tm^ ill' ( "''i'r''}' rm:f- si' !'*''■ !U3«4!!«aiis«iv8y:®)iisai*s4iiii«ti.^tisaii^ oand ileal 120414G Eangag Citp public Hihxaxv This Volume is for REFERENCE USE ONLY I t^iff^lg^lff'W^^l^^^ffl^i^ttffi^^ From the collection of the ^ m 0 PreTinger V JLjibrary t P San Francisco, California 2008 X-^ « ^^ ■J' J > '.« THE BELL SYSTEM TECHNICAL JOURNAL A JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION EDITORS R. W. King J. O. Perrine EDITORIAL BOARD W. H. Harrison O. E. Buckley O. B. Blackwell M. J. Kelly H. S. Osborne A. B. Clark J. J. PiLLiOD S. Bracken TABLE OF CONTENTS AND INDEX VOLUME XXIV 1945 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK • • *%• *-'*'*♦ •'#*.* ^-v *•• PRINTED m U. S. A. THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXIV, 1945 Table of Contents January, 1945 Intermittent Behavior in Oscillators — W.A.Edson 1 I'^valuating the Relative Bending Strength of Crossarms — Richard C. Egglcston 23 Mathematical Analysis of Random Noise (Concluded) — S.O.Rice 46 April, 1945 Piezoelectric Crystals in Oscillator Circuits — /. E. Fair 161 The Measurement of the Performance Index of Quartz Plates — C. W. Harrison 217 Lightning Protection of Buried Toll Cable — E. D. Sunde 253 July-October, 1945 Physical Limitations in Electron Ballistics — /. R. Pierce 305 Electron Ballistics in High-Frequency Fields — A. L.Samuel 322 Dynamics of Package Cushioning — Raymond D. Mindlin 353 iii 1204146 MAY 4 1946 Index to Volume XXIV Cable, Buried Toll, Lightning Protection of, E. D. Siinde, page 253. Crossarms, Evaluating the Relative Bending Strength of, Richard C. Egglestoii, page 23. Crystals, Piezoelectric, in Oscillator Circuits, /. E. Fair, page 161. Crystals: The Measurement of the Performance Index of Quartz Plates, C. H'. Harrison, page 217. Cushioning, Package, Dynamics of, Raymond D. Miiidlin, page 353. Edson, W. A., Intermittent Behavior in Oscillators, page 1. Eggleston, Richard C, Evaluating the Relative Bending Strength of Crossarms, page 23. Electron Ballistics, Physical Limitations in, /. R. Pierce, page 305. Electron Ballistics in High-Frequency Fields, A. L. Samuel, page 322. F Fair, I. E., Piezoelectric Crystals in Oscillator Circuits, page 161. H Harrison, C. IT., The Measurement of the Performance Index of Quartz Plates, page 217- High Frequency Fields, Electron Ballistics in, -1 . L. Samuel, page 322. L Lightning Protection of Buried Toll Cable, E. D. Suiide, page 253. M Mindlin, Raymond D., Dynamics of Package Cushioning, page 353. N Noise, Random, Mathematical Analysis of (Concluded), S. 0. Rice, page 46. O Oscillator Circuits, Piezoelectric Crystals in, /. E. Fair, page 161. Oscillators, Intermittent Behavior in, W . A. Edson, page 1. P Package Cushioning, Dynamics of, Raymond D. Mindlin, page 353. Pierce, J. R., Physical Limitations in Electron Ballistics, page 305. Q Quartz Plates, The Measurement of the Performance Index of, C. W . Harrison, page 217. R Rice, S. 0., Mathematical Analj-sis of Random Noise (Concluded), page 46. S Samuel, A. L., Electron Ballistics in High Frequency Fields, page 322. Sunde, E. D., Lightning Protection of Buried Toll Cable, page 253. VOLUME XXIV JANUARY, 1945 number i THE BELL SYSTEM TECHNICAL JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION Intennittent Behavior in Oscillators . . W. A. Edson 1 Evaluating the Relative Bending Strength of Crossarms Richard C.Eggleston 23 Mathematical Analysis of Random Noise (Concluded) S. O. Rice 46 Abstracts of Technical Articles by Bell System Authors 157 Contributors to this Issue . . * 159 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK 50c per copy $1.50 per Year BELL SYSTEM TECHNICAL JOURNAL Published quarterly by the American Telephone and Telegraph Company 195 Broadway i New York, N. Y. — — — ^■— <-^'« » > ■« EDITORS R. W. Zing 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 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, 1945 American Telephone and Telegraph Company PRINTED IN U. S. A. The Bell System Technical Journal Vol. XXIV January, ig4f; No. i Intermittent Behavior in Oscillators By W. A. EDSON Oscillators of all sorts ma\', for certain values of the parameters, show low- frequency disturbances. Usuall>- the disturbance takes the form of a low-fre- quenc>- interruption of the desired oscillation. By the method here i)resented it is possible to determine whether or not such intermittent behavior will occur in a given oscillator and what circuit modifications are required to promote stability. The intentional generation of a modulated wave by control of the low frequency behavior of an oscillator is also considered. Oscillators of the nega- tive resistance type are not considered. I. Introduction TT has been known for a long time that all kinds of oscillators are subject -*• to the trouble variously referred to as intermittent oscillation, motor boating, or squegging. In conventional circuits such as the Hartley the phenomenon is most likely to be observed if the grid leak and grid condenser are abnormally large. It is found that the time constant of this combina- tion must be reduced as the frequency is raised and as the Q of the resonant circuit is decreased. At frequencies above a few hundred megacycles the problem of producing a practical circuit with suitable margin of stability is quite difficult. With the advent of the oscillator having automatic output control the problem assumed a new aspect.^- - By application of an amplified control circuit a high degree of constancy of output together with low harmonic output is obtained. Satisfactory operation is secured, however, only when suitable attention is given to .the characteristics of the control circuit. The intentional generation of pulses by means of intermittent oscillations of relatively high frequency has been studied to some extent, and circuits of this kind are employed in some television systems. Usually the high- frequency oscillation is limited to a small portion of the low-frequency C3'cle, the charge stored during this period being allowed to dissipate itself relative^ slowly during the remainder of the cycle. In all of these circuits satisfactory performance depends upon a proper proportioning of elements not directly associated with the operating fre- 1 L. B. Argimbau, "An Oscillator Having a Linear Operating Characteristic," Proc. I.R.E., Vol. 21, p. 14, Jan. 1933. 2 J. Groszkowski, "Oscillators with Automatic Control of the Threshold of Regenera- tion," Proc. I.R.E., Vol. 22, p. 145, Feb. 1934. 1 2 BELL SYSTEM TECHNICAL JOURNAL quency. When continuous oscillation is necessary it is desirable to provide adequate margin against intermittent operation. When intermittent opera- tion is desired the opposite is true. In either case an understanding of the same general problem is necessary. The present analysis applies only to oscillators of the feedback type. Xo method of extending it to cover negative resistance oscillators such as the Dynatron and the Transitron has been found. Relaxation oscillators as such are not considered here inasmuch as they are seldom affected by inter- mittent operation. No specific frequency limits apply but it is sometimes difficult at very high frequencies to achieve desirable values of the constants. At very low frequencies oscillators employing automatic output control are relatively unsuitable because their performance tends to be unduly sluggish. The term linear oscillator is used to indicate an oscillator in which the range of operation is controlled within such limits that the harmonic content of the output is inappreciable. The general equation describing a simple amplitude-modulated wave is V = Vo{\ -\- m sin 2vjt) sin l-wFt This may be taken as defining the modulation factor m, a complex number which is limited to magnitudes between zero and one. II. General Theory of Oscillation It is found that three separate functions are necessary and sufficient for the operation of an oscillator of the feedback type.^ These are indicated in the block diagram of Fig. 1. The amplifier must be provided to overcome the losses of the rest of the system. The power output, if any, depends upon the fact that the output of an amplifier is greater than the input. Selectivity must be provided to insure that the output has a definite frequency. Ordinarily a tuned circuit of relatively high Q is used although some excellent oscillators employ resistance-capacitance networks. The term filter is employed as being sufficiently general to include these extremes. A limiter of some form is necessary to establish the level at which sustained oscillations occur. In many circuits the functions of amplifier and limiter are performed simultaneously in the vacuum tube. In an important class of oscillators the limiter is a thermal device such as a tungsten lamp. In the Meacham circuit the functions of limiter and filter are combined in a bridge employing a tuned circuit and a tungsten lamp. To simphfy the analysis it is convenient to assume that the amplifier of Fig. 1 is completely linear and operates with equal gain at all frequencies ^ This topic is discussed more fully in "Hyper and Ultra-High Frecjuency Engineering," R. I. Sarbacher, and W. A. Edson, John Wiley & Sons, Inc., 1943. INTERMITTENT BEHAVIOR IN OSCILLATORS 3 from zero to infinity. Similarly the filter is assumed to consist of linear circuit elements and to have a definite curve of loss versus frequency. Asso- ciated with this loss characteristic is some specific phase characteristic* The limiter is assumed to have a loss which is independent of frequency but which is explicitly related to the input (or output) voltage. Although amplifiers having the ideal performance indicated are not physi- cally realizable there are no new or unfamiliar concepts involved. Similarly the performance of passive networks, such as constitute the filter, has been extensively studied and is well understood. It is therefore appropriate to devote the following section to the third function. LIMITER Fig. 1 — Functional block diagram of an oscillator. III. Types of Limiters The limiters which are now in common use may be separated into four relatively distinct groups. 1. \'acuum tubes in which the gain is decreased by simple overload as the level of oscillation rises. This is the most common form of limiter. 2. \'aristors in which the impedance depends upon the instantaneous value of current. Copper oxide, thy rite, and electronic diodes are examples. 3. Thermistors in which the resistance depends upon the rms value of current but does not vary appreciably during any one cycle. Carbon and tungsten filament lamps are the most common examples. 4. Vacuum tubes in which the gain is reduced by application of a bias which depends upon the level of oscillation. Usually the bias is developed by rectifying a portion of the output. The limiters of the first two groups depend for their operation upon the generation of harmonic voltages and currents. The limiters of the second ^ H. W. Bode, "Relations Between Attenuation and Phase in Feedback Amplifier De- sign," Bell Sys. Tech. Jour., Vol. 19, pp. 421-457, July 1940. 4 BELL SYSTEM TECHNICAL JOURNAL two groups operate with very little harmonic distortion. The output of oscillators employing such limiters may, therefore, be made quite free from harmonic voltages. Oscillators of this sort are referred to as linear because the tube or tubes serve as simple Class A linear amplifiers. IV. Criterion of Self Modulation' The block diagram of Fig. 1 is characterized by the fact that the separate elements are connected to each other in the form of an endless ring. The output may be assumed to come from any of the three junctions. It is this fact of closure which complicates the problem of oscillator study. For purposes of analysis it is convenient to open the loop as shown in Fig. 2. For this example it makes no difference where we choose to make the cut, but in actual circuits some caution must be exercised. This matter is dis- TEST GENERATOR FILTER Zl Z2 LIMITER TEST DETECTOR ^^-^-—"^ Z2 Zl - the elements in the control circuit as well as those in the filter, and the performance will be generally poor. Because of its balance a push-pull rectifier is helpful in meeting the latter requirement. The principal require- ment is achieved by amplification and by the use of a constant counter emf or back bias. No bias is produced until the level of oscillation exceeds some threshold value. .Above this threshold the bias increases approximately volt for volt with the peak value of the signal. The same amplifier which is used to increase the control may be used advantageously as a buffer so that appreciable power outputs may be produced without degrading the frequency or amplitude stability. It will be assumed that a () of 100 is available in the coil and that a fre- quency of one megacycle is to be generated. The transmission of a modu- lated wave in terms of the sideband displacement through such a one-circuit filter is shown in Fig. 21. Because the cutoff occurs very slowly it will be convenient to incorporate a rapid cutoff in the auxiliary filter of the gain control, thus avoiding an excessive phase shift at any one frequency. The circuit features already discussed are shown in Fig. 22. A basic oscillator with a single tuned coil, a buffer amplifier having little selectivity and therefore contributing very little to the equivalent filter section, a source of biasing voltage, a balanced rectifier, and an auxiliary low-pass filter are shown. The condenser C is only large enough to allow the rectifier to be driven without serious loss at one megacycle. It has relatively little effect upon the modulation performance. It is assumed that the buffer-amplifier, rectifier, letc. are so chosen that a modulation of very low frequency of one part per million applied at the plate terminal of the oscillator will result in a modulation of one part in a thousand returned to that point. This is equivalent to saying that the envelope gain is 60 db at low frequencies, and corresponds to 60 db of negative feedback in a conventional amplifier. The auxiliary filter will be designed to approximate the attenuation and 18 BELL SYSTEM TECHNICAL JOURNAL 1.0 ^, N v^^ A -~->_^ 0 90" 60" 30' 0 10 f-KC 20 0 10 f-KC Fig. 21 — Envelope transmission through tuned circuit. — ^ ^ / 20 AUXILIARY FILTER Fig. 22 — Special .\-V-C oscillator. 60 40 Adb 20 1 j X / 1 I 0 >< / / y \ y f f A V / / y r^ y 0.1 120 80 r 40 10 100 1,000 10,000 100,000 1,000,000 f-'^/SEC. Fig. li — Characteristics of au.xiliarv filter. INTERMITTENT BEHA VIOR IN OSCILLA TORS 19 phase characteristics shown in Fig. 23. The choice of this particular shape is best explained by reference to Fig. 24 which presents the over-all envelope loop transmission of the system. It is seen that the phase shift is relatively constant at 90° over a wide band of frequencies and that the gain falls off appro.ximately linearly over the same band. In particular the gain becomes zero around 5000 cycles whereas the phase does not reach zero below 500, ()(K) 40 80 120 160 bO - ^^ / 40 Gdb 20 ^ / ^ / \ ^ / >_ ^ ,,,-^ 0 X ^ J \/ X s -20 / X ^ ' ^ "^ N 10 100 1,000 10,000 100,000\ 1,000,000 f-'N/SEC. Fig. 24 — Overall envelope transmission of Fig. 22. 200.000 w 100,000w Fig. 25 — Configuration of auxiliary filter. cycles. In terms of Nyquist's criterion this represents a very stable system which is little disturbed by transient effects. A system having even greater stability could be achieved by beginning the cut-off at lower frequencies. It would then be found that the output was somewhat sluggish in reaching a new equilibrium after being disturbed. Such a behavior is not uncommon but is generally undesirable. Elements which give approximately the characteristics called for in Fig. 23 are shown in Fig. 25. The peak of loss at one megacycle is contributed by the series resonant trap. The rest of the behavior is due to the 0.5 ni condenser in combination with the associated resistors. 20 BELL SYSTEM TECHNICAL JOURNAL XII. Auxiliary Control of Thermally Limited Oscillators In the Meacham and certain other oscillator circuits a thermistor is associated with reactive elements in a bridge circuit which functions as both limiter and filter. In these circuits a large increase in the frequency stabihty is observed. This may sometimes be conveniently expressed as a magnifica- tion of the effective Q of the filter. The advantages of great frequency stabihty and good amphtude stability of these systems are accompanied by an undesirable tendency toward intermittent operation. The thermal constants of the thermistor are not readily adjustable. Moreover adjustment of the reactances to secure Fig. 26 — Meacham circuit with auxiliary control. suitable envelope stability is likely to impair the frequency or amplitude stability for which the circuit is chosen. This dilemma may be resolved by the addition of an auxiliary network which does not affect the envelope transmission to very low frequencies but does modify the behavior at higher frequencies in such a way as to promote the stability of the system. A simple circuit illustrating the principle appears in Fig. 26. It will be noticed that the circuit is so arranged that the average bias applied to the tube is only that due to the cathode resistor. The steady voltage developed across Ci by the rectifier is unable to affect the bias because of the blocking condenser C2. Accordingly the rectifier circuit does not affect the normal operating condition, which is characterized by a bridge loss equal to the amplifier gain. The added elements come into play only if there is a tend- ency toward self -modulation. Then displacement currents of modulation frequency flow through Ci in such a magnitude and phase as to modify the tube gain and compensate the modulation returned from the bridge. INTERMITTENT BEHAVIOR IN OSCILLATORS 21 The exact nature of the control which must be added is best ascertained by opening the circuit at the plate of the tube. The loop transmission of a modulation envelope may then be determined, either experimentally or analytically. If instabiUty is found an auxiliary circuit must be designed to produce an over-all system which is stable. In general the elements of the auxiliary circuit are to be chosen so that the loop transmission is con- siderably less than unity in the region of zero phase. < This is ordinarily accomplished by increasing the fmal cutoff frequency at which the over-all loop envelope transmission is negligible. -xsmmh ■*— AyVVW- Fig. 27 — Self-modulating oscillator. XIII. A Self Modulated Oscillator The previous sections have been devoted primarily to the problem of preventing self-modulation in oscillators. Let us now consider an oscillator having envelope instability. The Nyquist diagram indicates that self- modulation will occur and tells the approximate frequency of the envelope wave. More detailed analysis of the circuit is necessary to determine the wave form of the envelope and the manner in which its amplitude is limited. If a circuit is to function well as an oscillator the Nyquist diagram for the operating frequency must loop the (1,0) point with considerable margin. This is necessary so that a small loss of gain will not stop oscillation. At the operating level the hmiter reduces the loop transmission to unity. In the region of (1, 0) amplitude stability is favored if the rate of change of gain 22 BELL SYSTEM TECHNICAL JOURNAL with respect to level is high. Similarly the frequency stability is favored if the rate of change of phase with respect to frequency is high. If a circuit is to function well as a self-modulated oscillator, the above conditions must be met and in addition the Nyquist diagram for the envelope must meet similar requirements. That is, there must be a limiter and tilter in addition to the effective amplifier in the envelope system. A circuit which meets these requirements is shown in Fig. 27. It is seen to be similar to that of Fig. 6 but to have a more complicated low-frequency path. The operation is best explained in terms of the