014 modes. However, the margin is small, and for larger P, the TE 02n modes become progressively poorer. Limitation on Frequency Range of Tunable Cavity as Set BY Ambiguity In the design of a tunable cylindrical resonant cavity intended for use in the TE 0\n mode, the requirements on Q may dictate a diameter large enough to sustain TE 02n' or TE 03n' modes. Also, the range of variation of cavity length may be such that the TE 01 (w + 1) mode is supported. As the cavity is required to tune over a certain range of frequency, the maximum frequency range possible in the TE 01« mode without interference from the TE 01 (w + l)t or any TE 02 or TE 03 modes is of interest. The interference from the TE 0\(n-\- 1) limits the useful range of the TE 01« by the presence of extraneous responses at more than one dial setting for a given frequency or more than one frequency for a given dial setting. In applications so far made, it has been possible to eliminate extraneous responses from the TE 02 and TE 03 modes, but crossings of these modes with the main TE Oln mode have not been permitted. No designs have had diameters sufficiently large to support TE 04 modes. The desired relations are easily obtained by simple algebraic manipula- tion of equation (1). For simplicity in presentation of the results, we in- troduce some symbols applicable to this section only: A = r^T B = r^T = 2.247 X 10=^" Ao = value of A for TE 01« modes = 13.371 X 10 / = A/Ao :Vo = (a/Ly at low frequency end of useful range of TE 01m mode maximum/ frequency range ratio = minimum /" The values of A and / depend upon the interfering mode under considera- tion. For the TE Oln modes, A = 44.822 X lO'", / = 3.3522. The two typical cases of interest are shown on Fig. 3. For case I, am- t It is easy to show that the extra,neous respo^nse from the TE 01 (m — 1) mode is not limiting. The proof depends on the inequality n* > (« -f 1) (w — 1). SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 421 / / / // ^ f\i / / (0 / • ' ' '.*. ■ • ' \>'^- ' • . .*."•.'■ •^^. ■.*•■*•■ C*- / •■'■'v^;^>i^-\'\'''-'-''i-y^'':-':?r//.'- II 3) m = 4/ / W'M^:-0B0--i-M n 3/ W00MiiM& 2 -- 'mUKti. — TE 02 n=4/ / 3 / ^- / u I i-^'^^'^ TE 01 £ ^ - — <_ — -""^ Xo Xo Fig. 3 — Mode chart illustrating types of interference with TE 01« mode, biguity from TE 01 (w + 1) mode, it is found that Curves of F for this case are shown on Fig. 4. The maximum value of F is obtained when Xo = oo and is i^ max — n + 1 422 BELL SYSTEM TECHNICAL JOURNAL ..'' / / TEOII^ / / / / / • 012^ 4 / / ^ ^ ■ 01^3 — ^ /, ^- — ^ ■ /f f / / / / / / / A / ^■ y 0.2 0.4 0.6 0.8 I.O t.2 1.4 1.6 1.8 2.0 2.2 2.4 -^ (minimum) Fig. -1 — Curves showing maximum value of frequency ratio without interference from TE 01 (?z + 1) mode (case I of Fig. 3). Table I. — Cast II: Maximum Frequency Range Ratio, t\ for TE Uln Mode wlien Limited by Mode Crossings with TE 02m and TE 02{m+I) Modes. n = 3 M =4 n = 12 F (''/■f-)min F ("/-^'min F («/i'min 1 1.198 1.323 1.086 0.966 1.008 0.313 2 1.242 1.080 1.013 0.316 3 1.019 0.322 4 1.027 0.331 5 1.037 0.343 6 1.051 0.360 7 1.071 0.384 8 1.104 0.418 9 1.168 0.471 10 1.345 0.564 SOME RESULTS OX CYLIXDRICAL CAVITY RESONATORS 423 For case II, range limited by mode crossings, it is found that A - .4o •To = F' = Bin'- - w'2) or - ■»/-)[»-/ - {n' + 1)'] Some values for this case are given in Table I. The formulas above are general and may be used for any pair of mode types by using the appropriate values for A and /. The Elliptic Cylinder In the design of high Q circular cylinder cavity resonators operating in the TE 01;/ mode, it is desirable to know how much ellipticity is tolerable, so that suitable manufacturing limits may be set. The elliptical wave guide has already been studied, notably by Brillouin^- and Chu,^^ but the results are not in suitable form or of adequate precision for the present purposes. More recently tables" have become available which permit the calculation of some of the properties of the elliptical cylindrical resonator. The elliptical cavity involves Mathieu functions, which are considerably more complicated than l^essel functions. ^^ The tables give the numerical coefficients of series expansions, in terms of sines, cosines, and Bessel func- tions, of the Mathieu functions up to the fourth order. These tables have been used for the calculation of some quantities of interest in connection with elliptical deformations of a circular cylinder in the TE 01« mode. The Ellipse All mathematical treatments of the ellipse (including the tables men- tioned above) use the eccentricity, e, as the quantity describing the amount of departure from the circular form. The eccentricity -is the ratio distance between foci e = . -. . major axis This is not a quantity subject to direct measurement, hence we here in- troduce and use throughout the ellipticity, E, defined as _ difference between major and minor diameters major diameter It is clear that the ellipticity is easily obtained directly. Again, many results are given in terms of the major diameter. Since we are interested in deform.ations from circular, and in such deformations the 424 BELL SYSTEM TECHNICAL JOURNAL perimeter remains constant, while the major diameter changes, we have expressed our results in terms of an average diameter, defined as _ perimeter Figure 5 shows the elHpse and various relations of interest. Y P=PERIMETER e=ECCENTRICITY = _ Co a E=ELLIPTICITY=^^^' . A= AREA = TTab D = "average" DIAMETER = £: TT b=aYi-e2 = a(i-E) A=Tra2'Yi-e2=Tra2 o-e) Fig. 5 — The ellipse Elliptic Coordinates and Functions The elliptic coordinate system is shown on Fig. 6. Following Stratton,'^ we have used ^ in place of the table's z, since we wish to use z as the coor- dinate along the longitudinal axis. Stratton also uses tj = cos if as the angu- lar coordinate; this is frequently convenient. Analogous to cos (6 and sin (d in the circular case, there are even and odd* angular functions, denoted by ^Sf{c, cos <f>) and °Sf{c, cos ^) which reduce to cos Id and sin Id respectively when c are even and odd* radial functions, denoted by 'Jf^c, k) and "Jfic, * For ^ = 0, only even functions exist. 0. Similarly, there SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 425 which both reduce to Jf(kip) when c -^ 0. In the above, c is a parameter related to the elHpticity.* The tables do not give values of the functions, but rather give numerical coefficients Di and Fi of expansions in series of cosine, sine and Bessel functions, which permit one to calculate the elliptic cylinder functions. The coefficients, of course, Fig. 6 — Elliptic coordinate system depend on the parameter c; the largest value of c in the tables is 4.5, which corresponds to an ellipticity of 39% in a cylinder operating in the TE 01// mode.** For this case, Bessel functions up to Jn(x) and Juix) are needed for calculating the radial function. It is clear that calculations on elliptic cylinders have not been put on a simple basis. * Not to be confused with c = velocity of electromagnetic waves; the symbol c is here carried over from the published tables. ** An ellipticity of 39% means that the difference between maximum and minimum diameters is 39% of the maximum diameter. For a given c, the ellipticity depends oii the mod^. 426 BELL SYSTEM TECHXICAL JOURNAL Field lujiialions The equations for the fields arc easily obtained from section 6.12 of Stratton's book, and are given in Table II, which is self-explanatory, except for the quantity c, which we now proceed to discuss. Resonaiil Frequencies The ellij)tic c\linder has the major diameter, 2a, and the focal distance; 2c[) . The equation of its surface is then cx{)ressed bv ^ = ^ — a. On this surface, £, must vanish. This requires that '"J f{c, a) ~ for TE modes and that '"J/ic. a) = for TM modes. The series expansions are in terms of c^ as variable. Let ca ~ rf,n or r^,,, be the roots of the above ^ r . equations. Then — = - (dropi)ing the subscripts f, m). Xow, in working out the solution of the differential equations, it turned out that c — Coki. , f Here ^i is one component of the wave number, kj. Hence ^i = - . Further- a more, the eccentricitv is e = — = - . The indicated procedure is: 1) choose a r a value of c; 2) laid the various values of r for which the radial function or its derivative is zero; 3) then calculate the corresponding eccentricity and resonant frequency. Notice that for a given value of c, the values of r will depend on the mode, and hence so will the eccentricity. We now wish to express our results in terms of the ellipticity and the average diameter. To convert eccentricity to ellipticity, we use £ = 1 - Vf ^^-• The perimeter of the ellipse is given by P = ■iaE(e) where E(e) is the com- plete elliptic integral of the second kind.tt In terms of the average diameter we find *-l 2r£(e) "[ 2s or calling the C[uantity in brackets s, A'l = -— . This is now in the same form as ki for a circular cylinder of diameter D. The quantity 5 is the recipro- cal of Chu's ■^. t It is recalled that 2ir / , r tiTT ^ = _ =, V)fe2 + k^ ; ki = - ; k, =— , X 1 ' a L tt This is tabulated as E(a) in Jahnke & Emde, p. 85, with a = sin-^e. SOME RESULTS OX CVLIXDKICAL CAVITY RESOXATORS 427 We liave calculated and give in Table III values of r, e, E and s for several values of c and for a few modes of special interest. For three cases, "TE 01, "TM 11 and "TM 11, we have determined an empirical formula to fit the calculated values of ^. These are also given in Table III. TE Modes TABLE II. Elliptic Cylinder Fields Et = —k i/ ^ S((c, r])J((c, sin k-.iZ cos cot r •Y/t2 _ \ Er, = k A/- S(,{c, ri)j'({c, t) sin k:i z cos ut y e 1 \/>^ - 1 ^j = ^3 >5'^(c, t])] \{c, f) cos k>, z sin wt H.q = kz S({c, ri)J((c, ^) COS kiZ sin wt q 11 z = klSfic, Ti)J(,{c, t) sin hz sin ut TM Modes \/^2 — 1 E^ = —kz Siic, ri)J({c, sin k^z cos ut Q ■\/ 1 ~2 ■Et, = —^3 S'((c, r))J({c, sin ^3 3 cos wt 1 Ez = k'l S((c, 7))J ({c, l) cos hz cos (Jit H^ = —k 4 / - S'((c, ri)Jp{c, i:) cos ^3 z sin coi /-y/t2 _ J - "S^Cc, j/jZ/Cc, $) cos h z sin wi Notes: Derivatives are with respect to ^ and 77. Sf and // carry prefixed superscripts, e or 0, since they may be either even or odd. q = Co Vl^ — rf' c = coki Kl = «3 = 7" «- = ^1 + «j a L 2co is distance between foci of ellipse. a is the semi major diameter of the ellipse, r^ „, is the value of c$ that makes J l{c,^) — for ^-^ modes J'^ifyO = for TE modes. 428 BELL SYSTEM TECIIMCAL JOiRXAL TAULK 111 Rout Valiks ok Kauial Elliptic Cylinder Functions Mode c r e E i TEOl 3.8317 3.8317 0.2 3.8343 0.05216 0.001361 3.8317 0.4 3.8423 0.10410 0.005434 3.8318 0.6 3.8558 0.15561 0.012181 3.8324 0.8 3.8753 0.20643 0.021539 3.8337 1.0 3.9015 0.25631 0.033406 3.8366 1.2 3.9349 0.30496 0.047636 3.8417 1.4 3.9763 0.35209 0.064033 3.8500 1.6 4.0264 0.39738 0.082346 3.8624 2.0 4.154 0.4814 0.12351 3.902 3.0 4.634 0.6474 0.2378 4.101 4.0 5.29 0.756 0.346 4.42 4.5 5.66 0.795 0.393 4.62 5 = 3.8317 + 4.33 E^ + \.9E^ ^TM 11 3.8317 3.8317 0.2 3.8330 0.05218 0.001362 3.8304 0.4 3.8370 0.10425 0.005449 3.8265 0.6 3.8436 0.15610 0.012259 3.8201 0.8 3.8532 0.20762 0.021791 3.8113 1.0 3.8658 0.25868 0.034036 3.8003 1.2 3.8818 0.30913 0.048981 3.7874 1.4 3.9015 0.35884 0.066599 3.7727 1.6 3.9253 0.40761 0.086844 . 3.7568 4.5 5.13 0.878 0.520 3.91 3.8317 - 0.96£ + 1.1 /^^ ^TM 11 0.2 0.4 0.6 0.8 1.0 1.2 1.4 3.8317 3.8356 3.8474 3.8670 3.8944 3.9298 3.9731 4.0243 0.05214 0.10397 0.15516 0.20542 0.25446 0.30203 0.34788 0.001361 0.005419 0.012111 0.021326 0.032918 0.046701 0.062462 3.8317 + 0.95E + 2.2E^ 3.8317 3.8330 3.8370 3.8436 3.8530 3.8654 3.8809 3.8997 'TE 22 6.706 6.706 0.4 6.712 0.0596 0.00178 6.706 0.8 6.729 0.1189 0.00709 6.705 1.2 6.756 0.1776 0.01590 6.702 1.6 6.788 0.2357 0.02817 6.693 2.0 6.826 0.2930 0.04389 6.677 "TE 22 6.706 6.706 0.4 6.712 0.0596 0.00178 6.706 0.8 6.730 0.1189 0.00709 6.706 1.2 6.762 0.1775 0.01587 6.708 1.6 6.810 0.2350 0.02799 6.715 2.0 6.877 0.2908 0.04323 6.729 SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 429 Mode c r e E s •r£32 8.015 8.015 0.4 8.020 0.0499 0.00124 8.015 0.8 8.035 0.0996 0.00497 8.015 1.2 8.059 0.1489 0.01115 8.014 1.6 8.093 0.1977 0.01974 8.013 2.0 8.135 0.2459 0.03070 8.010 "TEH 8.015 8.015 0.4 8.020 0.0499 0.00124 8.015 0.8 8.035 0.0996 0.00497 8.015 1.2 8.060 0.1489 0.01115 8.015 1.6 8.097 0.1976 0.01972 8.018 2.0 8.146 0.2455 0.03061 8.022 'TMQ\ 2.4048 0.2 2.4090 0.08302 0.4 2.4216 0.16518 0.6 2.4431 0.24559 0.8 2.4739 0.32337 1.0 2.5149 0.39762 'TEn 1.8412 0.2 1.8416 0.10860 0.4 1.8430 0.21704 0.6 1.8452 0.32516 0.8 1.8484 0.43280 1.0 1.8527 0.53975 Notes: Superscripts e and o on mode designation signify even and odd. c is parameter used in the Tables (Stratton, Morse, Chu, Hutner, "Elliptic Cylinder and Spheroidal Wave Functions") r is the value of the argument which, for TM modes, makes the radial function zero and, for TE modes, makes its derivative zero. e is the eccentricity of the ellipse; _ distance between foc i major diameter E is the ellipticity of the ellipse; difference between major and minor diam. major diameter 5 is the root value, referred to the "average diameter"; it is related to r by: _ r perimeter IT major diameter The quantity 5 is also related to the cutoff wavelength in an elliptical wave guide according to: _ perimeter of guide cutoff wavelength Resonator Q Although the calculation of the root values is straightforward and not overly laborious, the same cannot be said for the integrations involved in the determination of resonator Q. The procedure is obvious: The field 430 BELL SYSTEM TECHNICAL JOURNAL equations are given; it is only necessary to integrate H^dr over the volume and IPda over the surface and get Q from 2 / ^'^' Q = I (16) j H^da with 5 = skin depth, a known constant. Unfortunately the integrations cannot at present be expressed in closed form. A numerical solution can be obtained by a combination of integration in series and of numerical integration. The calculations have been made for the ^TE 01 mode with c — 2.0, for which r = 4.154. This value of c corresponds in this case to an ellipticity of about 12%; in a 4" cylinder this would amount to 1/2" difference between largest and smallest diameters. Evaluation* of the integrals yields: H-dr = 12.307 k^L + 12.294 klL v H"d<7 = 49.228 k^ + 0.1619 kiHL + 6.6847 kiL s Substituting k] — and kg = — ^ o^^^ obtains, finally Q8 = 0.471 D 1 + 0.1622 nR" ,1 + 0.0039 «2i?2 ^ 0.1529 n'-R^ For a circular cyhnder, '1 + 0.1681 nR" Qc8 = 0.5 D 1 + 0.1681 n'-R Comparison of these two formulas for Qd shows that the losses in the end plates {n-R term) are less with respect to the side wall losses in the ellip- tical cylinder. The net loss in Q8, as described by the reduction in the mul- tiplier from 0.5 to 0.471, is thus presumably ascribable to an increase in side wall losses (stored energy assunied held constant). The additional term in n^R in the denominator is responsible for the difference in the attenuation- frequency behavior of elliptical vs circular wave guide as shown by Chu, Fig. 4. Incidentally, these results agree numerically with those of Chu. * Numerical integration was by Weddle's rule; intervals of 5° in ^ and 0.1 in x were used. The calculations were made bj^ Miss F. C. Larkej'. SOME RESULTS Oi\ CYLINDRICAL CAVITY RESONATORS 431 Corresponding expressions for the resonant wavelength are ttD 0.805 D X = - a/\ + hnD\ ^1 + 0-1622 «2i22 \2sL/ 0.820 D Vl +0.1681 w2/?2- As an example, take n = 1, R = 1, then (Circular) Qc5 = 0.500 D X^ = 0.759 D (Elliptical) Q8 = 0.473 D X = 0.747 D Ratio = 0.946 Ratio - 0.984. Conclusions The mathematics of the elliptic cylinder have not yet been developed to the point where the design of cavities of large ellipticity could be undertaken. On the other hand, sufficient results have been obtained to indicate that the ellipticity in a cavity intended to be circular, resulting from any reasonable manufacturing deviations, would not have a noticeable effect on the reso- nant frequencies or Q values, at least away from mode crossings. Full Cylindrical Coaxial Resonator The full coaxial resonator has been of some interest because of various suggestions for the use of a central rod for moving the tuning piston in a TE OUi cavity. The cylindrical coaxial resonator, with the central conductor extending the full length of the resonator, has modes similar to the cylinder. In fact, the cylinder may be considered as a special case of the coaxial. The indices /, m, n have much the same meaning and the resonant frequencies are determined by the same equation (1). However, now the value of r depends in addition (see Fig. 1) upon 77, where _ diameter inner conductor _ ^ diameter outer conductor a ' The problem now arises of how best to represent the relations between /, a, b and L. The r's depend on tj; so one possibility is to determine their values for a given 77 and then construct a series of mode charts, one for each value of 77. A more flexible arrangement is to plot the values of r vs 77 and allow the user to construct graphs suitable for the particular purpose in hand. An equivalent scheme has been used by Borgnis.^^ It turns out that as 77 — ^ 1, r(l — 77) —> ftiir, for the TM modes and the 432 BELL SYSTEM TECHNICAL JOURNAL TE Omn modes, and r{\ — rj) -^ {m — l)x for all other TE modes. For the former modes, r becomes very large as r; — > 1, that is, as the inner con- ductor fills the cavity more and more, the frequency gets higher and higher. For the TE (In modes, however, as the inner conductor grows, the f re- queue}' falls to a limiting value. This is discussed in more detail by Borgnis.^^ Figure 7 shows r(l — 77) vs 77, for a few of the lower modes; the scale for 77 between 0.5 and 1.0 is collapsed since this region does not appear to be of great engineering interest. A different procedure is used for the roots of the TE (hi modes. Figure 8 is a direct plot of r vs 77 for a few of the lower modes. In this case, r -^ f as 77 -h^ 1. Distribuiion of Normal Modes The calculation of the distribution of the resonant modes for the coaxial case follows along the lines of that for the cyhnder, as given previously. The difference lies in the distribution of the roots r, which now depend upon the parameter r,. The determination of this latter distribution offers difficulties. There is some evidence, however, that the normal modes will follow, at least to a first approximation, the same law as the cylinder, viz.: V N = 4.4 ^ Ao with some doubt regarding the value of the coefficient. - in Coaxial Resonator X The integrations needed to obtain this factor are relatively straightfor- ward, but a little complicated. The final results are given in Fig. 1. The defining equation is (16); the components of H are given in Fig. 1. The integrations can be done with the aid of integrals given by McLachlan^^ and the following indefinite integral : which can be verified by differentiation, remembering that y = Zi{x) is a solution of y" + - y' -f ( 1 - - ) y = 0. X \ x-J 7.0 \ 1- ^. \ ^. — : TM 12 _ 6.2 TM02_ 6.0 ^ ( 5.4 5.2 ^; \ \ \\ A 4.4 V \ \ 4.2 \ v\l \ \, 4.0 3.8 \ \ \ \ V \ TE12^ \, \ TM 21 3.6 ^ s ^ ^^ ^ 3.4 ^ TE 01 ..^ TM11 -^ ;:C K 3.2 — "■ - . J^^ ^ . — - — -^^^^ 3.0 2.8 2.6 TM 01 / ( 2.4 -\ — 1 — 1 — 1 — 1 — 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1.0 INNER CONDUCTOR DIAMETER I ~ OUTER CONDUCTOR DIAMETER Fig. 7 — Full coaxial resonator root values r^^ (1 — »?) 433 TE 41 — l~l- --I — 1- 5.2 5.0 4 6 " ^^ \ \ 4.6 4.4 4.2 4.0 3.8 3.6 e £^'3.4 3 3.2 \ \ TE3I \ "~-~ ■\ ^v^ \ V \ \ \ \ TE 21 \ O 3.0 O Ct 2.8 2.6 2.4 2.2 2.0 1.8 1.6 -^ "*-^^ ^^ ^^ V \ \ V \ ■ ■~~- --^ TEll ■-- ..^ "~~~ \ 1.0 1 — 1 — 1- ii. 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 INNER CONDUCTOR DIAMETER n~ OUTER CONDUCTOR DIAMETER Fig. 8 — Full coaxial resonator root values r. 434 SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 435 An investigation needs to be made of the behavior of the formulas as 77 — > before any conclusion may be drawn regarding their blending into those for the cylinder. For TE modes with ^ = 0, the term involving jj — disappears, hence no question arises. Consider then / > 0, and let X = Tjr for the discussion following. From expansions given in McLachlan, it is easy to show that, for small x J({x) = -<^)-^';""©' T \X/ X X^ Since, from Fig. 1, A = J'({r) _ Jiiv) _ Ji(x) 2i{( - 1) ! y'lir) y'ti-nr) Y({x) it is found, upon substitution of the approximations given above: That is, Zt{x) '~ x^ and hence — > as x ^ 0. Furthermore Zt{r) remains finite as t? -^ 0. Hence H -^ 0^^ and — '^ x^~^. Therefore, for / > 0, n — — > as 77 — > 0. Hence, the expression for Q - for the coaxial structure reduces to that for the cylinder, for any value of (, in the TE modes. For the TM modes, and for ^ > 0, an entirely similar argument shows that H' remains hnite as 7? — > 0. Hence, the expression for Q - for these A modes also reduces to that for the cylinder. For the TM modes, and with / = 0, we have Zo(x) = -7i(.r) + 7o(-t) F,(x-) For X — )■ 0, /i(.v) — > and Jq{x) -^ 1, hence for small x, yo{x) 436 BELL SYSTEM TECHNICAL JOURNAL Now substitute the approximate values of the I' for small x. The result is Since Zo(r) is tinite, it follows that •qH' ' — ' 1 a; log ('-3^ and it is easily shown that r)II' — > <» as r; -^ 0. On the other hand, rfH' -^ as ?7 -^ 0. Hence, () - — > as 77 — ^ 0. On the other hand, for tj = 0, a A 0.50 0.45 q: a. 0.40 0.30 Q 0.25 z o '-' 0.20 0.15 ^ 0.10 0.05 / r ^ A ^ y Q 4- =0.30 / X y X y y' y aaj. y ^^ y^ 0.40 ^ s ^'^ ^ \ \, ^ c 3.45 . ..^ \ ■ ' 0.50 ^ \ , ■ - ^ --^ ^ \ \ [max. 0.656 2 ^DU^ 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 _a_ L R=-r- Fig. 9 — Coaxial resonator. TE Oil mode Contour lines of ()- A perfect cylinder exists whose (J - is not zero. It is concluded that the ex- X pression for Q - does not apply for small 7/ for the TM modes with /" = 0. A s Thus it is seen that the expressions for the factor (() -) reduce to those A given for the cylinder, when t; = 0, except for TM modes with /* = 0. For these latter cases, the factor approaches zero as 7/ approaches zero, because 77//' increases without limit. This means that an assumption SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 437 liJ 0.40 1- 1- LiJ III < 5 < 0.35 Q Q O a n 0.30 1- 1- U o a n 0.25 z 7 o o o o 0.20 or QC Ul Ol 7 1- 0.15 o d 0.10 0.05 OJg, ^ "^ ^ ^ ^ ^ ' 0.14 — ' ' ' ^^ -^ x-'^ of = 0.16 -^ 18 ^ ^ - — , 1 0.20 ^ ~ 0.22 ^ / ^ J3.24 ■ / / ^ ^" 0.2 76 ■ ■— ■ 0.26 ^ 0.2 0.4 6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.S ^ L Fig. 10 — Coaxial resonator. TE 111 mode Contour lines oiQ_- 3 0.25 Q Z o O 0.20 cr 0.10 f' — n A. O / / 1 / / y ^ /d / / / / / 1 / 7 7 % / / / / / / / / Y 4 // / ^ J"' / /, / / / ^ ■^ ^ // / / y / ^^ >-^ r/ y / ^ ^ ^-^ 2i^ -- 0.16 ________ 1 1 ^ "^^^^i ::::^ ^ --- •:^ — 0.2 0.4 0.6 0.6 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Fig. 11 — Coaxial resonator. TM Oil mode Contour lines of Q, 438 BELL SYSTEM TECHNICAL JOURNAL which was made in the derivation of the Q values is not valid for small tj; that is, the fields for the dissipative case are not the same as those derived on the basis of perfectly conducting walls. The expressions for the factor are rather complicated, as it depends on several parameters. When a given mode is chosen, the number of param- eters reduces to two, 77 and R. Contour diagrams of () - vs 77 and R are A given on Figs. 9, 10, 11 and 12 for the TE Oil, TE 111, TMOll and TM 111 Fig. 12— Coaxial resonator. TM HI mode Contour lines of Qj- modes. As mentioned above, the true behavior of () - for the TM Oil mode for small rj is not given by the above formula, so this contour diagram has been left incomplete. Fins in a Cavity Resonator The suppression of extraneous modes is always an important problem in cavity design. Among the many ideas advanced along these lines is the use of structures internal to the cavity. It is well known that if a thin metallic fin or septum is introduced into a cavity resonator in a manner such that it is everywhere perpendicular to the £-lmes of one of the normal modes, then the field configuration and SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 439 frequency of that particular mode are undisturbed. For example, Fig. 13 shows the £-lines in a TE llw mode in a circular cylinder. If the upper half of the cylinder wall is replaced by a new surface, shown dotted, the field and frequency in the resulting flattened cylinder will be the same as NEW SURFACE PERPENDICULAR TO E- LINES LEAVES REST OF FIELD AND FREQUENCY UNALTERED Fig. 13 — E Lines in TE lire mode ^ORIGINAL CYLINDER t'- Fig. 14— "TE 01m" mode in half-cylinder before. Indeed, they will also be the same in the crescent-shaped resonator indicated in the figure. Except for isolated cases, all the other modes of the original cylinder will be perturbed in frequency since the old fields fail to satisfy the boundary conditions over the new surface. Furthermore, if the original cylinder was 440 BELL SYSTEM TECHNICAL JOURNAL circular, its inherent double degeneracy will be lost and each of the original modes (with minor exceptions) will split into two. Although the frequency and fields of the undisturbed mode are the same, the Q is not necessarily so. For example, Fig. 14 shows a ""TE 01« mode" in a half cylinder.