^^^4!^J^J|^t^lvi^ltT^f?^flrrti»T^i>7^iJ7^it7^t^iy^fr^^^^ From the collection of the ^ m 0 Prejinger 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 M. R. Sullivan O. E. Buckley O. B. Blackwell M. J. Kelly H. S. Osborne A. B. Clark J. J. PiLLioD S. Bracken TABLE OF CONTENTS AND INDEX VOLUME XXIII 1944 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK PRINTED IN U. S. A. THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXIII, 1944 Table of Contents January, 1944 The Discernibility of Changes in Program Band Width — D. K. Gannett and Iden Kerney 1 Use of the Etch Technique for Determining Orientation and Twiiming in Quartz Crystals— G. W. Willard 11 Modes of Motion in Quartz Crystals, the Effects of Coupling and Meth- ods of Design — R. A . Sykes ^2 Response of a Linear Rectifier to Signal and Noise — W . R. Bennett 97 Dielectric Constants and Power Factors at Centimeter Wave-Lengths — Carl R. Englund 1 14 April, 1944 Indicial Response of Telephone Receivers — E. E. Mott 135 Theoretical Analysis of Modes of Vibration for Isotropic Rectangular Plates Having All Surfaces Free — H. J. McSkimin 151 Principles of Mounting Quartz Plates — R. A. Sykes 178 The Magnetically Focused Radial Beam Vacuum Tube — A. M. Skellett 190 July, 1944 Effect of Telegraph Distortion on the Margins of Operation of Start- Stop Receivers — W . T. Rea 207 The Mounting and Fabrication of Plated Quartz Crystal Units — R. M. C. Greenidge 234 Effects of Manufacturing Deviations on Crystal Units for Filters — .4. R. D'keedene 260 Mathematical Analysis of Random Noise — S. 0. Rice 282 iii 1164153 FEB 2 8 mb iv bell system technical journal October, 1944 The Conquest of Distance by Wire Telephony — Thomas Shaw 337 Some Aspects of Powder Metallurgy — Earle E. Schumacher and A lexan- der G. Souden 422 ; . i * t • Index to Volume XXIII Alloys: Some Aspects of Powder Metallurgy, Karlc E. Sclmmnchcr and Alexattdcr (J. Soiidett, page 422. B Kand Widtli, I'rogram, The Discernibility of (;!liangcs in, />. K. Gatuuil and Idni Kcrney, page 1. Bennett, W. K., Response of a Linear Rectifier to Signal and Noise, page 97. Crystal Units for Filters, Effects of Manufacturing Deviations on, A. R. D'heedenc, page 260. Crystal Units, Plated Quartz, The Mounting and Fabrication of, R. M. C. i!recnidg,e, page 234. Crystals, Quartz, Use of the Etch Technique for Determining Orientation and Twin- ning in, G. W. Willard, page 11. Crystals, Quartz, Modes of Motion in, the Effects of Coupling and Methods of Design, R. A. Sykes, page 52. Crystals: Theoretical Analysis of Modes of Vibration for Isotropic Rectangular Plates having AU Surfaces Free, H. J. McSkimin, page 151. Crystals: Principles of Mounting Quartz Plates, R. A. Sykes, page 178. D D'Jieedene, A. R., Effects of Manufacturing Deviations on Crystal Units for Filters, page 260. Dielectric Constants and Power Factors at Centimeter Wave-Lengths, Carl R. Englund, page 114. E l''lectronics: The Magnetically Focused Radial T^eam Vacuum Tube, A. M. SkelleU, page 190. Englund, Carl R., Dielectric Constants and Power Factors at Centimeter Wave-Lengths, page 114. Filters, Effects of Manufacturing Deviations on Crystal Units for, A . R. D'heedene, page 260. Gannett, D. K. and Iden Kerney, The Discernibility of Changes m Program Band Width, page 1. Greenidge, R. M. C, The Mounting and Fabrication of Plated Quartz Crystal Units, page 234. K Kerney, Iden and D. K. Gannett, The Discernibility of Changes in Program Band Width, page 1. BELL SYSTEM TECHNICAL JOURNAL Loading: The Conquest of Distance by Wire Telephony (A Story of Transmission De- velopment From the Early Days of Loading To the Wide Use of Thermionic Re- peaters, Thomas Shaw, page 337. M McSkimin, H. J., Theoretical Analysis of Modes of Vibration for Isotropic Rectangular Plates having Ail Surfaces Free, page 151. Metallurgy, Powder, Some Aspects of, Earle E. Schumacher and Alexander G. Soiiden, page 422. Molt, E. E., Indicial Response of Telephone Receivers, page 135. N Noise, Response of a Linear Rectifier to Signal and, W. R. Bennell, page 97. P Powder Metallurgy, Some Aspects of, Earle E. Schumacher and Alexander G. Soiiden, page 422. Quartz Crystal Units, Plated, The Mounting and Fabrication of, R. M. C. Greenidge, page 234. Quartz Crystals, Use of the Etch Technique for Determining Orientation and Twinning in, G. W. Willard, page 11. Quartz Crystals, Modes of Motion in, the Effects of CoupHng and Methods of Design, R. A. Sykes, page 52. Quartz Plates, Principles of Mounting, R. A. Sykes, page 178. Quartz: Theoretical Analysis of Modes of Vibration for Isotropic Rectangular Plates having All Surfaces Free, H. J. McSkimin, page 151. R Radio: The Discernibility of Changes in Program Band Width, D. K. Gannett and I den Kerney, page 1. Radio: Dielectric Constants and Power Factors at Centimeter Wave-Lengths, Carl R. Englund, page 114. Rea, W. T., Effect of Telegraph Distortion on the Margins of Operation of Start-Stop Receivers, page 207. Rectifier, a Linear, Response of to Signal and Noise, W. R. Bennett, page 97. Rice, S. 0., Mathematical Analysis of Random Noise, page 282. Schumacher, Earle E. and Alexander G. Souden, Some Aspects of Powder Metallurgy, page 422. Shaw, Thomas, The Conquest of Distance by Wire Telephony (A Story of Transmission Development From the Early Days of Loading To the Wide Use of Thermionic Re- peaters, page 337. Skellett, A. M., The Magnetically Focused Radial Beam Vacuum Tube, page 190. Souden, Alexander G. and Earle E. Schumacher, Some Aspects of Powder Metallurgy, page 422. Sykes, R.A., Modes of Motion in Quartz Crystals, the Effects of Coupling and Methods of Design, page 52. Principles of Mounting Quartz Plates, page 178. Telegraph Distortion on the Margins of Operation of Start-Stop Receivers, Effect of, W. T. Rea, page 207. Telephone Receivers, Indicial Response of, E. E. Matt, page 135. INDEX vii Telephony, Wire, The Conquest of Distance by (A Story of Transmission Development From the Early Days of Loading To the Wide Use of Thermionic Repeaters), Thomas Sbmv, page 337. Thermionic Repeaters: The Conquest of Distance Ijy Wire Telephony (A Story of Transmission Development From the Early Days of Loading To the Wide Use of Thermionic Repeaters), Thomas Shan.', page 337. Transmission, Telegraph: Effect of Telegraph Distortion on the Margins of Operation of Start-Stop Receivers, W. T. Rea, page 207. Transmission Development: The Conquest of Distance by Wire Telephony (A Story of Transmission Development From the Early Days of Loading To the Wide Use of Thermionic Repeaters), Thomas Shaiv, page 337. Twinning in Quartz Crystals, Use of the Etch Technique for Determining Orientation and, G. W. Willard, page 11. Vacuum Tube, Radial Beam, The Magnetically Focused, A . M. SkeUett, page 190, Vibration for Isotropic Rectangular Plates having All Surfaces Free, Theoretical Analysis of Modes of, H. J . McSkimin, page 151, W Wave Filters: Effects of Manufacturing Deviations on Crystal Units for Filters, A. R. D'heedene, jiage 260. Willard, G. W., Use of the Etch Technique for Determining Orientation and Twinning in Quartz Crystals, page 1 1 . Wire Telephony, The Conquest of Distance by (A Story of Transmission Development From the Early Days of Loading To the Wide Use of Thermionic Repeaters), Thomas Shaw, page 337. VOLUME xxiii JANUARY, 1944 number i THE BELL SYSTEM TECHNICAL JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION The Discemibility of Changes in Program Band Width — D. K. Gannett and Men Kerney 1 Use of the Etch Technique for Determining Orientation and Twinning in Quartz Crystals — G. W. Willard 11 Modes of Motion in Quartz Crystals, the EfiEects of Coup- ling and Methods of Design . . . R. A. Sykes 52 Response of a Linear Rectifier to Signal and Noise —W. R. Bennett 97 Dielectric Constants and Power Factors at Centimeter Wave-Lengths Carl R. Englund 114 Abstracts of Technical Articles by Bell System Authors 130 Contributors to this Issue 133 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 50c per copy $1.50 per Year THE BELL SYSTEM TECHNICAL JOURNAL Published quarterly by the American Telephone and Telegraph Company 19S Broadway i New York, N. Y. EDITORS R. W. King J. O. Perrine F. B. Jewett O. E. Buckley S. Bracken EDITORIAL BOARD M. R. Sullivan A. B. Clark M. J. KeUy O. B. Blackwell H. S. Osborne F. A. Cowan ii» ■» i«ii« ■»■«■■ 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, 1944 American Telephone and Telegraph Company PRINTED IN U. S. A. The Bell System Technical Journal Vol. XXIII January, 1944 No. i The Discernibility of Changes in Program Band Width* By D. K. GANNETT and IDEN KERNEY One of the factors that should be considered in determininj^ how wide a transmission band is required for high fidelity broadcasting is the ability of people to perceive the effects of restricting the band to various limits, when listening to typical radio programs. Tests are de- scribed in which this was directly measured. The tests were concerned only with the physical ability to hear the differences in band width and disregarded the question of the enjoyment or aesthetic apprecia- tion of wider bands. It is concluded that changes in band width are detectable about twice as readily with music as with speech; that one must go from 8 to 15 kc. to obtain a change as readily detected as a change from 5 to 8 kc; and that both these changes, for speech, are just sufficient to have an even chance of being detected by listeners having experience in such tests. THE question of how wide a frequency band it is necessary to transmit to provide high fidelity broadcasting involves consideration of a num- ber of factors. Among these are the limits of hearing of the human ear, the spectra of program material, the aesthetic sensibilities of listeners, the effect of room noise in studios and homes, and the acoustic properties of rooms. A true engineering solution of the problem would attempt to assign nu- merical values to each of these factors, and then to combine them in some way to obtain a figure of merit versus band width. Sufficient information to do this in a complete and satisfactory manner is not available, however, and in practice the final answer is usually obtained by the exercise of judg- ment, bolstered by such technical data as can be found on the component factors. The first two of the above factors, the limits of hearing and the spectra of program material, have been separately investigated and the results pub- lished in the technical literature by a number of experimenters. Because of the intangibles involved, however, even these two sets of data cannot readily be combined, forgetting the other factors, with complete assurance that their contribution to the answer is established. The authors, there- fore, undertook a series of tests to measure directly their combined effect. * This paper is a publication, substantially without change, of a report prepared some time ago before work non-productive to the war effort was suspended. 2 BELL SYSTEM TECHNICAL JOURNAL These experiments tested the ability of critical listeners to hear changes in band width on direct comparison when listening to representative program material. The purpose of this paper is to present the data from these tests. Similar experiments have of course been done before. The excuse for this paper is that the experiments represent a complete set of data and the analy- sis of the data is believed to be in such form as to be useful in further con- sideration of the requirements of program fidelity. The circuit arrangements used for the tests are shown schematically in Fig. 1. The essential features are a source of program, a switch for con- necting into the circuit either of two low-pass filters, and a high-quality loud- speaker. Controls for adjusting levels, volume indicators, etc., are omitted from the diagram. The arrangements included a signal visible to the _--- o- A B LI i FILTE?! Fig. 1 — Arrangement for testing program band widths. listeners in which one of the letters, A, B, or C, could be illuminated. On a given test two of these letters were associated with the switch so that one letter was illuminated for one position and the other letter for the other position. The choice of letters among the three was varied more or less at random for different tests. Low pass filters were available to provide cut- offs of 3, 5, 8, 11 and 13 kc. When no filter was inserted the band was con- sidered to extend to 15 kc. as this was about the upper limit of transmission of the testing circuits and loud speaker. The lower limit of the transmitted band for all conditions was approximately 40 cycles. In conducting a test, a group of observers listened to comparisons between two of the available band widths, the conditions being switched every few seconds until a sufficient number of comparisons had been made. The CHANGES IX PROGRAM BAXD WIDTH 3 conditions were unknown to the observers, being designated to them only by the letters in the signal. At the conclusion of the test the observers were asked to mark on a ballot which letter appeared to coincide with the wider band (not which they preferred). A series of tests consisted of comparisons between substantially all of the possible band widths among those available. There were also included in some of the series as a check, one or two tests in which the band width was the same for both positions of the switch. Ten complete series of tests were carried out, two on each of five different programs. The programs consisted of a dance orchestra, two large symphony orches- tras, speech from a male speaker repeating a test sentence, and a radio dramatic sketch. The programs, except for the spoken test sentences, were obtained by special arrangement over direct wire lines from the studio or theater in which the performance took place. The entire system from microphones to and including the loud-speaker had a substantially flat trans- mission characteristic from 40 to 15,000 cycles, with no filters in the circuit. The loud-speaker was of the two-unit type and was one of a number built for the demonstration of auditor}^ perspective in 1933. The tests were conducted in the program laboratory of the Bell Telephone Laboratories where the acoustic noise level was about -\-30 decibels. The noise con- tributed by the electrical parts of the system was considerably below the acoustic noise. The loudness of the programs was adjusted to about unity reproduction, that is, to the volume that would be heard by listeners in a favorable position at the original performance. The observers were engineers having a considerable experience in tests of program quality. They were doubtless therefore considerably more critical than the average radio listener. The number of observers varied somewhat during the tests but averaged about sixteen. The ages of the observers were in the 30's and 40's so that neither very young nor very old ears were represented. The immediate outcome of the tests was some 2,000 ballots which were meaningless until analyzed. Before the analysis could be made, however, it was necessary to decide how to express the results. There are no familiar units to express fidelity or program quality. It was decided therefore to employ the very useful concept of the limen and the liminal unit. These terms have occasionally been applied to other subjec- tive data and may be roughly defined as the least change in a quantity which is detectable. In the present case, if the band widths being com- pared differ greatly, there will be a nearly unanimous agreement among the observers as to which is the wider. If they differ only slightly, however, many of the observers will vote wrongly for the narrower band and on suc- cessive repetitions of the test many will reverse themselves. An average of 4 BELL SYSTEM TECHNICAL JOURNAL a large number of votes will show a plurality for the wider band, the margin of choice increasing as the difference in band width is made greater. A significant measure of the detectable difference in band width will be taken to be that difference such that 75% of the observers correctly select the wider band and 25% wrongly select the narrower band. This difference in band widths will be designated one "difference limen." The sensory effect of a change of one difference limen will be called one "liminal unit". The significance of the vote of 75 to 25% is assumed to be as follows: On a particular test some of the observers can detect the difference between the conditions while the remainder will guess. Of the latter, half are likely to guess right and half wrong. When 25% vote wrongly they are assumed to be guessing and must be paired with another 25% who also guessed but happened to guess right. Therefore a vote of 75 to 25% is taken to indicate that 50% of the observers were guessing and the remainder could actually detect the difference. The difference limen may now be more specifically defined as that difference in band widths which is detectable to half the ob- servers. It may be commented that this attempt to explain the definition of ''lim- inal unit" is perhaps over-simple. The observers themselves are frequently uncertain whether they are guessing or are influenced in their choice by some minute difference. The test could be done with a single observer, repeated many times to obtain the same number of observations as with a group. When the conditions are nearly equal he will vote about as often one way as the other, but as the difference between the conditions is increased he will vote a larger per cent of the time correctly for the wider band, just as did the group. When the two conditions are separated by one difference limen he will vote correctly 75% of the time and wrongly 25% of the time, which may be said, in line with the argument given earlier, to indicate that he is guessing half the time and can discern the difference half the time. The difference limen could therefore be defined as that threshold difference for which there is an even chance of its discernment by a listener. Having chosen a method of expressing the results, the analysis can now be attacked. The first step is to group together all tests on similar types of program material, and to determine for each band width comparison the per cent of votes for the wider and narrower band, respectively. The data thus obtained for music and speech are shown by the solid curves of Figs. 2 and 3. A curve labeled 8 kc, for example, shows the per cent of the total votes which selected as the wider each of the other band widths to which 8 kc. was compared. The points, although somewhat irregular, fell syste- matically enough to permit drawing the smooth curves with the application of some judgment and having due regard to the necessary symmetry be- tween them. (For example, the 8 kc. curve at an abscissa of 5 kc. must CHANGES IX PROGRAM BAND WIDTH 5 agree with the 5 kc. curve at an abscissa of 8 kc.) A much larger volume of data would be needed to obtain points falling accurately on a smooth curve. To facilitate obtaining the best approximations, the curves were plotted on several kinds of coordinates, including rectangular, semi-logarithmic (shown in the illustrations), probability and logarithmic probability. The dotted curves were interpolated between the solid curves and progress in steps of 1 kc. The interpolation was readily accomplished with consider- 5 8 11 13 15 KC Fig. 2 — Music 5 8 11 13 15KC Fig. 3 — Speecli Figs. 2 & 3 — Detectability of changes in band width. able accuracy. For example, points for the 10 kc. curve are obtained from the values of each of the solid curves corresponding to an abscissa of 10 kc. From these curves, the difference limens for each band width were deter- mined by reading directly the bands corresponding to votes of 25% and 75%. The bands at which these votes occur therefore by definition differ from the reference band by one limen. The following table gives the intervals of one limen as thus derived from the curves. 6 BELL SYSTEM TECHNICAL JOURNAL Differences in Upper Limit of Program Band in KC, Corresponding to One Limen Music Speech 3— 3.6 3— 3.3 3.3— 4— 4.8 3.4— 4— 4.8 4.1— 5— 6 4.1— 5— 6.9 5 —6—7.4 4.6— 6— 9.4 5.8— 7— 9.3 5.1— 7—12.8 6.4— 8—11 5.5— 8 6.9— 9—12.2 5.8— 9 7.4—10—13.4 6.2—10 8 —11—15 6.4—11 9.8—13 7 —13 11 —15 7.6—15 The difference limens are seen to vary with the frequency of cut-off, in- creasing as the frequency increases. Since each difference Umen corre- sponds to a sensory effect of one Hminal unit, it is obvious that the reciprocal UPPER LIMIT OF PROGRAM BAND-KC ^ ^^ ^ SP :ecH ^ ^ r^ / r USIC / 1 / f / / / / / Fig. 4 — Ability to detect changes in program band width. of the difference hmen gives the rate of change of hminal units with changes of program band width in terms of hminal units per kilocycle. Therefore, curves of liminal units versus the upper limit of the program band may be constructed from the figures in the table. Such curves are plotted in Fig. 4. The actual mechanics of the process used to plot the curves was as follows, taking the data for "music" for illustration. The lowest frequency occur- ring in the table is 3 kc, and it is seen that raising the band width to 3.6 kc. will bring about a subjective increase of one liminal unit. Therefore, on an arbitrary scale, 3 kc. was plotted at 0 and 3.6 kc. at one liminal unit. Next a smooth curve was drawn through these points and the location of ^.Z kc. (next line of table) was determined by interpolation. Since 4 kc. is one liminal unit above i.Z kc, and 4.8 is one liminal un't above 4 kc, these points were plotted and the curve extended through them. By a similar process CHANGES IX PROGRAM BAXD WIDTH 7 the curve was extended step by step up to 15 kc. Finally, the origin was shifted so as to express the liminal curve with respect to 15 kc. instead of 3 kc. It was mentioned above that a number of tests were introduced without the knowledge of the observ^ers in which the conditions were not changed, the band width remaining constant while the illuminated letters were switched. This produced the most interesting psychological result that observers voted nearly two to one for the letter appearing in the right- hand position in the signal, on each of the six tests of this kind. This raises the question as to whether this effect impaired the results on the other tests. In the course of the tests, comparisons between each pair of band widths were presented 10 times, 6 times with music and 4 times with speech. The letters corresponding to the two conditions were assigned more or less at random from the three letters A, B, and C. Taking 11 of these groups of tests in which the narrower band was represented about as often by the right hand as b}' the left hand of the pair of letters chosen, the average vote for the right-hand letter was 51.1*:^ and for the left-hand letter was 48.9%. The difference between these two figures is too small to be significant. It is therefore concluded that when there was a real difference, the observers were not measurably influenced by their slight subconscious predilection for the right-hand letter. It would be interesting to correlate this phenomenon with the right or lef t-handedness of the observers. This point illustrates the extreme care that must be taken in conducting judgment tests of this sort to insure that no irrelevant factors affect the statistical result. The curves of Fig. 4 permit drawing the following conclusions: 1. Increases in band width can be detected up to 15 kc. for both music and speech. The fact that this is true for speech is rather surprising. However, above about 5 kc, changes in band width are twice as readily detectable on music as on speech. 2. It requires an increase in band width from 8 to 15 kc. to be as readily detected as an increase from 5 to 8 kc, for both speech and music. 3. The following intervals correspond to one liminal unit and are there- fore just discernible half of the time to the observers: Speech: 5 to 8 kc; 8 to 15 kc. Music: 5 to 6^ kc; 6^ to 8 kc; 8 to 11 kc; 11 to 15 kc In considering these conclusions, the fundamental assumption and limita- tions of the data should be borne in mind. First, the data were obtained from tests with a certain group of observers and on certain program mate- rial. Curves of somewhat different slope would doubtless be obtained with observers of different average age, experience, musical appreciation, etc. It is likely, however, that this would affect the absolute importance of the different intervals in liminal units rather than the relative values. As noted 8 BELL SYSTEM TECHXICAL JOURNAL earlier, the observers in these tests were considerably more experienced and critical than average radio audiences. The program material tested was representative of most of the programs on the air, but different results would be obtained with material markedty different in nature. This would prob- ably be particularly true of selected sound effects. Secondly, it should not be forgotten that the results are based onty on the ability of the ear to detect the changes, with no weighting for factors such as aesthetic values or per- Table I Musical Instruments 1. Flute 2. Snare Drum 3. Violin 4. Soprano Saxophone 5. Oboe 6. 14 in. Cymbals. . . . 7. Bass Clarinet 9. Piccolo 9. Bassoon 10. Cello 11. Bass Saxophone. . . 12. Clarinet 13. Trumpet 14. Bass Viol 15. Trombone 16. Bass Tuba 17. French Horn 18. Piano 19. Bass Drum 20. Timpani Speech Male Female Sound Effects Footsteps Handclapping Key Jingling Upper Frequency Limit Versus Unrestricted Band, Corresponding to One Liminal Unit 13,500 cycles 13,000 13,000 12,700 12,700 12,000 10,500 10,200 10,000 9,800 8,600 8,500 8,300 7.800 7,200 6,300 6,100 5,600 4,300 3,500 7,300 9,200 12,000 15,000 15,000 sonal preferences, or for the effects of room noise and other factors present in the practical case. Thirdly, it should be appreciated that comparison tests such as these are very sensitive tests, showing up differences that could not be detected under usual home listening conditions. It is of interest to compare the above results with previously published data. In a paper "Audible Frequency Ranges of Music, Speech and Noise, "^ W. B. Snow gave data for 20 musical instruments, certain noises, and 1 Jour. Acous. Soc. Amer., July 1931; Bell Sys. Tech. Jour., Oct. 1931. CHANGES IN PROGRAM BAND WIDTH speech. The data showed the frequency limitations as compared with un- Umited bands (about 15 kc.) which yielded a vote of 60 to 40%, and 80 to 20% among a considerable number of observations. In Table I these data have been interpolated to determine the limits that would correspond to a xoie of 75 to 25%, in line with the criterion assumed in this paper. In making the interpolation, it was assumed that the curve of per cent of ob- servers voting correctly for the wider band versus logarithm of the frequency is a straight line in the range of interest. Table II Musical Instruments 1. Bass Viol 2. Bass Tuba 3. Timpani 4. Bass Drum 5. Bass Saxophone. . . 6. Bassoon 7. Bass Clarinet 8. Cello 9. Snare Drum 10. Piano 1 1 . Trombone 12. French Horn 13. Clarinet 14. Trumpet 15. Soprano Saxophone 16. Violin 17. Oboe 18. Flute 19. 14 in. Cymbals 20. Piccolo Speech Male Female Sound Effects Footsteps Handclapping Key Jinghng Lower Frequency Limit Versus Unrestricted Band, Corresponding to One Liminal Unit 53 cycles 55 60 72 72 74 80 83 87 95 110 125 140 160 210 230 240 250 370 510 115 190 95 135 915 It is difficult to interpret these data from individual instruments in terms of results to be expected from whole orchestras and other music as usually heard. However, comparing Table I with Fig. 4, it will be seen that the frequency limit determined from the present tests as corresponding to one liminal unit for music falls about one third the way down the list of instru- ments in the table, and the Umit corresponding to two liminal units falls about two thirds down the table, which seems reasonable. Also the fre- quency limit found in the present tests to correspond to one liminal unit for 10 BELL SYSTEM TECHNICAL JOURNAL speech lies between the figures given in the table for male and female speech, which is a good check. The present tests did not include measurements on the lower end of the frequency band. However, some clue to the results that would be expected may be obtained from Mr. Snow's paper. Table II, derived from Mr. Snow's data in a manner similar to that just described, gives the lower limit of the frequency band corresponding to a degradation of one liminal unit compared with transmitting a much lower frequency. The frequency corresponding to one liminal unit for speech may be taken as the mean of the figures for male and female speech, or about 150 cycles. In the case of music, it may be expected that at the lower as well as the upper end of the frequency range one liminal unit for an orchestra should fall about one third the way down the list of individual instruments, and two liminal units about two thirds the way down the list. This would make one liminal unit for music correspond to about 80 cycles and two liminal units to about 150 cycles. This speculation leads to the interesting hypothesis that the relations are probably the same at the lower as at the upper end of the frequency scale, that is, changes in band widths are twice as readily de- tected for music as for speech, and that the frequency limit corresponding to one liminal unit for speech corresponds to two liminal units for music. CHAPTER V Use of the Etch Technique for Determining Orientation and Twinning in Quartz Crystals By G. W. WILLARD This paper is one of a series of papers dealing witli piezoelectric circuit elements and their manufacture.^ Certain parts of the paper are not new or original, but have been added for the sake of completeness and for the convenience of the reader. 5.1 Introduction THE manufacture of piezoelectric plates from cr>'stalline material in- volves orientation problems not encountered in the fabrication of objects from non-crystalline materials. The reason for this is that crystalline ma- terials have physical properties which vary with the orientation, or direction, in which they are measured. Since the operating characteristics (activity, fre- quency, and temperature-coefficient) of the finished piezoelectric plate depend, not only upon the shape and dimensions of the plate, but upon the physical properties (electrical, elastic and thermal) of the crystalline material, the fin- ished piezoelectric plate must have a specific orientation with respect to the material as well as a specific shape and dimensions. In the case of quartz piezoelectric plates the orientation problem is complicated by two factors. First, a large portion of the available natural quartz cr\^stals lack such natural faces as are required to determine accurately the structure-orienta- tion from the shape of the original stone. Thus the raw stones must be examined for structure orientation by physical instruments before even the first cuts may be made. Secondly, a large portion of natural quartz cr>^stals are twinned, i.e. not of the same structure orientation throughout the stone. The boundaries of the respective, homogeneous regions are not predictable, and cannot be completely located in the uncut stone. Thus the processing of quartz involves a step by step examination for twinning boundaries and orientation as the raw stone is cut into sections, the sections cut into bars or slabs, and the bars or slabs cut into blanks. Even when using untwinned stones the orientation must be redetermined and corrected at each cutting step when making such plate types as require ver>^ exact orientation. The most widely used methods of determining the structure orientation 1 See B.S.T.J., Vol. XXII: No. 2, July 1943 for Chaps. I and II; No. 3, Oct. 1943 for Chaps. Ill and IV. 11 12 BELL SYSTEM TECHNICAL JOURNAL of quartz are: (1) by optical efifects (birefringence and rotator}^ power), (2) by X-ray reflections from atomic planes, and (3) by the use of etch pits which are developed when the quartz surface is etched in fluorine compounds. Other methods are or may be used in rather special cases. For example, in finished plates of known orientation types, the electrical axis direction is distinguished from other directions by electrical polarity tests (on tension or compression), or a plate known to be one of several types may be tested in an electric circuit for activity, frequency and temperature-coefi&cient, to determine which type it is. The selective fracture characteristics of quartz offer another method of determining orientation. Microscopic fractures re- sulting from grinding a quartz surface ma}' be used for determining orienta- tion. Thus unetched, ground, Z-cut surfaces of quartz give a hexagonal figure, when examined by pinhole illumination, which may be used to de- termine the approximate orientation (but not sense) of the electric axes.- By optical methods (see Chapter II) it is possible to determine the orienta- tion of a quartz body relative to only one direction of the structure, the optic or Z axis. Thus optical methods are hmited to determinmg the angle be- tween the optic axis and a line or surface of the body (but not the rotation of that line or surface about the optic axis). Twinning of the "optical" vari- ety may be detected optically, even when located internally, but the deter- mination of its location in depth is approximate. By X-ray methods (see Chapter III) it is possible to determine the struc- ture orientation of a quartz body exactly and completely. However, this method is limited in application by the complexity of analysis, except when the approximate orientation is already known. Though twinning can be detected on the surface of the body, it is not generally feasible to explore the surface to locate twinning boundaries. Further, though positive or negative sense of angular orientation is obtainable by X-rays, this part of the complete determination is not reliable unless the specimen examined is known to be free of twinning, or unless the twinning boundary locations are known. Thus X-ray determinations of orientation are generally limited to deter- mining exact orientations in quartz bodies of approximately known orienta- tion (which includes the case in which only one axis is approximately known). The etch method of determining orientation is commonly used in con- junction with the optical and X-ray methods to give the information that those methods do not give. The etch method, as most commonly and prac- tically applied, does not give exact orientation angles, nor is it applied to specimens of entirely unknown orientation. However, when a surface of approximately known orientation is etched, it is possible to determine ap- proximately the complete orientation (including sense) of the specimen, and further to detect at this surface both electrical and optical twinning and to 2 See Fig. 5.20, and further explanation at the end of Sec. 5.53. ETCH TECHNIQUE 13 determine exactly the twinning boundaty locations. The detection of twin- ning and twinning boundaries by this method has been practiced for years. The determination or orientation and sense of orientation has been exploited only more recently. At present the etch methods play an important and extensive role in the processing of quartz plates, not only in the routine de- termination of orientation, but also in the detection of twinning so that the most economical cutting methods may be practiced.^ 5.2 Twinning (General) Although the problems related to twinning are largely those of determining orientation of the crystal structure, the nature and prevalence of twinning in crj^stal quartz presents a special group of problems that would be absent were the twinning absent, and hence are separately grouped as twinning problems. As pointed out in Chapter I\', there are only two common types of twinning in the commercial quartz used for piezoelectric plates, namely, electrical and optical twinning. A simplifying feature of both these types is that the structure axes (optic axis and electric axes) of all portions of a single crj'stal are parallel each to each. However, they are not of the same sense, or handedness. The difference between the two types is as follows: In a cr\'stal which is only ELECTRICALLY TWINNED, the cr}^stal is entirely of one handedness (either right or left), but one portion is of OP- POSITE ELECTRICAL SENSE to another portion, i.e., the electric axes are of opposite sense. In a cr>'stal which is only OPTICALLY TWINNED, one portion of the crA'Stal is of OPPOSITE HANDEDNESS, and electrical sense, to another portion. This twinning (but not electrical) is detectable by optical means (polarized light) and is named optical twinning for this reason. The extent of twinning that may be present in commercial cr}-stals is seen in Fig. 5.1, which shows both electrical and optical twinning boundaries at the top surface of some Z-cut (basal) sections of quartz (which were cut up for the manufacture of quartz oscillators). Though the cr^'stals are seldom entirely free of twinning, they do not on the average run as badly twinned as here shown. These views, taken by means to be described, correspond to what one sees when examining an etched quartz surface by reflection from a strong light. Since untwinned finished plates must be cut entirely from one twin or another (not across a boundary), and since the proper sense of angular orien- tation of the plate is opposite for two adjacent electrical twins, the economic utilization of twinned quartz is a difficult problem.^ It involves cutting the ' Etching is also used on finished plates for removing grinding debris, and for frequency adjustment. ■* As herein used, a tuin is one of the homogeneous, untwinned portions of a twinned crystal. 14 BELL SYSTEM TECHNICAL JOURNAL stone into separate parts when the twins are large enough to be utilized separately. Further, at some stage before reaching the finished plate all twin portions but one must be cut away.^ In this connection it is important to note a size and form difference be- tween electrical and optical twins. Fig. 5.2 shows the appearance of twin- ning boundaries when only ELECTRICAL TWINNING is present. Note that electrical twins are commonly large, hence may often be separated ap- Fig. 5.1 — Examples of ELECTRIC.\L and OPTICAL twinning, as exhibited at the etched surface of Z-cut sections. These examples are tj-pical of an appreciable portion of the quartz that is cut up for quartz plates. proximately along a boundary and both portions utilized. Fig. 5.3 shows the appearance of twinning boundaries when only OPTICAL TWINNING is present. Since optical twins are commonly small and in the form of thin laminations, it is seldom possible to cut optical twins apart and use both parts separately. The conventions here used, regarding handedness and axial sense, are ^ See Section 5.7 for the possibility of utilizing partially twinned finished plates. ETCH TECHNIQUE 15 according to those of the proposed "I. R. E. Standard."^ Figure 5.4 shows the relation of these conventions to the natural faces of right and left quartz, to the electric charges developed on compression and tension, and to the more common cuts of oscillator plates. Also given are the relations of handedness to the conoscope and the polariscope means of detecting handed- ness (Section 2.7, Chap. II describes these instruments). It is important to Fig. 5.2— E.xamples of ELECTRIC.\L twinning alone. Electrical twins are com- monly large, and hence may be cut apart and used individually. note that AT and CT plates are always cut at such an angular sense, relative to the Z and X axes, as to be roughly parallel to a minor pyramidal face, whereas the BT and DT plates are roughly parallel to a major pyramidal face. Thus a stone exhibiting these faces may be cut into any of these plates « "Proposed Standard Conventions for Exi^ressing the Elastic and Piezoelectric Proper- ties of Right and Left Quartz", Proc. I. R. E., Xov. 1942, p. 495. 16 BELL SYSTEM TECHNICAL JOURNAL without determining the handedness and electrical sense of the stone (if twinning is negligible). As will be seen later, a similar situation prevails when analyzing etched X-cut sections for cutting into plates. 5.3 Nature of Etch-Pits When crystal quartz is etched by contact with hydrofluoric acid (or other etching agents) the surface of the quartz is eaten away in such a manner as Fig. 5.3 — Examples of OPTICAL twinning alone. Optical twins are commonly small and interlayered, and hence may not be separated and used individually. to leave microscopic etch- pits (or hills). These etch-pits are formed of minute facets which are definitely related to the cr\^stal structure. The form of these pits and the orientation of the facets may be used to determine the orientation of the crystal structure at the etched surface being examined. The general appearance of four types of etch-pits is shown in the photo- micrographs of Fig. 5.5. These are the pits that are deyeloped on ground surfaces which are approximately parallel to the well known X-, Y-, and Z- cut surfaces of right hand quartz, by the action of hydrofluoric acid. It is ETCH TECHNIQUE 17 seen that the positive and negative X-surfaces produce different etch-pits, and are thus usable in determining electrical sense. Further, the pits on all surfaces have directional properties which allow them to be used for deter- mining the approximate directions of the axis which lie in the etched surface. However, to be able to determine orientations from etched surfaces of other / LEFT HAND QUARTZ^x / ^ ■ ^ \ / RIGHT HAND QUARTZES ON COMPRESSION BT^ 0- in conoscope : contracting rings (eyepiece rot clockwise) in polariscope-. analyzer rotated counter-clockwise in conoscope- expanding rings (eyepiece rot. clockwise) in polariscope : analyzer rotated clockwise Fig. 5.4 — The conventions of handedness, axes, natural faces, and angular sense-of-cut of common oscillator plates, together with the electrical and optical rules for determining these characteristics in unfaced stones. orientations than those shown above, requires a knowledge of the appear- ance of the etch-pits developed on such surfaces. A rather complete catalog of etch-pits on all possible surfaces of quartz was prepared by W. L. Bond,^ using an etched sphere of quartz (Figs. 5.5, 5.6 are from Bond). Thirty-sLx different types of etch-pits were obtained and their angular range of coverage was found (the X-, Y-, and Z- surface ^ "Etch Figures of Quartz," Z. Kristallogr. (a) 99, 1938, pp. 488-498. 18 BELL SYSTEM TECHNICAL JOURNAL pits are obtained only on surfaces within 6° to 8°, from the X-, Y-, and Z- surfaces, respectively). Since the development of good etch pits and their exact appearance is considerably affected by the preparation of the surface for etching (fineness of grind), and by the strength of the acid and the length of etching time, and by the manner of illumination when viewing, the v' t |^»s *"; * '. ^ X-CUT (-+-X up) X-CUT (^-X up) X i 4 ^ Y- CUT Z- CUT Fig. 5.5 — Photomicrographs of etch-pits on the etched surfaces of common orientations. As seen the etch-pits are deiinitely related to the structure axes of the quartz. figures shown here do not represent the exact appearance of pits obtained by other manners of development. However, such figures are reproducible. The use of etch-pits to determine the orientation of a perfectly general surface is complicated by the fact that some different surface orientations give pits not readily distinguished from each other. However, for the sur- faces most commonly encountered in quartz plate manufacture the etch- ETCH TECHNIQUE 19 pits are quite distinctive, when well developed. Use may be made of a microscope or a high powered projector to view the figures. The pit out- lines may be aligned with lines ruled on the eye-piece or on the screen, and a tLxed marking device may be used to mark the quartz surface with orienta- tion lines. Twinning may be detected by the appearance of different etch- pits as the specimen is moved about. For example, on an electrically twinned X-cut surface both X-cut views of Fig. 5.5 could be found. How- ever, the location and marking of twinning boundaries involves a tedious exploration of the surface, since only a minute portion is viewed at any one time. This exploration may be eliminated if the surface is first viewed by reflection methods where the whole surface and extent of twinning is at once seen (as in Fig. 5.1) and marked. i". -35 CUT + 35, AT-CUT Fig. 5.6 — Etch-pits on the etched surface of a +35° AT plate, and on an analogous but wrong sensed —35° plate. This difference in etch-pits may be used in the manufacturing process to determine the right and wrong sensed regions of twinned AT slabs. A special case where the microscope or projector method might be em- ployed is in the examination of thin AT, BT, CT or DT slabs for twinning and sense of cut. Here the slabs are known to be cut with a reference edge parallel to an electric axis, and with the major faces inclined at 35° to 55° (depending upon the variety of slab) from the optic axis, the sense of the inclination being positive for the AT and CT slabs, and negative for the BT and DT. The effect of electrical twinning on such etched surfaces is shown in Fig. 5.6. The etch-pits of the good +35° AT-portion of the slab are easily distinguished from the analogous —35° (bad) portions. This difference is similarly distinguishable in the other cuts. Actually, orientation and twinning are seldom analyzed by the method described above, i.e. by examining their appearance in the microscope, or by projection on the screen. The method appears to be far less practical than other methods which depend upon the gross effect, of hundreds of simi- 20 BELL SYSTEM TECHNICAL JOURNAL lar etch-pits, in bending a light beam. By the latter methods the indi- vidual etch-pits are never seen, nor does their nature need to be known. Nevertheless, the resultant optical effect of hundreds of similar etch pits is as characteristic of structure orientation as the individual pits themselves. 5.4 Optical Effect of Etch-Pits The gross optical effect of hundreds of similar etch-pits results from the fact that each of the pits has minute facets which are similarly inclined to those of all the other pits. Though the pits of Figs. 5.5 and 5.6 may not appear to be formed from groups of flat facets they are generally so regarded. ''Curved-facets" are theoretically considered to be made up of individual flat-facets which are parallel to possible atomic planes (and hence may be given index numbers as in Chap. III). This view is the same as that taken Fig. 5.7 — Reflection of light from a single set of similarly oriented etch-pit facets, A, is like that from a single mirror, B. Reflection from all three sets of facets of a Z-cut section will give a three-fold etch-figure on a screen, as in C. with regard to natural faces, which are of course produced by essentially opposite effects, i.e., acid corrosion m the case of etch-pits, and growth from solution in the case of natural faces. Actually, many "curved-facets" give optical effects showing no discernible evidence of individual flat facets. However, the question is academic, so far as use of the pits for orientation purposes is concerned, for such facets are still definitety related to the cr\-stal structure. Etch-pit facets may be used to reflect a light beam into specific patterns or to refract the beam on transmission through the material into similar (but not identical) patterns. The different basic optical means of using etch-pit facets are shown in Figs. 5.7, 5.8, 5.9. Included in each figure is a diagram of the effects obtained by illuminating an idealized Z-cut section. This idealized section is assumed to have only simple, equilateral, three-sided ETCH TECHNIQUE 21 pyramidal etch-pits, oriented relative to the X axes as shown in Fig. 5.5. The actual results obtained with Z sections are more complicated than this and thus indicate that the etch-pits are not exactly as idealized here. A B SCREEN LENS SOURCE Fig. 5.8 — Light transmitted thru a single set of etch-pit facets, A, is refracted as by a prism, B. The three sets of facets of a Z-cut section give a three-fold etch-figure, as in C. ^EYE ^EYE SOURCE TOP VIEW OF SECTION SOURCE A B • C Fig. 5.9— Light transmitted thru a pin-hole is refracted by a single set of facets, A, as it would be by a prism, B. A virtual image of the pin-hole P will be observed at P'. The etch-figure seen down in a Z-cut section is three-fold, as in C. 5.41 The Reflection Method Figure 5.7 shows the reflection method, where a parallel beam of light striking the etched surface of a Z-section is reflected from one of the three sets of facets as shown in A. Each single facet reflects part of the beam by 22 BELL SYSTEM TECHNICAL JOURNAL ordinary reflection laws, and the whole groups of facets act similarly to a single mirror surface at the same angle, as in B.^ The individual facets being very minute and of irregular size and spacing, however, cause appre- ciable diffusion of the beam. The resultant effect of all three sets of facets is shown in C, where light passing down through a lens and a hole in the screen is reflected back to three spots on the screen. These three spots are located at equal distances from the incident beam and at 120° intervals around the incident beam. If the quartz section be rotated on its table the spots rotate around the screen correspondingly. However, lateral motion of the section across the table (without rotation) does not change the position of the spots, if the section be untwinned. If the section is twinned (or more exactly, if the etched surface is twinned) the three-fold figure will shift to a different position (angularly) on crossing a twinning boundary, for the etch pits are oriented differently in the two twins. If the twinning boundary divides the illuminating beam, then both figures appear at once, giving six spots instead of three. It is clear then that twinning, as well as orientation of the section, may be determined from the figure on the screen. The angular relation between the spots and the X-axes of the section will be considered later, where figures of actual sections are shown. The long used method of examining etched quartz surfaces by simple reflection from a bright light, may also be explained from Fig. 5.7C. If a spot of light on the screen is viewed along the line E, and the screen then removed, the light from the associted etch-pits will fall on to the eye. The illuminated portion of the section will appear bright. If a twinning bound- ary crosses the illuminating beam and one of the sbc reflected beams falls on the eye, one of the two illuminated twins will appear bright and the other dark. As the section is rotated, first one twin and then the other will appear bright, and in each case the twinning boundary is sharply defined over the whole region covered by the illuminating beam (the appearance of twinned Z-cut surfaces examined by this means is shown in Figs. 5.1, 5.2, 5.3). Due to the greater complexity of etch-pits than here idealized, the reflected beams are not so sharply defined as to require exact location of the eye relative to the incident beam and the section. Further, when a broad unfocused light source is used, it is possible and convenient to detect twin- ning boundaries merely by holding the section in the hand and rocking it about in various directions until a brightness contrast is observed. Though the brightness contrast is usually not marked by this simple examination it suffices for many purposes. ^ That the effect of a group of facets is not identically the same as that of a single mirror, is of more concern where lenses are used for focusing. In this case the displacement of the mirror facets causes a displacement of the focus of the beam from each facet. For beams of small angular range this is of little importance. ETCH TECHNIQUE 23 5.42 The Transmission Method Figure 5.8 shows one form of the transmission metliod of examining Z-cut etched surfaces. A parallel beam of light passing normally up through the bottom poHshed surface and the top etched surface of a section will be bent by refraction only at the etched surface, as in A. Each facet refracts the light by ordinary laws of refraction, and the whole group acts similarly to a single refracting surface at this angle, as in B.^ The resultant effect of all three sets of facets is shown in C (where a lens is added for focusing the light beam). If the incident beam is not normal to the bottom surface there is an additional bending of the beam at this surface. If the incident surface is not polished (or rendered optically flat, with a cover glass and im- mersion fluid, for example) the diffusion at this surface will mask or com- pletely destroy the desired effect.^" 5.43 The Pinhole Transmission Method Figure 5.9 shows the pinhole form of the transmission method, as applied to the examination of Z-cut etched surfaces. Here a section with a top, etched surface is illuminated from below through a small hole with a wide angle of illumination. The light radiates upward in all directions from the pinhole, and in passing through the upper etched surface is refracted by a single set of etch facets as in A. With the eye placed above the pinhole (and section), certain of these rays will fall on the eye. The eye then sees a virtual image of the pinhole P displaced to P', elevated from the level of P, and along the line of the ray which enters the eye. The effect of a group of facets is similar to that of a single prism, as in B." The resultant effect of all three sets of facets of a Z-cut section is shown in C, where the section is viewed from directly above and no optical system is shown. Only the three virtual images of the pinhole are seen and they are located down in the quartz (roughly two-thirds of the way down). Though the desired effect is due entirely to the top, etched surface, the nature of the bottom surface may cause a deleterious masking effect, which must be considered in the design of an mstrument. Due to the diffusing effect of irregularities in the top surface it may act somewhat as a screen upon which the extended light source shown in Fig. 5.9A, B may be imaged by the pinhole. This extraneous image occurs if the bottom surface is polished, and to some extent if the surface is semi-polished, strongly etched, or oily. * See footnote 8. " Similar optics hold if the section is illuminated from the etched side instead of the polished side. " See footnote 8. 24 BELL SYSTEM TECHNICAL JOURNAL This difficulty may be entirely obviated by the introduction of a diffusion screen directly adjacent to the pinhole.^- It might be noted that if it be desired to project or photograph the pin- hole figure, one must focus on the virtual image which lies between the top and bottom surfaces of the etched specimen. In the simple case dia- grammed in Fig. 5.10, it is assumed that the camera lens is at a distance from the section and directly over the section, so that the rays to the lens are essen- tially normal to the section. For a section of thickness T, and index of refraction n, the elevation E of the virtual image from the bottom surface of the section is given by: E/T = 1 — Vl + R-/T-/n. Here R is the radial displacement of the virtual image from the axis of the pinhole and is readily observed and measured. Also, R may be calculated from the thickness of the quartz T, the angle 6 between the facets and the gross surface, and the in- Fig. 5.10 — The elevation E of the virtual image may be calculated from the thickness of the etched section T, the radial displacement of the image R, and the index of refraction n; or from T, n, and d, the angle between the facets and the gross surface. dex w, (or d may be calculated from T, R, n) by: R/T = tan {d — sin~H(sin~^ 6)/n\). Commonly, pinhole figures from quartz which is weakly to moder- ately etched (up to one hour in concentrated HF) have a maximum diameter (or double radial displacement) 2R, nearly equal to the thickness of the section. Since the elevation of the image, £, depends upon its displace- ment R, an extended virtual image is not in a single plane and cannot be exactly focused (the elevation is commonly about one-fourth to one-third of the thickness of the section). The diameter of the pin-hole must always be kept small compared to the thickness of the section to give sharp figures (and the length of the pinhole must be small compared to its diameter). '^ The diffusion screen may be a sheet of white paper placed over the pinhole, or a piece of flashed glass placed under the pinhole, with the flashed side against the pinhole. In either case it is usually necessary to increase the light intensity by focusing a concentrated light source onto the pinhole with a lens. ETCH TECHNIQUE 25 Choice of one of the four above methods of examining etched surfaces for twinning and orientation, depends upon many factors, as will be noted in the following section. The pinhole method is used wherever possible because of the simplicity of the optical system and the brilliance of the figures obtained. 5.5 Etch-Figure Instruments Herein are described several instruments which have been designed for shop use in determining orientation and twinning of etched quartz sections and slabs. Their basic principles of operation are as described above. The nomenclature of handedness, sense of axes, sense of cuts, natural faces, etc. is according to Fig. 5.4, as explained at the end of Section 5.2. The etch-figures and reflection patterns obtained on these instruments van*' with the preparation of the specimen (i.e. the type of grind and the type of etch). A complete study of these factors would include a variation of the grind from a ver\- coarse grind to polishing (and include saw-cut surface), and a variation of the etching time from short to very long, and the strength and kind of etching agent. Here chosen for illustration are the simplest practical preparations, namely, the coarsest grind usable, and the shortest etching time (in hydrofluoric acid). The etch-flgures are thus markedly different than some which have appeared in the literature. Further, the photographic reproduction of etch-figures on paper, is not exact due to the limited contrast range of the paper. Thus in the accompanying illustrations detail is lost in the brilliant portions of the etch-figures in order to show de- tails in the weaker portions, and vice-versa. ^^ 5.51 The Reflection Oriascope Fig. 5.11 shows diagrammatically a reflection "Oriascope", which may be used on specimens with a single flat etched surface. By the reflection prin- ciple of Section 5.41 figures are obtained on a viewing screen. Due to the relatively weak figures obtained by reflection from weakly etched surfaces, the viewing screen must be enclosed in a well blackened enclosure, and viewed through an eye chute. The screen is ruled with appropriate lines, relative to which the figure is aligned by turning the specimen on the table. The table is mounted so that when the specimen is properly oriented, the table may be slid to the right or left over a marking template, and marked through the template with appropriate lines to indicate the desired axial orientations of the specimen. When used with Z-cut sections it is necessary to have two marking templates, one for each handedness of the quartz, since the three-fold figures "Apparent shifts in etch-figure orientation, with etching time for example, are not to be considered as resulting from an orientation shift of the individual etch-pit-/ace/j, but as a shift in the relative areas of differently oriented facets. See Figs. 5.12 and 5.17. 26 BELL SYSTEM TECHNICAL JOURNAL obtained are not aligned with the electric axes of the specimen. They are shifted approximately 12° therefrom, and in opposite directions for the right and left varieties. Figure 5.11 shows a section of right quartz so positioned on the sliding table that the etch-figure therefrom will be properly aligned with three radial lines of the viewing screen. The section need not have natural faces as here shown. With the section so positioned the sliding table is moved over the right-hand marking template, and the section is marked with three radial lines. These lines on the section then give the approximate direction (within 5°) and the sense of the three electric axes of the quartz, positive X-outward. With left quartz the etch-figure is still aligned with the same lines on the viewing screen, but the section is marked through the left-hand marking template (the marking having the same meaning as be- ;SLIDING TABLE QUARTZ LHQ y TABLE 0 RHQ SIDE VIEW MARKING TEMPLATE TOP VIEW SCREEN VIEW MARKING TEMPLATE Fig. 5.11 — The reflection ORIASCOPE as applied to determining the direction and sense of the X (electric) axes in Z-cut sections. After the etch-figure is aligned on the screen the table and sections are moved over a marking template and the section marked from below with axes. fore). The section so marked is ready for laying out the approximate cut- ting directions, the sense of which may be found from Fig. 5.4. The exact cutting directions are obtained by X-rays. It might be noted that ordi- narily the handedness of the section is determined in the conoscope (see Section 2.7, Chap. II) before examination on the oriascope. Also the twin- ning boundaries are previously determined by examination of the etched surface in a spot-light beam. Figure 5.12A, B show the type of etch-figures obtained on Z-cut sections (in each case the figure is properly aligned with the rulings on the viewing screen). The simpler etch-figure A is obtained on a fine ground (400 car- borundum) surface by a weak etch (about 10 minutes in S0% HF). Though the three faint spots, about 40° clockwise from the rulings (for the left-hand ETCH TECHS IQVE 27 quartz of A) may be used for determining the handetlness of the section, it is usually considered more reliable to use the conoscope for handedness deter- mination. The counter-clockwise rotation of these sjwts in B indicates right-hand quartz. The more complicated etch-figure \^, results from etch- ing a fine ground surface too long," or from using a coarse instead of a line grind. With such figures it is difficult to know which ]-)ortion of the figure is to be aligned with the screen rulings. Hence the sections must be hne ground and the etching time closely controlled. The obvious disadvantages of the reflection oriascoj^e (the necessity of |)re- determining handedness and twinning, and the requirements of fine ground surfaces and closely controlled etching time) are largely overcome by the pin-hole oriascope, later described. However, the reflection oriascope is an Fig. 5.12 — Etch-figures obtained on the reflection oriascope with Z-cut sections (re- duced from 11 inches square). A is a good usable figure while B is difiicult to use due to its complexity. excellent explanatory' instrument for obtaining experimental etch-ligures from surfaces of any orientation, preliminary to devising a special instru- ment to most advantageously utilize the reflection characteristics found. This fact results from the large and symmietrical screen coverage, and from the fact that only one etch surface is encountered by the light beam (thick- ness and back surface shape is of no concern). 5.52 The Reflection Twinoriascope Figure 5.13 shows diagrammatically a reflection "Twinoriascope" designed especially for shop use in detecting and marking twinning boundaries and the sense of orientation in etched AT, BT, CT and DT slabs. When, for ex- " It appears that excessively strong etches (hours long) again give a simple, strong, and reliable figure. 