relative size of the various elements. The by-pass condensers Cx and d are comparatively small. The blocking condensers Cz and d are quite large. The choke Li is large. Thus these elements ser\'e as open or short circuits but do not enter into the setting of either of the frequencies. The stability tests are carried out by opening the mesh at the plate of the tube. At the operating frequency, as defined by the plate coil and condenser the loop gain is high at low levels. Thus the fundamental conditions for oscillation exist. The next step in the analysis is to supply a signal of suitable magnitude and frequency to reduce the loop transmission to (1,0). A small modula- tion of very low frequency is returned magnified and reversed in phase, as with previous systems. The phase of the envelope transmission changes with increase of modulating frequency until it is zero at the resonant fre- quency of Lf, and C5. At this frequency a considerable gain exists so that the Nyquist diagram for the envelope also loops the point (1,0). The tungsten lamp in conjunction with the other impedances of the bridge serves to limit the degree of self-modulation. The operating frequency may be set by means of Ce in conjunction with a suitable value of Le. The operating amplitude may be controlled by adjustment of the bias battery B. The frequency of the self-modulation is set by means of C5 in conjunction with 1,5. XI\'. Conclusions A method of applying known feedback theory to the problem of self- modulation in oscillators has been presented. Although the discussion has been limited to electrical circuits it is clear that the analysis is applicable .to other systems, such as electromechanical or mechanical oscillators. The analysis has been applied to several familiar oscillators to illustrate the method and to clarify some details of their operation. A sample design of a bias controlled oscillator is presented to show application to new designs. The application of bias control to thermistor stabilized oscillators is described. The design of a self-modulated oscillator is undertaken to show how intentional modulation mav be introduced and controlled. Evaluating the Relative Bending Strength of Crossarms By RICHARD C. EGGLESTON /^\'ER a million crossarms are produced annually in the United States. ^^ In the open wire lines of the Bell System alone there are no\^ about 20 million arms in use. It is natural, therefore, that public utility engineers should have an interest in the strength of such an important item of outside plant material; and, consequently, an interest in any tool or means of evalu- ating the strength of such material. It is believed that the moment diagram is a convenient and reasonably reliable tool for estimating the loads an arm will support, for measuring the effect of knots of various sizes and of pinhole locations on arm strength, and for answering similar questions relating to the bending strength of crossarms under vertical loads. Two moment diagrams are shown in Fig. 1 for Bell System Type A cross- arms; and in the pages that follow are presented the method used in con- structing the diagrams and a discussion of their use. While the calculation results apply particularly to the type and quality of arm referred to, they would also be of value as a time saving reference in future studies that may be proposed relating to the strength of the same or other types of arms involving different knot allowances. The resisting moment of a beam is the product of its section modulus by the unit stress on the remotest fiber of the beam. The section modulus of a beam of uniform cross-section is constant and readily determinable. The section modulus, however, of a beam of nonuniform cross-section, such as a crossarm, varies because of the different cross-sectional shapes and dimen- sions involved. In this study the following five different shapes were recognized: (1) Roofed section between pinholes (2) Roofed pinhole section (3) Roofed brace bolt hole section (4) Rectangular pole bolt hole section (5) Rectangular section without bolt holes The dimensions of the sections investigated were as follows: Section of Arm Dimensions Minimum Nominal Roofed section, except at end of arm Roofed section at end of arm Unroofed sections (Inches) 3^ X 4^ 3^x4 3Ax4A (Inches) 3ix4^ 3ix4A 3ix4i 23 24 BELL SYSTEM TECHNICAL JOURNAL Since there is little, if any, engineering interest in the strength of structural members of maximum size, no investigations were made of sections of maximum dimensions. 50 U 20 z GRAPH 1 1 ' \ 1 i 1 1 -_————— __^ ^ . 1 20 30 40 DISTANCE FROM CENTER OF ARM -INCHES 50 60 Tz: -^ 7^Z ^ ^ ^=71 ^r "T '^ Fig. 1 — Moment diagram for T}pe A southern pine and Douglas fir crossarms per Specification AT-7075: Graph 1 — Resisting moments of arms of nominal dimensions, straight grained and free from knots. (Fiber stress 5000 psi) Graph 2 — Resisting moments of arms of minimum dimensions, having maximum slant grain (1" in 8"), and containing knots of the maximum sizes permitted (viz., sizes shown at bottom of arm sketch). (Fiber stress 3250 psi) Graph 3 — Bending moments from a load of 50 pounds at each pin position. Section modulus calculations were made of each shape of minimum and nominal size, both with and without knots. Tests have shown that, be- RELATIVE BENDING STRENGTH OF CROSS A RMS 25 cause of the distortion of the grain that occurs around them, knots are fully as injurious to the strength of structural timbers as knot holes. ^ Therefore, in dealing with sections containing knots, it was assumed for 'the purposes of this study that the knot extended across the section in the same manner as a hole having a diameter equal to the diameter of the knot. It was also assumed that the knot was located in, or reasonably close to, the most damaging position in the arm section. In the calculations of the section modulus of all roofed arm sections, it was necessary first to compute the moments of inertia of the whole or parts of the top segments of such sections (viz. nominal and minimum sections Fig. 2 — Brace bolt hole section containing a f inch knot located immediately below the top segment (knot and bolt hole shaded). between pinholes, and nominal and minimum pinhole sections). Accord- ingly, four such computations were made and the results used in calculating the section moduU of all the roofed sections investigated. The details of the four computations are shown in the Appendix. To insure uniformity in the results, the degree of precision used in these computations was con- siderably greater than is ordinarily employed in dealing with timber prod- ucts. All of the work, however, was done on a computing machine, and it was just about as easy to carry the operations to eight decimal places (which was the capacity of the machine used) as to a lesser number. As a matter of interest in this connection, it was found by actual trial in Computation I that absurd results would occur if fewer than five decimal places were used. For convenience, all of the section modulus calculations were made in tabular form. In such form the procedure employed would not be readily 1 Pg. 6 Dept. Circular 295, U. S. Dept. of Agriculture, "Basic Grading Rules and Work- ing Stresses for Structural Timbers," by J. A. Newlin and R. P. A. Johnson. 26 BELL SYSTEM TECHNICAL JOURNAL apparent. Therefore, a sample calculation follows showing the method of finding the section modulus of the brace bolt hole section containing a | inch knot. Sample Calculation Referring to Fig. 2, it will be noted that the knot and bolt hole divide the section into three parts: the top segment (D and two rectangular portions {R\ and R2). The moment of inertia (/) of such a compound section about its neutral axis (at a distance c from M-M) is equal to the sum of the moments of inertia {IT, IR\ and IR2) of the component parts T, R\ and R2 about axes through their own centers of gravity, plus the areas of the com- ponent parts multipilied by the squares of the distances of their own centers of gravity from the neutral axis of the comi)ound section. The section modulus {S) of this section is found, of course, by dividing its moment of inertia l)y the distance (y) from the neutral axis of the section to the most remote fiber. Dimensions: Areas: Moments about M — M: b = 3.1875" (Width of Section) k = 0.7500" (Diameter of Knot) d = 3.7625" (See Computation I in Appendix) //I = 0.7000" (d - 2.125" - 0.1875" - k) hi = 1.9375" (2.125" - 0.1875") g = 0.1330" (See Computation I) / = 3.8955" {d + ?) n = 2.6625" (I hi '+ 2.3125") r2 = 0.96875" (^ //2) D = 4.09375" (Depth of Section) T = 0.7099 sq. ins. (See Computation I) Rl = 2.2313 " (bhl) R2 = 6.1758 " {bh2) Tt Rlrl R2r2 9.1170 sq. ins. 2.7654 5.9408 5.9828 Moments of Inertia: 14.6890 = 9.1170 c; and hence = 1.6112 IT = 0.0053 (See Computation I) IRl = 0.0911 (MP -4- 12) IR2 = 1.9319 (W/23 -=- 12) T{t - c)2 = 3.7043 Rl{rl - c)2 = 2.4661 R2ic - r2Y = 2.5490 Section Modulus: I = 10.7477 y = 2.48255 (D - c) S = 4.3293 y The same general procedure shown in this sample calculation was fol- lowed in dealing with the other cross-sectional shapes. For this reason, only the final results of the several calculations are presented; although, for RELATIVE BEXDIXG STRENGTH OF CROSSARMS Table 1. — Section Modulus of Roofed Sections between Pinholes Knot Diameter- -Inches No Knot } l 1 U li 2 2i 3 Calculation 1: (Knots located at top of section) Section Size*: Minimum 6.86 5.08 3.57 2.33 1.35 0.64 End. Min 4.78 2.13 0.53 Nominal 7.37 5.50 3.91 2.59 1.54 0.76 Calculation 2: (Knots located at bottom of section) Section Size*: Minimum 6.11 5.24 4.42 3.71 3.05 1.92 1.05 0.47 Calculation 3: (Knots located im- mediately below top segment) Section Size*: Minimum 8.03 5.45 4.56 3.86 3.34 2.95 2.50 2.34 2.37 End. Min 7.65 3.65 Nominal 8.60 4.16 Table 2. — Section Modulus of Roofed Pinhole Sections Knot Diameter— Inches No Knot 1 1 i 1 H 2 Calculation 4: (Knots vertical) Section Size*: Minimum. . . . End. Min. . . Nominal 4.50 4.29 5.11 3.84 3.21 2.63 2.50 2.88 2.25 Calculation 5: (Knots horizontal) Section Size*: Minimum. . . . End. Min. . . . Nominal 3.63 2.96 2.40 2.26 2.76 1.97 1.41 1.33 1.64 1.11 * Section Sizes: Minimum = 3^" x 4^" End. Min. = 3^" x 4" (viz. minimum at end of arm) Nominal = 3J" x 4^" convenience, reference is made to the calculations by number in the pages that follow. These results are shown in Tables 1, 2. .S anH 4 and a brief discussion of the scope and use made of them follows. 28 BELL SYSTEM TECHNICAL JOURNAL Table 3. — Section Modulus of Bolt Hole Sections Knot Diameter— Inches No Knot i f 1 f 1 U i§ Calculation 6: Brace bolt hole section Section Size*: Minimum . Nominal . . 7.97 8.55 6.47 5.28 4.33 4.71 3.58 2.62 2.78 Calculation 7: Pole bolt hole section Section Size*: Minimum . Nominal . . 9.25 9.74 7.42 5.63 6.05 3.24 3.61 2^" Knot 1.51 1.66 3" Knot .75 .85 * Section Sizes: Minimum = Sys" x 4^" End. Min. = 3ys" x 4" (viz. minimum at end of arm) Nominal = 31" x 4-^" "Table 4. — Section Modulus of Rectangular Section without Bolt Holes {Calculation 8) Section Size Knot Diameter Section Modulus Minimum (3^" x 4^") (No Knot) 9.32 1 4 8.24 h 7.22 3 6.28 1 5.40 u 3.84 2 2.54 2i 1.51 3 .75 Nominal (3J" x ^") (No Knot) 9.78 i 8.67 ^ 7.62 3 6.64 1 5.72 H 4.10 2 2.74 2h 1.66 3 .85 Roofed Sections Between Pinholes As indicated in Table 1, three tabular calculations were made for roofed sections between pinholes. In Calculations 1, 2 and 3 it was assumed that the knots present were located (1) at the top, (2) at the bottom, and (3) immediately below the top segment of the section, respectively. The re- sults relating to the 3^" respectively, in Fig. 3. X 4^'' section are plotted as Curves 1, 2 and 3, RELATIVE BEXDI.XG STREXGTII OF CROSS A RMS 29 With respect to the knot positions considered, it is apparent from an exam- ination of the three curves (Fig. 3) that knots up to approximately 1^" in diameter are most damaging when located immediately below the roofed portion of the arm; and that the worst position for knots over 1|" in diam- eter is at the bottom of the arm. However, since under usual loading I 1.5 2 KNOT DIAMETER- INCHES Fig. 3 — Sections between pinholes. Section modulus of crossarm sections containing knots of the sizes shown on the base line and located in the positions indicated. The data apply to sections of minimum size (3]^" x 4^")- conditions knots at the bottom of an arm section are in compression, and thus would have less influence on strength than they would have on the tension side,- it was felt that the strength value shown by Curve 2 may be ignored; and that the values shown by a smooth curve, combining the values 2 On Page 69 of U. S. Dept. of Agriculture Tech. Bui. 479, "Strength and Related Properties of Woods Grown in the United States" by L. J. Markwardt and T. R. C. Wilson, is the following statement: "Knots have appro.ximately one-half as much elYect on com pressive as on tensile strength." 30 BELL SYSTEM TECHNICAL JOURNAL of Curve 3 up to the \\" knot point with those of Curve 1 for 2" and larger knots, would be the practical minimum section moduli for roofed sections between pinholes. Accordingly, such a smooth curve was constructed and is shown as Curve 2 in Fig. 4. The results of Calculations 1 and 3 for nom- KNOT DIAMETER- INCHES Fig. 4— Sections between pinholes. Section modulus of crossarm sections containing knots of the sizes shown on the base line and located in damaging positions. inal and arm-end minimum sections were also plotted, and Curves 1 and 3 drawn for those sections. Roofed Pinhole Sections Two calculations were made for the pinhole sections: Calculation 4, in which the knots were assumed to be located adjacent to the pinhole in a RELATIVE BEXDfXG ST RES GT II OF CROSS A RMS 31 vertical position; and Calculation 5, in which the knots were assumed to be immediately below the top segment in a horizontal position. The results of these two calculations are shown in Table 2. It has heretofore been gen- erally assumed that in pinhole sections knots less than \" in diameter were more damaging in a vertical position than in a horizontal position. The results of Calculations 4 and 5, however, show that the horizontal knots immediately below the top segment are the more damaging. In order to compare the effect of knots so located with the effect of knots at the extreme JliMiiMti^ 3» ^iicURVE I FOR 3-ix44 SECTIONS^ I CURVE 2 FOR 3^'x 4^" SECTIONS i r| CURVE 3 FOR S-j^'x 4" SECTIONS | mn 0 .5 I IS 2 KNOT DIAMETER - INCHES Fig. 5 — Pinhole sections. Section modulus of crossarm sections containing knots of the sizes shown on the base line and located in damaging positions. top of the section, the following two computations assumed \" and 2" hori- zontal knots at the latter location: l'\ Knot at Section Top: ^5 J _ .02875 (3.09375)' = 1.48156 S = 2.9631 2" Knot at Section Top: kS = .92875 (2.09375)- = .67857 5 = 1.3571 32 BELL SYSTEM TECHNICAL JOURNAL As the section modulus {S) values for sections containing 1" and 2" hori- zontal knots located immediately below the top segment are 1.97 and 1.11, respectively, (Calculation 5, Table 2) it is apparent that in pinhole sections horizontal knots immediately below the top segment are the more dam- aging. The results of Calculation 5 were accordingly plotted in Fig. 5 and smooth curves drawn to show the section modulus for each of the three sections containing knots of any size. Roofed Brace Bolt Hole Section The worst position for knots in the brace bolt hole section was assumed to be substantially the same as in the roofed sections between pinholes; and in Calculation 6, the results of which are shown in Table 3, knots up to \\" in diameter were assumed to be so located, viz. immediately below the top segment. To check this assumption with respect to worst position, the following analysis was made of the minimum sections: Distance from top of section: To top of bolt hole 1.78" To bottom of bolt hole 2.16" Distance from bottom of top segment: To top of bolt hole 1.45" To bottom of bolt hole 1.83" It is apparent that any knot ranging in diameter from 1.78" to 2.16", when located at the top of the section, would enter the bolt hole. The section modulus of any section containing a knot within that size range would be the section modulus of the remaining portion of the section, or — , where b is the width of the section and d the depth below the bolt hole. 6 Thus c r • • ^ 3.1875 (1.9375)^ . 001:5 6 (mmnnum arm) = = 1.9943 6 It is also evident that any knot from 1.45" to 1.83" in diameter, when located immediately below the top segment, would likewise enter the bolt hole; and that the section modulus, on this basis of knot location, would be the same for any section containing a knot within the size range mentioned. Continuing the analysis the following tests were made: 2" Knot: The distance between the top segment and the bottom of the bolt hole of a minimum section is 1.83". Therefore, a 2" knot located immediately RELATIVE BENDING STRENGTH OE CROSS ARMS 33 below the top segment would extend beyond the hole; and its effect would be the same as in Calculation 3 (Table 1), where the section modulus of a section containing a 2" knot similarly located was found to be 2.50. On the other hand, since a 2" knot is within the limits 1.78" and 2.16", the section modulus of a section containing such a knot located at its top would be 1.99. 1.7S" Knot: A knot of this size immediately below the top segment would enter the bolt hole since it is within the 1.45" and 1.83" limits, and the section modulus value associated with it would be the same as shown in the Calculation 6 results (Table 3) for a section containing a H" knot, or S = 2.62. But, as evident from previous discussion, the section modulus associated with this knot, if located at the top of the section, would be 1.99. 1.5" Knot: It can be shown that the section modulus of a section containing a knot of this size located at the top of the section would be 2.55; and that the section modulus associated w'ith a similarly located 1" knot would be 4.55. The foregoing analysis for minimum sections may be summarized as follows : Knot Size Section Modulus Knot at Top Knot below Top Segment (Inches) 2.0 1.78 1.5 1.0 (Inches^) 1.99 1.99 2.55 4.55 (Inches') 2.50 2.62 2.62 3.58 A study of this summary shows that knots 1^" and over are more dam- aging when located at the section top; and that knots under 1§" are more damaging when located immediately below the top segment. The section modulus values associated with 2|" and 3" knots would be the same as shown in the Calculation 1 results (Table 1). By a similar analysis for arms of nominal size it can be shown: (1) That the more damaging position for knots 1|" and under is imme- diately below- the top segment; (2) That the more damaging position for any knot within the diameter range from 1.875" to 2.25" and all the larger knots is at the top of the section; 34 BELL SYSTEM TECHNICAL JOURNAL (3) That the section modulus associated with 1.875" to 2.25" knots would be ?:?5y#^' = 2.0334; and 0 (4) That the section modulus values associated with 2|" and 3" knots would be the same as shown in the Calculation 1 results (Table 1). KNOT DIAMETER Fig. 6 — Brace bolt hole sections. Section modulus of crossarm sections containing knots of the sizes shown on the base line and located in damaging positions. The results of Calculation 6 (Table 3), and of the foregoing analyses, together with the Calculation 1 results for 2|" and 3" knots, were plotted in Fig. 6 for both minimum and nominal sections. t RELATIVE BEXDIXG STREXGTII OF CROSSARMS 35 Rectangular Pole Bolt Hole Section The most damaging position for knots in the pole bolt hole section was assumed to be at the top of the section. 