* It is easy to calculate Q - for this case. The result is (1 + p'R'f" in which Ki = 1.290 A'2 = 0.653 (17) Here A'l and K2 are constants which account for the resistance losses in the flat side. For the full cavity, shown dotted in Fig. 14, eq. (17) holds with A'l = A'2 = 0. If the circular cavity has a partition extending from the center to the rim along the full length, (17) holds with the values of A'l and A'2 halved. If a tin projects from the rim partway into the in- terior, still other values of A'l and A'2 are required. It is a simple matter to compute these for various immersions; Fig. 15 shows curves of A'l and K2 . The following table gives an idea of the magnitudes involved: mode: r£ 0,1,12 R = 0.4 8 Fin, % a ^4 Ratio 0% 2.573 1.0 10 2.536 .985 20 2.479 .965 50 2.04 .79 100 1.47 .57 The question now is asked, "Suppose a longitudinal fin were used, small enough to cause only a tolerable reduction in the Q. Would such a fin ameliorate the design difficulties due to extraneous modes?" Some of the effects seem predictable. All modes with ^ > will be split to some extent, into two modes of different frequencies. Consider the TE I2n mode, for example. There will be one mode, of the same frequency as the original whose orientation must be such that its £-lines are perpendicu- lar to the fin. The Q of this mode would be essentially unchanged. There will be a second mode, oriented generally 90° from the first, whose £-lines will be badly distorted (and the frequency thereby lowered) in the vicinity * Solutions for a cylinder of this cross-section are known and all the resonant fre- quencies and Q values could be computed, if they had any application. SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 441 of the fin. It would be reasonable to expect the Q of this mode to be appre- ciably lowered because of the concentrated field there. If two fins at 90° were present, there would be no orientation of the original TE \2n mode which would satisfy the boundary conditions. In this case both new modes 0,0.40 (M 0.35 UJ / / / / / / .^ / /" / 7 ' K2/ / / / f / / / / / f / / / / / / / / / / ^ ^,. ' fin O 0.20 $ 0.15 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 WIDTH OF FIN RADIUS OF CYLINDER Fig. 15 — Constants for calculation of Q of TE Oln mode in cylinder with longitudinal would be perturbed in frequency from the original value. If both fins were identical, the perturbations would be equal and a double degeneracy ensue. Similar effects would happen to the other types of modes. The major advantage derivable from such effects would appear to be in extraneous transmissions. The fin serves to orient positively the fields in 4^2 BELL SYSTEM TECHNICAL JOURNAL the cavity, and the input and output couphng locations can then be appro- priately chosen. On the basis that internal couplings are responsible for mode crossing difficulties, one might hazard a guess that a real fin would increase such couplings. Another application of fins might be in a wave guide feed in which it is desired to establish only a TE Oni wave. In this case, Q is not so important and larger fins can be used. If these extended virtually to the center and x of them were present (with uniform angular spacing) all types of wave trans- mission having / less than x/2, x even or / less than x, x odd, would be sup- pressed. This use of fins is an extension of the wires that have been proposed in the past. Conclusion It is hoped that the foregoing, which covers some of the theoretical work done by the author during the war, will be of value to other workers in cavity resonators. There is much that needs to be done and hardly time for duplication of effort. Bibliography 1. E. I. Green, H. J. Fisher, J. G. Ferguson, "Techniques and Facilities for Radar Test- ing." B.S.T.J., 25, pp. 435-482 (1946). 2. I. G. Wilson, C. W. Schramm, J. P. Kinzer/'High Q Resonant Cavities for Micro- wave Testing" B.S.T.J., 25, pp. 408-434 (1946). 3. J. R. Carson, S. P. Mead, S. A. Schelkunoff, "Hyper-Frequency Wave Guides — Mathematical Theory," B.S.T.J., 15, pp. 310-333 (1936). 4. G. C. Southworth, " Hyperf requency Wave Guides — General Considerations and Experimental Results," B.S.T.J., 15, pp. 284-309 (1936). 5. W. W. Hansen "A Type of Electrical Resonator," Jour. A pp. Phys., 9, pp. 654-663 (1938). — A good general treatment of cavity resonators. Also deals briefly with coupling loops. 6. W. W. Hansen and R. D. Richtmyer, "On Resonators Suitable for Klystron Oscil- lators," Jour. A pp. Phys., 10, pp. 189-199 (1939). — Develops mathematical methods for the treatment of certain shapes with axial symmetry, notably the "dimpled sphere," or hour glass. 7. W. L. Barrow and W. W. Mieher, "Natural Oscillations of Electrical Cavity Reso- nators," Proc. I.R.E., 28, pp. 184-191 (1940). An experimental investigation of the resonant frequencies of cyhndrical, coaxial and partial coaxial (hybrid) cavities. 8. R. Sarbacher and W. Edson, "Hyper and Ultrahigh Frequency Engineering," John Wiley and Sons, (1943). 9. R. H. Bolt, "Frequency Distribution of Eigentones in a Three-Dimensional Con- tinuum," J.A.S.A., 10, pp. 228-234 (1939) — Derivation of better approximation formula than the asymptotic one; comparison with calculated exact values. 10. Dah-You Maa, "Distribution of Eigentones in a Rectangular Chamber at Low-Fre- quency Range," J.A.S.A., 10, pp. 235-238 (1939)— Another method of deriving an a^jproximation formula. 11. I. G. Wilson, C. W. Schramm, J. P. Kinzer, "High Q Resonant Cavities for Micro- wave Testing," B.S.T.J., 25, page 418, Table IK (1946). 12. L. Brillouin, "Theoretical Study of Dielectric Cables," Elec. Comm., 16, pp. 350- 372 (1938)— Solution for elliptical wave guides. 13. L. J. Chu, "Electromagnetic Waves in EUiptic Hollow Pipes of Metal," Jour. App. Phys., 9, pp. 583-591 (1938). 14. Stratton, Morse, Chu, Hutner, "Elliptic Cvlinder and Spheroidal Wave Functions," M.I.T. (1941). SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 443 15. J. A. Stratton, "Electromagnetic Theory," McGraw-Hill, (1941). 16. F. Jahnke and E. Emde, "Tables of Functions," pp. 288-293, Dover Publications (1943). 17. N. VV. McLachlan, " Bessel Function for Engineers," Clarendon Press, Oxford (1934). 18. F. Borgnis, "Die konzentrische Leitung als Resonator," Hochf: tech u. Elek:akus., 56, pp. 47-54, (1940). — Resonant modes and Q of the full coaxial resonator. For long abstract, see Wireless Engineer, 18, pp. 23-25, (1941). Additional Bibliography 19. J. J. Thomson, "Notes on Recent Researches in Electricity and Magnetism," Oxford, Clarendon Press, 1893, — §300 gives the resonant frequencies of the TE modes in a cylinder with a/L = 0; §315-316 consider two concentric spheres; §317-318 treat of the Q of the spherical cavity. 20. Lord Rayleigh, "On the passage of electric waves through tubes or the vibrations of dielectric cylinders" Phil. Mag.; 43, pp. 125-132 (1897) .^Considers rectangular and circular cross-sections. 21. A. Becker, " Interf erenzrohren fiir elektrische Wellen," Ann. d. Phys., 8, pp. 22-62 (1902)— Abstract in Set. Abs., 5, No. 1876 (1902)— Experimental work at 5 cm. and 10 cm. 22. R. H. Weber, " Elektromagnetische Schwingungen in Metallrohren," Ann. d. Phys., 8, pp. 721-751 (1902)— Abstract in Set. Abs., 6A, No. 96 (1903). 23. A Kalahne, "Elektrische Schwingungen in ringformigen Metallrohren," Ann. d. Phys., 18, pp. 92-127 (1905).— Abstract in Sci. Abs., 8A, No. 2247 (1905). 24. G. Mie, "Beitrage zur Optik triiber Medien, spezieU kolloidaler Metallosungen," Ann. d. Phys., 25, pp. 337-445 (1908) — A part of this article deals with the solution of the equations for the sphere; also shown are the E and H lines for the lowest eight resonant modes. 25. H. W. Droste, " Ultrahochfrequenz-Ubertragung langs zylindrischen Leitern und Nichtleitern," TFT, 27, pp. 199-205, 273-279, 310-316, 337-341 (1931)— Abstract in Wireless Engr., 15, p. 617, No. 4209 (1938). 26. W. L. Barrow, "Transmission of Electromagnetic Waves in Hollow Tubes of Metal," Proc. I.R.E., 24, pp. 1298-1328 (1936)— A development of the equations of propa- gation together with a discussion of terminal connections. 27. S. A. Schelkunoff, "Transmission Theory of Plane Electromagnetic waves," Proc. I.R.E., 25, pp. 1457-1492 (1937)— Treats waves in free space and in cylindrical tubes of arbitrary cross-section; special cases; rectangle, circle, sector of circle and ring. 28. L. J. Chu, "Electromagnetic Waves in Elliptic Hollow Pipes of Metal," Jour. App. Phys., 9, pp. 583-591 (1938) — A study of field configurations, ci;itical frequencies, and attenuations. 29. G. Reber, "Electric Resonance Chambers," Communications, Vol. 18, No. 12, pp. 5-8 (1938). 30. F. Borgnis, " Electromagnetische Eigenschwingungen dielektrischer Raume," Ann. d. Phys., 35, pp. 359-384 (1939). Solution of Maxwells equations for rectangular prism, circular cylinder, sphere; also derivations of stored energy and Q values. 31. W. W. Hansen, "On the Resonant Frequency of Closed Concentric Lines," Jour. App. Phys., 10, pp. 38-45 (1939). — Series approximation method for TM OOp mode. 32. R. D. Richtmyer, "Dielectric Resonators," Jour. App. Phys., 10, pp. 391-398 (1939). 33. H. R. L. Lamont, "Theory of Resonance in Microwave Transmission Lines with Discontinuous Dielectric," Phil. Mag., 29, pp. 521-540 (1940).— With bibliography covering wave guides, 1937-1939. 34. E. H. Smith, "On the Resonant Frequency of a Type of Klystron Resonator," Phys. Rev., 57, p. 1080 (1940).— Abstract. 35. W. C. Hahn, "A New Method for the Calculation of Cavity Resonators," Jour. App. Phys., 12, pp. 62-68 (1941). — Series approximation method for certain circularly symmetric resonators. 36. E. U. Condon, "Forced Oscillations in Cavity Resonators," Jour. App. Phys., 12 pp. 129-132 (1941). — Formulas for coupUng loop and capacity coupling. 37. W. L. Barrow and H. Schaevitz, "Hollow Pipes of Relatively Small Dimensions," A.I.E.E. Trans., 60, pp. 119-122 (1941). — Septate coaxial wave guide and cavity resonator, based on bending a fiat rectangular guide into a cylinder. 444 BELL SYSTEM TECBNICAL JOURNAL 38. H. Konig, "The Laws of Similitude of the Electromagnetic Field, and Their Appli- cation to Cavity Resonators," Wireless Engr., 19, p. 216-217, No. 1304 (1942). "The law of similitude has strict validity only if a reduction in dimensions hy the factor \/m is accompanied by an increase in the conductivity of the walls bv the factor w." Original article "in Ilochf; tech u. Elek:akus, 58,' pp. 174-180 (1941). 39. S. Ramo, "Electrical Conce[)ts at Extremely High Frequencies," Electronics, Vol. 9, Sept. 1942, pp. 34-41, 74-82. A non-mathematical description of the physical phenomena involved in vacuum tubes, cavity resonators, transmission lines and radiators. 40. J. Kemp, "Wave Guides in Electrical Communication," Jour. I.E.E., V. 90, Pt. Ill, pp. 90-114 (1943).- — Contains an extensive hsting of U. S. and British patents. 41. H. A. Wheeler, "Formulas for the Skin EiTect," Proc. I.R.E., 30, pp. 412-424 (1942)— Includes: a chart giving the skin depth and surface resistivit}- of several metals over a wide range of frequency; simple formulas for H.F. resistance of wires, trans- mission lines, coils and for shielding effect of sheet metal. 42. R. C. Colwell and J. K. Stewart, "The Mathematical Theory of Vibrating Mem- branes and Plates," J.A.S.A., 3, pp. 591-595 (1932) — Chladni figures for a square plate. 43. R. C. Colwell, "Nodal Lines in A Circular Membrane" J.A.S.A., 6, p. 194 (1935)— Abstract. 44. R. C. Colwell, "The Vacuum Tube Oscillator for Membranes and Plates," J.A.S.A., 7, pp. 228-230 (1936) — Photographs of Chladni figures on circular plates. 45. R. C. Colwell, A. W. Friend, J. K. Stewart, "The Vibrations of Symmetrical Plates and Membranes," J.A.S.A., 10, pp. 68-73 (1938). 46. J. K. Stewart and R. C. Colwell, "The Calculation of Chladni Patterns," J.A.S.A., 11, pp. 147-151 (1939). 47. R. C. Colwell, J. K. Stewart, H. D. Arnett, "Symmetrical Sand Figures on Circular Plates," J.A.S.A., 12, pp. 260-265 (1940). 48. V. O. Knudsen, "Resonance in Small Rooms," J.A.S.A., 4, pp. 20-37 (1932)— Ex- perimental check on the values of the eigentones. 49. H. Cremer & L. Cremer, "The Theoretical Derivations of the Laws of Reverberation," J.A.S.A., 9, pp. 356-357 (1938)— Abstract of Akustische Zeits., 2, pp. 225-241, 296-302 (1937) — Eigentones in a rectangular chamber. 50. H. E. Hartig and C. E. Swanson, "Transverse Acoustic Waves in Rigid Tubes," Pliys. Rev., 54, pp. 618-626 (1938) — Experimental verification of the presence of acoustic waves in a circular duct, corresponding to the TE and TM electromag- netic waves; shows an agreement between calculated and experimental values of the resonant frequencies, with errors of the order of ± 1%. 51. D. Riabouchinsky, Comptes Rendus, 207, pp. 695-698 (1938) and 269, pp. 664-666 (1939). Also in Science Abstracts A42, j^364 (1939) and A43, 7^1236 (1940).— Treats of supersonic analogy of the electromagnetic field. 52. F. V. Hunt, "Investigation of Room Acoustics by Steady State Transmission Meas- urements," J.A.S.A., 10, pp. 216-227 (1939). ,53. R. Bolt, "Standing Waves in Small Models," J.A.S.A., 10, p. 258 (1939). 54. L. Brillouin, "Acoustical Wave Propagation in Pipes," J.A.S.A., 11, p. 10 (1939) — Analogy with TE waves. 55. P. E. Sabine, "Architectural Acoustics: Its Past and Its Possibilities," J.A.S.A., 11 pp, 21-28, (1939). — Pages 26-28 give an illuminating review of the theoretical work in acoustics. 56. R. H. Bolt, "Normal Modes of Vibration in Room Acoustics: Angular Distribution Theory," J.A.S.A., 11, pp. 74-79 (1939). — Eigentones in rectangular chamber. 57. R. H. Bolt, "Normal Modes of Vibration in Room Acoustics: Experimental Investiga- tions in Non-rectangular Enclosures," J.A.S.A., 11, pp. 184-197 (1939). 58. L. Brillouin, "Le Tuyau Acoustique comme Filtre Passe-Haut/' Rev. D'Acoiis., 8, pp. 1-11 (1939). — A comparison with TM waves; some historical notes, tracing the inception of the theory back to 1849. 59. E. Skudrzyk, "The Normal Modes of Viijration of Rooms with Non-Planar Walls," J.A.S.A., 11, pp. 364-365 (1940).— Abstract of Akustische Zeits., 4, p. 172 (1939).— Considers the equivalent of the TAl 00/) mode. 60. G. M. Roe, "Fre((uency Distribution of Normal Modes," J.A.S.A., 13, pp. 1-7 (1941). — A verification of Maa's result for a rectangular room, and an extension to the cylinder, sphere and several derived shapes, which leads to the result that the number of normal modes (acoustic) below a given frequency is the same for all shapes. SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 445 61. R. S. Bolt, H. Feshbach, A. M. Clogston, "Perturbation of Sound Waves in Irregular Rooms," J.A.S.A., 14, pp. 65-73 (1942) — Experimental check of eigentones in a trapezoid vs calculated values. Abstracts of Foreign Language Articles in Wireless Engineer 62. H. Gemperlein," Measurements on Acoustic Resonators," 16, p. 200, No. 1504 (1939), 63. M. Jouguet," Natural Electromagnetic Oscillations of a Cavity," 16, p. 511, No. 3873, (1939). 64. M. S. Neiman, " Convex Endovibrators," 17, p. 65, No. 455, (1940). 65. F. Borgnis, "The Fundamental Electric Oscillations of Cylindrical Cavities," 17, p. 112, No. 905, (1940). See also Sci. Abs., B43, No. 343 (1940). 66. H. Buchholz, "Ultra-Short Waves in Concentric Cables, and the "Hollow-Space" Resonators in the Form of a Cylinder with Perforated-Disc Ends," 17, p. 166, No. 1301 (1940). 67. J. Aliiller, "Investigation of Electromagnetic Hollow Spaces," 17, p. 172, No. 1379 (1940).— Sci. Abs., B43, No. 857 (1940). 68. V. I. Bunimovich, "An Oscillating System with Small Losses," 17, p. 173, No. 1380 (1940). 69. M. S. Neiman, "Convex Endovibrators," 17, p. 218, No. 1743 (1940). 70. M. S. Neiman, "Toroidal Endovibrators," 17, p. 218, No. 1744 (1940). 71. H. Buchholz, "The Movement of Electromagnetic Waves in a Cone-Shaped Horn," 17, p. 370, No. 3009 (1940). — Cavity formed by cone closed by spherical cap. 72. O. Schriever, "Physics and Technique of the Hollow-Space Conductor," 18, p. 18 No. 2, (1941).— Review of history. 73. F. Borgnis, "Electromagnetic Hollow-Space Resonators in Short- Wave Technique," 18, p. 25, No. 61, (1941). 74. T. G. Owe Berg, "Elementary Theory of the Spherical Cavity Resonator," 18, p. 287, No. 1843 (1941). 75. F. Borgnis, "A New Method for measuring the Electric Constants and Loss Factors of Insulating Materials in the Centimetric Wave Band," 18, p. 514, No. 3435 (1941). — An application of the cylindrical cavity resonator. 76. V. I. Bunimovich, "The Use of Rectangular Resonators in Ultra-High-Frequency Technique," 19, p. 28, No. 65 (1942). Use in 17 cm oscillator. 77. V. I. Bunimovich, "A Rectangular Resonator used as a Wavemeter for Decimetric and Centimetric Waves," 19, p. 37, No. 176 (1942). 78. M. Watanabe, "On the Eigenschwingungen of the Electromagnetic Hohlraum," 19, p. 166, No. 927 (1942). 79. F. Borgnis, "The Electrical Fundamental Oscillation of the Cylindrical Two-Layer Cavity," 19, p. 370, No. 2306 (1942). Considers cylindrical resonator with two concentric internal cylinders of different dielectric constant. 80. W. Ludenia, "The Excitation of Cavity Resonators by Saw-Tooth Oscillations," 19, p. 422-423, No. 2641 (1942). 81. Ya. L. Al'pert, "On the Propagation of Electromagnetic Waves in Tubes," 19, p. 520, No. 3181 (1942). — Calculation of losses in a cylindrical wave guide. 82. V. I. Bunimovich, "The Propagation of Electromagnetic Waves along Parallel Con- ducting Planes," 19, p. 520, No. 3182 (1942). — Equations for Zo and attenuation of rectangular wave guide, and resonant frequency and Q of rectangular cavity. 83. C. G. A. von Lindern & G. de Vries, "Resonators for Ultra-High Frequencies," 19, p. 524, No. 3206 (1942). — Discusses transition from solenoid to toroidal coil to "single turn" toroid, i.e., toroidal cavity resonator. Abstracts in Science Abstracts 84. L. Bouthillon, "Coordination of the Different Types of Oscillations," A39, No. 1773 (1936) .^General theory of mechanical, acoustic, optical and electric oscillations. 85. Biirck, Kotowski, and Lichte, "Resonance Effects in Rooms, their Measurement and Stimulation," A39, No. 5226 (1936). 86. G. Jager, "Resonances of Closed and Open Rooms, Streets and Squares," 40A, No 306 (1937). 87. K. W. Wagner, "Propagation of Sound in Buildings," A40, No. 2199(1937).— Trans- mission through a small hole in a wall. 88. M. Jouguet, "Natural Electromagnetic Oscillations of a Spherical Cavity," 42A, No. 3822 (1939). 89. H. R. L. Lament, " Use of the Wave Guide for Measurement of Micro-wave Dielectric Constants," 43 A, No. 2684 (1940). Precision Measurement of Impedance Mismatches in Waveguide By ALLEN F. POMEROY A method is described for determining accurately the magnitude of the reflection coeflicient caused by an inipe-iance mismatch in waveguide by measuring the ratio between incident and reflected voltages. Reflection coeflicients of any value less than 0.05 (0.86 db standing wave ratio) can be measured to an accuracy of ± 2.5%. TONG waveguide runs installed in microwave systems are usually -*— ' composed of a number of short sections coupled together. Although the reflection at each coupling may be small, the effect of a large number in tandem may be serious. Therefore, it is desirable to measure accurately the very small reflection coefficients due to the individual couplings. A commonly adopted method for determining reflection coefficients in phase and magnitude in transmission lines has been to measure the standing wave ratio by means of a traveling detector. Such a system when carefully engineered, calibrated and used is capable of good results, especially for standing waves greater than about 0.3 db. Traveling detectors were in use in the Bell Telephone Laboratories in 1934 to show the reactive nature of an impedance discontinuity in a wave- guide. A traveling detector was pictured in a paper^ in the April 1936 Bell System Technical Journal. Demonstrations and measurements using a traveling detector were included as part of a lecture on waveguides by G. C. Southworth given before the Institute of Radio Engineers in New York on February 1, 1939 and before the American Institute of Electrical Engineers in Philadelphia on March 2, 1939. Methods for determining the magnitude only of a reflection coefficient by measuring incident and reflected power have been developed by the Bell Telephone Laboratories. A method used during World War II incorporated a directional coupler^. The method described in this paper is a refinement of this directional coupler method and is capable of greatly increased accu- racy. It uses a hybrid junction^ to separate the voltage reflected by the mismatch being measured from the voltage incident to the mismatch. Each is measured separately and their ratio is the reflection coefficient. The problem to be considered is the measurement of the impedance mismatch introduced by a coupling between two pieces of waveguide due to differences in internal dimensions of the two waveguides and to imperfec- tions in the flanges. The basic setup might be considered to be as shown in Fig. 1. The setup comprises a signal oscillator, a hybrid junction, a 446 MEASUREMENT OF IMPEDANCE MISMATCHES 447 calibrated detector and indicator, a termination Z', a piece of waveguide EF (the flange E of which is to be part of the couphng BE to be measured) and a termination Z inserted into the waveguide piece EF so that the reflection coefl&cient of the couphng BE alone will be measured. In addi- tion a fixed shorting plate should be available for attachment to flange B. Four cases are considered : I. Termination Z and Z' perfect, only one coupling on hybrid junction. II. Termination Z imperfect, termination Z' perfect, only one coupling on hybrid junction. III. Termination Z perfect, four couplings on hybrid junction. IV. Termination Z imperfect, four couplings on hybrid junction. SIGNAL OSCILLATOR TERMINATION Z' A 1 HYBRID JUNCTION ^N\ '■ D C CALIBRATED DETECTOR & INDICATOR TERMINATION Z \AA/ rr Fig. 1 — Block schematic for cases I and II. It is assumed in all cases that: 1. The hybrid junction has the properties as defined in the discussion of case I. 2. The signal oscillator absorbs all the power reflected through arm A of the hybrid junction. 3. The calibrated detector and indicator absorb all the power transmitted through arm C of the hybrid jimction. 4. The oscillator output and frequency are not changed when the hybrid junction arm B is short-circuited. 5. The attenuation of waveguide may be neglected. I. Termination Z and Z' Perfect, Only One Coupling on Hybrid Junction In this case the hybrid junction, termination Z' and termination Z, as shown in Fig. 1, are all considered to be perfect. This means for the hybrid junction that its electrical properties are such that the energy from 448 BELL SYSTEM TECHNICAL JOURNAL the oscillator splits equally in paths AD and AB. The half in AD is com- pletely absorbed in the perfect termination Z' . The half in AB is partly reflected from the impedance mismatch due to the waveguide coupling BE and the remainder is absorbed in the perfect termination Z. Again due to the properties of the perfect hybrid junction, the impedance presented by the arm B when arms A and C are perfectly terminated is also perfect, and the reflected energy from waveguide coupling BE splits equally in paths BA and BC. The part in BA is absorbed by the oscillator. The part in BC representing the voltage reflected from the coupling BE is meas- ured by the calibrated detector and indicator. The magnitude of the inci- dent voltage may be measured when the waveguide piece EF is replaced by the fixed shorting plate. It is convenient to measure voltages applied to the calibrated detector and indicator in terms of attenuator settmgs in db for a reference output indicator reading. Then the ratio expressed in db between incident and reflected voltages (hereafter called W) is W2 (due to the coupling BE) = Ai - A2 (1) where Ai is attenuator setting for incident voltage and A2 is attenuator setting for reflected voltage. Both reflection coefficient and standing wave ratio may be expressed in terms of 11'. For if X = voltage due to incident power (2) and Y = voltage due to reflected power, (3) Y then reflection coefficient = — (4) and voltage standing wave ratio = .—^. p— - (5) Since Widb) = 20 logio .