28 BELL SYSTEM TECHNICAL JOURNAL ample, CT slabs are to be examined the tiltable mounting-table is clamped in the 38° position, and the slab placed crosswise on the table (X-axis normal to line of sight, and beveled edge as shown). Upon moving the viewing screen to position 1, only lamp 1 is lighted, and the slab is viewed by re- flected light at a preferred angle. If the slab be twinned, one portion of the slab will exhibit a bright sheen while the other portion is dull by contrast, see two examples in Fig. 5.14, Test /. The twinning boundary is now pen- ciled in. The viewing screen is then shifted to position 2 which lights only lamp 2, and the cr\'stal moved to right or left so that only one twin is illu- minated. On the screen^^ will be seen an etch-figure similar to one of the four shown in Fig. 5.14, Test 2. If either of the two positive-cut figures are observed the illuminated portion of the slab is usable, since the CT plate LAMP HOUSE I/eye chuteV ARM X TRAY OPENING s® ARM t, TRAY OPENING SIDE VIEW. SECTION FRONT VIEW Fig. 5.13 — The reflection TWINORIASCOPE for detecting twinning (using lamp 1 and no viewing screen, position 1) and for determining the orientation or sense-of-cut (using lamp 2 and the viewing screen in position 2), of AT- BT-, CT-, or DT-cut slabs. The "cut angle" is set for a CT slab. miist have a positive 38° orientation. The negative-cut, "golf-club", figures are produced by the unusable portion of the plate. The same procedures are followed with the AT, BT and DT plates, in each case resetting the table to the proper tilt, 35°, 49° and 52°, respectively. The reflection view of Test 1 is the same for all cuts, and the etch-figures of Test 2 are nearly the same (being almost identical for the negative-cut por- tions of the slabs). However, in the case of AT and CT slabs the positive- figures represent good portions (since these are positive cuts), and in the case of BT and DT slabs, the negative-figures represent good portions. The basic principle of this mstrument is as described in section 5.41. As here used, the two optical systems (including the eye and the slab) are so disposed as to obtain the best reflection-contrast in Test /, and the most dis- 1^ An excellent screen consists of two sheets of thin sandblasted cellulose acetate. ETCH TECHNIQUE 29 tinct portion of the etch-ligures in Test 2. That the observations are so similar for this 20° range of cuts indicates that the nature of the etch-pits on these cuts is very similar, (see Fig. 5.6 for the nature of the etch-pits on AT slabs). The angular arrangement of the Test / optical system makes use of strongly developed facets which are approximately parallel to the X- axis and inclined at an angle of —57.6° to the Z-axis of the quartz. Within experimental error these facets are parallel to the 01.2 atomic planes and hence are called the 01.2 facets. It is also these facets that give the enlarged POSITIVE-CUT FIGURES TEST 1 (twinning) TEST 2 (sense OF CUT) NEGATIVE-CUT FIGURES Fig. 5.14 — The appearance in the twinoriascope of twinning in Test 1 (two examples) and of the four possible etch-figures in Test 2. The observance (in Test 2) of either of the positive-cut figures indicates that the illuminated portion of the slab is a positive cut, while either negative-cut figure indicates a negative cut. These etch-figures for a CT slab, are not markedly different than those for AT, BT, and DT slabs. head of the golf-club, negative-cut figures. The right and left handedness of quartz results in two figures each for the positive and the negative orien- tation. Though it is commonly of no interest, it is possible to determine from the etch-figure observed, both the handedness and the electrical sense of the illuminated portion of the slab. The handedness is as indicated by L and R in each etch-figure of Fig. 5.14, and the electric axis is ± to the right or left as indicated by the -|- and — signs. Best etch-figures are obtained m the twinoriascope with fine ground (400 carborundun) slabs which have been given a strong etch (40 minutes in 50% 30 BELL SYSTEM TECHXICAL JOURNAL HF). Stronger etching is not deleterious. Very strong etching gives mod- erately good ligures with sawn or coarse ground slabs. For Test 1, alone, weaker etches would suffice. Under properly controlled conditions of slab preparation and instrument operation Test 2 might be eliminated, for under such conditions the negative-cut portion of the slab is bright, the positive-cut portion is dark. Under shop conditions this means of detecting sense of cut appears to be not reliable, especially with untwinned slabs (which are either all bright or all dark). The addition of Test 2, however, gives EYE Z - AXIS f TURN LJ TABLE Fig. 5.15 — The direction and sense of the electric axes of a sand-blasted and etched raw quartz stone may be determined by reflection of light from the 0.21 facets. These same facets are utilized in Test / of the twinoriascope, Figs. 5.13, 5.14. complete reliability, for if etch-figures are obtained the sense of cut is ob- vious, if no figures are obtained the slab can be returned for further etching. The principle of Test 7, above, has been applied by W. L. Bond to a lab- oratory instrument for determining the direction and sense of the X-axes in raw quartz stones prepared with a sand-blasted and etched surface. With the stone mounted rotateably about its Z-axis (previously determined by conoscope or inspectoscope), and a light beam properly projected onto the stone, reflection of the light beam to an eye piece or viewing screen will occur whenever the 01.2 facets come into proper angular position, see Fig. 5.15. The approximate direction and sense of the electric axis, or the sense of cuts ETCH TECHNIQUE 31 to be made from the stone, ma\' be determined from tliese reflecting posi- tions of the stone, and twinning may be partially explored. Thus if the stone appears to be not badly twinned, it may be cut up at once into slabs of proper sense of cut, without previously sectioning for further examina- tion, 5.53 The Pin-Hole Oriascope Figure 5.16 shows a "Basic Pin-Hole Oriascope" with auxiliary attach- ments for shop examination of etched Z-cut sections, and Fig. 5.18 the same Fig. 5.16— The BASIC PIN-HOLE ORIASCOPE with matching and marking arms for use on Z-cut sections. Twinning, and the direction and sense of the X (electric) axes may be determined and marked on the section. for X-cut sections. The optical principle of this instrument is according to Section 5.43. Light from a concentrated-filament lamp within the central ventilated housing, is projected horizontally forward by a pair of condenser lenses and reflected upward by a mirror in the forward housing, onto a dif- fusion-disk placed directly against the pin-hole. i" The latter is centrally located in the inclined mounting table. Etched quartz sections are placed over this pin-hole and viewed from above. The section may be moved about and examined for twinning boundaries, which are then penciled in. 1^ See footnote 12. 32 BELL SYSTEM TECHNICAL JOURNAL The section is then examined through the ruled window of a matching arm, one of which is shown in use in Fig. 5.18. The section is rotated on the table until the etch-figure seen in the quartz is properly aligned with the Unes on the window. Without moving the sections, the viewing arm is replaced with a marking arm, one of which is shown in place in Fig. 5.16. The section is t+x +x t LHQ RHQ B + x IN LHQ MATCHING WINDOW MARKING TEMPLATE + IN RHO MATCHING WINDOW Fig. 5.17 — Etch- figures obtained with the pin-hole oriascope in Z-cut sections; A for a line ground surface and B for a coarse grind. The relation of the etch-figures to the struc- ture orientation of the section is shown in C. marked through the template of this arm with the desired axes or cutting directions. Figure 5.17A, B shows the etch-figures obtained with the pin-hole oria- scope, on Z-cut sections. Figure 5.17A is for a fine ground surface (600 carborundum) while Fig. 5.17B is for a coarse ground surface (100 carborun- ETCH TECHXIQUE 33 dum), and in both cases a moderate etch, (20 to 30 minutes in 50% HF). It is noted that the spiralhng, outer tails of the etch-hgures (as well as other features) denote the handedness of the quartz. Such handedness features are not as marked with line ground surfaces, nor with weaker etches. The central triangular portion of these ligures is used for alignment of the section with the rulings on the marking arm windows. Since this triangular iigure is misaligned with the X-axes of the quartz by approximately 12°, and in an opposite sense for the two kinds of handedness, there are provided two match- ing arms. One is to be used for left quartz and the other for right quartz. The diagram of Fig. 5.1 7C shows the orientation arrangement of a combina- tion of matching windows and marking template, that results in the section being marked with three radial lines which correspond to the positive X-axes of the quartz. Though this is the most obvious manner of marking Z-cut sections, it is of advantage in practice to obtain a reversed marking on left- hand quartz (by using an oppositely ruled left-hand matching window). By so marking the quartz no further attention need be paid to handedness, see Section 2.4, Chap. 11.'^ In either case the relation of the various plate cuts to the axis markings obtained above, may be determined from Fig. 5.4. Since the etch-tigures give only approximate orientation X-rays are used for the linal determination. That X-rays are not used for the whole determina- tion is as explained in Section 5.1. With X-cut sections, having a coarse grind (100 carborundum) and a strong etch (30-45 minutes in 50% HF), the etch-iigures obtained are like those of Fig. 5.19. Here the positive face of the section gives an entirely different ligure than the negative face, as would be expected from the nature of the etch-pits shown in Fig. 5.5. Opposite-handedness gives re- versed ligures. The four possible ligures are oriented with respect to the Z-axis and the major cap face direction of the section "r" as shown in Fig. 5.19A and B. The non-parallelism of the Z-axis and the parallel sides of the etch-figures amounts to three to five degrees. This disposition of figures (relative to quartz axes) is taken into account in the design of the matching and marking arms shown in Fig. 5.18, and diagrammed in Fig. 5.19C. The etched X-cut section is rotated on the mounting table, with the central matching arm in position, until the long straight sides of the "parallelogram" figure, or the long parallel lines of the "H" figure, are parallel to the two parallel-lines ruled on the window of the matching-arm (the parallelogram figure is shown so aligned in C). The figure thus used is compared with the four figures sketched on this matching-arm, to determine which of the two marking arms is to be used for marking (note arrows giving this indi- cation). The proper marking arm is lowered onto the section and used to ^^ The instrument of Fig. 5.16 has a still different arrangement of matching and marking arms. 34 BELL SYSTEM TECHNICAL JOURNAL mark a long line approximately parallel to the optic axis and a short line in- dicatmg, in the case shown, the approximate direction and the sense of cut of a BT-plate. It is to be noted, here, that neither handedness nor elec- trical sense need be individually determined or considered, as such, for the sense of cut is directly obtained. The size of an etch-tigure depends upon the thickness of the section being examined, as explained in Section 5.43. For the etch-figures here presented the size of the figure relative to the thickness of the section, may be estimated Fig. 5.18— The BASIC PIX-HOLE ORIASCOPE with matching and marking arms for use on X-cut sections. Twinning, and the direction of the Z axis, and the direction and sense of cut may be determined and marked on the section. from a knowledge of the ratio, A', of the total diameter of the view to the thickness of the section giving that view. For Fig. 5.17A and B, N = 1.3; for Fig. 5.19A and B, .V = 2.7; for Fig. 5.20, N = 1.7; for Fig. 5.21, T = 2.5. The pin-hole oriascope may be used in a variety of other wa}-s for exam- ining any cr}-stal cut with at least one etched surface. When used with sections as described above the bottom flat surface may be very small, just large enough to cover the pin-hole. However, this restricts the inspection ETCH TECHNIQUE 35 to an area directly over the bottom surface. This restriction may be elim- inated, and no flat bottom surface need be used at all, if the bottom surface RHQ +X UP RHQ -X UP LHQ-XUP B LHQ+XUP BT marking template (for a) MATCHING WINDOW MARKING TEMPLATE (for B) Fig. 5.19 — Etch-figures obtained with the pin-hole oriascope in X-cut sections. After an etch-figure is aligned with the rulings on the matching window, as in C, the section is marked thru a marking template fin this case the one on the left) with the direction of the Z axis and the direction of cut of the desired plate (in this case the BT). of the section be immersed in a transparent dish of immersion fluid (whose refractive inde.x matches that of quartz) placed over the pin-hole. Here the 36 BELL SYSTEM TECHNICAL JOURXAL size of the etch-figure depends on the whole distance from the pin-hole to the etched top-surface, and hence, may be made as large as desired, by raising the section and fluid level. Xtry thin sections, slabs or plates may be ex- amined similarly, with the bottom surface contacting the immersion fluid, or the plates may be wet with immersion fluid and placed on thick glass plates and placed over the pin-hole. In either case the top etched-surface must be kept dr\'. By this means the twinoriascope examinations described in Section 5.52 might be performed on the pin-hole oriascope, (a disadvan- tage being the necessity of using an immersion fluid). Usually etch-figures are obtained from flat etched surfaces whose orienta- tion is known within 5°. However, if the surface be 10° to 20° off'-orienta- tion the etch-figure will be plainly distorted. If now the section be viewed at an angle to the normal position, or if the section be tilted in the fluid- A B Fig. 5.20 — CLEAVAGE-FIGURES may also be observed on the pin-hole oriascope in ground but unetched specimens, in this case a Z-cut section. Here the direction of the X axes but not their sense (nor handedness, nor twinning) may be determined. bath method described above, the undistorted figure may be observed. The direction and amount of misorientation of the surface may be thus esti- mated. By provision of suitable mounts and scales the misorientation could be measured to 5°. It might be added that m some cases unetched, ground (or sawn) quartz surfaces give "cleavage-figures." Thus with Z-cut sections which have been ground, but not etched, there may be observed on the pin-hole oria- scope cleavage-figures like those shown in Fig. 5.20. The difference be- tween the two views is mainly a difference in focusing and in photographic reproduction. The cleavage-figure indicates that there are preferential cleavage planes in quartz, which are parallel to the X-axes, and correspond approximately to the natural cap faces. Further, there is no indicated dif- ference between the major and minor planes. Thus, the cleavage-figure is ETCH TECHNIQUE 37 six-fold and may not be used to determine electrical sense or twinning. It ma)', however, be used to determine approximately the orientation of the X-axes. Cleavage-figures are seldom strong, but appear to be best with coarse grinding. ^^ 5.6 The Process of Etching Quartz Few factors related to the chemical process of etching quartz have been extensively studied. Much of the information here presented is taken from preliminary' reports of L. Egerton of the Laboratories, who has undertaken an investigation of the etching process. Though the information mainly regards hydrofluoric acid etching, some data is given on etching with hydro- fluoric gas, and bifluoride mixtures. The reaction of quartz, which is silicon dioxide (Si02), with hydrofluoric acid (HF) is given by the following equations: Si02 + 6HF ^ SiF4 + 2H2O + (2HF) i=± HoSiFe + 2H2O. Since the hydrofluoric acid is a solution of HF gas in water, the reaction of the acid with quartz results in a reduction of the concentration of HF. At the same time there is produced silicon tetrafluoride (SiF4) which reacts with more HF to give fluosilicic acid (H2SiF6) in solution. It is common practice to start with about 50% HF acid and to continue etching until the HF concentration is down to 20 or 25%, at which time there should also be a 30% to 35% concentration of H2SiF6, if all the depletion of HF were due to reaction with the quartz. Actually much smaller concentrations of HoSiFe are found, and this discrepancy is mainly due to the large continuous loss of HF from the solution by gassing. Further, the etching powder of this used acid is not the same as w^ould be obtained with a solution of 20%-25% HF alone in water. However, this dift'erence is hardly noticeable except with weak etches. Through the useful life of the acid, starting with 50% HF and depleting to about 20% HF, practically identical etch-figures may be obtained by properly adjusting the etching time. Means of testing the etching power of the acid to determine the proper etching time are compHcated by the pro- duction of HoSiFe in the solution, and by the irregular loss of HF by gassing. Further, the power of the acid to produce useable etch-figures is not the same as its power to remove quartz, or to etch glass, or as its concentration of HF or HoSiFe. For these reasons any indirect method of measuring etching-power must be correlated empirically with the etching-time required to give the desired etch-figures. An indirect method of testing the etching-power, developed by Dr. W. Hoft" of Western Electric, Hawthorne, involves the etching of sand blasted ** Scrubbing the surface with soap, water, and brush sometimes improves the figure. 38 BELL SYSTEM TECHXICAL JOIRXAL microscope slides for a standard length of time. The lead-glass slides be- come coated with a white lead-fluoride deposit to a depth dependent mainly upon the HF content of the acid. The optical density of this deposit is measured with a specially adapted photometer. The photometer readings are correlated with required etching-times to give the desired etch-figures; a different etching-time being required for different kinds of sections, slabs, etc. Use of this means of controlling the etching time has greatly improved the regularity with which good etch-figures are produced in the shop. Commercial hydrofluoric acid from a number of different suppliers has been analyzed for purity, and tested for the development of etch-figures. It appears that when such acids are brought to the same concentration (by addition of water if necessar^^) there is no difference in their effectiveness, nor are they inferior to pure reagent acid. Commonly the acid is supplied as 48% solutions in lead or hard rubber drums, or as 60% in steel drums (usually the concentration is a few per cent higher than labeled). The dif- ference in packaging is of no importance in the results obtained, provided the concentration is properly reduced. There are two important factors regarding the starting concentration of hydrofluoric acid baths. In the first place, acids stronger than 50%, though reacting vigorously with the quartz (and removing material rapidly), do not give good etch-figures. Secondly, strong acids not contained in sealed containers lose strength very rapidly by gassing of the HF gas. Hence un- used fresh acid should be kept well stoppered. Before use the acid should be diluted to a concentration of 45% to 50%. This may be accomplished by adding about | volume of water to one volume of 60% acid, or | volume of water to one volume of 55% acid. Concentrated hydrofluoric acid loses HF by gassing more rapidly than it loses water by evaporation. This preferential loss of HF continues until the HF concentration is reduced to 35% or less,^^ and is not completely over- come by covering the bath without sealing. In fact, in practice, it appears that about as much HF is lost by gassing as is used in etchmg the quartz. Thus the bath should be kept as tightly covered as is practicable. Whereas, in the past only lead and hard rubber have been used for fabri- cation of acid baths and racks, it appears that for concentrations not greater than 50% HF, copper, nickel, and brass may be used as well (steel is inferior at low concentrations). Lead-tin solders may not be used, but silver solder is satisfactory'. Thus shop acid equipment ma\' be easily fabricated out of common fabricating materials.-^ " At room temperatures there appears to be a constant-concentration mixture at some concentration below the 35% concentration of the constant boiling mixture, the exact value depending upon the temperature of the solution and the ambient humidit\^ 2" PolystjTene is a good material for use in fabrication of vessels for handling HF and its reaction products in the laboratorj-. ETCH TECHNIQUE 39 While agitation of the acid bath during etching does speed up the re- moval of quartz from the surface, it does not appear to speed up the de- velopment of the etch-figures here considered. However, moderate agita- tion does improve the uniformit>- of etch from one crystal to another, and even over the surface of single large surfaces (especially when such sur- faces are close together). Uniformity of etch is important in examining for twinning. The surfaces to be etched should never be placed in contact with each other, or with other surfaces, so that the acid cannot flow between them (the separation should be at least ^ of an inch). The effect of temperature on the etching process appears to be small for the range of room temperatures normally encountered in practice. A word of caution chould be added regarding the handUng of hydrofluoric acid and other fluorine etching materials. The dangers are of two kinds. First, fluorine poisoning may result from contact with any fluorine com- pounds, the effects of which may be cumulative. Special care should be taken to prevent inhalation of vapors from all etching baths containing fluorine. Some persons are especially sensitive to fluorine poisoning. Secondly, hydrofluoric acid baths, or any baths containing free HF, may pro- duce acid burns. Commonly such burns are attended by fluorine poisoning. For these reasons etching with all fluorine compounds is preferably carried out in ventilated hoods (with strong air suction through the door), with con- tinually running water for washing, and with rubber gloves, tongs, racks, etc. for handhng the quartz. Etching compounds other than hydrofluoric acid have been widely used in etching glass, as is evidenced b}^ the variety of formulae presented in the "Chemical Formulary."-^ Solutions of ammonium bifluoride (NH4HF2), with additions of various amounts of free hydrogen fluoride, sodium bi- fluoride, sugar, and other materials have long been used on glass. One of the possible advantages of such formulae for etching quartz is the elimination of the dangers of acid burns and strong fumes that may be obtained with hydrofluoric acid (care must still be maintained to prevent fluorine poison- ing). A number of these formulae have been made up and tested on quartz. The preliminary conclusions are as follows. The etch-figures that may be developed by the bifluoride compounds on Z and X-cut sections of quartz are not the same as those developed by hydro- fluoric acid. The results approach each other, however, for excessively long etching in both cases. To obtain usable etch-figures on X-cut sections with the bifluoride requires considerably longer etching time than with hydro- fluoric acid, or an elevation of the bath temperature to about 45°C. The addition of hydrofluoric acid to the bifluoride formulae speeds up the de- velopment, but partly negates the safety advantage of the bifluoride bath. ^1 Published by the Chemical Publishing Co., Brooklyn, N. Y. 40 BELL SYSTEM TECIIXICAL JOURNAL The figures produced on Z-cut surfaces are small and complex, (hardly usable) unless a considerable amount of free HF acid is added. Etch-figures here considered are those produced on the pin-hole instrument, and are us- able only if they have such character as will permit of their use in determin- ing quartz axes. Fig. 5.21 shows the type of usable etch-figure obtained on X-cut sections with an ammonium bifluoride and sugar solution (the sugar is here effective mainly in preventing creepage of the solution). It might HQ+XUP RHQ -X U LHQ.+X U HQ,-XUP B Fig. 5.21— Etch-figures obtained on the pin-hole oriascope with X-cut sections which have been strongly etched in bifluoride mixtures, or e.xcessively etched in hydrofluoric acid. These etch-figures differ from those of Fig. 5.19 (for a moderate etch in hydrofluoric acid) but are obviously usable. be noted that a similar figure is obtained with hydrogen fluoride gas, and with excessively long etching (several hours) in hydrofluoric acid. When the bifluorides are used only to develop reflection contrast in the detection of twmning, their effectiveness appears to be about the same as hydrofluoric acid, under equivalent process conditions. The etching power of the bifluorides may be maintained nearly constant over a long period of use by maintaining an excess of the salt in solution, a distinct advantage over the acid. The metals copper, nickel, brass and stainless steel may be used in fabricating tanks and racks, lead and steel are inferior. Finished quartz surfaces are sometimes etched to remove surface debris ETCH TECHNIQUE 41 (fragments of quartz loosened by grinding, and grinding refuse embedded in microscopic surface irregularities), and to remove predetermined small amounts of the surface for frequency adjustment. It is common for these purposes to use weaker etching solutions, since very small amounts of quartz are to be removed. With hydrofluoric acid, weak solutions (less than 20% HF) have an advantage in that their concentrations are little reduced by exposure to the air. In fact with very weak solution the con- centration may increase slightly by exposure, and thus partly compensate for the HF lost by reaction. Weak ammonium bifluoride solutions may also be used, provided no deposit forming material is added. 5.7 The Effect of Twinning in the Finished Plate While it is commonly considered that electrical and optical twinning are not allowable in a finished oscillator plate, it cannot be unconditionally stated that small amounts of twinning will too seriously affect the properties of all types of oscillator plates. The allowance of even small amounts of twinning in the finished plate would save quartz and simplify the processing procedures. Hence, consideration must be given to the factors which would affect the utilization of twinned material, and the effect of twinning on the operating characteristics of the finished plate. Consideration will first be given to the nature and distribution of electrical and optical twins- in the raw quartz. The analysis of twinning in raw quartz has been carried out by the ex- amination of numerous, etched Z-cut surfaces. By the method to be de- scribed it is possible to detect the handedness, and the axial orientation and sense, of each homogeneous portion, twin, appearing at the etched surface of a twinned specimen. Both electrical and optical twins may be analyzed by this method. It might be added that electrical twinning boundaries and orientation are only detectable at an etched surface, and that while interior optical twinning may be detected by polarized light, its exact analysis is only possible at an etched surface. Figure 5.22 E shows the optical arrangement used for examining twinning in etched Z-cut sections. The sections (prepared with a fine grind and weak etch) were mounted on a turntable, illuminated from an elevation of about 30° to the horizontal etched surface by a spot lamp, and viewed (or photo- graphed) from vertically above the section according to principles of Section 5.41). With the section properly aligned on the table (with the predeter- mined electric axes parallel to the table-lines joining diametrically opposite fiducial marks), the table was successively turned into positions about 12° to the right or left of the plane of illumination and reflection (as indicated by the R and L marks and the index pointer). Four of these positions of ^^ See footnote 4. 42 BELL SYSTEM TECHNICAL JOURNAL illumination of a given section are suf&cient to determine the nature of the four possible twins in the section. The four corresponding photographic views of the section have been arranged in a special manner to simplify their explanation. This arrangement, as shown in Fig. 5.22A, B, C and D, is equivalent to what would be observed if one looked down on a single, sta- tionary section, and illuminated the section from the four different direc- FROM PLANE or I AND R Fig. 5.22— Reflection patterns of the twinned, Z-cut sections shown in Figs. 5.23. 5.24, 5.25 and 5.26 were obtained by the means shown in E. A, B, C, and D are a key to the four equivalent directions of illumination of a single stationary section. tions shown in the figure. For each direction of illumination there is a cor- responding view, the outline of the section (and any cracks, chips or other flaws) being identically positioned in each view. However, when the four types of twins are present in a given section, each view will show a different region, or regions, of brightness. For each view, the interpretation of handedness and electrical sense of the bright portion of the view is according ETCH TECHNIQUE 43 to the labeling of this particular view, only. Thus if a section is entirely right quartz and of the electrical sense shown at A the whole surface of the section will appear bright in view A and dark in all other views. If a section is all right quartz, but partly of the electrical sense shown in A and partly that shown in B, then part of the surface will appear bright in A and the other part will be bright in B (the whole surface will be dark in C and D) Fig. 5.23 — The four possible conditions of handedness and electrical sense in a single section are shown here. In each view the handedness and sense is for only the bright portion of that view. The a and b regions are seen to be both of right quartz but of opposite electrical sense, hence electrical twins. (Flaws indicated by/ are to be disregarded). A section containing all four possible twins would exhibit bright regions in each view, and a different bright region in each view. All bright regions would fit together to make a complete map of the surface. Only the bright portion of each view has the handedness and electrical sense indicated for that view. Figure 5.23 shows a Z-cut section containing twins of the four possible 44 BELL SYSTEM TECHNICAL JOURNAL conditions of electrical sense and handedness. The two large, bright re- gions a and b (appearing in views A and B respectively) are both right quartz but of opposite electrical-sense. Hence the surface is mainly of electrically twinned right quartz. The small dark regions within the borders of a (view A) are bright in view D. Hence these small, triangular and line re- gions are left quartz and of opposite electrical sense to the large region a containing them. They are then optical twins of the large a region. Simi- larly the dark regions of b (view B) are found from view C to be optical twins of the b region. (Flaws labeled/ are cracks, chips, etc.) If the whole section were cut up to make AT plates, for example, and at the proper angu- lar sense according to the a portion of the section, then those plates coming from the b region|,would be of wrong angular sense. Those crossing a boundary between. the a and b regions would be of both senses, i.e., elec- trically twinned. Those few plates which contained some left quartz would be optically twinned. To make the most economical use of this section it should be separated, by cutting along a line approximating the a to & bound- ary, so that each half of the section may be cut at the correct sense of orien- tation. Even when so cut, some of the plates will contain optical twinning and remnants of electrical twinning. This section is typical of much of the raw quartz that must be used for manufacturing piezoelectric plates. Figure 5.24 shows a section which is mainh' of left quartz as exhibited by the large bright c and d regions of views C and D. The large c region is optically- twinned to a small extent by the line regions b of view B. One of the d regions is badly optically twinned by the small striated a regions, as seen in A. Such a section would be very uneconomical to process, since separating the larger electrical twins is not feasible. If processed at all, it should probably be entirely cut according to the handedness and sense of the large c portion, the wrong-sensed regions and twinning being cut away at a later stage (after inspection of the slabs in the twinoriascope, for ex- ample). It might be noted that only the optical twinning could have been observed in the initial polarized-light, raw quartz inspection, where such a stone would be passed as moderately good. Fig. 5.25 shows an unusual section that is mainly composed of left quartz, regions c and d. The right quartz regions shown in view B are of both opposite-handedness and electrical-sense to the c region inclosing or bordering them. This is the common and expected conditions. The unusual condition is exhibited by the regions c and a, where twins of opposite-handedness but same electrical-sense have a common boundary. Since this boundary could be detected by optical means, the a and c regions might be de- scribed as optical- twins, of an "uncommon variety". However, by con- vention optical twinning has long been used to denote twinning exhibiting both opposite-handedness and opposite-electrical-sense (crystallographically. ETCH TECHNIQUE 45 Brazil twinning). Further, twinning exhibiting both opposite-handedness and same-electrical-sense, combines the cr\'stallographic twinning laws of Brazil twinning and Dauphine (electrical) twinning. Hence this uncom- mon variety of twinning may preferably be called combined electrical and optical twinning, or just COMBINED TWINNING. Thus, the boundary Fig 5 24— Regions d are electrical twins of the region c. The striated regions a are of opposite handedness and electrical sense to the d region enclosing them, hence optical twins of d. The b regions are small optical twins of c, and/ are flaws. between the a and c twins separates combined twins. Note also that the a twin bounds the b twin and the h twin bounds the c twin. Thus, a and b are true electrical twins, and b and c are true optical twins." 23 It is possible that growth conditions are such that combined twinning cannot occur bv itself without the presence of true optical twinning and true electrical twinning. That is, a region of given handedness and sense can not be entirely bordered by a region of opposite-handedness and same-sense. 46 BELL SYSTEM TECHNICAL JOURNAL Figure 5.26 shows an unusual section which is mainly composed of left quartz, of the electrical sense shown in D, region d. The region c is an elec- trical twin of d. The region/ is a flaw in the quartz and is to be disregarded. The region a is an optical twin of d, and is uncommonly large for an optical twin (note: region a contains within it, two small optical twins). Since Fig. 5.25 — Regions c are electrical twins of the adjacent d regions, a is an electrical twin of b, and a is also an optical twin of d. An uncommon condition of twinning is pre- sented by the adjacent a and c regions which are of opposite handedness but the same electrical sense, thus exhibiting COMBINED-TWINNING. optical twins are usually very small (except for the one major surrounding twin), it is seldom possible to cut them apart and use each twin individually. Figures 5.1, 5.2 and 5.3 were obtained by the means above described, and all sections shown in these figures (except Fig. 5.2A and C) actually ex- hibited both electrical and optical twinning. Thus Fig. 5. 3D was obtained from Fig. 5.24A, and Fig. 5.2F from Fig. 5.24C, etc., by trimming the latter ETCH TECHNIQUE 47 named figures to give the sections simulated natural faces. Figures 5.2 and 5.3 are of particular use in learning to distinguish between electrical and optical twinning when examining etched surfaces by reflection. Note that electrical twins are usually large and separated by irregular boundaries, Fig. 5.2. Optical twins are usually separated by straight-line boundaries Fig 5 26— Since this section exhibited no bright regions (except flaws/) in view B (i.e. no ri-^ht quartz of electrical sense B) it was not reproduced in view B. The c region is an electrical twin of the adjacent d region, while a is an optical twin of d. It is uncommon for a minor optical twin to be as large as a. parallel to natural faces, thus forming triangular, parallelogram, and straight line insets. Fig. 5.3. Optical twins (except for the one major, surrounding twin) are usually very small and often interlayered (with the major twin). Large interlayered regions are entirely unusable and hence are cut away at the earliest possible stage to save the labor of processing worthless material. Small optical twins and small electrical twins (or remnants of electrical 48 BELL SYSTEM TECHNICAL JOURNAL twins left after cutting electrical twins apart) may be isolated or removed in an intermediate or late stage of processing, where they are detected by the etch technique. Commonly the final rejection of material twinned in either way is delayed until after the final blanks are cut out. These may be etched and examined by reflection, one at a time under a spot lamp, and those showing twinning (and other imperfections) sorted out and rejected. Another possible method of rejecting twinning which is of sufficient amount to be harmful is by making electrical tests on the finished (or semi- finished) plates, at which time those plates failing to meet the electrical tests for any reason (including twinning), are rejected. While this method of rejection does not assure that twinning will be entirely absent from the accepted plates, neither does any other method assure complete absence of twinning. Further, except for imperfections which may affect the useful life of the plate, acceptance of finished oscillator plates is not illogically based Table I. — Constants for Plates of Correct and Incorrect Sense of Cut Cut, Angle Frequency Constant (fxd. in Kc. mm.) Temperature Coefficient (parts/108 /C.°) AT +35° (-35°) 1670 (2400) 0 (+30) CT +38° (-38°) 3080 (2100) . 0 (-30) BT -49° (+49°) 2560 (1880) 0 (-55) DT -52° (+52°) 2060 (2850) 0 (+45) upon their meeting the desired electrical operating characteristics, i.e., frequency, temperature-coefficient, activity and internal damping (all de- terminable by electrical tests) .-^ It does not appear that twinning will afifect the useful life of the plate. Its effect upon the electrical operating charac- teristics of the plate depend upon many factors. An important factor regarding twinning in the finished plate is that optical twinning introduces a less important variation in the physical properties of the plate than does electrical twinning. Thus, in the case of optical twinning alone, both portions of the plate are of the same sense of cut, though still being of opposite electrical sense. This may be understood from an ex- amination of Fig. 5.4, the second and third views taken together represent optical twinning. In the case of electrical twinning the two portions of the 2^ With filter plates additional operating characteristics must be met. The ratio of capacities (see Chap. I, Appendix A.3) is greatly affected by the opposed electrical sense of twinning. ETCH TECHNIQUE 49 plate are of both opposite sense of cut and opposite electrical sense, as may be observed from the third and fourth views of Fig. 5.4. The effect of this difference in sense of cut for the two types of twinning is brought out by Table I, which gives the approximate frequency constants and temperature coefficients for the common cuts of oscillator plates, together with those for the analogous, oppositely (and hence wrong) sensed cuts. In the case of a CT plate, for example, both portions of an optically twinned plate (cut at +38°) will be of the same +38° orientation. The plate is elastically the same throughout and hence should exhibit the frequency and low temperature-coefficient desired. However, the opposed electrical senses of the two portions will cause a reduction in the electrical activity. The amount of this reduction will depend upon the relative size of the two portions and upon their placement relative to the vibration nodes of the plate. On the other hand, when a CT plate is electrically twinned one portion of the plate will be of the correct +38° orientation while the other portion is of the incorrect —38° orientation. The two portions of the plate have widely different elastic properties, as is exhibited in the table by the different fre- quency constants and their respective temperature-coefficients. Resulting from this difference alone, the plate will exhibit operating characteristics (if operable at all) intermediate between the two listed in the table (usually near one of these two), and its activity will be reduced. The activity will also be reduced by the opposite electrical senses in the two portions. The degree to which the frequency, temperature-coefficient, and activity are affected, again depends upon the relative sizes of the two portions of the plate and their placement relative to the ''nodes" of the plate. Thus, for equivalent proportions and placement of twinning, electrical twinning will cause a much greater change in the operating characteristics of the plate than will optical twinning.-^ A note may be inserted regarding the electrical testing of plates, some of which may be twinned while others may be untwinned but of incorrect sense of cut. As seen from Table I, untwinned plates of the correct sense of cut are easily distinguished from those of the incorrect sense of cut by their frequency. This distinction between sense of cut holds as well for plates containing very little twinning. The presence of appreciable twinning in the plate is easily distmguished b\- the activity of the plate. While ordinar- ily a plate would be electrically tested in the mode of vibration it is intended to be operated in, it is sometimes of advantage to test it in a different mode. -5 In the case of the uncommon ■"combined-twinning" the two portions of the plate are of opposite sense of cut but of the same electrical sense. The effect on the operating char- acteristics will be like that for electrical twinning, except that the activity may not be as greatly reduced. 50 BELL SYSTEM TECHNICAL JOURNAL Thus the high-frequency mode plates (AT and BT) might be tested in their low frequency modes (corresponding roughly to the CT and DT modes, respectively). A further discussion of this matter will be found in a later chapter by I. E. Fair. 5.8 Conclusions In the processing of quartz, consideration must be given to the nature of twinning and to its characteristic distribution in the raw stone. There are only two common types of twinning that need be considered, namely elec- trical twinning and (true) optical twinning (''combined- twinning" and other types are a rarity). Due to the characteristically large size (and the nature) of electrical twins, a stone must be examined for electrical twinning (by the etch technique) at an early stage of processmg so that the electrical twins may be observed and cut apart before the angular cuts (AT, BT, CT, DT, etc. slabs, bars, or wafers) are made. Otherwise, some of the large electrical twins will be entirely cut up with the incorrect angular sense, and hence wasted. On the other hand optical twins are characteristically small and inter- layered, or small and scattered. The interlayered regions are entirely un- usable. Hence processing labor will be saved by inspection of the raw stones (by the polarized light means of Chapter IV), and of the first sections at least (by the etch technique) for large regions of interlayered optical twin- ning. Scattered optical twins and small electrical twins, or remnants of elec- trical twins which have been cut apart, may be cut away in an intermediate processing stage, or in a later stage plates containing such twinning may be separated out. In either case the etch technique may be used to detect the twinning. An alternative method of eliminating small electrical twins (or remanents thereof) and of small optical twins (most of which are characteristically very small) is by electrical tests on the linished plate. This method has merit in that if the twins are sufficiently small, and not disadvantageously placed in plate, they may not harmfully efifect the desired operating charac- teristics of the plates. The degree of the effect depends not only upon the size of the twin and its location in the plate, but upon whether the twinning is electrical or optical; optical twinning being considerably less harmful than electrical twinning. The effect of the twinning further depends upon the type of plate being considered, i.e. its size and mode of operation, and use. It is probable that twinning is more tolerable in low-frequency mode oscil- lators (CT and DT) than in the high frequency modes (AT and BT), and of course more tolerable in plates of low requirements on the operating charac- teristics (activity, frequency and temperature-coefficient). Twinning is ETCH TECHNIQUE 51 probably least tolerable in filter plates, which have to meet very special re- quirements.-^ Detailed experimental studies of allowable amounts of twin- ning are of little value since to use the results in a manufacturing process would require a careful inspection of each plate and a difficult classification into groups depending upon the variety, amount, and placement of the twinning. Acceptance or rejection of finished plates on the basis of their final electrical operating characteristics appears to be the only practical means of separating usably twinned plates from unusably twinned plates. This method of selection does not determine whether the rejected plates con- tain twinning or other imperfections (or are misoriented or misdimensioned) and is therefore of little use in analyzing the processing methods to deter- mine best practices. This disadvantage may be eliminated by etching the rejected plates and examining them for twinning (and such other imperfec- tions as show up best after etching). The effects of crystal imperfections other than twinning were discussed in Chapter IV, Section 4.9. -" See footnote 24. CHAPTER VI Modes of Motion in Quartz Crystals, the Effects of Coupling and Methods of Design By R. A. SYKES 6.1 Introduction WITH the recent extended use of Quartz crystals in oscillators and electrical networks has come a need for a comprehensive view of the various types of crystal cuts. In addition there has been a need for illus- tration of some of the methods employed in choosing the proper cut for a given requirement, the manner in which quartz crystals vibrate and the basic principles governing the choice of a design to use certain cuts most advan- tageously. In particular one of the greatest problems associated with the recent large scale production of cr3^stals for oscillator purposes has been that of obtaining crystals the activity and frequency of which would not vary to any large degree over a wide range in temperature. It is the intention of this chapter to present a physical picture of the man- ner in which quartz crystals vibrate in their simplest forms and then to show what has been learned from these simple forms that will apply to the more complex combinations of motion. The motion of a bar or plate is deter- mined almost wholly b}' its dimensions and the particular type of wave gen- erated, or frequency applied, and very little upon the driving system if the coupling to the driving system is small. In the case of quartz the coupling between the electric and mechanical system is small and hence we may study the motion of rods and plates without always considering the effect of changes due to the method of excitation (i.e., piezo-electric). However the ease of exciting and measuring a particular mode does depend on the piezo-electric constant driving it. Basically only three t}-pes of motion will be considered ; flexural, extensional and shear. These three types of motion or combina- tions of these can be considered to represent most of the cases with which we will concern ourselves. In additioft, the frequency equations will be given for common types of motion and the effect of coupling between various modes of motion. Finally the general rules relating to the dimensioning of oscillator plates wiU be presented. 6.2 Types of Motion in Quartz Rods and Plates 6.21 Flexural The motion associated with flexure will be discussed first because this is the type of motion that we see more commonly in nature. This motion is 52 MODES OF MOTION IX QUARTZ CRYSTALS 53 the type which presents itself in the xylophone, the chime type door bell, and various other vibrating reeds or bars. Fig. 6.1 shows the general type of motion of a bar free to vibrate in flexure. The displacement takes place in the direction of W and the wave is propagated along the length. A flexure mode is one in which the center line does not change length. The type of motion associated with the first order, or fundamental, of a bar free to vibrate on both ends is shown in Fig. 6.1 with a dotted figure superim- 1 ST 2 NO 3 RD Fig. 6.1 — Motion of a bar in free-free flexure. posed to show the motion in the opposite phase. The straight bar then w^ould be distorted first in one direction and then in the direction of the dotted figure. In the case of the second mode of vibration, it will be noticed that it consists essentially of two of the fundamental mode types joined end to end. This is not strictly the case, but serves to illustrate the motion. The dots shown at various points on the bar show positions of zero motion or nodes. In the case of the fundamental mode, there are two nodes and in the second and third there are three and four respectively. One point of 54 BELL SYSTEM TECHNICAL JOURNAL interest in flexure vibration as seen in Fig. 6.1 is that the ends of the bar will be vibrating in the same direction for odd order modes and the motion of the two ends will be in opposing directions for even order modes. The frequency of a bar vibrating in flexure may be easily computed for low orders when the width is small in comparison with the length. When the width is appreciable other factors must be considered as will be shown later. In general, the flexure frequency of a bar will be the lowest frequency of vibration. In the case of a plate where we are concerned with flexural vibrations propagated along the length with motion in the direction of the thickness it Fig. 6.2 — Motion of a plate in free-free flexure. is necessary to consider also the width. As noted in Fig. 6.1, our concern was only for a bar of small third dimension. When considering the case of a plate in flexure along its length and thickness, then the third dimension must also be considered for more complicated types of motion. In a manner somewhat similar to the vibration of a bar, we can consider a plate vibrating in its thickness-length plane. Since a plate also has width, we must also consider this dimension. The simplest type of motion would be that of a simple flexure which would bend the plate into the shape of an arch. If now, the third dimension is permitted to flex, the distortion of a plate shown in Fig. 6.2 could be illustrated by a flexure in the t-t plane and in the MODES OF MOT I OX IX QUARTZ CRYSTALS 55 w-i plane. Considering the motion of the plate as a flexure vibration along the length vibrating in the thickness, then we may also have a distortion along the width and thickness corresponding to similar or higher types of flexure motion. The illustration at the bottom of the figure shows a plate vibrating in its second order flexure along the length and thickness and the fourth order flexure along the width and thickness. The effect of these higher orders in the uf-t plane is to slightly modify the frequency of the (-W mode. A thorough treatment of this t}'pe of double flexure in plates will be given in Chapter VHI by H. J. McSkimin. 1ST 2.ND • • • 3RD Fig. 6.3 — Motion of a bar in free-free extension. 6.22 Extensional The extensional or sometimes termed longitudinal motion of a bar free to vibrate is shown on Fig. 6.3. This motion is somewhat simpler than the flexure motion and consists simply of a displacement in the direction of the length of the bar of a wave propagated along the length. This means that the first mode of vibration will be simply an expansion and contraction of all points with respect to the center of the bar. This motion will be along the length. The displacements along the bar will then be in proportion to the sine of the angular distance from the center. The distortion of a free bar in its simplest mode is then illustrated in Fig. 6.3 labeled 1st. Since the 56 BELL SYSTEM TECHNICAL JOURNAL motion must be dynamically balanced, a node will appear at the center of the bar, and the bar will grow longer and shorter as shown by the solid and dotted lines. In the case of the second order of motion, as shown in Fig. 6.3, it consists essentially of two 1st order modes joined together at their ends and of opposite phase. That is to say, when one half of the bar is expanding, the other half is contracting. In the case of the 3rd mode, as can be seen from Fig. 6.3, the central element is contracting while the exter- nal elements are expanding. From this we may state generally, that for odd order types of motion, the extreme ends of the bar will be expanding or contracting in phase and for even order modes, the extreme ends will be expanding or contracting in opposite phase. Fig. 6.3 illustrates extensional motion in its simplest form. In a practical case an extension in one direction is accompanied by a contraction in one or both of the other two dimensions. This of course is due to elastic coupling and will be considered more in detail later. If we consider a rectangular plate it is not difficult to imagine that it would have three series of extensional modes of vibration due to the three principal dimensions. 6.23 Shear The low frequency of face shear type of motion of a plate is somewhat more complicated than either the flexure or longitudinal and, as shown in Fig. 6.4, consists simply of an expansion and compression in opposite phase along the two diagonals of the plate. This motion is shown in Fig. 6.4 labeled m = I, n = 1. The two phases are shown, one a solid curve and the other a dotted curve to illustrate the distortion with respect to the original plate. One peculiarity of shear motion in plates is that it may break up into motions similar to its fundamental along either the length or the width. For example, if we take the motion associated with w = 1, 11= 1, and superimpose two of these in opposite phase on the same plate, we would get the tynpe of motion illustrated by m = 2,n = I. In a similar manner, the motion may reverse its phase any number of times along either the length or the width. One particular case is shown for m = 6, n = 3. As can be seen from the case of m = 1, n = 1, the distortion is not that of a parallelogram as it is in the static case because here we are concerned only with the dynamic case. While the equation of motion of a free plate vibrat- ing in shear has not been completely solved, a microscopic analysis indicates that the actual motion of the plate edges appear to be somewhat as shown for the case m = 1, « = 1 when driven in this mode. The shear mode of motion in the case of a thin plate is somewhat diflferent for the high frequency case than for the low frequency case. In the case of high frequency shear modes of motion in thin plates, the motion of a particle is at right angles to the direction of propagation which in this case would be MODES OF MOTIOX IN QUARTZ CRYSTALS 57 the thickness. The simplest type of motion for high frequency shear is shown in Fig. 6.5 where the top of the plate is displaced in the direction along i with respect to the bottom of the plate. This would then be termed the length-thickness shear. When viewed from the edge of the plate, the motion is very similar to that shown in Fig. 6.4 for the case of ni — l,n = 1. In a manner similar to the previous case of shear the front edge of the plate may be divided into segments along C and along /. For example, we may get ■nn = 1 n = l ■m = 2 T1=1 fT^=6 71 = 3 Fig. 6.4 — Motion of a plate in low frequency shear. a double shear along ^ with a single shear along /. This case is illustrated in Fig. 6.5 for m = 1, n = 2 and p = l. In general, m and n may assume any integral value. As in the case of flexure we must also consider the third dimension. The motion associated with the third dimension may be repre- sented by simple reversals of phase as before. For example, in Fig. 6.5 the case for m = 1, n = 1, p = 2 is shown which simply means that the high frequency shear on the front half of the plate is out of phase with that of the 58 BELL SYSTEM TECHNICAL JOURNAL back half of the plate. This discussion relates only to the case of the high frequency shear commonly assumed to be a single shear along the length and thickness of the plate. Similar statements can be made if we consider the high frequency shear as being along the width and thickness. m = 1 n = 1 P =2 Fig. 6.5 — Motion of a plate in high frequenc}^ shear. 6.24 Type of Motion for Some Standard Filter and Oscillator Plates To get a more complete picture of the applications of the various types of motion, we will now take specific cases. The various crystals as com- monly used for oscillators or filters are shown in Fig. 6.6. At the top of Fig. 6.6 are shown the various types of shear plates with their relative position with respect to the crystallographic axis. The AT and BT plates are termed high frequency shear plates and the motion associated with them is that of a length-thickness shear as shown in Fig. 6.5. Their use is found for the control of radio frequency oscillators in MODES OF MOTION IN QUARTZ CRYSTALS 59 L V AT 3 BT I I ( / I • / ( r 1 • • > + 5^ Fig. 6.6 — Motions of typical cuts of quartz. the range from 1 to 10 megacycles. The AT is most useful in the lower range and the BT in the upper range since it has a higher frequency constant. 60 BELL SYSTEM TECHNICAL JOURNAL Considerable use for the .4 T plate has been found for filters on pilot channels for the coaxial telephone system. The CT and DT are analogous to the AT and BT but are termed low frequency shear plates. The motion associated with these cuts is that of a face shear as illustrated in Fig. 6.4. The CT and DT cuts are useful for both filter and oscillator applications in the frequency range from 60 kilo- cycles to 1000 kilocycles. Here again the DT would be most useful in the lower range and the CT the upper range due to the higher frequency constant for the CT cut. The GT is similar to the CT except that it is rotated by 45° about the nor- mal to the plate so that instead of a face shear type of motion there are two extensional modes similar to that shown in Fig. 6.3. These two modes are coupled to each other resulting in one of them having a zero temperature coefficient over a wide range of temperature. This crystal is most useful in the range from 100 kilocycles to 500 kilocycles for a primary standard of frequency and in filter networks having extreme phase requirements. The filter plates commonly called the — 18° cut and 5° cut are shown with their relation to the crystallographic axes in the central part of Fig. 6.6. The — 18° cut commonl}' used in filters employs a simple extensional motion along its length with small coupling to an extensional motion along its width and practically zero coupling to a face shear type of motion. Since the width is usually the order of half the length these modes are not troublesome. The + 5° cut is useful in filter work because it has a low temperature coefficient and in spite of its strong coupling to the plate shear, it has been found quite useful in both its extensional mode and its flexure mode. The —18° cut is used over the frequenc}' range from 60 kilocycles to 300 kilocycles and forms the basic crystal used in the channel filters of the coaxial telephone system. When driven in flexure the 5° cut may be made to operate as low as 5 kilocycles and is used in oscillator and filter circuits. The NT cut is shown at the bottom of Fig. 6.6 with its relation to the crystallographic axis. This is obtained by a rotation of +8.5° about the X axis with a second rotation of ± 60° about the resulting Y' axis. The pur- pose of the second rotation is to give the shear modulus a positive coefficient. This modulus enters into the equation for the flexure frequency and there- fore the effect of the second rotation is to change the temperature coefficient of the flexure mode from a negative value to zero. This crystal has been used to some extent as a low frequency oscillator. Its main purpose so far has been for the control of frequency modulation broadcast transmitters and for low frequency pilot channel filters. Another crystal called the MT which is cut in a manner similar to the NT but with angles of 8.5° and 36° respectively has been used for filter work where an extensionally vibrating crystal of zero temperature coefficient is MODES OF MOTION IX QUARTZ CRYSTALS 61 required. The motion associated witli this crystal is similar to that shown for the +5° cut of Fig. 6.6. The low temperature coefficient is obtained through coupling to, and the effects of, a shear mode of positive temperature coefficient. Its use has been mainly for pilot channel filters of rather narrow frequency bands. 6.3 Frequency Equations for Flexurel, Extensional and SHE.A.R Motions In determining the motion and resonant frequencies of a particular type of vibrating system it is customary to consider an isolated type of motion in order that the solution shall be in a simple enough form to be practical even though it may not be too accurate. The more accurate type of solution is often so complex that its use for practical solutions might be small. Since any solutions so far obtained are not complete in every detail, it is usually necessary to resort to experimentall}' determined frequencies in any case, and the solution can only be regarded as a guide to the complete result. In the following treatment it will be assumed that the frequency equations are given for isolated modes of motion and it will be later shown which of these forms are coupled and the effect of the coupling. 