'rhe\- were so ligured in Calcula- gip^^flHIHHgtaSHHfH^^ ^- 35 a o Z o I- 4 '\ 0 Ft:U:::^rt: CURVE I FOR 3:i:"» 4^" SECTIONS CURVE 2 FOR 3^"x 4^" SECTIONS I 2 KNOT DIAMETER -INCHES Fig. 7 — Pole bolt hole section. Section modulus of crossarm section containing knots of the sizes shown on the base line and located in damaging positions. tion 7, the results of which are shown in Table 3 and plotted in Fig, 7 for both minimum and nominal arms. 36 BELL SYSTEM TECHNICAL JOURNAL Rectangular Sections without Bolt Holes Here too the most damaging position for knots was assumed to be at the top of the section. In Calculation 8 the section moduU of sections contain- ing knots from I" to 3" in diameter were determined for both minimum and nominal sections. The results are shown in Table 4. As section modulus values for sections containing knots of other sizes than those shown may be found so simply by the formula for rectangular sections, 6* = -r- , no curves 6' of the results of this calculation were drawn. MoiiENT Diagrams From the results of this study as shown in Table 4 and in Figs. 4, 5, 6 and 7, section modulus values for clear arms and for arms containing knots of various sizes may be read and multiplied by appropriate fiber stresses to determine the resisting moments throughout the length of such arms. For example, the section moduli of clear arms of nominal dimensions, and of arms of minimum dimensions with the maximum knots permitted under the current Bell System crossarm specification (AT-7075) are as follows: Section of Arm Pole bolt hole Brace bolt holes Pole pinholes Other pinholes in middle section^. . End pinholes Other pinholes in end sections^. . . . Unroofed part of middle section. . . Roofed part of middle section Solid part of brace bolt hole zones'* Between pinholes in end sections. . Extreme ends Arms of nominal size Arms of minimum size with and free from knots ma.xiraum knots Section Modulus Section Modu- lus Diameter of Max. Knots 9.74 5.63 3" 8.55 4.33 3" 5.11 3.28 r 5.11 2.38 3// 5.11 1.33 H" 5.11 1.41 H" 9.78 3.84 H" 8.60 2.95 U" 8.60 4.56 an 4 8.60 2.17 2" 8.60 2.03 2" These section modulus values were used in preparing the moment dia- grams shown in Fig. 1. The clear arm of nominal dimensions was also assumed to be straight grained. The fiber stress factor used for it was 5000 psi, which is the ultimate fiber stress value that has been employed in the Bell System for many years for sawn southern pine and Douglas fir. The ^ For the purposes of specifying knot limitations, crossarms under Specification AT-7075 are divided into a middle section (between brace bolt holes) and end sections (beyond brace bolt holes). ^ Where a brace bolt hole zone is less than four (4) inches from a pinhole zone, these zones and the portion of the arm between them are considered as a single zone. RELATIVE BENDING STRENGTH OF CROSS ARMS 37 fiber stress factor used in computing the resisting moments for the arm of minimum size with maximum slant grain and maximum knots was vS250 psi, which is simply vSOOO psi discounted 35% to allow for slant grain of \" in 8", which is the maximum permitted by Specification AT-7075. A dis- count is, of course, unnecessary for the presence of knots, since allowance for their effect on strength was made in the section modulus values used. Since the 5000 psi value is an ultimate fiber stress and not a working stress, and since the arms were assumed to be made of clear, straight grained material, Graph 1 (Fig. 1) represents an idealized condition. The resisting moments shown are probably the maximum that may be expected from any commercial lots of southern pine or Douglas fir crossarms,* notwith- standing the fact that the dimensions of some of the arms may exceed the nominal specified. With respect to Graph 2 (Fig. 1), the objection may be raised that 35% is not a sufiicient discount for a 1" to 8" slant of grain and that the 3250 psi value makes no allowance for the effect of long continued loading. On the other hand, the graph assumes the simultaneous occur- rence of the maximum knot in a most damaging position in every section of an arm of minimum dimensions and having the maximum slant of grain allowed. Since the probability of such simultaneous occurrence of these defects and conditions is extremely small, it is felt that the resisting moments of Graph 2 represent the minimum strength of any arm of the two species concerned that may be furnished under Specification AT-7075. Under the assumptions made. Graphs 1 and 2 (Fig. 1) may be regarded as the upper and lower limits of the bending strength of specification cross- arms. On the same diagram may be plotted the graph or graphs of the moments resulting from any given load at each pin position, or any single load concentrated at any point on the arm. As an illustration. Graph 3, showing the bending moments from a load of 50 pounds per pin, is shown in the diagram (Fig. 1). A load of 50 pounds per pin is calculated to be the load of size 165 wire coated with ice having a radial thickness of \ inch in span lengths of 235 feet, or of wire of the same size in 100 foot spans where the radial thickness of the ice coating is | inch. Since Graph 3 is wholly below Graph 2, even an arm of lowest specification quality would support the assumed loads wdth some margin of strength to spare. This margin or factor of safety, would, of course, be increased greatly if the quality of the arm under consideration approached the quality assumed in Graph 1. As previously indicated, the probabiUty is extremely remote that any single arm will ever be furnished of a quality as low as assumed in Graph 2. It * Graphs 1 and 2 (Fig. 1 ) are for southern pine and Douglas fir crossarms. It is estimated that the resisting moments of comparable graphs for the other woods included in Specifi- cation AT-7075 should be about 20% lower. 38 BELL SYSTEM TECHMCAL JOURNAL follows, therefore, that the average strength of any lots of southern pine or Douglas fir arms produced under Specification AT-7075 may be expected to lie well above the Graph 2 limit. Graph 2 and a bending moment graph for vertical loads at each pin position are of considerable value to the material design engineer, since the degree of parallelism between the two will show whether a consistent strength relationship exists throughout the length of the crossarm. As a matter of interest in this connection, moment diagrams were used as a guide in setting the knot limitations shown in Specification AT-7075. Resisting and bending moment graphs may also be used to determine the location of the critical section of a crossarm by noting the point of coinci- dence between a maximum bending moment graph and the resisting moment graph for a clear arm. It can be shown by such graphs that this point in all types of Bell System crossarms, designed for vertical loads, is located at the pole pinholes. If the comparison were made between a maximum bending moment graph and the resisting moment graph of an arm containing all of the maximum defects permitted, the location of the point of coincidence between the graphs might or might not fall at the pole pinholes, depending on the magnitude and location of the defects allowed. It should be noted, however, that for such arms the critical section locations so determined apply only when the arms are actually of the assumed minimum quality; and, since the probability of such being the case is so extremely remote, it is concluded that the maximum stress or critical section locations in arms of that quality are of academic interest only, and that for all practical purposes the critical section of any 3j" x 4|" x 10' crossarm is located at the pole pinhole. This conclusion does not mean that every arm broken in service or under test will break at the pole pinhole; for, obviously, if some other section is rela- tively weaker because of some hidden defect which reduces its section modulus or its fiber strength, it will break at such section regardless of any mathematical determination of the break location. But the conclusion does mean that, generally speaking, when a crossarm breaks the break will occur at, or be closely related to, the pole pinholes. To check the accuracy of this conclusion, an examination was made of all available crossarm strength test data in which the break locations were recorded. The exam- ination revealed that, out of 258 arms tested, the breaks in 219, or 85 per cent, were either at, or directly related to, the pole pinholes. Six per cent of the breaks were located between the two pole pinholes, and 9 per cent at points outside the pole pinholes. As an illustration of another use to which such a moment diagram may be put, the following specific example is cited. Before the present standard Bell System specification for crossarms was drafted, it was decided to RELATIVE BEXDIXG STRENGTH OF CROSSAR.US 39 include a new type ("W6") with 16 i)in positions. It was felt that, if the additional pin holes in the type W6 did not unduly weaken the arm, it could not only replace the old t^-pe "JW" arm with 8 pin positions but also be used in installations where greater flexibility in wire spacings might be required. 10 20 30 40 50 60 distance: from center of arm -inches ^ ° 0 JW f c, 1 0 VV6 Fig. 8 — Resisting moments and maximum bending moments for clear JW and W6 crossarms. In order to obtain an estimate of the strength relationship between the two types, strength tests were made of 10 matched arms of each type. The test arms were made of air-seasoned, clear Douglas fir. The dimensions of the crossarm blanks were 3J" x 4|" x 20'. In selecting the 10 blanks from which the test arms were made, only straight grained pieces free from 40 BELL SYSTEM TECHNICAL JOURNAL evidence of manufacturing and other defects were chosen. Each blank chosen was cut into two 10' lengths, one of which was made into a JW arm and the other into a W6 arm, making 10 matched arms of each type. The tests were made on an Amsler testing machine. The average breaking load at the end pinholes was 1159 pounds for the JW arms and 1002 pounds for W6 arms. At the same time an estimate was made of the theoretical strength rela- tionship between the two types by means of the moment diagrams shown in Fig. 8. In this figure are shown the graphs of the resisting moments (fiber stress factor — 5000 psi) of clear JW and clear W6 arms, together with the graphs of the bending moments due to the maximum loads these arms would withstand when the loads are concentrated at the end pinholes. These maximum loads were determined by dividing the moments at the points of coincidence between the graphs (critical pole pinhole sections) by the dis- tances to the end pinholes. The maximum loads, so determined, are 608 pounds for the JW arm and 532 pounds for the W6 arm. The fact that these loads are low as compared with the actual breaking loads shows, of course, that the average ultimate fiber stress developed by these selected arms was considerably greater than 5000 psi, which is not surprising in view of their exceptionally high quality. However, so far as the information sought is concerned — namely, to determine not the actual strength but the strength relationship between the two types — the result would be the same regardless of the fiber stress factor used in the moment diagram. The ratio of the strength of the W6 arm to that of the JW arm as shown both by the actual strength tests and by the moment diagrams was as fol- lows: 1002 ,, ,^„ Actual strength tests - —r^ X 100 = Moment diagrams — ttt^ X 100 = OOo Strength Ratio W6 to JW (Per cent) 86.5 87.5 These ratios show a remarkably close agreement between theory and ac- tuahty and justify the belief that the crossarm moment diagram may be employed to obtain reasonably accurate estimates of relative bending strength. Summary The results of this study may be summarized as follows: 1. The moment diagram is a useful guide in setting specification limitations on defects. RELATIVE BENDING STRENGTH OF CROSS A RMS 41 2. It is shown that the critical section of a crossarm is located at the pole pinholes. The practical value of this observation is that it emphasizes the need for keeping the pole pinhole sections and the portion of the arm be- tween them reasonably free from strength reducing defects. 3. Only by breaking tests can the actual bending strength of crossarms be determined. The relative bending strengths, however, of two or more arms of different types or quality may be estimated with sufficient accuracy by means of the moment diagram, regardless of the fiber stress used in its construction. 4. If the fiber stress factor employed is dependable, the moment diagram may be used to estimate the minimum factor of safety that would obtain for an arm of any type or any assumed quality. In this connection, it is believed that the strength of Bell System crossarms is well above the mini- mum required to support the loads ordinarily carried. 5. The section modulus curves of Figs. 4, 5, 6 and 7 will simplify the con- struction of moment diagrams for arms of the same sizes shown in the figures but dififering with respect to type and quality. The uses listed lead to the general conclusion that the crossarm moment diagram is a convenient and reasonably reliable engineering tool. APPENDIX Computation I. Moment of Inertia of Top Segment of Minimum (J^" x 4^") Section between Pinholes: The moment of inertia {IT) of a segment {T) with respect to an axis through its center of gravity and parallel to its base may be found by the formula IT = Ibb - Ax'' where I bb is the moment of inertia of the segment about the axis BB,A the area of the segment and .v the distance between the two axes. The values I BB, A and x are given by: 1 sfi 1 2 sin^ a cos a 1 , . Ibb = \Ar 1 H -. (1) [_ a — sm a cos a J A = Ir^ (2a - sin 2a) (2). x = i'^^ (3) 42 BELL SYSTEM TECHNICAL JOURNAL Fig. 9 — Crossarm section between pinholes. The significance of ;- and a in these formulae, and of the other symbols used in the computations that follow will be clear from a glance at Fig. 9. 1/2 b D = 4.09375" b = 3.1875" ^b = 1.59375" r = 4" r- = 16.000000 (1/2 by- = 2.540039 p- = 13.459961 p = 3.668782" d A Sin a = = 0.39843750 a = 23° 28' 49.93" a = 0.40981266 radians 2 a = 46° 57' 39.86" Sin^ a = 0.063252925 Sin 2 a = 0.73089017 Cos a = 0.91719548 Sin a Cos a = 0.36544507 • = /)+(/)-;-) = 3.7625" = 0.7099 sq. ins. [Area of T by Formula (2)] = 3.8018" [By Formula (3)] = .V - /> = 0.1330" I BB= 10.2654 [By Formula (1)] Ax- = 10.2601 IT = 0.0053 (Note: WTiile the results of this and the following computations are shown to four decimal places, the actual work was done by machine and carried to eight decimal places as mentioned in the text.) Since the width of the section in this computation and the radius of its roof is the same as for the minimum 33^" x 4" section at the end of the arm, the top segments of the two are identical, and the only value that will differ RELATIVE BEXDLXG STRENGTH OF CROSS A RMS 43 will be the depth (d) of the rectangular j^jortion of the section, which for the smaller will he p + (D — r), ov 3.6688 + (4 - 4) = 3.6688" Computation II. Moment of Inertia of Top Segment of Nominal {3\" x 4ys") Section between Pinholes: As this computation was made in exactly the same manner as Computa- tion I, only the results are here shown: d = 3.8593" g - 0.1317" .4 = 0.7168 sq. ins IT = 0.0053 Computation III. Moment of Inertia of Top Segment of Minimum {3y&" x 4^_") Pinhole Section: It will be noted in Fig. 10 that the top segment is divided into four parts: the small segment {Ti) at the top of the pinhole, the rectangular portion Fig. 10 — Crossarm pinhole section. Ri, with a width of bi and a depth of di, and two portions designated Tc. The purpose of this computation is to determine the moment of inertia of one of the Tc portions with respect to its gravity axis parallel to its base. The moment of inertia of the two Tc portions about the axis BB ma}' be 44 BELL SYSTEM TECHNICAL JOURNAL found by deducting the moments of inertia of Ti and Ri about this axis from the moment of inertia of the entire top segment about the same axis. D = 4.09375" Sin a = ^-^^ = 0.16625 r hi = 1.33" a = 9° 34' 11.49" i Z>i = 0.665" a = 0.16702554 radians r = 4.00" 2 a = 19° 8' 22.98" ^2 = 16.000000 Sin^ a = 0.0045949941 (1/2 biY- = 0.442225 Sin 2 a = 0.32787285 Pi^ = 15.557775 Cos a = 0.98608364 pi = 3.9443345" Sin o Cos a = 0.16393640 d (Computation /) = 3.7625" di = pi-\- (D-r) - d = 0.2756" r, = pi- l/2di = 3.8065" Area Ri = h\di = 0.3665 sq. ins. A 1 = 0.0494 sq. ins. [Area of Ti by Formula (2) ] .vi = 3.9666" [By Formula (3)] By Computation I, IT bb = 10.2654 ITiBB [Formula (1)] — 0.7777 IRiBB= ^' + i?i;-i- = 5.3126 ' 6.0903 IITcbb = 4.1751 The moment of inertia of the 2 Tc areas with respect to the axis through their own centers of gravity is given by 21 Tc = IITcbb - ITcz^ where 2 Tc is the area of the two Tc portions of the top segment and is given by 2Tc = A - Ui^Ri) in which .4 is the area of the entire top segment as shown in Computation I; and where, by the principle of moments, Tx — TiX\ ~ Riti z = 2Tc in which Tx, TiXi and RiTi are the moments of the areas of T, Ti and Ri, respectivel}^ about the axis BB. (Tx = Ax of Computation I.) Thus 2Tc = 0.2940 sq. ins. z = 3.7680" RELATIVE BENDING STRENGTH OF CROSSARMS 45 As previously shown, IITcbb — 4.1751 2Tc s2 = 4.1738 llTc = 0.0013 ITc - 0.0007 D - r = 0.09375" s = 3.7680" 3.8618" d = 3.7625" g for Tc - 0.0993" The results of this computation apply also to the minimum Sys" x 4" pin- hole section at the ends of the arm. The depth (d) of the rectangular por- tion of the end pinhole sections will be the same as at the extreme ends of the arm, viz. 3.6688". Computation IV. Moment of Inertia of Top Segment of Nominal (Jj" x 4ys'') Pinhole Section: Since this computation was made in the same manner as Computation III, only the results are here shown: d = 3.8593" g = 0.1019" Tc — 0.1630 sq. ins. ITc = 0.0008 Mathematical Analysis of Random Noise BY s. o. RICE {Concluded from Jidy 1944 issue) PART III STATISTICAL PROPERTIES OF RANDOM XOISE CURRENTS 3.0 IXTRODUCTIOX In this section we use the representations of the noise currents given in section 2.8 to derive some statistical properties of /(/). The first six sec- tions are concerned with the probabihty distribution of /(/) and of its zeros and maxima. Sections 3.7 and 3.8 are concerned with the statistical prop- erties of the envelope of /(/). Fluctuations of integrals involving /'(/) are discussed in section 3.9. The probability distribution of a sine wave plus a noise current is given in 3.10 and in 3.11 an alternative method of deriving the results of Part III is mentioned. Prof. Uhlenbeck has pointed out that much of the material in this Part is closely connected with the theory of Markoff processes. Also S. Chandrasekhar has written a review of a class of physical problems which is related, in a general way, to the present subject." 3.1 The Distribution of the Noise Cureent^^ In section 1.4 it has been shown that the distribution of a shot effect current approaches a normal law as the e.xpected number of events per second, v, increases without limit. In Une with the spirit of this Part, Part III, we shall use the representation .V HO = Z) («" cos CO,,/ -\- bn sin aj„0 (2.8-1) to show that /(/) is distributed according to a normal law. This is obtained at once when the procedure outlined in section 2.8 is followed. Since <7„ and bn are distributed normally, so are a„ cos ooj and bn sin a)„/ when / is regarded as fixed. /(/) is thus the sum of 2X independent normal variates and consequently is itself distributed normally. 22 Stochastic Problems in Physics and Astronomy, Rei\ of Mod. Phys., Vol. 15, pp. 1-89 (1943j. 23 An interesting discussion of this subject by V. D. Landon and K. A. Norton is given in the I.R.E. Proc, 30 (Sept. 1942j pp. 425-429. 