^l (6) W 1 + antilog — then in db, standing wave ratio = 20 logio (7) W -1 + antilog — Standing wave ratio plotted versus W is shown in Fig. 2. Reflection coeffi- cient versus W can be found in any "voltage ratios to db" table. II. Termination Z Imperfect, Termination Z' Perfect, Only One Coupling on Hybrid Junction In Fig. 1, if the termination Z is not perfect, there will be two reflected voltages from branch B. The vector diagram of the voltage at C might be MEASUREMENT OF IMPEDANCE MISMATCHES 449 represented as in Fig. 3, where vector 0-1 represents the voltage reflected from couphng BE and vector 1-2 represents the voltage reflected from the termination Z. To make measurements, termination Z should be movable 4-0 30 40 W IN DB Fig. 2 — Standing wave ratio (SWR) versus W. and the magnitude of its reflection coefiicient be the same at a given position of rest for either direction of approach, and be the same for positions of rest over an interval of a half a w^avelength in waveguide. The reflected voltage is measured twice, once for minimum output as the position of the termination Z is adjusted and again for maximum output. Then Fn^in = Fb - F. and V^^ = Vb+V, (8) 450 BELL SYSTEM TECHNICAL JOURNAL where Vb is voltage reflected from coupling BE and V ^ is voltage reflected from termination Z. Equations (8) can be solved for Vh and V ^ for V 4- V ■ V — V F' max 1^ ' mm i t' ' max ' mm /rvN 5 = and V^ = (.9) The incident voltage is measured as before. Therefore, using equation (6) W = 20 log 1^1 and W" = 20 log L^' (10) where W is due to coupling BE, W" is due to termination Z and Va is incident voltage. ^ -*•"" "^"^^ y ^v • / N / / \ / / \ / / \ / 1 1 / 1 \ \ 'l 1 / / \ / \ / \ / \ y Fig. 3 — Vector diagram of voltages reflected from coupling BE and termination Z. A more practical solution involving only addition, subtraction and the use of the characteristics in Fig. 4 is now presented. The settings of the detector attenuator for incident voltage, minimum output and maxi- mum output might be yli , Az and Ai^ . Then Wz = ^i - ^3 and 1^4 = A^- A, (11) But Wz = 20 log j^° and W, = 20 log , „ | ^ , ° „ . (12) T and Wz-W, = 20 log ' 7 ^ '/ = 20 log % - 1 + antilog 2Q (13) where 20 log 'y^' = T = W" - W (14) MEASUREMENT OF IMPEDANCE MISMATCHES 451 40 20 1 0.8 O 0.6 o 0.4 0.2 0.1 o.oe 0.0 6 0.04 0.02 0.01 i V i \ \ \ \. > L >. V s. \, \ ^ s. \ \ \ T \,\ I \ 1+ ANTILOG -^ \ \ S^a ■ -1 + ANTILOG -^ 1 ^ \^2- 20 LOG 1 1 ANTILOG >x ^ 20 N s. SS. \, NS. \, ^ \ V N ^ \ \ \ ^ K \ r, = 20L0G 1- \ \ ' T ANTILOG -^ X ^ \ V \, \ v \ \ V V \, \ \ N \ \ \ \ \ 10 20 30 40 50 TIN DB Fig. 4— F, , Fi and Wz - W^ There is an Fi{T) = 20 log /l + \ \ antilog 20> and an FiiT) = 20 log 1 - 1 antilog 20 such that W = Wi + Fi= W^ - F^ W" = T+W, + F,= T+ W^ - F2 and Fi-\- F2= W3 - Wi 60 70 (15) (16) 452 BELL SYSTEM TECHNICAL JOURNAL Figure 4 shows Fi , F2 and their sum TT'3 — TI'4 plotted versus T. It may be noted that Wz — Tr4 versus T has the same values as SWR versus W in Fig. 2. Using equations (16) and Fig. 4, TI'' and W" may be evaluated for the particular values of Ws and Wi in equation (11). In the evaluation, if there is uncertainty as to which reflection coefficient belongs to the wave- guide coupling BE and which belongs to the termination /., a termination with a different magnitude of reflection coefficient should be used and the technique repeated. The reflection coefficient which is the same in the two cases is of course that due to the waveguide coupling BE. SIGNAL OSCILLATOR A~| p TERMINATION Z MOVABLE SHORTING PISTON 1 1 — 1 1— VARIABLE ATTENUATOR 1 1 HYBRID JUNCTION ^11 ''l 1 "1 vw ' 'e ' 'g ' _ _C CALIBRATED DETECTOR & INDICATOR Fig. 5 — Block schematic for cases III and IV. It is assumed in the above solution that multiple reflections between the two impedance mismatches are inconsequential. Appendix A outlines a procedure for evaluating the maximum probable error due to multiple reflections. III. Termination Z Perfect, Four Couplings on Hybrid Junction In this case the setup might be as shown in Fig. 5. This setup differs from that shown in Fig. 1 in that the hybrid junction has four couplings shown, termination Z' has been replaced by a variable attenuator and a movable shorting piston, and the waveguide coupling FG is to be measured instead of coupling BE. The hybrid junction and the termination Z are assumed to be perfect as defined for case I. Since it is the object of the measuring method to measure impedance mismatches in branch B, it is desirable to make the voltage at C depend only on power reflected from branch B. This is accomplished by adjusting MEASUREMENT OF IMPEDANCE MISMATCHES 453 branch D so that the voltages due to the flanges of the hybrid junction are cancelled. The vector diagram of the voltage at C might be represented as in Fig. 6. Vector 0-1 represents the voltage at C when input is applied to A , due to the impedance mismatch at the coupling BE. Vector 1-2 represents that due to the mismatch at coupling D. Vector 2-3 represents that due to the mismatch at the variable attenuator, (which will usually change in magnitude and probably in phase for different settings). Vector 3-0 repre- sents the voltage at C due to the cancelling voltage from the branch D. Its phase can be varied by changing the position of the movable shorting piston. Its magnitude can be varied by changing the setting of the variable attenuator. When the adjustment is accomplished effectively no power reaches the detector. It is necessary that the reflection coefficients of Fig. 6 — Vector diagram of voltages at terminal C. couplings A, B, and C be small so that multiple reflections caused by them will not affect the accuracy of measurement. The reflected power from coupling FG may be measured when wave- guide GH is connected to waveguide EF as shown in Fig. 5 and termination Z is located within waveguide GH. The detector attenuator setting might he A5 . The incident power may be measured as before when termination Z is withdrawn from the waveguide EF and the piece of waveguide GH is replaced by a fixed shorting plate. Wi, (due to reflection coefficient of the coupling FG) = Ai — A^ (17) IV. Termination Z Imperfect, Four Couplings on Hybrid Junction In Fig. 5 if the movable termination Z is not perfect, there will be two reflected voltages in branch B when the adjustment is being made. The vector diagram of the voltage at C might be as in Fig. 7. This is the same as Fig. 6 except that a new vector 0-5 represents the voltage due to the mismatch of the movable termination Z. The adjustment is accomplished the same as in the last section except that the criterion is to have no change in detector output as the movable termination Z is moved axially over a 454 BeIl system TECHNICAL JOURNAL range of a half a wavelength in waveguide. As for the last case it is neces- sary that the reflection coefficients of the couplings A, B and C be small if good accuracy is desired. When measuring the coupling FG the procedure and evaluation are the same as for case II. Part of a laboratory setup as used at about 4 kilomegacycles is shown in Fig. 8. It includes a hybrid junction, a variable attenuator, a movable shorting piston, a straight section of waveguide and a movable termination which consists of a cylinder of phenol resin and carbon with a tapered section at one end. It is mounted in a phenolic block so that it may be moved axially in the wave guide. Fig. 7 — Part of a laboratory setup as used at 4 kilomegacycles. In cases III and IV if the hybrid junction has "poor balance" so that voltage appears at C when input is applied to arm A even though B and D are perfectly terminated, the adjusting procedure will cancel this voltage as well. Measuring accuracy will not be impaired provided the other assumptions are fulfilled. Measuring TI'— A Fitting Which Does Not Admit of Measuring Each End Separately A piece with a configuration unsuited to the preceding technique may be measured by connecting it between two straight pieces of waveguide such as between flanges F and G in Fig. 5. The IT due to the vector sum of the reflection coefficients of the coupling at one end, any irregularities and the coupling at the other end, is measured. Due to the distance between the mismatches, the vector sum will vary over the frequency band of interest. MEASUREMENT OF IMPEDANCE MISMATCHES 455 m. r:- IP o 456 BELL SYSTEM TECHNICAL JOURNAL Accuracy There are three important sources of error. The first is lack of proper adjustment. The second is that due to the detector attenuator calibration. The third is that due to multiple reflections. Experience and care can almost eliminate the first source. The second source may have a magnitude of twice the detector attenuator calibration error. In equations (1) and (17) this is readily apparent. The evaluation of W using equations (16) introduces negligibly more error provided IFs — Wi is made large by proper choice of the magnitude of the reflection coefficient of the termination Z. The possible errors due to multiple reflections be- tween the waveguide impedance discontinuity being measured and an imperfect termination are discussed in Appendix A. If the impedance presented by the arm B of the hybrid junction is not perfect, energ>^ re- flected from the hybrid junction will be partly absorbed in the termination and cause an error in the measurement. If the magnitude of this reflection coefficient is known, the maximum error may be computed. If a detector attenuator calibration error of ±0.1 db is assumed to be the only contributing error, it is possible to measure the W due to an impedance mismatch to an accuracy of ±0.2 db provided the W is greater than 26 db. These numbers correspond to measuring a standing wave ratio of any value less than 0.86 db to an accuracy of ±0.02 db or reflection coefficients of any value less than 0.05 to an accuracy of ±2.5%. APPENDIX A Maximum Probable Error Due to Magnitude of Reflection Coefficient Being Measured When Measuring a Waveguide Coupling The purpose of this appendix is to derive equations so that the maximum probable error due to multiple reflections may be calculated. The assump- tions may not be rigorous but the mathematical treatment appears to represent a reasonable approximation. It is assumed that there is no dissi- pation in waveguide EF, waveguide GH and in coupling FG. The electrical relations of the coupling FG and the movable termination Z might be represented as in Fig. 9, where Ka = characteristic impedance of waveguide EF and Kh = characteristic impedance of waveguide GIL The first few multiple reflections from the two discontinuities, coupling FG and termination Z, can be illustrated as in Fig. 10. Evaluation of the magnitudes of the reflections can be accomplished as outlined in paragraph 7.13, page 210 in the book "Electromagnetic Waves"* by S. A. Schelkunoff. * Published by D. Van Nostrand, Inc., New York City, 1943. MEASUREMENT OF IMPEDANCE MISMATCHES 457 FG Z < > ^^b wv Fig. 9 — Relation between coupling FG and termination Z. Vq ^c Vb ^f ^h ^g Vci Vn Vp ^k h where r — Fig. 10 — Multiple reflections from two planes of discontinuity. Va = Incident voltage Vb = rVa Kb — Ka Kb + Ka Vo= Va-\- Vb^ Vail + r) V,= e~'^''Vc = e-'^' Vail + r) Ve = zVd = ze-'^"- Vail + r) where z = Z - Zb Z + Zb -i2^L Ve = ze-^'"- Vail + r) Vn = —rVf = ze -i2pL F„(l + r)i-r) (18) (19) (20) (21) (22) (23) (24) (25) (26) 458 BELL SYSTEM TECHNICAL JOURNAL ^^''' -' = KTTKb (27) Vh = Vf+V„ = 2^'=^^^ Va(l + r)(l - r) (28) V, = e-<P^V, = ze-''^'V.{l + r){-r) (29) F„. - zV, = z'^e-''^'^ Va(l + /•)(-;-) (30) F„ == e-'"^^7„. = sV'^^ F.(l + r){-r) (31) Kp = -rVn = 2^r''^^F„(l + r)(-r)'^ (32) V, = V„+ Vp= 2V'*^^Fa(l - 0(-r) (33) For purposes of analysis it is now assumed that further multiple reflections are negligible. 13? Fig. 11 — Vector voltage diagram for maximum vector sum. 3 2 1 * « ^ Fig. 12 — Vector voltage diagram for minimum vector sum. Equations (19), (28) and (33) are the reflected voltages that combine vectorially to be measured. If ^L = 0, 7r, 2x , ■ • • nw then the vector voltage diagram might appear as in Fig. 11. If BL =-, — , — , • • • — then the vector voltage diagram might appear as in Fig. 12. The followmg example illustrates the calculations involved in computing the errors due to the magnitude of the reflection coefiicient being measured. The assumptions are such that an appreciable error is computed. If one assumes r = 0.316 and z = 0.282, then from equation (6) TIV = 10 db and T^. = 11 db. In Figs. 11 and 12, vector 0-1 = r, vector 1-2 = z(l — r'-), vector 2-3 = rs-(l — r-) (34) then TFo_i = 10 db, IFi-2 = 11.00 + 0.92 = 11.92 db, and IFo-s = 10.00 +22.00 + 0.92 = 32.92 db (35) In order to evaluate vector 0-2 in Fig. 11 (the vector sum of vectors 0-1 and 1-2), one calculates their difference T. T = 11.92 - 10.00 = 1.92 db (36) For T = 1.92 db, /'i = 5.10 db (37) therefore W0-2 = 10.00 - 5.10 = 4.90 db (38) MEASUREMENT OF IMPEDANCE MISMATCHES 459 In order to evaluate vector 0-3 in Fig. 11 (the vector difference of vectors 0-2 and 2-3) one calculates their difference T. T = 32.92 - 4.90 = 28.02 db (39) For T = 28.02 db, F^ = 0.36 db (40) therefore TFo-3 = 4.90 dz 0.36 = 5.26 db = TF4 (41) In order to evaluate vector 0-2 in Fig. 12 (the vector difference between vectors 0-1 and 1-2), one uses T from equation (36). For T = 1.92 db, Fo = 14.10 db (42) therefore I['o-2 = 10.00 + 14.10 = 24.10 db (43) In order to evaluate vector 0-3 in Fig. 12 (the vector difference between vectors 0-2 and 2-3), one calculates their difference T. T = 32.92 - 24.10 = 8.82 db (44) For T = 8.82 db, F. = 3.93 db (45) therefore TF0-3 = 24.10 + 3.93 = 2S.03db = IF3 (46) Using equation (16) Ws-Wi= 22.77 db, r = 1.24 db, Fi = 5.40 and therefore W = 9.66 db. Since we started by assuming Wr = 10 db, the error amounts to 0.34 db. References 1. Page 120, "Transmission Networks and Wave Filters," T. E. Shea. Published by D. Van Nostrand, Inc., New York City, 1929. 2. "Hyper-frequency Waveguides — General Considerations and Experimental Results," G. C. Southworth, Bell System Technical Journal, April, 1936. 3. "Directional Couplers." W. W. Mumford, Proceedings of the Institute of Radio Engineers, Februar>' 1947. 4. "Hybrid Circuits for Microwaves," W. A. Tyrrell. A paper accepted for publication in the Proceedings of the Institute of Radio Engineers. 5. "Note on a Reflection -Coefficient RIeter," Nathaniel I. Korman, Proceedings of the InstiliUe of Radio Engineers and Waves and Electrons, September 1946. 6. "Probe Error in Standing-Wave Detectors," William Altar, F. B. Marshall and L. P. Hunter, Proceedings of the Institute of Radio Engineers and Waves afid Electrons, January 1946. 7. Pages 20 to 24, "Practical Analysis of UHF Transmission Lines — Resonant Sections — Resonant Cavities — Waveguides," J. R. Meagher and H. J. Markley Pamphlet published by R. C. A. Service Company, Inc., in 1943. 8. "Microwave Measurements and Test Equipments," F. J. Gaffney, Proceedings of the Institute of Radio Engineers and Waves and Electrons, October 1946. Reflex Oscillators By J. R. PIERCE and W. G. SHEPHERD Table of Contents I. Introduction 463 II. Electronic Admittance — Simple Theory 467 III. Power Production for Drift Angle of (m + f) Cycles 470 IV. Effect of Aiijiroximations 479 V. Special Drift Fields 480 VI. Electronic Gap Loading 482 VII. Electronic Tuning — Arbitrary Drift Angle 484 VIII. Hysteresis 493 IX. Effect of Load 512 A. Fixed Loads 513 B. Frequency Sensitive Loads — Long Line Effect 523 C. Effect of Short Mismatched Lines on Electronic Tuning 531 X. Variation of Power and Electronic Tuning with Frequency 537 XI. Noise Sidebands 542 XII. Build-up of Oscillation 545 XIII. Reflex Oscillator Development at the Bell Telephone Laboratories 550 A. Discussion of the Beating Oscillator Problem 550 B. A Reflex Oscillator with an External Resonator — The 707. . 553 C. A Reflex Oscillator with an Integral Cavity — The 723 558 D. A Reflex Oscillator Designed to Eliminate Hvsteresis — The 2K2^ 563 E. Broad Band Oscillators— The 2K25 '. 570 F. Thermally Tuned Reflex Oscillators— The 2K45 577 G. An Oscillator with Wave-Guide Output— The 2K50 597 H. A Millimeter— Range Oscillator— The 1464 603 I. Oscillators for Pulsed Applications— The 2K23 and 2K54 607 J. Scope of Development at the Bell Telephone Laboratories 620 Appendices I. Resonators 622 II. Modulation Coeflficient 629 HI. Approximate Treatment of Bunching 639 IV. Drift Angle as a Function of Frequency and Voltage 643 V. Electronic Admittance — Non-simple Theory 644 VI. General Potential Variation in the Drift Space 656 VII. Ideal Drift Field 660 VIII. Electronic Gap Loading 663 IX. Losses in Grids 673 X. Starting of Pulsed Reflex Oscillators 674 XI. Thermal Tuning 677 Symbols A A measure of frequency deviation (9.20). B Bandwidth (Appendix 10 only, ij-3)). B Susceptance Bi Reduced susceptance (9.7). Be Electronic susceptance. C Capacitance C Heat capacity (A'-l). D Reduced gap spacing (10.3). 460 REFLEX OSCILLA TORS 461 Ea Retarding field in drift space. F Drift effectiveness factor (5.4). G Conductance Gi, G-2 Reduced conductances (9.6), (9.12). Ge Gap conductance of loaded resonator. G< Electronic conductance. Gl Conductance at gap due to load. Gr Conductance at gap due to resonator loss. H Efficiency parameter (3.7). Hm Maximum value of // for a given resonator loss. / Radio-frequency current. h Current induced in circuit by convection current returning across gap. h D-C beam current. A' Resonator loss parameter (3.9). A" Radiation loss in watts/(degree Kelvin)'' {k-2). L Inductance. M Characteristic admittance (a-8). Ml Characteristic admittance of line. Ml/ Short line admittance parameter (9.38). N Drift time in cycles. N Length of line in wavelengths (Section IX only). N Transformer voltage ratio. P Power. Q Equation (a- 10). Qe External (? (a-11). Qo Unloaded Q (a-12). R Surface resistance (o-2). 5 Scaling factor (9.17). T Temperature. V Radio-frequency voltage. V Potential in drift space (Appendix VI only). I'o D-C beam voltage at gap. Vr The repeller is at a potential (— I'r) with respect to the cathode. W Work, energy (Appendix I). W Reduced radian frequency (10.5). A' Bunching parameter (2.9). V Admittance. Yc Circuit admittance. I\ Electronic admittance. Y L Load admittance. Y R Resonator admittance. Z Impedance. Zl Load impedance. a Distance between grid wire centers. d Separation between grid planes or tubes forming gap. e Electronic charge (1.59 X 10''^ Coulombs). / Frequency. / Factor relating to effective grid voltage (b-37). i Radio-frequency convection current. 72 Radio-frequency convection current returning across gap (c-4). {12) f Fundamental component of /•> . j V-1 k Boltzman's constant (1.37 X 10^^ joules/degree A). k Conduction loss watts/degree C (yfe-14). / Length. m Mutual inductance. ;« Electronic mass (9.03 x 10"-' gram sevens). n Repeller mode number. The number of cycles drift is n -{- -} for "optimum drift". p Reduced power (9.5). r Radius of grid wire, radius of tubes forming gap. t Time, seconds. Uf, D-C velocity of electrons. 462 BELL SYSTEM TECHNICAL JOURNAL v Total velocity (A]:)pen(li.\ VIII only). V Instantaneous gap voltage 'ii' Real part of frequency (12.1). X Coordinate along heani. y :\ rectangular coordinate normal to .v. _v Half separation of planes forming s\mmelrical gap. 3'c Magnitude of small signal electronic admittance. z A rectangular coordinate normal to x (Appendix II). c A variable of integration (Appendix VI). a Negative coeflicient of the imaginary part of frequency (12.1). /3 Modulation coeflicient. /3o Average value of modulation coefScient. /3o Modulation coeflicient on axis. /3r Modulation coefficient at radius r from axis. 0s Root mean squated value of modulation coefficient. /3y Modulation coeflicient at distance y from axis. 7 7 = oi/iio. e Dielectric constant of space (8.85 x 10~" farads/cm). 6 Drift angle in radians. 6g Gap transit angle in radians. X Wavelength in centimeters. <!> A phase angle. i Reduced potential (g-13). a Voltage standing wave ratio. T Transit time. T Time constant of thermal tuner. TH Cycling time on heating. Tc Cycling time on cooling. \}/ Magnetic flux linkage. w Radian frecjuency. THE reflex oscillator is a form of long-transit-time tube which has distinct advantages as a low power source at high frequencies. It may be light in weight, need have no magnetic focusing lield, and can be made to operate at comparatively low voltages. A single closed resonator is used, so that tuning is very simple. Because the whole resonator is at the same dc voltage, high frequency by-pass difhculties are obviated. The frequency of oscillation can be changed by several tens of megacycles by varying the repeller voltage ("electronic tuning")- This is very ad- vantageous when the reflex oscillator is used as a beating oscillator. The electronic tuning can be used as a vernier frequency adjustment to the manual tuning adjustment or can be used to give an all-electrical autcmatic frequency-control. Electronic tuning also makes reflex oscillators serve well as frequency mcdulated sources in low power transmitters. The single resonator tuning property makes it possible to construct (iscil- lators whose mechanical tuning is wholly electronically controlled. Such control is achieved by making the mechanical motion which tunes the cavity subject to the thermal e.xjiansion of an element heated by electron bom- bardment. The efficiency of the reflex oscillator is generally low. The wide use of the 707li, the 723A, the 726A and subsequent Western Electric tubes shows that this defect is outweighed by the advantages already mentioned. REFLEX OSCILLATORS 463 The first part of this paper attempts to give a broad exposition of the theory of the reflex oscillator. This theoretical material provides a back- ground for understanding particular problems arising in reflex oscillator design and operation. The second part of the paper describes a number of typical tubes designed at the Bell Telephone Laboratories and endeavors to show the relation between theory and practice. The theoretical work is presented first because reflex oscillators vary so widely in construction that theoretical results serve better than experi- mental results as a basis for generalization about their properties. While the reflex oscillator is simple in the sense that some sort of theory about it can be worked out, in practice there are many phenomena which are not included in such a theory. This leaves one in some doubt as to how well any simplified theory should apply. Multiple transits of electrons, different drift times for different electron paths and space charge in the repeller region are some factors not ordinarily taken into account which, neverthe- less, can be quite important. There are other effects which are difficult to evaluate, such as distribution of current density in the beam, loss of elec- trons on grids or on the edges of apertures and dynamic focusing. If we could provide a theory including all such known effects, we would have a tremendous number of more or less adjustable constants, and it would not be hard to fit a large body of data to such a theory, correct or incorrect. At present it appears that the theory of reflex oscillators is important in that it gives a semi-quantitative insight into the behavior of reflex oscilla- tors and a guide to their design. The extent to which the present theory or an extended theory will fit actual data in all respects remains to be seen. The writers thus regard the theory presented here as a guide in evaluating the capabilities of reflex oscillators, in designing such oscillators and in understanding the properties of such tubes as are described in the second part of this paper, rather than as an accurate quantitative tool. Therefore, the exposition consists of a description of the theory of the reflex oscillator and some simple calculations concerning it, with the more complicated mathematical work relegated to a series of chapters called appendices. It is hoped that this so organizes the mathematical work as to make it assimilable and useful, and at the same time enables the casual reader to obtain a clear idea of the scope of the theory. I. Introduction An idealized reflex oscillator is shown schematically in Fig. 1. It has, of course, a resonant circuit or "resonator."^ This may consist of a pair of grids forming the "capacitance" of the circuit and a single turn toroidal 1 For a discussion of resonators, see Appendix I. It is suggested that the reader consult this before continuing with the main work in order to obtain an understanding of the circuit philosophy used and a knowledge of the symliols employed. 