6.31 Flexural Resonant Frequencies The simplest equation relating the resonant frequencies of a rod vibrating in tlexure is given by 7H^ k 2^? f-'^J.' 6.1 where v = velocity of extensional propagation = -x/Fq/p k = radius of gyration of cross section Yq = Youngs modulus ^ = length w — (n + l/2)7r for free-free modes = (n — l/2)7r for clamp-free modes (» > 1) n = order of mode (1, 2, 3, etc.) This equation holds only for the case of a long thin rod. Measurements of the resonant frequencies of a quartz crystal vibrating with both ends free has shown the above equation to be true where m is defined approximately as {n + l/2)x provided — is less than .1. For values greater than this the measured values are somewhat lower than that predicted. When the di- mension in the direction of vibration is appreciable in comparison with the 1 Raj'leigh, Theory of Sound, Vol. 1, Chapter VIII. 62 BELL SYSTEM TECHNICAL JOURNAL length, Mason has shown that it is necessary to consider the effects of rotary and lateral inertia. His solution leads to the same frequency equation as 6.1 but with a different evaluation of the factor m which is obtained from the transcendental equations tan m X = K tanh mX' for even modes tan m X = —-r^ tanh mX' for odd modes A 6.2 where X Equation 6.2 holds only for the case of a rod free to vibrate on both ends. The case of a clamp-free rod is somewhat more complicated since it cannot be given by separate solutions for the even and odd modes. The interpretation of m given in equations 6.2 will result in the same value as before [m = (n + h)ir] for values of — less than .05 but decrease considerabh' for larger values and ultimately as the bar becomes wider the effects of rotary inertia result in the flexure frequency approaching the extensional mode as an asymptote. As stated before measurements on quartz bars vibrating in flexure departed from that predicted by the simple definition of m when the width of the bar was such that — > .1. By using the value of m defined by nw equation 6.2 it is possible to predict the frequency for widths as great as — = .5. For widths greater than this, experiment shows a frequency lower than that predicted by equation 6.2. This then leads one to believe that the effect of shear plays an important part in the flexure of bars with appreciable width. An investigation of the effect of shear on the flexure frequencies of beams has been made by Jacobsen^ and his results lead to the same frequency equation as 6.1 and to the same transcendental equations derived by Mason (6.2) but with different values of A', X' and A' to account for the shearing - \V. P. ^lason, "Electromechanical Transducers and Wave Filters," Appendix A. D. Van Nostrand Company, Inc. ^Jour. Applied Mechanics, March 1938. MODES OF MOTION IN QUARTZ CRYSTALS 63 effect. These values are given bv -i[(--i^"(i-)y-f(i-)] " A' = where Cj, is the shear constant in the plane of motion su is the elastic constant in the direction of propagation. While it is true that these values will result in a lower value of w than those associated with equation 6.2 and hence fit the actual measured results more closely for bars wider than — = .5, there is some doubt in the minds of various investigators as to the actual amount of correction necessary to apply to compensate for the shear. The solution of equation 6.2 using the constants of equation 6.3 is a lengthy process and could only be applied to a given orientation since the elastic constants vary with direction in quartz. While the results of Jacobsen's work are difficult to handle for intermediate values of — where the correc- tion of rotary and lateral inertia do not tit the measured results it does imply that for large values of — that the lie.xure frequencies will be mainly a function of the length alone. Therefore when we are concerned with very high orders of He.xure in plates such as the case of high frequency A T and BT shear crystals we may assume the interfering modes due to flexures will be essentially harmonic in nature. Restating the general problem of determin- ing flexure frequencies in quartz rods or plates we may assume that the ratio of width to length is the controlling factor in deciding which method of nw attack is to be employed. For values of — less than .1 equation 6.1 will . tiiv give quite accurate results. For values of — up to .5 equation 6.1, using the values of m determined by equation 6.2 will give satisfactory results. While the values of m determined by using equation 6.3 will give more accurate results for the range .4 to .6, it is not desirable to carry it further because, while 6.2 does take into consideration the effect of shear it does not account till' for coupling to the shear mode of motion. Hence for values of — > .6 64 BELL SYSTEM TECH. MCA L JOURNAL it is best to depend upon experimental measurements if accurate results are a factor. 6.32 Extensional Frequencies The resonant frequencies of a bar vibrating along its length, commonly called an extensional mode of motion is derived quite easily from the wave equation in one dimension and is given by U y Sin '--kVt, " where ( = length Sii = elastic constant in the direction of propagation p = density n — 1, 2, 3, 4, etc. This is the case when the length is the greatest dimension. When we con- sider extensional modes along the thickness of a plate, it can be shown that the c constants be employed to account for the lateral inertia in the two directions at right angles to the direction of propagation, (provided that the resulting motion is nearly along the thickness direction). Hence, for thin plates As an example of the use of the above equation an X-cut bar vibrating along its length would result in a series of resonant frequencies defined by equation 6.4. An X-cut plate vibrating along its thickness would result in a series of frequencies defined by equation 6.5. Applying the appropriate constants 2Y y 127.9 X 2.65 V77 — r n kilocycles 6.6 7(cm) and , _ ^ /86.05 X 10^ ^' IX y 2.65 285 X(cm) n kilocycles 6.7 This shows that although Young's Modulus is the same in the two directions the resulting frequency constants are different because of the conditions at the boundaries. MODES OF MOTION IN QUARTZ CRYSTALS 65 6.3v3 Shear Resonanl Frequencies As shown in seclion 6.23 the low frequency face tyjic shear mode results in a doubly infinite series of frequencies due to the manner in which the plate may break up into reversals of phase along its length and width. While a solution for the low frequency shear motion that satisfies the boundary condition of a free edge has not yet been accomplished, several approximate solutions for the frequencies are available. A modification of the equation developed by Mason will give results which verify experimental data. where p — density Sjj — shear modulus in dv plane m, n = 1, 2, 3, etc. ^ = length of plate IV — width of i)late The value of k so far remains exi)erimental and for low orders of m and it may be assumed unity. Its use is mainly for high orders of m and n where Young's modulus is different in the ^and w directions. Experimental data in the case of BT plates indicates that it should be 1.036 to account for the difference in velocity in the two directions. When m or n is large the velocity /T Z^- component, namely A/ — - should be replaced by A/ — for reasons ex- V P^ii V P plained for the extensional case. Equation 6.8 holds for the case of a plate vibrating in low frequency shear in regions where no highly coupled exten- sional or jflexural resonant frequencies exist. As will be shown later, these regions are few. By assuming the frequencies are given by these equations and then applying the normal correction for coupled modes, a fairly accurate result will be obtained. The high frequency case of a plale vibrating in shear is somewhat similar to the face shear or low fre(|uency case with the exception that three dimen- sions must be considered since two are large compared to the third (the main frequency controlling dimension). An experimental formula hn" this case is given by /=2t^VF + ^;^ + *'^^^' where Cjj = shear modulus in plane of motion p = density ■C,w,t= length, width and thickness '' "Electrical Wave Filters Kniplm inj,' Ouartz Crystals as Elements," W. 1'. Mason, B.S.TJ. July, 1934. 66 BELL SYSTEM TECHNICAL JOURNAL m, n and p represent reversals of phase along the three directions and may be termed overtones. The values of k and ki are inserted to correct for the change in shear velocity resulting from a change in Young's modulus in the three directions. For most work with oscillator crystals where the length and width are large compared to the thickness, the following simplification of equation 6.9 is most useful. f-l^ 6.10 When high frecjuency shear type crystals are used in connection with selec- tive networks, it is necessary to make use of equation 6.9 to determine where the next possible pass regions will occur. 6.34 Effects of Rotation About the CKystallographic Axes on, the Resonant Frequencies and Coupling between Modes of Motion Several of the elastic constants have been used in equations expressing the resonant frequencies. Since most of the crystal cuts now in use are rotated at some particular angle about the -Y crystallographic axis, it is of int:rest to know the effect of this rotation upon the elastic constants since they determine the resonant frequencies and the coupling between certain of the modes of motion. The general stress-strain equations for an aeolo- tropic body are given in equation A.l of Appendix A together with their definitions. In the case of quartz where the axes of the finished plate are aligned with the crystallographic axes the constants reduce to 7 and are shown in equation A. 8. Examination of these equations shows that there are extensional and shearing strains resulting from dissimilar extensional and shearing stresses through the elastic constants Sij and Cj,-. This results in coupling between modes of motion where a so-called cross strain exists. These couplings may be made zero or small by proper orientation of the crystal plate about the X crystallographic axis. The mathematics of this operation is simplified by the use of matrix algebra . Upon performing this ojieration a new set of elastic constants are obtained and are plotted graphically together with the piezoelectric constants on Fig. 6.7. From this figure we may see that the coupling resulting from the S2i constant will be zero if the crystal plate is orientated by — 18.5° about X with respect to the crystallographic axis. This constant determines the coupling between the extensional mode along the length (I^' dimension) and the face shear mode (F'Y' dimensions). This analysis resulted in the use of the —18.5° cut in the channel filters of the coaxial system. Two other crystal cuts resulting in low coupling between different modes of motion are the AC and 5 "The Mathematics of the Physical Properties of Crystals," W. L. Bond, B.S.T.J., Jan. 1943. MODES OF MOTION IN QUARTZ CRYSTALS 67 / ! j± r — ' N (^ o II \ i 10 — -o 1 \ \ ^ N 2 \ o -A. II o II - ■a 1 — J ^C\J / "O ^ \ m ^i"/ O O II - 73 / - -TV / L V (- o (M II s (M O II - N 73 - : vX — j X. 73 — J o II O II 73 - ^ -^ 68 BELL SYSTEM TECHNICAL JOURNAL BC cuts. The 556 constant determines the coupHng between the face and thickness shear modes. As shown in Fig. 6.7 this constant passes through zero at two values, namely +31° and —59° and the resulting angles have been termed the AC and BC cuts. These angles are very close to the AT and BT cuts and hence they also possess the benefits of low coupling between modes. In addition to making the cross coupling constants zero, a rotation of the crystal plate with respect to the crystallographic axes also results in a change in the e.xtensional ?.nd shear elastic constants. Notice that these pass through maxima and minima at the zero values for the cross coupling con- stants. This of course affects the resonant frequencies of isolated modes. Changes as great as 50% increase in frequency constants may be obtained by choosing the proper rotations. The equations relating the elastic con- stants as functions of orientation are given in appendix B for more com- plete use. 6.4 Coupling between Modes of Motion As pointed out in the previous section, the frequency equation of a given mode of motion will give accurate results only in the case where the mode of motion is isolated. This is very rarely the case since most quartz crystals in common use are in the form of plates where the frequency determining di- mension is not large in comparison with all other dimensions. Only in the case of a long thin rod vibrating in length-thickness flexure of the first order would this be true. It was also shown that the coupling between different modes of motion could be related to the mutual clastic constants (.9,-; and c,/) and that some of these could be made zero by the proper choice of orientation of the finished crystal plate. The elastic constants s^ and dj only relate to the coupling between the extensionals, the shears and the extensional to the shear. For example 523 relates to the coupling between the extensional modes along the Y and Z axes, 556 relates to the coupling between the low and high frequency shear modes of a F cut plate and .^24 relates to the cou- pling between an extensional mode along the Y axis and a shear mode in the YZ plane. One other important coupling condition occurs and that is between the flexure and the shear modes. There is at present no mathe- matical theory relating this form of coupling except from simple assumptions that may be drawn from the fact that the shear modulus enters as a control- ling factor in determining the frequency of a bar vibrating in flexure and from the similarity of the two types of motion near the boundaries. Since it is possible to have a definite coupling between extensional and shear modes there must be coupling between the extensional and flexure modes. It would be expected that it would be proportional to the coupling between the extensional and shear "modes. MODES OF M 01 ION IX QUARTZ CRYSTALS 69 6.41 Extensional to Shear and Extensional to Flexure Coupling The coupling between the extensional and shear motion can best be illus- trated by taking the case of an X cut plate the length of which hes along the Y axis and the width along the Z axis. This is shown in Fig. 6.8 together with two other cases, one in which the plate is rotated about the X axis by — 18° and the other a similar rotation but +18°. Also in Fig. 6.8 is shown an enlarged view of the change in the elastic constants and frequency con- stants as a function of the rotation of the plate about the electric or X axis. For the case of an X cut plate the strains resulting from an applied exten- "irir::^^ FREQUENC 1 ; ' P^OF^'Z y6 - — - y'z — , — ■ ^r ^ ^^^ \ ^ ^ ^'2?^- -^ j^^^^ +18 0 -18 ROTATION ABOUT X-AXIS IN DEGREES Fig. 6.8 — ]\Iotion in an X cut plate for different orientation about the X crystallographic axis, sional stress along the length according to equation A.8 would be Xi = Sl2 Yy Vy = 522 Y„ Zz. = 523 Yy yz = Suly where .v^ is an extensional strain along the thickness y/' " " " " " length 2, " " " " " " width yz " a shear strain in the length-width plane 6.11 70 BELL SYSTEM TECHNICAL JOURNAL If the plate is thin we may neglect the .Tx strain as far as its efifect on the resonant frequencies associated with the length and width are concerned. From the plot of the elastic constants on Fig. 6.8 we may determine the strains resulting from a stress along the length of an X cut plate for various orientations about the A'' axis. In addition to the expected extension along the length we have for a +18° cut, a large amount of length-width or y^ shear strain due to 524 and very little width or s^ strain. For the 0° cut there is also large length-width or y^ shear strain and a width or z^ strain. In the case of the — 18° cut the shear strain vanishes due to 524 being zero, leaving in addition to the expected length or yy strain a width or z^ strain. These relationships are more clearly shown if we plot the resonant frequencies resulting from the three modes of motion namely, the extensional modes along the length and width and the shear mode in the length-width plane \ u 3S0 _) o >- o o =) 300 A / A V A \ / f N <^ / \ 0 2 0.4 0.6 0.8 I.O c vK Tx / . T "^ 0.2 0.4 0.6 O.f w (i,= Y= lomm.' 0.2 0.4 0.6 0.8 :.o Fig. 6.9 — Effect of rotation about the .Y axis on the resonant frequencies of an X cut plate. A plot of measured resonances is shown in Fig. 6.9 for the above described three cases as a function of the change in width. The resonant frequencies for these three types of motion are given in section 6.3 as — , extensional along I PS22 fz' = — A/ —r > extensional along w 2w y pssz J_ /ill ^su y /2 + ^2 ' shear in tw plane 6.12 6.13 6.14 These equations specify only the uncoupled modes and do not take into consideration the effect of coupling to other modes of motion. In the case of Fig. 6.9 it is shown that when only the width is changed the extensional MODES OF MOTION L\ QUARTZ CRYSTALS 71 mode along the length (the Vy mode) is unafTected only in the case of the — 18° cut. The effect of coupling between the extensional and shear is clearly shown in the case of the 0° cut by the change in the length-extensional frequency. This is more pronounced in the -f-18° case not because of more coupling but because the frequency constants of the two modes are more nearly alike as indicated in Fig. 6.8. The mode of motion associated with the line intersecting the e.xtensional Xy mode is that due to the second length-width flexure mode. As, mentioned before it is strongly coupled to the shear mode in the same plane. The coupling between this flexure and the extensional mode is directly related to the coupling between the shear and the extensional mode. This is borne out by Fig. 6.9, for in the case of the — 18° cut, 504 is zero and as can be seen the change in frequency of the extensional mode is very slight even when the flexure mode is nearly identical in frequency. We may state generally that the change in frequency of a particular mode of motion from that of its uncoupled state is dependant on two factors; the coupling to and the proximity to other forms of motion. This follows well established mathematical procedures but to solve the case just discussed would require the solution of a four mesh network with mutual impedances the values of some of which are at best only approximate. This will serve to illustrate that the use of formulae such as given in section 6.3 may be used more as a guide in establishing certain modes of motion rather than for accu- rate determinations of resonant frequencies. 6.42 Flexure to Shear Coupling 1. Lon' Frequency Shear As previously indicated there is no simple means of mathematically determining the coupling between flexure and shear types of motion as there is between the extensional and extensional to shear modes. Here we must base our assumptions upon observed experimental evidence and simple rea- soning. The relation between flexure motion and shear motion can be illus- trated by the figures associated with Fig. 6.10. The forces that are necessary to produce flexure and shear motion are shown by arrows in Fig. 6.10. When the two arrows point toward each other, it indicates a compression and when the arrows point away from each other, it indicates tension. The diagrams on the left of Fig. 6.10 illustrate the conditions for flexure motion and the diagrams on the right indicate the conditions for shear motion. Notice that in the case of the first flexure and the second shear that the forces applied to the top and bottom of the plate are similar. Also in the case of the second flexure and third shear, they are similar. Here again we have certain similarities which in this case are important to remember. 72 BELL SYSTEM TECHNICAL JOURNAL The motion of the ends of the plate in the case of the first flexure are similar to those of the second shear. In the case of the second flexure the similarity is observed in the case of the third shear. The end motion in the case of the third shear is also the same in the case of the first or any odd shear. Like- wise, the end motion of the first flexure is similar to the second shear or any even shear. We may then generalize and say that it is very likely that an odd order flexure would be coupled to an even shear; and also an even flexure would be coupled to an odd shear. T w Jl \ ST FLEXURE 2 ND SHEAR 2 ND FLEXURE 3 RD SHEAR ODD FLEXURE EVEN SHEAR EVEN FLtXURE ODD SHEAR Fig. 6.10 — Similarities in shear and flexure motions in a bar. To illustrate the coupling between flexure and shear type motions, the frequencies of flexure and shear modes in a Z-cut quartz plate as shown in Fig. 6.11 have been measured. These measured frequencies are shown by the solid lines for various widths of the plate. It will be seen that there are no observed resonances following an unbroken continuous line to represent the shear frequency, but they are interrupted by several other frequencies which we must interpret as being various even modes of the flexure in the plane of the plate. It is clearly shown here that only even order flexures are MODES OF MOT 10 X IX QUARTZ CRYSTALS 73 strongly coupled to the fundamental or odd shear. The strong coupling shown between the Xy shear and the second Xy flexure explains why the frequency equations given in section 6.3 for the frequency of flexure and 240 200 8 32 16 24 Y IN MILLIMETERS Fig. 6.11 — Shear and flexure resonances in a Z-cut quartz plate. shear modes will not give even approximate results if applied to this case for a square crystal. It will be shown later that if account is taken of coupling, the shear mode for a square crystal of this type may be more accu- 74 BELL SYSTEM TECHNICAL JOURNAL rately determined. Fig. 6.12 is a more detailed representation of the conditions shown broadly in Fig. 6.11 except in this case an .4C-cut quartz plate was used and most of the observable resonant frequencies are shown Fig. 6.12 — Shear and flexure resonances in an ,4C-cut quartz plate. W I for various values of — . The plate shear is labeled Z^ shear and occurs at the frequencies predicted by equation 6.8 except in the regions where a flexure in the same plane exists. This is the type of motion shown in Fig. 6.4 for the case of m = I, n = 1. It can be seen that as the difference in order of modes becomes greater the effect on the shear frequency is less MODES OF MOTION IN QUARTZ CRYSTALS 75 except where they are coexistant. We can then state generally that ev^en though there is coupling between particular modes of motion, if the difference in order is great, the approximate frequencies may be computed as though they were isolated. This is more clearly shown in the case of thickness shear modes. The modes that are shown coupled to the face shear mode are Zx flexures propagated in the direction of the length or X axis. The lower orders can be shown to follow the general frequency equation discussed w . . in section 6.3 but the higher orders for a given — , it will be noticed, are regu- larly spaced in frequenc}' and show the eflfect of shear. The Xy flexure modes determined by the length and thickness are shown as nearly horizontal lines since only the width was changed. Since these two groups of flexure modes are propagated in the same direction, it would be expected that the (-■f=0 difference in frequency for the same ratio of dimension ( i.e., -7=7) would be due to the differences of the shear coefficients in the two planes of motion. The vertical dotted line indicates the ratio of thickness to length. When the ratio of width to length is equal to this value it can be seen that the flexure modes in the width-length plane are in all cases higher than the same order flexures in the thickness-length plane. An examination of Fig. 6.7 shows that for an .4C-cut crystal the shear modulus in the width-length plane ( /-j- \ is greater than that in the thickness-length plane ( /-;- J . This is in agreement with the observation made above. One other generality may be drawn from the experimental data shown in Fig. 6.12. The coupling between flexure modes and shear modes in planes at right angles to each other is very small in comparison with that between modes in the same plane. As mentioned before the eflfect of coupling between modes of motion is greatest when the orders are more nearly similar. In this particular crystal this effect can be shown between the fundamental width-length Zx shear and the second order width-length Z^ flexure. This is shown in Fig. 6.13 which is an extension of the data shown in Fig. 6.12 for a crystal nearly square and shows the frequency range covered only by the second flexure and the fundamental plate shear. A computation of the uncoupled second flex- ure mode propagated along the length and the first plate shear mode are shown by the solid lines // and /« respectively. Inserting the appropriate constants the formulae of section 6.3 become . 1 ^ /7.85 X 10" 2Z' ... •^^ = 2^ y 12 X 2.65 ^^ ^''' , 1 /71.8 X 101° /-I— f ^' = 2V 2.65 VX^ + Z^ '-'' 76 BELL SYSTEM TECHNICAL JOURNAL In evaluating 7n, account was taken only of the rotary and lateral inertia so that some error is expected at the larger ratio of axes. The curve of flexure crosses the shear curve at - = .76, a condition which we know to be non- l compatible since these two motions are coupled. From the theory of coupled circuits we can determine the displacement of two uncoupled frequencies as a result of the coupling, through the relation fU = Wl +/; ± V(/i - fff + ^kYj}] 6.17 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0 01 02 0 3 04 05 06 0 7 08 09 10 Fig. 6.13 — Effect of coupling on the plate shear and the second flexure mode in an ylC-cut quartz plate. where f, = uncoupled shear frequency, // = " flexure " k = coefficient of coupling. The coefi&cient of coupling in this case may be defined as the ratio of the mutual to the square root of the self compliances of the two vibrating sys- tems. As mentioned before no derivation has yet been made to indicate the relation between the coupling between these two forms of motion and the physical constants of the medium in which the vibration occurs. It is necessary to assume some coupling factor which will produce that observed \ I \ e = -31° t=Y'=300MM 1=X= 38 MM W-Z' fs \ • , \ ^ 1 •^ V i 1 1 1 ! N So 1 ! 1 ^ ^ !,"•-( fl i 1 I ■^ ' " -1 — f-i^4-. — ■ <1 r^ f— ^ r^ 1 > • L_« ^2 1 — 1 — -i- •• / ^ /] "f. / / , MODES OF MOTION IN QUARTZ CRYSTALS 77 by experiment. Applying a coupling coefficient of 35% and computing the values of fi and /j from equation 6.17 the results are the dotted curves shown in Fig. 6.13. The observed points follow the computed values to a fair degree of accuracy for all frequencies below 180 kilocycles. Above this range there is a strong coupling to the fourth flexure and this would require separate consideration. Based upon these results the equation for the low frequency or face shear given in section 6.3 would not give the observed results for a nearly square plate because of the high coupling to the second flexure mode. For an approximately square plate, cut near the ylC-cut the plate shear frequency including the effect of coupling would be given by f=-?^./X, 6.18 2(/ Y pS56 where d = i(A^ + ^0 and .849 is the factor resulting from the use of equation 6.17. For crystal cuts far different from the above it would be necessary to consider the flexure and shear as uncoupled and then apply equation 6.17 to determine the appropriate factor for square plates. 2. High Frequency Shear The motion associated with flexure has been shown in Fig. 6.1 and in order to determine the frequency of higher order flexures, measurements were made on an ylC-cut crystal. The results of these measurements are shown in Fig. 6.12. The first flexure motion to be expected with this crystal would be a flexure in the plane of the length and width. The various orders of these flexures are shown by the curved lines labeled second z'x fourth, sixth, etc., all radiating from zero frequency (Primed values of z and y indicate that these are not crystallographic axes). The equation commonly determining the frequency of flexure states that the frequency should be proportional to the width and inversely proportional to the square of the length. If this were true, these curved lines representing the resonances of this type flexure shown on Fig. 6.12 would then be straight lines. Since the actual conditions show a wide departure from this, we must assume that this departure is due to rotary and lateral inertia and the effects of shear. It will be noticed that as we progressively increase the order of the harmonic, that the actual frequency spacing for a given value of — is very nearly linear instead of a square law. This point is more clearly seen when we examine the frequency of higher orders of the flexures in the length thickness or xy' plane. As shown on Fig. 6.12 these frequencies 78 BELL SYSTEM TECHNICAL JOURNAL labeled 6th Xy, etc., change very little and are nearly horizontal straight lines. Here again they appear to be simple harmonics of some common low fre- quency. Also it will be noted that the coupling between the Zx flexures and the Zx shear is quite appreciable and in general decreases as the difference in order of the two modes becomes greater. This plot of the various flexure frequencies tells us a great deal about the behavior of progressively higher order of flexure type motion. The important effect to be noticed is that for w high orders, and a fixed ratio of — , the flexure may be treated as though it were harmonic so far as frequency is concerned. Some variations to this rule will be observed and special cases will be discussed. So far we have discussed the case of flexure modes of relatively low order. In the case of high frequency shear modes of motion, we would expect that the order of flexure which would interfere with this type of motion would be rather high. Figure 6.14 shows a plot of these flexure modes as observed in an .4r-cut plate. These are shown by dashed lines. The dots indicate actual meas- ured resonances. This figure also shows the various other resonant fre- quencies observed in this type of plate as discussed in section 6.2. The solid lines labeled mnp represent the type of shear motion shown in Fig. 6.5. Here again we may observe certain statements made before with respect to the coupling between shear and flexure type motions. Notice in this case that the coupling between an even order flexure and an odd order shear is high and increases as the orders more nearly approach each other. For example, the 38th flexure mode is coupled to the fundamental shear labeled niiHipi has very little coupling to the second order shear mifiipi, and again is strongly coupled to the third shear niinzpi and correspondingly higher coupling to the fifth shear. When we speak of higher order shears, such as W2W3«6, they are not higher order in the sense of harmonics, but do differ by a small amount in frequency. In the case of a plate where I is not great compared to t, these differences will be greater. In actual practice in the case of AT plates, we are usually concerned mainly with the fundamental high frequency shear and high even order flex- ures along the length. This case is shown in Fig. 6.15 which gives experi- mental results of measurements on actual AT plates. It will be noticed that the flexure frequencies show a rather regular displacement as the ratio of the length of the plate to its thickness is changed. In this case only the odd order modes of shear and the even modes of flexure are shown. It will be observed that as the ratio of the length to thickness decreases, the cou- pling between these modes is quite high. This some state of affairs is illus- trated again in the case of the third harmonic of high frequency shear and is shown in Fig. 6.16. The near vertical dashed lines represent even order MODES OF MOTION IN QUARTZ CRYSTALS 79 flexure frequencies and the curve labeled wz3«i and the curve labeled W3W3 correspond to two different values of the high frequency shear near its com- monly called third harmonic. An examination of Figs. 6.14 and 6.15 indicates that a regular pattern is formed of the ratios of axes at which the high frequency shear and succes- 5 I 660 5 9 1 620 1 520 1 500 t 46 0 I 460 - >.. t \ \ S • Lv ___»^ ^ \ \ \ \ \ > •••. • ' -M^N^P, \ • \ \ '•.\ \ • \ • t • \ ^ 1 ^ p77 • c '••-r 5 — ' - M,N 3P1 > FUND. • \ 'z^^-^^ ^ ^**^ 1 — _ M|N 2^ xy's HEAR \— Vi 1 — - M,N 1^1 \ t [ 1 N 38 \ \ \ « \ \ V \ \ \ ' '^37 \ \ \ \ \ \ « \ \ \ \ \ \ i 1 \ •v > 36 t / /I \ i *. \ \ \ / /^ L — I/w ^ > ^ V 1 ^ k— I ^ \ ^ V35 W= Z'= 32 00 MM ' • s ^ • • 1 = X 30 31 X Y' Fig. 6.14 — High frequency flexure and shear resonances in an ^T-cut quartz plate. sive even orders of the length-thickness flexure coincide. Rather than define these points on the basis of specific ratios of axes it is more convenient to place them on a frequency basis. Therefore we may say that for a given size plate there will be specific frequencies at which some mode of the fle.xure motion along the length will be the same as the high frequency thickness 80 BELL SYSTEM TECHNICAL JOURNAL shear. For the case oiAT plates experiment has shown these to be given by kilocycles 6.19 1338.4 fxf = -^i7- ^'f X where X = length of X axis in millimeters, Uxf = order of flexure along X axis = 1, 2, 3, 4, etc. In this equation as well as those of a similar nature to follow it is assumed that the thickness is such as to result in the same frequency for the high Fig. 6.15 — High frequency flexure and shear resonances in an ^T-cut quartz plate. frequency Xy shear mode. As shown in Fig. 6.14 only the even orders are strongly coupled to the fundamental thickness shear. The coupling between high even orders of the flexure along the X axis and the high frequency shear in the case of BT-c\xi plates is similar to that for .4r-cut plates. Fig. 6.17 shows the various resonant frequencies observed in a BT-axt crystal as a result of changing the ratio of the length or X axis to the thickness or Y' axis. The curve niifii represents the high fre- quency Xy, shear. Curves m\nz, min^, Wi«7 and min^ represent other Xy, shear modes as discussed in section 6.23 resulting from higher orders along the length or X axis. The dashed hues represent even order flexure modes along the X axis. The same regularity is observed here as in the case of the MODES OF MOTION IN QUARTZ CRYSTALS 81 .4r-cut. When placed on a frequency rather than a ratio of axis basis the frequencies at which flexure modes along the X axis would coincide with the 5.00 ,1 4.95 4.90 t = Y'= 1 .00 MM W= Z'= 15.90 MM X= X M = 3 N=3 M"3 N-1 N. i ,<^ ^ ^) t ZIl^) t i — \ — -_1^^ *«v«>->.^^^ ^^ 3RD Xv^' SHEAR Fig. 6.16 — High frequency flexure and shear resonances in an .4r-cut quartz plate near the third harmonic shear mode. fundamental A^^- shear mode are experimentally given by . 1818 ,., , /j/ = — — - Wi/ kilocycles 6.20 82 BELL SYSTEM TECHNICAL JOURNAL where X is given in millimeters. In this case it will be noticed also that only even order flexures are strongly coupled to the fundamental Xy' shear. The dependence of the flexure frequency on the shear coefficient can be seen from these two cases. The direction of propagation is the same in both cases (along the X axis) but the direction of particle motion is nearly at right angles. It would be expected then that the frequency constant would be highest for the case of the highest shear coefficient. Examination of equa- Fig. 6.17 — High frequenc\' flexure and shear resonances in a BT -c\xt quartz plate. tions 6.19 and 6.20 shows this to be true. In addition, the change in the frequency constant is about the order of magnitude of the change in the shear modulus in the respective planes of motion. 6.43 Coupling between Low Frequency Shear and High Frequency Shear From an examination of Fig. 6.7 it can be seen that the coupling between the low frequency shear (Zi) and the high frequency shear Xy' is related by the s^f, constant. In the AC and BC-c\xts this reduces to zero but for the AT and ^T-cuts it has a finite small value. According to section 6.3 the frequencies of the plate shear modes are given by equation 6.8 but this holds only for the case where ;;/ and n are small. When the third dimension MODES OF MOTION IX QUARTZ CRYSTALS 83 becomes appreciable in comparison with a half wave length along w or / it becomes necessary to use the c constants. When considering high orders of the low frequency shear equation 6.8 is modified to / = 6.21 Equation 6.21 shows that high orders of the low frequency or plate shear are dependent upon both the length and width dimensions and it might be as- sumed that this would lead to very complicated results in so far as analysis of experimental data is concerned. The coupling between these modes and the high frequency shear is a result of coupling in the mechanical as well as 5 1 640 > 1560 FUND. XljSHEAR \ I I \ i-J, \ \ l\ \ \l \ i.l t=Y = .293 MM. W= Z' \- X- 11.16 MM \ i. ^ \ \ \ \ \ '\^ \ \ \ \ \ \ \ \ I \ \ * V \ \ \ \ \ \ \ \ \ \ \ -\- \ \ \ FT \ \ \ \ \ \ \ \ \ \ \ \ \ \i \ \ \ to 14 18 22 26 30 34 38 42 46 J/ y Fig. 6.18 — High frequency shear resonances in an .4r-cut plate. the electrical systems. The strongest coupling with reference to the length axis would then be for high odd orders of w and unity for n with successively smaller coupling for higher orders for n if the driving potential extends over the complete surface of the crystal. In a similar manner when considering high orders of plate shear along the width axis the highest coupling will result from unit order for m. Based on these assumptions then to a first approximation we can assume these modes to be functions of length and width alone. Equation 6.21 then reduces to ffs - 1 f =^ P 1 Cjj tlwa p W 6.22 6.23 where Ugf = order of shear mode along ^ axis, nsw = order of shear mode along w axis. 84 BELL SYSTEM TECHNICAL JOURNAL These modes have been measured in AT and ^T-cut crystals. Fig. 6.18 shows the points at which these modes intersect the fundamental high fre- quency shear mode in A T-cut plates. This is the case for high orders along the Z' or width axis. A similar set of resonances can be shown to exist when the X or length axis is varied. Experiment has shown that these frequencies of coincidence between high order plate shear modes and the fundamental high frequency Xy' shear mode for the case of .4r-cut plates is given by 254 2 fxs = -^TT- ftxa kilocycles 6.24 fz'a = —^ «z'« kilocycles 6.25 where X and Z' are given in centimeters. Only odd orders are strongly coupled if the crystal plate has a symmetrical contour with respect to an applied equipotential electrode. Upon substitution of the value of C55 for an .4 T-cut crystal in equation 6.22 there results /. X / = ^ /^-^ = \ j/^^^l^ = 251.0 kilocycle - cm. 6.26 which is within 1 per cent of that found experimentally. Since Young's modulus is nearly the same along the A" and Z' axis the value of k in equation 6.23 is essentially unity. Fig. 6.19 shows measured values of high order Z^ shear modes near the high frequency A^- shear mode in a i^T-cut crystal for various values of the width or Z' axis. More detailed measurements have been made of the high order Z^ plate shear modes in ^T-cut plates along the X axis. Fig. 6.20 shows both the shear and flexure modes along the X axis near the vicinity of the high frequency Xy' shear mode. Since the frequency constant for the Z^ shear modes is different from that for the Xy' flexures there are regions where, if no coupling existed, all three modes would be at the same frequency. It is obvious from Fig. 6.20 that this is not the case. Therefore, we must assume that not only are the high order Zx shears and Xy> flexures coupled to the high frequency Xy- shear but that they are coupled to each other. While it is difficult to see from Fig. 6.20 the relative coupling of flexures to the Xj^' shear, experiment has shown the flexure modes along A' to have X . the greater coupling to the A'^' shear. This is true when the ratio — , is such that the flexure modes along A' and high order Zx shear modes along A" have their maximum separation. When these modes approach each other MODES OF MOTION IN QUARTZ CRYSTALS 85 and the X„' shear such as is shown in Fig. 6.21 at ^ = 31.35 the relative coupling of each to the X„' shear is about equal. This arises from the fact that the mutual coupling between them increases the apparent coupling 5 ? 2540 >. 24 60 I lit FUND. Xq'SHEAR \ \ t- Y'=.450MM. l = X= 17.50MM. V^^ \ \ » ^ -'V^n.l 1_L_A_1 \ \ \ \ \ -4-+ \ \ \ 12 16 20 24 28 32 36 40 44 48 y Fig. 6.19 — High frequency shear resonances in a BT-cut plate. Xl| shear mode Zx SHEAR MODE ALONG X (UNCOUPLED) Xq' FLEXURE MODES ALONG X(UNCOUPLED) • MEASURED RESONANCES t r Y'= .450 MM W= Z'= 18 00 MM l^X ^ Fig. 6.20 — High frequency thickness shear and flexure and shear resonances along the A" axis in a BT-cut quartz plate. between the Xy' shear and high orders of Zx shear along X. From this it would appear advisable to avoid such regions in the dimensioning of crystals for oscillator use over wide temperature ranges. Determination of the flex- ure as well as high order Zx shears then must be made in regions where 86 BELL SYSTEM TECHNICAL JOURNAL they are spaced so that the effect of coupHng between them will not influence the frequency constant that is determined experimentally. These regions have been investigated and the result for the flexure modes is that shown Fig. 6.21 — Flexure resonances in a GT-cut quartz plate. in equation 6.20. From Fig. 6.19 the high order Zx shears along Z' will be coincident with the high frequency Xy' shear at frequencies given by fz's = ', fig't kilocycles 6.27 From Fig. 6.20 high orders of the same Zj shear along X will be coincident with the high frequency Xy shear at frequencies given by MODES OF MOTION IN QUARTZ CRYSTALS 87 - 163.514 , •, , ^ oo fx» = — ^ — Wx, kilocycles 6.28 Upon substitution of the value of Css for a ^T-cut in equation 6.22 there results u ^p 1 /^ 1 , /30.3 X 1010 /f*X-C=-i/— = -/*/ ^-55 = 169.0 kilocycles - cm. which is 3.3% greater than that observed in equation 6.28 and 1.6% greater than that shown in equation 6.27. The apparent difference in the observed shear modulus in the X and Z' directions for the ^T-cut can be explained from the fact that Young's modulus is quite different in the two directions for the BT-cut while it is nearly the same for the AT-cnX. as verified by equation 6.24 and 6.25. From the discussion in this section it can be seen that a single theory that would relate all the now known resonances in quartz plates together with the effects of coupling would be prodigous indeed. In order to reduce the design of quartz plates to a simple engineering basis it is necessary to take specific examples and investigate the region in the vicinity of the frequency to be used based on general theory and then apply approximations that fit the specific cases. 6.5 Methods for Obtaining Isolated Modes of Motion 6.51 GT Type Crystals In the case oi GT type crystals the modes that cause the greatest concern are flexure modes in the two planes of the length and thickness and the width and thickness. The desired mode is that of an extensional mode along the width. To produce a low temperature coefl&cient it is also necessary that this mode be coupled to an extensional mode along the length, a fixed fre- quency difference from it. Therefore it will be necessary to prevent flexure modes from occurring at either of these two frequencies. Fig. 6.21 shows the frequency of various flexure modes that would be observed in Gr-cut plates for different ratios of thickness to length. In the case of the Gr-cut the elastic constants in the length and width directions are the same and therefore it is only necessary to determine the flexures in one plane to get a determination in both. From the plot of frequencies shown in Fig. 6.21, it would be very easy to determine the proper thickness for any given GT plate. Since in all practical cases there is a definite relation between the length and width of this type of plate, it would be necessary to examine the flexures in these two directions as a function of the change in thickness. 88 BELL SYSTEM TECHNICAL JOURNAL Fig. 6.22 shows a plot of this for the case of a GT crystal designed to operate at 164 kilocycles. All the information shown in this figure is obtained directly from Fig. 6.21. Since a change in thickness will not have any effect upon the length and width extensional modes of vibration and only 120 t ^=^ ^ / f^ J ' '^ I =20.05 MM W= 23.20 MM ^^ ^ ^ -"^H L-t ^7THW-t "^ L MODE Y"^ ^^ 6THW-t,,^^ -"^H L-t ^^ W MODE ^^ ^^ ^ ^^^ ^ 5TH W-t '-"^THL-t --^ --^ ""^ ___ "-^ ,-^ ^ 4TH W-t FLEXURE 0.075 0.08 t IN CENTIMETERS Fig. 6.22 — Flexure and extensional resonances in a 164 kc Gr-cut quartz plate. changes the flexure frequencies, it would be reasonable to suppose that some thickness could be obtained where no flexure along the length or width would be of the same frequency as the length or width extensional mode. Examining the curves of Fig. 6.22, we find that a thickness of .06 cm., .075 cm. or .085 cm. would meet these conditions. MODES OF MOTION IN QUARTZ CRYSTALS 89 6.52 BT Type Crystals As discussed in Section 6.4 the modes showing the greatest coupling to the high frequency thickness shear are of two types: high orders of Xy> flexure propagated along the X axis and high order Z, shears along the X and Z' axes independently. Complex orders of the flexure and plate shear as illus- trated in Fig. 6.2 and Fig. 6.4 do cause considerable difliculty and their analysis calls for special treatment and is not within the scope of this text. For the case of the ^T-cut the three primary interfering series of modes are given by /*/ = —3^ tixf kilocycles fx» = — ^ — «z« kilocycles 6.30 fz; = ' w«', kilocycles where X and Z' are given in centimeters and Jxj is the frequency at which integral orders of flexure modes along the X axis would coincide with the high frequency thickness shear mode. In a similar manner fxs and fz's relate the same conditions for integral orders of the plate shear modes. These equations are true only in the case where the thickness is of such a value as to place the high frequency thickness shear mode at the same fre- quency as the computed interfering mode. In most practical cases for oscil- lator use the electric field is applied to the crystal by means of a flat electrode on each side of the crystal plate. Under this condition only odd order Xy- shear modes along the X axis are excited and hence the strongest couplings to the Xy> flexure modes will be only for even order values of nxf in equation 6.30. In a similar manner the greatest interference between the Xy' shear mode and high orders of the Z, shear modes along both X and Z' will occur for odd orders. Therefore the strongest interference from these modes will occur only for odd integers of fixs and w^'j in equation 6.30. These assump- tions of only even flexures and odd shears showing appreciable coupling are based upon a crystal plate cut precisely along its proper axis and of uniform contour assembled in a holder using electrodes of uniform air gap. Deviations from these conditions will of course alter the ideal results de- pendent upon the amount and type of deviation. The relationships shown in equation 6.30 may be more clearly seen when plotted graphically. Assuming a BT-oxi crystal plate 1 centimeter square we may determine the frequencies at which an interfering mode will coincide with the high frequency Xy> shear by assigning even integers to «,/ and odd 90 BELL SYSTEM TECHNICAL JOURNAL integers to Hxs and fiz's. Fig. 6.23 shows a plot of these three types of inter- fering modes on a folded frequency scale covering the range from 5 to 15 megacycles for a plate 1 centimeter square. Each abscissae covers a range of one megacycle with dots at three levels. The first level shows the fre- quencies at which successive even orders of flexure along the X axis occurs. The second level shows successive odd Zj shear modes along X and the third level successive odd Z^ shear modes along Z' . The circles shown on the three levels indicate the results of actual measurements on i^T-cut crystals as resonating elements. It will be noticed that the circles and dots coincide for most frequencies, the regions of departure occur only when a high order shear mode and a high order flexure mode along the -Y axis approach each 13 :--J« 'e-» ~:*J tiifl -_-_- =.-: z:^- _-g)t o-» L=& zrzr. r-- _^- z- ^Q z--- i^<> ^J 14 13 - — --- Z~^L -"« -dl z=- -zEi *Lrl "£f ~-^ -«: L— r ^ zr:^; IZ.1 -31- '.nr^ >•- i aEr -Z-z €F- :-- r^J L"l_~ L~~ *F-' ITZ r:*: ■~ZJ r£¥ 'SZSZ =Ji ~^ 1^- :®r-- !1.~! ^^ inz 12 '^y r'lL- ■--1 y^- '-e ~-^— ^: "="f. irr =_-• :~z. -^ ---i o-^ '.~Z' TJrs Jt: :3£ u „ u z©?: ----^ Z 1 ■-z\ ;--J :—- Erri Fii: 'rl-E :f£z -_-f; ■-9 r=- r=^ L££ Vsl 'z~z -«=; ■ -€ • — •Oz J_ < O 9 ^ ^^ »r_-: ■^£: L-~J _^' L~Z z.-^. IZ.- :££ £-J i£l£ -z. 'K :£!- J :r_-_- --» -_-i .:«z ;V3 »— , ^&* ^^ ::f£ rJt -1-J — I:^< **rij: "-<> :££■ i®-_ £^ 2 a 1 1 — '-<>■ JL~ LITJ fzr. -_:r ^"■ :~~ ?isE ~-rs: r^J> "^^ zzri OA' ~?^ £.zr zEz ■j#r ZZZ- --e^ 7 6 z.~~: tr rl-C- »•-- ""*1 r— z z-^- .-_-^. 7-^ '—-. --» '—9 :~- r^"- LTL.- Qz?. :-- r*-_ :-E£ ^ "£z"; » — zO» ■JVl 9 r_~: :~z; .i-i: ~ z; /¥£ 13 J ~z: rzi~ ;~~ S --»■ Fig. 6.23 — Frequencies at which the Z',. shear along A', the Z'x shear along Z' and the Xy flexure along A' coincide with the high frequency A'j, shear in ST-cut crystals. other in frequency. The reason for this is obvious from the previous dis- cussion on the coupling between flexure and shear modes of motion. The chart of Fig. 6.23 is of course not limited to a crystal 1 centimeter square or for that matter even a square crystal. In reality it relates the product of the frequency and X and Z' dimensions. For example a flexure mode interferes with the high frequency shear mode at a frequency of 9.45 megacycles for a plate with X dimension equal to 1 centimeter. If the X and Y' dimensions were doubled the same situation would exist at one half the frequency. In determining the dimensions for a crystal at a given fre- quency we know that the product of the frequency and X dimensions as well as Z' dimension must not result in a frequency close to those given by the circles of Fig. 6.23. In addition other interfering modes as previously mentioned must be avoided. These at present may be determined experi- mentally by choosing regions on the chart clear of the known flexure and shear modes. MODES OF MOTION IN QUARTZ CRYSTALS 91 On the abscissae are shown certain discreet frequencies as well as frequency ranges which have been found to result in crystal units having no serious dips in activity over a wide range in temperature. These are for square crystals in the 18 millimeter size range and have been obtained by Mr. G. M. Thurston of the Bell Laboratories and Mr. F, W. Schramm of the Western Electric Company. It will be noted that no so-called ok regions have been found at the frequencies of the three principal coupled modes. While the use of the chart shown in Fig. 6.23 will often lead directly to the proper .Y and Z' dimensions for a given oscillator it cannot be overem- phasized that only the three principal interfering modes are shown and only the odd orders for the shears and only the even orders for the flexure modes. Since the even order shear modes are excited due to slight variations which would produce wedge shaped air gaps or quartz blanks, it is advisable to avoid these regions also. Complex combinations of the three principal modes as shown in Figs. 6.2 and 6.4 are also driven. Therefore when it is necessary to produce a crystal unit possessing the highest activity for a given area of quartz plate over an extended temperature range it is necessary to scan the supposed desirable regions shown in Fig. 6.23 by complete meas- urements on finished units of a given size and varying frequency or of con- stant frequency and varying size. As an illustration the region shown in Fig. 6.23 between 10.025 and 10.080 megacycles was determined in this manner with the use of crystal plates approximately 18 millimeters square. The use of crystals with other than square dimensions could undoubtedly have increased the range of this region but their use is undesirable from a manufacturing standpoint. Assuming that the electrodes and crystal holder permit a variation in size of the quartz plate from 17.20 millimeters to 18.20 millimeters this approved region will immediately specify the dimensions of crystals to cover the frequency range from 5508 to 5727 kilo- cycles. This also assumes crystal blanks cut to precise orientations with controlled contours and electrodes of uniform flatness and constant airgap. WTiile the theory would indicate that the frequency range given above could be expanded to considerably higher values by utilizing a smaller crystal blank this has not been proven so far since most crystals produced by the Western Electric Company require large area plates to meet high activity requirements. As an illustration of the effect on the behavior of oscillators of changing the X and Z' dimensions of ^T-cut quartz plates measurements have been made of the activity, in a conventional tuned plate circuit with the crystal connected between grid and cathode of quartz plates of constant thickness and varying X and Z' dimensions. Fig. 6.24 shows the effect of changing the X dimension of a quartz plate on its activity as an oscillator. By taking the product of the frequency and dimension we can determine the dimen- 92 BELL SYSTEM TECHNICAL JOURNAL Xs V Xs Xf Fig. 6.24 — Effect of change in A' dimension on the activity of a BT-c\xX. quartz plate in an oscillating circuit. X =y'=. 448MM w=Z' 1 = X= I 1 05MM ^ 7 Z 6 ^ 5 10.0 10.2 10.4 10.6 10.6 11.0 11.2 11.4 116 118 Z' .'v M M Fig. 6.25 — Effect of change in Z' dimension on the activity of a BT-cut quartz plate in an oscillating circuit. sions from Fig. 6.27 for this case where the Xy' flexures and Z^ shears will interfere to produce poor characteristics. These are shown in Fig. 6.24 for flexure modes as Xf and for the shear modes as Xs and do in general cor- MODES OF MOTION IN QUARTZ CRYSTALS 93 respond to the dimensions resulting in low or no activity. This illustrates quite clearly the necessity for grinding the edges of plates not dimensioned for a specific frequency. Fig. 6.25 shows the same conditions when only the Z' dimension is changed. In this case the dimensions shown at regular intervals as Z, were derived from Fig. 6.25 as before and correspond to the zero activity dimensions found experimentally. It will be noticed that low activity regions are found halfway between the dimensions designated as Zs. These correspond to even orders of the Zx shear and are the result of a slight wedge in the airgap. This was intentional to show the existence of this condition. Figures 6.24 and 6.25 show the necessity for avoiding certain dimensions for oscillator plates at specific frequencies. This can be accomplished by fv;^ --SZ zrr. -.ZT- :r-^ iv •--" zr~z :zz. :~zz — »- :rr ---.-_ ~z. rzz : — - • rz:- ..^4- ---z '~~Z - — • nnr -^ --. ZZ— ~— ZiT^Z :^i zzzzz zzzz ~-i'- z~ • -zrzz '^Z .---- --ZZZ ^'^ 533 —-. ^itZZ- — -^ -—•■' ' zzz z~z rz. ~~zz ZiZZ --ZZ ■•— rzrz- — r zi^'f£-. '--: L"ir_ szzz =z" :--» t.-: ~«^- ~~s '~^ "-" -5e -^^ -— - zzzz rz'-Z !rJ£ :=z ziTz^-' izr-.:ri,z' ::--: _~_~ 'jr~ --•- _-^» i"i ~ ; :zzr- ~" '-'-'-] -^ r^: _zz. ZZZ- ^--•! - -• _-zz .__J /\ ►-- — j-— ^ -~Z.~ zrz~ ~-"1 "-V X^- '.- — zVz ~1 :=^ '— z-S^ :j= ■_-:- -£"; ^ '£F: ~=z r£z > :—Z. Z^" :?^ * — -~rl 7-~ ^^ ;- ~ zzzz -zq -_T- - — « :~-Z z^": zz: --Z r£z _ y^ T~Z "J7~ — •- -"- 1^— • — - -^^ :^" .— 'Z r^- ~-i-" -L-. — •- -Sz "^^ ZZZ4 ?I-z zzz i-i-z: :zz TZZZ -3-:- « — '- -•- SZZZZ 'zzs: jir_ zzz irr: "z-'z -Jl z^ — , -•— 0 Fig. 6.26 — Frequencies at which the Z'x shear along .Y, the Zi shear along Z' and the Xy flexure along X coincide with the high frequency A'j, shear in ,4r-cut plates. individually adjusting the X and Z' dimensions by hand grinding of each plate or by predetermining the proper dimensions and using mass production methods of precise machine grinding. The advantages of predimensioned crystal units is the insurance of proper operation over a wide temperature range and uniformity of activity. The experience of most manufacturers of low frequency crystal units in the broadcast range and high frequency crystals requiring high activity over a wide temperature range has been that it is necessary to use specific dimensions to insure low rejects in the final tests. 6.53 AT-Type Crystals The modes of motion encountered in the A T-cut crystal are the same as that of the ^T-cut. The effects of coupling between most modes is greater 94 BELL SYSTEM TECHNICAL JOURNAL due to the increased piezo electric constant for this particular cut, and the frequency constants are different due to the change in angle with respect to the crystallographic axes. The three series of interfering modes as de- scribed for the BT -c\xi have been measured for this crystal and as shown in Section 6.4 are /./ 133.84 u 254.20 6.31 _ 254.00 Jt't — ^7 ^^2'» In a manner similar to the BT case a chart has been developed of a folded frequency scale showing the frequencies at which even order X„' flexure modes propagated along X and odd order Zx shear modes along X as well as odd order Zx shear modes along Z' will interfere with the high frequency Xy' shear mode for a crystal 1 centimeter square. This is shown in Fig. 6.26. Its use is the same as that described for the BT case. Insufficient experi- mental work has been done to indicate the relative shift in the flexure and shear modes along the X axis when they approach each other in frequency. Also, most of the use of square plates and experimental work has been con- fined to the ^T-cut crystals and hence no ok regions are shown for this chart. APPENDIX B Equation of elastic and piezoelectric constants for rotation of axes about the A^ axis. (5 = sin &; c = cos &) I C\\ — C\\ C22 = C\\C + C335* + 2(2C44 + ^13)5 c -|- 4cu5C C33 = CvJ" + C33C 4- 2(2C44 + ^13)5 C — 4Ci45 c c'u = C44 + (Cll + C33 — 4C44 — 2Ci3)5V — 2ci4(c" — s') sc C55 = CiiC -f- C665 -r -^CuSC C66 == Ci\S 4" cmc — 2cusc Cu = C12C -\- C13S — 2cusc Cu = cns^ + cizc 4- 2cusc cn = Ch(c — s) -\- (ci2 — cn)sc MODES OF MOTION IN QUARTZ CRYSTALS 95 C23 = cisic* + s*) 4- (di + C33 — 4^44)^^ — 2cu{c^ — s )sc C2i = Cu{As'^ — \)C^ + [CnC^ — C335^ — (2C44 + C13)(C — s )]sc Czi = —cui'ic^ — 1)^" + [cns' — C33C' -\- i2c44 + Cn)(c — s )]sc C&6 = Cu{c — 5 ) + (f66 — Cii)sC <^16 — ^16 — ^26 — ^26 — ^35 — f36 — ^45 — C46 — U S\\ = Su 522 = SnC + 5335 + (544 + 2513)5 C + 25145C 533 = ^11-^^ + -^33^ + (-^44 + 2513)5 C — ISuS^C 544 = -^44 + 4(5ii + -^33 — -5^44 " 25i3)5V — 45u(c — ^ )SC S%h = -^44^ "f" -^ee^ + 45i45f ■^66 = ■^44-^ "f" Sn^C — 45i45C ' 2,2 5i2 — ^^12^ I S\Z^ — 5l45C ' 2 I 2 I 5i3 = 5i25 4- 5i3C + 5i45C 5(4 ^ Su{c — 5^) + 2(512 — S]s)SC Si3 = Snic + 5^) + (511 4- ^33 — .^44) A^ — 5i4(c — 5 )sC 524 = ."^14(45^ — 1)C^ + [2(5iiC^ — 5335^) — (^44 + 25i3)(c^ — S'y\SC ^34 = — 5i4(4c^ — 1)5^ + [2(5u5^ — 533C^) + (^44 + ISy^if — 5^)]5C ^56 = 25i4(<:^ — 5^) + (566 — ^44)^^ •^16 = -^16 = -^25 — -^26 — -^36 — -^36 — -^45 — 546 — U d\\ = Jii d\i = — {dxi^s + d\\c)c dvi — (diiC — dus)s du = d\i{c — 5 ) — 2di\sc dih — — (duC + 2dns)c dit = {d\\S — 2d\\c)c d'zi = — {duc + 2^115)5 dz^ — {dus — 2di\c)s d\b = d\z = C?21 = d22 = <^23 = <^24 = <^31 = <^32 = <^33 = ^^34 = 0 96 BELL SYSTEM TECHNICAL JOURNAL en = en eii = — {lenS -\- enc)c 1 eu = (2euc — eiis)s 1 f'14 = eii{c^ — /) —ensc / ^25 = - {eiiC + ens)c / ^28 = (eiiS — enc)c / ^35 = -(euc-jr ens)s / ^36 = {eus — enc)s e\h = ei6 Cil = ^22 = ^23 = ^24 = esi = ^32 f33 = ^34 = 0 Response of a Linear Rectifier to Signal and Noise* By W. R. BENNETT WHEN the input to a rectifier contains both signal and noise com- ponents, the resultant output is a complicated non-linear function of signal and noise. Given the spectra of the signal and noise input waves, the law of rectification, and the transmission characteristics of the input and output circuits of the rectifier, it should, in general, be possible to describe the spectrum of the resultant output wave. Before discussing the solution of the general problem, we shall derive some results of a simpler nature, which do not require a consideration of the distribution of the signal and noise energies as functions of frequency. I. Direct-Current Component of Output A quantity of considerable importance is the average value of the output amplitude. This is the quantity which would be read by a direct-current meter. Calculation of the average or d-c response can be performed in terms of the distribution of instantaneous output amplitudes in time. The dis- tribution of output amplitude can be computed from the distribution of instantaneous input amplitudes and the law of rectification. As an example, we shall compute the average current obtained from a linear rectifier when the input to the rectifier consists of a sinusoidal signal with random noise superposed upon it. The probability density function of the signal voltage is first determined, and the result given in (3). The corresponding probability density for the voltage of the noise is well known and is given in (4). The distribution of occurrence of the resultant in- stantaneous amplitudes of the combined noise and signal voltages is then computed by the rules of mathematical probability, and the result is shown in (7). The assumption that the rectifier is linear then leads directly to an integral which yields the average current obtained from the rectifier. Let the signal voltage, £«, be given by Es = Po cos co/. (1) The possible angular values of oit are uniformly distributed throughout the range 0 to lir. The range Es to E^ + dE^ corresponds to the range of values of w/ comprised in the interval. Ea ^ , ^ E, -\- dEs ,^. arc cos — < a/ < arc cos (2) ■to ^o •Published in Acous. Soc. Anier. Jour., Jan., 1944. 97 98 BELL SYSTEM TECHNICAL JOURNAL The angular width of this interval is {PI — El)~^'^dEs. There are two such intervals in the range 0 < coi < 2ir. Values of Es outside the range —Po to Po do not exist. Hence, the probability that the signal voltage lies in the interval c?£« at any particular Es is given by *.(£.)i£. = {^^ If ll >.f-'n ^^j^^^ I £^ I < p] dE. (3) Random noise as discussed in this section may be characterized by the fact that the instantaneous ampUtudes are normally distributed in time; that is, if $„ (s) dz is the probabiUty that the noise amplitude lies in the amplitude interval of width dz at z, $n(s) = -^ e-''""' (4) (TV iTT where a is the root mean square noise amplitude. The mean noise power dissipated in unit resistance is given by TT',, = a . The corresponding mean signal power is given by W^ = Pol 2- Let 4>,(2) represent the probability density function of the instantaneous sum of the signal and noise ampli- tudes. Then ^r{z)dz = dz [ ^M *n(s - \)d\ (5) or By the substitution X = Po cos 6, we may convert the integral to the form $^3) = -^ r r^^-^" ^°^ '''"'" dd (7) TTO" v27r J o Suppose we insert a half-wave linear rectifier in series with the source of signal and noise, so that the current 7 is given in terms of the resultant instantaneous voltage E by fO, £ < 0 l-\ (8) \aE, £ > 0 Then the average value of current flowing in the circuit is I = a I z^riz) dz Jo = -^ r zdz r,-(-^«-s«w^^ TTCrV 217 Jo Jo RESPONSE OF RECTIFIER TO SIGNAL AND NOISE The value of this mtegral is shown in Appendix I to be 99 / = ayj/|^"e-«'^""|/oaF./2fF„) + pHw)-"m (10) This form is particularly convenient for calculation since Watson's Theory of Bessel Functions, Table II, gives c~'[Jz) and f""-/i(c) directly. 2.0 > _) z o i.e -I < z ^ 1.6 10 X 1- u O ^,.2 > < 1.0 _/ / y / / / y / /^ / / / ^- / JQlx ABSCISSA 0.6 0.8 1.0 1.2 RMS. NOISE INPUT RMS. SIGNAL INPUT 1.4 1.6 l.a 2.0 Fig. 1 — Variation of direct-current component in response of linear rectifier with ratio of noise input to signal input. Limiting forms of this equation may be expressed in terms of series in powers of 11^/11',, when the signal power is small compared with the noise power and in powers of TF,, Tr,. when the noise power is small compared with the signal power. The ascending series for small signal is: K-i) {w,f V^"[ ^" 1 + 1 w + 2(1!)MF„ 22(2 !)2 (PF„) l(-l)(-3) {Wjf 2^(3 !)2 (IFn) ■] = VI"'^'(t-.'^^-) (11) The asymptotic series, which is available for computation when the signal is large, is (-1)^.1=^ (IFn)^ ^ (-1)^1^3^ i^y ^ 2! (4IF,) (_1)2. 12.32.52 i^Wr)' (12) 3! (4IF.) 4! (4IF,) + Curves of I have been plotted in three ways. Fig. 1 shows the ratio of 7 to Iso = aPo/TT, the average current in the absence of noise, as a function 100 BELL SYSTEM TECHXICAL JOURNAL of ratio of rms noise input to rms signal input. Figure 2 shows the ratio of / to I no = oKr/y/lir, the average current in the absence of signal, as a function of ratio of rms signal input to rms noise input. Figure 3 shows I 1.6 /., /, / / / / / / / / / y / ^'asymptote '-L X ABSCISSA VTT /^ y / ^ / f 0.6 0.8 1.0 1.2 1.4 RM.S. SIGNAL INPUT RMS. NOISE INPUT 2.0 Fig. 2 — Variation of direct-current component in response of linear rectifier with ratio of signal input to noise input. O 12 o z 10 t- 2 8 LJ a. ° o ? 