46 MATHEMATICAL ANALYSIS OF RANDOM NOISE 47 The average value of /(/) as given by (2.8-1) is zero since dn = bn - 0: l{t) = 0 (3.1-1) The mean square value of /(/) is X J'iO — X^ {'^'n COS" COn/ + b'n Sin C0„ /) n = l V = E ^K/JA/ (3.1-2) w{f) df = ^^(0) = ^ In writing down (3.1-2) we have made use of the fact that all the c's and i's are independent and consequently the average of any cross product is zero. We have also made use of which were given in 2.8. \P(t) is the correlation function of /(/) and is related to iv(f) by v., ^ ,A(t) = [ w(f) cos 27r/T df (2.1-6) as is explained in section 2.1. In this part we shall write the argument of \P(t) as a subscript in order to save space. Since we know that I(t) is normal and since we also know that its average IS zero and its mean square value is \po , we may write down its probability density function at once. Thus, the probability of /(/) being in the range I, I -j- dl is This IS the probabihty ot finding the current between 7 and I -\- dl a.i a. time selected at random. Another way of saying the same thing is to state that (3.1-3) is the traction of time the current spends in the range /, / + dl. In many cases it is more convenient to use the representation (2.8-0} 1(0 = L Cn COS (O^nt - x I{t) becomes distributed according to a normal law. In order to make the limiting process definite we first choose .Y and A/such^that iVA/ = F where r wU) df 0 as .V -^ =c , and consequently the central limit theorem* may be used if wif) = 0 for f > F. Since we may make F as large as we please by choosing e small enough, we may cover as large a frequency range as we wish. For this reason we write =o in place of F. Now that the central hmit theorem has told us that the distribution of I(t), as given by (2.8-6), approaches a normal law, there remains only the problem of finding the average and the standard deviation: .V 1(0 = Zl <^n cos (cOn/ — 4/ [ fwif) df = -^'o' Jo Ml2 = ^V ~ ~Z^(^~n W„ cos (oJn h — W"^27rfi I A^ I and the interval over which zeros are produced is given by 2A/ = il!I^ ir/i MATHEMATICAL ANALYSIS OF RANDOM NOISE 57 The number of zeros is this multiphed by 2/2 . Since there are 2/i such intervals per second the number of zeros per second is TT This differs from the result given by our formula by a factor of 2/7r. This discrepancy is due to our representing the two bands by the sine waves h and I2. From this example we obtain the picture that when the integral for \J/o converges corresponding to .1 — ^ 0, while at the same time the integral for ^0 diverges, corresponding to /o ^ =c in such a way that Afo — > =c ^ the noise current behaves something like a continuous function which has no derivative. It seems that for physical systems the integrals will always cGn\-erge since parasitic effects will have the effect of making w(f) tend to zero rapidly enough. The frequency which represents the region where this occurs is of the order of the frequency of the microscopic wiggles. So far we have been considering the formulas of this section in the most favorable light possible. There are experiments which indicate the possi- bihty of the formulas breaking down in some cases. Prof. Uhlenbeck has pointed out that if a very broad band fluctuation current be forced to flow through a circuit consisting of a condenser, C, in parallel with a series com- bination of inductance, L, and resistance, R, equation (3.3-11) says that the expected number of zeros per second of the current, 7, flowing through R (and L) is independent of R. It is simply -(LC)~^^. The differential TT equation for / is the same as that which governs the Brownian motion of a mirror suspended in a gas^°, the gas pressure playing the role of R. Curves are available for this motion and it is seen that their character depends greatly upon the pressure^\ Unfortunately, it is difficult to tell from the curves whether the expected number of zeros is independent of the pressure. The differences between the curves for various pressures indicates that there may be some dependence*. 3.4 The Distribution of Zeros The problem of determining the distribution function for the distance between two successive zeros seems to be quite difficult and apparently ^^ For example, by putting the circuit in series with a diode. ^^ This problem in Brownian motion is discussed by G. E. Uhlenbeck and S. Goudsmit, Phys., Rev., 34 (1929), 145-151. 31 E. Kappler, Annalen d. Phys., 11 (1931) 233-256. * Since this was written M. Kac and H. Hurwitz have studied the problem of the ex- pected number of zeros using quite a different method of approach which employs the "shot-effect" representation (Sec. 3.11). Their results confirm the correctness of (3.3-11) when the integrals converge. When the integrals diverge the average number of elec- trons, per sec. producing the shot effect must be considered. 58 BELL SYSTEM TECHNICAL JOURNAL nobody has as yet given a satisfactory solution. Here we shall give some results which are related to the general problem and which give an idea of the form of the distribution for the region of small spacings between the zeros. We shall show (in the work starting with equation (3.4-12)) that the probabiUty of the noise current, /, passing through zero in the interval TjT -\- dr with a negative slope, when it is known that / passes through zero at r = 0 with a positive slope, is di 27 I [^J [f ^] (^0^ - ^ir'\^ + H cor\-H)] (3.4-1) where M^i and Miz are the cofactors of /i22 = — lAo and na = —ypj in the matrix ^0 0 ^; h 11 II 1 0 —Wo -^r -^r ^: -/; l" —Vo 0 Jr -^: 0 1^0 M = H = M23[Ml2 - Mis] (3.4-2) -1/2 We choose 0 < cot~^ ( — H) < r, the value tt being taken at t = 0, and the value 7r/2 being approached as r -^ co . It should be remembered that we are writing the arguments of the correlation functions as subscripts, e.g., — \}/r is really -Vir) = 47r' [ fw{f) cos lirfTdf (3.3-8) Jo As T becomes larger and larger the behavior of / at r is influenced less and less by the fact that it goes through zero with a positive slope at r = 0. Hence (3.4-1) should approach the probability that, for any interval of length dr chosen at random, I will go through zero with a negative slope. Because of symmetry, this is half the probabiUty that it will go through zero. Thus (3.4-1) should approach, from (3.3-10), ^r^'r (3.4-3) 27rL ^0 J oc . It actually does this since M approaches a diagonal matrix and both M23 and H approach zero with M23/H low pass filter cutting off at/t (3.4-3) is M2 drfb^ -1/2 — i/'oi/'o- For a (3.4^) The behavior of (3.4-1) as r — >• 0 is quite a bit more difficult to work out. 8 L — t/'o'Ao J MATHEMATICAL ANALYSIS OF RANDOM NOISE 59 1/22 and If 23 go to zero as r , M22 — M23 as r , and consequently // goes to infinity as t~ . The final result is that (3.4-1) approaches (4) ,"2- dr T ■ '"""■ — tAcAo as r — ^ 0, assuming \l/ exists. Here the superscript (4) indicates the fourth derivative at r = 0, rp'o*' = 16/ f Mf) df (3.4-6) For a low pass filter cutting off at/t (3.4-5) is dr ^ {lirf.f (3.4-7) WTien (3.4-1) is applied to a low pass filter, it turns out that instead of r the variable ip = lirfbT, dip = lirfb dr (3.4-8) is more convenient to handle. Thus, if we write (3.4-1) as p((p) d CO Pi^) "^ ^ ^^ *^ (3.4-9) p{ip) has been computed and plotted on Fig. 1 as a function of (p for the range 0 to 9. From the curve and the theory it is evident that beyond 9 p{ip) oscillates about 0.0919 with ever decreasing amplitude. We may take p{(p) d\p to be the probability that / goes through zero in ( \ N \ 0.^0 / '«; V ^ ^ y 0.10 / \ \ \ \ \ 1 ;/ / \ \ \ / • 9 0.05 y / 1^= TT/3- 0-' o EV PC Rl MENTAL POINTS o Fig. 1 — Distribution of intervals between zeros — low-pass filter j'xA(p is probability of a zero in Xtp when a zero is at origin. yn^v is probability of a zero in A(p when a zero is at origin and slopes at zeros are of opposite signs. 3'b — p{v)ifb = filter cutoff, r = time between zeros. through zero in t, t -\- dr when it is known that / passes through zero at T = 0 is where the notation is the same as in (3.4-1) and — - < tan H < - . This curve should always lie above p() is equal to D~^'^. Applying our transformation to the exponent: xi = yi — aU^^'^yi a;2 = 0 + D^^'^y^ Di= \ - a Since Xi runs from 0 to oo so must y0 gives 0 < 0 < tt yi ^ aDJ^'^yi gives cot 6 > aD^^'^ dyi dyi = p dpdd and obtain / JO r - 2 dd j pe " dp Jo = ^D7"' cot-^ iaD^'^') where the arc-cotangent lies between 0 and tt. This may be written in the simpler form T l/i 2\— 1/2 —1 1 J = ^(1 — a ) cos a = ^ip CSC cp where a = COS +i "" "" \ 2 ' 2 w?, n odd As was mentioned earlier, the method used to evaluate the double inte- grals may also be applied to similar triple integrals. Here we state two results obtained in this way. «Q0 rtOO «GO / dx I dy dz exp [—x^ — y — z — Icxy — Ihzx — 'layz\ Jo Jo Jo *00 /.GO ^QO / dx I dy dz yz exp [—x — y — z' — Icxy — Ihzx — layz] Jo Jo Jo \/Tr[l+ a -b-c (^ - be 1 ,2 c '7\ = m L l + a - ^^ (« + /5 + T - -) J (3.5-7) where 0 and 7 are obtained by cyclic permutation of a, b, c from a-cb ^ . _: r D^ Y (1 - c2)i/2(l - ^,2)1/2 ^'"^ L(l - c')(l - b^)j a = cos _i a — ic = cot ^1/2 where a, (3, 7 all lie in the range 0, t and where D, = 1 c b c 1 a b a 1 = 1 + 2 abc —a — b — c" For reference we state the integrals which arise from the definition of the normal distribution given in section (2.9) dxi • • • I dxn exp — X) «rs .^V •^*« = I — \\ /+00 /.+00 r « "IF" 'V-l- A dxi • • • I dXnXtXu exp | — 2Z arsXrX^ = I — 1^ ^-~ (3.5-8) MATHEMATICAL AX A LYSIS OF RANDOM NOISE 71 where the quadratic form is positive definite and | a | is its determinant. A lu is the cof actor oi atu . Incidentally, these may be regarded as special cases of [^ dx, ■■■ j clxjrZ OrsXrxA F (T, brxA 7 r n-l-|l/2 ^+00 ^« 2 /•/ 2 , 2-. fix + y ) (3.5-9) 2^ Arsbrbs 1/2^ i ( which is a generalization of a result given by Schlomilch.* 3.6 DlSTRIBUTIOX OF MAXIMA OF NOISE CURRENT Here we shall use a result similar to those used in sections 3.3 and 3.4. Let 3'^be a random curve given by (3.3-1), y = F{ai '•' Gn ; x). (3.3-1) If suitable conditions are satisfied, the probability that y has a maximum in the rectangle (xi , xi + dxi , ji , yi + dyi), dxi and dvi being of the same order of magnitude, is " —dxi dyi I p(yi, 0, f)f d^ (3.6-1) and the expected number of maxima of y in a < x < b is obtained by in- tegrating this expression over the range — =<= < yi < ^ , a- '^ xi < b. /'(s> Vy D is the probability density function for the random variables ^ = F(ai , • • • , c.v ; Xi) = ('!) \dx /i=xi (3.6-2) c ^ ydx^),=,. *Hoheren Analysis, Braunschweig (1879), Vol. 2, p. 49-1, equ. (29). ^- Am. Jour. Math.. Vol. 61 (1939) 409-416. A similar problem has been studied by E. L. Dodd, The Length of the Cycles Which Result From the Graduation of Chance Elements, Ann. Math. Stat., Vol. 10 (1939) 254-264. He gives a number of references to the literature dealing with the fluctuations of time series. 72 BELL SYSTEM TECHNICAL JOURNAL In our application of this result we replace x and yhy t and I as before. Then ^ = 7 = 2 C„ COS (a'«< — ^«) 1 where the primes denote differentiation with respect to /. According to the central limit theorem the distribution of ^, ??, f approaches a normal law. The second moments defining this law may be obtained either from the above definitions of ^, 77, ^, or may be obtained from the correlation function as was done in the work following equation (3.4-13). 1^ = V'o, rf = —^0 , ^»7 = 0 ^ = /'(/)/''(/) = Limit I [ I'{t)r\t) dt T~*oa I Jo = Lirmt ^ [I'\T) - l"m = 0 U = Limit i f mnt) dt T-* 1 Jo . . .6 \1/(t) // = Limit , , = lAo Y' = Limit i [ /"(/)/"(/) ^/ 7'-»oo i Jo = Limit i [ I^'\t)I{t) dt T-*oo 1 Jo = 1^0 where the superscript (4) represents the fourth derivative. The matrix M of the moments is thus M = 0 -^Po 0 j/'o 0 i/'o _ The determinant | M \ and the cofactors of interest are \M\ = -^o(M^'' - yp?) (3.6-3) MATHEMATICAL ANALYSIS OF RANDOM NOISE The probability density function in (3.6-1) is p(I,0,t) = (2Tr"'\M\-"'exp 73 L 2 \M\ (Mnl' + M33r' + 2Mi3/r) ] (3.6-4) and when this is put in (3.6-1) and the integration with respect to f per- formed we get dl dl s,-3/2 r (3.6-5) for the probabihty of a maximum occurring in the rectangle dl dt. As is mentioned just below expression (3.6-1), the expected number of maxima in the interval /i , /2 may be obtained by integrating (3.6-1) from h to t^ after replacing x by /, and / from — oo to + °° after replacing y by /. "When we use (3.6-4) it is easier to integrate with respect to / first. The expected number is then Mn (4) d^ = (/2-/x)'^Vr=^^r^,i 27r 27r L-i^oJ Hence the expected number of maxima per second is 27rL-^d / A(/) df / Mf) df (3.6-6) For a band pass filter, the expected number of maxima per second is 11/2 uii-fVi (3.6-7) where fb and fa are the cut-off frequencies. Putting /„ = 0 so as to get a low pass filter, W /ft - = .775/6 (3.6-8) 74 BELL SYSTEM TECHNICAL JOURNAL From (3.6-8) and (3.6-5) we may obtain the probability density function for the maxima in the case of a low pass filter. Thus the probability that a maximum selected at random from the universe of maxima will he in I, I -\- dl is dl 3v 2x^0 _ where 2,-9.^/8 ^(StT 1/2 ye I 1 +erfj(- l/2\ - (3.6-9) 1 k \ 1= OUTPUT NOISE CURRENT 1 \ ■/ijr=RMS VALUE OF - 1 0.3 / h y = I - - /0 2 0.1 Pi(y) ^ k — Fig. 2^Distribution of maxima of noise current. Noise through ideal low-pass filter. -7= dl = probability that a maximum of / selected at random lies between I and I + H- When y is large and positive (3.6-9) is given asymptotically by dl \/5 -J,2;2 — =. -^^ — ye V'/'o 3 If we write (3.6-9) as pi{y) dy, the probability density pi{y) of y may be plotted as a function of y. This plot is shown in Fig. 2. The distribution function P{I„u,^ < ys/xj/o) defined by P(/,nax < yV^o) = J Pi(y)-dy and which gives the probabihty that a maximum selected at random is less than a specified y\/\l/o = I, is one of the four curves plotted in Fig. 4. If / is large and positive we may obtain an approximation from (3.6-5). We observe that \M (4) 'Ao'/' MATHEMATICAL AX A LYSIS OF RANDOM NOISE 75 SO that when / is large and positive ^-Af 11/2/21 A/ 1 ^^ ^-/2/2^0 Also, in these circumstances the 1 + erf is nearly equal to two. Thus re- taining only the important terms and using the definitions of the M's gives the approximation to (3.6-5): \^T--' From this it follows that the expected number of maxima per second lying above the line 7 = /i is approximatel}' when 7i is large, 27rL 'Ao J (3.6-11) _ ^-iiiHo y i[the expected number of zeros of / per second] It is interesting to note that the approximation (3.6-11) for the expected number of maxima above /i is the same as the exact expression (3.3-14) for the expected number of times I will pass through /i with positive slope. 3.7 Results on the Envelope or the Noise Current The noise current flowing in the output of a relatively narrow band pass filter has the character of a sine wave of, roughly, the midband frequency whose amplitude fluctuates irregularly, the rapidity of fluctuation being of the order of the band width. Here we study the fluctuations of the envelope of such a wave. First we define the envelope. Let fm be a representative midband fre- quency. Then if 03m = 2irfrn (3.7-1) the noise current may be represented, see (2.8-6), by I = Z^ Cn cos (Uni — OOmt — (fn -\- C>^mi) n=l = Ic COS (i}mi ~ la sin COto^ where the components Ic and /« are (3.7-2) 7c = X/ ^n cos (cOnt — Oimi — S^n) n=l N Is = Z2 Cn sin (cO„ t — COmt — (p„) (3.7-3) ^ This expression agrees with an estimate made by V. D. Landon, Froc. I. R. E., 29 (1941), 50-55. He discusses the number of crests exceeding four times the r.m.s. value of /. This corresponds to I\ = IGiZ-o . 76 BELL SYSTEM TECHNICAL JOURNAL The envelope, R, is a function of t defined by R = [fc-\- fr (3.7-4) It follows from the central limit theorem and the definitions (3.7-3) of /« and Is that these are two normally distributed random variables. They are independent since IJt = 0. They both have the same standard deviation, namely the square root of 7! = 7! = 7 = r w{f) df = ,^0 (3.7-5) Jo Consequently, the probabiHty that the point (/c , h) lies within the ele- mentary rectangle dicdis is die dl. linpi -expf-^^n (3.7-6) 0 L 2^0 J In much of the following work it is convenient to introduce another ran- dom variable 6 where Ic = R cos e (3.7-7) I^ = R sin 6 Since Ic and /^ are random variables so are R and 6. The dififerentials are related by dIcdIs = RdBdR (3.7-8) and the distribution function for R and 6 is obtainable from (3.7-6) when the change of variables is made: dd RdR-R2i2^^ (3.7-9) lir \p, Since this may be expressed as a product of terms involving R only and 6 only, R and d are independent random variables, d being uniformly dis- tributed over the range 0 to Iv and R having the probability density ^0 e " ''*" (3.7-10) Expression (3.7-10) gives the probability density for the value of the en- velope. Like the normal law for the instantaneous value of I, it depends only upon the average total power ^0 = f wif) df Jq JO ^ See V. D. Landon and K. A. Norton, LR.E. Proc, 30 (1942), 425-429. MATHEMATICAL ANALYSIS OF RANDOM NOISE 77 . We now study the correlation between R at time t and its value at some later time / + r. Let the subscrij)ts 1 and 2 refer to the times t and t -\- t, respectively. Then from (3.7-3) and the central limit theorem it follows that the four random variables In , /*i , Ic2 , Is2 have a four dimensional normal distribution. This distribution is determined by the second^ mo- ments Id = Isl = Ic2 = Is2 = ^0 = Mil Ic\Ia\ — I dial — 0 Icihi = /»i/«2 = t; Z^ c„ cos (w„r — WmX) 2. n=l \ w{f) COS 27r(/ - /Jt df = mis (3.7-11) J AT Iclls2 — —IcilsX = T^^ZI Cn siu (w„ T — aj„. t) Z 71=1 / w{f) sin 27r(/ - /Jt df = nu Jo M = The moment matrix for the variables in the order Id , /«: , la , I»i is "Ao 0 M13 M14 0 l/'o — M14 M13 Mi3 — Mi4 iAo 0 .Ml4 Ml3 0 l/'o _ and from this it follows that the cofactors of the determinant | M \ are Mil = Mio = Mzz = Mu = }p(i{ypl — Mi3 — mh) = l/'o^, A = l/'o — Ml3 — Ml4 Mi2 = .¥34 = 0 Ml3 = M24 = — Ml3^ Mi4 = — lf23 = —H14A \M\ = A" The probability density of the four random variables is therefore ^^^v-^{Ui\ + il + il + il) (3.7-12) 2m13(/i/3 + 12/4) - 2m14(/i/4 - 12/3)] 78 BELL SYSTEM TECHNICAL JOURNAL where we have written Ii , h , h , h for Id , Tsi , Ic2 , I$2 . We now make, the transformation /i = Ri cos ^1 Is = R2 cos 02 I2 = Ri sin 61 Ii = R2 sin 62 and average the resulting probabiUty density over di and 62 in order to get the probability that Ri and R2 lie in dRi and dR2 . It is R\ R2 dR\ dRo f ddi \ dd2 exp 0 •'o 4:TV- A Jo {^oR\ + lAoi?' - 2(jiizRiR2 cos {do - di) - 2fxuRiR2 sin (^o - di)] lA Since the integrand is a periodic function of Bo we may integrate from Q2 = 61 to 62 = 61 -\- 27r instead of from 0 to lir. This integration gives the Bessel function, /o , of the first kind with imaginary argument. The result- ing probabiUty density for i?i and Ro is R1R2 r 1R1R2 r 2 , 2 il/2\ "Ao /T32 , „2n /, - . ,x [mi3 + M14J 1 exp - —- {Ri + R2) (3./-13) ^^'\-A A \ A ^" '^ / ^ 2.4 where, from (3.7-12), .,222 A = \f/o — His — fJLu His and nu are given by (3.7-11). Of course, Ri and R2 are always positive. For an ideal band pass filter with cut-offs at/a and/s we set fm = ^^^, -^(f) = ^^0 for fa- results which seemed worth sal- vaging at the time were given in reference^* cited in Section 3.3. 