464 BELL SYSTEM TECHNICAL JOURNAL coil forming the "inductance" of the circuit. Such a resonator behaves just as do other resonant circuits. Power may be derived from it by means of a couphng looj) hnking the magnetic field of the single turn coil. An electron stream of uniform current density leaves the cathode and is shot across the "gajV' between the two grids, traversing the radio-frequency held in this gaj) in a fraction of a cycle. In crossing the gaj) the electron stream is velocity modulated; that is, electrons crossing at different times gain ZERO — EQUIPOTENTIAL SURFACE OUTPUT LINE Fig. 1. — An idealized reflex oscillator with grids, shown in cross-section. different amounts of kinetic energy from the radio-frequency voltage across the gap." The velocity modulated electron stream is shot toward a negative repeller electrode which sends it back across the gap. In the "drift space" between the gap and the repeller the electron stream becomes "bunched" and the bunches of electrons passing through the radio frequency lield in the gap on the return transit can give up power to the circuit if they are returned in the proper phase. ^ The most energy any electron gains is jiV electron volts, where V is the peak radio frequency voltage across the gap and /3 is the "modulation coelTicicnt" or "gap factor", and is always less than unity, /i dci)ends on gap configuration and transit angle across the gap, and is discussed in Appendix II. REFLEX OSCILLATORS 465 The vital features of the reflex oscillator are the bunching which takes place in the velocity modulated electron stream in the retarding field be- tween the gap and the repeller and the control of the returning phase of the bunches provided by the adjustment of the repeller voltage. The analogy of Fig. 2 explains the cause of the bunching. The retarding drift field may Fig. 2. — The motion of electrons in the repeller space of a reflex oscillator may be lik- ened to that of balls thrown upward at different times. In this figure, height is plotted vs time. If a ball is thrown upward with a large velocity of I'l at a time Ti, another with a smaller velocity at a later time To and a third with a still smaller velocity at a still later time Ti the three balls can be made to fall back to the initial level at the same time. be likened to the gravitational field of the earth . The drift time is analogous to the time a ball thrown upwards takes to return. If the ball is thrown upward with some medium speed Vo , it will return in some time /o . If it is thrown upward with a low speed Vy smaller than ro , the ball will return in some time /i smaller than /o . If the ball is thrown up with a speed ^2 greater than Vq , it returns in some time /o greater than /o . Now imagine three balls thrown upward in succession, evenly spaced but with large, 466 BELL SYSTEM TECHNICAL JOURNAL medium, and small velocities, respectively.^ As the ball first thrown up takes a longer time to return than the second, and the third takes a shorter time to return than the second, when the balls return the time intervals between arrivals will be less than between their dei)artures. Thus time- position "bunching" occurs when the projection velocity with which a uniform stream of particles enters a retarding iield is progressively decreased. Figure 3 demonstrates such bunching as it actually takes place in the retarding field of a reflex oscillator. :\n electron crossing the gaj) at phase A R-F VOLTAGE ACCELERATING FOR ELECTRONS FROM CATHODE R-F VOLTAGE RETARDING FOR ELECTRONS FROM CATHODE FOR ELECTRONS RETURNING TOWARD / CATHODE \ T T Fig. 3. — The drift time for transfer of energy from the bunched electron stream to the resonator can be deduced from a plot of gap voltage vs time. is equivalent to the first ball since its velocity suffers a maximum increase, an electron crossing at phase B corresponds to the ball of velocity ^o where for the electron Vq corresponds to the d.c. injection velocity, and finally an electron crossing at j^hase C corresponds to the third ball since it has suffered a maximum decrease in its velocity. The electrons tend to bunch about the electron crossing at phase B. To a tirst order in this process no energy is taken from the cavity since as many electrons give up energy as absorb it. The next step in the process is to bring back the grou])ed electrons in such a phase that they give the maximum energy to the r.f. field. Now, f of a cycle after the gap voltage in a reflex oscillator such as that shown in Fig. 1 is changing most rai)i(lly from accelerating to retarding for electrons ^ The reader is not advised to try this experimentally unless he has juggling experience. REFLEX OSCILLATORS 467 going from the cathode, it has a maximum retarding value for electrons leturning through the gap. This is true also for If cycles, 2f cycles, n + f cycles. Hence as Fig. 3 shows if the time electrons spend in the drift space is 11 + f cycles, the electron bunches will return at such time as to give up energy to the resonator most effectively. II. Electronic Admittance — Simple Theory In Appendix III an approximate calculation is made of the fundamental component of the current in the electron stream returning through the gap of a reflex oscillator when the current is caused by velocity modulation and drift action in a uniform retarding field. The restrictive assumptions are as follows: (1) The radio-frequency voltage across the gap is a small fraction of the d-c accelerating voltage. (2) Space charge is neglected. Amongst other things this assumes no interaction between incoming and outgoing streams and is probably the most serious departure from the actual state of affairs. (3) Variations of modulation coefficient for various electron paths are neglected. (4) All sidewise deflections are neglected. (5) Thermal velocities are neglected. (6) The electron flow is treated as a uniform distribution of charge. (7) Only two gap transits are allowed. An expression for the current induced in the circuit (/3 times the electron convection current) is (0Vd\ j{ut-6) i = 2h^J,[^^Je^'^'-'\ (2.1) Here the current is taken as positive if the beam in its second transit across the gap absorbs energy from the resonator. The voltage across the gap at the time the stream returns referred to the same phase reference as the current is v — Ve~^ "'"" ' . Hence the admittance appearing in shunt with the gaps will be _ 21 (,13 (^Vd\ ,((W2)-9) (r. r.\ For small values of V approaching zero this becomes _ h^'O j((,r/2)-e) _ J((ir/2)-9) /^ ,, i es — r>-[T ^ Jef^ \^-Jj ZV 468 BELL SYSTEM TECHNICAL JOURNAL Here we define 1%, as the small signal value of the admittance, and }v as the magnitude of this quantity. If we plot the function Yes in a comple-v admittance plane it takes the form of a geometric spiral as shown in Fig. 4. CONDUCTANCE, G Fig. 4. — The negative of the circuit admittance (the heavy vertical line) and the small signal electronic admittance (the spiral) are shown in a plot of susceptance vs conductance. Each position along the circuit admittance line corresponds to a certain frequency. Each position along the spiral corresponds to a certain drift angle. Such a presentation is very helpful in acquiring a qualitative understanding of the operation of a reflex oscillator. In Appendi.x I it is shown that the admittance of the resonant circuit in the neighborhood of resonance is very nearly Vh = Gr + i2MAco/w (2.5) where Gr is a constant. The negative of such an admittance has been plotted in Fig. 4 as the vertical line A'B'. Vertical positjon on this line is REFLEX OSCILLATORS 469 proportional to the frequency at which the resonator is driven. The condi- tion for stable oscillation is W + F, = 0. (2.6) I.U 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 ■^ X > \ \ \ \ > \ \ \ \ V \ \ 1.0 1.5 2.0 2.5 3.0 BUNCHING PARAMETER, X Fig. 5. — Relative amplitude of electronic admittance vs the bunching parameter X The bunching parameter increases linearly with radio frequency gap voltage so that this curve shows the reduction in magnitude of electronic admittance with increasing voltage. We may rewrite (2.2) for any given value of 6 as where F{V) = 2/i (13 Vd) (2Fo) _ 2/i(X) 2Fo X The quantity X = 2Fo (2.7) (2.8) (2.9) is called the bunching parameter. A plot of the function F{V) vs X is shown in Fig. 5. For any given value of 6 and for fixed operating conditions 470 BELL SYSTEM TECHNICAL JOURNAL it is a function of V only and its action is clearly to reduce the small signal value of the admittance until condition (2.6) is satistied. It will be observed that this function affects the magnitude only and not the phase of the admittance. Thus, as indicated in Fig. 4, when oscillation starts the admittance is given by the radius vector of magnitude jc , terminating on the spiral, and as the oscillation builds up this vector shrinks until in accordance with (12.6) it terminates on the circuit-admittance line A'B', which is the locus of vectors (— Vr). The electronic admittance vector may be rotated by a change in the repeller voltage which changes the value of 6. This changes the vertical intercept on line A'B', and since the imaginary component of the circuit admittance, that is the height along A'B', is proportional to frequency, this means that the frequency of oscillation changes. It is this property which is known as electronic tuning. Oscillation will cease when the admittance vector has rotated to an angle such that it terminates on the intersection of the spiral and the circuit- admittance line A'B'. It will be observed that the greater is the number of cycles of drift the greater is the electronic tuning to extinction. \Miile it is not as apparent from this diagram, it is also true that the greater the number of cycles of drift the greater the electronic tuning to intermediate power points. Vertical lines farther to the left correspond to heavier leads, and from this it is apparent that the electronic tuning to extinction decreases with the loading. By sufficient loading it is possible to prevent some repeller modes (i.e. oscillations of some n values) from occurring. Since losses in the resonant cavity of the oscillator represent some loading, some modes of low n value will not occur even in the absence of external loading. III. Power Production for Drift Angle of (« + |) Cycles Now, from equation (2.2) it may be seen that Ye will be real and negative for d = On = (n + 4)27r. Because 6 also appears in the argument of the Bessel function this value of 6 is not exactly the value to make the real component of Ye a maximum. However, for the reasonably large values of n encountered in practical oscillators this is a justifiable approximation. Suppose, then, we consider the case of n + f cycles drift, calling this an opti- mum drift time. Using the value of n as a parameter we plot the magni- tude of the radio-frequency electron current in the electron stream returning across the gap given by equation (2.1) as a function of the radio-frequency voltage across the gap. This variation is shown in Fig. 6. As might be expected, the greater the number of cycles the electrons drift in the drift space, the lower is the radio-frequency ga]) voltage required to ])r(){luce a given amount of bunching and hence a given radio frequenc)- electron current. It may be seen from Fig. 6 that as the radio-frequency ga}) voltage REFLEX OSCILLATORS 471 is increased, the radio-frequency electron current gradually increases until a maximum value is reached, representing as complete bunching as is possible, after which the current decreases with increasing gap voltage. The maxi- mum value of the current is approximately the same for various drift times, but occurs at smaller gap voltages for longer drift times. The radio-frequency power produced is the voltage times the current. As the given maximum current occurs at higher voltages for shorter drift POWER DISSIPATED X CIRCUIT AND LOAD/ / / / / Q^S RADIO-FREQUENCY GAP VOLTAGE, V' Fig. 6. — Radio frequency electron convection current / and the radio frecjuency power given U]3 by the electron stream can be plotted vs the radio frequency gap voltage V for various drift times measured in cycles. Maximum current occurs at higher voltage for shorter drift times. For a given number of cycles drift, maximum power occurs at a higher gap voltage than that for maximum current. If the power produced for a given drift time is higher at low voltages than the power dissipated in the circuit and load (dashed curve), the tube will oscillate and the amplitude will adjust itself to the point at which the power dissipation and the power production curves cross. times, the maximum power produced will be greater for shorter drift times. This is clearly brought out in the plots of power vs. voltage shown in Fig. 6. The power dissipated in the circuits and load will vary as the square cf the radio-frequency voltage. Part of this power will go into the load coupled with the circuit and part into unavoidable circuit losses. A typical curve of power into the circuit and load vs. radio-frequency voltage is shown in Fig. 6. Steady oscillation will take place at the voltage for which the power production curve crosses the power dissipation curve. For instance, in Fig. 6 the power dissipation curve crosses the power production curve for 472 BELL SYSTEM TECHNICAL JOURNAL If cycles drift at the maximum or hump of the curve. This means that the circuit impedance for the dissipation cur\'e shown is such as to result in maximum production of power for If cycles drift. For 2f cycles drift and for longer drifts, the power dissipation curve crosses the power production curves to the right of the maximum and hence the particular circuit loading shown does not result in maximum power production for these longer drift times. This is an example of operation with lighter than optimum load. The power dissipation curve might cross the power production curve to the left of the maximum, representing a condition of too heavy loading for production of maximum power output. The power dissipation curve in Fig. 6 lies always above the power production curve for a drift of f cycles. This means that the oscillator for which the curves are drawn, if loaded to give the power dissipation curve shown, would not oscillate with the short drift time of f cycles, corresponding to a very negative repeller voltage. In general, the conclusions reached by examining Fig. 6 are borne out in practice. The longer the drift time, that is, the less negative the repeller, the lower is the power output. For very negative repeller voltages, how- ever, corresponding to very short drift times, the power either falls off. which means that most of the available power is dissipated in circuit losses, or the oscillator fails to operate at all because, for all gap voltages, the power dissipated in circuit losses is greater than the power produced by the elec- tron stream. Having examined the situation qualitatively, we want to make a some- what more quantitative investigation, and to take some account of circuit losses. In the course of this we will find two parameters are very important. One is the parameter X previously defined by equation (2.9), which ex- presses the amount of bunching the beam has undergone. In considering a given tube with a given drift time, the important thing to remember about X is that it is proportional to the r-J gap voltage V . For 6 = 6,, expression (2.2) is a pure conductance and we can express the power produced by the electron stream as one half the square of the peak r-f voltage times the cir- cuit conductance which for stable oscillation is equal to the negative of the electronic conductance given by (2.2). This may be written with the aid of (2.9) as . P = 2(hVo/en)XJ,(X). (3.1) Suppose we take into account the resonator losses but not the power lost in the output circuit, which in a well designed oscillator should be small. If the resonator has a shunt resonant conductance (including electronic loading) of Gr , the power dissipated in the resonator is P, = V'Gr/2. (3.2) REFLEX OSCILLATORS 473 Then the power output for dn is P = 2(/oFo/0„)X/i(X) - V^Ga/2. {3,.i) The efficiency, 77, is given by P 2 ■n = From (2.4) and (2.9) Hence we may write V = P DC ^^^'W-"^]- (^-^^ 'i^ = ~ X'. (3.5) 2/0 Fn ye ^)lH«-ff]- (w + 3/4) (3.6) TT Let us write r] = ~ where N = (» + f). We may now make a generalized examination of the effect of losses on the efficiency by examining the function H = (l/7r)[AVx(X) - iG,/ye)Xy2]. (3.7) Thus, the efficiency for 6 =0„ is inversely proportional to the number of cycles drift and is propotional to a factor H which is a function of X and of the ratio Gnlye , that is, the ratio of resonator loss conductance to small signal electronic conductance.'* For w + f cycles drift, the small signal electronic conductance is equal to the small signal electronic admittance. For a given value of Gn/ye there is an optimum value of X for which H has a maximum Hm ■ ^^'e can obtain this by differentiating (3.7) with respect to A^ and setting the derivative equal to zero, giving XJo(X) - (Gn/ye)X = (3.8) Jo{X) = (G,/ye). If we put values from this into (3.7) we can obtain Hm as a function of Gnhe ■ This is plotted in Fig. 7. The considerable loss of efficiency for values of Gn/ye as low as .1 or .2 is noteworthy. It is also interesting to note that for Gnlye equal to \, the fractional change in power is equal to the fractional change in resonator resistance, and for Gs/ye greater than \, the fractional power change is greater than the fractional change in resonator resistance. This helps to explain the fall in power after turn-on in some tubes, for an increase in temperature can increase resonator resistance considerably. ^ An electronic damping term discussed in Appendix VIII should be included in resona tor losses. The electrical loss in grids is discussed briefly in Appendix IX. 474 BELL SYSTEM TECHNICAL JOURNAL In the expression for the admittance, the drift angle, 6, appears as a fac- tor. This factor plays a double role in that it determines the phase of the admittance but also in a completely independent manner it determines, in part, the magnitude of the admittance. 6 as it has appeared in the foregoing analysis, which was developed on the basis of a linear retarding field, is the actual drift angle in radians. As will be shown in a later section, certain special repeller fields may give effective drift action for a given angle greater than the same angle in a linear field. Such values of effective drift angle may have fractional optimum values although the phase must still be such as to give within the approximations we have been using a pure con- ductance at optimum. In order to generalize the following work we will speak of an effective drift time in cycles, N e = FN, where N is the actual drift time in cycles, n -\- f , and F is the number of times this drift is more effective than the drift in a linear field. Suppose we have a tube of given /3^, 7o , Fo and resonator loss Gr and wish to find the optimum effective drift time, FN, and determine the effect on the efl&ciency of varying FN. It will be recalled that for very low losses we may expect more power output the fewer the number of cycles drift. How- ever the resonator losses may cut heavily into the generated power, for short drift angles. With short drift angles the optimum load conductance becomes small compared to the loss conductance so that although the generated power is high only a small fraction goes to the useful load. There is, therefore, an optimum value which can be obtained using the data of Fig. 7. We define a parameter K = |^«G. (3.9) which compares the resonator loss conductance with the small signal elec- C K tronic admittance per radian of driftan gle. Then in terms of A', — = -— . Je B Hence, for a fixed value of K, various values oi 6 = lirFN define values of /^ — . When one uses these values in connection with Fig. 7 he determines Je the corresponding values of //„, and hence the efficiency, r] = — ^ . These values of r] arc plotted against FN as in Fig. 8 with values of A' as a param- eter. In this })lot A' is a measure of the lossiness of the tube. The opil- mum drift angle for any degree of lossiness is evident as the maximum of one of these curves. The maximum power outputs in various repeller modes, « = 0, 1 etc. and the repeller voltages for these various power outputs correspond to discrete values of n and FN lying along a curve for a particular value of A'. REFLEX OSCILLATORS 475 Thus, the curves illustrate the variation of power from mode to mode as the repeller voltage is changed over a wide range. Changes in resonator loss or differences in loss between individual tubes of the same type correspond to passing from a curve for one value of K to a curve for another value of K. 0.50 0.40 E I qT 0.10 o t3 0.08 < u. >0.06 g 0.05 o ,7 0.04 LU 0.03 v Hrr> -— -- ■^ \45° '^"^- FN ^""«».„^^ \ \ ^ \ ^ S - \ V - - \ \ \" ) \ \ \ 1 1 \ 0.05 0.1 ye Fig. 7. — Efficiency factor Hm vs the ratio of resonator loss conductance to the small signal electronic admittance. Efficiency changes rapidly with load as the loss conductance approaches in magnitude the small signal electronic admittance. The efficiency is in- versely proportional to the number of cjxles drift. It will be observed from this that although, from an efficiency standpoint, it is desirable to work at low values of drift time such low drift times lead to an output strongly sensitive to changes in resonator losses. Perhaps the most important question which the user of the oscillator may ask with regard to power production for optimum drift is; what effect does the external load have upon the performance? If we couple lightly to the oscillator the r-f voltage generated will be high but the power will not be extracted. If we couple too heavily the voltage will be low, the beam will not be efficiently modulated and the power output will be low. There is 476 BELL SYSTEAf TECBNICAL JOURNAL apparently an optimum loading. Best output is not obtained when the external load matches the generator impedai.ce as in the case of anamplifier. 6.0 5.5 Z 5.0 O a. ff4.5 Z ^4.0 o 2 UJ G 3.5 IL U. UJ 2.0 1.5 1.0 0.5 \ \ \\ >lo''''" ye \ \ \ s. \ \ L K-O \ Os \ \ \ \ •v \ S,5 \^ \, / X \ \ \^ \ ^ ^ / .m^ V J / ^ / 15 ~~ / '' 20 / f y / / '^ / r / ^ 123456789 10 EFFECTIVE DRIFT, FN, IN CYCLES PER SECOND Fig. 8. — Efficiency in per cent vs the effective cycles drift for various values of a para- meter A" which is proportional to resonator loss. These curves indicate how the power output differs for various repeller modes for a given loss. Optimum power operating points will be represented l)y points along one of these curves. For a very low loss resona- tor, the power is highest for short drift times and decreases rapidly for higher repeller modes. Where there is more loss, the power varies less rapidly from mode to mode. We return to equation (3.7) for // and assume that we are given various n values of — . With these values as parameters we ask what variation in efficiency may be expected as we vary the ratio of the load conductance, C C Gl, to the small signal admittance, y«. When 1 = 1 oscillation ye ye REFLEX OSCILLATORS 477 will just start and no power output will be obtained. We can state the general condition for stable operation as 1% + Fc = 0, where Y c is the vector sum of the load and circuit admittances. For the optimum drift time this becomes Gc ye 2/i(X) X (3.10J ye / ~^ V Y N I Y \ \ / 0.