4 m " 2 0 /y // / // ^ R.M.S. ADDITION ^ / / ^/ / y ^ r A '/ LINEAR RECTIFIER y. y ^ ^ y __, ___: ':^ -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 NOISE INPUT POWER IN DB ABOVE SIGNAL INPUT POWER Fig. 3 — Variation of direct-current component expressed in decibels, showing compar- ison between linear rectification and power addition of signal and noise. the increment in d-c power output in decibels as varying amounts of noise expressed in decibels relative to the signal are added. The correspond- ing result for power addition is given for comparison. RESPONSE OF RECTIFIER TO SIGNAL AND NOISE 101 IT. Spectrum of Output A much more powerful method of attack on this problem is obtained by the use of multiple Fourier series. In this section we shall use Fourier analysis to obtain not only the direct-current output of the rectifier, but also the spectral distribution of the sinusoidal components in the output of the rectifier. We represent the input spectrum by N E = Fo cos Pot 4- E ^« cos pj (13) n=l This representation is more general than that given by (4) in that a frequency spectrum as well as an amplitude distribution is defined; it may be shown that the probability density for the sum of N sinusoidal waves with incom- mensurable frequencies approaches (4) when N is large. The first term represents the sinusoidal signal; the mean power which would be dissipated by this signal in unit resistance is Ws = Pill. (14) The noise is represented by a large number A^ of sinusoidal components with incommensurable frequencies (or commensurable frequencies with random phase angles) distributed along the frequency range /i to ji in such a way that the mean noise power in band width A/ is: «'(/)A/ =1 E P\- vLjP\j)ll (15) Here v is the number of components per unit band width and /*(/) represents the amplitude of a component in the neighborhood of frequency /". Note also that the mean total noise input power, Wn , is given by I^„ = jf w{j) df=^l P\f) df (16) The linear rectifier is specified by the current- voltage relationship (8), which is equivalent to It Jc z^ where C is an infinite contour going from — «; to + co vvith an indentation below the pole at the origin. We may expand / in the multiple Fourier series 1 Bennett and Rice, "Note on Methods of Computing Modulation Products," Phil. Mag., Sept. 1934. The present application represents an extension to N variables of the theory there given for two. 102 BELL SYSTEM TECHNICAL JOURNAL 00 00 i ^ ' ' ' ' ' ' ' ' ^TOomi"' .„,^ COS wqXo cos m\X\ where mo=0 mi=0 iBjv=0 (18) • • • COS DIn Xn Xk = pkt, k = 0,\,2,"' N (19) = '"° "'";,''/ '"'^ r dxo f dxv ■k"'^^ Jo Jq j I cos nio Xo COS nii xi- • • cos W/vXat dxif (20) Jo '■mo»n- • -TOAf (21) The response of the rectifier is thus seen to consist of ah orders of modula- tion products of signal and noise. In a typical case of interest the band of input frequencies is relatively narrow and centered about a high frequency while the output band includes only low frequencies. In such a case the important components in the output are the beats between signal and noise components and between noise components. The d-c component is present in the output only if the pass band of the system actually includes zero frequency; we have already computed its value in Section I, but we will derive it again by the method used here as a check. The amplitude of the d-c component is in fact: Ooo a / „=i , {22) ...0 = - TT Zi dz, 2ir J c 2 on substitution of the expression for E in the integral representation of 7, substituting the result in (20) and interchanging the order of integration. When N is large, P„ is small, hence the principal contribution to the integral occurs near small values of z, where Jo{Pnz) is nearly equal to unity, since the product of a large number of factors, all less than unity, will be small indeed unless each factor is only slightly less than unity. We therefore replace Jo{Pnz) by a function which coincides with it near z — Q and goes rapidly to zero as we depart from this region. Such an approximation (Laplace's process ) is MPnz) = e-'"^'"" (23) 2 Watson, "Theory of Bessel Functions," p. 421. RESrOXSE Of KECriFlER TO SIGNAL AND NOISE 103 which is correct for the first two terms in the Taylor series expansion near G = 0. Therefore, when /*„ approaches zero as N approaches infniity, N a f ^ - 2 '-nz^i dz ooo.-.o = / = — w- \ Jo{PoZ)e " = i — IttJc z^ ^-f [ Mr„,)e-" ■">''!! (24) Zir J c 2- 'J'he contour integral cannot be replaced by a real integral directly ])ecaiise the integrand goes to inhnity at the origin. However, since Jo{u) ^ _ Jiju) _ d^ Joju) . V u^ u du ti MPz) J,{Pz) d MPz) JiiPz) 1 d JoiPz) PV d{Pz) Pz PH"" P^dz (26) we can substitute (26) in the integral and perform an integration by parts to give the result. 7 = « f .-'■-^/^ \PoMPoz) ^ „.^^^^(p^,)] ^, 7r Jo L 2 J by Hankel's formula.^ But it may be shown that (see Appendix II) iF^{hU-M)-e-'"'lo(^^ (28) iFi (i; 2; - u) = e-"" [h («/2) - h(u/2)] (29) Hence, 7^«y/^"e----{/e(nV2irj4-|P; (30) [/c(nV2If^n) + Il{Ws/2]Vn)][ which is identical with the result of Section I, noting that a = \/Wn • We point out that a resistance-capacity coupled amplifier will not pass this component since there is no transmission at zero frequency. ^ Watson, "Theory of Bessel Functions," p. 393. As pointed out by Watson, in a foot- note, the difficulty with singularities at the origin could be avoided by expressing Hankel's formula in terms of a contour integral instead of an ordinary integral along the real axis. Tliis procedure would lead directly to the hypergeonietric function given in (11). 104 BELL SYSTEM TECHNICAL JOURNAL The amplitude of the typical difference product between the signal and the rih. noise component is Asn — 2 ^100.--010---0 J,{I\z)J,{l\z)J,{P^z)-' ■Ji{P,.z)-- -MPnz) W T J dz Z" Using the same process as before, we replace J\{Pnz) by UPnz) =^-^6-"'^''" ^ (32) and obtain in the limit as N becomes indefmitely large TT Jo 2 aP ^2 Relations between the i/''i function and liessel functions are discussed in Appendix II. The shape of the spectrum of the beats between Pi) and the noise input evidently consists of the superjjosition of the noise spectra above and below po , so that if we write n'snif) A/ for the mean energy from this source in that part of the filter output lying in the band of width A/ at/, ■Wsnimi = ^^ [(Atnf + (^7n)'1 (34) Asn — l^f »nJp„=po+2ir/ (35) Asn ^^ [Asn]pn=:po—27rf W^) P„ = ^^ (37) "■•"^^ = W„ ^ Hiwj + '\2W..)] (3S) X W/o+/) +w(r„- f)] The total noise from this source in the output of a particular filter of transfer admittance Y(f) is obtained by integrating Wsn{j)Y{f)df throughout the band of the filter. In the particular case in which the original band of noise is RESPONSE OF RECTIFIER TO SIGNAL AND NOISE 105 symmetrical about/,, and occupies the range/, — fa to/o +/a and an ideal low pass iilter cutting ofif at / = /<, is used in the rectifier output, the total noise output from beats between signal and noise is ]]\„ =^ 2 r u,,Xf) df = ^^ e-''-''''"[hin\/2Wn) + Ii{W,/2W„)f (39) Jo -iTT Next we shall calculate the spectrum of the energy resulting from beats between individual noise components. We write Ann — 2 ^flO- •■010---010-..0 ^a r ^^/o(Poz)/o(Pis)- • -JliPrZ)'- -MPsZ)-' 'MPxZ) T J c dz = —^ — / Jo{PoZ)e " dz Itt Jq (40) 2V2TrWn \2' ' Wn iVlirWn To find the resulting spectrum Wnnij)df produced at / by the resultant of all such components, we note that we may sum over all components by beating each component of the primary band with the frequency / above it and adding the resultant power values. The result is '^'n„(/) = tV e-'''"'^ll{Wj2Wr.) f w{\)w{\ + f) d\ (41) 47rlr„ Jo In the particular case of a flat band of energ\^ extending from /i to /z , (42) [ w{\)wi\ +f)d\= r ,,^\,. d\ = ^) ^\ / wi Jo Jfi (72 — jir (72 — jir 0 < / < /2 - /l 47r (/2 - fiY (43) 0 /2Wn) 106 BELL SYSTEM TECHNICAL JOURNAL provided /b < f2— fi ■ The spectrum is confined to the region 0 < f < fi — /i . If /ft is equal to/2 — /i so that the output filter passes all the noise of this type, we have WUh - /i) == Wnn = "^" .-"'-'"'" /^(T.TV2Pr„) (45) This result seems to hold approximately for a considerable range of input spectra. For example, if we assume that the original noise is shaped like an error function about /o , i.e., Wnif) = WnV^ g-(f-fo)' (46) with / taken from — sc to + =0 with small error for large /« , f 00 w (X)w(X +/) d\ = II'; Va/2:r e-"-'" (47) [ df [ wi\)ui\ ^ f)J\ = ir';/2 (48) which is in agreement with (45). The output of a half-wave linear rectifier contains fundamental compon- ents and all even order modulation products. In general, the amplitudes of the higher order products are small compared with the lower order. In a particular problem some consideration of where the principal products fall in the frequency band is required. The products just considered give a fair approximation for the problem of detection of a radio frequency band of signal and noise followed by audio amplification. Contain other products should also be added to obtain higher accuracy. We have calculated the products of order zero and two; the next ones of importance are the fourth order, since the third order products vanish in a perfectly linear rectifier. The fourth order products in this case which fall in the audio band are of frequency 2po — pr — ps .po + pq — p,- — ps , and p„ + p,j — p,- — ps , where the subscripts n, q, r, s refer to the original noise component frequencies. The latter is, however, less important than the sixth order product 3po — Pq — pr — ps , which involves only three noise components. Expressions for the contributions from these products are given in Appendix III. Figure 4 shows computed curves for the noise produced in an audio band by the various components. Curve A is Wsn + Tr„„ and includes what are usually regarded as the principal contributors, the difference frequencies between signal and noise, and between individual noise components. Curve B is obtained by adding to Curve A, the contribution from the fourth order products 2po — pr — ps and po + pq — pr — ps and the sixth order products ipo — pq — pq ~ ps ■ Thus all products which include three or less noise fundamental components are included. The curves are plotted in terms of RESPOXSE OF RECTIFIER TO SIGXAL AND NOISE 107 fraction of noise power received compared to the limiting noise when the mean signal input power is made indefinitely large compared to the mean input noise power. Some experimental points given by Williams are shown for comparison. ^^'iUiams gives the intercept at zero signal power as 35%; the theoretical value deduced here is tt/'S or 39.27%. It will be noted that the inclusion of the higher order products improves the agreement between experimental and theoretical curves, even though the value of the intercept is unaffected bv them. It shold also be stressed that our analysis applies < 0.9 O 0.5 o > B [lirr: / 1 v/*^^ ^ ""■""a i^ / y / A: SIGNAL- NO / B: CURVE A + / O WILLIAMS' E SE AND NOISE -NOISE COMPONENTS 2 XSIGNAL- (noise + NOISE) SIGNAL- (noise + NOISE -NOISE) 3 X SIGNAL - (noise + NOISE + NOISE) xperimental data ' 3 4 5 6 7 MEAN SIGNAL INPUT POWER MEAN NOISE INPUT POWER Fig. 4 — Calculated noise power in audio band of output of linear rectifier when noise and signal are applied in a relatively narrow high-frequency band. The direct-current component is excluded. strictly to purely resistive networks. The conventional radio detector circuit (which was used b}' Williams), in which a condenser is shunted across a resistance in series with a diode, departs from the conditions here assumed because of the reactive element, the condenser. The customary approxima- tion made in treating this circuit is that the condenser has infinite impedance in the audio frequency range and zero impedance at the radio frequencies. This leads to a bias on the detector which depends on the signal. The methods given here may be applied, but the resulting formulas are much more difficult from the standpoint of numerical computation. A recent paper by Ragazzini^ gives an approximate solution based on ^F. C. Williams, "The Response of Rectifiers to Fluctuation Voltages," Journal I. E. E., 1937, Vol. 80, pp. 218-226. ^John Ragazzini, "The Effect of Fluctuation Voltages on the Linear Detector," Proc. I. R. E., June 1942, Vol. 30, p. 277-288. 108 BELL SYSTEM TECHNICAL JOURNAL expanding the envelope of the input wave by the binomial theorem and retaining only the first two terms. The validity depends on the noise amplitude being small compared with the sum of signal and noise, and hence the result should agree with our solution in the neighborhood of TT'„/tT's = 0, which it does. WTien TT',, U',, is small, the error is appreciable. Ragazzini's result (Equation 15 of the paper) expressed in our notation is ^•' + " "" = 7= 1 + w./w. ^'^^ It will be seen by comparing the limiting values for Trs/Tr„ = 0 with that of Trs/Tr„ = ^ from (49) that the intercept of the curve of Fig. 4 would be 50% instead of our value of 39.27%. The results given in the present paper have been compiled from unpub- lished memoranda and notes by the author extending back as far as 1935. Discussions with colleagues have been of great aid, and in particular ac- knowledgment is made to Messrs. S. 0. Rice and R. Clark Jones for many helpful suggestions. APPENDIX I EVALUATION OF INTEGRAL FOR I Interchanging the order of integration in (9), we have 7 = .^ r dd f e-'--'^ ""^ '''"'''^ z dz (50) 7rv27rlKn •'0 •'0 By substituting 2 = Po cos 0 + n \/2lF„ , we may evaluate the second inte- gral in terms of the error function, obtaining T = ^f dd f _ e""'(M \/2?F"n + Po COS d) du TT'- Jo J-P cos «/\/2Wn _ a -ylVn r -Pq^ cos2 ei2W„ ^Q IT l-K Jo + ^ [ erf (Po cos d/VWn) cos 6 27r Jo dd _ a y/Wn g-P§/4W„ C ^-co3 29/4Jr„ ^^ TT l-K Jo RESPONSE OF RECTIFIER TO SIGNAL AND NOISE 109 aPo + 2 (eri?^^)de do \ V2Wn/ ^ a_ -yWn ^-W,/2W„ r" ^-W, cos */2If„ j^ 2ir lir Jo = " i/r-" '"■'''"'" (wv./2w„) + 2j [7o(T7./2TF„) + h{Wj2Wn)]\ (10) In the above we have made use of the relations V erf s = ^ f e"'' Jz (51) TT Jo r " ^-^cos* ^^g ^^^^ ^^ ^ (-)'"27r7„(2) (53) Jo APPENDIX II RELATIONS BETWEEN HYPERGEOMETRIC AND BESSEL FUNCTIONS The modulation coefficients appearing in the linear rectification of noise are expressible in compact form in terms of the hypergeometric function: (54) ^ r(£) y V{a + m) ^ . V{a) £^oV{c-\-m)m\^ ^ The iFi function is a limiting case of the more familiar Gaussian hypergeo- metric function ^Fi (o, h; c; z), viz. iFi(a; c, z) = Limit 2Fi(a, b; c; z/b) (55) 6=00 In certain special cases this function may be expressed in terms of ex- ponential and Bessel functions. For example, by a formula given by 110 BELL SYSTEM TECHXICAL JOURNAL Campbell and Foster, Fourier Integrals for Practical Application, Bell System Monograph B-584, p. 32 (also Watson, Theory of Bessel Functions, p. 191), we may show that iFi\v + -;2v+ 1; -z\ = ^—- Iv{-z/2) (36) or setting v = 0 ^F^{h;l;-z)^e-'''h{z/2) (57) which is one of the functions appearing in our work. We have also encountered the function iFi (1/2; 2; — s) which is not directly reducible by the above formula. The reduction may be effected in a number of ways. By making use of the relation obtained from (56) by setting v = 1, iFi(3/2;3;-s) = "^^ e~"' h{z/2) (58) z and noting that :Fi(l/2; 2; -2) - ,F,(l/2; I; -z) ^ 1 Y T{m + 1/2) ^ 1_ ^ T{?n + 1/2) ,^ r(l/2)^ow!(^ + 1)!^ ^^ r(l/2)^o (m!)2 ^ ^^ — 1 '<^ T(m 4- l/2)w , .„ 2^ ...w... I tx, (-2) r(l/2)iri m\{m + 1)1 ' "^^ (59) 2 Y^ r(w + 3/2) E ^^ ,vr (-^r r(l/2) ^0 (m + 2)lm[ = |iFi(3/2;3;-2), we find that iFi (1/2; 2; -s) = e"" [Io{z/2)+h(z/2)] (60) It may also be verified by integrating the series directly that r iFi(l/2; 1; -z) dz = 2iFi(l/2; 2; -z) (61) Jo Combining this relation with (57) and (60) above, we deduce the indefinite integrals ^ The relation (60) was brought to the attention of the author by Mr. R. M. Foster. RESPONSE OF RECTIFIER TO SIGNAL AND NOISE 111 f e'hix) dx = xe'ihix) - h{x)] f e-'Io(x) dx = xe-Vo(x) + L(x)] . \ (62) j e'hix) dx = e'[{l - x)h{x) + xh{x)] f e-^'hix) dx - ^-^(1 + x)Io(x) + xh(x)] These integrals may be derived by differentiating the right hand members, and could, therefore, serve as a basis for an alternate derivation of (60). In addition it was noted in Eq. (11) that the constant term in the modula- tion spectrum could be expressed in terms of iFi (—1/2; 1; —z); from the equations given, it follows that we must have the relation: iFi (- 1/2; 1 ; -z) = e-'" [(1 + z) h{z/2) + z hiz/l)] (63) Another interesting set of formulas which can be obtained as a by-product from (62) by setting x — iy is: / /o(y) cosydy = >'[/o(y) cos y + /i(y) sin y\ I Joiy) sin ydy = y[Jo(y) sin y — Ji(y) cos y] I Jiiy) cos ydy = yJi{y) cos y — Jo{y){y sin y — cos y) I Ji(y) sin y dy = yJiiy) sin y + Jo(y)iy cos y — sin y) The hypergeometric notation is particularly convenient in determining series expansions for the coefficients to be used for calculation when the variable s is either very small or very large. For small values of z, the form (54) suffices; for large values of s, we may use the general asymptotic expan- sion formula for the real part of z positive : (64) iFiia; c; -z) = Tie) T{c — fl)2" iFoia, I + a — c; 1/z) r(c - a) r a(l + g L 1!2 -c) (65) a(g+l)(l + a-c)(2 + a-c) "^ 2!22 "^ Copson, "Functions of a Complex Variable," pp. 264-5. 112 BELL SYSTEM TECHNICAL JOURNAL The series expansions required here could also be obtained from the appropri- ate series for Bessel functions. It will be noted, however, that the typical modulation coefficient can be expressed in terms of either a single iFi function or several Bessel functions, so that manipulations must be performed on the series for the latter to give the final result. The Bessel functions on the other hand are more convenient for numerical computations because of the excellent tables available. Reduction formulas for certain other h}^ergeometric functions are needed in evaluating the higher order products. They are: iFi(3/2; 1; -2) = e-'"[{l - z)h{z/2) + h{z/2)] (66) iF^{Z/2- 2; -2) = e-'"[h{z/2) - Uz/2)] (67) iFi(5/2; 4; -2) = \ e-'" \^^ + 1)71(2/2) - 7o(z/2)] (68) Derivation of these is facilitated by the use of the easily demonstrated relations: iFi(a; 1; -2) = -^ [2iF:(a; 2; -2)] (69) az 2ziFi(a; 2; -2) = ^ [z\Fiia; 3; -2)] (70) az iFa(3/2; 3; - 2) - :F:(3/2; 2; -2) = | iFx(5/2; 4; -z) (71) APPENDIX III HIGHER ORDER PRODUCTS The methods described in Section II may be applied to calculate the gen- eral expression for the general modulation coefficient. The result is for the amplitude of the term cos mpot cos pnj cos pnJ • • • cos pnu^' OT+Af ^ (-) ^ ^ Pn.Pn, ■" Pn^ ^ /^ + M - l\ {W sT" -KiWnny^-'^i^'ml \ 2 ) {Wn) (72) ^ ^ (m + M - \ ,. -W\ X.F:(^ -2 '" + ^'177; The coefficient of the term cos {mpa ± ^„i ± Z?,,, ± ... pnj^) ^ is amM divided by 2"~^ em • The number of terms of a particular t>TDe falling in a particular frequency interval can be calculated by a method previously described by RESPONSE OF RECTIFIER TO SIGNAL AND NOISE 113 the author.* Under the assumed conditions that the original noise spectrum is either flat throughout a limited range, or falls off like an error function, and that the audio amplifier passes all the difference components in question, we find the following results: (73) (74) 2p0 — pr — pi'. TF,,.„n = ^"6-"-'/"^"/?(IF,/2TF„) OTT pQ-\- Pi — Pr — Ps'- Win.nr. = ^T^^/"" [/o(IF./2IF„) - h{Wj2Wn)f 327r 3^0 — Pq — Pr — P» ' TF3..„„n = "^ e-'^-/'^- [(1 + mjW.)L(W,/2Wn) (75) - Io{W,/2Wr.)f This includes all beats containing not more than three noise fundamentals. The reductions of hypergeometric functions to exponential and Bessel func- tions given in Appendix II have been used in deriving the above results. 8 Bennett, "Cross-Modulation in Multichannel Amplifiers," BellSys. Tech. J our., Oct. 1940, Vol. XrX, pp. 587-610. Dielectric Constants and Power Factors at Centimeter Wave-Lengths By CARL R. ENGLUND The theory underlying the measurement of dielectric constants and power factors, by means of resonant lengths of coaxial transmission line, is developed, apparatus used for such measurements is illustrated and the measurement routine described. A table of typical results is ap- pended together with an "-Y tan A'" table for aid in the calculations. Introduction THERE are two instrumentalities available for measuring dielectric constants and power factors at centimeter wave-lengths. These are, coaxial conductor lines and wave guides. Which one is, for any condition, the more favorable one depends a great deal upon the wave-lengths used. Under the conditions encountered in this work the coaxial line appeared to have the practical superiority, down to something like 10 cms. wave-length, anyway. Below this, the wave guide is ver}-- manageable and has several advantageous features. When this work was begun, tlie most easily available and practicable vacuum tube which would oscillate around 20 cms. wave-length was the W. E. Co. 368A. This could be pushed down to something below 19 cms. wave-length but was undependable there and as a practical compromise 22.5 cms. wave-length was finally chosen. Later another tube became available and as it could be operated down to at least 9 cms. it was used in the more recent work. Thus, while the bulk of the measurements made were at 22.5 cms. wave-length, a good share of the samples investigated were also measured at approximately 10 cms. wave-length. Any measurements made at these wave-lengths must be made in the form of transmission line measurements and the dielectric must be physically part of the coaxial line. There are various transmission line quantities definable and measurable, such as series impedance per unit length, shunt admit- tance per unit length, surge impedance, impedance transformation factor, voltage and current step-up factors, resonance selectivity or "Q", etc. The first two are measurable directly only at long wave-lengths, the last two are properties of space resonant line elements. Of these the "Q" was the most advantageous in the present instance. 114 DIELECTRIC CONSTANTS AND POWER FACTORS 115 "(2" Definition At low frequencies the resonance selectivity factor of lumped circuits is identified as the "Q" and is defined as — . It is measured by a detuning process. For a length of transmission line with negligible shunt conduc- tance losses this process gives -— as for a coil; when this process is applied R to complex circuits the physical embodiment of the "Q" becomes difficult to realize and it is preferable to define the "Q" in terms of the detuning process itself. This is equally true for tlie resonant, centimeter wave, line element and we proceed as follows: For this element some current or voltage ampli- tude, conveniently measurable, is selected and three values of it are measured as the line tuning is varied. This variation may be either in generator fre- quency for constant line length or in line length for constant generator fre- quency. Thus, for example, Q = T^f^ where /. > /o > /i h ~ h Q = J , where 4 > ^ > A At = Al = ^ (1) ^2 — /l with ^0 as the resonant amplitude. For low-loss lines the two definitions will give the same results in practice. Neither is ideal for second order accuracy since there is a variation of line constants with frequency in the first and a variation in total attenuation in the second. For practical reasons it is usually preferable to excite and observe the line resonance in terms of the current at one end, this end shorted. The ele- mentar}' line lengths are then the quarter and the half-wave ones, the former with open circuit far end, the latter with shorted far end. The latter is the more nearly ideal unit. In order to short effectively the input end, the in- put and output couplings must be made as loose as possible. As these couplings are reduced the observed "Q" will asymptotically approach the line "Q"- At the present moment the line variation in length is the most convenient process, the chief trouble being the micrometric measurement of the tiny length changes involved. Thus for 10 cms wave-length and a half-wave coaxial line, a "()" of 1000 involves a plunger movement of .0019 inches. Theory of Measurement It is shown in the appendix that the "Q" of a given resonant line segment can be broken up into parts representing the equivalent "^'s" of the ter- minal impedances and the line itself. Thus 116 BELL SYSTEM TECHNICAL JOURNAL JL _ J. J. J. (2) where "Q" is the actually measured quantity, Qq is the part due to the line itself, ^0 and Qf the parts due to the near and far end terminations, respec- tively. If we now take a quarter-wave line segment, with near end shorted through a movable plunger and far end open, we may make two "^" meas- urements without and with the far end loaded with a dielectric segment, and obtain ^1 = 14-1 = ^1 Q' Q, Qo x/4 (3) A _ A 1 1 _ A I <2 " <2. "^ <2o "^ (2^ " x/4 1 1 _ 1 _ d_-^ ^^^ Q~Q'-Qr X/4 with d' and d equal to the widths of the resonance curves halfway down in power. These two ^'s are, of course, directly measurable. When the line is loaded with a dielectric segment the loaded part of the line can be represented as an impedance Z^ connected to the unloaded re- mainder of the line. The effect of the loaded segment upon the imloaded V D /Z. line (See appendix, eq. 4) appears in the form - - where A/ -is the surge impedance of the unloaded line, with "Z" and "Z)" the series impedance and shunt admittance, respectively, for unit length of this line. If we put we have tanh d = tanh {a^ + ib^) = f A _ ^ - ^' _ ^df. \Q^~ X/4 -V ZTT Zi (4) (5) where A^ is the measured plunger movement necessary to retune the line, after adding the dielectric loading, and "/" is the length of the dielectric segment. DIELECTRIC CONSTANTS AND POWER FACTORS 117 z /i Now, the power factor of "Z/' is the same as that of -^ — , as long as Z( l/I — is substantially a resistance, and since we have . . sinh 2a/ + i sin 2be ... tanh {ai + thf) = / — -* , (6) cosh 2a( -\- cos 26^ power factor Z( = p.f. = . (7) sin 2b( Substituting eq. (5) in (7), sinh ^{d - d') p.f. = -^ , (8) sin ^ (A^ + t) A which is the power factor of the loaded line segment in terms only of meas- urable lengths. This does not complete the theory, however. We are interested in the power factor of the dielectric itself and it is evident that except for very short dielectric segments, the variation of the standing electrical field along the dielectric segment will result in a calculated power factor smaller than the true one. We also wish to determine the dielectric constant. The impedance of the dielectric line segment, open circuited at the far end, can be written as Z( = . ' '^ .-. (9) tanh where "a" is the attenuation per unit length and "e" is the dielectric constant. Hence tanh (c^ + ibi) = ■\/~t tanh [a-\-i — — ^ j / and sinh lat sm — e — i 118 BELL SYSTEM TECHNICAL JOURNAL an alternative expression. Now when "/" is very small the functions of the angles become equal to the angles and we write, for the dielectric power fac- tor itself , 2at "'"■ = wr:- <"' Dividing this expression by eq. (10) P.F. = pi X 2at 47r'\/e i sinh 2ai X and as the last term is always very nearly unity we have, if we put 4X, 4xV'e ^ sinh ^ (d — d') . , ^ P.F. = —^ '^. (12) sin ~ {AC + t) A Ordinarily the "sinh" is very closely equal to the angle. The reactance of the dielectric segment of line is necessarily equal to the reactance of the part of the original line which it displaces, since space resonance occurs in both cases. Hence, AC + t /-^ VTt f.^. tan TT — - — = Ve tan tt ^ — [16) A X which we can rewrite to wt A( + 1 \/7t ^ VTt — • tan TT — - — = X ^ — • tan x -— — . X X X X Putting f TT/ A^ + / ' y = — tan tt — - — X X _ we have y = X tan X, (14) "y" is directly determinable by measurement and this gives X from the X ^~ tan X table suppHed.^ The value of e = — follows and P.F. is immediately irt _\ calculable. This completes the reduction of the observation. 1 As no X tan X table to the necessary subdivision was available, one was calculated from the Hayashi tan A' tables. .0000 0000 .0001 0000 .0004 0006 .0009 0027 .0016 0085 .0025 0209 .0036 0433 .0049 0802 .0064 1369 .0081 2194 .0100 3347 .0121 4904 .0144 6952 .0169 9S85 .0197 2907 .0226 7028 .0258 2071 .0291 8166 .0327 S4S2 .0365 4077 .0405 4201 .0447 5991 .0491 9627 .0538 5297 .0587 3201 .0638 3548 .0691 6560 .0747 2470 .0805 1521 .0865 3971 .0928 0088 .0993 0153 ,1060 4461 .1130 3321 .1202 7054 .1277 5997 .1355 0503 .1435 0937 .1617 7683 .1603 U42 .1691 1729 .1781 9879 .1876 6047 .1972 0704 .2071 4343 .2173 7478 .2279 0643 .2387 4397 .2498 9320 .2613 6019 .0000 0100 .0001 2100 .0004 4106 .0009 6131 .0016 8194 .0026 0326 .0037 2562 .0060 4949 .0065 7539 .0083 0393 .0102 3683 .0123 7186 .0147 1287 .0172 5984 .0200 1381 .0229 7589 .0261 4731 .0295 2939 .0331 2351 .0369 3119 .0409 5402 .0451 9369 .0496 5200 .0543 3084 .0592 3222 .0643 6825 .0697 1117 .0752 9329 .0811 0709 .0871 5513 .0934 4012 .0999 6488 .1067 3237 .1137 4669 .1210 0803 .1285 2292 .1362 9373 .1443 2421 .1626 1821 .1611 7973 .1700 1296 .1791 2228 .1885 1223 .1981 8765 .2081 5321 .2ia4 1434 .2289 7633 .2398 4477 .2510 2650 .2626 2462 .0000 0400 .0001 4401 .0004 8408 .0010 2435 .0017 6604 .0027 0644 .0038 4893 .0051 9298 .0067 3911 .0084 8796 .0104 4023 .0126 9672 .0149 6829 .0176 2691 .0203 0063 .0232 8359 .0264 7602 .0298 7923 .0334 9464 .0373 2377 .0413 6820 .0466 2966 .0501 0992 .0548 1093 .0597 3468 .0643 8330 .0702 5903 .0758 6421 .0817 0131 .0877 7292 .0940 8176 .1006 3065 .1074 2269 .1144 6067 .1217 4815 .1292 8342 .1370 8503 .1451 4169 .1634 6325 .1620 6076 .1709 1140 .1800 4867 .1894 6684 .1991 7097 .2091 6594 .2194 6691 .2300 4929 .2409 4869 .2621 6099 .2636 9227 .0000 0900 .0001 6901 .0005 2909 .0010 3940 .0013 5014 .0028 1163 .0039 7426 .0063- 3849 .0069 0486 .0086 7402 .0106 4668 .0128 2363 .0152 0576 .0177 9404 .0205 8954 .0236 9339 .0268 0683 .0302 3120 .0338 6791 .0377 1849 .0417 8466 .0460 6780 .0605 7006 .0552 9325 .0602 3939 .0654 1061 .0708 0913 .0764 3744 .0822 9787 .0883 9308 .0947 2579 .1012 9886 .1031 1527 .1151 7814 .1224 9074 .1300 6649 .1378 7893 .1469 6180 .1543 0898 .1629 2460 .1718 1259 .1809 7767 .1904 2430 .2001 6730 .2101 3162 .2205 0249 .2311 2532 .2420 5574 .2532 9966 .2648 6319 .0000 1600 .0001 9601 .0005 7611 .0011 5645 .0019 3725 .0029 1884 .0041 0160 .0054 8602 .0070 7264 .0083 6212 .0108 6616 .0130 5259 .0154 5529 .0180 6425 ,0208 8063 .0239 0528 .0271 3975 .0306 8629 .0342 4332 .0381 1637 .0422 0307 .0465 0814 .0510 3240 .0567 7779 .0607 4634 .0669 4020 .0713 6163 .0770 1299 .0828 9679 .0890 1662 .0963 7224 .1019 6960 .1088 1041 .1168 9811 .1232 3587 .1308 2712 .1336 7644 .1467 8466 .1651 5838 .1633 0097 .1727 1657 .1819 0968 .1913 8464 .2011 4654 .2112 0028 .2215 6110 .2322 0442 .2431 6693 .2544 4152 .2660 3736 .0000 2600 .0002 2502 .0006 2513 .0012 2550 .0020 2637 .0030 2306 .0042 3096 .0066 3657 .0072 4245 .0090 5226 .0110 6670 .0132 8361 .0157 0639 .0183.3653 .0211 7360 .0242 1927 .0274 7479 .0309 4161 .0346 2087 .0385 1441 .0426 2377 .0469 5067 .0514 9696 .0662 6457 .0612 5555 .0664 7207 .0719 1638 .0775 9087 .0834 9805 .0396 4064 .0960 2109 .1026 4257 .1095 0802 .1166 2057 .1239 8363 .1316 0032 .1394 7455 .1476 0997 .1560 1048 .1646 8017 .1736 2331 .1828 4432 .1923 4784 .2021 3869 .2122 2191 .2226 0272 .2332 8661 .2442 7925 .2556 8669 .2672 1481 .0000 3600 .0002 6602 .0006 7615 .0012 9666 .0021 1749 .0031 3928 .0043 6234 .0057 8715 .0074 1429 .0092 4442 .0112 7827 .0135 1668 .0159 6055 .0186 1088 .0214 6876 .0245 3536 .0273 1193 .0312 9986 .0360 0066 .0389 1561 .0430 4664 .0473 9540 .0519 6373 .0567 5353 .0617 6702 ,0670 0621 .0724 7343 .0781 7107 .0841 0166 .0902 6783 .0966 7234 .1033 1809 .1102 0810 .1173 4564 .1247 3373 .1323 7610 .1402 7628 .1434 3302 .1668 6527 .1666 6211 .1745 3283 .1837 8188 .1933 1391 .2031 3377 .2132 4661 .2236 6738 .2343 7189 .2453 9673 .2667 3487 .2683 9607 .0000 4900 ,0002 8903 .0007 2918 ,0013 6963 .0022 1063 .0032 6262 .0044 9573 .0059 4075 .0076 8815 .0094 3862 .0114 9289 .0137 6181 .0162 1628 .0188 8731 .0217 6601 .0243 6364 .0281 5119 .0316 6032 .0353 8239 .0393 1396 .0434 7169 .0478 4232 .0524 3271 .0572 4483 .0622 8074 .0675 4263 .0730 3278 .0787 6361 .0847 0763 .0908 9761 .0973 2601 .1039 9606 ,1109 1066 ,1180 7303 ,1254 8647 ,1331 5446 .1410 8062 .1492 6873 .1577 2275 .1664 4679 ,1764 4514 .1347 2227 .1942 8287 .2041 3178 .2142 7409 .2247 1508 .2354 6025 .2466 1535 ,2578 8636 .2696 7952 8 ,0000 6400 .0003 2403 .0007 8420 .0014 4470 .0023 0577 .0033 6778 .0046 3114 .0060 9637 .0077 6405 .0096 3486 .0117 0966 .0139 8899 .0164 7407 .0191 6582 .0220 6534 .0261 7383 .0284 9256 .0320 2292 .0367 6637 .0397 2448 .0438 9892 .0482 9144 .0529 0391 .0577 3831 .0627 9673 .0680 3133 .0735 9444 .0793 3847 .0853 1596 .0915 2967 ,0979 8210 .1046 7646 ,1116 1569 ,1188 0302 .1262 4175 .1339 3539 ,1418 8768 .1501 0210 .1685 8293 .1673 3421 .1763 6023 .1866 6550 .1962 5470 .2051 3272 .2163 0466 .2257 7681 .2365 5172 .2476 3813 .2690 4107 .2707 6680 9 .0000 8100 .0003 6104 .0008 4124 .0015 2177 .0024 0292 .0034 8504 .0047 6867 .0062 6402 .0079 4198 .0098 3315 .0119 2828 .0142 2823 .0167 3393 .0194 4640 .0223 6677 .0264 9619 .0288 3606 .0323 8765 .0361 6260 .0401 3216 .0443 2832 .0487 4275 .0533 7733 .0582 3404 .0633 1497 .0636 2232 .0741 6842 .0799 2567 .0859 2665 .0921 6403 .0936 4060 .1063 5930 .1123 2321 .1195 3662 .1269 9959 .1347 1891 .1426 9716 .1509 3313 .1694 4582 .1682 2437 .1772 7311 .1866 1156 .1962 2942 .2061 3661 .2163 3822 .2268 3960 .2376 4629 .2487 6409 .2601 9902 .2719 5737 DIELECTRIC CONSTANTS AND POWER FACTORS 119 The above tlieory applies to the quarter wave line. This is a rather difficult practical one; it is best to add anotlier quarter wave to make a half- wave resonator, shorted at both ends, with the dielectric positioned exactly in the center. From conditions of symmetry we then employ the above equations, taking half of our measured quantities. Or, in terms of the actually measured four lengths which constitute an observation on a half -wave line, (d-d'), f, M and X, we have. sinli - (d — d') P.P. = sin ^ (A^* -\- t) A sin2X 2.Y — •tan TT A M + t = X tan X (15) which are the expressions used in this work. In practice the dielectric plug is pushed into the half-wave line and the line is tuned. The line center is then calculated and the plug reset to this. Retuning checks the correct location. Two trials are always sufficient if the plug was nearly centered originally. There are several shortcomings affecting this theor\'. The Q of the un- loaded line depends partly on' metal power loss along the line. When the line is shortened by the dielectric plug, part of this loss disappears and part is transferred to the dielectric plug. Fortunately these losses are small since they are metal losses at a current node, but for long dielectric plugs or plugs of high dielectric constant the need for correction can arise. The necessary calculations have not yet been reduced to a simple form. Again, the calculation of half-wave results by means of a quarter wave theor\' is safe only for a high Q situation. It is easy to show, experimentally, that the maximum line shortening results when the dielectric plug is exactly centered in the line but the calculated power factor is not a maximum here, as might be expected. In the meantime, experience shows that results can be duplicated from day to day and at other frequencies and that over a reasonable range of plug thickness no change in dielectric constant and power factor values, greater than the unavoidable errors of measurement, is ob- tained. Description of Apparatus The apparatus can be divided into three parts for purposes of description. The high frequency generator consists of a small ''relay rack" assembly. DIELECTRIC CONSTANTS AND POWER FACTORS 119 The above tlieory applies to the quarter wave line. This is a rather difficult practical one; it is best to add another quarter wave to make a half- wave resonator, shorted at both ends, with the dielectric positioned exactly in the center. From conditions of symmetry we then employ the above equations, taking half of our measured quantities. Or, in terms of the actually measured four lengths which constitute an observation on a half -wave line, (d-d'), /, AC and X, we have, sinli - (d — d') P.F. = sin ^ (Af -\- t) A sin2X 2X — •tan TT A M + i = A^ tan X e = ~X' 2 _x_ ^ (15) which are the expressions used in this work. In practice the dielectric plug is pushed into the half-wave Hne and the line is tuned. The line center is then calculated and the plug reset to this. Retuning checks the correct location. Two trials are always sufficient if the plug was nearly centered originally. There are several shortcomings affecting this theor\'. The Q of the un- loaded line depends partly on metal power loss along the line. When the line is shortened by the dielectric plug, part of this loss disappears and part is transferred to the dielectric plug. Fortunately these losses are small since they are metal losses at a current node, but for long dielectric plugs or plugs of high dielectric constant the need for correction can arise. The necessary calculations have not yet been reduced to a simple form. Again, the calculation of half-wave results by means of a quarter wave theon.' is safe only for a high Q situation. It is easy to show, experimentally, that the maximum line shortening results when the dielectric plug is exactly centered in the line but the calculated power factor is not a maximum here, as might be expected. In the meantime, experience shows that results can be duplicated from day to day and at other frequencies and that over a reasonable range of plug thickness no change in dielectric constant and power factor values, greater than the unavoidable errors of measurement, is ob- tained. Description of Apparatus The apparatus can be divided into three parts for purposes of description. The high frequency generator consists of a small "relay rack" assembly. 120 BELL SYSTEM TECHNICAL JOURNAL including 60-cycle power panel, rectifier panel, meter and control panel and centimeter wave oscillator panel with coaxial conductor output jack. All high-frequency connectors are coaxial conductor units with plug tips. The measuring unit is shown in the two photographs; Fig. 1, assembled and Fig. 2, disassembled. Two combination input-output heads are shown in Fig. 2. These heads and tubing together with center conductor and plun- ger are of coin silver. While the highest possible conductivity metal is desirable, pure silver is mechanically too poor for spring fingers and bearing surfaces and the alloy must be used. The good sliding contact properties of silver are preserved but the conductivity is no better than that of copper. Both heads are drilled, for input and output connections, flush with the bottom of the cylindrical cavity terminating the tubing. Head ^1, shown attached in Fig. 1 and detached in lower right-hand corner of Fig. 2, has a silicon crystal, mounted and insulated in a small cylindrical holder which carries a tiny pickup loop, one side of which is grounded to the cylinder. The total length of pickup conductor including loop and crystal "whisker" is about one centimeter and no tuning is neces- sary. The loop pickup can be adjusted by moving the holder in or out. The d-c circuit is from an insulated pin on the holder through crystal to apparatus body. The current input connection is through a coaxial plug which is tapped across a fraction of a tunable half-wave line. This fraction consists of a Y coaxial conductor terminated in a tiny feed loop; the remainder of the line is an ordinary \" coaxial with sliding plunger. The line is used, well off tune, as an input current amplitude control. The coupling with the cavity in head is adjusted by moving the feed loop in or out. By inverting another half-wave coaxial with feed loop, so as to put the crystal where the feed jack was, it is possible to use an externally mounted crystal as in head ^ 2. For this head the input current ampUtude control is obtained by using, as a feeder, a short \" coaxial tipped with a tiny loop and a coaxial jack, at opposite ends. This coaxial is mounted in a spring clamped bearing so as to permit a rotation of the plane of the loop. All coaxials, except the measuring unit itself, are 72-ohm ones. There is no essential difference in operation between these two heads; they are interchangeable. However, head ^ 1 is more convenient in ma- nipulation, during the disassembly required to insert the dielectric sample. (This sample is always positioned in the piece of tubing connecting to the head.) An ordinary model 301 microammeter, low resistance, served as indicat- ing instrument. By replacing the cr\'stal holder of head ;^ 2 with a loop tipped coaxial and plug, a conventional double-detection radio receiver with DIELECTRIC CONSTANTS AND POWER FACTORS 121 bo 122 BELL SYSTEM TECHNICAL JOURNAL output meter could be used instead. The cr>'stal type detector is by far the most convenient but with the power available wouldn't give workable outputs when bad dielectrics were to be measured. With the amplification available in the double detection set, any dielectric could be measured, while retaining the necessary attenuation between generator-resonator and resonator-receiver to keep these elements electrically independent of each other. It is necessan.^ to maintain an electrical isolation of this sort to get a high apparatus Q. The equivalent Q of all good dielectrics being high, the measuring apparatus Q must be of the same order to give favorable meas- uring conditions. And, further, unless the generator-resonator couphng is weak, the act of var>'ing the resonator tune will drag the generator fre- quency around and will also vary the generator output amplitude. The crystal plus microammeter required something like 80 millivolts for full scale deflection and this could be obtained with the present apparatus with coupUngs giving a resonator Q of 1500, while having enough power in reserve to measure any of the good dielectrics. However, most of the dielec- trics with power factor greater than .01 were measured with the d.d. re- ceiver. All the 10 cm wave-length measurements were made with this re- ceiver. For the latter measurements a shorter tube was substituted for the tubes shown screwed into the two heads in the disassembly photo. The cr}-stals were calibrated at 60 cycles by means of a 70-ohm y/l attenuation pad." With full scale deflection this pad was introduced and the new scale deflection read. This \/2 ratio was, as far as was possible to check, maintained in the kilo megacycle range. For calibration the cr}'stal was tapped across 4 ohms in the attenuator pad output. A 15 mf electroly- tic condenser was permanently connected across the meter terminals and, by means of a pair of switches, calibration could be checked in a few seconds, during a measurement run. The calibration process, using the d.d. set, was to adjust the output to a convenient meter deflection and then calibrate the meter by throwing in 3 db in the IF attenuator. The resonator itself constitutes an accurate wave meter when corrected for the change in diameter at the moving plunger. The method of operation was then as follows. The plunger vernier, which allowed reading to 0.01 cm., was set at the desired wave-length. The osciUator was then turned on and after it had attained temperature equilibrium, was adjusted if necessary to resonance at this value. This adjustment was infrequently necessary- and always sUght. The apparatus Q was then determined by traversing the plimger across the resonance setting by means of the micrometer. This 2 Exact, not 3 db. DIELECTRIC CONSTANTS AND POWER FACTORS 123 U^ 124 BELL SYSTEM TECHNICAL JOURNAL "mike" read to the ten-thousandth of an inch and could be estimated to one-fifth of this. Initially, by means of the amplitude control, the micro- ammeter deflection had been adjusted to the desired scale value at the reso- nance point. The traverse was observed between the two -v/2 microam- meter deflections and was repeated in the opposite direction. When successive round trips showed consistency the value of d' was noted. The dielectric sample, after thickness measurement, was then introduced, cen- tered by cut and try and the Q traverses repeated. This gave d and, after noting At, the change in plunger setting for resonance, the measure- ment was complete. During the measurement the generator had to be protected from drafts and, usually, it was necessary to traverse rapidly, the power line voltage not being stable. Settings could usually be reproduced to 1 per cent, with ade- quate care. A sample observation on a good dielectric is the following: July 28, 1941 Polystyrene plate, all dimensions in cms. / = 1.28 d' = .0084 X = 22.42 At = 1.79 d = .010 P.F. = .00028, e = 2.49 The dielectric samples were machined on a precision lathe, dimensions being held to .001 inch. The nominal dimensions were O.D. .640 inch, I.D. .174 inch. A favorable thickness, from the standpoint of ease of measure- X I in em's. Cleanliness in handling was carefully observed. ment, is lOe After a lapse of several days the interior bearing surfaces of the resonance cavity would have to be cleaned with fine French crocus cloth. The plun- ger bearing surfaces also had to be smoothed up, fine scratches being polished off. Dirt was immediately noticeable when the plunger contacted it, and when microscopic bits of silver were rolled up under the plunger springs cleaning was necessary. Othenvise no particular treatment or smoothing up of the contacting surfaces was required. A table of dielectric power factors and constants is a very desirable piece of information. Unfortunately, experience tends to the conclusion that such a table does not exist. The organic plastics in particular, are rather variable from sample to sample and a table of values is merely a table for particular specimens. Where a great number of samples are available "best", "worst" and "most common" values can be established. The accompanying list of observed values must be interpreted in the light of the above statements. As a large number of measurements of certain special materials had to be made, dielectrics in general were rather neglected and the tabulated values are more or less incidental. It was noted that for the low loss, sub- TABLE 1 Material Ceramic BTL F3 Mg. Silicate type "Dielectene" Glass, Corning Gl,lead G8, lime, annealed G12, lead 199-1 702EJ, Pyrex 702P 704EO 705BA 707DG Glyptal Lucite Mycalex Red White Phenolics Cast specimen Bakelite sheet ^" Polyethylene Worst Most common Best Polystyrene VVorst Most common Best Poly vinylcarbazole Rubber Hard, brown Hard, black Soft, black Resin Styralloy Xo. 10 Desig. Unknown Xo. 22 Stvramic E1689 Tenite H Mnylite V Wax Paraffin Boler Superla 5.83 4.30 6.38 6.08 8.70 6.35 4.70 4.42 3.80 4.69 3.38 2.58 5.91 5.74 2.26 2.45 2.87 3.15 2.32 2.49 2.49 2.40 2.78 2.17 2.17 2.26 10 cms. 3.39 4.8 3.36 2.56 4.63 3.57 2.77 2.69 2.50 2.55 2.95 2.61 2.26 P.F. 22.5 cms .00023 .0049 .0102 .0035 .0019 .0067 .0053 .0033 .0011? .0037 .030 .0090 .0030 .0033 .00229 .00060 .00031 .00090 .00070 .00028 .0040 .0058 .0018 .0036 .0019 .0047 .0076 .00019 .00019 .00019 10 cms. .0038 .0036 .036 .0087 .139 .080 .0041 .0059 .00105 .00087 .031 .0068 .00015 125 126 BELL SYSTEM TECHNICAL JOURNAL stituted paraffin-type, carbon chain dielectrics no difference, greater than experimental error, exists between the 22.5 and 10 cm. measurements. Acknowledgement The measurements by means of the double-detection set were made by my co-worker, Mr. W. E. Eckner, whose valuable assistance I am glad to acknowledge. To Mr. C. F. Mattke, also of the Bell Laboratories, I am indebted for assistance in getting my crude original apparatus into its final finished form. APPENDIX The typical ultra high-frequency transmission line can be represented as in Fig. 3 (Eo.io) (Ex.ix) (Ei,Ii) Fig. 3 — Equivalent circuit of transmission line and the equations describing it are Z^ cosh ^ DZ ^l - X) ^ \/ £x = Fo Z^ cosh ^DZ (^ - x) + 4/^ sinh ■\/ DZ {I - x) Fo (Zo + Z^) cosh ^DZ I + (^oZ^i/l + J^A sinh VdZ I cosh ^m. {I - x) -\r ^lA/^ sinh ^DZ {t - x) (Zo + Zi) cosh ^/dZ I + f ZoZ^i/| + |/| j sinh ^DZ i X ~7~ — £. /z ^^ "^ ^/\ ^^^ ^^^ ^^ ~ ^^ J^- -f Z( tanh ^DZ {t - x) Ep — Zpli (1) £0 = Fo — ZqIo , -c\ A/ —\ and Ra<^\ A/ -^\- The condition of reasonable short- * \ y D\ \ y D\ ening of the line (or lengthening) by the terminal reactance implies that ^f^\]/W'^'^ ]/l)\- Hence weshallhave |Z^| »|y^||, I ^0 1 « I y^ Fo tanh {VdZ ^ + 6) If we put -?-- — = tanh 6, we get (4) J\ 1 + ~^tanh (V£>Z I + ^) We now make the assumption that "Zo" is a pure resistance (which is no limitation on the measurement to be discussed) and put ^ = a^ + ihi . 128 Then, BELL SYSTEM TECHNICAL JOURNAL y / tanh^ {al + a^) + tan f^ + bA tanh {af + a^) + tanh (aC + fl^) (5) ex^^') This expression cycles, as "(" is varied, and has its maximum or "tuned" value of V, /l/1 + tanh {al + a^) (¥ + *^ for tan ( — - + J/ j = <» or 27r^ , - (2w + l)7r ^ . ^ -T- + J/ = -^ TT-^ « = 0, 1, 2, • . • The resonant length is thus ^ = t ( 1 ~ (-^-^) for n = 0. Note that successive resonances differ b.v a line length of -; the reactive termination has merely- shortened, by the amount of M = -—^, the first resonant element preceding ZTT it. WTien, therefore, we measure the "Q" of this line segment by line- length tuning we use ^ = - in the Q process definition. The Q process now follows. Putting ^ = 4 =fc 5^ where 4 is the actual observed resonance length, we have 27r^ ^. 6, = ?^- + ft, d= lirSC = ^ ± lirSe Then tan f _ + *,j = tan (2 ± "JT j = ^^ l.U ^"^ DIELECTRIC CONSTANTS AND POWER FACTORS 129 /o V. 1 + tanh" {a€ + af) • tan IttSC V I / r ^0 ,1 rj + tanh {aC + flf) Ye 2 + Forming the current values | /m | = | /02 1 = 1 + — -Tj tanh {at + ' with low L/C ratios to get oscillations but an improvement in Q alone may give poorer stability. To get the fullest measure of stability with low L/C and high Q calls for slight adjustments in the circuit and possibly the provision of loose coupling to the frequency-determining circuit. Modern Spedrochemical Analysis.^ Edwin K. Jaycox. The spectro- graph, originally developed by the physicist, has become a most useful tool in the hands of the analytical chemist. Today few large analytical labora- tories are without one. The instrument, with its attendant accessories, provides a rapid method for analyzing metals, alloys, minerals, ores, liquids, and gases, particularly for their metallic constituents and in some cases for their anions. Both emission and absorption spectra are important to the analyst. Important applications of the spectrograph to the analytical problems of research and industrial organizations are discussed. The spectrograph did not come into general use as an analytical tool until the early 1920's, although Kirchhof and Bunsen saw the practicability of the method in 1860, when they published their paper entitled, "Chemical Anal- ysis by Means of Spectral Observations." During the intervening years only a few enthusiasts like Lockyer, Roberts, Hartley, Leonard, Pollack, and de Gramont, kept the art alive. In spite of their persistent efforts to influence chemists to use spectrographic methods, they were quite generally ridiculed and the value of the method was recognized by only a few workers. In 1922, Meggers, Kiess, and Stimson published their paper "Practical Spectrographic Analysis" and modern spectrochemical analysis was born. Under the stimulus of this paper and the backing of a high caliber scientific organization like the Bureau of Standards, the use of the spectrograph as an analytical tool increased rapidly. This is evidenced from the Index to the Literature on Spectrochemical Analysis by Meggers and Scribner. In 1920, for example, only five papers were published concerning spectrochemical analysis, four of which were by de Gramont; whereas in 1930, 33 papers were published and in 1939, 170 papers, indicating an increasing interest in and use of spectrochemical analysis in industrial and research organizations. ^Jour. Applied Physics, December 1943. 132 BELL SYSTEM TECHNICAL JOURNAL Determination of Small Amounts of Arsenic, Antimony, and Tin in Lead and Lead Alloys.^ C. L. Luke. A new method for the determination of small amounts of arsenic, antimony, and tin in lead and lead alloys consists of separation of the three metals from the lead by a double co-precipitation with manganese dioxide, reduction of arsenic and antimom^ to the trivalent state, separation of the arsenic by distillation as chloride, titration of the arsenic and antimony separately by the method of Gyory, and reduction of tin with lead and titration with standard iodine solution. Determination of Total Sulfur in Rubber} C. L. Luke. A new volumetric method has been developed for the determination of sulfate sulfur. The sulfate is reduced to sulfide by treatment with hydriodic acid and the h}-dro- gen sulfide is distilled oflf and titrated iodometrically. The new method has been applied to the determination of total sulfur in natural and synthetic rubber. Machine Screws. Fastening Strengths in Various Materials.^ A. C. Millard. Although standard machine screws in the numbered sizes have been widely used as fastenings for many years, very little has been published concerning their strength of fastening in various metals and non-metals. Numerous articles have appeared regardmg the strength of bolts and ma- chine screws for j in. and larger sizes, but very little, if any, published in- formation is available on the strength of machine-screw fastenings in the numbered sizes. The need for machine-screw fastening-strength information has increased recently due to the use of more compact designs and the shortage of mate- rials. The use of substitute materials has accentuated the lack of machine- fastening-strength information in making fastenings in such materials, as well as in the more commonly used materials. Frequently, it is desirable to know the load-carr>ang capacity of screw fastenings of various diameters, as well as the length of thread engagement in the weaker materials needed to develop either the full strength of the screw, or the strength of fastening required of the assembly. The purpose of this paper is to make available to designers the results of fastening-strength tests of machine-screw fasten- ings in a number of materials, which were carried out by the author at the Bell Telephone Laboratories, Inc. The work is by no means complete but is hoped that the data offered will prove to be of some use in its present form. ^ Indus. & Engg. Chemistry, October 1943. ^ Indus. & Engg. Chemistry, September 1943. « Mech. Engg. October 1943. Contributors to this Issue W. R. Bennett, B.S., Oregon State College, 1925; A.M., Columbia Uni- versity, 1928. Bell Telephone Laboratories, 1925-. Mr. Bennett has been engaged in the study of the electrical transmission problems of com- munication. Carl R. Englund, B.S. in Chemical Engineering, University of South Dakota, 1909; University of Chicago, 1910-12; Professor of Physics and Geolog>', Western Maryland College, 1912-13; Laboratory Assistant, Uni- versity of ]Michigan, 1913-14. Western Electric Company, 1914-25; Bell Telephone Laboratories, 1925-. As radio research engineer Mr. Englund is engaged largeh' in experimental work in radio communication. D. K. Gannett, B.S. in engineering. University of Minnesota, 1916; E.E. University of Minnesota, 1917. American Telephone and Telegraph Com- pany, Engineering Department, 1917-1919; Department of Development and Research, 1919-1934; Bell Telephone Laboratories, Inc. 1934-. Prior to October 1942 Mr. Gannett, as Toll Transmission Engineer, was concerned with the transmission requirements of toll systems including program cir- cuits. Since then, as Circuit Research Engineer, he has directed a group engaged in research and development on war projects. Iden Kerney, B.S. Harvard University, 1923. American Telephone and Telegraph Company, 1923-1934; Bell Telephone Laboratories, Inc. 1934-. Before the war Mr. Kerney was in charge of the laboratory in which experi- mental work on program transmission was conducted. Since early in 1942 he has been engaged full time on war projects. R. A. Sykes, Massachusetts Institute of Technology', B.S. 1929; M.S. 1930. Columbia L^niversity, 1931-1933. Bell Telephone Laboratories, Research Department, 1930-. Mr. Sykes has been engaged in the applica- tions of quartz crystals to broad-band carrier systems as filter and oscillator elements. Other work has included the application of coaxial lines as ele- ments of filter networks and more recently the design and development of quartz crystals for radio frequency oscillators. G. W. WiLLARD, B.A., University of Minnesota, 1924; M.A., 1928; In- structor in Physics, University of Kansas, 1927-28; Student and Assistant, University of Chicago, 1928-30. Bell Telephone Laboratories, 1930-. Mr. Willard's work has had to do with special problems in piezo-electric crystals. 133 VOLUME XXIII APRIL, 1944 NUMBER 2 ' THE BELL SYSTEM TECHNICAL JOURNAL DEVOTED TOi^THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION Indicial Response of Telephone Receivers . E. E. Moit 135 Theoretical Analysis of Modes of Vibration for Isotropic Rectangular Plates Having All Surfaces Free —H. J. McSkimin 151 Principles of Mounting Quartz Plates . . R. A. Sykes 178 The Magnetically Focused Radial Beam Vacuum Tube —A. M. Skelleit 190 Abstracts of Technical Articles by Bell System Authors 203 Contributors to this Issue 206 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 50c 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 F. B. Jewett O. E. Buckley S. Bracken EDITORIAL BOARD M. R. Sullivan A. B. Clark M. J. KeUy O. B. Blackwell H. S. Osborne F. A. Cowan »»««»■»■«■■ 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, 1944 American Telephone and Telegraph Company PRINTED IN U. S A. The Bell System Technical Journal Vol. XXI H April, 1944 No. 2 Indicial Response of Telephone Receivers Ey E. E. MOTT A method of analyzing telephone receiver characteristics by indicial response is discussed and illustrated by oscillograms. The indicial response of a telephone receiver is the instantaneous response of the receiver to a suddenly applied electro- motive force. This type of response is of particular fundamental interest because it furnishes a key to the solution of transient problems such as are involved in the response to speech waves. Oscillograms of indicial response, together with the more familiar steady-state frequency response characteristics, are shown for different types of receivers. The relationships existing between the two types of measurements are discussed. From the standpoint of most faithfully reproducing transients, indicial response data indicate that a receiver having a limited range of frequency response should have a frequency response characteristic which droops gradually rather than abruptly near the upper end of the range. Introduction THE use of indicial response analysis as an outgrowth of the Heaviside operational calculus' has been extended to a number of different fields. The indicial admittance as defined by J. R. Carson^ in his analysis of the sub- marine cable and other transmission problems has been an efifective tool in the study of transients. More recently, a similar type of measurement has been used as an indication of performance of amplifiers^, television equip- ment\ and audio frequency transformers^. In the field of telephone receivers^ an analysis by means of impressed square waves has been found useful as a measure of transient response. In the transmission of speech, so much emphasis has been placed upon steady- state frequency response as an indication of performance, that it seems in order to consider the possible advantages of a transient method of analysis, as obtained by measuring the indicial response. Only recently has the technique of such measurement been made feasible by the improvement at low frequencies of amplifiers and related apparatus. The Indicial Response The indicial response of a telephone receiver may be defined as the in- stantaneous sound pressure generated by the receiver in a closed air chamber due to a suddenly-applied unit voltage. This term differs from Carson's indicial admittance only in that sound pressure rather than current response is used. The sound pressure in an air chamber of pure stiffness is a measure 135 136 BELL SYSTEM TECHNICAL JOURNAL of the volume displacement, and as such it is proportional to the transfer displacement admittance of the system. W^en we are interested in the charge rather than in the current, the admittance takes the form of a dis- placement admittance, related to the ordinary admittance by a factor of the frequency co. That Carson's original equations apply to such a system with little if any change may be easily demonstrated. The term A{t) may be used to denote any of these forms of indicial admittance or indicia! response. The form of the applied voltage assumed is shown by Fig. 1. This form, defined by Heaviside as the unit function, is a function of time equal to zero before, and unity after the time t = 0. More properly, however, it may be regarded as an increment in voltage closely analogous to Isaac Newton's concept of infinitesimal elements of rectangular area, the summation of which forms the basis of the integral calculus. The successive application of small increments of voltage Hkewise forms the basis of the operational calculus, or more particularly, the basis of the Carson extension theorem. TIME AXIS THE UNIT FUNCTION Fig. 1 The Carson Extension Theorem Having obtained the indicial response, either experimentally or theoreti- cally, we have the key to the more general problem where the applied voltage e{t) may be of any form, such as that of speech waves. Let e(/). Fig. 2, be any arbitrary voltage wave corresponding to speech^. Let a series of con- secutive increments of voltage, differing in time by Ar be applied, of such magnitude as to build up the form of the curve e{t). By analyzing each of these components in terms of the indicial admittance A{i), and synthesizing them again, the instantaneous sound pressure may be related to the voltage producing it and the indicial admittance A{r) by the Carson extension equation^: Pit) =jj A(T)e(t-T)dT I When the above integration is carried out, the term t disappears and is replaced by /. The above sound pressure p(t) represents the sound pressure generated by the receiver in a closed coupler due to an applied voltage e(t). INDICIAL RESPONSE OF TELEPHONE RECEIVERS 137 ^eCT)AT Fig. 2 — Method of derivation of Carson's extension formula. 1,000 2,000 TIME (micro -seconds) Fig. 3 — Indicial admittance of two types of telephone receivers. 3,000 138 BELL SYSTEM TECHNICAL JOURNAL From the above, it is evident that the ideal form of receiver response to a suddenly-impressed voltage would be a copy of the unit function shown in Fig. 1, and that any deviation from this form will cause distortion. If the building blocks of the curve e{l) are undistorted, the curve itself will like- wise be reproduced free from distortion of wave form. Thus, the more closely the indicial response can be made to approach the form of the unit function, the more closely the receiver sound pressure p{t) will be a copy of any arbitrary speech wave e{t). Curve 1, Fig. 3, shows the indicial response of a receiver having a frequency range of 8000 cps, which comes rather close to this ideal. On the other hand, the further the indicial response departs from this ideal form, the more it will deviate from any impressed transient, such as speech waves. Thus curve 2, Fig. 3, corresponds to a receiver of narrow range, which contains resonant oscillations, and rises much later in time than the other receiver. Conversion Formulae The indicial response is as fundamental in character as frequency response, and may be converted into frequency and phase response if the proper in- tegrations are carried out for any particular system, as follows: Indicial Response A{t) ^ [Frequency Rcsponsel ^ [_ -j- Phase Response J j t. ^ / where A (w) is the transfer admittance of the system. In order to carry out these conversions, certain integrations must be performed, either mechani- cally or theoretically. The following are conversions^ which may be used to carry out this process: TT Jo sm Oil aui 0 a; A{t) = P(0) -{-- f 9^ COS oitdoi T Jo OJ ^ = f AiOsinuitdi 0) Jo QM = I [A{1) - Am^osoitdt CO Jo Where P(a;) and Q{(xi) are the real and imaginary parts of the frequency reponse, A (co) is expressed in terms of pressure reponse*, while the indicial response A {t) is expressed as an instantaneous sound pressure. The integra- tions are difficult to carry out, but serve to show how the two systems of INDICIAL RESPONSE OF TELEPHONE RECEIVERS 139 measurement are related, and how they may theoretically be converted one into the other, provided in the case of frequency response the magnitude and phase are both known. General Applications The use of indicial response as a tool in telephone receiver studies is par- ticularly adapted to the study of transients. Since all voice and sound trans- mission, particularly that of orchestra! music, may be regarded as essentially a transient problem, it is appropriate that we visuaUze the effects on the complex wave forms of any distortions which may be present in the trans- mission apparatus. The indicial response will, in general, depart from the ideal square form, and the amount of this departure may be regarded as indicative of the relative faithfulness of wave form reproduction by ap- paratus having different frequency characteristics. An examination of these departures should therefore be helpful as a supplementary method of appraising the relative merits of different frequency response characteristics. The effect, for example, of small resonance peaks or dips upon transients is very forcefully shown in the form of the indicial admittance. The de- parture from squareness of a particular system may often be improved by use of the proper shape of frequency characteristic. The use of a closed coupler when measuring telephone receivers is par- ticularly adapted for such studies, because the disturbing effects of de- ficiencies at the low frequencies due to leakage may thus be eliminated. Interpretation by inspection then becomes a matter of observation of the various types of departures at the higher frequencies from the ideal form. Since listening tests do not always agree with interpretations of physical measurements of steady-state frequency response, it often becomes a matter of interest to obtain different criteria of judgment in which the weight given to the various frequencies may be judged by the relative effects of irregulari- ties in various parts of the frequency spectrum upon the indicial response. Apparatus and Method of Testing Various forms of apparatus may be used for receiver testing with square waves. Square-wave generator circuits have been published both for audio^ and video' frequency use, involving vacuum tube circuits which overload at low voltages. For low speeds using low-frequency waves of the order 60 cps, a simple mercury switch operated by an oscillator gives very satis- factory results. The square-wave voltage is introduced across a small part of the resistance termination as shown in Fig. 4, the whole resistance termination being matched to the magnitude of the receiver impedance at 800 cps. The re- 140 BELL SYSTEM TECHNICAL JOURNAL ceiver is then operating from an idealized resistance source having an im- pedance which matches that of the receiver approximately, over the range of interest. The receiver is coupled acoustically to a small-diameter condenser micro- phone by means of a closed coupler^. The condenser microphone has a substantially uniform characteristic up to a frequency of 10 kc. The SQUARE WAVE GENERATOR •TEST RECEIVER STRING OSCILLOGRAPH CONDENSER MICROPHONE -6 CO. COUPLER Fig. 4 — Circuit diagram of apparatus for indicial response measurements. 