80 BELL SYSTEM TECHNICAL JOURNAL X1X2. = lol's = X w(/n)A/27r(/„ — fm) = ^1 1 «4^5 = hic = —bi N 'cc^, = Icic = -L^^(/)A/'47r'(/„ -fmY = -62 1 X4X6 = Igls = "^2 0C2XZ = Isle = —bz X^Xf, = Icis = bz All of the other second moments are zero. The moment matrix M is thus bo bi - -62 0 0 0 bi bo - -bz 0 0 0 M = — &2 —bs 0 0 bi 0 0 bo 0 -b. 0 -b2 0 0 0 -bi bo bz 0 0 0 -b2 bz bi_ The adjoint matrix is Bo Bi — B2 0 0 0 Bi B22 —Bz 0 0 0 — B2 —Bz Bi 0 0 0 0 0 0 Bo - -5i - -B2 0 0 0 -B, B22 Bz _ 0 0 0 -B2 Bz Bi_ Bo = (b^bi - bl)B B22 = (bobi - bl)B B\ = — (bibi — bibzjB Bz = — (b(^z - bib2)B B2- = (bibs — b2)B Bi = (b^o -b\ )B (3.8-3) B = bobibi -\- 2 bib2bz — 62 — bobz — bibi I M I = 5' where B is the determinant of the third order matrices in the upper left and lower right corners of^M. jjjAs in the earlier work, the distribution oi Xi , • • - , Xe is normal in six dimensions. The exponent is — [2 j M | ]~ times Bo{xi + Xi) + 25i(xiX2 — XiXi) — IBiixiXi + XiX^) + -622(^1^2 + Xi) — 2B3{X2Xz — XaXo) (3.8-4) + Bi{xl + xl) MATHEMATICAL ANALYSIS OF RANDOM NOISE 81 In line with the earher work we set Xi = Ic = R cos 6 Xi = Is = R sin 6 X2 = l[ = R' sin ^ + i? cos dd' x^ = l[ = R' cosd - R sin 66' .x-3 = I'J = R" cos 6 - 2R' sin 66' - R cos 66'- - R sin 66" X6 = I's = R" sin 6 + 2R' cos 66' - R sin 66'^ + R cos 00" The angle 6 varies from 0 to lir and 6' and 6" vary from — oo to + oc . By forming the Jacobian it may be shown that dxi dx2 • • • dxe = R^ dR dR' dR" dd d6' d6" Also, the quantities in (3.8-4) are xl + xl = R^ 0:1X3 + XiXs = RR" - R^d'^ X1X2 — XiX^ = R'6' X2 + xl = R'' + R''6'' X2Xz - x^xg = RR"6' - 2R'~6' - R'R6" - Ri'6'^ xl + xl = R'" - 2RR"6'- + AR'V + 4RR'6'd" + i?'0'* + R^d"^ The expression for p(R, 0, R") is obtained when we set these values of the x's in (3.8-4) and integrate the resulting probability density over the ranges of 6, 6', 6": ^(^' "' ^") = 8^ i '' L "' L """ ^'-'-'^ exp -^^[BqR^ + 2BiR:6' - 2B.XRR" - R^6'^) + B22R-6'- - 2B3R6'{R" - R6'-) + B,(R"' - 2RR"6'^ + R'6" + R'd'")] The integrations with respect to 6 and 6" may be performed at once leaving p(R, 0, R") expressed as a single integral which, unfortunately, appears to be flifficult to handle. For this reason we assume that w(f) is symmetrical about the mid-band frequency /« . From (3.8-2), Z»i and bs are zero and from (3.8-3), Bi and B3 are zero. 82 BELL SYSTEM TECHNICAL JOURNAL With this assumption (3.8-5) yields p{R, 0, R") = R\2Tr"'BT"' f dd' (3.8-6) J—oo exp —^[B,F: + R{[Bo^ + IB-ARS" - 2B,R") + B,{R" - RO'^] The probability that a maximum occurs in the elementary rectangle dR dt is, from (3.8-1), p{t, R) dR dt where p{t, R) = -! p{R, 0, R")R" dR" (3.8-7) We put (3.8-6) in this expression and make the following change of variables. r1/2 „1/2 X = -^ Re'\ y = -4^ R" V2B -^ V2B z = ^ R = -4^ i? (3.8-8) \/254 B V2Bi ^ ^ _(^22 + 2^o) 25 6; 2 _ Bo 2Bi _ bobi 3 Z»o64 .2 ~ 2^J = K3 - a' where we have used the expressions for the B's obtained by setting bi and bz to zero in (3.8-3). Thus Pit, R) = -1- (PY r y dy f X-'" dx (3.8-9) bobl \27r/ ^0 ^0 exp [ — a" 2^ + 2bzx + 2zy — (x -{- y)^] As was to be expected, this expression shows that p(t, R) is independent of /. A series for p(t, R) may be obtained by expanding exp 23(y + bx) and then integrating termwise. We use [ dy [ dxx'y'^e-^'^"'" = Jq Jq Vtt r(7 + i)r(M + 1) 2M+7+2 r (._t^3) which may be evaluated by setting X = p cos* if, y = p~ sin' (p MATHEMATICAL ANALYSIS OF TLiNDOM NOISE 83 The double integral in (3.8-9) becomes _a=.2 /t f> (2sr Y nib"" T(m + ^Wn - m + 2) n — 0^2^ 2 e (M) n=0 where Aq = 1 and - (*)(!) ••• (>n - h) ^^^ Y: ^'^^^^ "T (^' - '" + I)*'"' 0 < « (3.8-10) m=0 WZ ^„ ~ {n + 1)(1 - b)-'" - ^ (1 - by"\ n large The term corresponding to m = 0 in (3.8-10) is » + 1. We thus obtain Pit, R) = -— -, '—J^ E -7 ^ An \2 ^V (3.8-11) -a222 t1/2 4'\/7r ^0 n=0 (M) We are interested in the expected number, .V, of maxima per second. From the similar work for /, it follows that N is the coefficient of dt when (3.8-1) is integrated with respect to R from 0 to 20 . Thus from (3.8-7) and dR = VWibfdz = (2boBy"b7'^'dz = [26o(a- - \)f"dz we find N = [ pit, R) dR Jo ^ ja' - If (b^Y f ^ (I + 4) A^ i2ayi- \Trbo/ h (n . l\ a- (3.8-12) Equations (3.8-11) and (3.8-12) have been derived on the assumption that ivii) is symmetrical about /„, , i.e. the band pass filter attenuation is 84 BELL SYSTEM TECHNICAL JOURNAL symmetrical about the mid-band frequency. We now go a step further and assume an ideal band pass filter: W{f) = Wo fa po) = probability of / being less than yy/-\pQ . Similarly C = P{R < B = P{I max < yy/xpa) = probability of random maximum of / being less than yv ^o • Similarly D = F(R max < yVTo)- MATHEMATICAL AX A LYSIS OF RANDOM NOISE 87 The asymptotic expression for puiy) may be obtained from the integral (3.8-9) for pit, R). Indeed, replacing the variables of integration x, y in (3.8-9) by x' = X y' = X + y, integrating a portion of the y' integral by parts, and assuming b < I (a' > 1, by Schwarz's inequality, so that 6 < 1 always) leads to '--(09;-e;-') when R is large. If, instead of an ideal band pass filter, we assume that w(f) is given by ""'^^^ ^ ^V^ e-'^-^'"^''^''^ /„ » a (3.8-16) we find that h= 1 hi = 4:ir'a~ b[ = 167r -ia a- = 3,b = 0 An = ill + 1) Some rough work indicates that the sum of the series in (3.8-12) is near 3.97. This gives the expected number of maxima of the envelope as N = 2.52(7 (3.8-17) per second. The pass band is determined by a. It appears difficult to compare this with an ideal band pass filter. If we use the fact that the filter given by .a)=».exp[-,(^— ^J_ passes the same average amount of power as does an ideal band pass filter whose pass band is fb — fa , we have fb — fa = (T^/2ir and the expression for N becomes 1.006 (fb — fa)- 3.9 Energy Fluctuation Some information regarding the statistical behavior of the random vari- able rll+T E = / /'(/) dt (3.9-1) 88 BELL SYSTEM TECHNICAL JOURNAL where /(/) is a noise current and ti is chosen at random, has been given in a recent article. Here we study this behavior from a somewhat different point of view. If we agree to use the representations (2.8-1) or (2.8-6) we may write, as in the paper, the random variable E as /r/2 I\t) dt (3.9-2) r/2 where the randomness on the right is due either to the a„'s and bnS if (2.8-1) is used or to the <^„'s if (2.8-6) is used. The average value of £ is Wj- where, from (3.1-2), /r/2 -.T/2 P{t) dt = / i/'(0) dt = TiPo r/2 J—T/2 = T [ w{f) df Jo (3.9-3) Jo The second moment of E is /r/2 »r/2 dti / dt2P{ti)Pit2) (3.9-4) 7-/2 J—T/2 If, for the time being, we set ^2 equal to /i + t, it is seen from section 3.2 that we have an expression for the probability density of I(ti) and /(/i + t) arid hence we may obtain the required average : ^2 = A f ^^1 f dhlUlexp ZtA J-ao •'-00 A' = 4^1 -rr, h = /(/i), h = m + r) = m) The integral may be evaluated by (3.5-6) when we set (3.9-5) /. = ,..^. /. = .4,^ ^pT = — 'Ao COS

^ (3.9-9) Jo Jo L 7^2 (/i 4-/2)2 sin' Tjfi - f2)Tl ^Kh-hY J If this formula is applied to a relatively narrow band-pass filter and if T{fb —fa) > > 1 the contribution of the/i +/2 term may be neglected and we have the approximation (3.9-10) 2 = / wodfi 1 Wo J fa J-00 df2 sin^ x(/i -/2)r = wlTift -fa) = Wq niT 90 BELL SYSTEM TECHNICAL JOURNAL where, from (3.9-3) Mr = WoTifb - fa) (3.9-11) The third moment E^ may be computed in the same way. However, in this case it pays to introduce the characteristic function for the distribution of I(ti), lih), I{k). Since this distribution is normal its characteristic function is Average exp [izili + izili + izzh] = exp - y (zi + 22 + zl) + ypih - h)ziZ2 .^ ^_^^. ] ■i- 4'ih — tijZiZs + Xpits — /2)Z2 23 From the definition of the characteristic function it follows that 2 2 2 Illlll= -coeff. Of ^-i|^; in ch. f. = l/'O + 2\po{\p2i + V'31 + ^32) + S\p2l4^Sl4^32 where we have written i/'2i for i/'(/2 — ^1), etc. When (3.9-13) is multiplied by dti dti dtz , the variables integrated from 0 to T, and the above double integral expression for ar used, we find (E - Ef = 2\t [ dh [ dU [dh^l^.2lhl^■s2. Jq Jq Jo Denoting the triple integral on the right by / and differentiating, ^ = 3 [ dh f dhHi2 - h)i{T - h)i{T - h) dl Jo •'0 = 3 / dx \ dy\p{x — y)4'{x)\p{y) Jo Jo = 6 / dx i dy\l/(x — y)rl/(x)4'(y) Jo Jo In going from the first line to the second ti and 1-2 were replaced by T — x and T — y, respectively. In going from the second to the third use was made of the relations symbolized by ' dx \ dy = I dx I dy -{- I dx I dy 0 ^0 Jo Jo Jq *'x = / dx I dy -\- I dy I dx Jo Jo Jo Jo MATHEMATICAL ANALYSIS OF RANDOM NOISE 91 and of the fact that the integrand is symmetrical in x and y. Integrating dJ/dT with respect to T from 0 to T\, using the formula r dT I fix) dx ^ r {Ty - x)f{x) dx, Jo Jo Jo noting that / is zero when T is zero, and dropping the subscript on Ti finally gives (E - £)' = 48 f dx [ dy(T - x)^p{x)^p{y)^P(x - y). Jo Jq E* may be treated in a similar way. It is found that (E - EY - 3{E -£)'' = 3!2' [ dh [ dU [ dh [ dU^ly,,rPnh2h:i Jo Jo Jq Jo which may be reduced to the sum of two triple integrals. It is interesting to note that the expression on the left is the fourth semi-invariant of the random variable E and gives us a measure of the peakedness of the dis- tribution (kurtosis). Likewise, the second and third moments about the mean are the second and third semi-invariants of E. This suggests that possibly the higher semi-invariants may also be expressed as similar multiple integrals. So far, in this section, we have been speaking of the statistical constants of E. The determination of an exact expression for the probability density of E, in which T occurs as a parameter, seems to be quite difficult. When T is very small E is approximately / (t)T. The probability that E lies in dE is the probability that the current lies in — /, — / —dl plus the probability that the current lies in I, I -\- dl: 2dl P E Vm. ^^P '^r (2-^«£r)-" exp -• — dE (3.9-14) where E is positive, r = {fj\ di==l{ETr"dE and T is assumed to be so small that /(/) does not change appreciably during an interval of length T. Wlien T is very large we may divide it into a number of intervals, say n, each of lengtli T/n. Let Er be the contribution of the r th interval. The energy E for the entire interval is then £ = £i + £2 + • • • + £» If the sub-intervals are large enough the £r's are substantially independent random variables. If in addition n is large enough E is distributed nor- 92 BELL SYSTEM TECHNICAL JOURNAL mally, approximately. Hence when T is very large the probability that E lies in dE is dE (E — Wr)- exp - .2 (3.9-15) where ^«r = r [ W(f) df Jo al = T f w\f) df Jo (3.9-16) the second relation being obtained by letting T -^ °o in (3.9-9). The analogy with Campbell's theorem, section 1.2, is evident. WTien we deal with a band pass filter we may use (3.9-10) and (3.9-11). Consider a relatively narrow band pass filter such that we may find a T for which Tfa >> It but T{fb —fa) < < -64. Thus several cycles of fre- quency/„ are contained in T but, from (3.8-15), the envelope does not change appreciably during this interval. Thus throughout this interval /(/) may be considered to be a sine wave of amplitude R. The corresponding value of E is approximately 2 where the distribution of the envelope R is given by (3.7-10). From this it follows that the probabihty of E lying in dE is dE E dE -Elmr fin 1'7\ -— exp - -— = — e ^ (3.9-17) when E is small but not too small. When we look at (3.9-14) and (3.9-17) we observe that they are of the form n+l pn ^ ^ -"^ dE (3.9-18) T{n + 1) Moreover, the normal law (3.9-15), may be obtained from this by letting n become large. This suggests that an approximate expression for the dis- tribution of E is given by (3.9-18) when a and n are selected so as to give the values of Wr and ctt obtained from (3.9-3) and (3.9-9). This gives a = ^4^ «+l=^ (3.9-19) MATHEMATICAL ANALYSIS OF ILiXDOM NOISE 93 and if we drop the subscript T and substitute the value of a in (3.9-18) we get (f)" exp(-!^)^(^-^), n = i-l (3.9-20) r(« + 1) '^ \ a- / \a- /' a An idea of how this distribution behaves may be obtained from the following table: n T(f,-fa) ^.28 --V.oO X.-i X.io 0 0 .29 .695 1.39 .415 2.00 1 1.45 .96 1.68 2.69 .572 1.60 2 2.4 1.73 2.67 3.94 .647 1.47 3 3.4 2.54 3.67 5.12 .692 1.39 5 5.4 4.22 5.67 7.42 .744 1.31 10 10.5 8.63 10.67 13.02 .808 1.22 24 25 21.47 24.67 28.17 .870 1.14 48 50 44.1 48.7 53.5 .905 1.10 where n is the exponent in (3.9-20). The column T(fb —fa) holds only for a narrow band pass filter and was obtained by reading the curve yu in Fig. 1 of the above mentioned paper. The figures in this column are not very accurate. The next three columns give the points which divide the dis- tribution into four intervals of equal probability: ^.25 = — ~ , -E.25 = energy exceeded 75% of time ^.50 = — ^ , £.50 = energy exceeded 50% of time X 75 = - — -^ , £ 75 = energy exceeded 25% of time The values in these columns were obtained from Pearson's table of the in- complete gamma function. The last two columns show how the distribu- tion clusters around the average value as the normal law is approached. For the larger values of n we expected the normal law (3.9-15) to be approached. Since, for this law the 25, 50, and 75 per cent points are at f)i — .675cr, m, and m -f- .675cr we have to a first approximation x.,0 = % = (n + l) ^ T{f, - fa) a- m ._ , A7C /— (3.9-21) ^.25 = -, (w — .6/5cr) = x.oo — .o75v^£o ^.75 = X,50 -f .675\/x.5o This agrees with the table. 94 BELL SYSTEM TECHNICAL JOURNAL Thiede has studied the mean square value of the fluctuations of the integral A{t) = f l\r)e-"''-'UT (3.9-22) The reading of a hot wire ammeter through which a current / is passing is proportional to A{t). a is a constant of the meter. Here we study A{t) by 2.00 AT 0 -^ ^^ y L 0.403 ATO _^ /a V 1 . 1 U -r . _ 1 1 y,-' ' jz 0.5 l/T(fb-fa) PROBABILITY DENSITY 5 6 8 10 30 40 50 60 Fig. 5* — Filtered thermal noise — spread of energy fluctuation •''1 P{t) dt, li random, / is noise current. ^'i = Ejb/E.io , y2 = E.2i/E.io ■ fb — fa = band width of filter. first obtaining its correlation function. This method of approach enables us to extend Thiede's results The distributed portion of the power spectrum of A(t) is given by (3.9- 30). When the power spectrum w(f) of /(/) is zero except over the band fa < f < fb where it is Wo , the power spectrum of A (t) is and is zero iromfb — fa up to 2/o . The spectrum from 2fa to 2fb is not zero, and may be obtained from (3.9-34). The mean square fluctuation of A{t) is given, in the general case, by (3.9-28) and (3.9-32). For the band pass case, when (/& — fa)/oi is large, r.m.s, Ajt) - A A [-J1/2 2(/6-/a)j 5 Elec. Nachr. Tek., U (1936), 84-95. This is an excellent article. Note added in proof. The value of >'2 at 0 should be .415 instead of .403. MATHEMATICAL AX A FAS IS OF RAX DOM XOISE 95 We start by setting t — t — u which transforms the integral for A{t) into A{t) = I I\t - u)e~"''du (3.9-23) In order to obtain the correlation function ^(t) for A(t) we multiply A{t) by A{t + t) and average over all the possible currents ^(t) = A{t)A{t + r) = f e~"" du [ e~"" dv ave. l\t - u)l\t + t - v) Jo Jo Just as in (3.9-4) the average in the integrand is the correlation function of /■(/), the argument being t -}- t — v — t -\- u = t -\- u — v. From (3.9-7) it is seen that this is ypl + 2\P~{t + u — v) where \1/{t) is the correlation function of /(/). Hence ^(t) =tl + 2 I du I ^z; e-""-"V'(r + u- i) (3.9-24) a- Jo Jo From the integral (3.9-23) for A{t) it is seen that the average value of ^(0 is A = ^- = ^" (3.9-25) where we have used h = 4^(0) = [ w(f) df = p Jq Jo Using this result again, only this time applying it to A{t), gives .42(7) = ^(0) r" r (3.9-26) = A +2 du dv e-""~"V'(« - v) Jo Jo The double integrals may be transformed by means of the change of variable « + z' = x, u ~ v = y. Then (3.9-24) becomes ^(r) = A' +\ [ dy f dx + I dy f dx e~"' yp\T + y) \_Jo Jy •'-00 J—y J (3.9-27) = ^" + i [ e-"'[4^\r +y) + rl^'ir - y)] dy a Jq 96 BELL SYSTEM TECHNICAL JOURNAL WTien we make use of the fact that \p{y) is an even function of y we see, from (3.9-26), that the mean square fluctuation of A{t) is {A{t) - Af = Yif) - A' =- [ e-""xly\y) dy (3.9-28) a Jo '^(t) may be expressed in terms of integrals involving the power spectrum wif) of I{t). The work starts with (3.9-24) and is much the same as in going from (3.9-8) to (3.9-9). The result is ^(r) = A' + [ df, [ dfow{fi)wif2) Jo Jo r cos 2x(/i + /2)r cos 2ir{fi - fijr 1 la' + [2x(/x + f2)f "^ «2 + [2x(/i - /2)PJ It is convenient to define 'w(—f) for negative frequencies to be equal to ■w(f). The integration with respect to /2 may then be taken from — »: to + oc and we get Jo J-« a- -\- [lTr{ji — J2)\ The power spectrum W(f) of ^4 (/) may be obtained by integrating "^(r) : Wif) = 4 [ ^(t) cos 27r/r dr Jo Let us concern ourselves with the fluctuating portion A{t) — A oi A{t). Its power spectrum Wdf) is Wcif) = 4 / (\^(t) - A') cos Itt/t dr Jo The integration is simplified by using Fourier's integral formula in the form / dr / #2F(/2) cos 27r(« -/2)t = |F(w) Jo J-« We get Wcif) = 2 ,\ 2,2 [ df,[wif,)wif+f,) +w(/x)w(-/ + /0] (3.9-30) = aN^^I« ^(/0-(/-/i)^/: The simplicity of this result suggests that a simpler derivation may be found. If we attempt to use the result wif) = Limit 2MZli (2.5-3) r-»oo -/ MATHEMATICAL AX A LYSIS OF RANDOM NOISE 97 where S{f) is given by (2.1-2) we find that we need the result Limit -J [ dt, f dt.J'"^'''-'''^ I-{h)l\t-^ 7'-»M T Jo Jo +00 (3.9-31) = I w(f,)w(f-fOdf, where / > 0 and /(/) is a noise current with w(/) as its power spectrum. This may be proved by using (3.9-7) and »CO -.+00 8 / \P'{t) cos lirfr dr = I w(x)w{f — x) dx Jo J-« which is given by equation (4C-6) in Appendix 4C. An expression for the mean square fluctuation of A (/) in terms of w(f) may be obtained by setting r equal to zero in (3.9-29) (A{t) -Ay = ^(0) -A •+»\, w{h)w{^) (3.9-32) Jo J-oo OC ^' + 4t'(/i - /2)' The same result may be obtained by integrating Wdf), (3.9-30), from 0 to cc : r df r+°° / 2 ■ ' 2.2 dfMfiMf-fi) (3.9-33) Jo cc -f- 47r / J_oo Although this differs in appearance from (3.9-32) it may be transformed into that expression by making use of w(— /) = w(f). Suppose that /(/) is the current through an ideal band pass filter so that -d)(f) is zero except in the band /a fb , A = - (fb - fa) (3.9-34) a ^ 2wl{f, - fa - f) 0 ^ In this case it is simpler to obtain the probability density directly from (3.10-1) instead of from the characteristic function. MATHEMATICAL ANALYSIS OF RANDOM NOISE 99 Now suppose that we have a noise current In plus a sine wave. By com- bining our representation (2.8-6) for /jv with the idea of ipp being random mentioned above we are led to the representation /(/) = I = 1, + h^ .If = P cos (Upt — (^p) + ^ Cn COS {C0,J — iPn), (3.10-4) 1 c; = 2w{fn)^f where (p^ and v'l , • • • (fM are independent random angles. If we note / at the random times ti , to • • • how are the observed values distributed? Since Ip and /jv may be regarded as independent random variables and since the characteristic function for the sum of two such vari- ables is the product of their characteristic functions we have from (3*. 1-6) and (3.10-2) ave. e'" = ave. e'^^^^^+^v) /-./'oA (3.10-5) = MPz) e.xp [^~-) which gives the characteristic function of /. The probabiHty density of I • 37 is 1 f ^°° ,--/-(^o-'^/2) j^^p^^ ^2 ^ ^^^^ r ^-(/-p cos e^v-2,0 ^Q (3_io_6) 27r J-oo 'K\' iTrxpo Jo In the same way the two-dimensional probability density of (/i , 72), where /i = /(/) is a sine wave plus noise (3.10-4) and I2 = I{t + r) is its value at a constant interval r later, may be shown to be {^l - ^IV r-^ r Bid) 1 where B{d) = Uih - P cos ef + (A - P cos {6 + co,,r))'] - l^Prill - P cos d){l2 - P cos {d + WpT)) The characteristic function for 1\ and I1 is ave. g'"^i+"'^2 _ j^[p-^^ii _|_ ^2 _|_ 2uv cos Wpx) X exp — y {ll' + V) — lArWZ^ '^ A different derivation of this expression is given bv W. R. Bennett, Jour. Aeons. Soc. Amer., Vol. 15, p. 165 (Jan. 1944); B.S.T.J., Vol. 23, p. 97 (Jan. 1944). 100 BELL SYSTEM TECHNICAL JOURNAL Sometimes the distribution of the envelope of I = Pcospt -{- Im (3.10-9) is of interest. Here we have replaced Wp by p and have set >fp to zero. By the envelope we mean R{t) given by R\t) ^ r' ^ {P + I,f + 7^ (3.10-10) where Ic is the component of I^ "in phase" with cos pt and Is is the com- ponent "in phase"' with sin pt: Ic = ^ Cn cos [{(Jin — P)t — ^Pn] /j = 2^ c„ sin [(co„ — p)t — ^„] ^ In = Ic cos pt — Is sin pt In = I'c ^ I's ^ ypo Since I c and /« are distributed normally about zero with a variance of i/'o , the probability densities of the variables are (27n/'o) (27n/'o)' X= P-^Ic y = Is ■'" ex-p - 2h ■'" ex-p - y" respectively. Setting X = R cos d y = Rsmd and using these distributions shows that the probability of a point {x, y) lying in the ring R, R-\- dR is RdR 27n/'o r Jo exp - -~ {R' + P' - 2ypo 2RP cos 6) dd _RdR R: exp — — ' + P 2h ■]'•(? where /o is the Bessel function with imaginary ' argument. h{z) = n=0 22»;z! n\ (3.10-11) MATHEMATICAL ANALYSIS OF RANDOM NOISE 101 and is a tabulated function. Thus (3.10-11) gives the probabiUty density of the envelope R. The average value of R" may be obtained by multipl}-ing (3.10-11) by R" and integrating from 0 to -x . Expansion of the Bessel function and term- wise integration gives ^ = (2^.r«r(« + l).