^ — ^ s \ \ \ / V \ \ f) / 04 V \ \ \ \ 1/ / N \, \ \ \ ¥ \ \ \ \ 0.1 0.2 0.3 0.4 0.5 6 0.7 0.8 0,9 10 ye Fig. 9. — EiEciency parameter U vs the ratio of load conductance to the magnitude of the small signal electronic admittance. Curves are for various ratios of resonator loss conductance to small signal electronic admittance. The curves are of similar shapes and indicate that the tube will cease oscillating {U = 0) when loaded by a conductance about h' ice as large as that for optimum power. where ye Gl + Gr (3.11) G c Hence for a given value of — we may assume values for — between zero ye ye Q and 1 — — and these in (3.11) will define values of X. These values of X ye ^ substituted in (3.7) will define values of H which we then plot against the assumed values for — , as in Fig. 9. Thus we have the desired function of ye the variation of efhciency factor against load. 478 BELL SYSTmr TkCSNlCAL JOURNAL From the curves of Fig. 9 it can be seen that the maximum efficiency is obtained when the external conductance is made equal to approximately half the available small signal conductance; i.e. ^{je — Gr). This can be seen more clearly in the i)l()t of Fig. 10. Equation (3.8) gives the condi- tion for maximum efficiency as Gn - /o(X). Gl ye — ^-y^ 0.4 0.5 0.6 ye Fig. 10. — The abscissa measures the fractional excess of electronic negative conductance over resonator loss conductance. The ordinate is the load conductance as a fraction of electronic negative conductance. The tube will go out of oscillation for a load conductance such that the ordinate is equal to the abscissa. The load conductance for optimum power output is given by the solid line. The dashed line represents a load conductance half as great as that required to stop oscillation. If we assume various values for — these define values of Xo which when substituted in ^ Je Gc ye Gr ye 2/i(Zo) Xn - /o(Xo) (3.12) give the value of the external load for ojitimum power. We plot these data against the available conductance 1 - Gr = 1 MX) (3.13; as shown in Fig. 10. In Fig. 10 there is also shown a line through the origin of slope 1/2. It can thus be seen that the optimum load conductance is slightly less than half the available small signal or starting conductance. This relation is independent of the repeller mode, i.e. of the value FN. This does not mean REFLEX OSCILLATORS 479 that the load conductance is independent of the mode, since we have ex- pressed all our conductances in terms of je , the small signal conductance, and this of course depends on the mode. What it does say is that, regard- less of the mode, if the generator is coupled to the load conductance for maximum output, then, if that conductance is slightly more than doubled oscillation will stop. It is this fact which should be borne in mind by the circuit designer. If greater margin of safety against "pull out" is desired it can be obtained only at the sacrifice of eflficiency. ye I.U 0.9 0.8 ^ -^ ^^ y^ 0.7 0.6 ^ y- y y 0.5 y^ 04 O.b Gr ye Fig. 11. — The ratio of total circuit conductance for optimum power to small signal electronic admittance, vs the ratio of resonator loss conductance to small signal electronic admittance. An equivalent plot for the data of Fig. 10, which will be of later use, is shown in Fig. 11. This gives the value of — for best output for various values of — . ye IV. Effect of Approximations The analysis presented in Section II is misleading in some respects. For instance, for a lossless resonator and N = \ cycles, the predicted efficiency is 53%. However, our simple theory tells us that to get this efficiency, the radio-frequency gap voltage V multiplied by the modulation coefficient /3 (that is, the energy change an electron suffers in passing the gap) is I.OI8V0 • This means that (a) some electrons would be stopped and would not pass the gap (b) many other electrons would not be able to pass the gap against a retarding field after returning from drift region (c) some electrons would 480 BELL SYSTEM TECHNICAL JOURNAL cross the gap so slowly that for them /3 would be very small and their effect on the circuit would also be small (d) there might be considerable loading of the resonator due to transit time effects in the gap. Of course, it is not justifiable to apply the small signal theory in any event, since it was derived on the assumption that /ST' is small compared with Fo . In Appendix IV there is presented a treatment by R. M. Ryder of these Laboratories in which it is not assumed that /3r«Fo . This work does not, however, take into account variation of /3 with electron speed or the possibility of electrons being turned back at the gap. For drift angles of If cycles and greater, the results of Ryder's analysis are almost indistinguishable from those given by the simple theory, as may be seen by examining Figs. 128-135 of the Appendix. His curves approach the curv-es given by the simple theory for large values of n. For small values of n, and particularly for f cycles drift, Ryder's work shows that optimum power is obtained with a drift angle somewhat different from n + f cycles. Also, Fig. 131 shows that the phase of the electronic admittance actually varies somewhat with amplitude, and Fig. 130 shows that its magnitude does not actually pass through zero as the amplitude is increased. The reader is also referred to a paper by A. E. Harrison. The reader may feel at this point somewhat uneasy about application of the theory to practice. In most practical reflex oscillators, however, the value of w is 2 or greater, so that the theory should apply fairly well. There are, however, so many accidental variables in practical tubes that it is well to reiterate that the theory serves primarily as a guide, and one should not expect quantitative agreement between experiment and theory. This will be apparent in later sections, where in a few instances the writers have made quantitative calculations. V. Special Drift Fields In the foregoing sections a theory for a reflex oscillator has been developed on the assumption that the repeller field is a uniform retarding electrostatic field. Such a situation rarely occurs in practice, partly because of the diffi- culty of achieving such a field and partly because such a field may not return the electron stream in the manner desired. In an effort to get some in- formation concerning actual drift fields, we may extend the simple theory already presented to include such fields by redefining X as X = ^VFe/2Vo. (5.1) Here the factor F is included. As defined in Section /// this is the factor which relates the effectiveness of a given drift field in bunching a velocity ^ A. E. Harrison, "Graphical Methods for Analysis of Vrlocitv Modulation Bunching." Proc. I.R.E., 33.1, pp. 20-32, June 1945, REFLEX OSCILLATORS 481 modulated electron stream with the bunching effectiveness of a field with the same drift angle 6 but with a linear variation of potential with distance. Suppose, for instance, that the variation of transit time, r, with energy gained in crossing the gap V is for a given field dr/dV (5.2) and for a linear potential variation and the same drift angle (dr/dV),. (5.3) Then the factor F is defined as F = (dT/dV)/(dT/dV),. (5.4) In appendix V, F is evaluated in terms of the variation of potential with distance. The efficiency is dependent on the effectiveness of the drift action rather than on the total number of cycles drift except of course for the phase re- quirements. Thus, for a nonlinear potential variation in the drift space we should have instead of (3.7) ■n = H/FN. (5.5) In the investigation of drift action, one procedure is to assume a given drift field and try to evaluate the drift action. Another is to try to find a field which will produce some desirable kind of drift action. As a matter of fact, it IS easy to find the best possible drift field (from the point of view of efficiency) under certain assumptions. The derivation of the optimum drift field, which is given in appendix VH, hinges on the fact that the time an electron takes to return depends only on the speed with which it is injected into the drift field. Further, the varia- tion in modulation coefficient for electrons returning with different speeds is neglected. With these provisos, the optimum drift field is found to be one in which electrons passing the gap when the gap voltage is decelerating take IT radians to return, and electrons which pass the gap when the voltage is accelerating take l-rr radians to return, as illustrated graphically in Fig. 136, Appendix VH. A graph of potential vs. distance from gap to achieve such an ideal drift action is shown in Fig. 137 and the general appearance of electrodes which would achieve such a potential distribution approximately is shown in Fig. 138. With such an ideal drift field, the efficiency of an oscillator with a lossless resonator is Vi = (2/7r)(/3F/Fo). (5.6) 482 BELL SYSTEM TECHNICAL JOURNAL For a linear potential variation in the drift space, at the optimum r-f gap voltage, according to the approximate theory presented in Section III the efficiency for a lossless resonator is r? = (.520)(/3F/Fo). (5.7) Comparing, we find an improvement in efficiency for the ideal drift tield in the ratio ■m/r) = 1.23, (5.8) or only about 20%. Thus, the linear drift field is quite effective. The ideal drift field does have one advantage; the bunching is optimum for all gap voltages or, for a given gap voltage, for all modulation coefficients since ideallv an infinitesimal a-f voltage will change the transit time from tt to 27r and completely bunch the beam. This should tend to make the efficiency high despite variations in /3 over various parts of the electron flow. The hmitation imposed by the fact that electrons cannot return across the gap against a high voltage if they have been slowed up in their tirst transit across the gap remains. ' This last mentioned limitation is subject to amelioration. In one type of reflex oscillator which has been brought to our attention the electrons cross the gap the first time in a region in which the modulation coefficient is small. If the gap has mesh grids, a hole may be punched in the grids and a beam of smaller diameter than the hole focussed through it. Then the beam may be allowed to expand and recross a narrow portion of the gap, where the modula- tion coefficient is large. Thus, in the first crossing no electrons lose much energy (because /3 is small) and in the second crossing all can cross the gap where /3F is large and hence can give up a large portion of their energy^ \T. Electronic Gap Loading; So far, attention has been concentrated largely on electronic phenomena in the drift or repeller region. To the long transit time across the gap there has been ascribed merely a reduction in the effect of the voltage on the electron stream by the modulation coefficient /3. Actually, the long transit across the gap can give rise to other effects. One of the most obvious of these other effects is the production of an elec- tronic conductance across the gap. If it is positive, such a conductance acts just as does the resonator loss conductance in reducing the power out- put. Petrie, Strachey and Wallis of Standard Telephones and Cables have treated this matter in an interesting and rather general way. Their work, in a slightly modified form, ap])cars in Ajipcndix \'III, to wliicli the reader is referred for details. REFLEX OSCILLATORS 483 The work tells us that, considering longitudinal iields only, the electron flow produces a small signal conductance component across the gap 7 = -- (6.2) Here ^ is the modulation coefficient and Uo is the electron speed. 7o and Vq are beam current and beam voltage. If the gap has a length d, the transit angle across it is 6g = yd and (6.1) may be rewritten It is interesting to compare this conductance with the magnitude of the small-signal electronic admittance, ye ■ In doing so, we should note that the current crosses the gap twice, and on each crossing produces an elec- tronic conductance. Thus, the appropriate comparison between loss con- ductance and electronic admittance is IGehlje ■ Using (6.3) we obtain Usually, the drift angle Q is much larger than the gap transit angle Qg . Further, if we examine the curves for mcdulation coefficient /? which are given in Appendix II, we find that {dl3^/ddg)/l3''^ will not be very large. Thus, we conclude that in general the total loss conductance for longitudinal fields will be small compared with the electronic admittance. An example in Appendix VIII gives {IGehlj^ as about 1/10. It seems that this effect will probably be less important than various errors in the theory of the reflex oscillator. Even though this electronic gap leading is not very large, it may be in- teresting to consider it further. We note, for instance, that the conductance GeL is positive when jQ" decreases as gap transit angle increases. For paral- lel fine grids this is so from Qg = to ^^ = 27r (see Fig. 119 of Appendix II). At Qg = Itt, where /3 = 0, dfS'^/ddg = 0, and the gap loading is zero. In a region beyond dg — 2x, d^'^/ddg becomes positive and the gap conductance is negative. Thus, for some transit angles a single gap can act to produce oscillations. For still larger values of dg , Gcl alternates between positive and negative. Gap transit angles of greater than lir are of course of little interest in connection with reflex oscillators, as for such transit angles /3 is very small. For narrow gaps with large apertures rather than fine grids, d^^/ddg 484 BELL SYSTEM TECHNICAL JOURNAL never becomes very negative and may remain positive and the gap loading conductance due to longitudinal fields be always positive. In such gaps, however, transverse fields can have important effects, and (6.3) no longer gives the total gaj) conductance. Transverse fields act to throw electrons approaching the gap outward or inward, into stronger or weaker longitudinal tields, and in this manner the transverse felds can either cause the electrons to give up part of their forward velocity, transferring energy to the reso- nator, or to pick up forward velocity, taking energy from the resonator. An analysis of the effect of transverse fields is given in Appendix VIII, and this is applied in calculating the total conductance, due both to longitudinal and to transverse fields, of a short gap between cylinders with a uniform cur- rent density over the aperture. It is found that the transverse fields con- tribute a minor part of the total conductance, and that this contribution may be either positive or negative, but that the total gap conductance is always positive (see Appendix \TII, Fig. 140). The electron flow across the gap produces a susceptive component of admittance. This susceptive component is in general more difficult to cal- culate than the conductive component. It is not very important; it serves to affect the frequency of oscillation sHghtly but not nearly so much as a small change in repeller voltage. Besides such direct gap loading, the velocity modulation and drift action within a gap of fine grids actually produce a small bunching of the electron stream. In other words, the electron stream leaving such a gap is not only velocity modulated but it has a small density modulation as well. This convection current will persist (if space-charge debunching is not serious) and, as the electrons return across the gap, it will constitute a source of elec- tronic admittance. We find however, that in typical cases (see Appendix VIII, (h59)-(h63)), this effect is small and is almost entirely absent in gaps with coarse grids or large apertures. Secondary electrons produced when beam electrons strike grid wires and grid frames or gap edges constitute another source of gap loading. It has been alleged that if the frames supporting the grids or the tubes forming a gap have opposed parallel surfaces of width comparable to or larger than the gap spacing, large electron currents can be produced through secondary emission, the r-/ field driving electrons back and forth between the opposed surfaces. It would seem that this phenomenon could take place only at quite high r-f levels, for an electron would probably require of the order of 100 volts energy to produce more than one secondary in striking materials of which gaps are usually constructed. VH. Electronic Tuning — Arbitrary Drift Angle So far, the "on tune" oscillation of reflex oscillators has been considered except for a brief discussion in Section II, and we have had to deal only with REFLEX OSCILLATORS 485 real admittances (conductar.ces). In this section the steady state operation in the case of complex circuit and electronic admittances will be discussed. The general condition for cscillaticn states that, breaking the circuit at any point the sum of the admittances looking in the two directions is zero. Par- ticularly, the electronic admittance Ye looking from the circuit to the electron stream, must be minus the circuit admittance Yc , looking from the electron stream to the circuit. Here electronic admittance is used in the sense of an admittance averaged over a cycle of oscillation and fulfilling the above condition. It is particularly useful to consider the junction of the electron stream and the circuit because the electronic admittance Ye and the circuit admit- tance Yc have very different properties, and if conditions are considered elsewhere these properties are somewhat mixed and full advantage cannot be taken of their difference. The average electronic admittance with which we are concerned is a function chiefly of the amplitude of oscillation. Usually its magnitude decreases with increasing ampUtude of oscillation, and its phase may vary as well, although this is a large signal effect not shown by the simple theory. In reflex oscillators the phase may be controlled by changing the repeller voltage. The phase and magnitude of the electronic admittance also vary with frequency. Usually, however, the rate of change with frequency is slow compared with that of the circuit admittance in the vicinity of any one resonant mode. By neglecting this change of electronic admittance with frequency in the following work, and concentrating our attention on the variation with amplitude and repeller voltage, we will emphasize the im- portant aspects without serious error. However, the variation of electronic admittance with frequency should be kept in mind in considering behavior over frequency ranges of several per cent.^ The circuit admittance is, of course, independent of amplitude and is a rapidly varying function of frequency. It is partly dependent on what is commonly thought of as the resonator or resonant circuit of the oscillator, but is also profoundly affected by the load, which of course forms a part of the circuit seen from the electron stream. The behavior of the oscillator is determined, then, by the electronic admittance, the resonant circuit and the load. The behavior due to circuit and load effects applies generally to all oscillators, and the simplicity of behavior of the electronic admittance is such that similarities of behavior are far more striking than differences. We have seen from Appendix I that at a frequency Aw away from the resonant frequency wo where Aw<<Ca;o , the admittance at the gap may be expressed as: Yc = Gc + i2MAa;/a;o. (7.1) * Appendix IV discusses the variation of phase with frequency and repeller voltage. The variation of phase of electronic admittance with frequency is included in Section IX A. 486 BELL SYSTEM TECHNICAL JOURNAL Here the quantity M is the characteristic admittance of the resonator, which dej^ends on resonator shape and is unaffected by scaling from one frequency to another. Gc is the shunt conductance due to circuit and to load. Ye as given by (7.1) represents to the degree of aj^proximation re- quired the admittance of an)^ resonant circuit and load with only one resonance near the frequency of oscillation. It is ])rofitable to consider again in more detail a complex admittance plot similar to Fig. 4. In Fig. 12 the straight vertical line is a plot of (7.1). -Ye = ye(2J,(X)/x)e-J^Q Ae UJo = (LC)-'/2 Y = G+j2MAuj/u)o CONDUCTANCE, G *■ Fig. 12. — The resonator and its load can be represented as a shunt resonant circuit with a shunt conductance G. For frequencies near resonance, the conductance is nearly constant and the susceptance B is proportional to frequenc\', so that when susceptance is plotted vs conductance, the admittance Y is a vertical straight line. The circles mark off equal increments of frequency. The electronic admittance is little affected by frequency but much affected by amplitude. Tne negative of an electronic admittance Y ^ having a constant phase angle \6 is shown in the figure. The dots mark off equal amplitude steps. Oscillation will occur at a fref[uency and amplitude specified by the intersection of the curves Y and — Ye ■ The circles mark equal frequency increments. Now if we neglect the varia- tion of the electronic admittance with phase, then the negative of the small signal electronic admittance on this same plot will be a vector, the Iccus of whose termination will be a circle. The vector is shown in l-ig. 12. The dots mark off admittance values corresponding to equal amplitude incre- ments as determined by the data of Fig. 5. Steady oscillation will take place at the frequency and amplitude repre- sented by the intersection of the two curves. If the phase angle 16 of the — Ye curve is varied by varying the repeller voltage, the point of intersection will shift on both the I'c curve and the — !'« curve. 'I'hc shift along the REFLEX OSCILLATORS 487 I'(. curve represents a change in frequency of oscillation; the shift along the — Yc curve represents a change in the amplitude of oscillation. If we know the variation of amplitude with position along the — 1% curve, and the varia- tion of frequency with position along the Y ,■ curve, we can obtain both the amplitude and frequency of oscillation as a function of the phase of — 1% , which is in turn a function of repeller voltage. From (2.3) and (2.7) we can write — Ye in terms of the deviation of drift angle M from n + f cycles. - Fe = yXlJ^)/Xy^\ (7.2) The equation relating frequency and Ad can be written immediately from inspection of Fig. 12. 2MAco/coo = -Gc tan Ad Aco/wo = -{Gc/2M) tan A0 (7.3) Aco/wo = - (1/2(3) tan M. Here Q is the loaded Q of the circuit. The maximum value of Ad for which oscillation can occur (at zero ampli- tude) is an important quantity. From Fig. 12 this value, called A^o , is obviously given by cosA^o = Gc/ye = {Gc/M)(M/ye) (7.4) = (M/ye)/Q. From this we obtain tan A^o = ± {Q'(ye/My - 1)\ (7.5) By using (7.3) we obtain (Aa,/coo)o = ± (h) iye/M) (1 - {M/yeQYf (7.6) or (Aco/a;o)o - ±(§) (y./M) (1 - {Gc/yeYf. {1.1) These equations give the electronic tuning from maximum amplitude of oscillation to zero amplitude of oscillation (extinction). The equation relating amplitudes may be as easily derived from Fig. 12 Gl + (2MAa,/co)2 = y; {2J,{X)/xy (7.8) at Ao) = let X = Xo . Then Aco/a'o = {ye/2M) {{2J,{X)/XY - {2J ,{X ,) / X ,Y)\ (7.9) -188 BELL SYSTEM TECHNICAL JOIRNAL It is of interest to ha\'e the value of Aw wo at half the i)o\ver for Aw = 0. At half power, X = A'o/\/2, so (Ac., o;o)i = (:ye/2M)((2/i(Xo/V2)/(Xo/\/2))- - {IJ ,{X ,) / X ,y)\ (7.10) For given values of modulation coefficient and Fn , X is a function of the r-f gap voltage V and also of drift angle and hence of A0, or repeller voltage (see Appendix IV). For the fairly large values of d typical of most reflex oscillators, we can neglect the change in A^ due directly to changes in M, and consider X as a direct measure of the r-J gap voltage V, Likewise Ve is a function of drift time whose variation with A0 can and will be dis- regarded. Hence from (7.9) we can plot (X/A'o)- vs. Aw/coo and regard this as a representation of normalized power vs. frequency. Let us consider now what (7.3) and (7.9) mean in connection with a given reflex oscillator. Suppose we change the load. This will change Q in (7.3) and A'o in (7.9). From the relationship previously obtained for the condition for maximum power output, Gn/ye = /o(Xo), we can find the value of A^o that is, A' at Ao; = 0, for various ratios of GrIj^ . For Gr — ^ (zero resonator loss) the optimum power value of A^o is 2.4. When there is some resonator loss, the optimum total conductance for best power output is greater and hence the optimum value of A^o is lower. In Fig. 13 use is made of (7.3) Aw/wo in plotted vs. A0 (which decreases as the repeller is made more negative) for several values of (), and in Fig. 14, (7.9), is used to plot (A7.A0)" vs. (2M/ye)Aw/ajo , which is a generalized electronic tuning variable, for several values of Xo . These curv^es illustrate typical behavior of frequenc}- vs. drift angle or repeller voltage and power vs. frequency for a given reflex oscillator for various loads. In practice, the S shape of the frequency vs. repeller voltage curves for light loads (high Q) is particularly noticeable. The sharpening of the amplitude vs. frequency curves for light loads is also noticeable, though of course the cusp- like appearance for zero load and resonator loss cannot be reproduced ex- perimentally. It is important to notice that while the plot of output vs. frequency for zero load is sharp topped, the plot of output vs. repeller volt- age for zero load is not. Having considered the general shape of frequency vs. repeller voltage curves and power vs. frequency curves, it is interesting to consider curves of electronic tuning to extinction ((Aa'/a-o)o) and electronic tuning to half power ((Aw/coo)i) vs. the loading parameter, {MjyeQ) = Gdye . Such curves are shown in Fig. 15. These curves can be obtained using (7.7) and (7.10). In using (7.10) X can be related to Gdye by the relation previously derived from 2J\iX)/X = Gc/ye and given in Fig. 5 as a function of A'. It is to be noted that the tuning to the half power point, (Aoo/a'o)> , and the tuning to the extinction point, (Aa)/coo)o , vary quite differently with loading. REFLEX OSCILLATORS 489 10^ X4 \: s \ \ Ql = ioo 200 N^ V150 \ X r>^ \ ^--^ ^ ^ ;^.- •^^ \ :?^>- \ \ \ \ \ \ -60 -50 -40 -30 -20 -10 10 20 30 40 50 60 ANGLE, AG, IN DEGREES Fig. 13. — A parameter proportional to electronic tuning plotted vs deviation from optimum drift angle M for various values of loaded Q. For lower values of Q, the fre- quency varies rapidly and almost linearly with M. For high values of Q, the frequency curve is S shaped and frequency varies slowly with A^ for small values of A5. 1.0 ^ / \ \ Xo = 2.40, (|B- = 0.43) MAX. POWER WITH ZERO/' RESONATOR LOSS/ .'/ ;^ V ^. x| y/. Xo= (t-) \^ Xo = 1.6 {^-A ^\ I \ -1.0 -0.8 -0.6 -0.4 -0.2 /2M'\ Au 0.4 0.6 Fig. 14. — The relative power output vs a parameter proportional to the frequency deviation caused by electronic tuning, for various values of load. For zero loss and zero load, the curve is peaked. For zero loss and ojjtimum load, the curve has its greatest width between half power points. For zero loss and greater than optimum load, the curve is narrow. 