10,000 (B) Fig. 5 — Frecjuency response (A) and indicial response (B) of measuring apparatus. microphone voltage is then amplified to the point where it can be measured by an oscillograph. Either the cathode-ray oscilloscope or a rapid-recording string oscillo- graph^ may be used, but in the latter case it is necessary to equalize the string oscillograph to a frequency of about 10 kc in order to cover the audio frequency range. The choice of these instruments depends somewhat upon whether a permanent record is desired or whether a visual indication is sufficient. INDICIA L RESPONSE OF TELEPHONE RECEIVERS 141 The amplifier must be compensated at low frequencies in order to main- tain a strictly square-wave output. The entire system characteristic is shown in Fig. 5 and covers a range of 1 to 10,000 cps with a substantially uniform frequency response. The indicial response of the system is also shown to be reasonably free from irregularities. Such irregularities as do exist are due largely to the sharp cut-off of the s\'stem at 10 kc which was necessitated by the limitations of the string oscillograph. Indicial vs. Frequency Response The calculated pairs of curves for telephone receivers in Fig. 6 show the relations between the frequency response and the indicial response. Since the characteristics of receivers measured on a closed coupler of known volume are readily amenable to calculation if the constants of the receiver are known, such a procedure is often useful in predetermining the design of a receiver. The upper three curves, Fig. 6, are the characteristics of a moving coil receiver calculated for three different frequency ranges, being otherwise similar in shape, the curve being shifted in frequency by an arbitrary factor K. The effect on the indicial admittance is to shift it in time by the same factor without change of shape, if the plot is logarithmic as shown. In gen- eral, if the cut-off frequency is divided by the factor K, the corresponding time delay will be increased by the factor K. This is an application of a theorem by Carson'^ that: where p = jui is proportional to frequency, and / is the time, , , is the L{kp) frequency response, and A (t/k) is the indicial response. In other words, the curve may be shifted in frequency by a simple transformation and the effect on the indicial admittance curve is very similar except that the shift is in a direction opposite to the change in frequency, and is inversely proportional to the change in frequency scale. The second group of curves. Fig. 6, relates to the effect of damping on an early magnetic type of receiver, showing the freely resonant condition, a moderately damped, and a highly damped receiver. The curves of indicial response show the effects of free resonance to be very detrimental, and the ringing of the diaphragm is sustained over such a long period that any speech waves would have superposed on them a continual train of sine waves. If the rate of decay of these waves is increased, as shown by the damped curves, a noticeable improvement results. By using critical damping as in the highly damped curve, all oscillations can be eliminated, but the time of pickup is degraded and the departure from a square wave is somewhat greater than for the moderately damped condition. 142 BELL SYSTEM TECHNICAL JOURNAL imiciAL response: FREQUENCY RESPONSE CAI^CULATCD /P£C£VI^€/P CHAI?ACT£I?/STICS lOOCO (Mf^v-sec) / ^N^ / / ff — 1 — '—- ^^-U TJMe {M/C/Pa-S£C.) /\ A A A A« y N / V ' V /\ r \ ]\ \\ / ^ V s / 1 Ot «B TME (wen-sec.) FeequENcr CJ'.S. J i\ h" y ■"^ \ ' , \ ^ < r' ,u. ^ « / / / 1 Tff4E (WC/?0~S^) = E = = = = = 3 E b\ — ~ ^ - ^ - -■ — - 7 / A E E : = = E^ — = l'_ : / ^ /. ^ — = =; ^ -^ = = = = = - z i- ^q r:: = =: = : : ^ ^ / ^^ y " vo 77 \te I'l^ceo-SE -) ' XXX) I. INDICIAL KSPO/JSE MB hs FUEOuEtJcy /e^spOKse of A P£SIST/lfJC£ CcrjreOLLED OF A MASS CONTROLLED /?ECEIVEe. 3. iTIFFNESS COHTfiOLLeDy RECEivelf '°° TIK£( »'CA 0- SEC, lOOO _, ^ / / > 1 TtME fMKXO-SEc) FeeQuehjcr- cj?s. — ^ > "' \ S'- \ \ rREQUENCY- C.P.S. ^eeQcf£NCY-CPs. i Fe£Qf/£/^r-ap.s. FRSQUEf^r- C.PS Fig. 6 — Calculated indicial response versus calculated frequency response of various types of telephone receivers. INDICIA L RESPONSE OF TELEPHONE RECEIVERS 143 The indicial response shows more emphatically than frequency response, the importance of damping and the oscillations which arc to be avoided, or reduced to a minimum. It also shows that the effect of delay is closely re- lated to attenuation of the higher frequencies, and that frequency of cut-off is inversely proportional to the time delay, for a given t\pe of receiver circuit. There is a noticeable similarity between the appearance of the frequency response and the indicial response curves, and in many cases one curve is approximately the image of the other. As an cxamj)lc of this, the three pairs of linear curves show the similarity of indicial and frequency response for constant velocity, constant acceleration, and constant am.plitude devices, as depicted by the three curves denoted by 1, 2, and 3 in which the three moving-coil instruments are assumed to be controlled by (1) a predominance of acoustic resistance behind the diaphragm, (2) a mass controlled system, and (3) a stiffness controlled system. In either case, the fundamental shape of the curves is such that the indicial response is the image of the frequency response in its general character. The two lower curves. Fig. 6, indicate the effect of a sharp cut-off versus a gradual one. In terms of indicial response, the gradual cut-off appears to be the better of the two, a principle which is widely accepted in television and telegraph transmission. Experimental Measurements The oscillographic measurements of indicial response, together with cor- responding frequency response measurements of telephone receivers, are shown in Figs. 7, 8, and 9. The oscillograms on the left. Fig. 7, show the type of data which constitute indicial response as compared with the more familiar frequency response on the right. Curve 1, Fig. 7, represents a moving-coil receiver similar to that calculated in Fig. 3, and constitutes the standard of performance which can be obtained by this particular system of measurement. Each division of the oscillogram represents .001 second, a somewhat faster film speed than is usual for the string oscillograph. * Curve 2 shows the characteristics of a magnetic bipolar type of receiver having a frequency range of 3000 cps with a fairly sharp cut-off at this fre- quency. The acoustic circuits of this receiver serve to damp the resonance of the diaphragm and extend the range from 1600 up to 3000 cps. The oscillogram shows a partially damped but still somewhat oscillatory condi- tion which is due to the receiver. With all damping circuits removed, we obtain the characteristic of curve 3, a simple diaphragm resonance, which is similar to the earlier type of re- ceivers of the magnetic type. Curve 2 represents a real improvement over 144 BELL SYSTEM TECHNICAL JOURNAL curve 3j both as regards introduction of damping and extending the fre- quency range. INDICIAL RESPONSt . FREQUENCY RESPONSE M£AS>U/?£D RECEI^EfS CHAt?ACT£ISlSTICS miinifl! IIP™"" >s< liBSSRI!!!! UHI r.=Ef=====i: >s< r (MJcn-secoNos) n^QUENCr (cp!^ 1 il , Ul 1 Ttr "" I » ffcouef/crrc^j --LUi FSCQUENCr (Cf?S.) ftlEQUENCy (CPS.) ! ^. \ mcquENcy fc.Rs°) "1 ^ \, \ S iVE^eNcr (cesf y\ ^ "" -' ^ fn?E ?ME AO- (CPS r 1 1 ^ ~ - ' ^ " /=ee (Sat 'r (c^ 1 1 ^ " ^ ffiS out -JJ -rfii •m 1 1 !^ \ \ m -qu, re* \ tuM L FIfEQuefJCr (c. ffSi Fig. 7 — Measured indicial response versus measured frequency response of various types of telephone receivers and electrical filters. The effects of further increases in damping are shown by curves 4, 5, and 6. Such changes in the shape of the curve are brought about by relatively simple INDICIAL RESPONSE OF TELEPHONE RECEIVERS 145 changes of the constants of the acoustic circuits. The oscillograms indicate a marked improvement as regards oscillations, which is to be expected with increased damping. The time delay is eventually degraded with further increases of damping, however, and the optimum damping is a matter of compromise. •20 •h 15 + 10 + 5 0 9-10 5 L ''-15 -20 -25 -30 -35 -40 A /\ A FREQUENCY RESPONSEs^^ 'y^'v \ •^H ■^^-^ ^^ ^ ■\ A INDICIAL RESPONSE issBsm ] ' * ^ B ' \ \ 1 mtM IIV9 \ \ b 1 Br -i„ ^_,-l 100 1,000 FREQUENCY (C.RS.) Fig. 8 — Three types of hearing aid receivers — frequency response and indicial response The effects of a low-frequency cut-off characteristic are shown by curves 7,8, and 9, Fig. 7. The absence of a d-c component makes these curves very difficult of interpretation. Curve 7, taken with the same receiver as curve 2, except with coupler leakage, shows a loss at low frequencies which is typical of cases where the receiver cap does not make a perfect seal with the ear. The effect on the indicial response is that of a large pulse followed by a few oscillations at the frequency of the leak circuit. Curve 8 is a similar condition except taken on a high-quality receiver 10,000 146 BELL SYSTEM TECHNICAL JOURNAL circuit. This also shows a similar effect. The initial pulse contains most of the receiver characteristic, while the curve which follows is mainly de- pendent on the leakage constants. Curve 9 is taken on a high-pass filter of the characteristic shown. It may- be proved that this curve is the inverted image of the corresponding low-pass filter characteristic, of which a similar curve is shown as curve 10. +5 0 -5 D -10 -15 -20 1 ■^ S ^ S N > \ S 100 1000 IO.C FREQUENCY (c.P.S.) FREQUENCY RESPONSE 2 NDICIAL RESPONSE Fig. 9— String oscillograph characteristics — frequency response and indicial response with different amounts of damping. The curves 7, 8, and 9 show that when the low frequencies are absent, the indicial response becomxs too difficult to interpret. We must restrict our measurements to systems which are ideal at the low frequencies in order to interpret the indicial admittance by inspection. Curves 10 and 11, Fig. 7, are low-pass filter characteristics, the former being a measured curve of a typical filter, while the latter is a calculated curve for an ideal filter. The two curves check reasonably well and indicate the effect of a very sharp cutoff as compared to those of the receivers shown INDICIAL RESPONSE OF TELEPHONE RECEIVERS 147 above. This indicates the oscillatory nature of any system having a sharp cutoff at the upper frequencies. FREQUENCY RESPONSE OF TELEPHONE RECEIVERS 2 I T lI 'n\ L-Ui I , f\ ' 1 -\ i ' \ J_ . Jl L ^ III 1 1 J _ _ _ A- _ _ IJJj FRtputKCrCCP.S) rREflUCNCYCCRS) f REOUCNU (CPS) SQUARE WAVE RESPONSE '■ ^ 11 (VI «~- 1 ifl/io/^ 1 i L f\\^ no llf\A n n 11 \\J\ (1 n lu - y Ij fl n, i. 11 1 ^ w Vu (1 w ii[/v u TIME (SEC ) 50 WAVES PER SEC. jT r ~ JU \ \ " r 1 1 Ny ■Xy -x ( \y V r\ I KJ \j \ j A. \ L/ \/ r /\ r \y r V, f \, / f \ / \ 1 \ / 1 \ / \ J \ J \ Jiuuu A 1 1 A J 1 I / \ \ 1 V '1 \ V. _J M- om Of ," , u — TfMECSEC) 300 WAVES PER SEC. (^ ^ ^ ^ \ i 1 /\ u 1 \ A ^ ^ / \ r ^ ^ / \ / \ \ \ y^ f \ . / N, \ •J \/ V J \ / \ / / \ \ y / ^ / / / s^ s / ' / / \ / / TIME (src) 500 WAVES PER SEC. f \7 ~ ^ p \ ■ a 1 » i / + 1 1 1 / \ c\ / / N v 1 \ / \ § 1 s \ ^ s \ ^ 2 1 \ / ^ S / ' \ / ' \ i / / \ / y \ 1 \ / / g sooo \ y N / \ \ r \ / 3™: \ / \, / \ / s s / t«0 \ / \ / TIME (SEC) 1000 WAVES PER SEC. Fig. 10 — Transient response to square waves of three different types of telephone receivers denoted by Nos. 1, 2 and 3, whose frequency response characteristics are shown above. Note the change in each type of pattern as the frequency of the square waves is increased. 148 BELL SYSTEM TECHNICAL JOURNAL Figure 8 shows a group of curves of the frequency response and indicial re- sponse of a group of receivers used as hearing aids. Curve 1 shows a very efficient but resonant receiver. Curve 2 is somewhat damped but still contains oscillations. Curve 3 is comparatively much better than either of the others from an indicial response viewpoint, and has a drooping fre- quency response characteristic, and demonstrates the advantages of this form of curve. Figure 9 shows the effect of adding damping to the system of the string oscillograph when subjected to an ideal square wave. Curve 1, which has a virtually flat characteristic from 1 to 10,000 cps, is characterized by a sharp oscillatory peak in the indicial response. Curve 2 contains some oscil- lations, while curve 3 is substantially free from oscillations. The trend of these curves also shows the more faithful reproduction of transients obtained with a drooping frequency response. Figure 10 shows the response to square waves of three receivers having different frequency response characteristics. The low-frequency waves of 50 cps are similar to the indicial response of the three receivers whose frequency characteristics are shown at the top, Fig. 10. As the frequency of these waves is increased to 300 cps, a noticeable departure from the square form is apparent in receiver No. 3. Receiver No. 2 shows a slight departure, while No. 1 is virtually a perfect reproduction. As the frequency of the square waves is increased to 500 cps, the receiver No. 1 still shows very Httle departure from the original form. Receiver No. 2 maintains a fair approximation, while receiver No. 3 has lost all resem- blance to the square form. At a frequency of 1000 cps, only the first receiver maintains an approxi- mately square form. Receivers Nos. 2 and 3 have both lost their identity and have become practically pure sinusoids. For all higher frequencies of the square waves, these two receivers will exhibit practically pure sinusoidal forms, due to the relatively sloping character of the frequency response at these frequencies, and the absence of harmonics. The same will be true of receiver No. 1 beyond a frequency of 3000 cps. It will be realized, of course, that the patterns were obtained with square waves repeated at frequencies of 50, 300, 500 and 1000 cycles per second. While some speech waves approximate square waves in character such waves, when they occur, are repetitive only at the lower range of these fre- quencies. The above patterns were therefore obtained under conditions much more severe than are involved in the reproduction of speech waves and are included primarily for the purpose of illustrating the sensitivity of this form of analysis when applied to repeated square waves. IXDICIAL RESPONSE OF TELEPHONE RECEIVERS 149 Conclusions To summarize these data, it seems evident that square wave analysis may be applied in some fields of acoustics for both theoretical and practical applications. In theory, the indicial response forms a somewhat different approach to the problem of obtaining the optimum characteristics of telephone receivers at the upper end of the frequency range. The greatest value of the square wave analysis lies in the fact that it gives us an entirely different conception of the behavior of an ideal sound system in terms of the unit function. The frequency response characteristic is ordinarily interpreted on the theory that any transient, such as an interval of conversation, may be represented by a Fourier series of sinusoidal frequencies of constant intensity lasting over the entire interval. If these equivalent component frequencies are to be repro- duced in their true proportions, the ideal sound system must have mathe- matically uniform response for all single frequencies. On the other hand, the indicial response characteristic is judged from the Carson extension theorem, which shows that the more closely this characteristic approaches the unit function, the more perfect will be the reproduction of any given transient. Thus, the unit function and the sinusoid may be used as mutually complementary tools of analysis to show different aspects of the same type of problem. In sound systems which are not ideal, due to inherent physical limitations, we tend to apply the Fourier Theorem out to a certain frequency, just as if it were an ideal system out to this frequency, and then beyond this fre- quency we do not attempt to sustain the higher frequencies. For most faithful reproduction of transients, it would seem that such practices might be altered somewhat to advantage by allowing the frequency response to drop off more gradually wherever it seems feasible to do so. The exact shape of the ideal curve under these circumstances is a matter of compromise between excessive delay on the one hand and excessive oscillations on the other. In practice, however, a fairly good picture is soon formed when curves such as the last in Figs. 6, 8, and 9 are found to approach the ideal more closely than those of other forms. Such listening tests as have been made tend to confirm these views, but cannot be regarded as being more than an indication. Square wave analysis is somewhat limited in its practical applications to cases which may be interpreted by inspection. Systems having only a single cutoff frequency, or in the case of an additional low-end cutoff, ratios of the upper and lower cutoff frequencies /2//1 of 100 or more, seem necessary to interpret the results by inspection. ISO BELL SYSTEM TECHNICAL JOURNAL The use of indicial response is not necessarily limited to any particular coupler or method of response measurement, since frequency response and indicial response are so closely related that one is a function of the other. The choice of a closed coupler measurement does, however, permit some in- terpretation of the results to be made by inspection, whereas other types of measurement may require laborious mathematical means to obtain an in- terpretation. Other types of vibration instrum.ents, such as recorders, vibration pickups, crystal phonograph reproducers and carbon transmitters, which sustain their response down to zero frequency, should lend themselves to such methods of analysis. In conclusion, the writer wishes to acknowledge the assistance of Mr. T. J, Pope in connection with the oscillographic work of this paper, and to express his sincere appreciation. Bibliography 1. Oliver Heaviside, "Electromagnetic Theory." 2. J. R. Carson: a. "Transient Oscillations of Electrical Networks and Transmission Systems," Trans. A I EE, 1919, p. 445. b. "i-lectric Circuit Theory and the Operational Calculus," McGraw-Hill. 3a. Gilbert Swift, "Ampliher Testing by Means of Square Waves," Communications , Vol. 19, No. 2, Feb. 1939. 3b. Bedford and Frehendahle, "Transient Response of Multi-Stage Video Frequency Amplifiers," Froc. I. R. E., Vol. 25, No. 4, April 1939. 4. H. E. Kallman, "Portable Equipment for Observing Transient Response of Television Apparatus," /. R. E. Froc, Vol. 28, No. 8, August 1940. 5. L. B. Arguimbau, "Network Testing with Square Waves," General Radio Experi- menter, Vol. XIV, No. 7, Dec. 1939. 6. W. C. Jones, "Instruments for the New Telephone Sets," B. S. T. J. Vol. XVII, No. 3, p. 338, July 1938. 7. V. Bush, "Operational Circuit Theory," Wiley and Sons, p. 176. 8. F. F. Romanovv, "Methods for Measuring the Performance of Hearing Aids," Acotis. Soc. Am. Jour., Vol. 13. p. 294, Jan., 1942. 9. A. M. Curtis, "A Oscillograph for Ten Thousand Cycles," B. S. T. J., Vol. XII No. 1, January 1933. CHAPTER VII Theoretical Analysis of Modes of Vibration for Isotropic Rectangular Plates Having All Surfaces Free By H. J. McSKIMIN 7.1. Introduction The comparatively recent advent of crystal controlled oscillators and of wave filters employing piezoelectric elements has resulted in an extensive study of the ways in which plates made of elastic materials such as quartz or rochelle salt can vibrate. Of special interest have been the resonant frequencies associated with these modes of motion. As will be indicated in subsequent paragraphs, the general solution to the problem of greatest interest is quite complex, and has not been forthcoming, (i.e., as applied to rectangular plates completely unrestrained at all boundary surfaces). For this reason numerous approximate solutions have been developed which yield useful information in spite of their limitations. Several of these solutions will be discussed in the following sections. The three general types of modes (i.e., the extensional, shear, and flexural) will be analyzed in some detail. Also, as a preliminary step the formulation of the general problem along classical Hnes will be developed. For the most part, the solutions obtained here are limited to those for an isotropic body. However, such solutions provide considerable guidance for the modes of motion existing in an aeolotropic body such as quartz. 7.2. Method of Analysis In order to set up the desired mathematical statement of our problem it will be necessary to consider first of all two very fundamental relationships. The first of these is the well known law of Newton which states that a force /acting on a mass m produces an acceleration a in accordance with the formula / = m-a The second relationship which we shall need is Hooke's law relating the strains in a body to the stresses. If forces are applied to the ends of a long slender rod made of an elastic material such as steel (Fig. 7.1) a certain amount of stretching takes place. If the forces are not too great, a linear 151 152 BELL SYSTEM TECHNICAL JOURNAL relationship between the applied stress and ensuing strain is found to exist. Expressed as an equation — = £ in which Xx is the force per unit area, Xx is the strain per unit length, and -E is a constant known as Young's Modulus. (Refer to Section 7.7 for further definition of terms). In an analogous manner, shearing stresses applied to an elastic solid as shown by Fig. 7.2 produce a shearing strain such that — = A, the shear modulus. Xy In general there will be contributions to a particular strain from any of the stresses which may happen to exist. For example, when an isotropic Fig. 7.1 — Bar under tensional stress Fig. 7.2 — Bar under shearing stress bar is stretched, there will be a contraction along the width which has been produced by a stress along the length. A statement of these relationships (known as Hooke's Law) is given by the equations of Section 7.8. It is now of interest to consider the conditions of equilibrium for a very small cube cut out of the elastic medium which in general is stressed and in motion. Reference to Fig. 7.3 will help to visualize the stresses which may exist on the faces of this cube. Since these stresses vary continuously within the medium, a summation of the forces acting on the cube along each of the major axes can be made with the use of differential calculus. From Newton's Law previously cited, it is apparent that any unbalance of these forces will result in an acceleration inversely proportional to the mass of our small cube. Three equations may *hen be derived, one for each major direction. If only simple harmonic n\)tion is considered (i.e. all displace- ^ Refer to "Theory of Elasticity" by S. Timcshenko or to any standard text on elasticity. MODES OF VIBRATION 153 ments are proportional to sin w/ where co = 27r times frequency) the following simplified equations result. Fig. 7.3 — Stresses acting on small cube dXx , dXy dXz dx dy dz dYy dX_y dY, dy dx dz dZ, dXz SYz dz dx dy — po) u = — poj V > = —p; = 0, X^ = E—- will always equal zero. ox TT . IT - or any whole number multiple of - the extensional Furthermore, if k stress will likewise reduce to zero at x = ^. The desired solution will then be as follows, / being the resonant frequencies. U = cos CO a/ -^ 1* 0) = 27r/ = WT VI s^ m = 1, 2, 3, etc. (7.8) -* t Fig. 7.5 — Thin plate The plate of Fig. 7.5 will now be considered. Here it can no longer be assumed that the Xx stress is the only one of importance. Instead, the displacements v and w will be considered zero and the displacement u a function of x only. This means that the shear stresses Xy , Xz , Yz vanish, so that the equilibrium equations 7.2 reduce to or . 5 M , „5 W 2 d U — pu> U dx^ A -{- B (7.9) (7.10) This is seen to be of the same form as equation (7.7) previously discussed, and will again have the solution w = cos kx with k = o) A/ - — j — - y A -\- Jj The MODES OF VIBRATION 157 boundary condition on the Xx stress will be met if ^ = — so that the following solutions result. ^ /p(l + e' = ''"' A (7.14) This equation may be solved graphically to yield values of frequency corresponding to given values of k. For our discussion of the length ex- tensional mode of vibration, the first root only will be considered. Fig. 7.8 shows a plot oi d-b against b-k assuming that Poisson's ratio is .33.^ If ^ = 1, and b = I, for example, 6 = a/ -00 = 1.62. The equations for the displacements when determined as explained in Section 7.9 become: u = Vi [cosh -344 y -f- .402 cos 1.278 y] cos x (7.15) V = Ui [.344 sinh .344 y + .315 sin 1.278 y] sin x All three stresses Xx, Yy, and Xy may be calculated from the above equations. If the length of our plate is made equal to niT, where m is an integer, the extensional stress Xx will equal zero regardless of y at the boundaries x = 0 and x = t since X^a sin x = 0 when x = mr. Also it can be shown by calculation that Xy is so small in comparison to the exten- sional stresses as to be entirely negligible; hence our solution is complete. If ^ = TT, the plate will be vibrating in its fundamental longitudinal mode. The distortion which results is shown by Fig. 7.9. It is seen that most of * Plotted in this way, the same curve results regardless of the value of b chosen for the purpose of solving Eq. 7.13. 160 BELL SYSTEM TECHNICAL JOURNAL the motion is along the x axis, though there is a certain amount of lateral contraction as the plate elongates. The second harmonic will have the same resonant frequency if ^ = lir, the third if ^ = Stt, etc. 1^ J3 Fig. 7.9 — Distortion of plate vibrating in first longitudinal mode In addition to harmonic modes along the length just considered there will be those for which the motion breaks up along the width. In general, the distortion of the plate may be quite complex with simultaneous variations along both dimensions. Similarly, for plates such as shown in Fig. 7.5 MODES OF VIBRATION 161 there will be many extensional modes which have resonant frequencies somewhat above those given by Eq. 7.11. Analysis of the motion shows that for these modes the displacement along the thickness varies periodically (or "breaks up") along the major dimensions of the plate. There again the distortion pattern of the plate may become very complex. 7.4. Shear Vibrations The second class of vibrations which will now be considered is the shear. This type of mode is of special importance because of the fact that piezo- electric plates vibrating in shear are widely used for frequency control of oscillators. For example, the AT quartz plate which is so much in demand utilizes a fundamental thickness shear mode in which particle motion is principally at right angles to the thickness. The distortion of the plate will be similar to that shown in Fig. 7.2. A simple, yet very useful formula for the resonant frequencies associated with the above type of displacement has been derived on the assumption Fig. 7.10 — Orientation of thin plate that the length and width of the plate are very large in comparison to the thickness. For the xy shear mode, the displacement u is assumed to be II = U cos ky, all other displacements being equal to zero. The only stress that need be considered then, is the Xy shear which is proportional to sin ky. Boundary conditions on this stress at the major surfaces of the plate are easily satisfied by choosing k such that Xj, = 0 at y = 0 and y = t. (Refer to Fig. 7.10.) This will be the case if ^ = • — , where m is any integer, and t t is the thickness of the plate. By using the simplified equilibrium equation as reduced from equations 7.1, a formula for the resonant frequencies is obtained in much the same manner as for extensional thickness modes. 0, = 27r/ = — 4/? w = 1, 2, 3, etc. (7.16) t y p In this formula the shear modulus A appears instead of Young's modulus as in the case of longitudinal modes. Harmonic modes are given by values of m greater than unity. 162 BELL SYSTEM TECHNICAL JOURNAL In addition to the resonant frequencies predicted by the foregoing analysis, there will be others corresponding to shear vibrations in which the principal shear stress varies periodically along the length and width of the plate. A formula which yields the approximate frequencies for these modes is developed in Section 7.9. It is shown that if the length and width are large in comparison to the thickness, the following expression may be used: co = 27r/=:ry-y,,,^+ -^ + ^ (7.1/) In this formula which has been derived for xy shears the c constants are the standard elastic constants for aeolotropic media. For isotropic plates such as have been considered up to this point Cl\ = TT-o - A + 2/i 1 — 2a^ — (J and ^55 = c^6 = A, the shear modulus (7.18) Various combinations of the integers m, n, and p may be chosen, with the restriction that neither m nor n can equal zero. It is seen that if € and w are very large the formula reduces to that of Eq. 7.16 which was derived on precisely that basis. Also, it is seen that the more complex modes all lie somewhat above the fundamental shear obtained by setting m = n = I and p = 0. Plate shear modes are also of considerable interest, particularly the one of lowest order. For a plate having a large ratio of length to width a formula similar to that given by equation 7.17 (but for two dimensions only) may be developed. If the plate is nearly square, however, this formula no longer yields sufficiently accurate values for the resonant frequencies. Coupling to other modes of motion* complicates the problem so much that only experimental results have been of much practical consequence. Fig. 7.11 shows in an exaggerated way the distortion of a nearly square plate vibrating in the first shear mode. 7.5. Flexural Vibrations 7.51. Plale Flexures One of the most studied types of vibrations has been the flexural. Perhaps this is true because it is the most apparent and comes within the realm of ex-perience of nearly everyone. The phenomena of vibrating reeds, xylo- phone bars, door bell chimes, tuning forks, etc. are quite well known. * It is found ejcperimentally that odd order shears are strongly coupled to even order flejcures; similarly, even order shears and odd order flexures are coupled. MODES OF VIBRATION 163 Beam theory has been used quite extensively to derive the equations which yield the resonant frequencies and displacements for bars vibrating in flexure. To obtain reasonably accurate results for ratios of width to length approaching unity, however, the effects of lateral contraction, rotary inertia, and shearing forces must be considered. This leads to a rather complicated solution which is much more accurate than that derived by the use of simple beam theory only, though it is still approximate in nature. For two dimensional plates free on all edges a method of analysis may be used which is similar to that described under extensional modes. While it is somewhat involved it yields direct expressions for the two displacements u and V, so that all stresses may be calculated, and the extent to which boundary conditions are satisfied determined. Fig. 7.11 — Distortion of plate in first shear mode Solutions for u and v are assumed to be of the form u = U sin Cv ccs kx (7.19) V = V cos {y sin kx For the infinite strip previously considered a transcendental equation is obtained which is the same as equation 7.13 with the exception that the left-hand expression is inverted. (Refer to Eq. 7.14 also.) ^ This is an extension of DoerfTler's analysis used to obtain harmonic flexure frequenceis for plates — "Bent and Transverse Oscillations of Piezo-Electrically Excited Quartz Plates"— Zeitschrift Fiir Physik, v. 63, July 7, 1930, p. 30. Also refer to "The Distribu- tion of Stress and Strain for Rectangular Isotropic Plates Vibrating in Normal Modes of Flexures" — New York Univ. Thesis by Author, June 1940. 164 BELL SYSTEM TECHNICAL JOURNAL The lowest order solution to this equation is found to correspond to flexure vibrations in the infinite strip. A calculation of stresses, however, reveals that boundary conditions cannot be satisfied properly even for the case of a long narrow plate. It can be shown, however, that another solution may be derived for the same value of frequency by letting k become imaginary. This simply means that the u and v displacements become hyperbolic functions of x instead of sinusoidal. The two complete solutions for the infinite strip may then be superimposed and parameters adjusted so that for definite values of length corresponding to fundamental and harmonic modes the proper stresses reduce essentially to zero on the ends of the plate. For plates having a ratio of width to length less than .5, this method gives very accurate expressions for displacements and stresses. If only the resonant frequency is required, ratios up to unity and beyond (for the fundamental mode) may be considered. An example has been worked out to provide a complete picture of the displacements for a bar of width = \,k = 1 and c = .2)2>. Use of equation 7.20 yields the quantity 6- = — - = .166 from which the resonant frequency may be obtained. Using this value of 6'^, one finds that k"^ = —.800 also satisfies equation 7.20. By making the total length of the bar equal to 4.50 the Xx extensional stress and the Xy shear stress may be made essen- tially zero on the ends of the plate regardless of y. The following expressions for u and v are obtained: u = (sinh .9132 y — 1.02 sinh .9718y) sin x -.160 (sin .98283' - .9568 sin .9250y) sinh .8944:*; V = (-1.094 cosh .9132y + .9915 cosh .9718) cos x -.160 (.9095 cos .9828y - .990 cos .9250^) cosh .8944x (7.21) Fig. 7.12 shows the distortion of the plate as calculated from the above expressions. It is seen that there will be two points at which there is no motion in either the x or y directions. These nodal points can be used in holding the plate, since it may be clamped firmly there without altering the displacements or resonant frequency. For the example shown, these nodes are positioned a distance of .211^ from the ends of the plate as com- pared to .224f for a long thin bar. ' A graphical solution to determine t is most convenient in which parameters are / h f adjusted so that X^ = 0 at x = ±- and y = ±^; Xy = 0 at a; = i^ and y = 0. These IV . stresses will remain essentially zero for all values of y if the ratio of j is not too great. MODES OF VIBRATION 165 Figures 7.13 and 7.14 show the distribution of the principle stresses as a function of position along the length. It is seen that for the particular DISPLACEMENT 5CALE Fig. 7.12 — Distortion of bar vibrating in first free-free flexure mode 0.05 0.04 0.03 X x ^ 0.02 0.01 0 ■ 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 X Fig. 7.13 — Distribution of longitudinal stress for free-free bar vibrating in first flexure mode I 2 0.001 "3 0.006 0 0.2 04 0.6 0.8 1.0 12 1.4 1.6 1.8 2.0 2.2 2.4 X Fig. 7.14 — Distribution of shear stress for free-free bar vibrating in first flexure mode example cited, the maximum shear stress is only about one-tenth the maximum Xx extensional stress. Both of these stresses reduce to zero at 166 BELL SYSTEM TECHNICAL JOURNAL the ends of the plate as they should in order to satisfy the boundary condi- tions. As the ratio of - is increased the shear stress becomes of greater importance. 7.52. Thickness Flexures The final analysis to be considered in this paper is for thickness flexures along the width or length of a thin plate. These modes are of particular interest in connection with the dimensioning of quartz plates for which it is desirable to utilize the fundamental thickness shear mode. (AT plate, for example.) It is found experimentally that even ordered thickness flexures are coupled to this shear to such a degree that at certain ratios of dimensions the operation of the plate as an oscillator or filter component is impaired. The two-dimensional solution derived in the preceding paragraphs can be used to predict certain harmonic thickness flexures ; however, in order to obtain a complete picture it is necessary to extend the theory to three dimensions. This has been done by the author with the following transcen- dental equation as a result (refer to Section 7.93). 2 _ — 2A^2^a /y 22^ ~~rh~[aB{(\+a')-^Al\\[(l-a'] ^' ^ tan I2 - Solutions to this equation are exact in nature for a plate of thickness h and of infinite extent in both the x and 2 directions. The quantity a- is equal to the sum of the squares of k and m which appear in the expressions for displacements as follows: u = U fi(y) sin kx cos mz V = V f'Ay) cos kx cos mz ^ (7.23) 10 = W fsiy) cos kx sin mz Also in equation (7.22) /2 - ff' ^ 2 fl2 pOi (7.24) >1 The lowest order solution to equation (7.22) with a- positive again cor- responds to flexure vibrations, as in the two dimensional case. Fig. 7.15 shows a plot oi 6-b against a-b calculated for a = .3. MODES OF VIBRATION 167 For reasonably high order flexures it may be reasoned that the true dis- ])lacements will be very nearly the same as those for the doubly infinite I)late as derived by the above method since the correction necessary to fulfill the boundary conditions will only apply very close to the edges of the plate. It will then be sufficient to choose values for k and m such that k = p ^ and m — — where p and q are integers. I w The values of a~ obtained in this way determine the corresponding resonant frequencies. 5 4 3 ab=bVkV^ Fig. 7.15 — d-b versus «•& for thickness flexures If it is desired to solve for the ordinal xy flexures, for example, m should be set equal to zero. The displacements in this case will be independent of the z dimension. When q is assigned values other than zero however, the resulting modes may be considered as xy flexures which vary or break up along the third major dimension. If q is small the resonant frequencies will lie only slightly higher than that of the corresponding ordinal flexure for which q — 0. Fig. 7.16 shows a few of the resonant frequencies as calculated for values of shear modulus and density corresponding to AT quartz. The effects of couphng to the fundamental thickness shear are shown by dotted lines for the 14th xy flexure. As might be expected there is similar coupling between 168 BELL SYSTEM TECHNICAL JOURNAL the 14th flexure which breaks up once along the z dimension and the shear which breaks up once along z — etc. A few of these flexures which break up along s are shown for the 16th ordinal flexure. 2200 2100 2000 1900 2 1600 If) uj 1700 y 1600 1500 1400 1300 1200 \ H 0 ^ V \ ^> • ^ ^ \ ^.^^ ^ ^ 1ST XY SHEAR MODE / \^ -■^-..^^ ^ ^ \ -* S N, 16^ ^ K S Vs ^ ^ >^. 14 \ V X ^ =^ s. S \ X ^" 10 13 14 LENGTH THICKNESS 16 Fig. 7.16 — -XY thickness flexure modes for square plate 7.6. Summary Three main classes or families of vibrational modes are found to exist in rectangular elastic plates free on all surfaces; namely, the extensional, the shear, and the flexural. In general, the associated displacements are functions of all three dimensions and may vary in such a manner as to make the distortion of such plates quite complex. For certain limiting cases, approximate solutions for the resonant fre- quencies and displacements (from which strains and stresses may be cal- culated) can be derived. Though there are a number of methods that can be used for specific problems, it has been found very convenient to utilize the classical formulation. For this reason the basis of this method has been discussed briefly. In essence it requires that displacements and stresses occurring within the elastic solid satisfy conditions of equilibrium as de- rived from Newton's Law. At the boundaries, certain other relations must be satisfied in order that conditions of clamping might be fulfilled. For plates entirely unrestrained the latter requires that all forces (tractions) acting through the free surfaces must vanish. For thin rectangular plates (such as quartz crystal oscillator plates) the modes of greatest practical consequence are plate modes, for which all MODES OF VIBRATION 169 stresses are essentially coplanar and independent of the thickness, and thickness modes, for which all dimensions must be considered except in limiting cases. Because of their great utility, simplified formulae have been derived for the resonant frequencies associated with long, narrow bars vibrating longi- tudinally, thin plates with extensional motion along the thickness dimension, and thin plates vibrating with shearing motion at right angles to the thickness. Exact solutions for the infinite strip have been derived, and used in obtaining the displacements and resonant frequencies for flexural and longitudinal modes. Such solutions take account of the fact that the width of the plate may become appreciable. WTiile limiting cases of plate shear w may be analyzed, solutions for ratios of — approaching unity have not proved very satisfactory. This is attributable to the fact that coupling to flexural modes is severe. Thickness flexural modes which exhibit displacement variations along both length and width dimensions of the plate have been analyzed by extending the "infinite strip" theory to three dimensions. Solutions obtained are fairly accurate if the harmonic order of the flexure is sufficiently great. 7.7. Nomenclature p = density E = Young's modulus cr = Poisson's ratio A = Shear modulus = E = M 2(1 + cr) B = -7- — ; — r-7- --7 = X + M for 3 dimensions 2(1 + (r)(l — Iff) for plane stress 2(1 - ff) o) = angular velocity = 27r/ 6^ = 4 A u, V, w = displacements in r, y and z directions _ du . dv , dw dx dy dz 170 BELL SYSTEM TECHNICAL JOURNAL _2 T , . d , 0 . d V = Laplacan =_ + _ + - % 'V 2 1 ' ^ ' ^ > unit strain components Xy J Xz J yz J > unit stresses 7.8. Stress-Strain Equations for Isotropic Media Xx = r- {Xx — (rYy — aZz) Jy = -j^ {Yy — (^^x — (tZz) Zz = — (Zg — (jXx — (tY'^ shear strain = - X shear stress A X^ = 2{aBe+ A X = A (~ 4- —\ ^ \dy dxj X = A (— 4- — ^ \dz dx/ For plane stress in xy plane 1 X =ix Xy . -^y MODES OF VIBRATION X. E 1 - / ,2 , ,?v 7« J must not be zero, for in simplifying equation (7.51) it was used as a divisor. Equation (7.51) may be rewritten to give h '^^ ^2 ^ -2U.A{k' + m') ^ . b [(7B{]^ + 't\ + m') + ACl\[Cl - k' -m^] tan (3- In the above U + B) {k^- + /"I + m-) = fx,-] 2 (7.53) 2 By letting 6- = —— and k- + m- = a- equations (7.52) and (7.53) above become ''"^^2* -2^.^3^a^ ^_ .6 [-stal surface. An enlarged view of Fig. 8.2 is shown in Fig. 8.3 and shows how the rod would vibrate. Figure 8.3A shows the type of motion for the first mode of a clamp- Fig. 8.2 — Cantilever type mounting. I D N^' A B Fig. 8.3 — Type of motion in cantilever support mountings. free bar. Figure 8.3B shows the type of motion of the sam.e bar vibrating in its second mode. This would indicate that for a given length of bar we could use it at several difi"erent frequencies by simply using higher orders of vibration. By using a clamp type mounting where the clamping rods are designed as shown in Fig. 8.2, we may now have a mounting which at the crystal frequency will allow the crystal to vibrate unrestricted but at the same time provide a very secure clamp thus preventing the crystal from moving about in its holder. To prevent rotation of the crystal about the axis of the clamped points, more than two can be used provided they are of the proper design. The frequency of a clamp-free rod in flexure is given by equation (8.1) where m now has values different than in the case of free free flexure. PRINCIPLES OF MOUNTING QUARTZ PLATES 181 where v = velocity in cm. /sec. d = diameter in cm. I = length in cm. m = 1.875 for first mode = (n-l/2)Tr for 2nd, 3rd, etc. From this we can compute the length necessary for a given rod at a given frequency and use this for the design of the clamping rods. This length is given in equation (8.2) for the case of a 100-kc crystal using phosphor bronze rods 1 millimeter in diameter /^ c = 1.8/:) 87r X 10^ (8.2) = ,225 cm This corresponds to the case of Fig. 8.3A. For the case of Fig. 8.3B, the length is given by I = .567 cm Using this same diameter rod, if we should go to a considerably higher fre- quency, for example 5 megacycles, the value of / would be extremely small even for the case of Fig. 8.3A and would be somewhat sm.aller than the diameter of the rod. As mentioned before in Chapter VI, the simple for- mulae that apply in the case of fiexure are only for the case of a long thin rod. When the length becomes equal to or less than this diameter, it is very prob- able that the support member should be designed as though it were vibrating in shear. These follow well-known rules and are only mentioned here in case designs for high-frequency crystals are contemplated using this method. The design of rod-supported crystals following this procedure has not been carried on to a large extent in these laboratories because, at present, the wire-supported crystal appears to have many advantages. A great deal more of the work in regard to resonating supports has been done for the case of the soldered lead type^ 8.2 Wire Type Supports The theory of resonating supports involving soldered leads on crystals is very similar to that just discussed for the case of rods. There are two additional elements that we have here that are not present in the case of the rod, these elements being the actual solder connections that fasten the wire ' The presence of standing waves on the lead wires of CT crystals was found experi- mentally by Mr. I. E. Fair. 182 BELL SYSTEM TECHNICAL JOURNAL to the crystal and the coupHng between the crystal and wire vibrating sys- tems. Considerable work has been done in regard to the amount of solder necessary and the most desirable shape for the solder cone. The complete assembly of a wire support for a crystal is shown in Fig. 8.4. The shape of the solder cone shown in Fig. 8.4 has proved to be the most desirable and has been termed as "bell-shaped." This type of cone formation allows the wire to be twisted in handling and still not break away the top of the cone and form an appreciable crater. For the purposes of analysis we may then as- sume that the cone becomes part of the crystal and moves with it so that when computing the length of a wire vibrating in flexure, this length should be determined from the top of the cone. The amount of solder used in the cone since it is part of the crystal must be kept at a minimum in order that the constants of the crystal equivalent circuit will not be modiiied too much by it. One established fact of the effect of the solder in the cone on the Fig. 8.4 — Soldered lead type movinting. equivalent circuit is to raise the resistance in the equivalent circuit for the crystal and this resistance increases considerably with an increase in tem- perature. The amount of solder permissible in the cone would then be determined by the maximum temperature at which the crystal is to be oper- ated and the minimum Q allowable. The type of motion that the crystal would generate in the support wire when oscillating is that shown in Fig. 8.4 by the dotted line. The solder ball shown to the right of the figure acts as the clamp for the wire. This solder ball may be placed at any point along the wire corresponding to a node. The diameter of this ball need only be sufficient to act as a clamp. In general, this will be in proportion to the wire diameter. For example, at 200 kc it was necessary to use a solder ball 60 mils in diameter on a 6.0-mil diameter phosphor bronze wire. The spacing between the solder ball and the head of the cone may be readily computed from equation (8.1). In practice, it has been found that in most all cases this distance is slightly greater than that given by the formula due to the I PRINCIPLES OF MOUNTING QUARTZ PLATES 183 fact that the free end is restricted to zero slope and for a given crystal and support wire it should be determined experimentally using the values ob- tained from equation (8.1) as a guide in the design. The diameter of the solder ball that acts as a clamp may also be determined experimentally by ^, Itt^ jTffttifl^>>*«itf -^^d* * -i^tu*-^ * Fig. 8.5 — FT-241 crystal mounting. increasing its size until the standing waves on the wire to the right of the ball are sufficiently reduced. A practical application of this type of support is shown in Fig. 8.5. The top view shows the small wires soldered to the crystal as well as the solder balls that are spaced at points corresponding to the second node on the lead wire from the crystal. These solder balls act 184 BELL SYSTEM TECUXICAL JOURNAL as mechanical termination for the lead wires and also as connection to larger size spring wires forming the rest of the shock-proof m.ounting. Another type of wire support that has found considerable practical use and is superior to the straight lead and solder cone type of connection is that of the headed wire. This is shown in Fig. 8.6. A headed wire is similar to that of common pin and may be connected to the crystal by sweating the head to the crystal as shown. This has certain advantages over the solder cone in that the head of the wire being a machined part is always constant and the distance d, as shown in Fig. 8.6, is the same for all mountings. The amount of solder necessary to sweat the head to the crystal is considerably less than in the case of the cone and hence this type of mounting will have less dissipation at the higher tem.peratures. One other factor not men- tioned above is that the coupling between the vibrating system of the wire and the vibrating system of the cr3Stal is considerably reduced by the use of ^1 ^ Fig. 8.6 — Headed wire t^-pe mounting. the headed wire. This is an important factor in reducing what may be termed a double system of standing waves on the wire. One standing wave system would result from reflections from the clamped end of the wire, while the other would result from reflections between the clamped wires coupled through the crystal. This may be reduced by a reduction of coupling be- tween the crystal and wire vibrating systems. Measurements have been made^ on the efi"ect of clamping the wire-sup- ported crystal at various points, on the activity and frequency of several different crystals used in oscillators and filters. Figure 8.7 shows the efi"ect of clamping a 500-kc CT type crystal such as now used in the FT-241 holder. Figure 8.8 shows the same condition for a 370-kc CT crystal. It will be noted that in these two cases with the decrease in frequency of the cr>'stal that the coupling between the wire and crystal has decreased, as shown by a smaller change in frequency and also, that for the lower frequency crystal the change in activity is modified only when the clam.p is very close to a loop of motion on the wire. The mountings of these crystals were of the type PRINCIPLES OF MOUNTING QUARTZ PLATES 185 shown in Fig. 8.4 where the amount of solcJer in the cone equals that of a solder pellet 20 mils in diam.etcr and 12 mils high. Figure 8.9 shows the change in frequency as a result of clamping one wire of a four-wire mounting of a GT-cut crystal designed for use as a filter ele- 0 0.020 0.040 0.060 0.080 0.100 0.120 0.140 DISTANCE FROM CRYSTAL SURFACE IN INCHES Fig. 8.7 — Effect on frequency and activity of clamping one lead of 500 kc. CT-cut crystal t 0.40 A /^ 4 y t / - A / 1 / 1 ^"^ - -10 -20 0.040 0.060 0.080 0.100 0.120 DISTANCE FROM CRYSTAL SURFACE IN INCHES Fig. 8.8 — Effect on frequency and activity of clamping one lead of 370 kc. CT-cut crystal. ment at 164 kilocycles. This change is shown for the lower resonance at 143 kilocycles since this mode would be more aflfected by clamping. The large deviations in frequency correspond to clamping at the loops of the wire as shown in Figs. 8.8 and 8.9 but the small sudden changes in frequency are a result of a second system of standing waves as previously described. This 186 BELL SYSTEM TECHNICAL JOURNAL second system of standing waves results from too much coupling between the crystal and the two oppositely disposed lead wires. It may be reduced by first placing the wires closer to the nodal point and second, using a smaller amount of solder in the cone to attach the lead wire to the crystal. Measure- ments on this same type of crystal when the above conditions were fulfilled showed practically no effects of secondary standing waves. It is important to keep the energy transmitted to the lead wires low since a soldered connec- tion near a loop of motion resulting from secondary standing waves on the wire will act as a clamp and will materially decrease the resulting Q of the crystal. This is probably the best reason for the use of the headed wire type of lead wherever practical. + 300 + 1 50 0 -150 j J t / ^ r -4 ^ ""^ y" / ^' ^ J /^ ^ / 1 / -300 1 \ 0.020 0.040 0.060 0.080 0 100 0.120 DISTANCE FROM HEAD OF SOLDER CONE IN INCHES Fig. 8.9 — Effect on the frequency of lower resonances of clamping one lead of 164 kc. GT-cut crystal. 8.3 Air-Gap Type Supports A third form of mounting for quartz cr>'Stals is that of the airgap type shown in Fig. 8.10 where the crystal plate is held between two fiat electrodes. Two forms of the airgap type of mounting are shown. In Fig. 8.10A the crystal is free to vibrate between two flat electrodes held together to produce a definite airgap of thickness /. In Fig. 8.10B small lands are left on the corners of the electrodes to produce a uniform airgap on each side of the crystal as well as to clamp the crystal plate. This type of mounting has found its greatest use for oscillator crj^stals of the AT and BT type. The factor that determines the choice of mount is the ratio of length to thickness of the crystal. For example, when the length is less than 20 times the thickness, clamping the corners of AT and BT type crystals will decrease the activity in proportion to the clamping pressure. This is apparent from a study of the type of motion for these crystals de- scribed in Chapter VI. This then indicates that AT andBT t}T3e cr^^stals for broadcast frequencies should employ a mounting with the crystal un- restricted as shown in Fig. 8.10A while the higher radio frequency crystals may be clamped as shown in Fig. 8.10B. The clamping pressure will be PRINCIPLES OF MOUNTING QUARTZ PLATES 187 dependent upon the area of the crystal, its frequency and the amount of activity required. One advantage of the clamped type support lies in the fact that many of the unwanted modes of motion are restricted or dampened to the extent that they will not cause serious dips in the activity character- ELEC TRODE ^ ^ 1 ^ ^ J_ T ^ y ) t // /' // '/ / '/ / ELECTRODE CRYSTAL B Fig. 8.10 — Air gap type mounting. A — Crystal free. B — Crystal clamped at corners. L| C, fW^ — If— 1 L, C HRP — |f-| Co Co A B Fig. 8.11 — Equivalent circuit of a quartz crystal in an air gap type mounting. istic over a wide temperature range. This explains in part the necessity for accurate control of the length and width dimensions for crystals of low radio frequencies using the type of mounting shown in Fig. 8.10A. The effect of the airgap on the constants of the crystal equivalent circuit may be determined from Fig. 8.11. In Fig. 8.1 lA is shown the usual crystal equivalent circuit in series with a capacity Ca which represents the capacity 188 BELL SYSTEM TECHNICAL JOURNAL of the airgap. This may be reduced to the circuit of Fig. 8.1 IB where the constants are given by Co = c; = Ca + Co Co cj {Ca + Co)(Ci + Ca + Co) Ci The circuit of Fig. 8.1 IB is the same form as that of the original cr^'stal and therefore we may assum.e that the effect of the airgap is to produce a similar A A A/2 3A/4 0.3 0.2 03 0.4 AIRGAP IN M.M. Fig. 8.12 — Effect on frequency of the air gap thickness on a 550 kc. AT-cut crj'stal. crystal of reduced capacity and reduced effective piezoelectric coupling. In the case of oscillatory crystals the effect of the airgap is to reduce the activity and decrease the range of frequency adjustment with parallel capacity. For filter applications the effect of the airgap is to produce narrower transmission bands and higher characteristic impedance. One other effect of the airgap results from the propagation of acoustic waves from the crystal. It is known that most any type of crystal in a vibrating condition will produce acoustic waves in air and if an object capable of reflecting these waves is the proper distance away, these acoustie waves may be reflected back to the crystal surface. The reflections from distances corresponding to even quarter wave-lengths will cause considerable dam^ping while the re- flections from distances corresponding to odd quarter wave-lengths will PRINCIPLES OF MOUNTING QUARTZ PLATES 189 cause very little. The wave-length of a sound wave in air may be readily computed, and since we arc interested in multiples of one-quarter wave- length, it is desirable to determine these for a given frequency. This can be computed readily from equation S.3, 4 4/ where v is the velocity of sound in air at room temperature and pressure and equals 33,000 centimeters per second. For example, a quarter of a wave- length at 5 miCgacycles is given by X 33,000 ._.,. 7 = A ., - -.y .f^r = .0016:) cm 4 4 X :) X 10^ which indicates that if / of Fig. 8.10 were made equal to this or odd multi- ples, there would be very little efifect of the electrode on the crystal and if / corresponded to even multiples of a quarter wave-length, we would expect considerable damping. Some measurements of this effect have been made with a low frequency A T-cut quartz crystal and are shown in Fig. 8.12. The sound wave generated by an /IT-cut probably results from flexure waves generated by the high-frequency shear wave. It will be noted that when the airgap is equal to even multiples of a quarter wave-length, the activity is considerably reduced. Further, it will be noticed that airgaps in the order of 1/8 of the wave-length may be used and produce very little effect. Since a large airgap reduces the piezoelectric coupling it is desirable to keep this about 1/8 of a wave-length as a maximum unless, in special cases, a reduction in piezoelectric coupling may be tolerated. The Magnetically Focused Radial Beam Vacuum Tube By A. M. SKELLETT A new type of vacuum tube is described in which a flat radial beam of elec- trons in a cylindrical structure may be made to rotate about the axis. Features of the tube are its absence of an internal focusing structure and resultant sim- plicity of design, its small size, its low voltages, and its high beam currents. The focusing of the beams and their directional control are accomplished by the magnetic fields in small polyphase motor stators. A time division multiplex signaling system for 30 channels using these tubes is brieflj' described. IT HAS long been recognized that the substitution of electron beams for mechanical moving parts would offer decided advantages in many applica- tions in the field of communications. The high voltages .equired for the usual cathode-ray type of tube and the very low currents obtainable there- from prevent their use in most such proposals; their complicated guns and their large sizes are also undesirable features. The kind of tube described herein has no focusing structure, is small in size, requires only low voltages, utilizes the cathode power efficiently, and produces beam currents of the same order of magnitude as the space currents of ordinary vacuum tubes. Figure 1 shows the elementary tube structure. It consists, in the simplest case, of a cyhndrical cathode of the sort in common use in vacuum tubes, sur- rounded by a cylindrical anode structure. When this structure is made positive with respect to the cathode and there is no magnetic field in the tube, the electrons flow to the anode structure in all directions around the axis. When a uniform magnetic field is applied with its direction at right angles to the axis, the electrons are focused into two diametrically opposite beams as shown. The beams are parallel to the fines of force of the magnetic field so that if the field is rotated the beams move around with it. Thus the magnetic field serves both to focus the electrons and to direct the resulting beams to different elements of the anode structure. If ordinary commercial cathodes are used with anode structures an inch or two in diameter, 100 volts or less on the anode will draw the full space current for which the cathode was designed. The application of the mag- netic field will then focus from 85 to 90 per cent of this electron current into the two beams, the remaining 10 or 15 per cent being lost at the cathode due to an increase in the space charge which the magnetic field produces. Some of the smaller tubes produce beam currents of more than 5 milliamperes with only 50 volts on the anode structure, and in some of the tubes with larger cathodes beam currents of 50 milliamperes or more are easily obtainable. The magnetic field strengths range from 50 to 300 gauss. 