— .F,g + M;|J = (2^.)-rg + l),F.(-|;l;-^j (3.10-12) where iFi is a h\^ergeometric function. In going from the first line to the second we have used Kummer's first transformation of this function. A special case is R2== p^ ^ 2rPo (3.10-13) When only noise is present, P = 0 and R = (2^0)^^^ r(|) = (^f^)"' \ 2 / (3.10-14) R^ = 2iAo Before going further with (3.10-11) it is convenient to make the following change of notation ^ = 7172 ' ^^ = 7T72 ' ^ = Tm (3.10-15) V'o 'Ao Wo "a" is the ratio (sine wave amplitude)/(r.m.s. noise current). Instead of the random variable R we now have the random variable v whose probability density is p(v) = V exp ■"4^1 ^"^""^^ (3.10-16) Curves of p{v) versus v are plotted in Fig. 6 for the values 0, 1, 2, 3, 5 of a . Curves showing the probabiUty that v is less than a stated amount, i.e., dis- tribution curves for v, are given in Fig. 7. These curves were obtained by integrating p(v) numerically. The following useful expression for this probability has been given by W. R. Bennett in some unpublished work. jf" piu) du = exp f-'^lAn |; h\ i^^av) (3.10-17) ^ Curves of this function are given in "Tables of Functions", Jahnke and Emde (1938), p. 275, and some of its properties are stated in Appendix 4C. 102 BELL SYSTEM TECHXICAL JOURNAL This is obtained by integration by parts using \Mien av >> 1 but 1 << a - r, Bennett has shown that (3.10-17) leads to r . ^ ^ ( -^ Y' 1 / p{u) ail ~ \ - — I exp Jo \27ra/ a - V ^ (v - af ( 1 - 3(fl + v)~ — 4i'" 8av{a — v)- (3.10-18) 0.6 ^ , ! i 1 / \a=o 5 V- f^ / / i \ V^'n. -^ Od ^'-'3 I. > a. t:a^:^ \ z Q - 0.2 /iu.^r \ < m o tr t^ 0.1 ITZi^lT 1 \ v_ "V.^ Fig. 6 — Probability density of envelope R of I{t) = P cos /)^ + /_v This formula may also be obtained by putting the asymptotic expansion (3.10-19) for p(v) in (3.10-17), integrating by parts t\\ice, and neglecting higher order terms. Wben av becomes large we may replace Io{av) b}' its asj'mptotic expres- sion. The expression for p(v) is then Thus when either a becomes large or v is far out on the tail of the probability density curve, the distribution behaves like a normal law. In terms of the original quantities, the normal law has an average of P and a standard devia- tion of 1^0 "• This standard deviation is the same as the standard deviation MATHEMATICAL AX A LYSIS OF RANDOM XOISE 103 of the instantaneous values of /.v. When av » 1 and (7 » v — a | we may expand the coefficient of the exponential term in (3.10 19) in powers of 99.90 , /// /// / /// / /// / /// / /// / 95 90 /// / J// / W / / 70 > 60 V / / / / z 5 ttao A / a /// / /// w / — I 2a\/2x _ 1 - 4a a 1 + (z) — a)-' 8a- exp (v - a)n 104 BELL SYSTEM TECHXICAL JOURNAL WTien I consists of two sine waves plus noise I = Pccspt-{-Q sin qt + /.v , (3.10-20) where the radian frequencies p and q are incommensurable, the probability density of the envelope R is R / rJo{Rr)Jo{Pr)Jo{Qr)e-'^'''" dr (3.10-21) where xpo is I'\ . \Mien Q is zero the integral may be evaluated to give (3.10-11). When both P and Q are zero the probabiUty density for R when only noise is present is obtained. If there are three sine waves instead of two then another Bessel function must be placed in the integrand, and so on. To define R it is convenient to think of the noise as being confined to a relatively narrow band and the frequencies of the sine waves lying within, or close to, this band. As in equations (3.7-2) to (3.7-4), we refer all terms to a representative mid-band frequency /„» = Wm/27r by using equations of the t}'pe cos pt = cos [(p — 03,,,)/ + 0}J] = cos (p — Oim)t cos OJmt — Sin (p — (jim)t Sln 0)mt. In this way we obtain V = A cos o^a - B sin wj = R cos {wj + 6) (3.10-22) where A and B are relatively slowly var}-ing functions of t given by A = P cos {p — Um)i + Q cos {q — 0^,n)t + Z^ C„ cos (a3„/ — Wmt — (pn) B = P sin {p — o}m)t + Q sin (g — Urn)i + 2_/ ^n sin {uni — Wmt — cpn) (3.10-23) and R^ = A^ + B'-, R > 0 tan d = B/A (3.10-24) As might be expected, (3.10-21) is closely associated with the problem of random flights and may be obtained from Kluyver's result by assuming 39 G. X. Watson, "Theory of Bessel Functions" (Cambridge, 1922), p. 420. MATHEMATICAL AXALYSIS OF RANDOM XOISE 105 the noise to correspond to a very large number of very small random dis- placements. Another way of deriving (3.10-21) is to assume (p — co,,.)/, (q — Wm)t, — avj/ + Q sin ( 0 (4.1-9) tan^ = B/A. This delinition of R has also been given in equations (3.10-22, 23, 24). The envelope of V is R and the output current is I = aR: \ + \ cos (2co„J + 2^) (4.1-10) Since i? is a slowly varying function of time, so is R\ The power spectrum of R' is confined to frequencies much lower than 2fm and consequently the power spectrum of R' cos {2w„4 + 26) is clustered around 2/,,, . Thus the only term in / contributing to the low frequency output is aR'/2 which is what we wished to show. We now return to the statistical properties of Iti . First, consider the case in which T' consists of noise only, T' = Vn , so that the probabihty density of the envelope R is R ^-fi^/^if-o where Hence ^ e-'"'"' (3.7-10) xPo = [rms V^■f = Vl (4.1-11) 7 ai?2 ^0 2 i^-u aR' R .-^2/2^0 arpo 0 4^0 (4.1-12) a\J/Q 112 BELL SYSTEM TECHNICAL JOURNAL Second, consider the case in wliich V = Fjv + P cos pt (4.1-13) where P/2t hes near the noise band of IV • The probabiUty density of the envelope R is -[-'-^1^3 R From this and equations (3.10-12), (3.10-13), we find y aR- aP /. 1 1 A\ ^dc = ^ = «^o -1- -^ (4.1-14) I'U = ^ i?^ = a' o P'' 2^0 + 2PVo + J I}f = ru - Idc = (x-[h + P-]h (4.1-15) In (4.1-14) xpo is the mean square value of V^ and P"/2 is the mean square value of the signal. These two equations show that Idc and the rms value of 7^/ are independent of the distribution of the noise power spectrum in IV as long as the input V is confined to a relatively narrow band. In other words, although this distribution does affect the power spectrum of the output, it does not affect the d.c. and rms 7^/ when xf/o and P are given. That the same is also true for a large class of non-Unear devices was first pointed out by Middleton (see end of Section 4.9). When the voltage is V = TV + P cos pt -\- Q cos qt, (4.1-4) p 9^ q, we obtain from equation (3.10-25) 2 lU=-fR' (4.1-16) l}^ =a'Ul + P'h + Q'h + p^ 2 ^ These results are special cases, obtained by assuming no audio frequency filter, of formulas given by F. C. Williams, Jour. Inst, of E. E., 80 (1937), 218-226. Williams also discusses the response of a linear rectifier to (4.1-4) when P ^ Q -\- F,v • An account of WilUaras' work is given by E. B. MouUin, "Spontaneous Fluctuations of Voltage," Oxford (1938), Chap. 7. MATHEMATICAL ANALYSIS OF RAX DOM XOISE 113 4.2 Low Frequency Output of a Linear Rectifier In the case of the hnear rectifier I 0, T^ < 0 \aV, V > 0 the low frequenc}' output current, assuming no audio frequency filter, is aR (4.2-1) ia = (4.2-2) This formula, like its analogue (4.1-6) for the square law device, assumes that the applied signal and noise lie within a relatively narrow band. It may be used to compute the probability density and statistical properties of 1 1( when the corresponding information regarding the envelope R of the applied voltage is known. The truth of (4.2-2) may be seen by considering the output /. It con- sists of the positive halves of the oscillations of aT'. The envelope of / is the same as that of a]'. However, the area under the loops of / is only about I/tt of the area under aR, this being the ratio of the area under a loop of sin X to the area of a rectangle of unit height and length 27r. From the low frequency point of view these loops of / merge into a current \yhich varies as aR/ir. When 1' is a sine wave plus noise, V = Fjv + P cos pt (4.1-13) the average value of /^^is^ '-=='-(£)"' .'.(-1--I.) (£)"■.-"[(. + •)'.© (l+a;)7o(^j + xA(| (4.2-3) where h , 7i are Bessel functions of imaginary argument and _ P _ ave. sine wave power 2h ave. noise power (4.2-4) *^ This result was discovered independently by several investigators, among whom we may mention W. R. Bennett and D. O. North. The latter has appUed it to noise measure- ment work. He has found that the diode detector, when adapted to noise metering, is a great improvement over the thermocouple, and has used noise meters of this type satis- factorily since 1940. See D. O. North, "The Modification of Noise by Certain Non- Linear Devices", Paper read before I.R.E., Jan. 28, 1944. 114 BELL SYSTEM TECHXICAL JOURNAL xj/o being the average value of V'y . Equation (4.2-3) follows from the formulas (3.10-12) and (4B-9). WTien x is large the asymptotic expansion (4B-3) of the iFi gives 2P 8P^ Similarly, the mean square value of I if is I-f = -,R'= - [P- + 2^o) (4.2-5) (4.2-6) and the mean square value of the low frequency current I(f , excluding the d.c, is given by T- " Pif = I'd — I'dc ^0 2P2 \Mien X is large we have 4^0 — IT- i_ and when x = 0, 4, =^^„I2-;, IT- l_ iX (4.2-7) (4.2-8) Curves for Idc are given in Figures 1, 2 and 3 of Bennett's paper. He also gives curves, in Fig. 4, showing l}f versus .v. These show that the effect of the higher order modulation terms is small when Iff is computed by adding low frequency modulation products. \Mien V consists of two sine waves plus noise, V = Vn + P cos pt -{- Q cos ql, (4.1-4) the average value of Itf is, from (3.10-25), a sort of double iFi function: 1/2 00 00 / i\ Idc = - R = OL 1-K k=o m=o klklmlml EZ (4.2-9) where P^ 2tAo' y = 2^0 ' Piciz) = Legendre polynomial (4.2-10) If X is large and y < x, we have from (3.10-27) the asymptotic expression hc^-P± ^~^!^^~^^' 2F, (k-hk-h 1; -") (4.2-11) TT k=0 klx'' MATHEMATICAL AX A IAS IS OF RANDOM^NOISE 115 The 2^1 may be expressed in terms of the complete eUiptic functions E and A' of modulus v^'~x~^''^. Thus .F,(-J,-J;l;>:)=*£-?('l-0^. .V / TT X .F.(^,i;l;l) = lK (3.10-28) and the higher terms may be computed from the recurrence relation (3.10-29). The tirst term, ^ = 0, in (4.2-11) gives Idc when the noise is absent. The mean square value of 1 1( is 2 2 la = %R' = -. [2.Ao + P' + Q'] (4.2-14) From this expression and our expression for Idc , the rms value of the low- frequency current, If/ , excluding the d.c, may be computed. For example, when the noise is small, + ,,.(._.,,(_,, _.,;05)_ The term independent of xj/q gives the mean square low frequency current in the absence of noise. As Q goes to zero (4.2-15) approaches the leading term in (4.2-7), as it should. When P = Q our formula breaks down and it appears that we need the asymptotic behavior of "" In view of the questionable nature of the derivation given in Section 3.10 of equations (4.2-9) and (4.2-11) it was thought that a numerical check on their equivalence would be worth while. Accordingly, the values x = 4, y = 3 were used in the second series of (4.2-9). It was found that the largest term (about 130) in the summation occurred at ^ = 11. In all, 24 terms were taken. The result obtained was ^ = 2.5502 V2;/', « See \V. R. Bennett, B.S.TJ., Vol. 12 (1933), 228-243. ^5 This mav be done bv the method given l)v W. B. Ford, Asymptotic Developments, Univ. of Mich. Press (193'6), Chap. VI. 116 BELL SYSTEM TECHNICAL JOURNAL For the same values of x and _v the asymptotic series (4.2-11) gave 2.40 + 0.171 + .075 + 0.52 + •••• If we stop just before the smallest term we get 2.57 for the sum. If we include the smallest term we get 2.65. This agreement indicates that (4.2-11) is actually the asymptotic expansion of (4.2-9). WTien the voltage is of the form T' = Q{\ -\- kcos pt) cos qt + Vn we may use ^ = (2,o)-r(i + |)lf (4.2-16) iFir-|;l; -^(1 ^-kco&dAdd where R is the envelope with respect to the frequency g/27r and y is given by (4.2-10). The integral may be evaluated by writing iFx as a power series and integrating termwise using the result — / (1 + /fe cos ey cos md dd (4.2-17) where m is a non-negative integer, / any number, (a)„, = a(a + 1) • • • (a + m - 1), (a)o = 1, and (0)o = 1. The integral may also be evaluated in terms of the associated Legendre function. By applying the methods of Section 3.10 to (4.2-16) we are led to « " 1 (4.2-18) where the as}'mptotic series holds when _\' is ver\' large and k is not too close to unity. These expressions give /F/ ~ ^^ {q' f + U2 - (1 - kr"'] + • • •) (-1.2-19) MATHEMATICAL ANALYSIS OF ILiNDOM NOISE 117 The reader might be tempted to associate the coefficient of ^o in (4.2-19) with the continuous portion of the output power spectrum. However, this would not be correct. It appears that the principal contribution of the continuous portion of the power spectrum to Iff is aVo/Tr , just as in (4,2-7) when k is zero. The difference between this and the corresponding term in (4.2-19) seems to arise from the fact tliat the amphtude of the recovered signal is not exactly aQk/ir but is modified by the presence of the noise. This general type of behaAdor might be expected on physical grounds since changing P, say doubling it, in (4.2-7) does not appreciably affect the Iff in (4.2-7) (which is due entirely to the continuous portion of the noise spectrum). The modulating wave may be regarded as slowly making changes of this sort in P. 4.3 Some Statistical Properties of the Output of a General Non-Linear Device Our general problem is this: Given a non-linear device whose output / is related to its input T" by the relation I = — [ F{iu)e'''" du (4A-1) 27r Jc which is discussed in Appendix 4A. Let the input V contain noise in addi- tion to the signal. Choose some frequency band in the output for study. \Miat are the statistical properties of the current flowing in this band? It seems to be difficult to handle this general problem. However, it appears that the two following results are true. 1. As the output band is chosen narrower and narrower the statistical properties of the corresponding current approach those of the random noise current discussed in Part III (provided no signal harmonic lies within the band). In particular, the instantaneous current values are distributed normally. 2. When the input V is confined to a relatively narrow band the power spectrum of the output I is clustered around the 0 (d.c), 1st, 2nd, etc. harmonics of the midband frequency of T'. The low frequency output in- cluding the d.c. is Id = AoiR) = ^ [ F{iu)MuR) du (4.3-11) 2x Jc where R is the envelope of T'. The envelope of the nth harmonic of the output, when w > 0, is A^{R) =- [ F{iu)Jn{uR) du (4.3-1) 118 BELL SYSTEM TECHNICAL JOURNAL The mathematical statement is 00 / = Z) MR) cos {noirnt + nd) (4.3-9) where fm = com/(2Tr) is the representative mid-band frequency of V and 6 is a relatively slowly varying phase angle. The results of Sections 4.1 and 4.2 are special cases of this. Middleton's result that the noise power in each of the output bands (in the entire band corresponding to a given harmonic) depends only on \\- = ^0 and not on the spectrum of Vu , where V^ is the noise voltage component of V, may also be obtained from (4.3-9). We note that the total power in the n^^ band depends only on the mean square value of its envelope An{R), and that the probabiUty density of the envelope R of the input in- volves Vn only through xpo . The argument we shall use in discussing the first result is not very satis- factory. It runs as follows. The output current / may be divided into two parts. One consists of sinusoidal terms due to the signal. The other con- sists of noise. We shall be concerned only with the latter which we shall call In . The correlation between two values of In separated by an interval of time approaches zero as the interval becomes large. Let t be an interval long enough to ensure that the two values of In are substantially independent. Choose an interval of time T long enough to contain many intervals of length r. Expand In as a Fourier series over this inten-al. We have Lv = 2 + 2^ p« co^ ^^ + *« ^^^ ^Y~ 71=1 L — (4.3-2) 1 Jo dt Let the band chosen for study be/o — - to/o + - and let T (fo -f) = 'h, T (fo + 0 = ^2 (4.3-3) where Wi and «2 are integers. The number of components in the band is (w2 — wi). We suppose /3 is such that this is small in comparison with T/t. The output of the band is Jn = Z! \anCos^t + b„ sin -^ (4.3-4) MATHEMATICAL AX A LYSIS OF KAXDO.]f NOISE 119 where I Jo Wl + W2 , ^ fh + th r rj. , , . rj.. n = 2 — + « - 7y — = joT -V {n - j^T) (4.3-5) We choose the band so narrow that n. - wi « TJT or ^t « 1 (4.3-6) This enables us to write approximately In - ibn = Z e--^^(("/^)-/0)-| r e-''-'''l,{l) r=l i J <,r-\)T dt Ti = T/t, T being chosen to make n an integer. Suppose we do this for a large number of intervals of length T. Then /jv(0 will differ from interval to interval. The set of integrals for r = \ gives us an array of values which we regard as defining the distribution of a complex random variable, say xi . Similarly the set of integrals for r = 2 defines the distribution of a second random variable ;V2 , and so on to aVi . Because we have chosen t so large that /a'(/) in any one integral is practically independent of its values in the other integrals we may say that Xi , ^2 , • • • Xr^ are independent. We have ~i2iT(,(nlT)-/o)TT ^ e /, _ ^-A — V ^-'2T((ni+l/T)-/o)rr — tb„„_ = £ e" j2?r((no/r)-/o)rT and if no — ni « ri , as was assumed in (4.3-6), we may apply the central limit theorem to show that c„i , b,n , On^+i , • • ■ fln., , b,,., tend to become in- dependent and normally distributed about zero as we let the band width j8 ^ 0 and T —^ (and hence ri — > ^■-■^ ) in such a way as to keep n^ — Hi fixed. In this work we make use of the fact that Isit) is such that the real and imaginary parts of xi, X2, • • • Xr all have the same average and standard deviation. It is convenient to assume /oT' is an integer. Thus as the band width ^ approaches zero the band output Jn given by (4.3-4) may be represented in the same way, namely as (2.8-1), as was the random noise current studied in Part III. Hence Jn tends to have the 120 BELL SYSTEM TECHNICAL JOURNAL same properties as the random noise current studied there. For example, the distribution of J^ tends towards a normal law. In our discussion we had to assume that /3r 0, (4.1-8) where fm = oom/i^ir) is some representative frequency within the band and R and 6 are functions of time which vary slowly in comparison with cos comt. We call R the envelope of V. From equation (4A-1) I = -^ f F{iu)e''"' ""' ^"'"'+'' du (4.3-7) 27r J c We expand the integrand by means of ^ix cos V _ g ^jn ^^g n^j^{^x) (4.3-8) 71 = 0 where eo is 1 and €„ is 2 when w > 0 and Jr,{x) is a Bessel function. Thus eo I = Yj An{R) COS («w^/ + ne) (4.3-9) n=0 where Ar^iR) = €n^ [ F{iu)Jn{uR) du (4.3-10) 27r Jc Since i? is a relatively slowly varying function of time we expect the same to be true of An{R), at least for moderately small values of n. Thus from (4.3-9) we see that the power spectrum of / will consist of a suc- cession of bands, the n^^ band being clustered around the frequency «/,„ . If we eliminate all of the bands except the n^ by means of a filter we see that the output will have the envelope An{R) when n ^ 1. Taking n to be zero, shows that the low frequency output is simply A^{R) =^ [ F(iu)MuR)du (4.3-11) 27r J c MATHEMATICAL ANALYSIS OF RANDOM NOISE 121 Taking n to be one shows that the band around /„, is given by R (4.3-12) The statistical properties of the low frequency output and of the en- velopes of the output bands may be obtained from those of R. For ex- ample, the probabihty density of An(R) is of the form p{R)/ ^-^ (4.3-13) dR where p{R) is the probability density of R. In this expression R is con- sidered as a function of An . It should be noted that we have been assuming that all of the band surrounding the harmonic frequency nfn is taken. Wlien we take only a portion of it, presumably the statistical properties will tend to approach those of a random noise current in accordance with the first statement made at the beginning of this section. WTien we apply (4.3-11) to the square law device we have Ztti J (0+) 2m _ « I?2 When we apply (4.3-11) to the linear rectifier; F{iu) = — u + 00 J^juR) . ^oR where the path of integration passes under the origin. These two results agree with those obtained in Section 4.1 and 4.2 from simple considerations. As a final example we find the low frequency output of a biased linear rectifier in terms of the envelope R of the applied voltage. From the table of F{;iii) given in Appendix 4A we see that F{iu) corresponding to / = 0, V B 122 BELL SYSTEM TECHNICAL JOURNAL is —iuB F{iu) = — - Consequently, the low frequency output is A^{R) = -J- I e~""'Jn{uR)u'Uu 2tt J-oo where the path of integration is indented downwards at the origin. When B > R the value of the integral is zero since then the path of integration may be closed in the lower half plane by an infinite semi-circle This value also follows at once from the physics of the problem. When —R 0, {R is always positive) AoiR) = 0, R < B B B B I / (4.3-15) AoiR) = -^ + - arc sin ^ + -^ VR^ - B\ B < R 2 IT R IT MATHEMATICAL ANALYSIS OF RANDOM NOISE 123 and for i? < 0 it is Ao{R) = \B\, R <\B\ \B\ \B\ \B\ 1 / , , (4.3-16) A,(R) = +LeJ + L^i arc sin L„-' + - \/R' - B% \B\ < R 2 TT K IT where tlie arc sines lie between 0 and ir/l. Ao{R) and its first derivative with respect to R are continuous. From (4.3-15), the d.c. output current is, for i^ > 0, he = f T-f + - arc sin | + - VR- - bA p(R) dR (4.3-15) J H ]_ 2 IT K IT J where p(R) is the probability density of the envelope of the input V, e.g., p(R) is of the form (3.7-10) for noise alone, and of the form (3.10-11) for noise plus a sine wave. Similarly, the rms value of the low frequency current If/ , excluding d.c, may be computed from ilf = fa - lie where, if 5 > 0, T]l= f T-f + - arc sin I + - VW^=^^\ p(R) dR (4.3-16) J B [_ 2 TT K T J If T' consists of a sine wave of amplitude P plus noise T',v , so it may be represented as (4.1-13), and if P » rms V^ , the distribution of R is approximately normal. If, in addition, P — B ^ rms F.v > 0, (4.3-15), (4.3-16), and (3.10-19) lead to the approximations ^-l + ^ + ^I+P (4.3-17) 2 T 2irP Ti P- - B ^f ~ V¥^^ h The second expression for Idc assumes P » B. When B = Q, these re- duce to the first terms of (4.2-5) and (4.2-7). By using a different method Middleton has obtained a more precise form of this result. Incidentally, for a given applied voltage, /dc(+) for a positive bias | B \ is related to /dc( — ) for a negative bias — | 5 | by /dc(-) = \B\ -\- /do(+) (4.3-18) Also r.m.s. 7^/(+) is equal to r.m.s. Iff{ — ). Equation (4.3-18) follows from a physical argument based on the areas underneath a curve of I for 124 BELL SYSTEM TECHNICAL JOURNAL the two cases. Both of the above relations follow from formulas given by Middle ton when T' is the sum of a sine wave plus noise. They may also be derived from (4.3-15) and (4.3-16). 