490 BELL SYSTEM TECHNICAL JOURNAL The quantity (Aoj/coo) I has a maximum value at Gc/yc = .433(X = 2.40), which is the condition for maximum power output when tlie resonator loss is zero. In Fig. 11 we have a plot of Gc/jc vs. GR/je for optimum loading (that is loading to give maximum power for A0 = 0). This, combined with the ^ \ "v \ .... \ / ^^ "^ "~^ ^X^ K \ . \ \ \ N \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 M _ Go_ yeQL ye Fig. 15. — A parameter proportional to electronic tuning range vs the ratio of total circuit conductance and small signal electronic admittance. The electronic tuning to extinction (Aco/a)o)o is more affected by loading than the electronic tuning to half power points (Aaj/wo)f . curves of Fig. 15, enables us to draw curves in the case of optimum leading for electronic tuning as a function of the resonator loss. Such curves are shown in Fig. 16. From Fig. 16 we see thai with optimum loading it takes very large reso- nator losses to affect the electronic tuning range to half power very much, and that the electronic tuning range to extinction is considerably more affected by resonator losses. Turning back to Fig. 7, we see that power is affected even more profoundly by resonator losses. It is interesting to REFLEX OSCILLATORS 491 compare the effect of going from zero less to a case in which the less con- ductance is \ of the small signal electronic conductance (Gr = ydT). The table below shows the fraction to which the power cr efficiency, the elec- "^ ^ (AO)] ^ V \ \ ~"--. N \ ~'^^. \ ^ N \ N \ \ \ \ \ \ \ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 M _ ^ yeQo ye Fig. 16. — The effect of resonator loss on electronic tuning in an oscillator adjusted for optimum power output at the center of the electronic tuning range. A parameter pro- portional to electronic tuning is plotted vs the ratio of the resonator loss to small signal electronic admittance. The electronic tuning to extinction is more affected than the electronic tuning to half power as the loss is changed. tronic tuning range to extinction, and the electronic tuning range to half power are reduced by this change. Power, Efficiency (77) .24 Electronic Tuning to Extinction (Aw/a,o)o .76 Electronic Tuning To Half Power (Am/ojo) 1 From this table it is obvious that efforts to control the electronic tuning by varying the ratio — are of dubious merit. 492 BELL SYSTEM TECHNICAL JOURNAL One other quantity may be of some interest; that is the phase angle of electronic tuning at half power and at extinction. We already have an expression involving A^o (the value at extinction) in (7.4). By taking ad- vantage of (3.10) and (3.8) (F'igs. 5 and 11), we can obtain Ado vs. Gr/ye 70 50 30 LU 20 ^ ^-^ ^ ^^ *^^^^ "-\ ^0 *^^ \ \, 2 """-- \ "'"H^ 'X x\ \ 0.4 0.5 0.6 M _ Gr Qoy ye 0.8 Fig. 17. — The phase of the drift angle for extinction and half power vs the ratio o resonator loss to small signal electronic admittance. { = M/Qye) for optimum loading. By referring to Fig. 12 we can obtain the relation for A6i (the value at half power) Gc = ye [2Ji{Xo/V2)/{Xo/\/2)] cos A^j. However, we have at A^ = Gc = ye [2/,(A'-o)/Xo]. Hence cos AOi = JiiXo) V2MXo/V~2)- (7.11) (7.12) (7.13) Again, from (3.10) and (3.8) we can express A'o for optimum j^ower at Ad = in terms of Gc/ye • In Fig. 17, A^o and A6{ are plotted vs. Gc/ye for optimum loading. REFLEX OSCILLATORS VIII. Hysteresis 493 All the analysis presented thus far would indicate that if a reflex oscillator is properly coupled to a resistive load the power output and frequency will be single-valued functions of the drift time or of the repeller voltage, as illustrated in Fig. 18. During the course of the development in these labora- tories of a reflex oscillator known as the 1349XQ, it was found that even if NEGATIVE REPELLER VOLTAGE >■ Fig. 18. — Ideal variation of power and frequenc\- with repeller voltage, arbitrary units. the oscillator were correctly terminated the characteristics departed vio- lently from the ideal, as illustrated in Fig. 19. Further investigation dis- closed that this departure was, to a greater or less degree, a general charac- teristic of all reflex oscillators in which no special steps had been taken to prevent it. The nature of this departure from expected behavior is that the output is not a single valued function of the repeller voltage, but rather that at a given repeller voltage the output depends upon the direction from which the repel- 494 BELL SYSTEM TECHNICAL JOURNAL ler voltage is made to approach the given voltage. Consider the case illus- trated in Fig. 19. The arrows indicate the direction of repeller voltage vari- ation. If we start from the middle of the characteristic and move toward more negative values of repeller voltage, the amplitude of oscillation varies continuously until a critical value is reached, at which a sudden decrease in NEGATIVE REPELLER VOLTAGE »- Fig. 19.— A possible variation of power and frequency with repeller voltage when there is electronic hysteresis. The arrows indicate the direction of variation of repeller voltage. amplitude is observed. This drop may be to zero amplitude as shown or to a finite amplitude. In the latter case the amplitude may again decrease con- tinuously as the repeller voltage is continuously varied to a new critical value, where a second drop occurs, etc. until finally the output falls to zero. In every observed case, even for more than one drop, the oscillation always dropped to zero discontinuously. Upon retracing the repeller voltage varia- tion, oscillation does not restart at the repeller voltage at which it stopped but remains zero until a less negative value is reached, at which point the REFLEX OSCILLA TORS 495 oscillation jumps to a large amplitude on the normal curve and then varies uniformly. The discontinuities occur sometimes at one end of the charac- teristic and sometimes at the other, and infrequently at both. It was first thought that this behavior was caused by an improper load/ but further investigation proved that the dependence on the load was secondary and the conclusion was drawn and later verified that the effect had its origin in the electron stream. For this reason the discontinuous behavior was called electronic hysteresis. In any self-excited oscillator having a simple reasonant circuit, the os- cillating circuit may be represented schematically as shown in Fig. 20. Here L and C represent the inductance and capacitance of the oscillator. Gr is a shunt conductance, representing the losses of the circuit, and Gi is the conductance of the load. Henceforth for the sake of convenience we •Gr Fig. 20. — Equivalent circuit of reflex oscillator consisting of the capacitance C, induct- ance L, the resonator loss conductance Gr, the load conductance G^ and the electronic admittance W ■ will lump these and call the total Gl ■ Ye represents the admittance of the electron stream. Such a circuit has a characteristic transient of the form V = Voe" (8.i: where Ge+Gi 2C and Vlc' Oscillations will build up spontaneously if Geo + Gi < . (8.2) For stable oscillation at amplitude V we require Ge[V] + Gi = (8.3) (8.2) and (8.3) state that the amplitude of oscillation will build up until non-linearities in the electronic characteristics reduce the electronic con- ductance to a value equal and opposite to the total load plus circuit con- ductance. Thus, in general Ye = G,o/'i(F) + jBeoF^iV) (8.4) ' See Section IX. 496 BELL SYSTFAf TECHNICAL JOURNAL wliere Ve = G.0 + jBrO (8.5) is the admittance for vanishing; amplitude, wliicli is taken as a reference value. The foregoing facts are familiar to an}' one who has worked with oscillators. Now-, condition (8.,\) ma}- be satisfied although (8.2) is not. Then an oscillator will not be self-starting, although once started at a sulTiciently large amplitude its operation will become stable. An example in common experience is a triode Class C oscillator with fixed grid bias. In such a case ■ F{Vi) > /<(()) (8.6) holds for some Fi . As an example of normal behavior, let us assume that F(V) is a continu- ous monotonically decreasing function of increasing V, with the reference value of V taken as zero. Then the conductance, G> = G(oF{V) will vary with V as shown in Fig. 21. Stable oscillation will occur when the ampli- tude Vi has built up to a value such that the electronic conductance curve intersects the horizontal line representing the load conductance, Gi . G,o is a function of one or more of the operating parameters such as the elec- tron current in the vacuum tube. If w-e vary any one of these parameters indicated as X„ the principal effect will be to shrink the vertical ordinates as show-n in Fig. 21 and the amplitude of oscillation will assume a series of stable values corresponding to the intercepts of the electronic conductance curves with the load conductance. If, as we have assumed, F{V) is a monotonically decreasing function of F, the amplitude will decrease con- tinuously to zero as we uniformly vary the parameter in such a direction as to decrease Geo . Zero amplitude will, of course, occur when the curve has shrunk to the case where Gco = Gl . Under these conditions the power output, ^GlV-, will be a single value function of the parameter as shown in Fig. 22 and no hysteresis will occur. Suppose, however, that F{V) is not a monotonically decreasing function of V but instead has a maximum so that G,qF{V) appears as shown in Fig. 23. In this case, if we start with the condition indicated by the solid line and vary our parameter A' in such a direction as to shrink the curve, the ampli- tude will decrease smoothly until the parameter arrives at a value of A'5 corresponding to amplitude Fsat which the load line is tangent to the maxi- mum of the conductance curve. Further variation of A' in the same direc- tion will cause the amplitude to jump to zero. Upon reversing the direction of the variation of the parameter, oscillation cannot restart until X arrives at a value A'4 such that the zero amplitude conductance is equal to the load conductance. When this occurs the amplitude will suddenly jump to the REFLEX OSCILLATORS 497 , , ^"\X Ge = Geo (x) F M QJ <J) ^ 111 U z ^ N. \ < H o \ Z) \ a \^ \ z \^ \ o u ^ ^^\ \ \ o ^^^^^ > V ^V \ z ^^^^ >v \ \ NEGATIVE OF LOAD o ^\, N. N^ \^ CONDUCTANCE, -Gl K "^N. ^\. ^V \. u UJ ^^^ "^ N. \ \ _J ^s^ ^v ^v^ ^^ UJ "^^^ AMPLITUDE OF OSCILLATION, V *- Fig. 21. — A possible variation of electronic conductance with amplitude of oscillation for the general case of an oscillator. Arbitrary units are employed. Different curves correspond to several values of a parameter A' which determines the small signal values of the conductance. The load conductance is indicated by the horizontal line. Stable oscillation for any given value of the parameter A' occurs at the intersection of the elec- tronic conductance curve with the load line Gl- Fig. 22. 21 apply. BUNCHING PARAMETER, X *• -Variation of power output with the parameter X when the conditions of Fig. 498 BELL SYSTEM TECHNICAL JOURNAL value Vi . Under these conditions the power output will appear as shown in Fig. 24, in which the hysteresis is apparent. Let us now consider the conditions obtaining in a reflex oscillator. Fig. 1 shows a schematic diagram of a reflex oscillator. This shows an electron gun which projects a rectilinear electron stream across the gap of a resonator. y^ N. Ge ^ Geo (X) F (V) ^^x^ ^\ \ \\ NEGATIVE OF LOAD y^^ X^ \ \ \ CONDUCTANCE, -Gl 1%^ AMPLITUDE OF OSCILLATION, V — — *■ Fig. 23. — Variation of electronic conductance with amplitude of oscillation of a form which will result in hysteresis. The parameter A' determines the small signal value of the conductance. The horizontal line indicates the load conductance. After the beam passes through this gap it is retarded and returned by a uni- form electrostatic field. If we carry out an analysis to determine the elec- tronic admittance which will appear across the gap if the electrons make one round trip, we arrive at expression 2.2 which may be written Fe = lo^'eMX) [sin 6 -\- j cos d] (8.7) where X = REFLEX OSCILLATORS 499 This admittance will be a pure conductance if = 0o = (« + f ) 27r. As we have seen, in an oscillator designed specifically for electronic tuning, n usually has a value of 3 or greater and the variations M from 6 arising from l-l p4GlV2 BUNCHING PARAMETER, X »• Fig. 24. — A curve of power output vs parameter X resulting from the conductance curves shown in Fig. 23 and illustrating hysteresis. repeller voltage variation are sufficiently small so that the efifect of M in varying .Y may be neglected. In this case we may write Ge = -Je L„ COS ^^ cv ye = c = 2Fo M (8.8) The parameter which we vary in obtaining the repeller characteristic of the tube is Ad. The variation of this parameter is produced by shifting the repeller voltage Vr from the value Fro corresponding to the transit angle do . Since as is shown, Fig. 25a, ^ decreases monotonically as V increases, no explanation of hysteresis is to be found in this expression. Fig. 25b shows the smooth symmetrical variation of output with repeller voltage about the value for which A^ = which is to be expected. 500 BELL SYSTEM TECHNICAL JOURNAL Now suppose a second source of conductance Gei exists whose amplitude function is of the form illustrated in Fig. 26a. Let us suppose that for the 1.0 0.9 LU u Z 0.8 < I- o 3 0.7 Q Z o O0.6 O gO.5 cr '-'04 _I LU ,,,0.3 ====: ^ REPELLER VOLTAGES: K (a) -^ -H .^ ^ ^v. vro "^ ^v ^ -^ ^-4- NEGATIVE OF LOAD CONDUCTANCE ,-Gl " ■^ u^ [S \,N k ^I'oi ''^ >> N 1 sN ^ 1 s <: ^ s ^^ ^ Vsj voj ^ ^ •s^ 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 1.0 1.5 2.0 2.5 3.0 AMPLITUDE OF OSCILLATION IN ARBITRARY UNITS ^ >s (b) / / / / x AV|^]>) V / 1 ! 1 k---AVr2----->| \ / 1 1 L 1. Vr3- .i\ 1 1 ^\ / i i ! 1 ^ I* 1 AVr4-----]-- 1 1 1 ! \^ / 1 \ / Vroi \ / \ NEGATIVE REPELLER VOLTAGE »- Fig. 25. — a. Variation of electronic conductance with amplitude of oscillation for an ideal oscillator. The parameter controlling the small signal electronic conductance is the re])eller voltage wl'ich determines the transit angle in the repcilcr region. The horizontal line indicates the load conductance. b. The variation of power output witli tlie repeller voltage which results from the characteristics of Fig. 25a. value of ^0 assumed tlie phase of this coiukiclaiue is such as to oppose Gel , Gel may or may not be a function of Id. For the sake of simplicity let us assume that G^o varies with \d in the same way as Gei . The total conduc- REFLEX OSCILLATORS 501 0.8 V 0.6 5 ,0.2 '0.1 ;,.o I 8 0.6 RESELLER VOLTAGES:^ (a) /^^-^^ : ./ ^^v:"-/^ ^°^ N S, y ^.>-^ ■iWf N C^ y -^ f^-5"\.^^V NEGATIVE OF , ! 1 'V^ \ \ V ^ " LOAD CONDUCTANCE ,-Gl ^^ ?u>^ '4 P^ V^ ^<i<:i^ C3^ V4 V5 :vo 1 ^^ Gei "Ge2 (b) s^l ^ <, ^^^-"X^ 1 j ! , ! 1 i '0 1.5 2 2.5 3.0 3.5 AMPLITUDE OF OSCILLATION IN ARBITRARY UNITS 1- U-Z v§ (c) LU>- 1-a. 4 y y [""""'■^ N V -it acD << 3 / / \ N / Vroj N "'.^ • U<- -— - AV lA*"^ 4 0^ 1 vli i 1 1 \a.'\.'? <J 1 L_ r4 - f -»' NEGATIVE REPELLER VOLTAGE - Fig. 26. — a. Curve Ga shows the variation of electronic conductance with amplitude of oscillation for an ideal reflex oscillator. Curve Ge2 represents the variation of a second source of electronic conductance with amplitude. The difference of these two curves indicated Gei-Gti shows the variation of the sum of these two conductance terms with amplitude. h. Electronic conductance vs amplitude of oscillation when two conductance terms exist whose variation with repeller voltage is the same. c. Power output vs repeller voltage for a reflex oscillator in which two sources of con- ductance occur varying with amplitude as shown in Fig. 26b. 502 BELL SYSTEM TECHNICAL JOURNAL tance d = Ge\ — Gti will appear as shown. As the repeller voltage is varied from the optimum value the conductance curve will shrink in proportion to cos A0, and the amplitude of oscillation for each value of M will adjust itself to the value corresponding to the intersection of the load line and the con- ductance plot as shown in Fig. 26b. When the load line becomes tangent, as for amplitude F4 , further variation of the repeller voltage in the same direction will cause oscillation to jump from F4 to zero amplitude. Cor- respondingly, on starting oscillation will restart with a jump to Vz . Hence, two sources of conductance varying in this way will produce conditions pre- viously described, which would cause hysteresis as shown in Fig. 26c. The above assumptions lead to hysteresis symmetrically disposed about the optimum repeller voltage. Actually, this is rarely the case, but the ex- planation for this will be deferred. Fig. 27 shows repeller characteristics for an early model of a reflex oscil- lator designed at the Bell Telephone Laboratories. The construction of this oscillator was essentially that of the ideaUzed oscillator of Fig. 1 upon which the simple theory is based. However, the repeller characteristics of this oscillator depart drastically from the ideal. It will be observed that a double jump occurs in the amplitude of oscillation. The arrows indicate the direction of variation of the repeller voltage. The variation in the fre- quency of oscillation is shown, and it will be observed that this also is dis- continuous and presents a striking feature in that the rate of change of fre- quency with voltage actually reverses its sign for a portion of the range. A third curve is shown which gives the calculated phase A0 of the admittance arising from drift in the repeller field. This lends very strong support to the hypothesis of the existence of a second source of conductance, since this phase varies by more than 180°, so that for some part of the rangelhe repel- ler conductance must actually oppose oscillation. The zero value phase is arbitrary, since there is no way of determining when the total angle is {n + f)27r. Having recognized the circumstances which can lead to hysteresis in the reflex oscillator, the problem resolves itself into locating the second source of conductance and eliminating it. A number of possible sources of a second conductance term were in- vestigated in the particular case of the 1349 oscillator, and most were found to be of negligible importance. It was found that at least one important second source of conductance arose from multiple transits of the gap made by electrons returning to the cathcde region. In the case of the 1349 a de- sign of the electron optical system which insured that the electron stream made only one outgoing and one return transit of the gap eliminated the hysteresis in accordance with the hypothesis. REFLEX OSCILLATORS 503 Inasmuch as multiple transits appear to be the most common cause for hysteresis in reflex oscillator design, it seems worthwhile to obtain a more detailed understanding of the mechanism in this case. Other possible z9 100 90 80 70 60 50 40 30 20 to 50 40 30 20 10 -10 -20 -30 -40 ^ ^^^ .^ / ■\ N 1 f / / / / : 1 (a) \ \ / > 1 r ^^. ,^'' V, X ' ^ y;^ -^ Af^ x' ^' / ^ "^ .^' ^ (b) / ^> ■le / f / , ■'" <D < 7 80 1- UJ 60 a. T-UJ n> 40 n-"- H-> oo 20 't'- oo u,a -I -20 o in /iiJ < UJ -4 UJ u (/I UJ <n -60 Q-Z UJ > -80 ^7 1 20 NEGATIVE 130 140 REPELLER VOLTAGE Fig. 27. — Amplitude, frequency and transit phase variation with the repeiler voltage obtained experimentally for a reflex oscillator exhibiting electronic hysteresis. The arrows indicate the direction of variation of the repeiler voltage. mechanisms such as velocity sorting on the repeiler will give rise to similar effects and can be understood from what follows. In the first order theory, the electrons which have retraversed the gap are conveniently assumed to vanish. Actually, of course, the returning stream is remodulated and enters the cathode space. Unfortunately, the 504 BELL SYSTEM TECHNICAL JOURNAL conditions in the cathode region are very complex, and an exact analysis would entail an unwarranted amount of effort. However, from an approxi- mate analysis one can obtain a very simple and adequate understanding of the processes involved. Let us examine the conditions existing after the electrons have returned through the gap of the idealized reflex oscillator. In the absence of oscilla- tion, with an ideal rectilinear stream and ideally fine grids all the electrons which leave the cathode will return to it. When oscillation exists all elec- trons which experience a net gain of energy on the two transits will be cap- tured by the cathode, while those experiencing a net loss will not reach it, but instead will return through the gap for a third transit, etc. In a prac- tical oscillator even in the absence of oscillation only a fraction of the elec- trons which leave the cathode will be able to return to the cathode, because of losses in axial velocity produced by deflections by the grid wires and vari- ous other causes. As a result, it will not be until an appreciable amplitude of oscillation has been reached that a major proportion of the electrons which have gained energy will be captured by the cathode. On the other hand, there will be an amplitude of oscillation above which no appreciable change in the number captured will occur. The sorting action which occurs on the cathode will produce a source of electronic admittance. Another contribution may arise from space charge interaction of the returning bunched beam with the outgoing stream. A third component arises from the continued hunching , ^suiting from the iirst transit of the gap. From the standpoint of this third component the reflex oscillator with multiple transits suggests the action of a cascade amplifier. The situation is greatly complicated by the nature of the drift field in the cathode space. All three mechanisms suggested above may combine to give a resultant second source. Here we will consider only the third com- ponent. Consider qualitatively what happens in the bunching action of a reflex oscillator. Over one cycle of the r.f. field, the electrons tend to bunch about the electron which on its first transit crosses the gap when the field is changing from an accelerating to a decelerating value. The group re- crosses the gap in such a phase that the field extracts at least as much energy from every electron as it gave up to any electron in the group. When we consider in addition various radial deflections, we see that very few of the electrons constituting this bunch can be lost on the cathode. Although it is an oversimplification, let us assume that we have a linear retarding field in the cathode region and also that none of the electrons are intercepted on the cathode. To this order of ai)pr()ximation a modified cascade bunching theory would hardly be warranted and we will consider only that the initial bunching action is continued. Under these conditions, REFLEX OSCILLATORS 505 we can show that the admittance arising on the third transit of the gap will have the form F: = +7o ^' Al^ [sin e, + j cos d,] (8.9) where /o is the effective d.c. contributing to the third transit, dt = 6 -\- Be is the total transit angle made up of the drift angle in the repeller space, 6, and the drift angle in the cathode space dc . As before, assume that the small changes in dt caused by the changing repeller voltage over the elec- tronic tuning range exercise an appreciable effect only in changing the sine and cosine terms. Then we may write Y'e=G'e+ jB'e = y'e ^^^^ [siu Ot + j COS 9t] (8.10) where If Ad = di - dto Ci'e = y'e ^'^^jf^P [sin 0,0 cos ABt + cos 0,o sin A0,]. (8.11) C2 V Now AFr Ad = waT + Aw To Vr + V, Ada = AuTc (8.12) AVr Adt = CjOoT + ACOTO + AcOTc • Fr+ Fo We observe that the phase angle of the admittance arising on the third transit varies more rapidly with repeller voltage (i.e., frequency) than the phase angle of the second transit admittance. This is of considerable im- portance in understanding some of the features of hysteresis. Let us consider (8.11) for some particular values of ^ccr di . We remem- ber that 6 1 is greater than 6 and hence Co > Ci . Since this is so, the limit- . . . lAiaV) .... ^ , , ,,^,, 2/i(CiF) mg 1 unction — will become zero at a lower value ot l than — — . C2 F CiV We will consider two cases 6 1 — (« + 4)27r and dt — (// + f)2x. These 506 BELL SYSTEM TECHNICAL JOURNAL correspond respectively to a conductance aiding and bucking the conduct- ance arising on the first return. In case 1 we have ^, /2/i(C2F) (8.13) 2Jl(CiV) Ge - ye Ky 5 y ^e - ye C2V AMPLITUDE OF OSCILLATION. V Fig. 28. — Theoretically derived variation of electronic conductance with amplitude o^ oscillation. Curve Ge represents conductance arising from drift action in the repeller space. Curve Gi represents the conductance arising from continuing drift in the cathode region. G" represents the conductance variation with amplitude which will result if Ge and Ge are in phase opposition. and case 2 ^, , /2/i(C2F) C2 V (8.14) Figure 28 illustrates case (2) and Fig. 29 case (1). If cos M , and cos 16 varied in the same way with repeller voltage, the resultant limiting function would shrink without change in form as the repeller voltage was varied, and it is apparent that Fig. 28 would then yield the conditions for hysteresis and Fig. 29 would result in conditions for a continuous characteristic. If Fig. 28 applied we should e.xpcct hysteresis symmetrical about the opti- mum repeller voltage. We recall, however, that in Fig. 27 hysteresis REFLEX OSCILLATORS 507 occurred only on one end of the repeller characteristic and was absent on the other. The key to this situation lies in the fact that M t and A6 do not vary in the same way when the repeller voltage is changed and the fre- quency shifts as shown in (8.12). As a result, the resulting limiting function does not shrink uniformly with repeller voltage, since the contribution Ge changes more rapidly than G^ . Hence we should need a continuous series of pictures of the limiting function in order to understand the situa- tion completely. \^Gg = Ge+ Gg V ■ -^ r.