190 RADIAL BEAM VACUUM TUBE m For some applications it is desirable to eliminate one of the two beams and this may be accomplished by substituting a uniform electrical field in the tube for the cylindrical one described above. The uniform field may be obtained by applying to the anode elements a series of potentials that vary according to the sine of the angle taken around the axis. The line joining the maximum potentials (+ and — ) is maintained parallel to the magnetic field so that on one side of the cathode the potentials are all negative and the ANODE STRUCTURE Fig. 1. — Elementary tube structure showing focused beams. beam on that side is suppressed. The remaining beam will have somewhat less current than the corresponding one in the cylindrical field but the mag- netic field-strength required for focus is reduced. Cylindrical Electrical Field For the case of the cylindrical electric field the focus is obtained by ap- plying a magnetic field that is strong enough to reduce the radius of curva- ture of the spiral electron trajectories to a small value. There is not ob- tained an electron optical image of the cathode in the usual sense that for 192 BELL SYSTEM TECUXICAL JOURXAL each point on the cathode there is a corresponding point on the image. Tlie sharpness of the image may be increased by increasing the strength of the magnetic field and the field required for any degree of focus is not sharply critical. Figure 2 shows a series of drawings of the various electron images that were obtained as the magnetic field-strength was increased in a tube having Z7 HO SI ^9 t 07. tSLZ M6 ZOQ Z70 340 dcLuss Fig. 2. — Drawings of the patterns obtained with a fluorescent coating on the inside of the anode when the magnetic field strength is increased from zero to the focus values. a fluorescent coating on the inside cylinder. The cathode and anode diam- eters were 0.0625 and 2.5 inches, respectively, and the axial length was 2 inches. The anode was held at 150 volts. Only one-half inch of the cathode length, located centrally along the axis, was coated to emit electrons. The image at 340 gauss appeared to be one-half inch long. In attempting to interpret these patterns it should be remembered that on the two sides of the cathode at right angles to the plane of the beam the electrons follow RADIAL BEAM VACUUM TUBE 193 B Fig. 3. — Electron trajectories made visible with a small amount of gas. A. — Magnetic field lined up with active spots on the cathode. B. — Magnetic field at 45° with respect to the active spots. cycloid-like paths along the cathode, moving up on one side and down on the other. The photographs of Fig. 3 showing the trajectories were obtained by 194 ^ BELL SYSTEM TECHNICAL JO URN A L introducing argon at a pressure of about a micron into the tube. The elec- trons are emitted from only two spots of active material located at the op- posite ends of a diameter on the cathode sleeve. In Fig. 3a the line joining the spots is lined up with the magnetic field and in 3b this line is at an angle of about 45° with respect to the field. This arrangement does not reproduce exactly the space charge conditions in the tube as actually used but does serve to give a picture of the electron paths in a qualitative sort of way. As shown by the patterns of Fig. 2 above a minimum strength of magnetic field the shape of the focus does not change greatly. An approximate equa- tion may be derived for the beam width in terms of the magnetic field above this minimum value that is useful for predicting the performance of new designs. The electrons that leave the cathode at right angles to the beam require the strongest magnetic field to keep them in focus. Now because of the cylindrical structure the electric field is concentrated near the cathode and we will assume that after leaving the vicinity of the cathode the velocity does not change appreciably. Setting v equal to the component of this velocity at right angles to the magnetic field we have that the radius r of the spiral path is given by the relation mv where H is the magnetic field-strength and m and e are the mass and charge of an electron. We also write '= ]/ 2eKV where K is the fraction of the anode voltage corresponding to v. The width of the focus A is approximately equal to the cathode diameter D plus twice the maximum radius of curvature of the spiral paths 6.7\/KV A •= D + H where A and D are in centimeters and V is in practical volts. By substitu- tion in this formula we have found that the empirical constant K is about 0.7 for the tubes that have been made to date. A minimum value for H is obtained, again approximately, by setting the last term in the equation equal to D. Unifokm: Electric Field As mentioned above the uniform field is obtained by imposing potentials around the anode periphery varying as the sine of the angle. The cathode is RADIAL BEAM VACUUM TUBE 195 at a point of zero potential. In this case a real electron optical image of the cathode is obtained. Neglecting the distortion of the field in the vinicity of the cathode, the force equation for the electrons is d^x V m — = e — df R where V is the maximum anode potential, R is the radius of the anode struc- ture and X is measured in the direction of the fields. Since the acceleration is uniform the transit time t, neglecting space charge effects, may be obtained from the expression 1 /^\ .2 ^ 2\dty R R Combining these equations we get t = The condition for focus is that the electrons make one revolution around the lines of force in time t. The angular velocity of the electrons is given by the well-known expression He CO = — m Setting oit = 2ir we get ^--rV or in practical units ^ R — Since the effect of the magnetic field on the space charge has not been evaluated, we can only estimate the order of magnitude of the increase of transit time due to the space charge. On the assumption that this increase introduces a factor of 3/2* the above expression with space charge is rr _ 7.1V7 ^--R~ This formula has been found to check well experimentally. * The factor of 3/2 is the ratio of the transit times in a plane parallel diode with and without space charge. See for example JNIillman and Seely, "Electronics," Chapt. 7, p. 231. 196 BELL SYSTEM TECHNICAL JOURNAL These last two formulae are for the first focus. Focii will also be obtained for values of H equal to nH where n is an integer and equal to the number of electronic revolutions. Actually as the field is increased beyond that neces- sary for the first focus the beam does not get very badly out of focus because the radius of curvature of the spiral path is small and for still higher fields the beam remains in approximate focus for all values of H. In applications where the beam is rotated by means of a rotating mag- netic field this electrostatic field is made to turn by separating the anode structure into four or six elements (or groups thereof) and applying either two- or three-phase alternating potentials to them. Magketic Field Supply The stator of a two-pole pol^'phase alterating-current motor furnishes an excellent magnetic field for use with these tubes. The tube is inserted in place of the armature and when the polyphase currents are applied the beams are formed and rotate at the cyclic frequency. For applications where the beams are not rotated continuously, a two-phase stator may be used in which the currents through the two windings are adjusted to be proportional to the sine and cosine of the desired direction angle of the beam. Per- manent magnets of the horseshoe design have also been found to be suitable. The power consumed by a stator depends on its size and the strength of the field it produces and on the cyclic frequency if it is used to rotate the beam. At low frequencies, e.g., 20 or 60 cycles, the power consumed is primarily that due to the copper loss. At higher frequencies the losses in the core material become important. For some of the smaller tubes operating at a low frequency, the power consum.ed by the stator is less than three watts. This stator has the regular motor windings which do not completely fill the slots. Since a pol>T)hase source of power is not always readily available, it is sometimes advantageous to split single-phase power in the stator itself to produce the rotating field. This may be done by inserting a condenser in series with each winding so that the current through one phase winding lags by 45° and that through the other leads by an equal angle. Polyphase po- tentials for producing a rotating electrostatic field in the tube may then be taken from the windings of the stator if desired. Tube Design The particular design of tube depends on its application. The simple design shown in Fig. 1 has been found adequate for som.e purposes but more elaborate designs which increase the versatility of the tube are also needed. Figure 4 shows a tube with 30 anodes that incorporates various auxiliary elements. This tube is 2.25 inches in diameter. Figure 5 shows the internal RADIAL BEAM VACUUM TUBE 197 arrangement of the elements. Closely surrounding the cathode is a control grid that may be used for modulating the current density of the electron beams. Farther out is a cylindrical element with 30 windows that is main- tained positive and which by virtue of its similarity in position to the third element of a tetrode is called a screen. Immediately behind each window there is a pair of paraxial wires which because of its similarity in function to the fourth element of a pentode is called a suppressor grid. In back of each suppressor grid there is an anode. In this particular tube there are pro- Fig. 4. — Radial beam tube with 30 anodes and unwound stator used with it. jections like gear teeth on the back of the screen clement to prevent electrons, destined for one anode, from reaching an adjacent one. The control grid that is close to the cathode is biased negatively and con- trols the electron current in the same way that it would if the magnetic field were not present. The space current vs. grid potential curve is nearly identical for the two cases: with and without the magnetic field. The slight difference is due to the fact that the presence of the magnetic field increases the space charge near the cathode. Thus the tube may be used for amplification in the usual way when the electrons are focused. The pres- ence of this grid has no appreciable effect on the focusing of the electrons. 198 BELL SYSTEM TECHNICAL JOURNAL Since the screen element is in one piece there will be present two beams out to it. One of these may be suppressed after it has passed through the screen by the suppressor grids or by the anodes in the manner described below. These suppressor grids are generally operated at cathode potential or at a potential that is negative with respect to the cathode. They may be used for three purposes: to suppress secondaries from the anodes, to modulate the beam current to their particular anode, and to suppress one of the two beams. For the first of these functions they are biased at cathode potential. For the second they are biased negatively and have a modulation curve simi- lar to that of the suppressor grid in a pentode. Curve A of Fig. 6 shows the variation of beam current to one anode when the potential of the suppressor grid in front of it is varied. This curve is for a grid similar to the two paraxial SUPPRESSOR GRID CATHODE ELECTRON BEAM CONTROL GRID INDIVIDUAL ANODE Fig. 5. — Arrangement of elements in the tube shown in Figure 4. Only the operating beam is shown. wires in the tube shown in Fig. 5. For some applications a higher sup- pressor-anode transconductance or a lower cut-off is desirable and these may be obtained by welding lateral wires across this grid window to make the grid action more efifective. Curve B of Fig. 6 was taken with the same size window across which laterals were welded. The table below gives the data for this suppressor grid with and without the lateral cross wires. Without Laterals With Laterals Transconductance (mho) 100 250 Anode Resistance (ohms) 30,000 64,000 Amphfication Factor 3.5 16.0 Cut-Off Voltage -80 -20 It is apparent from these data that amplification of the signals applied to the individual suppressors may be readily obtained. RADIAL BEAM VACUUM TUBE 199 If the screen element is split to give a uniform electrostatic field to sup- press one beam, the beam current is only about half that of one beam of the cylindrical field case. This is because with the uniform electrostatic field the potential gradient at the cathode decreases with azimuthal angle away from the beam axis. If the unwanted beam is rejected by the suppressor grids, however, the beam current for the cylindrical case is obtained since the screen in this latter case supplies a cylindrical electrostatic field at the cathode and the unwanted beam is rejected between the screen and sup- pressor grids. 7 SCREEN 140 VOLTS ANODE 140 VOLTS GRID +24 VOLTS MAGNETIC FIELD 150 GAUSS 6 y y ^ f y / b/ / / / y / / / / j y / / 0 -80 -70 -60 -50 -40 -30 -20 SUPPRESSOR GRID POTENTIAL IN VOLTS Fig. 6. — Suppressor grid characteristics. A. — Without lateral wires. wires. B.— With lateral For this case the screen is maintained at the same positive potential re- quired for the two-beam condition and the suppressors are so biased that they are beyond cut-off on one side of the tube and at or near cathode po- tential on the other side. If the beam is rotated the suppressors are con- nected to the polyphase supply in groups in the same way that the screen elements would be connected except that the d-c. bias above and below which the a-c. potentials swing is made negative at a value near cut-off for the suppressors. When one beam is suppressed either by splitting the screen or by grouping the suppressors, the currents to the different anodes are not all exactly the same. For instance, maximum current will be received by an anode back 200 BELL SYSTEM TECHNICAL JOURNAL of the center of one of the screen elements or one of the suppressor groups and a minimum current will be received by an anode back of the junction of two such elements or groups. If two-phase supply is used (4 elements or groups) the ratio of maximum to minimum anode current will be 0.707 and for three- phase supply this ratio will be 0.866. There will be 4 or 6 maxima, respec- tively, around the tube. This variation may be effectively eliminated by varying the individual anode load impedances or in other ways. The anode characteristics are similar to those of a pentode if suppressor grids are used and to that of a tetrode if these grids are not used. There is still another method of effectively eliminating one beam. This consists in using an odd number of anodes so that when one beam is focused on an anode the opposite one falls on the screen in between two anode posi- tions. With this type of tube the effective rotational frequency is twice the cyclic frequency of the rotating field, that is, all of the anodes are con- tacted twice (once for each beam) per revolution of the field. Applications The many possible combinations of the tube elements just described per- mit a variety of applications. One of the simplest and most obvious is that of an electronic commutator which has the advantages over the correspond- ing mechanical device of speed and freedom from contact trouble. There is, however, a practical limitation to the speed of this electronic commutator that is set primarily by the alternating-current losses in the stator. This is estimated to be in the neighborhood of 10,000 cycles per second for ordi- nary stator and tube designs. The highest cyclic speed for a stator that has been used to date was 600 cycles per second which with utilization of both beams gave an effective cyclic frequency of 1200 cps. One of the earliest systems of multiplex telegraphy was based on time division using mechanical rotating commutators. A small portion of the time of one cycle of the moving brush was allotted to each channel. The usefulness of this system is limited because of the faults of the mechanical commutators. The substitution of these electronic commutators eliminates these difficulties and puts the time division system on a more practical basis. It has an advantage over the frequency division multiplex system (carrier system) in that the elaborate filters of the latter are not required. A 30-channeI multiplex system for signaling using two of the 30 anode tubes described above has been successfully tested over short distances in the metropolitan area in New York City. The tube at the transmitter had all of the anodes tied together and the signal from them was sent over the line. The 30 input channels terminated on the suppressor grids of this tube. At the receiver, the input was fed to the negative grid surrounding the cathode and each of the anodes was connected in scries with a small neon lamp for RADIAL BEAM VACUUM TUBE 201 an indicator. A signal on any one or signals on any group of the 30 input channels would actuate the corresponding lamp or lamps at the receiver. No amplification other than that provided by the receiver tube was needed. A single beam was used in each tube, the other one being rendered ineffec- tive in the transmitter by m.eans of two-phase potentials applied to the Fig. 7. — Circular trace oscillograph of transmitted signal when 3 out of 30 channels are in operation. Fig. 8. — Linear trace oscillograph showing transmitted signal with 3 channels in operation 2 of which are adjacent. suppressors in the manner described above and in the receiver by means of a combination of d-c. and two-phase a-c. potentials applied to the individual anodes. The potential of an anode was zero when the unwanted beam arrived and at or near 200 volts at the time of passage of the operating beam. The rotational frequency of the beam was sixty cycles and since both 202 BELL S YSTEM TECH NIC A L JOURNA L stators were tied into the same source of power, no separate synchronizing means was necessary. Figure 7 is a photograph of the cathode ray trace of the output of the transmitter tube when signals were being sent over three channels. A circular sweep circuit w^as used which distorted the signals somewhat. The shape of the pulses is shown better in Fig. 8 for which a linear sweep was employed. Signals were put on three channels, two of which were adja- cent. The double-humped top of the pulse is caused by the window in the screen being slightly narrower than the beam width so that as the beam crosses the window, the greater densities in the edges relative to the center give this shape. A flat-topped pulse may be obtained by making the win- dows wider than the beam. In conclusion the writer wishes to acknowledge his indebtedness to a number of his colleagues in the Laboratories for aid in the development of the tube. The 30-channel multiplex system was set up with the aid of Mr. W. H. T. Holden. Abstracts of Technical Articles by Bell System Authors A Modification of Halleii's Solution of the Antenna Problem} M. C. Gray. An alternative formula for the input impedance of a cylindrical antenna is derived from Hallen's integral equation. It is shown that the introduction of a variable parameter Z{z) in place of Hallen's S2 = log (4/-/a^) modiiies the numerical results considerably, and leads to much better agreement with experimental evidence. Motor Systems for Motion Picture Production? A. L. Holcomb. The various types of motor systems and speed controls used in motion picture production are reviewed, evaluated, and the basic theory of operation described. Motor drive systems are a fairly simple but important element in the production of motion pictures, but to many people who do not have direct contact with this phase of activities, the number of systems in use and their peculiarities are very confusing. Data on most of the different types of motors and motor systems in use have been published, but in different places and at different times so that no comprehensive reference exists. This paper is not intended as information on new developments or as a technical study, but rather as a review of all the major systems with an indication of their fields of greatest usefulness and with comments on both their desirable and undesirable features. A Dial Switching System for Toll Calls} Howard L. Hosford. At Philadelphia, on the night of August 21st and the early morning hours of August 22, 1943, the cutover of the new #4 System was no mere episode; it was one of the milestones of telephone history. IntertoU dialing in itself is not new but this joint project of the Bell Telephone Company of Penn- sylvania and the Long Lines Department is especially significant as it has been designed so as to extend the field of toll dialing by the operators to include the largest cities and joins together various types of dialing equip- ment. In its scope this project includes many points in an area reaching from Richmond, Va. to New York City and from Harrisburg, Pa. to At- lantic City, N. J. From a traffic standpoint the # 4 toll switching system actually comprises ^Jour. Applied Physics, January 1944. ^Joiir. S. M. P. E., January 1944. ^Bell Tel. Mag.. Winter 1943-44. 203 204 BELL SYSTEM TECHNICAL JOURNAL three units, the switching equipment itself which is wholly mechanical, together with the so-called # 4 and ;i^ 5 switchboards. The # 4 board is a cordless, key-typed call distributing board which is used in conjunction with the new switching system for such calls as must be given to an operator by offices not equipped for intertoU dialing. The operators at this board function as combined inward, through and tandem operators, thus eliminat- ing the provision of separate units to provide these particular services. In brief, there is no basic difference between the essential operation of the # 5 board and the conventional through board where delayed traffic is handled; however, operators handling calls at this board must make use of the new switching system to obtain both the calhng and called offices by dialing. Prior to the cutover the first trainees were given experience by handling some 300,000 test calls of every conceivable traffic characteristic. These were routed through the new system to break in the equipment and to shake down potential troubles. Two weeks prior to cutover a dress re- hearsal was held, at which time about ten per cent of the circuits were put through their paces. To provide information of value for future installations, arrangements were made for liberal provision of registers and meters to measure any and all phases of the various steps performed by the equipment. Some of these aids are not entirely new to telephone work but their application to toll, inward and through service is a departure. The ^4 System is running satisfactorily and both the equipment and the operators who use it deliver a high grade of service. Daily some 80,030 tandem, inward and through connections formerly handled by operators are routed through the equipment. In connection with postwar planning, studies are now being made to determine future installations in order to take advantage of the possibilities of the new system. It is confidently expected that this will provide faster service on outward, inward and through calls and that transmission will be improved. These advantages should result in overall economies in outside plant and operating. Theoretical Limitation to Transconductance in Certain Types of Vacuum Tubes.^ J. R. Pierce. The thermal-velocity distribution of thermioni- cally emitted electrons limits the low-frequency transconductance which can be attained in tubes in whose operation space charge is not important. A relation is developed by means of which this dependence may be evaluated for tubes employing electric and magnetic control. This relation is applied to deflection tubes with electric and magnectic control and to stopping- *Proc. I. R. E., December 1943. ABSTRACTS OF TECHNICAL ARTICLES 205 potential tubes. Magnetic control is shown to be inferior to electric control from the point of view of band-width and gain. Antenna Theory and Experiment J' S. A. Schelkunoit. This paper presents: (1) a comparison between several approximate theoretical formulas for the input im.pedance of cyhndrical antennas in the light of available experimental evidence; and (2) a discussion of the local capacitance in the vicinity of the input terminals, mathematical difficulties created by its presence, and m.ethods of overcoming these difficulties. No exact solution of the antenna problem is available at present and so far it is impossible to set definite limits for errors which may be involved in various approxi- mations. For this reason in appraising these approximations one is forced to rely on one's judgment and on experimental evidence. It is hoped that this paper will aid in correlating theory and experiment to the advantage of both. ^ Jour. Applied Physics, January 1944. Contributors to this Issue H. J. McSkimin, B.S. in Electrical Engineering, University of Illinois, 1937; M.S. in Physics, New York University, 1940. Bell Telephone Labo- ratories, 1937-. Engaged primarily in a study of electrical and electro- mechanical properties of piezoelectric crystals. E. E. MoTT, Massachusetts Institute of Technology, B.S. 1927; M.S. 1928. General Electric Company, 1926-28. Bell Telephone Laboratories, 1928-. Mr. Mott has been engaged in telephone instruments research and devel- opment, particularly in connection with various types of telephone receivers and related devices. Since 1941 he has been engaged on war projects. A. M. Skellett, A.B., 1924, M.S., 1927, Washington University; Ph.D., Princeton University, 1933; Instructor, 1927-28, Assistant Professor of Physics, 1928-29, Un versity of Florida. Bell Telephone Laboratories 1929-. Dr. Skellett, formerly engaged in investigations pertaining to the transatlantic radio telephone, is concerned with applications of electronic and ionic phenomena. R. A. Sykes, Massachusetts Institute of Technology, B.S. 1929; M.S. 1930. Columbia University, 1931-1933. Bell Telephone Laboratories, Re- search Department, 1930-. Mr. Sykes has been engaged in the applications of quartz crystals to broad-band carrier systems as filter and oscillator elements. Other work has included the application of coaxial lines as ele- ments of filter networks and more recently the design and development of quartz crystals for radio frequency oscillators. 206 VOLUME XXIII JULY, 1944 NUMBER 3 THE BELL SYSTEM TECHNICAL JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION Efifect of Telegraph Distortion on the Margins of Opera- tion of Start-Stop Receivers . . , . W. T, Rea 207 The Mounting and Fabrication of Plated Quartz Crystal Units R. M. C. Greenidge 234 Effects of Manufactiu*ing Deviations on Crystal Units for Filters A. R, D'heedene 260 Mathematical Analysis of Random Noise . . S. O. Rice 282 Abstracts of Technical Articles by Bell System Authors 333 Contributors to this Issue 336 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 50c 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, F. ■ » » » .^i^i^ii^-^iii^ EDITORS R. W. King J. O. Perrine F. B. Jewett O. £. Buckley S. Bracken EDITORIAL BOARD M. R. Sullivan A. B. Clark M. J. KeUy O. B. Blackwell H. S. Osborne F. A. Cowan » i« ««■■«>« »ii 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, 1944 American Telephone and Telegraph Company PRINTED IN U. S. A. The Bell System Technical Journal Vol. XXIII July, 1944 ^0. 3 Effect of Telegraph Distortion on the Margins of Operation of Start-Stop Receivers By W. T. REA Recent practical and theoretical investigations of the effect of signal dis- tortion on the margins of operation of start-stop telegraph receivers have led to the development of improved methods of testing and adjusting receivers, have enabled criteria of distortion tolerance to be set up for subscribers' and monitoring receivers and regenerative repeaters, and have made possible the application of more convenient and accurate standards of telegraph trans- mission. This paper describes the causes of distortion occurring both externally and internally to the receiver and the effects of such distortion on the operating margins. Methods of determining the internal distortion of a receiver are described and some of the more important considerations involved in establish- ing distortion tolerance criteria are discussed. DURING the past decade the proportion of Bell System telegraph service operated on a start-stop teletypewriter basis has shown a continuous increase. Whereas in 1930 about 65% of telegraph long- distance circuit mileage was manual Morse, the present proportion of teletypewriter and teletypesetter service stands at 92%. The rapid growth of teletypewriter switching facilities has been an important factor in this development. Naturally, this situation has made increasingly important a thorough understanding of the factors which affect the performance of start-stop receivers. In the present paper, an effort will be made to show some relationships between signal distortion and the operating margins of start- stop receivers. A properly designed start-stop telegraph receiver requires only a small portion of the time of each signal element to permit a selection to be made; i.e. to determine whether the signal element in question is marking or spacing. The remainder of the signal element gives an operating margin, and serves as a reserve to take care of imperfections in the receiver or distortions which the telegraph signals may suffer in their passage over Unes and through repeaters. The greater the signal distortion is, the smaller will be the margin which remains in the receiver to overcome the effect of such factors as wear of parts, variation of adjustments, or differences in speed between transmitter and receiver. A consideration of the effects of telegraph distortion on the margins of 207 208 BELL SYSTEM TECHNICAL JOURNAL operation of start-stop receivers may well begin with a brief review of the nature and causes of the various types of distortion commonly experienced by telegraph signals. Telegraph distortion is generally considered to be divided into three types or components: bias, characteristic distortion, and fortuitous distortion.^ The magnitude of the distortion is expressed in per cent of a unit pulse. The Components of Telegraph Distortion Bias, which is the simplest and most common component of distortion, may be positive (marking) or negative (spacing). Positive bias appears as a uniform lengthening of all marking pulses and an equal uniform shorten- ing of all spacing pulses. Conversely, negative bias appears as a uniform lengthening of all spacing pulses and an equal uniform shortening of all marking pulses. Bias is caused by an improper relation between the levels at which the relay or other receiving device responds and the steady-state marking and spacing levels of the signal. For example. Fig. 1(B) shows the signals of Fig. 1(A) as they might appear as a symmetrical wave on a line. With such a wave zero bias will be received when the currents at which the receiving relay operates from spacing to marking and from marking to spacing are symmetrically located with respect to the average of the steady- state marking and spacing currents. That is, zero bias will be received if the relay operates from spacing to marking and from marking to spacing at B-B, or if the relay operates from spacing to marking at A-A and from marking to spacing at C-C, or if the relay operates from spacing to marking at C-C and from marking to spacing a.t A-A. Negative bias will be received if the relay operates in both directions at A-A, and positive bias will be received if it operates in both directions at C-C. In Fig. 1 (C) is shown an unsymmetrical wave, in which the transient from space to mark is more rapid than that from mark to space. In this case, positive bias will result when the relay operates in both directions at B-B or at C-C, but no bias will result if the relay operates in both directions a.t A-A. In the remaining diagrams of Fig. 1 it is assumed that the relay operates in both directions at a level midway between the steady marking and spacing levels. Fig. 1(D) shows a wave in which the transients are of such duration that the steady-state value is not attained in the shortest pulse length. It will be seen that the operation of the relay is delayed less after a short pulse than after a long one, and that this is true whether the pulse be marking or spacing. This effect is known as negative characteristic distortion, and it tends to shorten short pulses and lengthen long pulses. When a series of unbiased dots (called telegraph reversals) is transmitted, a steady-state condition is reached, in which the delays become equal on all transitions. START-STOP RECEIVERS 209 Hence, the signals are received as sent. When biased reversals are trans- mitted, the longer pulses are further lengthened and the shorter pulses are further shortened, causing the bias of the received signals to be of greater magnitude than that of the transmitted signals. Fig. 1 (E) shows a wave in which the current overswings the steady-state value, and fails to complete the return to steady state within the duration of the shortest pulse. It will be seen that the operation of a relay will be (A) ' (C) A Fig. 1 — Signal diagrams illustrating causes of distortion. delayed more after a short pulse than after a long one, and that this is true whether the pulse in question be marking or spacing. This effect is known as positive characteristic distortion, and it tends to shorten long pulses and lengthen short ones. When unbiased reversals are transmitted, a steady- state condition is reached, in which the delays become equal on all transi- tions. Hence, the signals are received as sent. When biased reversals are transmitted, the shortening of the long pulses and lengthening of the short pulses causes the bias of the received signals to be less than that of the trans- mitted signals. 210 BELL SYSTEM TECHNICAL JOURNAL Fig. 1(F) shows a wave which performs a damped oscillation before settling to a steady state. This t>^e of wave tends to produce a negative characteristic effect on certain transitions and a positive characteristic effect on others. In general, if, on a given transition, the sum of all previous transients is such as to delay the operation of the receiving device, positive characteristic distortion is said to occur. If, on the other hand, the sum of all previous transients is such as to advance the operation of the receiving device, negative characteristic distortion is said to occur. Bias and characteristic distortion, considered together, are called "sys- tematic" distortion, because they occur with some regularity, and obey certain constant laws. There is another type of distortion that is not systematic. This is known as forluitoiis distortion. It may be caused by the effect of various interfering currents on the receiving device. Fig. 1(G) shows a wave upon which interfering currents have been superposed. It will be noted that, for a given magnitude of interfering current, the more sloping the wave is in the region of the operating level of the receiving device, the greater will be the resulting fortuitous distortion. Fortuitous distortion may also occur, in cases of extremely sloping wave- shape, due to the "indecision" of the receiving device, or, in other words, due to small variations of its effective operating level fiom signal to signal. Fig. 1(H) shows a wave that is affected by interfering currents and in which the mark-to-space and space-to-mark transients have different slopes in the region of the operating level of the receiving device. The interfering current therefore causes fortuitous distortion of different magnitudes on mark-to-space and space-to-mark transitions. It will be shown later that distortion of this type affects a start-stop receiver in a particular manner which differs from the effect of distortion of the type illustrated in Fig. 1(G). These, then are the generally-recognized components of telgraph dis- tortion. More complicated effects ensue when characteristic distortion occurs on waves having dissimilar transients in the mark-to-space and space-to-mark directions, but a consideration of such phenomena is outside the scope of an elementary explanation of telegraph distortion, and is not necessary to an understanding of the effects of distortion on the margins of operation of start-stop receivers. Start-Stop Displacements The basic principles of operation of start-stop receivers have been described in previous articles- •'. A brief review of these principles will, therefore, suffice here. The start-stop signal train consists of a start pulse, which is generally spacing, several selective pulses, each of which may be either marking or START-STOP RECEIVERS 211 spacing, and a stop pulse which is generally marking. The receiving mechanism is started by the transition at the beginning of the start pulse, and its speed is such that it arrives at the stop position before the end of the stop pulse occurs, and remains stopped until the succeeding start transition takes place. Thus any speed difference between the transmitter and receiver is prevented from cumulating for more than the duration of one signal train. Since the receiving device starts anew at each start transition, and the instants of selection of the selective pulses are spaced in time relative to the instant of starting, as shown in Fig. 2(A), the start transition acts as a basic reference point to which all other instants of time during the selective cycle may be referred. The advances and delays of the transitions of the start-stop signal train from their normal times of occurrence, relative to the start transition, are known as "start-stop displacements." Fig. 2(B) shows the four types of displacement that may occur: MB or "marking beginning displacement," which is the advance of a space-to-mark transition (beginning of a marking pulse) relative to the start transition; SB or "spacing beginning displace- ment," which is the delay of a space-to-mark transition relative to the start transition; SE or "spacing end displacement," which is the advance of a mark-to-space transition (end of a marking pulse) relative to the start transition; and ME or "marking end displacement," which is the delay of a mark-to-space transition relative to the start transition. Effect of Bias on Displacement Since bias affects all pulses alike, and since in the usual start-stop receiver the start transition is mark-to-space, the succeeding mark-to-space transi- tions of the signal train are not shifted relative to the start transition. Hence the total effect of the bias appears on the space-to-mark transitions. Positive bias causes MB displacement alone, as shown in Fig. 2(C). Nega- tive bias causes SB displacement alone, as illustrated in Fig. 2(D). The total range through which the selective periods may be shifted, relative to the start transition, without producing an incorrect selection is known as the orientation range of the receiver. Its limits are read on a scale calibrated from 0 to 100 in per cent of a unit pulse-length. Figure 3 is a graph of teletypewriter orientation range versus input signal bias, for a receiver whose range is from 10 to 90 on unbiased signals. Diagrams of this type are called "bias parallelograms." Effect of Characteristic Distortion on Displacement Characteristic distortion does not affect all pulses of miscellaneous signals alike, because, as explained above, the effect on each transition depends 212 BELL SYSTEM TECHNICAL JOURNAL upon the signal combinations that have previously been sent over the circuit. Hence the start transition and the transitions occurring between selective pulses are, in general, delayed by varying amounts. All four types of displacement shown in Fig. 2(B) occur, depending upon whether the transi- tion in question is mark-to-space or space-to-mark and whether it has been (A) (B) (C) (D) (E) (F) ~1 START 1 1 t 1 2 1 3 4 t n 5 1 STOP 1 + t t 1 1 1 ,,t, , , . 1 INS1 1"me| START 1 PANTS OF SELECTION 0 50 100 L mbJsbI m j^SBj iSE|MEJ -STOP DISPLACEMENTS jmbIsbJ 1 1 iJ MARKING BIAS 1 1 i 1 I i 1 i 1 BIAS 1 i 1 SPACING 1 1 T i 1 1 DISTORTION i 1 1 MARK NG "END i"i 1 1 1 1 SPACING "END DISTORTION" Fig. 2 — Diagrams illustrating start-stop displacements. 100 UJ X o \ z \, \, 50 z o \ \ \, z K O \ 0 -40 -20 0 +20 +40 BIAS OF RECEIVED SIGNALS PERCENT OF A UNIT PULSE Fig. 3 — The bias parallelogram. delayed more or less than the start transition. For example, if a space-to- mark transition is delayed less (on an absolute time basis) than the start transition, MB displacement occurs; if more, SB displacement. If a mark- to-space transition is delayed less than the start transition, SE displacement occurs, if more, ME displacement. 5 TA RT-STOP RECEI VERS 2 1 3 Maximum Displacements Caused by Characteristic Distortion The maximum MB displacement will occur when the start transition is delayed as much as possible and some space-to-mark selective transition is delayed as little as possible. This will take place, in the case of negative characteristic distortion, when as long a marking signal as is possible precedes the start transition and a combination of pulses as predominantly marking as possible precedes the space-to-mark transition in question. A marking signal sufficiently long to permit a steady state to be attained, followed by any signal train having the first selective pulse marking satisfies this condi- tion, as shown at "X" in Fig. 4(B), but it will be noted that the MB dis- placement extends into the start pulse, where, in the case of a start-stop receiver, no selection is made. Hence it will not affect the margin of operation of the receiver, provided it is not so large as to prevent the receiver from starting. This particular distortion will, however, affect a start-stop distortion measuring set^ or regenerative repeater which is so designed that measurements or selections are made during both the selective pulses and the start pulse. As far as a start-stop teletypewriter, in which no selection occurs during the start pulse, is concerned, the maximum MB displacement occurs on the fourth transition of the letter K following as long a marking sig- nal as possible, as shown at "F" in Fig. 4(B). This space-to-mark transition, being preceded by a spacing pulse of unit length which, in turn, was preceded by signals which are predominantly marking, is delayed for a short time, whereas the mark-to-space start transition, which was preceded by a long marking signal, is delayed for a longer time. Except in the case of unusual wave forms, there will be very little difference between the magnitudes of the displacements shown at "X" and "F" unless they are both very large, since the wave will usually attain steady state during the steady marking interval constituted by the first, second, third and fourth selective signal intervals. In the usual case of positive characteristic distortion, the maximum MB displacement will occur when the start transition is preceded by a combina- tion of pulses as predominantly spacing as possible, and some space-to-mark transition is preceded by the longest spacing signal possible in the start-stop code. These conditions are met by repeated, "BLANK" signal trains, shown in Fig. 4(E). The maximum SB displacement will occur when the start transition is delayed as little as possible, and some space-to-mark selective transition is delayed as long as possible. This takes place, in the case of negative char- acteristic distortion, when a combination of pulses as predominantly spacing as possible precedes the start transition and the longest possible spacing signal precedes the space-to-mark transition in question. As noted in the 214 BELL SYSTEM TECHNICAL JOURNAL ti < \ a. \ O s < UJ O z < -'< ; X / <->/ '/ 1- / cc / < / / □. o h- 01 n \, O s ) «f> / 1 (~ ujt;; ^ o ■< o: Soj u.t3 o ^2 > -1 •< z: VJ o UJ \l a: s, 1- \ tr < / Q. O 1— \ U1 N ^ V o \ s )- / o -1 < 1- z ui ^ SE 42 ME 44 Let the orientation setting be raised 2 per cent, to 51. Then the tol- erances are as follows: MB 42 SB 40 SE 40 ME 46 START-STOP RECEIVERS 221 The shift of orientation has increased the minimum tolerance (spacing bias) from 38 to 40. Any further shift would make the tolerance to spacing "end distortion" less than the tolerance to spacing bias. This setting is called the "center of fortuitous distortion tolerance," since at this point the receiver will tolerate the maximum amount of fortuitous distortion. If, on the other hand, bias is considered more probable than distortions which produce "end distortion" effects, the orientation might be adjusted to the point at which the tolerances to marking and spacing bias are equal. For example, suppose the orientation setting of the receiver under con- sideration were raised 1 per cent to 52. The tolerances would then be MB 41 SB 41 SE 39 ME 47 This setting is called the "center of bias tolerance," since at this point the receiver will tolerate the maximum amount of bias regardless of the sign of the bias. There is one more setting that is of interest. It is that at which the tolerances to marking and spacing "end distortion" are equal. Suppose the orientation of the receiver were lowered 4 per cent to 48. The tolerances would then be MB 45 SB 37 SE 43 ME 43 This setting is called the "center of end distortion tolerance," since at this point the receiver will tolerate the maximum amount of "end distortion" regardless of its sign. Calculation of Components of Internal Distortion Figure 6 illustrates how the components of internal distortion are deter- mined from measurements of distorted signals. Each diagram shows a portion of a teletypewriter character consisting of a start pulse, a marking selective pulse and a spacing selective pulse. The solid lines show an undis- torted signal. The dashed lines show the displacement of a transition due to internal bias. The shaded area defines the fortuitous effect which is skew; that is, the transition in question may fall anywhere within the shaded area during repeated transmission of the signal. The arrows below the figure show the extent of the displacement occurring on each transition due to the presence of a given displacement of the transmitted signals. The four types of displacement are of equal magnitude D. The arrows above the diagram designated Lb and Le show the lower limits of orientation with, respectively, 222 BELL SYSTEM TECHNICAL JOURNAL spacing bias and marking "end distortion" (SB and ME displacements). The arrow U b and U e show the upper limits or orientation with, respec- tively, marking bias and spacing "end distortion" (MB and SE dis- placements). Figure 6(A) shows the case of positive internal bias and positive skew; Fig. 6(B), positive bias and negative skew; Fig. 6(C), negative bias and posi- START I 2 -*»Ue -•>UB SE ME MBkJ-: (A) + BIAS +SKEW Hb' ^ "1 -Lc SB ■*^f SEi^ — Ub ■*Le \Mf MBi CB) + BIAS -SKEW ~L ►Ur — ^Lb _LL"i=^ SB SE IME ^ (C) -BIAS +SKEW -*-^B s^£|^= "I _jSB SEi< -*Le h^ -MiET" A (D) -BIAS -SKEW Fig. 6 — Use of distorted test signals in measuring inte nal distortion. tive skew; and Fig. 6(D), negative bias and negative skew. The following relationships hold, bearing in mind that MB = SB = SE = ME = D: Fig. Eias Skew Lb Ub Le Ue (A) -I- -I- D+s-b l-b-s-D D 1-Z» (B) + - D-h i-b-D D+(-s) ].-(-s)-D (C) - + D+s+i-b) l+(-b)-s-D D 1-D (D) - - D+(-b) l + (-b)-D D+(-s) l-(-s)-D START-STOP RECEIVERS 223 In any figure Lb — Le = s —b and U B — U E = —s—b Adding and subtracting, we find that: Internal bias = ^-— — " 2 2 oi Ue — Lb Ub — Lb Skew = — 2 2 U -\- L Any — - — is the center of an orientation range. Hence it may be stated that the internal bias is equal to the difference between the centers of toler- ance to "end distortion" and bias. It will also be noted that any — - — is half of an orientation range. When the test signal displacements determining the range limits are equal, the amount of tolerance equals — - — + D (assuming no curvature in the distortion parallelogram). Hence the skew is equal to the difference between the amounts of tolerance to "end distortion" and bias. For example, the receiver cited previously has the following characteristics Internal bias = 48 - 52 = -4% Skew = 43 - 41 = +2% Incidentally, this means that internal bias does not reduce the total bias tolerance of a receiver, but merely shifts the center of bias tolerance with relation to the center of "end distortion" tolerance. Hence the effects of internal bias may be compensated for, as far as the bias tolerance of the receiver is concerned, by setting the orientation at the center of bias tol- erance. However, internal bias does reduce the minimum "end distortion" tolerance of a receiver whose orientation is adjusted to the center of bias tolerance. "Switched" Bias When biased signals are produced by the action of a biasing current on a relay driven by a symmetrical wave, and the sign of bias is suddenly reversed during the transmission of a teletypewriter character, all the succeeding transitions of that character are affected, not by bias, but by "end dis- tortion." This is shown in Fig. 7, of which (A) shows the original unbiased signals, (B) shows the signals affected by bias which changes from positive to negative at time T, and (C) shows the effect on the same signals when the bias is changed from negative to positive. Signals such as these, in which the sign of bias is changed at intervals, are said to be affected by "switched bias." Since all four types of displace- 224 BELL SYSTEM TECHNICAL JOURNAL ment are present in equal magnitude in switched bias signals, the effect on a start-stop receiver resembles that of fortuitous distortion. Thus the center of switched bias tolerance is the center of fortuitous distortion tolerance and the amount of switched bias tolerance is the amount of fortuitous distortion tolerance. This center is also the center of orientation in a receiver having no curvature or symmetrical curvature of the displacement-vs.-orientation- limit characteristic. The switched bias tolerance is, of course, one-half the orientation range in a receiver having no curvature of the characteristic. In actual field practice, switched bias signals, applied at a central oflSce, are used as a test of tolerance of the teletypewriter at a subscriber station in combination with the subscriber loop. They provide a more accurate measure of transmission capabilities than an orientation range measurement with undistorted signals from the central office, since not only is the curva- ture of the distortion parallelogram taken into account, but the character (A) I START I CB) I START I [mB| 2 |_ (C) START I ;56 2 3 I 4 I 5 I STOP DECREASED CHARACTER LENGTH \ W\ 5 I STO^ [j INCREASED CHARACTER LENGTH 3 I 4 [mI| 5 I STOP j^ Fig. 7 — Switched bias. length changes in much the same manner as in signals affected with char- acteristic or fortuitous distortion. The components of internal distortion of a receiver may be estimated from bias and switched bias measurements, but they cannot be accurately specified thereby. Figure 8 illustrates the difficulty in separating bias and skew by means of measurements of the difference between the amounts and centers of tolerance to steady bias and switched bias. Figure 8(A) shows the bias and end displacement parallelograms of a receiver having -}-24 per cent bias and + 16 per cent skew. The center of tolerance to switched bias is 4 per cent above the center of steady bias tolerance and the steady bias tolerance is 4 per cent greater than the switched bias tolerance. Figure 8(B) shows the parallelograms of a receiver having +4 per cent bias and —4 per cent skew. Again, the center of tolerance to switched bias is 4 per cent above the center of steady bias tolerance and the steady bias tolerance is 4 per cent greater than the switched bias tolerance. Of course, the components of internal distortion can be measured by START-STOP RECEIVERS 225 observing both ends of the orientation range with positive and negative bias rather than observing the upper end with positive bias and the lower end with negative bias. This type of measurement is merely equivalent to using a fairly large percentage of bias and zero per cent of end distortion. The dis- advantage of this measurement is that no account is taken of the curvature (A) / \ ^^ ' y \ Cb= center of bias = 26 Tr= TOLERANCE TO BIAS =34 BIAS PARALLELOGRAM "END DISTORTION" PARALLELOGRAM DISTORTION TOLERANCE t 16 "To SKEW Cs= CENTER OF SWITCHED BIAS= 30 Ts= TOLERANCE TO SWITCHED BIAS = 30 Cc,-Cb= + 4 Ts-Tb=- 4 (B) Cb= CENTER OF BIAS = 4b Tb= TOLERANCE TO BIAS = +50 DISTORTION TOLERANCE + 4°7„ BIAS -4% SKEW Cs= CENTER OF SWITCHED BIAS = 50 Ts= TOLERANCE TO SWITCHED BIAS = 46 Cs- Cb = +4 Tc- Tn = -4 Fig. 8 — Switched bias measurements of the end displacement parallelogram, and hence the indicated values of tolerance may not be an accurate measure of the receiver's ability to receive distorted signals. Internal Fortuitous Distortion It is usually considered, in measurements of miscellaneous signals, that the difference between the maximum distortion tolerance and 50 per cent (the 226 BELL SYSTEM TECHNICAL JOURNAL latter being the tolerance of a perfect receiver) is due to internal fortuitous effects, even though part of it may be due to the effects of mternal char- acteristic distortion. Hence the internal fortuitous distortion is usually defined as the difference between 50 and the tolerance to bias or end dis- tortion, whichever of the latter may be the larger. For example, in the sample receiver considered on page 220, the internal fortuitous distortion is: 50 - 43 = 7 per cent ^ ^^ ^ V \ \ u o z N \ N, a. z o \l \ \ s 7* \ \ ^^ \ UJ a. o s \ \ \ \ — - — — +20 +40 BIAS OF TRANSMITTED SIGNALS PERCENT OF A UNIT PULSE ■WITHOUT CHARACTERISTIC DISTORTION ■WITH NEGATIVE CHARACTERISTIC DISTORTION Fig. 9 — Effect of negative characteristic distortion on bias parallelogram. Internal Characteristic Distortion In practice it is found that the relation between displacement and reduc- tion of margin is sometimes not strictly linear. Especially at large values of displacement, the reduction in margin is often greater than the displacement causing it. This effect is due to internal characteristic distortion, which causes an increase in the distortion of shortened pulses. Internal char- acteristic distortion, like any other form of characteristic distortion, is caused by the failure of some circuit or mechanical element to attain steady state before the occurrence of a succeeding transition. Figure 9 shows an example START-STOP RECEIVERS 227 of the bias parallelogram of a receiver sufltering from internal negative characteristic distortion. Some CoNSirs nations In\'Olved in the Measurement and Adjustment OF Start-Stop Receivers Because of the effects of characteristic distortion, it cannot be assumed that the ultimate tolerance of a receiver is equal to the sum of the displace- ment of the received test signals and one-half the remaining orientation range, especially if the latter is large. To attain accurate results, the ultimate tolerance must be measured with the orientation adjusted to the center of tolerance. For the same reason (the curvature of the "parallelogram" caused by- internal characteristic distortion) measurements of internal distortion on a receiver which is, itself, to be used to measure distortion should be made with displacements of approximately the same magnitude as the distortions which the receiver is to measure. In a receiver which is to be used to measure small distortions, we are interested in the properties of the linear portion of the parallelograms. Hence we measure the receiver's internal bias and skew using small amounts of displacement in the measuring signals. The internal fortuitous distortion may generally be neglected, since it does not affect the shape, but only the size, of the distortion-vs-margin char- acteristic. On the other hand, in a receiver which is to be used for receiving signals we are interested not so much in the shape of the characteristic as in the ultimate tolerance to telegraph distortion at an optimum setting of the orientation mechanism. For this reason, a receiver destined for service use is best tested with signals containing fairly large displacements. Internal fortuitous distortion is deleterious in such a receiver, since it decreases the tolerance to displacement of all kinds. Skew, depending upon its sign, affects the tolerance to either space-to-mark or mark-to-space displacements. It should be realized that the removal of skew does not necessarily improve a service receiver. In the case of bias or characteristic distortion the introduction of distortion of a given sign will remove internal distortion of the opposite sign, and thus improve the performace of the receiver. But since skew is the difference between two fortuitous distortion effects, it may be removed either by reducing the larger or increasing the smaller effect. The former procedure will increase the receiver's total tolerance to distor- tion, whereas the latter will reduce it. In practice bias tolerance is generally considered to be more desirable than "end distortion" tolerance. The reason for this is that most transmission circuits suffer from some bias (of unpredictable sign and amount) which uses up some of the receiver's bias tolerance but none of its "end distortion" 228 BELL SYSTEM TECHNICAL JOURNAL tolerance. This is why the orientation of a service receiver is generally- adjusted to the center of bias tolerance, and small amounts of internal bias or negative skew are not considered objectionable, since they do not affect the tolerance to bias at the center of bias tolerance. By the same token, the presence of positive skew, which indicates a lowered bias tolerance, usually calls for a readjustment of the receiver to reduce the fortuitous effect on the space-to-mark transitions. As explained above, removing the skew by introducing a fortuitous effect on the mark-to-space transitions will not, of course, improve the bias tolerance. It is the present practice in the field to specify a minimum bias tolerance about 5 per cent greater than the minimum permissible "end distortion" tolerance, the orientation being adjusted to the center of bias tolerance for both measurements. Some Causes of Internal Distortion Up to this point internal distortion has been considered without regard to its probable causes. The more obvious causes will be found to be analogous to those which produce equivalent distortions in telegraph trans- mission circuits. Bias will result when an element (whether electrical, mechanical, or elec- tronic) of a receiver possesses dissymmetry toward marking or spacing. For example, a mechanical element may travel more slowly from spacing to marking than from marking to spacing and thus cause spacing bias, or its range of travel may be divided unequally into marking and spacing portions, thus producing an equivalent efi'ect. Characteristic distortion will result when an element (whether electrical or mechanical) of a receiver fails to attain a steady state before being acted upon by a succeeding transition, or otherwise depends, in its action, upon the previous history of the signal train. An example of characteristic distortion is found in the 20-milliampere holding magnet selector when it is equipped with a resistive shunt. In this type of selector the armature is actuated by a cam, which presents it to the pole-face at about the middle of each pulse, and then disengages it. The armature is then free to release or remain operated, according as the received pulse is spacing or marking. The shunt that is normally used presents so low an impedance to the magnet winding that the motional impedance effect which is produced by the sudden mechani- cal presentation of the armature to the pole-faces causes a sizeable reduction in the magnet current. In the case of a short marking pulse, the current fails to attain steady state before the next mark-to-space transition occurs. The magnet therefore releases sooner than it does at the end of a long marking pulse, during which the current has had time to attain steady state. It will be seen that this is really a characteristic distortion effect, since it is due to a failure to reach steady state and depends upon the previous history START-STOP RECEIVERS 229 of the signal train. However, when miscellaneous signals are being received the effect appears similar to a fortuitous distortion occurring on mark-to- space selective transitions, and hence it is usually thought of as negative skew. Fortuitous distortion will result when an element is irregular in its action, and if such action is more irregular on one type of transition than on the other, the result will appear as skew. For example, irregular action of the receiving clutch affects the selector alike in regard to all selective transi- tions, and appears as internal fortuitous distortion. Another source of internal fortuitous distortion is the period of indecision that occurs during the passage of a selective element past a locking member, at which time the choice between marking and spacing is largely fortuitous. A common cause of skew in teletypewriters may occur in the following man- ner: If the armature stops are so adjusted that, for example, the armature travel is greater on the marking side than on the spacing side of the armature lock, positive internal bias results. If, now, this bias is compensated for by so adjusting the armature air-gap and retractive spring tension as to cause the receiving magnet to operate in a negatively biased manner (rather than by correcting the improper armature travel), the armature will be forced to op- erate in a region of the operating wave that is more sloping than the region in which it releases. Hence, it will operate more irregularly than it releases, and thus will be affected by positive skew. Selector Action Over and above the sources of internal distortion which are analogous in effect to sources of distortion encountered in telegraph transmission cir- cuits, there is another whose action in causing internal distortion is not so obvious as those just described. This source of internal distortion may be termed "selector action,''^ and it depends upon the relation between the operating time of a selector element and the period of time allowed for said element to act. For the purpose of explaining the effect of time relations within the selector on internal distortion, selector mechanisms may be classified as of three basic types: M, S, and P. In a mechanism of type M each selector is initially in the spacing condition and either remains spacing or operates to marking when subjected to the action of the corresponding received signal element. When it attains the marking condition it becomes locked for the duration of the character. Early types of start-stop prin'.ers having an individual selector magnet for each pulse of the code and employing a separate receiving distributer,^ are illustrative of type M. In a device of type 5 each selector is initially in the marking condition and either remains marking or operates to spacing when subjected to the action of the corresponding received signal element. When it attains the 230 BELL SYSTEM TECHNICAL JOURNAL spacing condition it becomes locked and cannot again operate to marking during that character. The Siemens-Halske five-selector teleprinter^ is an example of this type. In a mechanism of type P, the selector may be in either the marking or spacing condition initially, according to the type of the previous signal ele- ment to which it has responded. When subjected to the action of a received pulse the selector may go in either direction, and it remains responsive to the action of the signal during the entire selecting interval. The No. 14 and No. 15 teletypewriters^ (not equipped with holding magnet selector) of the Teletype Corporation are examples of type P. ^^ k^ SE (A) TYPE M RECEIVER ■f^^^f K^ SE (B) TYPE S RECEIVER K^ m^ I -n -^^^ ""^ .H^^^^-f (C) TYPE P RECEIVER Fig. 10 — Effect of selector action on internal distortion. Figure 10 (A) illustrates the action of a type M selector. A portion of a teletypewriter character is shown, consisting of the spacing start pulse, a marking first selective pulse and a spacing second selective pulse. The undistorted signal is shown in solid lines. The maximum amounts of mark- ing and spacing bias that the receiver will tolerate are shown by dashed lines and are designated MB and SB. The limiting amounts of marking and spacing end displacement are shown by dotted lines and are designated ME and SE. Above the signal train is shown a schematic representation of the action of the selective system. The periods of time T are those during which the selector is subject to the action of the received signal, and / is the time that the selector must be subjected to the operative force in order that it START-STOP RECEIVERS 231 operate. The line A-A indicates the boundary between the marking and spacing positions of the selectors. In this type of receiver, as mentioned previously, when the selector crosses to the marking or upper side of line A-A it becomes locked and cannot again go to spacing even though the sig- nal should subsequently become spacing during the selective period T. It will be noted that the limits of end displacement tolerance occur at time t after the beginning of the selective period. This instant is sometimes called the "instant of decision for end displacement." On the other hand, the limiting tolerances to bias are determined at a time / before the end of the selective period, sometimes known as the "instant of decision for bias." If the selective periods were advanced relative to the start transition by lowering the orientation until the bias tolerances were equal, the instants of decision for bias would correspond with the center of bias tolerance. If, then, the selective periods were delayed, by raising the orientation, by an amount T — 2t, the instants of decision for end displacement would corre- spond with the center of end displacement tolerance. Since the difference between the center of end displacement tolerance and the center of bias tolerance is equal to the internal bias of the receiver, it will be obvious that the internal bias is also equal to the difference between the instant of deci- sion for bias and the instant of decision for end displacement. In this type of receiver the internal bias is T — 2/, and will be positive, zero, or nega- tive according as 2/ is less than, equal to, or greater than T. Figure 10 (B) shows the action of a type S selector. Here the instant of decision for bias occurs at time t before the end of the selective period and that for end displacement at time t after' the beginning of the selective period. Hence, the internal bias is equal to 2/ — T. The action of a type P receiver is illustrated in Fig. 10 (C). It is assumed in this figure that the selector operates toward marking at the same rate as toward spacing, since the effect of unequal rates of operation has been de- scribed previously. In a selector of this type, both instants of decision occur at time / before the end of the selective period and hence the internal bias is not dependent upon the relation between T and t. If, however, / is so long that the selector cannot pass from one extreme of travel to the other, attain a steady state, and return to the center position within time T, a sort of characteristic distortion occurs, in which the instant of decision de- pends upon whether the selector began the selective period in the same or the opposite condition from that finally selected. In measurements of miscellaneous signals this appears similar to a fortuitous effect, since it decreases all tolerances equally. Hence it is usually considered as internal fortuitous distortion. Receivers equipped with holding magnet selectors are of Type S, since the armature may be released, but not operated, by the magnet. In this 232 BELL SYSTEM TECHNICAL JOURNAL type of mechanism, the armature generally drives a subsidiary selective member, and the time T extends from the instant at which the armature is disengaged by its operating cam until the instant when the subsidiary selec- tor becomes locked. As this period is often long in relation to the magnet releasing time / 1 and the subsidiary selector operating time / 2, holding magnet selectors are often subject to negative internal bias. In those mechanisms in which the subsidiary selector is flexibly coupled to the magnet armature, the former's operation is of type P. It, therefore, may be subject to the characteristic distortion effect noted in the description of type P operation, except that the effect, when it occurs in this type of mechanism, affects only the instant of decision for end displacement and hence resembles negative skew rather than internal fortuitous distortion. An interesting, but somewhat unusual, effect occurs in any receiver, of whatever type, in which the lengths of selective period or selector operate time, or both, differ for the various selective pulses, or in which the spacing of the selective periods is improper. In a case of this sort, the receiver ex- hibits an internal bias equal to the difference between the average instant of decision for bias and the average instant of decision for end displacement, an internal fortuitous distortion equal to the variation of the instant of decision having the smaller variation, and a skew equal to the difference between the variations of the instant of decision for bias and the instant of decision for end displacement. Conclusions A working knowledge of the effect of telegraph distortion on the margins of operation of start-stop receivers is essential in dealing with a plant in which the use of teletypewriters, regenerative repeaters and start-stop dis- tortion measuring sets is as widespread as it is in the Bell System. When a major portion of the communication system operates on a start-stop basis, it is desirable that transmission measurements be made on the same basis. The knowledge of this subject that has been gained in recent years has made possible many improvements in technique both in the field and in the laboratory, and these have led to corresponding improvements in the mecha- nisms used in telegraph service. The analysis of new start-stop devices may now be carried out efficiently and accurately, and this often permits the formulation of suggestions leading to improved operation of the devices. The general level of service excellence has been raised by the setting up of criteria for the distortion tolerances of station teletypewriters, regenerative repeaters and other start-stop devices used in service, including those pro- vided for switching. The sources of distorted test signals that are now available are useful not only in measuring the tolerances of service receivers. START-STOP RECEIVERS 233 but also in determining the characteristics, and hence the accuracy, of start-stop distortion measuring sets and monitoring teletypewriters. Finally, there has resulted an improved ability to analyze and predict the performance of transmission links from the results of distortion measure- ments made on a start-stop basis. References 1. "Measurement of Telegraph Transmission," H. Nyquist, R. B. Shanck, S. I. Cory, Jour., A. I. E. E., March 1927, p. 231. 