4.4 Output Power Spectrum The remainder of Part IV will be concerned with methods of solving the following problem: Given a non-linear device and an input voltage con- sisting of noise alone or of a signal plus noise. WTiat is the power spectrum of the output? In some ways the answer to this problem gives us less information than the methods discussed in the first three sections. For example, beyond giving the rms value, it tells us very httle about the probabiUty density of the current corresponding to a given frequency band of the output. On the other hand, this rms value may be found (by integrating the power spectrum) for any band we choose to study. The methods described earlier depended on the input being confined to a relatively narrow band and gave information regarding only the entire band corresponding to a given har- monic (0th, 1st, 2nd, etc.) of the input. There was no way to study the output when part of a band was eliminated by filters except by obtaining the power spectrum of some function of the envelope. At present there appear to be two general methods available for the determination of the output power spectrum each with its own advantages and disadvantages. First there is the direct method which has been used by W. R. Bennett*, F. C. Williams**, J. R. Ragazzini"^ and others. The noise is represented as the sum of a finite number of sinusoidal components. The typical modulation product is computed and the output power spectrum is obtained by considering the density and amplitude of these products. The chief advantage of this method lies in its close relation to the known theory of modulation in non-linear circuits. Generally, the lower order modulation products are the only ones which contribute significantly to the output power and when they are known, the problem is well along towards solution. The main disadvantage is the labor of counting the modulation products falling in a given interval. However, Bennett has developed a method for doing this.^^ The fundamental idea of the second method is to obtain the correlation function for the output current. From this the output power spectrum may be obtained by Fourier's transform. The correlation function method and its variations are of more recent origin than the direct method. They have * Cited in Section 4.0. Also much of this writer's work on interference in broad band communication systems may be carried over to noise theory without any change in the methods used. ** Cited in Section 4.1. «Proc. I.R.E. Vol. 30, pp. 277-288 (June 1942), "The Effect of Fluctuation Voltages on the Linear Detector." "^.S.TJ., Vol. 19 (1940), pp. 587-610, Appendix B. MATHEMATICAL AX A LYSIS OF RAX DOM XOISE 125 been discovered independently and at about the same time, by several workers. In a paper read before the I.R.E., Jan. 28, 1944, D. O. North described results obtained by using the correlation function. J. H. Van Meek and D. Middleton have been using the two variations of the method which we shall describe in Sections 4.7 and 4.8, since early in 1943. A primitive form of the method of Section 4.8 had been used by A. D. Fowler and the writer in some unpublished material written in 1942. Recently, I have learned that a method similar to the one used by Fowler and myself had already been used by Kurt Franz in 1941. The correlation function method avoids the problem of counting the modulation products. However, in some cases it becomes rather unwieldy. Probably it is best to have both methods in mind when investigating any particular problem. The direct method will be illustrated by applying it to the square law detector. Two approaches to the correlation function method will then be described and applied to examples. 4.5 Noise Through Square Law-Device Probably the most direct method of obtaining the power spectrum W(f) of /, where / = aV\ (4.1-1) V being a noise voltage, is to square the expression M V = Vif = ^ C,n cos (oJm t — Ol'A/ / w(f)w(Jh - f) df I 71=1 4 n=l Jo and this leads to the second term in (4.5-7). \\lien the voltage T" applied to the square law device is the sum of a noise voltage T> and a sine wave : F = P cos ^^ + TV, (4.1-13) we have Y- = P^ cos^ pi + IPVxcos pt + Vl (4.5-10) From the two equations 2 1 1 cos pt = -z -{• - cos 2pt ave. Vl = ^cl,--^ / w{f) df 1 2 JQ we see that 7, or aV , has a dc component of + a f wif) df (4.5-11) Jo i2 2 JO which agrees with (4.1-14), and a sinusoidal component ^ cos 2pt (4.5-12) The continuous power spectrum Wc(f) of the remaining portion of I may be computed from 2PVn cos pi + T'a- . MATIIEMAriCAL AXALV^IS OF RAX DOM NOISE 129 Using the representation (2.8-6) we see 2PVx cos pt = Pj^ f,Jcos (co„,/ -{- pi - v^«) + cos {o)„J - pt - ^„0] 1 For the moment, we take p = l-wfL^J. The terms pertaining to frequency fn = iiAf are those for which Wm + /> = 27r/n \(J^m — P \ = 2irfn m -\- r = n \ m — r\ = n m = n — r m = r ± n where only positive values of m are to be taken: If n > r, then m \s n — r or r + n. If n < r, then m is r — n or r + n. In either case the values of m are \n — r\ and « + r. The terms of frequency /„ in 2PTV cos ^/ are therefore PC\n-T\ cos (2x/„/ — (p\n-r\) -\- PCn+r COS {IrfJ — iPn+r) and the mean square value of this expression, the average being taken over the ip's, is - (c5„-r| + cl+r) = P^ Af[w{f\n-r\) + w{fn+r)] where fp denotes ^/27r. By combining this with the expression (4.5-5) which arises from V^ we see that the continuous portion WdJ) of the power spectrum of / is Wcif) = a~P\uif - /,) + w{f+fp)] •+" _ (4.5-13) dx /-l-oo w(x)w{f — x) •00 where ■zc'(— /) has the same value as «'(/). Equation (4.5-13) has been used to compute Wc(f) as shown in Fig. 8. The input noise is assumed to be uniform over a band of width j8 centered at fp , cf . Filter c, Appendix C. By noting the area under the low frequency portion of the spectrum we find Wcif) df = a'fiwoiP' + /3wo) Jo Since the mean square value of the input W is i/'o = |Swo , it is seen that this equation agrees with the expression (4.1-15) for the mean square value of Iff , the low frequency current, excluding the d.c. If audio frequency 130 BELL SYSTEM TECHNICAL JOURNAL filters cut out part of the spectrum, Wdf) may be integrated over the re- maining portion to give the mean square value of the corresponding output current. This idea is mentioned in the footnote pertaining to equation (4.1-6). If T' consists of W plus two sinusoidal voltages of incommensurable fre- quencies, say V = P cos pt + Q cos qt + TV , CONTINUOUS PORTION OF OUTPUT SPECTRUM OF SQUARE LAW DEVICE INPUT zz P COS 2nr„t + NOISE a-^wjf) OUTPUT D,C.= a(_p2/2+p Wg) LET p w2=C INPUT SPECTRUM INPUT NOISE C/2 c/2 2f--^ • FREQUENCY Fig. 8 2fp+p the continuous portion Wdf) of the power spectrum of / may be shown to be (4.5-13) plus the additional terms a'Q\w(f - /,) -F iv(f + /,)] (4.5-14) where /g denotes q/2ir. When the voltage applied to the square law device (4.1-1) is'*^ V(t) = Qil + kcos pi) cos qt + Vn Ok Ok = Q cos qt + ^ cos ip + q)i + '±- cos (p - q)t -f Kv the resulting current contains the dc component ^ <3' (1+ I) + « j[ Mf) df (4.5-16) *^ A complete discussion of this problem is given bj- L. A. MacCoU in a manuscript being prepared for publication. MATHEMATICAL ANALYSIS OF RAXDOM NOISE 131 The sinusoidal terms of 7 are obtained by squaring (3(1 + k cos pt) cos qt and multiplying by a. The remaining portion of / has a continuous power spectrum given by fQ'[iv(f- Wcif) =cx'Q'\ wif - U) + w{f + /,) w{x)w(f — a:) dx 00 where /p denotes p/lir and/g denotes (7/27r. 4.6 Two Correlation Function Methods As mentioned in Section 4.4 these methods for determining the output power spectrum are based on finding the correlation function ^(r) for the output current. From this the power spectrum, W(f), of the output cur- rent may be obtained from (2.1-5), rewritten as Wif) = 4 [ ^(t) cos 27r/T dr (4.6-1) •'0 It will be recalled that W(f)Af may be regarded as the average power which w^ould be dissipated by those components of / in the band/,/ + A/if / were to flow through a resistance of one ohm. The input of the non-linear device is taken to be a voltage V(t). It may, for example, consist of a noise voltage F.v(/) plus sinusoidal components. The output is taken to be a current I(t). The non-Unear device is specified by a relation between V(t) and /(/). In this work /(/) at time / is assumed to be completely determined by the value of V{t) at time /. Two methods of obtaining ^(r) will be described. (a) Integrating the two-dimensional probability density of V(t) and V(t -f r) over the values allowed by the non-linear device. This method, which is especially direct when applied to noise alone through rectifiers, was discovered independently by Van Vleck and North. (b) Introducing and using the characteristic function, which for the sake of brevity will be abbreviated to ch. f., of the two-dimensional prob- ability distribution of V(t) and V{t -f r). 132 BELL SYSTEM TECHNICAL JOURNAL 4.7 Linear Detection of Noise — The Van Vleck-North Method The method due to Van Vleck and North will be illustrated by using it to determine the output power spectrum of a linear detector when the input consists of noise alone. The linear detector is specified by ^« - \i-(o, Y(o > 0, (■'■'-'^ which may be obtained from (4.2-1) by setting a equal to one, and the input voltage is V(t) = ]\-(t) (4.7-2) where VN(t) is a noise voltage whose correlation function is i/'(t) and whose power spectrum is w(f). The correlation function ^(r) is the average value of I(t)I(t + r). This is the same as the average value of the function r/T' T- ^ /^ i^'2, when both Vi, V2 > 0 /, - -.n ^(''■'^'=\0, all other T's, ^^'''^^ where we have set T'l = y(t) V2 = V(t + r) The two-dimensional distribution of T'l and V2 is given by (3.2-4), and from this it follows that the average value of any function F(Vi , F2) is C '"'' C '''' T[Mw "^p [-2W1 (^» '^' + ^» ''' - '-*' '- '^=>] (4.7-4) where \M\=xkl-^l.l. For the Hnear rectifier case, where F{Vi, V2) is given by (4.7-3), the integral is \M\~"'^l dVij^ dV2V,V2exp[--^{hVl+^PoVl-24^rViV2)j = ^([.2 -.;]- + ., cos- [^']) MATHEMATICAL ANALYSIS OF RANDOM NOISE 133 where we have used (3.5-4) to evaluate the integral. The arc cosine is taken to be between 0 and tt. We therefore have for the correlation func- tion of /(/), ^(r) = 1 ([^0^ - ^T' + h cos-^ [^^]) (4.7-5) The power spectrum ]V(f) may be obtained from this by use of (4.6-1). For this purpose it is convenient to write (4.7-5) in terms of a hypergeo- metric function. By expanding and comparing terms it is seen that 4 27r \ ;/,-/ = — + — + -— + terms mvolving xf/r , ^r , etc. 4 Zir Airxf/o (4.7-6) As will be discussed more fully in Section 4.8, a constant term A" in \1/(t) indicates a direct current component of /(/) of ^4 amperes. Thus I{t) has a dc component equal to r^ I = -^ X rms value of V(t) (4.7-7) LzttJ 'v'lir This agrees with (4.2-3) when the P of that equation is set equal to zero. Integrals of the form Gn(f) = I ^T cos 27r/r dr Jo which result w^hen (4.7-6) is put in (4.6-1) and integrated termwise are discussed in Appendix 4C. From the results given there it is seen that if we neglect i/'^ and higher powers we obtain an approximation for the con- tinuous portion Wdf) of W{f): Wcif) = Gi(/) + ^-^ ■KXf/Q W{f) , 1 1 r , N ,. .. = -^ -f -— •- / w{x)w(J - X) dx where iv{—f) is defined as w{f). When VN(t) is uniform over a relatively narrow band extending from fa to fb so that w(/) is equal to wo in this band and is zero outside it, we may use the results for Filter c of Appendix 4C. The /o and jS given there are related to fa and fb by /a = /o — 2 > /& = /o + 2 134 BELL SYSTEM TECHNICAL JOURNAL and the value of li'o taken there is the same as here and is i/'o//3. The value of Giif) given there leads to the approximation, for low frequencies: Woij) -iTT V h- fa) 7n/'o4/3 (4.7-9) when 0 < / < /b - /„ , and to W^f) -= 0 ior fb - fa < f < fa ■ By setting P equal to zero in the curve given in Fig. 8 for Wdf) corresponding to the square law detector, we see that the low frequency portion of the power spectrum is triangular in shape and is zero at / = /3. Thus, looking at (4.7-9), we see that to a first approximation the shape of the output power spectrum is the same for a linear detector as for a square law detector when the input consists of a relatively narrow band of noise. An approximate rms value of the low frequency output current may be obtained by integrating (4.7-9) = nfi—fa Jo Wcif) df Woifb - - fa) _ ^0 87r Stt rms low freq. current = ~y^ X rms applied voltage (4.7-10) It is seen that this is half of the direct current. It must be kept in mind that (4.7-10) is an approximation because we have neglected \pr and higher powers. The true value may be obtained from (4.2-8). It is seen that the coefficient (Stt)"^'^ = 0.200 should be replaced by K-i)"=«- = 0.209 Wcif) for other types of band pass filters may be obtained by using the corresponding G's given in appendix 4C. It turns out that (4.7-10) holds for all three types of filters. This is a special case of Middleton's theorem, mentioned several times before, that the total power in any modulation product (it will be shown later in Section 4.9 that the term i/'" in (4.7-6) corresponds to the n order modulation products) depends only on the total input power of the applied noise, not on its spectral distribution. 4.8 The Characteristic Function Method As mentioned in the preceding parts, especially in connection with equa- tion (1.4-3), the ch. f. of a random variable x is the average value of exp MA THEM A TIC A L . 1 .Y. 1 LYSIS OF RA NDOM XOISE 135 {inx). This is a function of u. The ch. f. of two random variables x and V is the average value of exp {iux -\- ivy) and is a function of u and v. The ch. f. which we shall use here is the ch. f. of the two random variables V{t) and Vit + t) where T'(/) is the voltage applied to the non-linear device, and the randomness is introduced by / being selected at random, t remaining lixed. We may write this characteristic function as 1 r g{u, V, t) = Limit - / exp [/«F(/) + ivV{t + r)] dt (4.8-1) r—w 1 Jo If T'(0 contains a noise voltage T'iv(Oj as it always does in this section, and if we use the representation (2.8-1) or (2.8-6) a large number of random parameters (dnS and 6„'s or (^,,'s) will appear in (4.8-1). In accordance with our use of such representations we may average over these parameters without changing the value of (4.8-1) and may thereby simpHfy the integra- tion. For example suppose V{t) = VM + F^(/) (4.8-2) where !'.,(/) is some regular voltage which may, e.g., consist of one or more sine waves. Substituting this in (4.8-1) and using the result (3.2-7) that the ch. f. of Fjv(/) and Vx{t -f t) is gx(u, T, -) = ave. exp [iuVM{i) + ^vV^'(t -f- r)] — ^ («' + V') - ^rUV = exp -— {u -]- V-) - ip \p^ = \1/(t) being the correlation function of Fv(0> we obtain for the ch. f. of T'(/) and F(/ + t), g{u, V, t) = exp ■y (//" + V') — XprtlV X Limit ]- [ exp [iuVs(t) + ivV,{t + r)] dt ^^'^'^^ = A'.v(", V, T)g,{u, V, t) In the last line we have used gs{u, v, r) to denote the limit in the line above: 1 r'' g^{ti, V, t) = Limit - / exp [iuVsiO + ivVs{t + r)] dt (4.8-5) 7'_oo I Jq The principal reason we use the ch. f. is because quite a few non-linear devices may be described by the integral I = ^ f Fiiiije'"'' du (4.\-l) 27r J c 136 BELL SYSTEM TECHNICAL JOURNAL where the function F{iu) and the path of integration C are chosen to fit the device. Examples of such devices are given in Appendix 4A. The corre- lation function ^(r) of I{t) is given by dt dv ^(r) = Limit 1 \ I{t)I{t + t) = Limit j\- [ dt f F{iu)e'"''^'^ du f F{iv)e''"'^'^'^ T-*oo 4:ir^ I JQ Jc Jc = -^ f F(iu) du [ F{iv) dv (4.8-6) 47r~ J c J c 1 r"" Limit - / exp [iuV{i) + ivV{t + t)] di = — -, / F{iu) du I F{iv)g{u, v, t) dv At" Jc J c This is the fundamental formula of the ch. f. method. \Mien ]'{t) is the sum of a noise voltage and a regular voltage, as in (4.8-2), (4.8-6) becomes ^(r) = — [ Fiiu)e-^'^°"^''' du f F{iv)e-^'^''-'''' Att-Jc Jc (4,8-7) e-'^r^" g,(u, V, t) dv where gs(u, v, r) is the ch. f. of Vs{t) and Vs(t + r) given by (4.8-5). This is a definite expression for ^(t). All that follows is devoted to the evalua- tion of this integral and to the evaluation of W{f) = 4 /" ^(t) cos IwfT dr (4.6-1) Jo for the power spectrum of /. Quite often /(/) will contain dc and periodic components. It seems con- venient to deal with these separately since they correspond to terms in ^(t) which cause the integral (4.6-1) for W{f) to diverge. In fact, from Section 2.2 it follows that a correlation function of the form ,2 . C A' + ~ cos 27r/oT (2.2-3) corresponds to a current ^ -f C cos (2x/o/ - if) (2.2-2) MATHEMATICAL ANALYSIS OF RANDOM NOISE 137 where the phase angle tp cannot be determined from (2.2-3) since it does not affect the average power. Consider the correlation function for V{i) = Vs{t) + TatC/) given by (4.8-2). It is .7" Limit il f Vs(t)Vs(t + r)dt+ [ VM)VAt + r) dt + jf VAi)Vs(t-h T)dt + f^ VAi)VAi + r)dt\ (4.8-8) Since Vs(t) and T'iv(0 are unrelated the contributions of the second and third integrals vanish leaving us with the result Correlation function of T'(/) = Correlation function of Fs(/) + Correlation function of T a'(/). Now as T -^ =0 the correlation function of 1^(0 becomes zero while that of Vs{t) becomes of the type (2.2-3) given above. Hence the correlation func- tion of the regular voltage Vs{t) may be obtained from V{t) by letting r — > cc and picking out the non-vanishing terms. Although we have been speaking of V{t), the same results hold for I(t) and this process may be used to pick out those parts of ^(r) which correspond to the dc and periodic components of I(t). Thus, if we look at (4.8-7) we see that as r -^ cc , i//^ — > 0, while the gs {u, V, t) corresponding to Vs{t) given by (4.8-5) remains unchanged in general magnitude. This last statement may be hard to see, but examina- tion of the cases discussed later show that it is true, at least for these cases. Thus the portion of ^(r) corresponding to the dc and periodic components of /(/) is, setting i/'^ = 0 in (4.8-7), ^^{r) = ^J F{m)e-'^'>""'' du [ F{iv)e-'^'"'''" gs{u, v, r) dv (4.8-10) 47r" J c *' c where the subscript =o indicates that ^oo(t) is that part of ^(t) which does not vanish as t -^ co . We may write (4.8-9), when applied to I{t), as ^(r) = M^«(t) -I- M^e(T) (4.8-11) where ^c{t) is the correlation function of the "continuous" portion of the power spectrum of /(/). Incidentally, the separation of ^{t) into the two parts shown in (4.8-11) may be avoided if one is willing to use the 8(1) functions in order to interpret the integral in (4.6-1) as explained in Section 2.2. This method gives the proper dc and sinusoidal components even though (4.6-1) does not con- verge (because of the presence of the terms leading to ■^oo(t)). 