^,,< 2J, (C2V) .V^ \ "^ ^^^^^^"'^ AMPLITUDE OF OSCILLATION, V >- Fig. 29. — Theoretically derived curves of electronic conductance vs amplitude of oscil- lation. Curve G" shows the variation of the resultant electronic conductance when the repeller space contribution and the cathode space contribution are in phase addition. Suppose we consider Fig. 29 and again assume in the interests of simplicity that Mt and A0 vary at the same rate. In this case we observe that in the region aa' the conductance varies very rapidly with amplitude. This would imply that in this region the output would tend to be independent of the repeller voltage. If we refer again to Fig. 27 we observe that the output is indeed nearly independent of the repeller voltage over a range. We see that these facts all fit into a picture in which, because of the more rapid phase variation of 6 1 than 6 with repeller voltage, the limiting function at one end of the repeller voltage characteristic has the form of Fig. 28. accounting for the hysteresis, and at the other end has the form of Fig. 29, 508 BELL SYSTEM TECHNICAL JOURNAL accounting for the relative independence of the output on the repe'.Ier voltage. In what has been given so far we have arrived qualitatively at an explana- tion for the variation of the amplitude. There remains the explanal i' ,-, for the behavior of the frequency. In this case we plot susceptance as a function of am])litude and, as in the case of the conductance, there will be several contributions. The primary electronic susceptance will be given by Be = ye ^-^-^ sin e. (8.15) Hence, as we vary the parameter M by changing the repeller voltage the susceptance curve swells as the conductance curve shrinks. The circuit condition for stable oscillation is that Be + 2iAcoC = 0. (8.16) A second source of susceptance will arise from the continuing drift in the cathode space. Referring to equation (8.10) we see that this will have the form Be = ye—p^-j^— c^&Qt (8.1/) C2 V and corresponding to equation (8.11) we write B'e = y'e ' ^ ' ^ [cos 0,0 cos ^^ i - sin dt^ sin ^^t\. (8.18) C2 V Consider the functions given by (8.18) for values oi 6 1 — (n + l)2r and (« + f)27r as functions of V. These are the extreme values which we considered in the case of the conductance. The ordinates of these curves give the frequency shift as a function of the amplitude. In case 1 we have Be = —ye ' T/ ^^" ^^' (.^-l^) C2 y and case 2 „/ /2/i(C2F) . /o lr»\ Be = ye ' „ sm Adt . (8.20) C2 V The total susceptance will be the sum of the susceptance appearing across the gap as a result of the drift in the repeller space and the susceptance which appears across the gap as a result of the cascaded drift action in the repeller region and the cathode region. If sin Adt and sin Ad varied in the same way with the repeller voltage, the total susceptance would expand REFLEX OSCILLATORS 509 or contract without change in form as the repeller voltage was varied. In Figs. 30 and 31a family of susceptance curves are shown corresponding respectively to cases 1 and 2 above for various values of A0( , assuming that Ml and A0 vary in the same way with the repeller voltage. As the (J 1 (a) Ae.L=4),= o^-;::;:=:- ^r:;::::;-^^^ V 57^ --^^ ___ — ■ — ' ^ ^\r "^^^ V5 V4 V3V2V, AMPLITUDE OF OSCILLATION, V Fig. 30.- — a. Theoretical variation of electronic conductance vs amplitude of oscillation in the case in which two components are in phase opposition. The parameter is the re- peller transit phase. It is assumed that the two contributions have the same variation with this phase. h. Susceptance component of electronic admittance as a function of amplitude for the case of phase opposition given in Fig. 30a. The parameter is the repeller phase. The dashed line shows the variation of amplitude with the susceptance shift. repeller voltage is varied the amplitude of oscillation will be determined by the conductance Umiting function. In the case of the susceptance we cannot determine the frequency from the intersection of the curve with a load line. The frequency of oscillation will be determined by the drift angle and the amplitude of oscillation. The amplitude variation with 510 BELL SYSTEM TECHNICAL JOURNAL angle may be obtained from Fig. 30a, which gives the conductance family. This gives the frequency variation with angle indicated by tlie curve con- AMPLITUDE OF OSCILLATION, V *■ Fig. 31. — Theoretical variation of the susceptaiice components of electronic admittance vs amplitude of oscillation for the case in which two components of electronic susceptance are in phase addition. necting the dots of Fig. 301). On the assumption that A0, and A0 vary at the same rate with repeller voltage a symmetrical variation about A0 = will occur as shown in Fig. 30b. However, from the arguments used con- REFLEX OSCILLATORS 511 cerning the conductance the actual case would involve a transition from the situation of Fig. 30b to that of Fig. 31. If a discontinuity in amplitude occurs in which the amplitude does not go to zero, it will be accompanied by a discontinuity in frequency, since the discontinuity in amplitude in general wall cause a discontinuity in the susceptance. If this discontinuity in susceptance occurs between values of the amplitude such as Va and Vh of Fig. 30, we observe that the direction of the frequency jump may be opposite to the previous variation. We also observe that if the rate of change of susceptance with amplitude is greater than the rate of change of susceptance with Ad, then in regions such as that lying between zero ampli- tude of Vb the rate of change of frequency with A0 may reverse its direction. One can see that because of the longer drift time contributing to the third transit the conductance arising on the third transit may be of the same order as that arising on the second transit. In oscillators in which several repeller modes, i.e., various numbers of drift angles, may be displayed, one finds that the hysteresis is most serious for the mcdes with the fewest cycles of drift in the repeller space. One might expect this, since for these mcdes the contribution from the cathode space is relatively more important. Some final general remarks will be made concerning hysteresis. One thing is obvious from what has been said. With the admittance conditions as depicted, if all the electronic operating conditions are fixed and the load is varied hysteresis with load can exist. This was found to be true ex-peri- mentally, and in the case of oscillators working into misterminated long lines it can produce disastrous effects. Where hysteresis is severe enough, it will be found that what we have chosen to call the sink margin will be much less than the theoretically expected value. An illustration of this is given in Fig. 109. The explanation which we have given for the hysteresis in the reflex oscillator depends upon the existence of two sources of conductance. This was apparently a correct assumption in the case studied, since the elimina- tion of the second source also eliminated the hysteresis. It is possible, however, to obtain hysteresis in a reflex oscillator with only a single source. This can occur if the phase of the electronic admittance is not independent of the amplitude. Normally, in adjusting the repeller voltage the value is chosen for the condition of maximum output. This means that the drift angle is set to a value to give maximum conductance for large amplitude. If the drift angle is then a function of the amplitude, this will mean that for small amplitude it will no longer be optimum. Thus, although the limiting function ^ tends to increase the electronic conductance as the ampli- tude declines, the phase factor will oppose this increase. If the phase factor depended sufficiently strongly on the amplitude, the decrease in Gr caused by 512 BELL SYSTEM TECHNICAL JOURNAL the phase might outweigh the increase due to the function ^ '^ . Asa CiV result the conductance niiglit have a maximum value for an amplitude greater than zero, leading to the conditions shown in Fig. 23, under which hysteresis can exist. The first order theory for the reflex oscillator does not predict such an effect, since the phase is independent of amplitude. The second order theory gives the admittance as _ ^ihO 2Ji(X) y(e_(^/2)) /., _ 1 .. (8.21) ■ I i\-(A- + 1) - X-' ^-^ - -^^ (2 - A-) - X ^1^ The quantity appearing outside the brackets is the admittance given by the first order theory. The second order correction contains real and imaginary parts which are functions of A" and hence of the amplitude of oscillation. Thus, for fixed d-c conditions the admittance phase depends upon the am- plitude of oscillation and hence hysteresis might occur. It should be ob- served that the correction terms are important only for small values of the transit angle 9. In particular, this explanation would not suffice for the case described earlier since the design employed which eliminated the hys- teresis left the variables of equation (8.21) unchanged. IX. Effect of Load So far we have considered the reflex oscillator chiefly from the point of view of optimum performance; that is, we have attempted chiefly to evaluate its performance when it is used most advantageously. There has been some discussion of non-optimum loading, but this has been incidental to the general purpose of the work. Oscillators frequently are worked into other than optimum loads, sometimes as a result of incorrect adjustment, some- times through mistakes in design of equipment and quite frequently by intention in order to take advantage of particular properties of the reflex oscillator when worked into specific non-optimum loads. In this section we will consider the effects of other than o])timum loads on the performance of the reflex oscillator. We may divide this discussion into two major subdivisions classified according to the type of load. The first type we call fixed element loads, and the second variable element loads. The first type is constructed of arbitrary passive elements whose constants are independent of frequency. The second category includes loads con- structed of the same tyi)e of elements but connected to the oscillator by lines of suflicient length so that the frequency variation of the load admit- tance is appreciably modified by the line. REFLEX OSCILLATORS 513 A. Fixed Element Loads In this discussion it will be assumed initially that M, the phase angle of — Ye , is not affected by frequency. The results will be extended later to account for the variation of A0 with frequency. A further simplification is the use of the equivalent circuit of Fig. 118, Appendix I. Initially, the output circuit loss, R, will be taken as zero, so the admittance at the gap will be Yc = Gr^ 2jM^oi/oi + Yl/N\ (9.1)8 Here, Gr is the resonator loss conductance, M is the resonator characteristic admittance, and Fj, is the load admittance. We will now simplify this further by letting Gk = F. = 2iMAco/co + Yl/N\ (9.2) From Fig. 12 we see GJN^ = yA2Ji(X)/X] cos Ad (9.3) B, ^ 2MAC. ^ _y^i2MX)/X] sin Ad . (9.4) Now it is convenient to define quantities expressing power, conductance and susceptance in dimensionless form. p = X^G^/2.Smye (9.5) Gi = GjWye (9.6) ^1 = Bz./7V2y«. (9.7) The power P produced by the electron stream and dissipated in G^, is related to p e-^>- P = (^-^7 P- (9.8) In terms of p and Gi , (9.3) can be written p = (l/1.25)(2.5/>/Gi)~Vi[(2.5/^/Gi)1 cos A0. (9.9) By dividing (9.4) by (9.3), we obtain Aco/coo = (-Gi/2A'W) tan A^ - BlI2X'^M (9.10) - (2M/ye)Aco/a'o = Gi tan AQ -^B,. (9.11) * To avoid confusion on the reader's part, it is perhaps well to note that we are, for the sake of generality, changing nomenclature. Hitherto we have used F/, to denote the load at the oscillator. Actually our load as the appendix shows is usually coupled by some transformer whose ecjuivalent transformation ratio is 1/A'^, so that the admittance at the gap will be YiJN^. 514 BELL SYSTEM TECHNICAL JOURNAL Equations (9.9) and (9.11) give the behavior of a reflex oscillator with zero output circuit loss as the load is changed. It is interesting to plot this behavior on a Smith chart. Such a plot is known as a Rieke diagram or an impedance performance chart. Suppose we iirst make a plot for A^ = 0. This is shown in Fig. 32. Constant p contours are solid and, as Fig. 32. — Theoretical Rieke diagram for a reflex oscillator operating with optimum drift angle. The resonator is assumed lossless. Admittances are normalized in terms of the small signal electronic admittance of the oscillator so that oscillation will stop for unity standing wave. can be seen from the above, they will coincide with the constant conductance lines of the chart. Constant frequency curves are dashed and, for M = 0, they coincide with the locii of constant susceptance. The numbers on the frequency contours give values of (2M/ye)(Aco/wo). The choice of units is such that Gi = 1 means that the load conductance is just equal to small signal electronic conductance which, it will be recalled, is the starting condi- tion for oscillation. Hence, the d = 1 contour is a zero power contour. Any larger values of Gi will not permit oscillation to start, so the Gx contour 'P. H. Smith, "Transmission Line Calculator," Electronics, Jan. 1939, pp. 29-31 REFLEX OSCILLATORS 515 bounds a region of zero power commonly called the "sink," since all the frequency contours converge into it. The other zero power boundary is the outer boundary of the chart, Gi = 0, which, of course, is an open circuit load. The power contours on this chart occur in pairs, except the maximum power contour which is single. These correspond to coupling greater than and less than the optimum. The value of Gi for any given power contour for A0 = may be deter- mined by referring to Fig. 9. We are assuming no resonator loss so we use the curve for which Gulje = 0. From (9.5), ii p = 1 we have Gt/N^ye = 2.5/X- which, substituted in (9.3), gives XJi{X) = 1.25. This is just the condition for maximum power output with no resonator loss. From this it can be seen that we have chosen a set of normalized coordinates. Hence, in using Fig. 9, we have p = H/Hm, where Hm = .394 is the maximum gen- eralized efficiency. Thus, for any given value of p we let H in Fig. 9 have the value .394/> and determine the two values of Gi corresponding to that contour. From Fig. 32 we can construct several other charts describing the per- formance of reflex oscillators under other conditions. For instance, sup- pose we make M other than zero. Such a condition commonly occurs in use either through erroneous adjustment of. the repeller or through inten- tional use of the electronic tuning of the oscillator. We can construct a new chart for this condition using Fig. 32. Consider first the constant power contours. Suppose we consider the old contour of value pn lying along a conductance line Gin . To get a new contour, we can change the label from pn to pm = pn cos A0, and we move the contour to a conductance line Gn = Gm cos A0. That this is correct can be seen by substituting these values in (9.9). Consider a given frequency contour lying along Bi . We shift each point of this contour along a constant conductance line Gi„ an amount B^ = Gin tan M. It will be observed that this satisfied (9.11). In Fig. 33 this has been done for tan M — \, cos A0 = ■s/ll'l. Now let us consider the effect of resonator loss. Suppose we have a shunt resonator conductance Gr . Let G. = Gnhe. (9.12) Then, if the total conductance is G„ , the fraction of the power produced which goes to the load is / = {Gn - G,)/Gn = Gi/(Gi + G,) (9.13) accordingly, we multiply each power contour label by the fraction/. Then we move all contour points along constant susceptance lines to new values G„. = Gn- G2 (9.14) In Fig. 34, this has been done to the contours of Fig. 32, for G-z = .3. 516 BELL SYSTEM TECHNICAL JOURNAL The diagrams so far o])tained have been based on the assumption that A0 has been held constant. To obtain such a diagram experimentally would be extremely difficult. It would require that, as the frequency changed through load puUing, and hence the total transit angle d = IttJt changed, an adjust- ment of the repeller voltage be made to correct the change. In actual practice, Rieke diagrams for a reflex oscillator are usually made holding the LOAD POWER AG LOAD POWER Ae= Fig. 33. — A transformation of the Rieke diagram of Fig. 32 showing the effect of shifting the drift angle away from the optimum l)v 45°. transit time r constant or in other words, with fixed operating voltages. What this does to the basic diagram of Fig. 32 is not difficult to discover, I)rovided that bd is sufficiently small so that we may ignore the variations of the Bessels functions with bd. We will tirst investigate the effect of fixed repeller voltage on the constant frequency contours. To do this we will rewrite (*X11), rei)lacing A0 by A^ + bd and expand. Aco ACOT = COoT Wo (9.15) REFLEX OSCILLATORS 517 POWER INTO LOAD FOR 62= 03 MAX. POWER INTO LOAD FOR 62= 0.3 LOAD POWER G2 = 0.3 ___ A = (2M1 (AOJ^ Fig. 34. — A transformation of the Rieke diagram of Fig. 32 to show the effect of the resonator loss if the phase angle is assumed to be optimum. In rewriting (9.11) we will also replace Gi by Gi + G^ , to take resonator loss into account. We obtain for very small values of hd -(2M/3;,)(Aco/a'o) = ((Gi + G2) tan A^ + B,)S (9.16) S = 1/(1 + (Gi + G2)wor/(2M/>;,) cos^ A^) S = 1/(1 + wor/2() cos- A^). (9.17) Q is the loaded Q of the oscillator. To obtain the new constant frequency contours in the case of A^ = we shift each point of the old contour from its original position at a sus- ceptance B,, along a constant conductance line G^,, to a new susceptance line B,n = B„/S. This neglects a second order correction. It will be observed that for small values of the conductance Gi near the outer boundary, the frequency shifts will be practically unchanged, but near the sink where the 518 BELL SYSTEM TECHNICAL JOURNAL conductance Gi is large the effect is to shift the constant frequency contours along the sink boundary away from the zero susceptance line to larger sus- ceptance values. Hence, the constant frequency contours no longer coincide with the constant susceptance contours, not even for A0 = 0. The change in the power contours is considerably more marked. As the frequency of the oscillator changes the transit angle is shifted from the optimum value by an amount bd = (Aco/coo)c<;or. Thus the electronic conductance is reduced in magnitude by a factor cos — coot. In particular. Wo for the sink contour where the load conductance is just equal to the elec- tronic conductance we see that when the repeller voltage is held constant the power contour lies not on the Gi = 1 — G2 contour but on the locus of Ao) values Gi = cos — wot — d . In order to determine the power contours when the transit time rather than the transit angle is held constant we make use of (9.3) with addition of resonator loss. In normalized coordinates ((9.6) and (9.12)) and for a phase angle of electronic admittance 86 we have Gi + G2 = '^^^^ cos 89 . (9.18) From (9.5) and (9.13) we have for the power output Gi 2XJi{X) ,_ . . Along any constant frequency contour 86 is constant and has the value given by (9.15) in terms of wo and coqt. Hence, it will be convenient to plot (Gi + G2) vs X for various values of 86 as a parameter. This has been done in Fig. 35. The angle 86 has been specified in terms of a parameter A which appears in the Rieke diagrams as a measure of frequency deviation. ^=^^ (9.20) ye Wo In terms of the parameter A 86 = (y,/2A/)(coor)/l . (9.21) Once we have the curves of Fig. 35 we can find the power for any point on the impedance performance chart. We may, for instance, choose to find the power along the constant frequency contours, for each of which A (or 86) has certain constant values. We assume some constant resonator loss G2 . Choosing a point along the contour is merely taking a particular value of Gi . Having 86, G2 and Gi we can obtain A^ from Fig. 35. Then, knowing A^, we can calculate the power from (9.19). REFLEX OSCILLATORS 519 In constructing an impedance performance chart we want constant power contours. In obtaining these it is convenient to assume a given value of G2 . We will use G2 = -3 as an example. Then we can use Fig. 35 and a95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 I 0.50 0.45 0.40 0.35 0.30 0.25 11^ xN \, ^^ N . N^=o \\ \ K\ ^ ^^ N >N\\ V N ^" \^ \ .WW ^" \ \^ N, \ \ i 3^67, v ^ \ \^ \^ ^^ 4.36 .\ vV '^ ^ ^ rv 1 G2 = 0.3 i "^ ^\ ^^ ^ sV ^^ X ^ \x m ^ ^ 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 BUNCHING PARAMETER, X Fig. 35. — Curve of load plus loss conductance vs bunching parameter X for various values of a parameter A which gives the deviation in the drift time from the optimum time. The load and loss conductance are normalized in terms of the small signal elec- tronic admittance. The horizontal line represents a loss conductance of G2 = .3. (9.19) to construct a family of curves giving p vs Gi with A (or 86) as a parameter. In a particular case it was assumed that M/y, = 90 COOT = 27r(7 + f). 520 BELL SYSTEM TECHNICAL JOURNAL These values are roughly those for the 2K25 reflex oscillator. Figure 36 shows p vs Gi for the particular parameters assumed above. The curves were obtained by assuming values of Gi for an approj)riate .1 and so obtain- ing values of .V from Fig. 35. Then the power was calculated using (9.19) and so a curve of j)ower vs d for a })articular value of .1 was constructed. Figure 37 shows an impedance performance chart obtained from (9.16) and Fig. 36. In using Fig. 36 to obtain constant power contours, we need merely note the values of Gi at which a horizontal line on Fig. 36 intersects the curves for various values of A. Each curve either intersects such a horizontal (constant power) line at two points, or it is tangent or it does not intersect. The point of tangency represents the largest value of A at which the power can be obtained, and corresponds to the points of the crescent shaped power contours of the impedance performance chart. The maximum power contour contracts to a point. Along the boundary of the sink, for which p — 0, X = and we have from (9.18) Gi = cos bd - Gi. (9.22) The results which we have obtained can be extended to include the case in which Id 9^ 0. Further, as we know from Appendix I, we can take into account losses in the output circuit by assuming a resistance in series with the load. In a well-designed reflex oscillator the output circuit has little loss. The chief effect of this small loss is to round off the points of the constant power contours. In actually measuring the performance of an oscillator, output and fre- quency are plotted vs load impedance as referred to the characteristic impedance of the output line. Also, frequently the coupling is adjusted so that for a match (the center of the Smith chart) optimum power is obtained. We can transform our impedance performance chart to correspond to such a plot by shifting each point G, B on a contour to a new point Gi = G/Gxaax Bi = B/Gmas where Gmax is the conductance for which maximum power is obtained. Such a transformation of Fig. 37 is shown in Fig. 38. It will be noted in Fig. 38 that the standing wave ratio for power, the sink margin, is about 2.3. This sink margin is nearly independent of the resonator loss for oscillators loaded to give maximum power at unity stand- ing wave ratio, as has been discussed and illustrated in Fig. 10. If the sink margin must be increased or the pulling figure must be decreased^" the coup- '" The pulling figure is arbitrarily defined as the maximum frequency excursion pro- duced when a voltage standing wave ratio of v 2 is presented to the oscillator and the phase is varied through 180°. REFLEX OSCILLATORS 521 0.46 ^=0 0.42 i ^ 1 fe k 0.40 0.38 0.36 0.34 f- KS A ^ \ Y \ \ \^ 1 \ > \\\ \ \\ V \ \ ,\ 0.32 / 2 \ \\ w / \, \ y \\\ 030 ^ / \ \ \ \\\ / s \ \ AV i 0.28 a ^-0.26 / \ y \\\ \ f \ \ \v \ \ \ \ \ LU 2 0.24 \ V \ \^ \\ 2.725 \ \ \ \\ \ Q UJ N 0.22 -J < cr O20 o z ■ ill V \ \ \ \\\ 1 / \ \ \ u\ \ \ V ^ \ \\\ 0.18 ill \ i \ \ \ I \ III 1 \ \ \ \\ \ 0.16 III 1 \ \ \ \\ \\ 1/ 3.335 \ \ \ \^ w 0.14 / / ji^ ~V. \ \ \ \ \ / / / \ \ \ \ \\\ 0.12 / / / \ \ \ Y \\\ // // \ \ \ \\\ 0.10 // 1/ \ \ 1 \ \ /// / \ ^ \ \ W \ 0.08 /// 1 \ \ \ \ w \ //// 3.87 \ \ \ \ \\ \\ 0.06 0.04 III 1 / \ \ \ \ \ w If f V \ ^ \ \ \ i 0.02 IL ^.36 i \ ^ \ — \ \i r \ \ \ 1 \ \\\i 0.05 0.10 015 0.20 0.25 0.30 0.35 0.40 0.45 0.50 NORMALIZED LOAD CONDUCTANCE, Gi 0.55 0.60 0.65 0.70 Fig. 36. — Normalized power vs normalized load conductance for various values of the parameter A which gives the deviation in drift time from the optimum drift time. These curves are computed for the case G2 = .3. Optimum drift angle equal to 15.5 n radians and a ratio of characteristics resonator admittance to small signal electronic admittance of 90 is assumed. 