2. "Fundamentals of Teletypewriters Used in the Bell System," E. F. Watson, Bell Sys. Tech. Jour., October 1938, p. 620. 3. "A Transmission System for Teletypewriter Exchange Service," R. E. Pierce and E. W. Bemis, Bell Sys. Tech. Jour., October 1936, p. 529. 4. "Recent Developments in the Measurement of Telegraph Transmission," R. B. Shanck, F. A. Cowan, S. I. Cory, Bell Sys. Tech. Jour., January 1939, p. 143. 5. "Der Spielraum des Siemens — Springschreibers," M. J. de Vries, Telegraphen-und Ferns prech-Technik , January 1934, p. 7. CHAPTER XIII* The Mounting and Fabrication of Plated Quartz Crystal Units By R. M. C. GREENIDGE 13.1 Introduction THIS paper is one of a series on piezoelectric quartz plates and deals primarily with the methods employed in mounting crystal plates operating up to approximately one megacycle for practical utilization in communication equipment. The theoretical aspects of mounting crystals have been covered in Chapter VII. The discussion is confined to plates^ having definite nodal lines or points, such as +5° and —18° 25' X cuts, GT, CT, DT, MT and NT cuts. The mounting of high-frequency crystal plates such as AT and BT cuts, which vibrate in thickness shear modes, is not included. It should also be noted that the subject matter is treated descriptively and that no attempt is made to go into the more intricate details of design or to give performance characteristics. These matters will be dealt with fully in a later paper. The designs and methods outlined are up to date for each type of unit, the results of many years of development on the part of Bell System engineers to evolve practical designs for commer- cial manufacture and use. Expanding on the contributions of the early investigators mentioned by W. P. Mason in Chapter I,^ these engineers had, in the ten years prior to 1939, worked out practical designs and devel- oped suitable tools and processes for wide commercialization in telephone appUcations. In the last five years, under the impetus of war, further improvements have been made in the design and manufacture of crystal units, particularly those for use by the Armed Forces. The term "Crystal Unit", originally adopted by the Bell System to desig- nate the complete assembly of a crystal plate in its mounting and case, has now been standardized quite generally in the art, replacing a variety of names by which these devices were formerly called. The basic design fea- tures of a crystal unit involve the use of: 1. Electrodes, on or near to the crystal surfaces for impressing voltage across the plate, * Chapters IX, X, XI and XII, which will be included in a forthcoming volume are omitted from the Technical Journal because they deal largely with details of manufacturing operations. 1 "Quartz Crystal Applications", W. P. Mason, B.S.T.J., Vol. XXII, Page 191, July 1943. 234 PLATED QUARTZ CRYSTAL UNITS 235 2. Supports for holding the crystal plate in its mount, and 3. A sealed outer case having the necessary terminals, and provisions for incorporating the unit electrically and mechanically into the apparatus. Two distinct types of crystal units have been evolved, one embodying the use of pressure pins or anvils for supporting and holding the crystal plate and the other involving the suspension of the cr}-stal plate by means of fine wires.- These designs are known, respectively, as the Pressure T^'pe and Wire Supported Type, and will be discussed later under these headings. However, there are several details of fabrication common to both types which can best be discussed at this point. Fig. 13.1 — Pressure-type holders. Irrespective of the type of mounting, realization of the desired perform- ance in a crystal unit depends to a considerable extent on the processing of the quartz plate itself. Previous articles^' ^ have brought out the signifi- cance of such factors as the precision of angular orientation and linear dimensions on the fundamental characteristics of the plate. The plate must also be virtually free of impurities or imperfections.^ In the prepara- tion of a quartz plate it must be lapped using increasingly finer abrasive materials until the final dimensions are reached. Depending upon the type of crystal unit. No. 400, No. 600 carborundum or finer abrasives are now * A. W. Ziegler, Patent 2,275,122, March 3, 1942. ' "The Use of X-Rays for Determining the Orientation of Quartz Crystals", W. L. Bond and E. J. Armstrong, B.S.T.J., Vol. XXTI, Oct. 1943. ■> "Raw Quartz, Its Imperfections and Inspection," G. W. VVillard, B.S.T.J., Vol. XXII, Oct. 1943. 236 BELL SYSTEM TECHNICAL JOURNAL employed for the final stage grinding. Following this, the plate is thor- oughly cleaned by acid treatments or by the use of solvents and detergents followed by copious washing. It is then etched in commercial hydrofluoric acid to remove all loose particles of quartz that might have remained on the surfaces or in the crevices after cleaning. Etching also smooths off the roughness of the ground surfaces. The effect of this treatment reduces energy dissipation in the plate itself and increases by many times the effi- ciency of the crystal units. Etching also improves the stabiUty of perform- ance of the crystal unit. Standard designs of crystal units require etching of the plates, uniformly on all surfaces, for a period of thirty to forty min- utes. For units of highest precision and efficiency longer etching periods are employed. The electrodes employed with types of crystal units being described con- sist of metallic coatings, generally aluminum, silver, or gold deposited over the major surfaces of the crystal plate. These coatings are applied by the evaporation process which results in an extremely thin and uniform coating of metal having excellent adherence to the quartz. With reference to the mechanical supporting members for the plate, it has been brought out that such supports should be confined as closely as possible to the nodal points or nodal lines where the motion for all practical purposes is zero. It is common practice for the supporting members to be made of metal so that they will serve also as a means of making electrical connections to the electrode coatings on the surfaces of the plates. 13.2 Pressure Type Crystal Units This type of crystal unit was initially developed for use in telephone filters. Up to about five years ago it was employed in virtually all com- mercial designs of filters. Depending upon the mode of vibration and size of the crystal plate the design of the mounting varies. However the prin- ciples employed for clamping are essentially the same in all cases. Where small longitudinal or face shear plates are involved one pair of pressure pins is used unless two are required for electrical reasons as explained later. For medium size plates of the same type or for face flexure plates, two pairs of pins are employed. In the case of large low-frequency longitudinal plates double anvils are used instead of pins in order to obtain firmer clamp- ing of the plate to prevent translation or rotational movement which might cause wear in the electrode surface at the pressure point with resultant variations in frequency and resistance. The blocks are usually composed of molded steatite and the springs for exerting the necessary pressure are of ph osph or-br onze . Figure 13.1 (B) shows a pressure mounting for holding four crystals which have single coatings on each of their major surfaces. The main require- PLATED QUARTZ CRYSTAL UNITS 237 ^" 238 BELL SYSTEM TECHNICAL JOURNAL ments which must be met for such a mounting are small areas, accurate alignment, and adequate pressure to hold the plate in place. Some de- signers make slight indentations in the quartz at the point of contact to improve the mechanical stability of the plate. For crystal plates of the order of one-half inch square or smaller, points having an area of about 10 mils in diameter are employed and the pressures used for holding the plate range from one to two pounds. For larger plates correspondingly larger areas of points and increased pressures are employed. The accuracy of alignment required for ten mil points is of the order of two or three mils. This is obtained in the mountings shown in Figure 13.1 by using con- centric sleeves for holding the points which are brought into alignment by means of a straight rod and then cemented in place. One of the points is fixed while the other point slides in its sleeve and the pressure required is obtained by the spring which presses on the outer end of the sliding point. In balanced filter structures it is desirable to use crystal plates with the coating on each side divided into two equal areas. This reduces to one-half the number of plates that would otherwise be required. In mounting plates with divided coating, it is necessary to provide a mounting which makes double contact on each side of the plate. Figure 13.1 (A) shows a pressure type mounting which accomplishes this. This is the mounting which has been used for several years in holding the plates used for the 75-type crystal channel filters^ for the standard terminal common to all broad-band tele- phone systems. The crystal is mounted in the holder in such a way that the two pairs of points clamp the crystal along the nodal line. The rectangu- lar dimensions of the points used for this type mounting for crystals oper- ating in the frequency range from 60 kc up to 120 kc are about 35 mils long in the direction of the nodal line and from 10 to 15 mils wide. A very important requirement for such a mounting is that the flat area of the points on each side of the plate fall in the same plane. This is accomplished by a precise milling operation after the points are assembled in the mounting. The pressure applied to the pair of points is furnished by the flat spring shown and is equalized by the action of the roller centrally located under the springs. The pressure employed is of the order of four to five pounds for each pair of points. The most commonly used coating for crystal plates held in pressure-type mountings is aluminum.^ Aluminum has been found to be most satisfactory for this type of unit because its hard surface is more resistant to wear at the points of clamping than other metals such as silver, or gold. Except for a few designs, which are mounted in sealed metal or glass * "Crystal Channel Filters for Carrier Cable Systems," C. E. Lane, B.S.T.J., Vol. XVII, Page 125. ^ The details of processing aluminum-coated crystals are similar to those described for silver-coated crystals in paragraphs 13.42 and 13.43. PLATED QUARTZ CRYSTAL UNITS 239 conlainers, pressure-type crystal units are not sealed in individual con- tainers. However, the entire filter in which they are employed is dried and sealed off after filling with dry air. In pressure-type units of this type, variations in frequency of the order of .01% may be expected if crystals are transferred from one mounting to another or relocated in the same mounting. Consequently, if high-frequency precision is desired, it is necessary to make the final frequency adjustment with the crystal located in its final position. Due to the inherent difficulties of adjustment, coupled with the close manufacturing tolerances on parts, and precision adjustments necessary for the holders during assembly, these designs have been virtually discarded in favor of wire supported designs. However, more recent developments by J. F. Barry on pressure-mounted- type units have brought forth some new ideas which might prove in for wider future application. 13.3 Wire Supported Crystal Units This type of mounting is being used extensively on crystal applications and has superseded the earlier pressure-type units. Various designs of this type of crystal are shown on Fig. 13.2. The wire-mounted crystal possesses the definite advantage in that after the supporting wires are attached to the plate, they remain fixed in position throughout the subsequent manu- facturing process thus facilitating adjustment of the frequency or the frequency-temperature characteristic. Moreover, the supporting wires can be formed so as to provide a spring mounting for the crystal plate which protects it from any shocks or vibration it may encounter in shipment or use. In the wire-supported design the suspension wire is also employed as a means of connecting the electrical circuit to the electrode plating on the crystal. By virtue of the solder bond between the wire and the electrode, this type of unit is free from the possibility of instability in frequency per- formance due to slight changes in position and variations in contact resist- ance prevalent in pressure type designs, and for this reason the stability and frequency precision of wire supported crystals is superior. From a manufacturing standpoint the wire-supported unit involves a greater number of processing operations than the pressure-type unit, but the adjustment operations are considerably easier. Moreover, it is possible to realize greater precision of frequency adjustment by a factor of at least two or even three. The crystals are also more uniform in their effective resistance. Since the mount or cage in which the crystal is suspended is comparatively inexpensive, there should in general be little difference in the manufacturing costs of the two types. Consequently, in the wire-supported crystal units a very appreciable improvement should be gained in perform- ance without increasing the cost of the unit. 240 BELL SYSTEM TECHNICAL JOURNAL 13.4 Fabrication of Wire Supported Unit 13.41 Silver Spotting 13.411 Application of Silver Paste Starting with the crystal plate, the first step in manufacture is to apply- silver spots to the surfaces of the plate. These spots serve as footings to which the supporting wires are ultimately soldered or sweated. They are placed on the nodal points or along the nodal lines of the plate in order to detract as little as possible from the intrinsic characteristics of the plate itself. The areas of the silver spots cover the range from about 40 to 90 mils in diameter depending upon the amount of solder to be used in attach- ing the wire to the plate. Before spotting the plates it is essential that they be free from any contamination such as grease or organic material that might affect the fusion of the silver spots into the surface of the quartz. One of the best methods to ensure cleanliness is to boil the plates in aqua regia, followed by copious rinsing in water. Detergents such as sodium meta-siUcate are also employed followed by a rinsing. The plates may finally be boiled in distilled water and carefully dried. Throughout the subsequent processes the plates should be handled with clean tweezers or gloved fingers and kept away from any source of contamination. Prefiring of the plates at 950°F prior to spotting has also been used as a positive way of ensuring freedom from any contamination that would affect the fusion of the spots, but this process is not necessary if the first mentioned process is properly controlled. In spotting, small quantities of a prepared silver paste are placed on the areas of the plate to which the wires will ultimately be attached. The paste consists of a compound of finely divided silver and low melting point glass (lead borate) thoroughly mixed with a suitable vehicle to facilitate appHca- tion. For spotting purposes it has been found that a paste having a specific gravity of between 2.3 and 2.6 gives best results. In use, the materials must be constantly agitated or stirred in order to prevent the solid ingredi- ents from settling out. This is important, for, unless the concentration of silver is maintained around 90 to 95 per cent of the solid matter, it will not be possible to obtain good wetting of the solder in making the wire attachment. The placement of the semi-liquid material on the plate is accomplished by means of a small stylus, the crystal plate being held in a clamp or vise and the stylus guided so as to place the material at the exact location on the plate as desired. A typical tool for doing this work is shown in Figure 13.3. The point of the stylus should have a slightly rounded end. With the rounded point the tendency of the paste to spread out is minimized and consequently the diameter of the spot is substantially the same as that of PLATED QUARTZ CRYSTAL UNITS 241 the stylus. The rounded stylus also results in a more uniform distribution of the material. The material is applied to the end of the stylus by spread- ing a small amount of the paste on a glass plate from which it is transferred to the stylus and then deposited on the crystal. The material on the glass plate should be wiped off and replaced quite often due to settling and drying out of the mixture, in order to insure uniformly good spots. Generally speaking, anywhere from two to six spots at the most should be possible from one loading of the transfer plate depending upon the speed of the operator. Regarding the character of the crystal surface, aside from cleanUness, and its effect on the ultimate strength of adhesion of the silver spot, experience ppiyi.ig aliVCf SpjLS. so far with all the various cuts of plates does not indicate that this is a factor. Spots have been found to adhere to poUshed surfaces as well as ground and etched surfaces with about the same degree of strength. 13.412 Firing oj Silver Spots Following the appUcation of the paste to the plates it is desirable to pre- dry the spots in order to remove the low volatile constituents of the vehicle prior to firing at high temperature. This may be done in a ventilated oven or over a hot plate at approximately 300°F for about 15 minutes. The plates are then placed in a furnace and heated up to between 975 to 1010°F and held at that temperature for a sufficient length of time to obtain good fusion of the spot to the plate. Ordinarily this reaction takes only a few minutes after the plate has reached the proper temperature. After firing, the plates are allowed to cool in the furnace to the point where they can be 242 BELL SYSTEM TECHNICAL JOURNAL removed without danger of the crystal cracking due to cold shock. It is essential to control the temperature of the furnace so that the temperature of the crystal plates does not reach 1063°F, otherwise the crystals may become electrically twinned and consequently useless. In order to avoid shattering of quartz plates due to thermal shocks while heating up and cooling, fairly long cycles have heretofore been specified. However, more recent experience has shown that much shorter cycles can be employed especially where small crystals are involved. Moreover, there are indica- tions that the faster heating, particularly during the last two or three hun- dred degrees temperature rise, results in better spots. During the firing Fig. 13.4 — Continuous-belt furnace for firing silver spots. operation the crystal plates may be placed on nichrome wire mesh trays or Pyrex dishes provided ample provision is made for air to circulate around the spots. This precaution is essential with the types of pastes employed as the baking reaction must take place in the presence of oxygen so that the lead borate will not be reduced to lead, leaving only a partially bonded mixture of silver and lead on the plate. This type of spot when encountered usually has a dull appearance after burnishing as compared with the bright surface obtained with a good silver spot, and is quite difficult to wet with solder. For firing silver spots, ventilated continuous-belt-type furnaces with open ports are being used with very satisfactory results. Figure 13.4 shows a furnace of this type developed by C. J. Christensen for this purpose. After the firing operation, the spots should be examined to ensure that they are satisfactory. Besides a visual inspection, it is desirable that a PLATED QUARTZ CRYSTAL UNITS 243 small percentage of the plates be used as a control sample to which mcjunt ing wires are attached and pull tested. Satisfactory spots should withstand for a few seconds a force of at least two pounds. The average pull-off strength of commercial attachments using 6-mil hooked or headed wires is between three and four pounds. Unsatisfactory plates can be reclaimed at this stage by stripping off the spots by means of aqua rcgia and ammonium hydroxide, and reprocessing in the manner described. 13.42 Sihcr Plating At this point of the process the surfaces of the plates are coated with silver electrodes by the evaporation process previously mentioned. Four milligrams per square inch of silver is the weight of coating generally em- ployed, which amounts to a thickness of .024 mil. Except for harmonic and other special types of crystal units, these coatings are required only on the major surfaces of the plates. However, during the evaporation procees, the silver is deposited to some degree on the minor surfaces or edges as well, and it is necessary to remove it. This process called "edge cleaning", is done by lapping the edges of the crystal on a flat plate covered with a mixture of jiumice or a finely divided abrasive such as No. 600 carborundum and water or kerosene in the form of a paste. Rubbing the edge of the plate lightly over very fine abrasive cloth is also satisfactory. The pumice is preferable, however, since while it readily removes the silver, it is much softer than quartz and consequently does not remove any material from the plate. The use of harder abrasives has a tendency to chip the edge of the quartz unless the operation is performed very carefully. After the edge cleaning is completed the plates are washed, dried and inspected by testing for insulation resistance at 500 volts d.c. to make certain that no conducting material remains between the silver coatings on the major surfaces. 13.43 Dhision of Coating For circuit reasons all but a few types of crystal units require a balance.! pair of electrodes on each side of the plate. Division of coatings along the longitudinal axis is essential on flexure mode crystals in order to make the plate vibrate in flexure. Typical divisions of coating can be noted on the crystals shown in Figure 13.2. In the case of the flexural crystal the divid- ing line is carried around the wire attachments in such a manner that each of the divided surfaces is connected to one of the wires. One method for dividing coating involves the use of a low voltage (two to three volts) impressed between the coating to be divided and a stylus.'^ When the fine point of the stylus is brought in contact with the silver plating and moved along the desired line of division, the silver is burned away, leaving a smal' ' W. L. Bond, Pat. M 2,248,057. 244 BELL SYSTEM TECHNICAL JOURNAL gap in the plating between 8 and 18 mils wide depending upon the point of the stylus. Following the burning operation, the plate is immersed in a photographic hypo solution to remove all traces of the burned residue after Fig. 13.5 — Tool for dividing plating (electric stylus). which it is carefully washed and rinsed in water and dried. In this process it is important that the plates are not kept in the h^-po solution for longer than two or three minutes, otherwise discoloration of the silver coating will PLATED QUARTZ CRYSTAL UNITS 245 result. The gap in then tested for presence of metallic particles by grad- ually impressing voltages up to 1000 volts a-c across it. If no fiashover occurs the division is satisfactory. If flashover occurs the voltage is maintained until the slivers are burned out. The hypo and burning treat- ments are repeated until a good division is obtained. Figure 13.5 shows an electric dividing tool developed for this purpose. In using this method it has been found that the arc at the point of the stylus may cause twinning of the quartz to a minor extent along the dividing hne. This effect is usually insignificant although it may be objectionable especially where pre- cise values of crystal inductance or frequency-temperature performance are required. Where more complicated divisions are necessary, as in the case of face flexure and harmonic plates, the electric stylus method is employed, although methods and tools for performing this operation by other means to avoid twinning are being developed. 13.44 Attachment of Wire Supporting Leads Phosphor-bronze wire is employed in wire supported crystal units pri- marily because of its high tensile strength, and excellent fatigue resistance characteristics. Five- and six-mil diameter wires are the most widely used sizes, depending upon the mass of the crystal plate, the desired electrical performance, and the severity of treatment it is Hkely to encounter in use. To facihtate soldering the wires to the spot on the crystal plate and to the crystal support system the phosphor-bronze wire is given a heavy electro- tinned finish. 59.5-34.5 per cent tin-lead eutectic solder saturated with approximately 6 per cent silver at 570''F is employed for attaching these fine wires to the silver spots of the crystals. This solder solidifies at approxi- mately 360°F with practically no mushy stage. The reason for saturating the solder with silver is to discourage migration of the silver in the spot to the solder during the soldering operation. Even with this solder it is advis- able to limit the time for heating of the joint to a minimum. One method of attaching the wires to the crystal plate is by means of a special machine developed for the purpose. Such a machine is illustrated in the photograph on Fig. 13.6. The wire is fed from a spool through the head in the movable arm. The head contains a wire guide having a hole only slightly larger than the diameter of the wire and a small vise for firmly clamping the wire. The crystal plate is clamped in the vise on the hot plate which is thermostatically controlled at approximately 240°F. The position of the arm carrying the wire is lined up with respect to the crystal plate by means of guides so that the wire will be placed exactly on the nodal point or line of the crystal. In making the attachment, with everything fined up, the wire is fed through the guide until it touches the spot on the plate and the vise closed. Since the curvature of the wire can never be entirely 246 BELL SYSTEM TECHNICAL JOURNAL eliminated the distance between the tip of the guide and the crystal plate is kept as small as possible. A small disc of solder is then punched in the press at the left, the little disc remaining in a round slot whose position is also lined up with respect to the arm carrying the wire. The movable arm Fig. 13.6 — Wire-soldering machine for straight or hooked wires. is then rotated until the wire is directly over the solder disc at which point it falls into a guide and comes down and spears the disc. The arm is then Hfted, picking up the solder at the end of the wire. The solder and the end of the wire is then wetted with rosin-alcohol flux and the arm rotated PLATED QUARTZ CRYSTAL UNITS 247 until it falls into its original position over the crystal plate. The solder is then, fused to the wire and plate by means of a special aluminum tipped soldering iron as shown in the illustration or by a controlled hot air blast focused on the joint to melt the solder. In this operation a fillet or conical button is formed around the wire attaching it to the silver spot. To pro- mote good wetting of the solder, the spot should be clean and well burnished. Rubbing the spot on a hard polished metal surface or burnishing with a blunt pointed tool of agate, are the best methods found so far. The hot air blast has now replaced the iron entirely in commercial use. It consists of a tube through which air at about one inch water pressure is passed over a hot filament and through a nozzle directed at the solder. The head of the filament is adjusted so that the temperature of the blast is just hot enough to melt the solder and complete the attachment in 10 to 15 seconds time. In using this method with large plates care must be taken to insure that the temperature of the crystal has reached that of the hot plate and that the blast is brought up to the plate slowly, for otherwise the heat shock of the localized blast may cause the crystal to crack. Fcr very small plates the use of a hot plate may be dispensed with if the hot blast is brought up slowly enough to preheat the crystal plate. Other means of melting the solder such as a hot radiant wire or ribbon or the use of a minute flame have been considered, but so far no extensive trials of these methods have been made. The advantage of the hot blast over the other methods mentioned is that it can be better controlled since little is left to the judg- ment of the operator. If an iron is used it must actually be touched to the solder with the possibility of displacing the position of the wire. Moreover, as already mentioned, the iron must be equipped with a special aluminum tip to prevent removal of solder from the joint on withdrawal of the iron. Considerable maintenance is required to keep such irons in satisfactory operating condition. 13.45 Type of Wire Attachments The type of attachment described above wherein the part of the wire embedded in the solder cone is straight was used in the first designs of wire- supported crystals. However, it was found that with such attachments vibration of the crystal plate caused breakage of the bond between the solder cone and the wire with resultant failure of the attachments, especially in large plates. Because of this the use of straight wires is recommended only for small size plates. In order to eliminate the above difficulty a little hook has been placed at the end of the wire embedded in the solder in order to obtain a better anchorage. The hook is formed in the wire by means of a special tool affixed to the soldering machine. The basic methods described for straight wires are otherwise used for this type of attachment. Instead 248 BELL SYSTEM TECHNICAL JOURNAL of spearing the little solder discs as with the straight wire, the solder is punched in the shape of a horseshoe and squeezed in place on the hook or positioned by tool with the hooked wire in place on the spot. Hooked wire attachments will withstand pulls of the order of three to four pounds before pulling off. Under severe vibrat ion hooked-wire attachments have the same tendencies as straight wires towards breaking away of the wires from the top of the solder cone forming a small crater in the latter. However, the crater does not progress deeply enough into the solder cone to impair their strength or cause failure under ordinary conditions. Fig. 13.7 — Examples of headed phosphor-bronze wire. The most recent development for wire supports involves the use of headed phosphor-bronze wires as worked out by A. W. Ziegler. In this procedure individual wire lengths are cut and one end upset in a cold heading tool which provides a Uttle cone-shaped head with a base of about 22 mils as shown in Figure 13.7 for 6-mil diameter wires. The head is carefully pre- tinned, leaving a small globule of solder at the end. Depending upon the size of the crystal plate, globules of 1000, 3000 or 7000 cubic mils of solder are used. The attachments are made in a wire soldering tool which attaches the wires to both sides of the plate simultaneously. This tool is illustrated in Figure 13.8. The prepared wires are fed into positioning guides, which PLATED QUARTZ CRYSTAL UNITS 249 2 fe 250 BELL SYSTEM TECHNICAL JOURNAL are aligned with respect to the crystal plate, which is properly located and held between sliding jaws as shown. Prior to the operation the silver spots are burnished and fluxed. The wires in their guides are then slid into con- tact with the plate. The hot blasts are raised from below so as to aim directly at the work. The solder is melted and the attachment completed in about 10 seconds when the bjasts are withdrawn. Slight pressure is maintained on the wires during this operation by springs in the guides to force the head of the wire to seat on the spot. A small fillet is obtained around the head making the solder cover an area of about 3S to 40 mils diameter. Crystal units using headed wire attachments have many advantages over those made with straight or hooked wires. The pull-ofF strength is more uniform and averages shghtly better than that of hooked wires, despite the fact that only a fraction of the amount of solder used with hooked wires is employed. With this type of attachment the cratering effects encountered with previous methods have also been eliminated. The reduced quantity of solder on the face of the crystal plate effects a decided improvement in the temperature coefficient of the crystal unit as well as in its efficiency and stability. Although the heading of the wires and the subsequent cleaning and tinning operations involve more work, the process of making the attach- ments is simpler and quicker, since the use of individual wires is better adapted to making all the attachments in one operation. Headed wire attachments are more uniform in size and shape and give a more workman- like finish to the job. This type of attachment has now replaced those using straight and hooked wires in virtually all designs of telephone type crystal units using wire supports. While the potential advantages of a headed wire type of attachment for crystal support wires had been known for many years, the practical exploitation of the idea depended on finding commercial means for producing the headed wires. The development of a suitable machine for this purpose was carried out by the Western Electric Company in close collaboration with the Laboratories. Figure 13.9 shows such a machine. The fine wire is fed through the lower mechanism to a die in which it is firmly clamped with a predetermined amount extending above the plate. This part of the wire is then cold-worked by multiple punches in the head of the machine until a conical shaped head is formed in the die cavity. As the individually headed wires are foimed they are cut off to a definite length and expelled as the vise is released and the next wire brought into position. Cold heading of the wires is necessary in order to retain the elastic properties of the phosphor-bronze springs employed in the suspension. The operation of the tool is simple after the precise align- ments of the die and punches have been made. Fig. 13.9 — Tool for cold-heading phosphor-bronze wires. 251 252 BELL SYSTEM TECHNICAL JOURNAL 13.46 Mounting the Crystal Plate After the suspension wires are affixed to the plate, they are then bent to serve as springs and to permit soldering into the cages as illustrated in Fig. 13.2. Two different types of springs are used, one of them involving one bend and the other two. The direction of the bends and the distances between them have been worked out so that the crystal will be displaced to about the same extent in all three directions for equal forces. The cages are of simple construction being made up of mica stampings and metal rods. The assembly of these parts is performed by welding little AMPLITUDE OF WIRE VIBRATION VERSUS NUMBER OF COMPLETE EXCURSIONS FOR .0063 INCH DIA. PH. BR WIRE 10* 10' 10' 10° NUMBER OF COMPLETE EXCURSIONS Fig. 13.10 — Characteristic performance of phosphor-bronze spring wires. eyelets, which are staked into the micas, to the rods. In the structures shown, the inside micas are provided with rectangular slots which limit the sidewise movement of the crystal plate from 25 to 30 mils. The end micas are spaced so as to limit the movement of the plate in the lengthwise direc- tion by the same amount. Aside from being used as parts of the cage, the micas therefore serve as "bumpers" to prevent excessive displacement, and possible breakage of the wires or plate if the crystal units are subject to extreme vibration or shock. Figure 13.10 is an experimental curve showing the minimum number of excursions made by wire-mounted crystal plates vibrated at different amplitudes before wire failure occurs for 6.3 mil phos- phor bronze spring wires with single bends. On the basis of these data, the PLATED QUARTZ CRYSTAL UNITS 253 chosen spacing of 25-30 mils between the crystal plate and the bumper should ensure against any service failure of the unit in this regard. In order to center the crystal laterally and longitudinally in the bumper system, the plate is assembled first in the cage by means of spacers. The fine wires are then soldered to the vertical rods or "straights" as they are called, and the spacers removed leaving the plate suspended in position. In order not to set up any strains in the junctions of the wire to the straight which might tend to displace the plate after this operation, the spring wires are usually pre-formed to come within about 5 to 20 mils of the straight. The junction is made by immersing the intersection of the wire and straight in a ball of molten solder. As the wires are withdrawn the ball of molten solder comes with them, solidifying in the air and thus joining the fine wire to the straight without strain. It will be noted from Fig. 13.2 that the wires to the longitudinal crystal are equipped with little weights close to the plate. This practice has been found desirable on virtually all types of crystal units to alleviate problems of wire resonance^ which arise in occasional units thereby causing high resist- ance as well as a shift in the frequency of the plate. Initially, while these effects were noted to some e.xtenl in the course of laboratory developments, it was not thought that they would be prevalent enough to warrant taking precaution to eliminate them by loading the wires, since they can usually be corrected by refloating and resoldering the crystal plate thereby changing the effective length of the wire. However, it has turned out that in manu- facture a large enough percentage of crystals contain resonant wires to warrant the use of weights. For low-frequency crystals (up to 200 kc) solder balls are placed on the wire at the desired location using a method worked out in conjunction with the Western Electric Company. The process is performed in somewhat the same manner as that described above for con- necting the crystal support wires to the straights, except that the weight of the solder deposited and the distance from the plate is more critically controlled. For higher-frequency crystals above 200 kc in which more pre- cise positioning of the weight is essential, small metal discs are employed. They are threaded onto the mounting wire and held in the correct position by a definite amount of solder on the back to obtain the desired loading. Since the free length of wire must be accurately controlled, the manufac- turing aspects of this job have been greatly simplified by the use of headed wires in which the variation in height of the solder cones is very small. The chart shown in Fig. 13.11 shows the weights of solder balls or discs and the position they should take on crystals having frequencies up to about one megacycle. It should be noted that the chart covers .0063" phosphor 8 "Principles of Mounting Quartz Plates," R. A. Sykes, B.S.T.J., April 1944. 254 BELL SYSTEM TECHNICAL JOURNAL bronze wire. For any other diameter, d, of phosphor bronze wire, the new distance X' = X /: d U063 Earlier, it was mentioned that the supporting wires for the plates were formed with one or two bends. In addition to the function of suspension these bends also introduce changes in impedance along the wire thus mini- LOCATION OF WEIGHTS ON MOUNTING WIRES OF OUARTZ CRYSTALS TO SUPPRESS WIRE VIBRATION DISTURBANCE NOTE: Information shown is for 6.3 mil Phosphor Bronze Wire (For 3.5 mil P-b wire, weight should be multiplied by .50 and located at .75X) For 5 mil P b wire, same weight should be located at .89X For 8 mil Pb wire, weight should be multiplied by IR and located at M2X Weight 30 40 50 60 70 8090100 200 300 400 500 600 FREQUENCY • KILOCYCLES Fig. 13.11 — Graph for determining placement of weights on wires for damping vibration. mizing the possibility of trouble due to wire resonance. The use of a greater number of bends in the wire would tend to accomplish the same result as that of weights. However, the use of weights is considered more practical and has been adopted. As a result of this change, it is possible to employ wire supports having only one bend in virtually all crystals. In low- frequency crystals (below 2 kc) where the wave-length of the flexural wave in the wire is relatively long it is unnecessary to use weights since the wire length can be controlled adequately by the termination of the support wire at the straight. Depending upon the frequency, the desired length of wire is obtained by using either two or three direction bends. PLATED QUARTZ CRYSTAL UNITS 255 13.47 Housing of Crystal Units For the pressure-type units first discussed no provision was made for protecting or sealing them other than the hermetically sealed containers in which all the other associated components of the filter were enclosed. How- ever, the wire-supported designs have been worked out so that each unit is sealed in its own individual container. Fortunately, the sizes of virtually all crystal units are in the range which permits the use of relatively inexpen- sive radio tube parts for these housings. There are man}' obvious advan- tages to the individually sealed unit. After adjustment and sealing it can be handled more readily in subsequent assembly operations. It is not subject to variations due to changes in ambient humidity and consequently does not restrict the assembly of apparatus to conditioned space. It can be made up and stored or shipped as an individual unit. It has a higher degree of stability. There is one small effect, however, in the case of units which are sealed in vacuum. Due to the absence of any gaseous medium around the crystal, a slight change in frequency is encountered when the tube is evacuated. However, this change is always the same for each particular type and size of crystal and can be allowed for in the final adjust- ment before seaUng. Most designs of crystals can be sealed in an atmosphere of dry air although better performance results from the use of vacuum. Some crystals must be sealed in vacuum for this reason. A decided advantage in favor of vacuum-sealed crystals is the elimination of acoustic effects from air resonance. Both metal and glass tubes are used for housing crystal units. Initially it appeared that metal tube radio parts were ideally adapted to crystal use, and it was felt that, instead of welding the stem to tube, this seaUng opera- tion could be done by soldering. However, it was found that while sound solder joints could be obtained, extreme precautions were necessary to protect the button-type glass seals, through which the leads emerge, during the pre-tinning and soldering operations. Even with such precautions, it would have been essential to include in every vacuum type tube a means of detecting whether or not a leak had developed. For air-filled tubes at atmospheric pressure this would not have been necessary since minute leaks can be tolerated with little likelihood of the crystal being afifected over a long period of time. The possibility of welding as is done in the case of radio tubes was considered but did not appear justified on the basis of equipment cost. Moreover, even with welding there still appeared to be problems from leakage and outgassing of the metal since, after the crystal is enclosed, the assembly cannot be exposed to high temperature to drive off adsorbed gases during the evacuation process. In view of these draw- 256 BELL SYSTEM TECHNICAL JPU_RNAL backs the use of metal tubes has been discarded in favor of glass tubes except in the case of a few special designs. The procedure of mounting a crystal unit on a stem and sealing it in glass is much the same as for a radio tube. Figure 13.2 shows crystal units" mounted on stems ready for sealing and also shows units sealed in glass and based. The extensions of the straights through the bottom micas are welded to the formed wires emerging from the glass seals. In the glass-sealing operation care must be taken not to heat up the assembly to the point where the solder attachments will be melted or even softened enough to permit the crystal to change position. To accomplish this it is necessary to use hot, sharp-pointed fires localized to the region where the seal will be made. The use of oxygen-gas flames is virtually essential to accomplish the seal quickly. Having the fires strike the bulb at tangency is also desirable. The ordinary type of glass-seaUng head for use with gas-air fires is not well adapted to this work since the rotating pillars require the fires to be held too far away from the work thereby necessitating larger flames and consequently more heating up of the crystal unit assembly. The screening effect of the pillars as they revolve also slows up the work of the fires thus increasing the over-all heating of the assembly. A special glass-sealing machine developed for seaUng crystal units is shown in Fig. 13 . 12. Immediately following the seal- ing operation the glass units should be placed in a suitable annealing box or leer where they can cool off very slowly. A large wooden block equipped with holes to admit the individual bulbs is convenient. The holes ma}^ be covered with a cloth to prevent air circulation. After the units have cooled they are placed on a vacuum pumping station and evacuated. During the first half hour of pumping they are enclosed in a heated oven in which the ambient temperature is maintained at about 240°F. This drives off any traces of moisture that might have entered the tube prior to sealing. Following the heating interval, the tubes are pumped for another half-hour during which time they will have cooled down to room temperature. At this point the pressure in the tubes should be at the minimum of which the pump is capable of attaining. This value should be at most 20 microns and preferably less. However, with a six or eight-tube station better than 15-20 microns is not likely to be attained unless a Hquid nitrogen trap is employed in the system for eliminating moisture. After the pumping period, vacuum-type units are sealed off, with pump running, by melting the glass tubulation with a fine-pointed oxygen-gas flame as close to the stem as possible. If air is to be admitted, the pump is closed off from the system and dry air admitted to the tubes after which they are sealed off. After testing the crystal unit to see that it meets its requirements, the unit is equipped with a base in the same manner as fol- lowed for radio vacuum tubes. ^^^^liiM^I 1- .i^^^l Fig. 13.12 — Glass-sealing machine. 257 258 BELL SYSTEM TECHNICAL JOURNAL 13.48 Stabilization of Crystal Units Despite the close dioiensional tolerances applying to the manufacture of the indi\ndual crystal plates, the exact frequencies are rarely realized in the mounted crystals. To bring the crystal to frequency it is therefore neces- sary to grind off minute layers from the lengthwise or widthwise edges of the plate depending on the mode of vibration. This adjustment, which causes superficial disruption of the quartz areas affected, results in unstable operation of the unit with respect to frequency and resistance. Unless the crystal plate is properly treated after these operations, considerable drift in these characteristics will take place, particularly during the initial service life of the unit To alleviate this condition, the crystal units are first rough- adjusted to the approximate frequency and then heat-aged in an oven which subjects the units to several heating and cooUng cycles between 240°F and 75°F. The units are then mounted in their cages as previously described and fine-adjusted after which they are again aged. This operation also tends to drive off any moisture which might be troublesome. With this type of accelerated aging the crystals are stabilized to the point where changes in performance can be detected only by the use of the most precise measuring equipment over long periods of time. Crystals so stabilized may generally be depended upon, at any one temperature, maintaining their frequency indefinitely within two or three parts per million provided the>- are not subjected to excess voltage. 13.49 Cleaning of Crystal Units Throughout the manufacturing process it is essential that every precau- tion be taken to keep the crystal plate and associated parts absolutely free from contamination and dirt. The rigorous cleaning necessary before the spotting operation has already been discussed. In all the subsequent opera- tions care must be taken to prevent the plates coming in contact with sub- stances that might tend to cause corrosion. Any particles of foreign matter that may have accumulated on the plate or wires before rough and fine adjustments should be carefully washed off. Otherwise the performance and life of the crystal may be adversely affected. A suitable method for cleaning crystal units before seaUng consists of washing and rinsing in chemically pure carbon-tetrachloride or other suitable solvent to remove grease, followed by washing and rinsing in hot distilled water at about 150°F. To facilitate removal of unwanted substances, the parts should be scrubbed gently with a soft brush or agitated in the solution during this process. The use of pure alcohol (95%) in addition to carbon-tetrachloride is also good for this process, but is not essential. The cleansed crystal units should be carefully dried out and protected from further contamination prior to the sealing operation. PLATED QUARTZ CRYSTAL UNITS 259 13.5 Conclusion In ending I should like to acknowledge the aid and useful suggestions given me by Mr. C. E. Lane and my other associates in preparing this article. I should also like to reiterate the fact that the status of the art as described was reached after many years of pioneering development by many engineers. In some cases the names of individuals associated with specific contributions of a major nature have been mentioned. CHAPTER XIV Effects of Manufacturing Deviations on Crystal Units for Filters By A. R. D'HEEDENE 14.1 The Effect of Deviations in the Characteristics of Crystal Units on Filter Performance THIS chapter emphasizes primarily the need for close control in the manufacture of crystal units for use in filters. The first telephone use of crystal units in the commercial manufacture of filters was made by the Western Electric Company in about 1936. To make such commercial manufacture practical, it was necessary to establish accurate design informa- tion and allowable manufacturing tolerances. The quantitative data collected for this purpose provided the chief source of material for this chapter. While the data is quite extensive, it will be observed that there are still some factors which must be treated qualitatively. While filter crystal units are like oscillator crystal units in that they must have low internal dissipation and a close control of resonant frequency, they are different in tliat many additional characteristics of the filter crystal units must also be controlled accurately. Two typical illustrations will demonstrate how characteristics other than resonant frequency and Q may react on filter performance. The first characteristic considered is the slope of the reactance with frequency curve in the vicinity of the series resonant frequency. This slope is sometimes referred to as the impedance level of the crystal unit. A con- venient measure is the inductance of the equivalent electrical circuit. When this inductance departs from its nominal value, the performance of the filter using the crystal unit may undergo appreciable change. This is particularly true of filters in which the schematic contains a lattice or some other type of bridge circuit with crystal units contained in all the bridge arms. For example, in Fig. 14.1 the solid curve illustrates the transmission characteristic obtained from a lattice-tj-pe crystal filter, in which both the series branches and the diagonal branches contain two balanced crystal units. High loss results from a close impedance balance between the branches of the lattice. Wlien the inductance of any of the crystal units departs from its nominal value, the bridge balance is disturbed and the transmission characteristic of the filter is changed. The two dotted curves of Fig. 14.1 illustrates the characteristics that result when the inductance 260 MANUFACTURING DEVIATIONS IN CRYSTAL UNITS 261 values of the crystal unit in eitlier branch depart from their nominal values by about one per cent. A negative departure in one branch results in about the same efifect on performance as a positive departure in the other branch. The difference between the two curves shown on Fig. 14.1 is that one as- sumes a positive departure and the other a negative departure for the in- ductance of a branch. Due to the close impedance balance which is required for these filters, the effect of small departures in resonant frequency will produce rather large variations in the transmission characteristic. For example, departures of about 10 cycles per second in the crystal units of either branch will produce variations in discrimination of about tlie same type and magnitude as those NORMAL INSERTION LOSS SERTIQN LOSS WHEN NOUCTANCE or CRYSTAL UNITS ARE IN ERROR -r" -4 -3 FREQUENCY -2 -I C +1 +2 IN KILOCYCLES FROM CARRIER Fig. 14.1. — The insertion loss characteristic of a crystal band-pass filter as affected by deviations in the inductance of the crystal units. illustrated in Fig. 14.1 for departures in inductance. On the other hand, if the crystal units of both branches exhibit equal departures the entire transmission characteristic will be shifted by the frequency departure of the crystal units, and there will be no loss in discrimination. Another way in which deviations in the properties of crystal units may react on filter performance is illustrated by the schematic and curves shown in Fig. 14.2. The schematic is the equivalent electrical circuit of a narrow band filter, using two balanced quartz crystal units. The filter is designed to provide a passed band of about 10 cycles per second with distortion of less than 0.2 db. The insertion loss characteristics show that the desired transmission can be obtained for various magnitudes of effective resistance as long as the resistances in the series and diagonal branches are equal. 262 BELL SYSTEM TECHNICAL JOURNAL However, if the effective resistance in one branch is twice as large as that in the other branch a highly distorted characteristic results as shown by the middle curve of Fig. 14.2. Both of these illustrations show that filter performance is degraded rapidly if the crystal units of the lattice have characteristics which depart from their nominal values by different extents for the two branches. A similar effect is produced when the temperature coefficient of resonant frequency for the crystal units in one branch differs frcm the temperature coefficient of the units in the other branch. Deviations occurring in a single unit may also aflfect filter performance. Such deviations include the presence of unwanted resonances of even weak amplitude, inadequate insulation re- sistance between the metalUzed coatings or unbalance between the halves of plates on which the coating has been divided. r-W^ — TTZTTHh 5 0 5 10 C. P. S. FROM 92 KC. Fig. 14.2. — Effect of deviation in the effective resistance of crystal units on the distor- tion characteristic of a crystal filter. The importance of controlling the electrical characteristics of the crystal units is indicated from the above considerations. It is pertinent to correlate deviations in the mechanical properties of the crystal unit with the devia- tions in electrical characteristics. This is the subject of the succeeding sections. Consideration is restricted to the plates commonly used in filters, that is, X-cut plates, vibrating in extensional or flexural modes, and GT-cut plates. 14.2 The Effect of Deviations That Occur in the Manufacture of Quartz Plates Quartz is an anisotropic material. Accordingly, plates cut from a quartz crystal exhibit elastic and piezo-electric properties which depend on the orientation of the plates with respect to the principal axes of the crystal. MANUFACTURING DEVIATIONS IN CRYSTAL UNITS 263 For that reason, any deviation in the orientation of the plates from nominal will affect the electrical characteristics cf the crystal units. Tn addition, these characteristics are affected by imperfections in the plates due to deviations in linear dimensions, to the presence cf flaws, or to the condition of the surface of the plates. The effects cf these deviations differ for various cuts of crystal plates, for plates of various shapes and for the various modes of vibration. In the following paragraphs, each type of deviation will be considered in turn and data will be presented to show its effect on the charac- teristics of crystal units using the various t>'pes of plates. 14.21 Deviations in the Angle of Orientation Accurate information is available on the effect of deviation in angle of orientation on the characteristics of X-cut plates vibrating in the exten- -0=0 C = 4 -rr t FARADS ^ 2^J?^ L. = tt:? * 9x10" HENRIES 8 d c = e dig our jg CYCLES PER SECOND 9«I0" Fig. 14.3. — Equivalent electrical circuit of piezoelectric crystal. sional mode. The relation between the electrical characteristics of this type of vibration and the properties of the quartz are shown in Fig. 14.3. This information, with minor changes, is reproduced from a preceding pubUcation.^ In Fig. 14.3: IjW and / are the length, width and thickness respectively of the plate; K is the dielectric constant; p is the density; (/12 is the piezo-electric constant; and ^22 is the modulus of comphance (inverse of Young's modulus). All these individual quantities are ex- pressed in electrostatic units. The quantities which depend on the orien- tation of the plates are the piezo-electric constant and the modulus of com- pliance. The symbols for these quantities usually are primed when they are used for a generalized orientation. When unprimed, the symbols desig- nate quantities measured along the principal axes. For X-cut plates, devia- tions of the plane of the major surface from the YZ plane have relatively 1 "Electrical Wave Filters Employing Crystals with Normal and Divided Electrodes", W. P. Mason and R. A. Sykes, B. S. T. J., April 1940, page 222. 