522 BELL SYSTEM TECHNICAL JOURNAL ling can be reduced so that for unity standing wave ratio the load conduct- ance appearing at the gap is less than that for optimum power. Finally, in making measurements the load impedance is usually evaluated at a point several wavelengths away from the resonator. If performance is plotted in terms of impedances so specified, the points on the contours of LOAD POWER G2 = 0.3 MAX.LOAD POWER G2=0.3 LOAD POWER G2=0.3 MAX. LOAD POWER 6^=0 I ^<m^) ye POWER Fig. 37. — A Rieke diagram for a reflex oscillator having a lossy resonator, taking into account the variation of drift angle with frequency pulling. This results in closed power contours. Fig. 38 appear rotated about the center. As the line length in wavelengths will be different for different frequencies, ]:)oints on different frequency contours will be rotated by different amounts. This can cause the contoirs to overlap in the region corresponding to the zero admittance region of Fig, 38. With very long lines, the contours may overlap over a considerable region. The multiple modes of oscillation which then occur are discussed in somewhat different terms in the following section. REFLEX OSCILLATORS 523 Figure 39 shows the performance chart of Fig. 38 as it would appear with the impedances evaluated at a point 5 wavelengths away from the resonator. Figure 71 of Section XIII shows an impedance performance chart for 2K25 reflex oscillator. — -i^K^i Fig. 38. — The Rieke diagram of Fig. 37 transformed to apply to the oscillator loaded for optimum power at unity standing wave. B. Frequency — Sensitive Loads — Long Line Efect When the load presented to a reflex oscillator consists of a long line mis- matched at the far end, or contains a resonant element, the operation of a reflex oscillator, and especially its electronic tuning, may be very seriously affected. For instance, consider the simple circuit shown in Fig. 40. Here Mr is the characteristic admittance of the reflex oscillator resonator as seen from the output line or wave guide and Ml is the characteristic impedance of a line load I long, so terminated as to give a standing wave ratio, <r. 524 BELL SYSTEM TECHNICAL JOURNAL In the simple circuit assumed there are essentially three variables; (1) the ratio of the characteristic admittance of the resonant circuit, Af« to LOAD POWER Fig. 39. — The Rieke diagram of Fig. 38 transformed to include the effect of a hne five wave lengths long between the load and the oscillator. Mr M|_ Fig. 40. — Equivalent circuit of a lossless resonator, a line and a mismatched load. that of the line, Mi, . This ratio will be called the external Q and signified hyQ, Q, = Mn/M^ . (9.23) REFLEX OSCILLA TORS 525 For a lossless resonator and unity standing wave ratio, the loaded Q is equal to Qe ■ For a resonator of unloaded Q, Q» , and for unity standing wave ratio, the loaded Q, obeys the relation \/Q = \/Qe + 1/(3.. (9.24) 5-0.5 1.5 2.0 2.5 3.0 CONDUCTANCE, G Fig. 41. — Susceptance vs conductance for a resonator coupled to a 50 wave length line terminated by a load having a standing wave ratio of 2. Characteristic admittance of the resonator is assumed to be equal to 100 in terms of a line characteristic admittance of unity. The circles mark off relative frequency increments Aco coo 10-3, where coo is the frequency' of resonance. (2) the length of the line called 6 when measured in radians or n when measured in wavelengths, (3) the standing wave ratio a. Figures 41 and 42 show admittance plots for two resonant circuits loaded by mismatched lines of different lengths. The feature to be observed is the loops, which are such that at certain points the same admittance is achieved at two different frequencies. It is obvious that a line representing —Ye 526 BELL SYSTEM TECHNICAL JOURNAL may cut such a curve at more than one pohit : thus, oscillation at more than one frequency is possible. Actually, there may be three intersections per loop. The two of these for which the susceptance B is increasing with fre- quency represent stable oscillation; the intersection at which B is decreasing with frequency represents an unstable condition. The loops are of course due to reactance changes associated with varia- tion of the electrical length of the line with frequency. Slight changes in tuning of the circuit or slight changes in the length of the line shift the loops up or down, parallel to the susceptance axis. Thus, whether the electronic admittance line actually cuts a loop, giving two possible oscillating fre- quencies, may depend on the e.xact length of the line as well as on the ex- D-O.l BETWEEN POINTS k k^ J/ 0.5 0.6 0.7 CONDUCTANCE, G Fig. 42. — Susceptance vs conductance for line 500 wave lengths long terminated by a load having a standing wave ratio of 1.11. Circles mark off relative frequency increments of 10"''. Characteristic admittance to the resonator equals 100. istence of loops. The frequency difference between loops is such as to change the electrical length of the line by one-half wavelength. The existence or absence of loops and their size depend on all three pa- rameters. Things which promote loops are: Low ratio of Mr/M ^ or Qe Large n or 6 High 0- As any parameter is changed so as to promote the existence of loops, the Y curve first has merely a slight periodic variation from the straight line for a resistiveiy loaded circuit. Further change leads to a critical condition in which the curve has cusps at which the rate of change of admittance with frequency is zero. If the electronic admittance line passes through a cusp, REFLEX OSCILLATORS 527 the frequency of oscillation changes infinitely rapidly with load. Still further change results in the formation of loops. Further change results in expansion of loops so that they overlap, giving more than three intersections with the electronic admittance line. Loops may exist for very low standing wave ratios if the line is sufficiently long. Admittance plots for low standing wave ratio are very nearly cy- cloidal in shape; those for higher standing wave ratios are similar to cycloids in appearance but actually depart considerably from cycloids in exact form. By combining the expression for the near resonance admittance of a tuned circuit with the transmission line equation for admittances, the expression for these admittance curves is obtained. Assuming the termination to be an admittance I'V which at frequency wo is do radians from the resonator, 1 -\-j{Yt/Ml) tan 0o(l + Aco/wo) The critical relation of parameters for which a cusp is formed is important, for it divides conditions for which oscillation is possible at one frequency only and those for which oscillation is possible at two frequencies. This cusp corresponds to a condition in which the rate of change with frequency of admittance of the mismatched line is equal and opposite to that of the circuit. This may be obtained by letting Yt be real. Yt/Ml > 1, do = nir where n is an integer. The standing wave ratio is then a = Yt/Ml . (9.26) The second term on the right of (9.25) is then \1 +_;o- tan ^oAco/coo/ For very small values of Aco we see that very nearly 72 = MlW - i(cr2 - l)0oAco/a'o] • (9.28) Thus for the rate of change of total admittance to be zero 2Mh = Ml{c' - 1)60 % = 2{Mj,/ML)(a' - 1) = 2Q^/{a' - 1) . (9.30) Thus, the condition for no loops, and hence, for a single oscillating frequency, may be expressed 00 < IQeHo" - 1) (9.31) 528 BELL SYSTEM TECHNICAL JOIRNAL We will remember that ^o is the length of line in radians, a is the standing wave ratio, measured as greater than unity, and Qe is the external Q of the resonator for unity standing wave ratio. Replacing a given length of line by the same length of wave guide, we fnd that the phase angle of the reflection changes more rapidly with frequency, and instead of (9.31) we have the condition for no loops as e < 2(3^(1 - (X/Xo)2)/(a-^ - 1) (9.32) '^ < Vl +2Qe(1 - (X/Xo)2)/0o- Here X is the free space wavelength and Xn is the cutoff wavelength cf the guide. Equations (9.32) are for a particular phase of standing wave, tl at is, for relations of Yt and 6o which, produce a loop symmetrical abcve the C axis. Loops above the G axis are slightly more locped than Iccps belcw the G axis because of the increase of do with frequency. For reasonably Icng lines, (9.32) applies quite accurately for formation of loops in any position; for short lines locps are cf no consequence unless they are near the G axis. An imporant case is that in which the resonant lead is ccupled to the resonator by means of a line so short that it may be considered to have a constant electrical length for all frequencies of interest. The resonant load will be assumed to be shunted with a conductance equal to the charac- teristic admittance of the line. As the multiple resonance of a long mis- matched line resulted in formation of many locps, so in this case we would rightly suspect the possibility of a single loop. If the resonant load is |, f, etc. wavelengths from the resonator, and both resonate at the same frequency, a loop is formed symmetrical about the G axis. Figure 43 is an admittance curve for resonator and lead placed 5 wavelength apart. Tuning either resonator or load moves this loop up or down. If the distance from resonator to resonant load is varied above or below a quarter wave distance, the loop moves up or down and expands. This is illustrated by an eighth wavelength diagram for the same resonator and load as of Fig. 43 shown in Fig. 44. When the distance from the resonator lo the resonant load, including the effective length of the coupling loop, is 5, 1, 1^, etc. wavelengths, for frequencies near resonance the resonant load is essentially in shunt with the resonator, and its effect is to increase the loaded Q of the resonator. An admittance curve for the case is shown in Fig. 45. In this rase the loo])s REFLEX OSL'ILLA TORS 529 have moved considerably away in frequency, and expanded tremendously. There are still recrossings of the axis near the origin, however, as indicated in this case by the dashed line which represents 2 crossings, in this case about 4% in frequency above and below the middle crossing if the length of the line t is X/2. Mp=I 00 CD 0.25 \ 4^=0.5X10-3 BETWEEN POINTS \ \ \ \ \ /^ -^ ? "^^ _^ . i ! / I 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 CONDUCTANCE, G Fig. 43. — Susceptance vs conductance for two resonators coupled by a quarter wave line. The resonator at which the admittance is measured has a characteristic admittance of 100 in terms of a line characteristic admittance of unit}-. The other resonator has a character- istic admittance of 200 and a shunt conductance of unity. The circles mark off relative frequency increments of 5 X 10"' in terms of the resonant frequency. As a sort of horrible example, an admittance curve for a high () lead 50 wavelengths from the resonator was c( mputcd and is shown in Fig. 46. Only a few of the loops are shown. Admittance curves for more complicated circuits such as impedance trans- formers can be computed or obtained experimentally. 530 BELL SYSTEM TECHNICAL JOURNAL As has been stated, one of the most serious effects of such mismatched long line or resonant loads is that on the electronic tuning. For instance, consider the circuit admittance curve to be that shown in Fig. 47, and the minus electronic admittance curve to be a straight line extending from the origin. As the repeller voltage is varied and this is swung down from the -\-B axis its extreme will at some point touch the circuit admittance line r- - i=-S^-^ ■^ r-T > M = 1 00 M5=200 \ -^=0.5X10-3 BETWEEN POINTS \ ^^ V A ^ N \ \ / A / V J / \ V / ,: J ^- 0.25 0.50 0.75 1.00 1.25 CONDUCTANCE, G Fig. 44. — Susceptance vs conductance for the same resonators as of Fig. 43 coupled by a one-eighth wave line. and oscillation will commence. As the line is swung further down, the frequency will decrease. Oscillation will increase in amplitude until the — Ye line is perpendicular to the I' line. From that point on oscillation will decrease in amplitude until the — Ye line is parallel to the Y curve on the down side of the loop. Beyond this point the intersection cannot move out on the loop, and the frequency and amplitude will jump abruptly to correspond with the other intersection. As the — 1% line rotates further, REFLEX OSCILLATORS 531 amplitude will decrease and finally go to zero when the end of the — Ye line touches the V curve. If the — Ye line is rotated back, a similar phe- nomenon is observed. This behavior and the resulting electronic tuning characteristic are illustrated in Figs. 47 and 48. Such electronic tuning Mr=I 00 Ml=1 Ms=200 m 0.25 ' [ f^^ -0.5X10-3 ' 1^0 BETWEEN POINTS < 1 j OTHER CROSSINGS |,,--AT 2 ±47o 1* IN FREQUENCY ( 1 ( ( ' O.a.-^ 0.50 0.75 I.OO 1.25 1.50 1.75 2.00 CONDUCTANCE, G Fig. 45. — Susceptance vs conductance for the same resonators as of Fig. 43 coupled by a one-half wave line. The dash line indicates two other crossings of the susceptance axis, at frequencies ±4% from the resonant frequency of the resonators. characteristics are frequently observed when a reflex oscillator is coupled tightly to a resonant load. C. Effect of Short Mismatched Lines on Electronic Tuning In the foregoing, the effect of long mismatched lines in producing addi- tional multiplewesonant frequencies and possible modiness in operation has 532 !2 -0.5 2.5 BELL SYSTEM TECHNICAL JOURNAL h«- l=50X ^ G = :. Mr=IOO Ml=i Ms=200 / '■■'' / \ b. / \ j / ^ ^ \, \ / y \ . \ / \ \ \V / \ y\ /\ 7^ b \ \ h N L J 1 A 1 \ / \ \ / \ \ y ^ / \ \ V •^ ^ y / \ / \ V y / V ^ 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 CONDUCTANCE , G Fig. 46. — Susceptance vs conductance for the resonators of Fig. 43 coupled by a line 50 wave lengths long. been explained. The effect of such multiple resonance on electronic tuning has been illustrated in Fig. 48. Tf a short mismatched Hne is used as the load for a reflex oscillator, there REFLEX OSCILLA TORS 53^ may be no additional modes, or such modes may be so far removed in fre- quency from the fundamental frequency of the resonator as to be of little CONDUCTANCE , G • Fig. 47. — Behavior of the intersection between a circuit admittance line with a loop and the negative of the electronic admittance line of a reflex oscillator as the drift angle is varied (circuit hysteresis). REPELLER VOLTAGE ► Fig. 48. — Output vs repeller voltage for the conditions obtaining in Fig. 47. importance. Nonetheless, the short line will add a frequency-sensitive reactance in shunt with the resonant circuit, and hence will change the char- acteristic admittance of the resonator. Sii BELL SYSTEM TECHNICAL JOURNAL Imagine, for instance, that we represent the resonator and the mismatched line as in shunt with a section of Hne N wavelengths or 6 radians long mis- terminated in a frequency insensitive manner so as to give a standing wave ratio <r. If Ml is the characteristic admittance of the line, the admittance it produces at the resonator is Y,=M,f±4^^. (9.33) 1 + ja- tan 6 Now, if the frequency is increased, 6 is made greater and Y is changed. {1 -j- j(T tan&)2 We are interested in the susceptive component of change. If Vz. = Gl+JBj^ (9.35) we find »Bjm = M, " ~ "'Y'r ^ytf """ ' • (9-36) (1 + 0- nan^ 6) Now, if frequency is changed by an amount df, 9 will increase by an a mount 6(df/f) and Bl will change by an amount dB:^ = {dBJdd){2T,N){df/f). (9.37) We now define a parameter Mm expressing the effect of the mismatch as follows TidB^/dd) = Mm. (9.38) Then dBj^ = INMuidf/f). (9.39) If the characteristic admittance of the resonator is Mr , then the characteris- tic admittance of the resonator plus the line is M = Mji-\- NMm. (9.40) If, instead of a coaxial line, a wave guide is used, and Xo and X are the cutoff and operating wavelengths, we have dB^ = 2NMM{df/f)(l - (X/Xo)2)-^ (9.41) and ikr = M« + NMm(1 - (X/Xo)2)-^ (9.42) In Fig. 49 contour lines for Mm constant are plotted on a Smith Chart (reflection coefficient plane). Over most of the plane Mm has a moderate REFLEX OSCILLATORS 535 positive value tending to increase characteristic admittance and hence decrease electronic tuning. Over a very restricted range in the high admit- tance region Mm has large negative values and over a restricted range outside of this region Mm has large positive values. Fig. 49. — Lines of constant value of a parameter. Mm shown on a chart giving the con- ductance and susceptance of the terminating admittance of a short line. The parameter plotted multiplied by the number of wave lengths in the line gives the additional charac- teristic admittance due to the resonant effects of the line. The parameter Mm is of course for terminated lines (center of chart). This is an appropriate point at which to settle the issue: what do we mean by a "short line" as opposed to a "long line." For our present purposes, a short line is one short enough so that Mm does not change substantially over the frequency range involved. Thus whether a line is short or not depends on the phase of the standing wave at the resonator (the position 536 BELL SYSTEM TECHNICAL JOURNAL on the Smith Chart) as well as on the length of the line. Mm changes most rapidly with frequency in the very high admittance region. As a simple example of the effect of a short mismatched line on electronic tuning between half power points, consider the case of a reflex oscillator with a lossless resonator so coupled to the line that the external Q is 100 and the electronic conductance is 3 in terms of the line admittance. Sup- pose we couple to this a coaxial line 5 wavelengths long with a standing wave ratio cr = 2, vary the phase, and compute the electronic tuning for various 100 50 0.04 0.06 008 010 0.12 QW ai6 0.18 0.20 022 Q24 0.26 VOLTAGE STANDING -WAVE RATIO PHASE IN CYCLES PER SECOND Fig. 50. — The normalized load conductance, the characteristic admittance of the resona- tor and the normalized electronic tuning range to half power plotted vs standing wave ratio phase for a particular case involving a short misterminated line. The electronic tuning for a matched line is shown as a heav\' horizontal line in the |ilot of (Aw/coo)! . phases. We can do this by obtaining the conductance and Ml from Fig. 49 and using Fig. 15 to btain (Aw/wo)j . In Fig. 50, the parameters GlIJc (the total characteristic admittance including the effect of the line), A'', and, finally, (Aaj/wo)j have been plotted vs standing wave phase in cycles. (Ac<j/ajo)j for a matched load is also shown. This example is of course not tyi:)ical for all reflex oscillators: in some cases the electronic tuning might be reduced or oscillation might stop entirely for the standing wave phases which produce high conductance. 1 REFLEX OSCILLATORS 537 X. Variation of Power and Electronic Tuning with Frequency When a reflex oscillator is tuned through its tuning range, the load and repeller voltage being adjusted for optimum efficiency for a given drift angle, it is found that the power and efiiciency and the electronic tuning vary, having optima at certain frequencies. When we come to work out the variation of power and electronic tuning with frequency we at once notice two distinct cases: that of a fixed gap spacing and variable resonator (707A), and that of an essentially fixed resonator and a variable gap spacing (723A etc.); see Section XIII. Here we will treat as an example the latter case only. The simplest approximation of the tuning mechanism which can be ex- pected to accord reasonably with facts is that in which the resonator is represented as a fixed inductance, a constant shunt "stray" capacitance and a variable capacitance proportional to 1/rf, where d is the gap spacing. The validity of such a representation over the normal operating range has been verified experimentally for a variety of oscillator resonators. Let Co be the fixed capacitance and Ci be the variable capacitance at some reference spacing di . Then, letting the inductance be L, we have for the frequency CO = (L(Co + Ci d,/d))K (10.1) Suppose we chocse di such that Co = Ci. (10.2) Then, letting d/di = D (10.3>) a'l = (ILCor = 27r/i (10.4) w/a;i = IF. (10.5) IF = 2'(1 + \/D)~K (10.6) We find This relation is shown in Fig. 51, where D is plotted vs TF. It is perfectly general (within the validity of the assumptions) for a proper choice of refer- ence spacing di . We have, then, in Fig. 51a curve of spacing D vs re- duced frequency IF. The parameter which governs the power and eflicency is Gn/ye . We have Cs/jc = (G«/i8')(2Fo//o0). (10.7) As Fo , /o and 6 will not vary in tuning the oscillator, we must look for varia- ton in Gu and (3'^. 538 BELL SYSTEM TECHNICAL JOURNAL For parallel plane grids, we have l/)82 = (V2)Vsin2 {ej2) (10.8) where 6g is the transit angle between grids. We see that in terms of W and D we can write dg = diWD . (10.9) lU - \ // - - V \ / ^7 - \, /y / S ^ .02 J ''/ \ /; / \ \, / ^ /■ ^^ ^ w-i \ \ '\ ^ / ;^' - ,^ ^ - - y ^^ \ \, — — D^' V \ y' '* y \ ^ y / WD \ ^, y X / \ 0.1 / / / \ % V i 0.6 0.7 0.8 0.9 1.0 I.I 1.2 1.3 1.4 RELATIVE FREQUENCY, W Fig. 51. — Various functions of relative frequency W and relative spacing D plotted vs relative frequency. Here B\ is the gap transit angle at a spacing d\ and a frequency TFi . So that we may see the effect of tuning on 1//3-, WD has been plotted vs IF in Fig. 51 and l//3^ has been plotted vs Qg in Fig. 52. We now have to consider losses. From (9.7) of Appendix IX we see that the grid loss conductance can be expressed in the form Gg = GgyW^D^ (10.10) Here Ggi is the grid loss conductance a.t d = di and co = wi . Finally, let us consider the resonator loss. If the resonator could be represented by an inductance L with a series resistance R, at high frequencies the conductance would be very nearly REFLEX OSCILLATORS If R varies as co', we see that we could then write G^ = GnW-K Here Gli is the conductance at a frequency wi . S39 (10.11) (10.12) 1000 800 600 100 80 60 05 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 TRANSIT ANGLE, Gg , IN RADIANS Fig. 52.- — The reciprocal of the square of the modulation coefficient is a function of the gap transit angle in radians for the case of fine parallel grids. As an opposite extreme let us consider the behaviour of the input conduct- ance of a coaxial line. It can be shown that, allowing the resistance of such a line to vary as oj , the input conductance is Gt = ^C0*CSC2(C0//C). (10.13) Here t is the length of the line and C is the velocity of propagation. If Gl given by (10.12) and Gi of 10.13 give the same value of conductance at some angular frequency wi then it will be found that for values of t typical of reflex oscillator resonators the variation of G( with w will be significantly I less than that of Gl • Although typical cavities are not uniform lines I (10.13) indicates that a slower variation than (10.12) can be expected. It will be found moreover that the shape of the power output vs frequency i curves are not very sensitive to the variation assumed. Hence as a rea- sonable compromise it will be assumed that the resonator wall loss varies as 540 BELL SYSTEM TECHNICAL JOURNAL Suppose that at D — 1, i.e. Gs = GsxW~\ (10.14) In Fig. 51 ir~ has been plotted vs W. Now let us take an actual example. {d = d\, (j> = oji) 6 = 2 G,a = .inyye Gs, = .()95/ye The information above has been used in connection with Figs. 51 and 52 and ratio of resonator loss to small signal electronic admittance, Gr/jc, has been plotted vs IF in Fig. 53. A 2K25 oscillator operated at a beam Gr ye 1.0 \ ' / \, 1 t / 1 0.9 s. s . 1 08 s. / \, J 0.7 0.6 ^ V / X «s^ j / ' y 1 0,5 ; ^*****- ^-^^ 1 0.76 0.80 0.84 0.88 092 0.96 1.00 1.04 RELATIVE FREQUENCY, W ' Fig. 53. — Computed variation of ratio of resonator loss