264 BELL SYSTEM TECHNICAL JOURNAL small effect, while variations in the angle of rotation about the X-axis have a relatively large effect on these quantities. Mason has shown^ how the magnitudes of the piezo-electric constants and the moduli of compliance for any angle of rotation may be derived from their magnitudes along the principal axes of quartz. Using these equations and the magnitudes for the principal axes tabulated in a recent paper^ by Mason, dn and 522 have been calculated as a function of the angle of rotation of the plates about the X-axis. In turn, the frequency and inductance con- stants have been calculated as a function of the angle of rotation, using the relations shown in Fig. 14.3. Figure 14.4 is a plot of the frequency and in- ductance constants as a function of the angle of rotation for angles between about —70° and +70°. It shows how the inductance and resonant 3600 3500 3400 3300 3200 3100 3000 2900 2800 2700 2600 2500 \ / ^ s \ / / \ \ / 1 \ \ / /k \ \, / /' L \ \, / / 'l k. ^ / y / \ \ / / \ / r^ 280 260 ■^ I— o 220 i^;- I cc c 200 I 2 ^ 5 c/; I- ISO J;; 2 g 160 !40 2rO- 2700 z ^ 2680 z 8 2660 >- ^ 2640 o- 2620 "^ 2600 2580 2560 2540 2520 0 .1 .2 .3 .4 .5 .6 7 8 RATIO OF WIDTH TO LENGTH Fig. 14.5. — Frequency constant of the longitudinal mode of X-cut quartz plates as a function of their ratio of width to length. ^ \ \ +5* \ \ \ ^ N \ -18.5° ~^^ inductance with increase in width is much more rapid. With a ratio of a.xes of 0.6 the inductance decreases about as a square power, while with a ratio of 0.1 the decrease is about as the third power. The width dimension of the +5° plate has an appreciable effect on the temperature coefficient of the plate. Mason has shown* that while the temperature coefficient is zero for a long narrow bar, it increases quite rapidly as the width dimension increases, due to coupling between the face shear and the longitudinal modes. In the case of an —18.5° plate, coupling with other modes is relatively weak. Hence its temperature co- ^ "Motion of a Bar Vibrating in Flexure Including the Effects of Rotary and Lateral Inertia", W. P. Mason, Jour. Acous. Soc. America, April, 1935, pages 246-249. 268 BELL SYSTEM TECHNICAL JOURNAL efficient, which is about 25 parts per million per degree C, does not change appreciably with changes in width. For a +5° plate vibrating in its flexure mode, Fig. 14.7 illustrates measurements made on the variation of temperature coefficient with ratio of axes. For all these X-cut crystals, it mav be observed that deviations of 1% in the width dimension will not 10000 5000 2000 1000 500 200 100 50 20 10 +5° FLEXURE \ \ \ ^18.5° LONGI TUDINAL \ \ -^^^^\ \ V. \ \ +5.' .ONGIT UDINAL ^ ^ \ .15 .5 .6 .7 .8 .9 1. RATIO OF WIDTH TO LENGTH Fig. 14.6. — Inductance of the crystal units used in filters as a function of the cuts of the plates and their ratio of width to length. change the temperature coefficient by more than 5%. Such changes are usually negligible. 14.23 Internal Defects Internal defects in the quartz plates may have a large effect on their electrical characteristics. These defects vary so widely in type, size and concentration that it is impossible to predict the effects quantitatively. General comments regarding the results that may be expected for various MANUFACTURING DEVIATIONS IN CRYSTAL UNITS 269 defects are described in Chapter TV"'. The conclusions drawn there for oscillator plates are also applicable to filter plates. These are: (1) Evidence that a particular defect is perm'ssible in a given t.)npe of plate docs not prove that a similar defect is permissible in seme other type of plate, and (2) proof that a particular defect is permissible in a given type of plate can be obtained only by a statistical study. Seme qualitative statements can be made regarding the effect of mechan- ical flaws. Cracks result in instabilit)- of resonant frequency and effective resistance and must be avoided. The effect of inclusions or chips depends -18 - -14 -12 -10 / ^ l/ / / o -6 -4 -2 -0 .1 15 .2 .3 .4 .5 RATIO OF WIDTH TO LENGTH Temperature coefficient for +5° flexural crystal units as a function of the Fig. 14.7 ratio of width to length of the plates. on the size of these defects relative to the size of the finished plates and also on their location in the plate. Twinning in quartz may be either of the optical t>^e (Brazil twin) or of the electrical type (Dauphine twin)^. The effect of these two types of twinning on the performance of oscillator crystal units has been described thoroughly in Chapter V^. When optical twinning is present, the plate will exhibit the same elastic properties throughout, but the two portions of the plate will tend to expand 7 "Raw Quartz, its Defects and Inspection", G. VV. Willard, B. S. T. J., October 1943, pp. 338-361. * "The Properties of Silica", R. B. Sosman, Chapter XII. ^ "Use of the Etch Technique for Determining Orientation and Twinning in Quartz Crystals", G. \V. WiUard, B. S. T. J., January 1944, pp. 11-51. 270 BELL SYSTEM TECHNICAL JOURNAL and contract in opposite phase. Hence there is Httle change in frequency constant or temperature coefl&cient, but there will be a large change in in- ductance. Th change in inductance can be estimated roughly by compar- ing the twinned plate with an untwinned plate in which activity has been reduced by removing electrical charge from part of the surface. The area of the surface from which this charge is removed would be twice the twinned area and located at about the same position in the plate. It is believed that a small amount of electrical twinning is more serious than a similar amount of optical twinning, because the twinned areas are of opposite angular sense. Each of the two areas has a different modulus of compliance and the effective modulus of the plate has a value intermediate between the two different values of modulus. Therefore, the frequency constant of the plate will be intermediate between that of the desired cut and its electrical twin. For a small amount of twinning, the direction and rate of change of frequency can be estimated from the comparison shown on Table I between the standard filter cuts and their electrical twins. Table I Filter Plate • Frequency Constant— kc. m.m. Electrical Twin Frequency Constant — kc. m.m -18.5° +5° +51.1° (GT) 2560 2815 3280 + 18.5° -5° -51.1° 3120 2650 2610 This verifies the experimentally observed fact that for —18.5° X-cut plates, twinning increases the frequency, while for +5° X-cut and GT plates, twinning decreases the frequency. Even for small amounts of twinning the inductance will increase rapidly for plates of any orientation. When the amount of twinning becomes large, the equivalent inductance approaches infinity. That is, the crystal will not be set in motion by an applied voltage. The quantitative effect of twinning (probably electrical) has been meas- ured on one set of plates by R. M. Jensen. Figure 14.8 includes a photograph of the plates used, illustrating the extent of the twinning in each. All of the plates are —18.5° X-cut plates, having the dimensions 30.88 X 10.56 X .86 mm. The tabulation below the photograph compares the inductance and resonant frequency measured for each of the plates with the one, designated AN-3, which shows the least effect of twinning. While there is a good correlation between the amount of twinning in the plates and their change in electrical performances, it is not practical to estimate accurately the effect of a given amount of twinning. For this reason, crystal plates having any twinning should not be used for crystal units for filters. MANUFACTURING DEVIATIONS IN CRYSTAL UNITS 271 14.24 Etching The surface condition of the quartz plates also has some effect on crystal characteristics. This surface condition is determined in large part by tlie lapping operation used to obtain final dimensions. As described in Chapter Percentage Increase over Values Measured for AN-3 Plate Resonant Designation Inductance Frequency AN-1 +22.12 + .50 AN-2 +27.20 + .59 AN-3 0 0 AN-4 +3.03 + .04 AN-5 +3.12 + .01 AN-6 +276.54 + 5.01 AN-7 +1.58 -.03 AN-8 +14.21 + .37 AN-9 +738.00 +6.63 AN-10 +32.44 + .83 Fig. 14.8. — Effect of various degrees of twinning on the performance of — 18.5° X-cut quartz crystal plates. XIIIi", the plates are given a final lap with 400 or 600-mesh carborundum. This, in turn, is followed by an etching bath which removes foreign particles. A short etch, about eight minutes in 47% hydrofluoric acid, has been found adequate to ensure firm adherence of the metal coating to the quartz. On i" "The Mounting and Fabrication of Plated Quartz Crystal Units," R. M. C. Green- idge, this issue of the B. S. T. J. 272 BELL SYSTEM TECHNICAL JOURNAL the other hand, the use of a relatively long etch, 30 minutes or more, is desirable when a high Q is desired. The long etch also results in an im- proved stability of the resonant frequency as a function of current. This will be discussed in a subsequent paragraph. A disadvantage of a long etch is the difhculty of controlling the etching process within close toler- ances. The variations in rate of removing material may be sufiScient to affect the uniformity of the linear dimensions of the plates. These factors indicate that etching is an important process in preparing crystal plates. A close control must be maintained on the strength of the acid, the uniformity with which the surfaces of the plates are exposed and the duration of the exposure. 14.3 The Effects of Deviations during Fabrication of Wire- Supported Unit As described in Chapter XIII'", two types of mountings have been de- veloped for supporting crystal plates, the Pressure Type and the Wire- Supported Type. The wire-supported type of mounting is the more recent development and has resulted in crystal units which have a much higher degree of stability and can be reproduced within much closer tolerances than the units using the pressure type of mounting. Since this chapter is concerned chiefly with the problem of obtaining a high degree of precision in crystal units, the discussion is restricted to the wire-supported type of mounting. 14.31 Silver Spotting For the wire-supported type of mounting the first operation is to bake small silver spots on the surface of the crystal plates. In the application of these silver spots to the crystal plates three factors are of importance in their effect on the characteristics of the plate, namely, the size of the spot, its location, and the firing temperature. Since in all crystal designs to date the silver spots are applied at or near the nodal line of the crystal plate the principal effect of the spots is to increase the stiffness of the plate, so shghtly increasing the frequency of resonance. Variations of an appreciable magnitude in either the amount of silver paste used (that is, the size of the spot) or in the location of the spot with respect to the nodal line will change the resonant frequency of the plate. Such changes could be corrected later, when the plates are adjusted for resonant frequency, as long as the length is increased sufficiently to allow such adjustment. However, if the length be increased sufficiently to allow for extreme cases, avirage adjusting time will be increased materially, while if the allowance is insufficient some of the plates may be unusable. For this reason, close control of the size and location of the silver spots is well justified. MANUFACTURING DEVIATIONS IN CRYSTAL UNITS 273 I In baking the silver spots, care must be taken to prevent "heat" twinning. If the temperature of a quartz plate is raised above the inversion point (573°C) and then is reduced again, the plate will be electrically twinned.^ The firing temperature of the silver paste currently used for the spots is not many degrees below this inversion point. Hence, the firing temperature may easily become so high as to result in twinned plates. In addition, it has been observed that the twinning may occur at a considerably lower temperature if the plate is subjected to large thermal stress. For this reason, care must be taken to heat the plates uniformly during the baking operation. 14.32 Division of Coating The next operation is to evaporate a coating of silver on the surface of the quartz plates. The plates must be thoroughly cleaned before this coat- ing is applied in order to ensure firm adherence of the coating. Poor ad- herence may cause the coating to peel off the plate, changing all of the electrical characteristics of the plate. In many cases the coating must also be divided. ^'^^ Two methods are in general use for dividing the silver coating on crystal plates, namely, an abrasive method and an electrical stylus method. In general, the abrasive method of dividing the coating is superior to the electric stylus for all cases requiring a simple straight line division, but it has not been found practical for complicated divisions such as are desirable for harmonic longitudinal plates and fiexure plates. In using the abrasive method for dividing the coating only two factors are likely to change the characteristics of the crystal plate, these being the location and the width of the dividing line. Deviations in the location of the dividing Hne from the lengthwise center line for a longitudinal plate will affect the capacity and inductance balance between the two halves of the plate. Deviations in the width of a properly centered dividing line will cause changes in the inductance of the plate since for a given plate the inductance is a function of the ratio of the plated area to the total area of the plate. So, for a wide crystal plate deviations in the width of the divid- ing hne will be negligible while for narrow plates these deviations can cause an appreciable change in the inductance of the plates. When the electric stylus is used for dividing the coating, the location and the width of the dividing line again will aflfect the performance of the plates. In addition, varying amounts of twinning will occur along the division line apparently due to instantaneous high temperature gradients introduced by burning oi" the silver at the point of contact of the stylus. In measure- ^Loc. cit. * Loc. cit. *^ "Crystal Channel Filters for the Cable Carrier System", C. E. Lane, B. S. T. /., January 1938, pp. 125-136. 274 BELL SYSTEM TECHNICAL JOURNAL ments made on a group of —18.5° X-cut crystals on which the coating has been divided carefully with an electric stylus, the increase in the induc- tance of the plates ranged from 1.4% to 2.6%. Any twinning resulting from the dividing operation will also change the resonant frequency of the plates. 14.33 Soldering of Wires to Plates The next process, that of soldering the supporting wires to the crystal plate may have considerable effect on the performance of the unit. The deviations which may be introduced depend on the amount of solder used, the location of the solder button with respect to the nodal line of the plate, bO 30 »- "~^~- A -- ^^ 77 -- :::^ ^ 0 "^ :;.i;s Si^ ^*Ji» VOLUME or "«s "^ ^l,*^ WIRE SOLDER IN CUBIC MILS ^ X -X y -£> i -• -o ^ HOOKED 10,000 STRAIGHT 10,000 7.000 N -'^v. ■v^ ( -40 • — o X k \ t 30 20 -10 -20 -30 -40 -50 -20' 0 20° 40° 60° 80° 100" 120° 1- TEMPERATURE IN DECREES F Fig. 14.9. — Change of resonant frequency with temperature of -f 5° X-cut quartz crystal plates. The curves show that when the volume of solder used for joining the plates to the supporting wires is appreciable compared to the volume of the plates, the frequency-temperature coefl&cient characteristic is affected by the volume of solder. the shape of the solder button, and the possible twinning of the plate during the soldering operation. The amount of solder used in forming the joint of the wire to the plate becomes extremely important when the plate is small. For example, Fig. 14.9 illustrates the changes in the frequency-temperature characteristic resulting from the use of varying amounts of solder for a particular size of plate. The units on which these measurements were made used X-cut -1-5° plates of 16 mm x 6 mm x 0.5 mm. The types of wire referred to in the figure, that is, hooked, straight and headed, were described in Chapter XIIP". The frequency-temperature characteristic expected on the basis of measurements made on crystal units using larger plates is approximated " Log. cit. MANUFACTURING DEVIATIONS IN CRYSTAL UNITS 275 closely by the solid curve. This solid curve actually was obtained from measurements made on crystal units using plates supported with headed wires and using a very small amount of solder. The other curves indicate that the temperature coefficient may be increased appreciably due to the presence of a larger amount of solder. Further, when the larger amounts of solder are used, the characteristics depend on the exact amount of the solder, so that the characteristics represented by the dashed curves are hard to reproduce. The amount of solder used in this operation also affects the Q of the crys- tal unit and its resonant frequency. Measurements using several crystal plates of relatively small sizes have shown improvements in Q of as much as 25 per cent when headed wires are used over that obtained with other wires using larger amounts of solder. Variations in the consistency of the solder joint will, of course, afifect the adherence of the supporting wire to the plate. A poor joint will result in a high effective resistance for the crystal unit and will generally cause in- stability both in resistance and in resonance frequency. In soldering the supporting wire to the crystal plate two methods have been used for melting the solder; namely, the soldering iron, and the hot- air blast. With either method, lack of sufficient control can seriously change the electrical characteristics of the plate due to twinning. It has been observed that this twinning occurs when there is a large temperature gradient in the quartz, even at temperatures well below the inversion point. Experimental work by G. W. Willard has indicated that it may occur even when the temperature of the soldering iron is as low as 300°C. To avoid such twinning during the soldering operation, it has been found desirable to raise the temperature of the entire plate to just below the melting point of the solder. Twinning, when it occurs, will afifect the crystal plate by causing an in- crease in inductance, a change in the resonant frequency, increased effective resistance, and a change in the temperature coefficient, as stated previously. Also, in crystals with divided plating there will be an inductance unbalance between the two halves of the crystal plate set up due to unequal amounts of twinning. Several measurements made, using GT plates at 160 kc, showed that twinning during the soldering operation decreased the resonant fre- quency in a range from 200 to 100 cps and the temperature coefficient of the units ranged from 2 to 6 times that of units using untwinned crystal plates. 14.34 Effects due to Wire Resonance As described in Chapter VIII^^, the characteristics of crystal units may be changed due to vibrations set up in the supporting wires. When any 12 "Methods of Mounting and Holding Crystals", R. A. Sykes, B. S. T. J., April 1944. 276 BELL SYSTEM TECHNICAL JOURNAL one of the wires is not located exactly on a node of the plate, the plate will set the wire into vibration. For certain critical lengths of the wire, it will offer considerable resistance to this motion and there will be a rapid increase in effective resistance and some change in resonant frequency of the crystal plate. The effect of wire \abration can be described in terms of its electrical analogy. The vibrating wire, clamped at its far end, may be considered a rather special electrical transmission line open-circuited at its far end. WTien viewed from the crystal plate the impedance changes rapidly with X OR R /ZJ315V Zc=Ri+jXi Zw=R2+iX2 IMPEDANCE IMPEDANCE OF CRYSTAL OF WIRE PLATE RESONANCE RESONANCE Fig. 14.10. — Effect of wire resonance on the resonant frequency of a crystal unit. frequency in a succession of pronounced resonances and anti-resonances. In the vicinity of an anti-resonance the electrical equivalent of the vibrating wire may be approximated by a coil and condenser in parallel as shown by L2 and C2 of Fig. 14.10. This acts in series with the mechanical resonance of the quartz plate, represented by Zi and Ci. The impedance curves illus- trate the effect of the wire resonance on the crystal impedance. R\, the equivalent resistance of the crystal plate, is constant for frequencies in the vicinity of resonance. Xi, the equivalent reactance of the crystal plate, increases rapidly as the frequency departs frcm resonance. R2 and X2, the equivalent resistance and reactance of the wire resonance, are typical of an MANUFACTURING DEVIATIONS IN CRYSTAL UNITS 277 anti-resonant electrical network. The curve labeled (Xi -\- X2) shows the efifect of the wire resonance on the response of the crystal plate. It may be observed that the apparent resonance has been reduced by a small frequency decrement. The amount of frequency shift and the increase in effective resistance depend on the Q of the wire resonance, its frequency location compared with the resonance of the crystal plate, the mass of the wire rela- tive to that of the quartz plate, and the distance from the node to the point at which the wire is actually fastened to the plate. The slope of the frequency-reactance characteristic corresponding to the mechanical resonance of the quartz plate is very steep and the efifect of the wire resonance will be noticed only when an anti-resonant frequency of the wire is close to the resonant frequency of the plate. The changes in resonant frequency and effective resistance due to wire resonance have been measured for some filter crystal units and the measurements are tabulated in Table II. Table II Effect of Wire Vibrations on the Resistance of a Quartz Crystal Plate Crystal Type -f-5° X-Cut +5° X-Cut -18° X-Cut -18° X-Cut 5th Harmonic GT Mode of Vibration for Crystal Flexural Longitudinal Longitudinal Longitudinal Longitudinal Resonant Frequency 12 kc 164 kc 335 kc 552 kc 164 kc Crystal Mass in Grams .51 .142 .075 .068 .98 Distance of Wire Maximum from Nodal Frequency Line Shift CPS (N) .060" ±2.0 (N) .012" ±30 (M) .002" ±90 (N)0 .0" ±75 (N) .011" ±12 Maximum Increase in Resistance 250% 640% 360% 1100% 370% (N) Specified Dimension. (M) Measured Dimension. The relation between the length of a wire and the frequencies at which it will resonate in flexural modes is expressed by the following equation: I = m vLf where v is the velocity of sound in the wire d is the diameter of the wire / is the length of the wire / is the frequency of wire resonance in cycles per second w is a number that depends on the manner in which the ends of the wire can move. At a particular frequency and for wire of a particular material and diam- eter there is a series of critical wire lengths which must be avoided. The critical lengths are these which cause the wire to present a high impedance to the motion of the plate. This high impedance may be considered, from the electrical point of view, as corresponding to an anti-resonance of the wire. 278 BELL SYSTEM TECHNICAL JOURNAL The critical lengths are defined by the series of numbers m — in -\- \) tt where n takes the values 1, 2, 3, etc. and apply to successive modes of a bar clamped at both ends. Beyond the first mode, the critical wire lengths are spaced at equal intervals, corresponding to increments of m each equal to TT. There is also a series of wire lengths which will present minimum impedance to the motion of the plate. These may be considered as corresponding to a resonance of the wire. These minima of impedance are obtained for lengths of wire defined by the series of numbers m — {n — \) ir. They apply to a bar which is clamped at one end and, while free to vibrate at the other end, is constrained to a slope perpendicular to the plate. In selecting a desirable length for the supporting wire, it is not essential that this length be such as to cause the wire to present minimum impedance to motion of the plate. As a matter of fact, since the wire is of relatively low characteristic impedance a small departure from the critical length is sufficient to avoid trouble from wire resonance. In order to allow for as wide a manufacturing tolerance as possible the supporting wire is usually designed to have a length half-way between two successive critical lengths. For a 6.3-mil phosphor-bronze wire, the spacing between successive critical lengths ranges from about 58 mils at 100 kc to about 15 mils at 1000 kc. Hence, even at 100 kc the length of the supporting wire must be controlled within a tolerance of about 20 mils. These supporting wires are formed to have definite bends along their length and the location of these bends varies slightly from one wire to another. In addition, the wires are terminated by solder at both ends. Because of these complications it is impractical to meet such close tolerances on the effective length of the wires. Furthermore, a wire that does have a suitable effective length at room temperatures may exhibit sufficient change of properties with variations in temperature so that it becomes of critical length at some other operating temperature. Much of the difficulty due to wire resonance is avoided by use of a solder ball on the supporting wire, as described in Chapters VIII and XIII. The solder ball is located near the quartz plate. Since it serves as a clamp at that point, it makes the supporting wire short. By locating and forming the solder ball accurately, the length of the supporting wire is controlled within a close tolerance. Further, since the wire is shortened by use of the ball it is less affected by changes in temperature. Experience at about 500 kc indicates that a tolerance of about 10 mils in locating the solder ball is prac- tical and has provided satisfactory operation between — 40 C and -f 85 C. 14.4 Need For Cleanliness and Low Relative Humidity One of the most serious difficulties encountered in manufacturing quartz crystal plates is that of assuring sufficient cleanliness. Even minute par- MANUFACTURING DEVIATIONS IN CRYSTAL UNITS 279 tides of foreign matter will introduce appreciable changes in crystal per- formance. Usually, the presence of foreign matter will act to load the crystal and will reduce the resonant frequency but there are also instances where the added matter tends to stififen the plate and increase its frequency. The latter has been observed to occur as the result of the deposit of a thin film of rosin on the surface of the plate. In the presence of foreign matter on the surface of the plates, the performance will be unstable with time and temperature even after the plate is sealed into a container. Also, erratic variations are observed as the plate is shifted from a normal atmosphere to a container which is evacuated or filled with dry air. Experience has shown / \ J V -^ ^ M \, V RELATIVE HUMIDITV - LESS THAN 40".' 1 1 X 'X r' i' 1 1 RELATIVE HUMIDITY ~^1 3f 80 " K \\ -" ^1 ^ } CAR RIE R -4 -3-2-111 2 FREQUENCY KC TROM CARRIER Fig. 14.11. — Efifect of humidity on the discrimination of channel crystal filters. To prevent decrease of discrimination with increase of relative humidity, the crystal units must be hermetically sealed. that elaborate precautions for insuring cleanliness are justified by the time saved in the adjusting processes. The need for cleanliness is closely related to the effect of humidity on the insulation resistance of crystal units. As used in filters, crystal units must provide extremely high impedances at their anti-resonant frequencies. These impedances may be as high as 100 megohms. With clean crystal plates in relatively dry atmospheres, such insulation resistance can be main- tained up to 1000 kc. However, even a trace of salts or other t>'pes of con- tamination will make the insulation resistance highly sensitive to moisture in the adjacent air. While it is relatively difficult to measure insulation resistance at high carrier frequencies, the effect of the reduced insulation due to moisture is evident on inspection of the discrimination characteristics of the filters. For example, Fig. 14.11 illustrates the result of high relative 280 BELL SYSTEM TECHNICAL JOURNAL humidity on the transmission characteristic of a typical crystal filter. It may be observed that the discrimination almost disappears for a relative humidity of 80%. These measurements were made on a filter con- taining well cleaned crystal plates. It will be found frequently that an unsatisfactory discrimination characteristic is produced by considerably lower values of relative humidity when the plates are not so clean. Ex- perience has shown that it is impractical to let the relative humidity sur- rounding the crystal plate exceed 40% for satisfactory filter performance. When a high degree of accuracy is required, the plates are assembled -5 -10 -15 -20 -25 -30 II II ^ \^ a\ \ A- ETCHED 20 MIN, B- ETCHED BETWEE 40 AND 90 MIN. N \ \ \ i \ \ ' 10 20 50 100 200 MICRO AMPERES PER MM. OF WIDTH 500 1000 Fig. 14.12. — Change of resonant frequency of GT-cut plates due to increase of trans- mitted current. in a unit which is either evacuated or filled with air at a relative humidity of less than 5%. 14.5 Effect of Current Level Crystal units will undergo change in effective resistance and in frequency of resonance as the current transmitted is increased. Some change might be expected due to the heating of the plate by the dissipative loss associated with the transmission of current. However, the effects are not identical with those obtained with a change in ambient temperature. Appreciable MANUFACTURING DEVIATIONS IN CRYSTAL UNITS 281 changes have been observed even when using GT-cut plates adjusted to zero temperature coefficient as shown, for example, in the curves of Fig. 14.12. Also, it has been observed that after a plate has been driven hard and the transmitted current then reduced, the original resonant frequency is re- stored only after a considerable time interval. The data of Fig. 14.12 pro- vides a rough correlation between stabihty and current levels. For ex- ample, if the stability desired for a crystal unit using a GT t}^e plate be in the order of one part per million, the circuit design should be such as to keep the current level in the plate below about 10 microamperes per milli- meter of width. The parameter used in these paragraphs for measuring current levels is the current per unit of width. This appears to be useful as a common basis for comparing various plates of any one cut and mode of vibration. Theo- retically, in the case of a plate vibrating longitudinally, the current, /, per unit of width, w, is directly proportional to the elongation per unit of length, yy, as shown by following equation: I/w = K yy where K is a constant which depends on the cut of plate and mode. Figure 14.12 also illustrates the importance of the surface condition of the plates. Curve A is the average frequency-current characteristic for a group of crystal units using plates etched for twenty minutes in 47% hydro- fluoric acid and curve B the average characteristic for a group of crystal units using plates etched for over forty minutes but less than ninety minutes. Evidently, crystal units using plates which have been etched for a long period exhibit a frequency-current characteristic which is appreciably more constant than those using plates etched for a shorter period. Mathematical Analysis of Random Noise By S, O. RICE Introduction THIS paper deals with the mathematical analysis of noise obtained by passing random noise through physical devices. The random noise considered is that which arises from shot effect in vacuum tubes or from thermal agitation of electrons in resistors. Our main interest is in the sta- tistical properties of such noise and we leave to one side many physical results of which Nyquist's law may be given as an example. About half of the work given here is believed to be new, the bulk of the new results appearing in Parts III and IV. In order to provide a suitable introduction to these results and also to bring out their relation to the work of others, this paper is written as an exposition of the subject indicated in the title. When a broad band of random noise is applied to some ph5'sical device, such as an electrical network, the statistical properties of the output are often of interest. For example, when the noise is due to shot efifect, its mean and standard deviations are given by Campbell's theorem (Part I) when the physical device is hnear. Additional information of this sort is given by the (auto) correlation function which is a rough measure of the dependence of values of the output separated by a fixed time interval. The paper consists of four main parts. The first part is concerned with shot effect. The shot effect is important not only in its own right but also because it is a typical source of noise. The Fourier series representa- tion of a noise current, which is used extensively in the following parts, may be obtained from the relatively simple concepts inherent in the shot efifect. The second part is devoted principally to the fundamental result that the power spectrum of a noise current is the Fourier transform of its correlation function. This result is used again and again in Parts III and IV. A rather thorough discussion of the statistics of random noise currents is given in Part III. Probability distributions associated with the maxima of the current and the maxima of its envelope are developed. Formulas for the expected number of zeros and maxima per second are given, and a start is made towards obtaining the probability distribution of the zeros. When a noise voltage or a noise voltage plus a signal is applied to a non- ^ An account of this field is given by E. B. Moullin, "Spontaneous Fluctuations of Voltage," Oxford (1938). 282 MATHEMATICAL ANALYSIS OF RANDOM NOISE 283 linear device, such as a square-law or linear rectifier, the output will also contain noise. The methods which are available for computing the amount of noise and its spectral distribution are discussed in Part IV. Acknowledgement I wish to thank my friends for many helpful suggestions and discussions regarding the subject of this paper. Although it has been convenient to acknowledge some of this assistance in the text, I appreciate no less sincerely the considerable amount which is not mentioned. In particular, I am in- debted to Miss Darville for computing the curves in Parts III and IV. Summary of Results Before proceeding to the main body of the paper, we shall state some of the principal results. It is hoped that this summary will give the casual reader an over-all view of the material covered and at the same time guide the reader who is interested in obtaining some particular item of informa- tion to those portions of the paper which may possibly contain it. Part I— Shot Effect Shot effect noise results from the superposition of a great number of disturbances which occur at random. A large class of noise generators produce noise in this way. Suppose that the arrival of an electron at the anode of the vacuum tube at time / = 0 produces an effect F{t) at some point in the output circuit. If the output circuit is such that the effects of the various electrons add linearly, the total effect at time / due to all the electrons is /(/) = z nt- h) (1.2-1) where the k^ electron arrives at tk and the series is assumed to converge. Although the terminology suggests that /(/) is a current, and it will be spoken of as a noise current, it may be any quantity expressible in the form (1.2-1). 1. Campbell's theorem: The average value of /(/) is I(J) = V \ Fit) dt (1.2-2) •Loo and the mean square value of the fluctuation about this average is ave. [/(/) - 7(0]' = V £ F^{t) dt (1.2-3) 284 BELL SYSTEM TECHNICAL JOURNAL where v is the average number of electrons arriving per second at the anode. In this expression the electrons are supposed to arrive independently and at random. ve~*'' dt is the probability that the length of the interval between two successive arrivals lies between / and / + dt. 2. Generalization of Campbell's theorem. Campbell's theorem gives information about the average value and the standard deviation of the probabihty distribution of /(/). A generalization of the theorem gives information about the third and higher order moments. Let m = T.akF(t - 4) (1.5-1) — 00 where F(t) and tk are of the same nature as these in (1.2-1) and • • -ci , az , ■ • ' dk , • • • are independent random variables all having the same distribution. Then the n' semi-invariant of the probability density P{I) oil = I{t) is X„ = ,? f '^ [F{t)Tdt (1.5-2) J— 00 The semi-invariants are defined as the coefficients in the expansion of the characteristic function /(«/): \ogef{u) = S -: ("0" (1.5-3) where f{u) = ave. /'" = f " P(/)e''" dl J—eo The moments may be computed from the X's. 3. As V —^ the probabihty density P(I) of the shot effect current ap- proaches a normal law. The way it is approached is given by -1 (0)/- \ A3 . Let this curve be denoted by I{t). The correlation function of /(/) is i/'(t) which is defined as ^(r) = Limit I [ /(/)/(/ + r) dl (2.1-4) r-»oo 1 •'0 where the limit is assumed to exist. This function is closely connected with another function, the power spectrum, w(/), of /(/). /(/) may be regarded as composed of many sinusoidal components. If I(i) were a noise current and if it were to flow through a resistance of one ohm the average power dissipated by those components whose frequencies lie be- tween/and/ + df would be w(f) df. The relation between w(f) and \I/(t) is w{f) = ^ I rPir) cos It/t dr (2.1-5) Jo ^(t) = [ wif) COS IttJt df (2.1-6) Jo When /(/) has no d.c. or periodic components, w(f) = Limit ^'y^l' (2.1-3) where "-'"^' dt. su) = f me-'' Jo Jo The correlation function for /(O = ^ + C cos (27r/o^ - A and is •■('-'i') for 0 < I T I < A and the power spectrum is »W=^"C-^7 (2.7-9) MATHEMATICAL ANALYSIS OF RANDOM NOISE 287 5. There are two representations of a random noise current which are especially useful. The first one is N I{t) = S (^n cos unt -{• bn Sin a}„ /) (2.8-1) n-1 where c„ and &„ are independent random variables which are distributed normally about zero with the standard deviation ■\/w{fn)Af and where OJn = 27r/„ , /„ = wA/ The second one is AT ^(0 = Jl Cn cos (co„/ - 2.5 where 1/2 y = Tm , ^0 = r.m.s. value of /(/), the probability that a maximum of the envelope, selected at random from the universe of such maxima, lies between R and R -^ dR is approximately , 9 . .,2,9 dR 1.13(/ - l)e-''" -TTi Wo A curve for the corresponding probability density is shown for the range 0 < y < 4. Curves which compare the distribution function of the maxima of R with other distribution functions of the same type are also given. 8. In section 3.9 some information is given regarding the statistical behavior of the random variable : rh+T E = \ l\t) dt (3.9-1) where h is chosen at random and /(/) is a noise current with the power spectrum wif) and the correlation function i/'(t). The average value niT of E is T\pQ and its standard deviation cjt is given by (3.9-9). For a relatively narrow band-pass fi.lter (Tt t niT VT{fb — fa) when T(fb — /a) ^ 1. This follows from equation (3.9-10). An ex- pression which is believed to approximate the distribution of E is given by (3.9-20). 9. In section 3.10 the distribution of a noise current plus one or more sinusoidal currents is discussed. For example, if / consists of two sine waves plus noise: I = P cos pt -\- Q cos qt -\- In, (3.10-20) where p and q are incommensurable and the r.m.s. value of the noise cur- rent In is ypl ^, the probabihty density of the envelope R is R f rJo{Rr)MPr)MQr)e-'^°'''" dr (3.10-21) Jo where /o( ) is a Bessel function. MATHEMATICAL ANALYSIS OF RANDOM NOISE 291 Curves showing the probability density and distribution function of R, when Q = 0, for various ratios of P/r.m.s. Iff are given. 10. In section 3.11 it is pointed out that the representations (2.8-1) and (2.8-6) of the noise current as the sum of a great number of sinusoidal components are not the only ones which may be used in deriving the results given in the preceding sections of Part III. The shot effect representation /(0^= 2 Fit - O —00 studied in Part I may also be used. Part IV — Noise Through Non-Linear Devices 1. Suppose that the power spectrum of the voltage V applied to the square-law device I = aV^ (4.1-1) is confined to a relatively narrow band. The total low-frequency output current It( may be expressed as the sum la = Ida + Iff (4.1-2) where Idc is the d.c. component and !(/ is the variable component. When none of the low-frequency band is eliminated (by audio frequency filters) la = "f (4.1-6) where R is the envelope of V. If V is of the form V = V If ■{- P cos pt -{- Q cos qt, (4.1-4) where Fat is a noise voltage whose mean square value is ^o , then ll,=a' hi + P'h + OVo + ^'] (4.1-16) 2. If instead of a square-law device we have a linear rectifier, 1 = 1^ ^<^ (42-1) ^ \aV, V>0 ^^-^ ^^ the total low- frequency output is lU = - (4.2-2) 292 BELL SYSTEM TECHNICAL JOURNAL WTien F is a sine wave plus noise, Fa- + P cos pt, Tac = a^l^J ' iF,(-i; 1; -x) (4.2-3) 2 lit =-AP' + 2^o) (4.2-6) where i^i is a hypergeometric function and P Ave. sine wave power ,^„ ,, a: = — = — , ^ (4.2-4) 2^0 Ave. noise power When X is large 1-1...] (4.2-7) 2 aVo 1(! ~ TT If F consists of two sine waves plus noise, Idc consists of a h}'pergecmetric function of two variables. The equations running from (4.2-9) to (4.2-15) are concerned with this case. About the only simple equation is 2 fit = -. [21^0 + P' + Q'\ (4.2-14) 3. The expressions (4.1-6) and (4.2-2) for I ,( in terms of the envelope R of F, namely al^ , aR — - and — , 2 T are special cases of a more general result lit = Aq{R) = ~ f F{iu)Jo{uR) du. (4.3-11) Ztt J c In this expression Jq{uR) is a Eessel function. The path of integration C and the function F{iu) are chosen so that the relation between I and F may be expressed as I = ^ [ F{iu)e''"' du. (4A-1) 27r Jc A table giving F(iu) and C for a number of common non-linear devices is shown in Appendix 4A. If this relation is used to study the biased linear rectifier. / 0, V B MATHEMATICAL ANALYSIS OF RANDOM NOISE 293 for the case in which V is Viv + P ccs pt, we find h (4.3-17) . B ,P , B' + h 2 TT ZTrr" 72- P- - B, Iff ~ ■ — :r^^ ¥'0 when P » I J5 I , P" » i/'o where i/'o is the mean square value of Fat . 4. When V is confined to a relatively narrow band and there are no audio-frequency filters, the probability density and all the associated sta- tistical properties of Id may be obtained by expressing lu as a function of the envelope R of V and then using the probabiUty density of R. When V is Vf] + P ccs pi -{- Q cos qt this probability density is given by the in- tegral, (3.10-21) (which is the integral containing three Bessel functions stated in the above summary of Part III). When V consists of three sine waves plus noise there are four /o's in the integrand, and so on. Expres- sions for i?" when R has the above distribution are given by equations (3.10-25) and (3.10-27). \\T:en audio-frequency filters remove part of the low-frequency band the statistical properties, except the mean square value, of the resulting cur- rent are hard to compute. In section 4.3 it is shown that as the output band is chosen narrower and narrower, the statistical properties of the output current approach those of a random noise current. 5. The sections in Part IV from 4.4 onward are concerned with the problem: Given a non-Hnear device and an input voltage consisting of noise alone or of a signal plus noise. What is the power spectrum of the output? A survey of the methods available for the solution of this problem is given in section 4.4. 6. "Wlien a noise voltage TV with the power spectrum w(/) is applied to the square-law device / = aV^ (4.1-1) the power spectrum of the output current / is, when/ ?^ 0, Z+00 w{x)w{f - x) dx (4.5-5) where w{—x) is defined to equal wix). The power spectrum of / when V is either P cos pt + TV or Q(l + k cos pt) cos qt ■{■ Vff is considered in the portion of section 4.5 containing equations (4.5-10) to (4.5-17). 294 BELL SYSTEM TECHNICAL JOURNAL 7. A method discovered independently by Van Vleck and North shows that the correlation function ^(t) of the output current for an unbiased linear rectifier is *M = |' + |%F.[-i-J;J;|] (4.7-6) where the input voltage is Vn • The correlation function )^(t) of Vjf is denoted by ^t and the mean square value of Fa- is xpo . The power spectrum W{f) of / may be obtained from W{f) = 4 f ^(r) cos 2irfT dr (4.6-1) Jo by expanding the hypergeometric function and integrating termwise using Gn{f) = I 4^r COS 2tvJt dr. (4C-1) Jo Appendix 4C is devoted to the problem of evaluating the integral for G„(/). 8. Another method of obtaining the correlation function i/'(t) of /, termed the "characteristic function method," is explained in section 4.8. It is illustrated in section 4.9 where formulas for ^(r) and W{f) are developed when the voltage P cos pt + Vn is applied to a general non-Unear device. 9. Several miscellaneous results are given in section 4.10. The char- acteristic function method is used to obtain the correlation function for a square-law device. The general formulas of section 4.9 are applied to the case of a v^^ law rectifier when the input noise spectrum has a normal law distribution. Some remarks are also made concerning the audio-frequency output of a linear rectifier when the input voltage V is Q{\ + r cos pt) cos qt -{- Vn . 10. A discussion of the hypergeometric function iFi{a; c; x), which often occurs in problems concerning a sine wave plus noise, is given in Appendix 4B. TART I THE SHOT EFFECT The shot effect in vacuum tubes is a typical example of noise. It is due to fluctuations in the intensity of the stream of electrons flowing from the cathode to the anode. Here we analyze a simpUfied form of the shot effect. MA THEM A TICALANAL YSIS OF RA NDOM NOISE 295 1.1 The Probability of Exactly A' Electrons Arriving at the Anode in Time T The fluctuations in the electron stream are supposed to be random. We shall treat this randomness as follows. We count the number of electrons flowing in a long interval of time T measured in seconds. Suppose there are Ki . Repeating this counting process for many intervals all of length T gives a set of numbers Ki , K^ - • • K » where M is the total number of intervals. The average number v, of electrons per second is defined as V = Lim — (1.1-1) jvi-*oo MT where we assume that this limit exists. As M is increased with T being held fixed some of the A's will have the same value. In fact, as M increases the number of A's having any particular value will tend to increase. This of course is based on the assumption that the electron stream is a steady flow upon which random fluctuations are superposed. The probability of getting A' electrons in a given trial is defined as ,^^. _ . Number of trials giving exactly K electrons ,. . _. p{K) = Lim ^ ,/ (lA-2) Of course p{K) also depends upon T. We assume that the random- ness of the electron stream is such than the probability that an electron will arrive at the anode in the interval {t, t + A/) is vM where M is such that vM « 1, and that this probability is independent of what has happened before time / or will happen after time / + A/. This assumption is suflacient to determine the expression for p{K) which is p{K) = ^^e-'' (1.1-3) This is the "law of small probabilities" given by Poisson. One method of derivation sometimes used can be readily illustrated for the case iv = 0. T Thus, divide the interval, (0, T) into M intervals each of length M = —. A/ is taken so small that vA/ is much less than unity. (This is the "small probability" that an electron will arrive in the interval A/). The prob- ability that an electron will not arrive in the first sub-interval is (1 — vA/). The probability that one will not arrive in either the first or the second sub-interval is (1 — I'A/)^. The probability that an electron will not arrive in any of the M intervals is (1 — v^t)". Replacing M by T/A/ and letting A/ — ^ 0 gives 296 BELL SYSTEM TECHNICAL JOURNAL The expressions for /»(1), pi2), ■ • • p(K) may be derived in a somewhat similar fashion. 1.2 Statement of Campbell's Theorem Suppose that the arrival of an electron at the anode at time / = 0 produces an effect F(t) at some point in the output circuit. If the output circuit is such that the effects of the various electrons add linearly, the total effect at time / due to all the electrons is /(/) = t. Fit- /,) (1.2-1) where the k electron arrives at //.- and the series is assumed to converge. Campbell's theorem states that the average value of 7(0 is Uj) = V i F{t)dt (1.2-2) J—tc and the mean square value of the fiuctuation about this average is (/(O - mf = V [ F\i) dt (1.2-3) J— 00 where v is the average number of electrons arriving per second. The statement of the theorem is not precise until we define what we mean by "average". From the form of the equations the reader might be tempted to think of a time average; e.g. the value Lim 1 f /(/) dt (1.2-4) 7— oo 1 JQ However, in the proof of the theorem the average is generally taken over a great many intervals of length T with t held constant. The process is somewhat similar to that employed in (1.1) and in order to make it clear we take the case of /(/) for illustration. We observe /(/) fcr many, say M, intervals each of length T where T is large in comparison with the interval over which the effect F{t) of the arrival of a single electron is appreciable. Let nl{t') be the value of /(/), t' seconds after the beginning of the n^ in- terval, t' is equal to / plus a constant depending upon the beginning time of the interval. We put the subscript in front because we wish to reserve the usual place for another subscript later on. The value of /(/') is then defined as 7(0 = Lirrit i [i/(/') + J{t') + • • • + Ar/(/')] (1-2-5) M-*oo M and this limit is assumed to exist. The mean square value of the fluctua- tion of I{t') is defined in much the same way. 2 Proc. Camb. Phil. Soc. 15 (1909), 117-136. 310-328. Our proof is similar to one given by J. M. Whittaker, Proc. Camb. Phil. Soc. 2,2> (1937), 451-458. MATHEMATICAL ANALYSIS OF RANDOM NOISE 297 Actually, as the equations (1.2-2) and (1.2-3) of Campbell's theorem show, these averages and all the similar averages encountered later turn cut to be independent of the time. When this is true and when the M in- tervals in (1.2-5) are taken consecutively the time average (1.2-4) and the average (1.2-5) become the same. To show this we multiply both sides of (1.2-5) by di' and integrate from 0 to T: I{t') = Lin-.it -i- i; ^I{1') dl' M-*ao M 1 »i=l ''0 1 r^ ^^-^'^^ = Limit -— / 7(0 dl and this is the same as the time average (1.2-4) if the latter limit exists. 1.3 Proof of Campbell's Theorem Consider the case in which exactly /I electrons arrive at the anode in an interval of length T. Before the interval starts, we think of these K elec- trons as fated to arrive in the interval (0, T) but any particular electron is just as Ukely to arrive at one time as any other time. We shall number these fated electrons frcm one to K for purposes of identification but it is to be emphasized that the numbering has nothing to do with the order of ar- rival. Thus, if tk be the time of arrival of electron number k, the probability that tk lies in the interval (/, / -{- dl) is dl/T. We take T to be very large compared with the range of values of I for which F(l) is appreciably different from zero. In physical applications such a range usually exists and we shall call it A even though it is not very definite. Then, when exactly K electrons arrive in the interval (0, T) the effect is approximately K IkO) = lLFil- Ik) (1.3-1) the degree of approximation being very good over all of the interval except within A of the end points. Suppose we examine a large number M of intervals of length T. The number having exactly K arrivals will be, to a first approximation M p{K) where p(K) is given by (1.1-3). For a fixed value of / and for each interval having K arrivals, I Kit) will have a definite value. As M — > co , the average value of the TK(tys, obtained by averaging over the intervals, is Jo I Jq I k=i = E/ '-pFit-lk) k=i Jo I 298 BELL SYSTEM TECHNICAL JOURNAL and ifA oo , 7(0 = £ PiK)T^) = V / F{t) dt (1.3-4) and this proves the first part of the theorem. We have used this rather elaborate proof to prove the relatively simple (1.3-4) in order to illustrate a method which may be used to prove more complicated results. Of course, (1.3-4) could be established by noting that the integral is the average value of the effect produced by one arrival, the average being taken over one second, and that v is the average number of arrivals per second. In order to prove the second part, (1.2-3) of Campbell's theorem we first compute P{t) and use W) - nor = im - 2 /(/)/(o + /(o = 7^) - Wf (1-3-5) From the definition (1.3-1) of //c(/), Averaging this over all values of /i , ^2 , • - • iK with t held fixed as in (1.3-2), iiio=tt r^--- r ^-^ Fit - t,)F{t - u k=l m=l Jo 1 Jo 1 The multiple integral has two different values, li k = m its value is Jq dk JO T and ii k 7^ m its value is / ''<' - '') T 1 ^(' - '"'> T MATHEMATICAL ANALYSIS OF RANDOM NOISE 299 Counting up the number of terms in the double sum shows that there are K of them having the first value and A' —A' having the second value. Hence, if A < / < r - A we have Averaging over all the intervals instead of only those having K arrivals gives nt) = z p{K) i\{i) K=0 + 00 =./ F\t) dt + /(/)2 where the sunmiation with respect to K is performed as in (1.3-4), and after summation the value (1.3^) for /(/) is used. Comparison with (1.3-5) estabhshes the second part of Campbell's theorem. 1.4 The Distribution of /(/) When certain conditions are satisfied the proportion of time which /(/) spends in the range I, I -\- dl is P{I)dI where, as v -^ co , the probabihty density P{I) approaches ^ e-''-'^'"'] (1.4-1) 0. When we identify Xk with F{t — ti) we see that the average value of tzi.u , e is 1 r^ ^ exp [inF{t - k)\ dh 1 JQ 3 The essentials of this method are due to Laplace. A few remarks on its history are given by E. C. Molina, Bull. Amsr. Math. Soc, 36 (1930), pp. 369-392. An account of the method may be found in any one of several texts on probability theory. We mention "Random Variables and Probability Distributions," by H. CramSr, Camb. Tract in Math and Math. Phys. No. 36 (1937), Chap. IV. Also "Introduction to Mathematical ProbabiUty," by J. V. Uspensky, McGraw-Hill (1937), pages 240, 264, and 271-278. MATHEMATICAL ANALYSIS OF RANDOM NOISE 301 All of the K characteristic functions are the same and hence, from (1.4-3), PKiDdl is "^^ h ir '""" {f r ^'p ^'"^^^ ~ ^^^ ^o'' "^^ Although in deriving this relation we have taken A' > 0, it also holds for K = 0 (provided we use (1.4-4)). In this case Po{l) = 5(1), because 7 = 0 when no electrons arrive. Inserting our expression for P/r(/) and the expression (1.1-3) for p(K) in (1.4-2) and performing the summation gives Pil) = ^ [ expl-ilii - uT + V J exp {mF{t - t)] dAdu (1.4-5) The first exponential may be simplified somewhat. Using pT = u [ Jo permits us to write — vT+v I exp [iuF{t — t)] dr = u i (exp [iuF{i — t)] — 1) dr Jq Jo Suppose that A < / < T — A where A is the range discussed in connection with equation (1.3-1). Taking | /^(Z - t) | = 0 for | / - r | > A then enables us to write the last expression as u f^'k"'^^'^ - l]dl (1.4-6) Placing this in (1.4-5) yields the required expression for P(I): P{I) = ^ / exp (-z7« + V J [e'"^'" - 1] dljdu (1.4-7) An idea of the conditions under which the normal law (1.4-1) is ap- proached may be obtained from (1.4-7) by expanding (1.4-6) in powers of 11 and determining when the terms involving u and higher powers of ti may be neglected. This is taken up for a slightly more general form of current in section 1.6. dr jQ 302 BELL SYSTEM TECHNICAL JOURNAL 1.5 Extension of Campbell's Theorem In section 1.2 we have stated Campbell's theorem. Here we shall give an extension of it. In place of the expression (1.2-1) for the I{t) of the shot effect we shall deal with the current + 00 I m = E a,F{t - h) (1.5-1) where F{t) is the same sort of function as before and where • • • ai , a^ , • • • ak , • • • are independent random variables all having the same distribution. It is assumed that all of the moments a"^ exist, and that the events occur at random The extension states that the nth semi-invariant of the probabiUty density P{I) of /, where / is given by (1.5-1), is \n= v^- I [F{t)Tdt (1.5-2) J— 00 where v is the expected number of events per second. The semi-invariants of a distribution are defined as the coefficients in the expansion log. (ave. e'^") = E -! {iuf + o(w^) (1.5-3) n=i n\ i.e. as the coefficients in the expansion of the logarithm of the characteristic function. The X's are related to the moments of the distribution. Thus if Wi , W2 , • • • denote the first, second • • • moments about zero we have N ave. e = 1 + Z^ — i (*w) + o{u ) By combining this relation with the one defining the X's it may be shown that / = wi = Xi /2 = W2 = X2 + XiWi P = mz = X3 + 2X2W1 + X1W2 It follows that Xi = / and X2 = ave. {I — I) . Hence (1.5-2) yields the original statement of Campbell's theorem when we set n equal to one and two and also take all the a's to be unity. The extension follows almost at once from the generalization of expression (1.4-7) for the probability density P{I). By proceeding as in section 1.4 and identifying Xk with akF{t — tk) we see that ave, e"*" =7^,1 q{a) da I exp [iuaF{t — tk)] dtk 1 J-00 Jo MATHEMATICAL ANALYSIS OF RANDOM NOISE 303 where q(a) is the probabiHty density function for the a's. It turns out that the probability density P(I) of / as defined by (1.5-1) is 1 r^" / r^°° P{I) = TT- I exp I —iln + V I q{a) da 2ir J— 00 \ J— 00 The logarithm of the characteristic function of F(I) is, from (1.5-4), V [ q(a) da I " [e'"''''^'^ - \\dt J— 00 J— 00 n=l Wl J— 00 J— 00 Comparison with the series (1.5-3) defining the semi-invariants gives the extension of Campbell's theorem stated by (1.5-2). Other extensions of Campbell's theorem may be made. For example, suppose in the expression (1.5-1) for 1(1) that ti , h , - • ■ tk , • • - while still random variables, are no longer necessarily distributed according to the laws assumed above. Suppose now that the probability density p{x) is given where x is the interval between two successive events: t2 = h-\- xi (1.5-5) ^3 = ^2 + ^2 = ^1 + :Vi + X2 and so on. For the case treated above p(x) = ve-'\ (1.5-6) We assume that the expected number of events per second is still v. Also we take the special, but important, case for which F{t) = 0, / < 0 (1.5-7) F{t) = g-"', / > 0. For a very long interval extending from t = iitot = T •{- ti inside of which there are exactly K events we have, if / is not near the ends of the interval, /(/) = aiF(t - h) + a2F{t - h - x^) + - - • + aK+iF{t — h — xi'" — Xk) = aiF(/') + 02F(/' - .Ti) H + OK^iFit' - xi- ■■■ - Xk) 304 BELL SYSTEM TECHNICAL JOURNAL I\t) = a\F\t') + alF\l' - xi) + • • • + aK^xF\t' - x, Xk) + 2aia2F(l')F{t' - xi) + ■ ■ ■ + 2aiaK+iF{t')F{t' - x, ■ ■ ■ -xk) + 2a2aiF{t' - xi)F{l' - xi - X2) -\ 1 where i' = t — ti . If we integrate I (t) over the entire interval 0 < t' < T and drop the primes we get approximately I l\l)dt = (fl^ + . . . + aU,) , Xs 0" (6) / \ , P(/) ~ (T <^ U) - ^y-

oc . 1.7 The Fourier Components of /(/) In some analytical work noise current is represented as m = f + t (a. COS '-f' + b. sin f) (1.7-1) where at a suitable place in the work T and iV are allowed to become infinite. The coefficients a„ and i„ , 1 < w < iV, are regarded as independent random variables distributed about zero according to a normal law. It appears that the association of (1.7-1) with a sequence of disturbances occurring at random goes back many years. Rayleigh and Gouy suggested that black-body radiation and white light might both be regarded as se- quences of irregularly distributed impulses. * Einstein and von Laue have discussed the normal distribution of the coefficients in (1.7-1) when it is used to represent black-body radiation, this radiation being the resultant produced by a great many independent os- 5 See, for example, pp. 86-87, in "Random Variables and Probability Distributions" by H. Cramer, Cambridge Tract No. 36 (1937). '^Phil. Mag. Ser. 5, Vol. 27 (1889) pp. 460-469. 7 A. Einstein and L. Hopf, Ann. d. Physik 33 (1910) pp. 1095-1115. M. V. Laue, Ann. d. Physik 47 (1915) pp. 853-878. A. Einstein, ^nw. d. Physik 47 (1915) pp. 879-885. M. V. Laue, Ann. d. Physik 48 (1915) pp. 668-680. I am indebted to Prof. Goudsmit for these references. MATHEMATICAL ANALYSIS OF RANDOM NOISE 307 dilators. Some argument arose as to whether the coefficients in (1.7-1) were statistically independent or not. It was finally decided that the}' are independent. The shot effect current has been represented in this way by Schottky. The Fourier series representation has been discussed by H. Nyquist and also by Goudsmit and Weiss. Remarks made by A. Schuster are equiv- alent to the statement that a„ and b,, are distributed normally. In view of this wealth of information on the subject it may appear super- fluous to say anything about it. However, for the sake of completeness, we shall outline the thoughts which lead to (1.7-1). In Une with our usual approach to the shot effect, we suppose that exactly A' electrons arrive during the interval (0, T), so that the noise current for the interval is Ui) = E F(/ - 4) (1.7-2) k=i The coefficients in the Fourier series expansion of /«:(/) over the interval (0, T) are a„K and b„K where OnK - ibnK = ^ I] j F {t - 4) exp ~^ -y- r^ -It [y^t) exp [-i ^' (/ + 4)]^/ = Rne-'^-^te''"'" (1.7-3) fr=i In this expression ft - 2^^^* RnC-''" = C„ - iSn = ^ £" F{t)e-'''"'"Ut (1.7-4) In the earlier sections the arrival times /i , /2 , • ■ ■ Ik were regarded as K independent random variable each distributed uniformly over the interval (0, T). Hence the ^a's may be regarded as random variables distributed uniformly over the interval 0 to Itt. Incidentally, it will be noted that in (1.7-3) there occurs the sum of K randomly oriented unit vectors. When A' becomes very large, as it does Mkm. d. Physik, 57 (1918) pp. 541-567. ' Unpublished Memorandum, "Fluctuations in Vacuum Tube Noise and the Like," March 17, 1932. 1" Investigation of Hidden Periodicities, Terrestrial Magnetism, 3 (1898), pp. 13-41. See especially propositions 1 and 2 on page 26 of Schuster's paper. 308 BELL SYSTEM TECHNICAL JOURNAL when V -^ oo , it is known that tlie real and iiraginan- parts of this sum are random variables, which tend to become independent and normally dis- tributed about zero. This suggests the manner in which the normal dis- tribution of the coefBcients arises. Averaging over the 6k s in (1.7-3) gives when n > 0 dnK = hr.K = 0 (1.7-5) Some further algebra gives T~ _ 7Y~ _ K 2 QnK — OnK — ^ A^ 2 (1.7-6) anKbuK = dnKOmK = b„KbmK = 0 where n 9^ m and n, m > 0. So far, we have been considering the case of exactly K arrivals in our interval of length T. Now we pass to the general case of any number of arrivals by making use of formulas analogous to 5 = Z PiK)ZK (1.7-7) A = 0 as has been done in section 1.3. Thus, for w > 0, dn = hn = 0 an = On = — Rn = (Tn (1-7-8) Cnbn = anUm = bnb„, = 0, H 9^ M In the second Hne we have used (r„ to denote the standard deviation of c„ and bn . ^ by writing and bn . We may put the expression for (t„ in a somewhat different form /n = I = n^f, A/ = 1 (1.7-9) where /„ is the frequency of the «th component. Using (1.7-4), (tI = 2,/A/l [ F(/)e~"'^^"' dl (1.7-10) Thus, 00 . This appears to be quite involved. It will be noted that the integrand in the integral defining E is composed of N factors of the form exp [ipn cos {7td — }pn)] = /o(pn) + 2i COS {nd — \pn)Jl{pn) " 2 cos {2nd — 2)/'„)J2(pn) + • • • where 2 / 2 , 2 \ /^2 io2\ 2 2/2 1 2\ Pn = (Wn + Vn){Cn + 0„) = ^a„(Mn + Vn). As J/ becomes large, it turns out that the integral (1.7-11) for the prob- abiUty density obtains most of its contributions from small values of u and v. By substituting the product of the Bessel function series in the integral for E and integrating we find £ = n JoiPn) -\-A+B^C n=l where A is the sum of products such as -2i cos {\pk-^t — \pk - ^()J\{pk)J\{pt)J\{pk+() times A^ - 3 /o's in which Q < k 3'" — (Tn