138 BELL SYSTEM TECHNICAL JOURNAL 4.9 XoisE Plus Sine Wave Applied to Non-Linear Device In order to illustrate the characteristic function method described in Section 4.8 we shall consider the case of a non-linear device specified by / = J- f F{m)e''"' du (4A-1) Iir J c when V consists of a noise voltage plus a sine wave : Vit) = P cos pt + V^it) (4.1-13) As usual, F.v(/) has the power spectrum k'(/) and the correlation function ;^(r). \f/(r) is often written as xj/r for the sake of shortness. Comparing (4.1-13) with (4.8-2) gives F,(/) = P cos pt (4.9-1) Our first task is to compute the ch. f. gsiu, v, r) for the pair of random variables VsQ) and Vs{t -\- t). We do this by using the integral (4.8-5): 1 r gs{u, V, t) = Limit - / exp [luP cos pt + ivP cos p{t + r)] dt r-»oc T Jo (4.9-2) = Jo{P\/u' + V- -f 2uv cos Pt) where Jo is a Bessel function. The integration is performed by writing u cos pt -\- V cos p (t -\- t) = (u + V cos pr) cos pt — V sin pr sin pt = a/w- -\- V" -{- 2nv cos />r cos {pt -f phase angle) and using the integral Mz) =^ f Iir Jn The correlation function for (4.1-13) has also been given in Section 3.10. The correlation function "^(t) for /(/) may now be obtained by substi- tuting the above expressions in (4.8-7) ^(r) = A f du F{iii)e-^'^'"^"' [ dv F{iv)e ■iTT- Jc J C (4.9-3) e~'^'-'"'/o(P a/m^ + t)2 + 2uv cos pr) . ^oc{t), the correlation function for the d.c. and periodic components of /, may, according to (4.8-10), be obtained from this by setting 4't equal to zero. When we have a particular non-linear device in mind the appropriate F{m) may often be obtained from Appendix 4A. For example, F(iu) for a linear rectifier is —u~'. Inserting this value in (4.9-3) gives a definite MATHEMATICAL AXALYSIS OF RAX DOM XOISE 139 double integral for ^(t). If there were some easy way to evaluate this in- tegral then everything would be fine. Unfortunately, no simj^le method of evaluation has yet been found. However, one method is available which is closely related to the direct method used by Bennett. It is based on the expansion gs{u, V, t) = Ja{P\/ifi + v^ -\- 2uv cos pr) 00 = Z en{-TJn{Pu)Jn{Pv) C05 npr (4.9-4) eo = 1, en — 2 for ii > 1 This expansion enables us to write the troublesome terms in (4.9-3) as e~^^"Vo(P\/«2 + V' -{- 2uv cos pr) = 2^ 2^ { — ) en cos npT JniPu)Jn(Pv) n=0 k=0 kI The \drtue of this double sum is that it simplifies the integration. Thus, putting it in (4.9-3) and setting •n+k r> hnk= — / F{iu)u'j„(Pu)e-^^'>""''du (4.9-6) Ztt J c gives 00 oo ^ ^(t-) = Z E t7 ^''rhlten COS 7lpT (4.9-7) n=0 k=0 k\ The correlation function ^oo(t) for the dc and periodic components of / are obtained by letting t -^ 'x where xf/j — > 0. Only the terms for which k = 0 remain: 00 ^«(t) = Z) en hlo COS npT (4.9-8) Comparing this with the known fact that the correlation function of yl + C cos {2Trfot - 1 shows that Amplitude of dc component of / = //oo lip (4-9-9) Amplitude of ~- component of / = 2//„o 27r 140 BELL SYSTEM TECHNICAL JOURNAL Incidentally, these expressions for the amplitudes follow almost at once from the direct method of solution. This will be shown in connection with equa- tion (4.9-17). Since the correlation function "^c(t) for the continuous portion Wdf) of the power spectrum for / is given by ^e(r) = ^(r) - ^„(t), (4.8-11) we also have 00 00 A ^c(t) = Z Z x^ i^rhlken COS npr (4.9-10) 71=0 fc=i «! Wlien this is substituted in W we obtain (/) = 4 [ ^o(t) cos lirfr dr (4.9-11) Jo where Gicif) = [ 'Ar COS lirfr dr (4.9-13) Jo is the function studied in Appendix 4C. Gkif) is an even function of/. The double series (4.9-12) for Wc looks rather formidable. However, when we are interested in a particular portion of the frequency spectrum often only a few terms of the series are needed. It has been mentioned above that the direct method of obtaining the out- put power spectrum is closely related to the equations just derived. We now study this relation. We start with the following result from modulation theory : Let the voltage V = Po cos .vo + Pi cos .Vi + • • • -f- Pn cos Xn (4.9-14) Xk = pj, k = 0,\, ■•• N, where the ^/,'s are incommensurable, be applied to the device (4A-1). The output current is 05 00 i = Z_* ••• 2^ 2-^mo--m^^mo mo=0 mfj=0 (4.9-15) • • ' Cmjv COS moXo cos ftliXi • • • cos MffXN ^° Bennett and Rice, "Note on Methods of Computing Modulation Products," Phil. Mag. S.7, V. 18, pp. 422-424, Sept. 1934, and Bennett's paper cited in Section 4.0. MATHEMATICAL ANALYSIS OF RANDOM NOISE 141 where eo = 1 and e,,, = 2 for m > 1. When the product of the cosines is expressed as a sum of cosines of the angles ;wo xo ± ffii Xi • • • zLhinXn , it is seen that the coefficient of the typical term is /Imo-mjv > except when all the w's are zero in which case it is ^.lo-o • Thus ^yioo-.-o = dc component of / I Amo---mx I = amplitude of component of frequency (4.9-16) ;r- I niopo =b niipi ± • • • ± niffpff \ Ztt For all values of the m's, IT J c r=0 (4.9-17) M = mo -}- nil + " • + Mn Following Bennett's procedure, we identify V as given by (4.9-14), with V = Pcospt -\- V^ (4.1-13) by setting Po = P, po ^ p, and representing the noise voltage I'at by the sum of the remaining terms. Since this makes Pi , P^ all very small, Laplace's process indicates that in (4.9-17) we may put n MPrti) = exp - ^ (PI + • • . + Pi) r=i 4 (4.9-18) __ ^-l^ou2/2 — ^ o We have used the fact that \{/o is the mean square value of V^ . It follows from these equations that dc component oi I = ~ [ F{iu)MPu)e^~'^°'^^"^ du Component of frequency-^ = - / F{iu)Jn{Pu)e''^°" '' du ZTT TT Jc These results are identical with those of (4.9-9). The equations just derived show that h„Q is to be associated with the n harmonic of p. In much the same way it may be shown that hnk is to be associated with the modulation products arising from the n harmonic of p and k of the elementary sinusoidal components representing IV . We consider only combinations of the form pi ± p2 ± ps , taking ^ = 3 for ex- ample, and neglect terms of the form 3pi and 2pi ± p2 . The former t>'pe is much more numerous, there being about N of them while there are only about N and N^ , respectively, of the latter type. 142 BELL SYSTEM TECHNICAL JOURNAL We again take k = Z and consider Wi , m^ , niz to be one, and mi , • • • my to be zero, corresponding to the modulation product np ± pi ± p2 dz ps . By making the same sort of approximations as Bennett does we find vl„,i,i,i,o.o...o = — / F{tu)Jn{Pu)u e ^'^ du It 8 J c PiP^Psj = 4 ^'nZ When any other modulation product of the form np ± p^ ± pr., ± prs is considered we get a similar expression in which P1P2P3 is replaced by PriPr^Prz • This may be done for any value of k. The result indicates that hnk , and consequently also the (n, k) terms in the double series (4.9-10) and (4.9-12) for "^dr) and Wdf), are to be associated with the modulation products of order {n, k), the n referring to the signal and the k to the noise components. We now may state a theorem due to Middleton regarding the total power in the modulation products of a given order. For a given non-linear device (i.e. F(iu) is given), the total power which would be dissipated by all of the modulation products which are of order {n, k) if / were to flow through a resistance of one ohm is *.,(0)=!^';,L='4pl* (4.9-19) The important feature of this expression is that it depends only on the r.m.s. value of T'.v and on F{iu). It depends not at all upon the s'pectral dis- tribution of the noise power in the input. The proof of (4.9-19) is based en the relation ^nkiO) = f Wnkif) df Jo between the total power dissipated by all the (n, k) order products and the corresponding correlation function obtained from (4.9-7). This theorem has been used by Middleton to show that when the input is confined to a relatively narrow frequency band, so that the output spec- trum consists of bands, the power in each band depends only on V^ and not on the spectrum of IV • 4.10 Miscellaneous Results Obtained by Correlation Function Method In this section a number of results which may be obtained from the theory given in the sections following 4.6 are given. MATHEMATICAL ANALYSIS OF RANDOM NOISE 143 When the input to the square law device / = aV^ (4.1-1) consists of noise only, so that 1' = I'a- , the correlation function for I is ^(r) = a'[^l + 2^p;] (4.10-1) where \pT is the correlation function of Vx . This may be compared with equation (3.9-7). \Mien V is general, ^(r) = ave. /(/)/(/ + t) = Sive. a' V\t)V\t + t) 2 s/ r- «; • . Amfiiv?. . . (4.10-2) = a X Coeflhcient of — - — — — ni power series expansion of ch. f. of V{t),V{t + t) where we have used a known property of the characteristic function. An expression for the ch. f., denoted by g{n, v, r), is given by (4.8-4). For example, when V consists of a sine wave plus noise, (4.1-13), the ch. f. is obtainable from (4.9-3). Hence, ii" if ^(t) = Coeff. of — ^ in expansion of 4 (4.10-3) a Jo{P\/h- -\- v^ -\- 2uv cos pr) X exp — y {u' + v^) — \prUV = a- [^ + rPoj + J cos 2pr + 2PVr COS pr + 24^1 The first two terms give the dc and second harmonic. The last two terms may be used to compute Wdf) as given by (4.5-13). Expressions (4.10-1) and (4.10-3) are special cases of results obtained by Middleton who has studied the general theory of the quadratic rectifier by using the Van Vleck-North method, described in Section 4.7. As an example to which the theory of Section 4.9 may be applied we con- sider the sine wave plus noise, (4.1-13), to be applied to the f-law rectifier 7 = 0, T' < 0 (4.10-4) 7 = Y\ V > 0 From the table in Appendix 4.4 it is seen that F(hi) = T(p + \)({u)""' 144 BELL SYSTEM TECHNICAL JOURNAL and that the path of integration C runs along the real axis from — oo to oo with a downward indentation at the origin. The integral (4.9-6) for hnk becomes •n+k—v—l /• = -^^ r(. + 1) u'—'jn{Pu)i ZTT J C :n-\rk—v—\ ink = — ~ 1 ir -r J.; ; 11 Jn\i"ii)e " " du (I \(i'-A;)/2 - i , \ . ^,J^±^.,n+V,-.) (4.10-5) P' where the integration has been performed by expanding J„(Pii) in powers of u and using l e "" u'^ ^ du = ie ^^"^ a ^ sin Xxr(X) ^ (1 - e-'-'nviX) (4.10-6) tire a^T{l - X) it being understood that arg w = 0 on the positive portion of C. From (4.9-9), the dc component of / is P-p'-i-'r--) hoo= ^/ ' \(^") i/^i(-^;l; --t) (4.10-7) 2r which reduces to the expression (4.2-3) when v = 1 for the Unear rectifier (aside from the factor a). When the input (sine wave plus noise) is confined to a relatively narrow band, and when we are interested in the low frequency output, consideration of the modulation products suggests that we consider the difference products from the products of order (0, 0), (0, 2), (0, 4), • • • (1, 1), (1, 3), • • • (2, 0), (2, 2), • • • etc. where the typical product is of order (n, k). The orders (0, 0) and (2, 0) give the dc and second harmonic and hence are not con- sidered in the computation of Wdf). Of the remaining terms, either (0, 2) or (1, 1) gives the greatest contribution to the series (4.9-12) and (4.9-10) for Wcif) and "^dr). The remaining terms contribute less and less as n and MATHEMATICAL ANALYSIS OF RANDOM NOISE 145 k increase. The low frequency portion of the continuous portion of the output power spectrum is then, from (4.9-12), Wcif) = ^^hl.G^if) + ^,/4G4(/) + ••• + j^j hUCrif - /o) + Gi(/ + /o)] + |, hUC^if - /o) (4.10-8) + Gz(f + /o)] + |j hUG^if - 2/o) + G2(f + 2/o)] + • • . From Table 2 of Appendix 4C we may pick out the low frequency portions of the G's. It must be remembered that Gm(x) is an even function of x and thatO r) X Jo{Q\^U' -\- v- -{- luv cos ^r) From equations (4.9-16) and (4.9-17) it is seen immediately that /%o =^ ~ I F{iu)MPu)MQu)e-'"'"^^° du (4.10-11) is the d.c. component of / when the applied voltage is P cos pt -\- Q cos qt + IV . (4.1-4) J. R. Ragazzini has obtained an approximate expression for the output power spectrum when the voltage V = Vs+V^ (4.10-12) Vs = Q(l -{- r cos pt)cos qt is impressed on a linear rectifier. In terms of our notation his expression for the continuous portion of the power spectrum is (for low frequencies) (4.10-13) , . _ 1 ^, [Wcif) given by equation l^c(;) - _._./r.. , .., ^ X 1^^^ 5_j. 7r2a^((22 -I- 2\^o) LC-^-S-l^) for square law device_ The a' is put in the denominator to cancel the a' in the expression (4.5-17). We take the linear rectifier to be and replace the index of modulation, k, in (4.5-17) by r. ^^ Equation (12), "The Effect of Fluctuation Voltages on the Linear Detector," Proc. I.R.E., V. 30, pp. 277-288 (June 1942). MATHEMATICAL AX A LYSIS OF RANDOM NOISE 147 Ragazzini's formula is quite accurate when the index of modulation r is small, especially when y — Q'/{2\{/o) is large. To show this we put r = 0 in (4.10-13) and obtain Wcif) ^ T^^Q- + 2^o) L _ (4.10-15) + / w{x)w{f — x) dx where fq = q/{2ir). This is to be compared with the low frequency por- tion of ITc(/j obtained by specializing (4.10-8) to obtain the output power spectrum of a linear rectifier when the input consists of a sine wave plus noise. The leading terms in (4.10-8) give Wcif) = hlMf, -f) + ^(A +/)] •+« (4.10-16) o 1 f^ + /?o2 -. I w{x)w{f — x) dx 4 J— 00 The values of the /?'s appropriate to a linear rectifier are obtained by set- ting V = 1 in (4.10-5) and noticing that Q now plays the role of P. hn = \(^^'\F^{h2;-y) y = Q'/(2h) Incidentally, the first approximation to the output of a Hnear rectifier given by (4.10-16) is interesting in its own right. Fig. 9 shows the low fre- quency portion of Wdf) as computed from (4.10-16) when the input noise is uniformly distributed over a narrow frequency band of width I3,fq being the mid-band frequency, //n and //02 may be obtained from the curves shown in Fig. 10. In these figures P and .v replace Q and y of (4.10-17) in order to keep the notation the same as in Fig. 8 for the square law device. These curves may also be obtained from equations {33) to (43) of Bennett's paper. The following values are useful for our comparison. When X = 0 When x is large //n = 0 hn = I/tt (4.10-18) A02 = (27n/'o)~'^' //02 = l/iirQ). The values for large x are obtained from the asymptotic expansion (45 — 3) given in Appendix 45. 148 BELL SYSTEM TECHNICAL JOURNAL wc(f) LOW FREQUENCY OUTPUT OF L I NEAR RECTIFIER APPROXIMATION -SECOND ORDER PRODUCTS ONLY INPUT- V= Pcos zrrfpt + noise jo V < 0 I ouTPUT = i= j^; y>oj» OUTPUT D.C.= P*'„+ P Wg h(,2 ik. vvn Kf. ^ = '-.='' &) P/2 P FREQUENCY Fig. 9 INPUT SPECTRUM INPUT NOISE T""i I ' 1 \ 0.3 Pho 1 n i / !____ r - n V 4xy 0.2 / -^ / / ^, 1 1 // / 1 1 1 1 '/ i 1 x = 1.5 2.0 2 5 Ave SINE WAVE POWER P^ AVE NOISE POWER 2 P Wq Fig. 10 — Coefficients for linear detector output shown on llg. 9 P//02 = ^/- iFiG; 1; -.v) //n = \\ / - iFi(h 2; -x) We make the first comparison between (4.10-15) and (4.10-16) by letting Q ^ oc . It is seen that both reduce to Wcif) = \ [wif, - /) + w{f, + /)] (4.10-19) MATHEMATICAL ANALYSIS OF RjiNDOM NOISE 149 which shows that the agreement is perfect in this case. Next we let Q = 0. The two expressions then give i+OO Wc(f) = ./ . / w(x)w(f — x) dx 1 /• =^) = To~r / '^ix)wU - x) AZttxI/o J-.oo where .1 = x for Ragazzini's formula and A = 4 for (4.10-16). Thus the agreement is still quite good. The limiting value for (4.10-16) may also be obtained from (4.7-8). Even if the index of modulation r is not negligibly small it may be shown that when Q —^ cc Wdf) still approaches the value given by (4.10-19). Ragazzini's formula gives a somewhat larger answer because it includes the additional terms, shown in (4.5-17), which contain k /4, but this difference does not appear to be serious. If the Q + Ixpo in the denominator of (4.10- 13) be replaced by Q' -\- ^Q k" -\- 2\f/o the agreement is improved. APPENDIX 4A T.4BLE OF Non-linear Devices Specified by Integrals Quite a number of non-linear devices may be specified by integrals of the form 1 = ^1 F(iu)e''''' du (4A-1) Zir J c where the function F(iu) and the path of integration C are chosen to fit the device.* The table gives examples of such devices. Some important cases cannot be simply represented in this form. An example is the limiter I = - aD, F < -D I = aV, -D < V < D I = aD, D B V any positive number otTiv + 1) ,„B {iuy+^ j'th power recti- fier with bias 7 = 0, F < 0 I = aV, 0 < F < Z> 7 = aZ), D 0 F{p) = / e-PV(0 '^^ ^0 APPENDIX 4B The Function iFi(a; c; x) In problems concerning a sine wave plus noise the h3^ergeometric func- tion ,F,(a; .; .) = 1+ -, + ^^^-^ -, + (4B-1) arises. Here we state some of its properties which are of use in the theory of Part IV. Curves of iF\{a; c; z) are given for a = — 4, — 3.5 • • • , 3.5, 4.0 and c = - 1.5, - .5, + .5, 1, 1.5, 2, 3, 4 in the 1938 edition, page 275, of "Tables of Functions", by Jahnke and Emde. A list of properties of the function and other references are also given. In addition to these refer- ences we mention E. T. Copson, "Functions of a Complex Variable" (Ox- ford, 1935), page 260. If c is not a negative integer or zero iFi(a; c; z) = e\Fi{c — a; c; — z). (4B-2) MATHEMAIICAL AX A LYSIS OF RASDOM NOISE 151 \\'hen R (z) > 0 we have the asymptotic expansions i){c — a) r ( ^ r(c)g-' fi . (1 - «) \z , (1 - a){2 - a){c - a)(c - a + 1) "1 2!s- i- •••J P / N r(<;) r, , a(l + a iFi(a; c; -s) ~ — — ^^ 1 + -^ — - r(c — a)2" |_ l!z -c) (4B-3) , a(a + 1)(1 + a - c){2 + a - c) . 212- ] Many of the hypergeometric functions encountered may be expressed in terms of Bessel functions of the first kind for imaginary argument. The connection may be made by means of the relation^^ iFi ^. + ^ 2. + 1; z^ = f'r{u + l)z~V"lJ^ (4B-4) together with the recurrence relations Fa+ F^ Fc+ Fc- F 1. a (a - c) c — 2a — z 2. ac {c — a)z — c(a + z) 3. a 1 - c c — a — 1 4. — c — z c 5. a — c c- 1 I — a — z 6. (c - a)z c{c - 1) cil- c- z) For example, the first recurrence relation is obtained from line 1 as follows aF{a + 1; c; 2) + (a — c)F{a — 1; c; s) + (c - 2a - z)F{a; c; g)= 0 (4B-5) These six relations between the contiguous i^'i functions are analogous to the 15 relations, given by Gauss, between the contiguous 2F1 hypergeometric functions and may be derived from these by using {a, b; c- fj iFi{a; c; z) = Limit 2FA a, b; c; -) (4B-61 A recurrence relation involving two i/'\'s of the type (4B-4) may be ob- tained by replacing a by a + 1 in the relation given by row four of the table " G. N. Watson, "Theory of Bessel Functions" (Cambridge, 1922j, p. 191. 152 BELL SYSTEM TECHNICAL JOURNAL and then eliminating iFi(a + 1 ; c; s) from this relation and the one obtained from row 3 of the table. There results iFi(a; c; z) = ,Fi{a; c - 1; z) + _ F{a + 1; c + I; z) (4B-7) C{^1 c) Setting V equal to zero and one in (4B-4) and a equal to §, c equal to 2 in (4B-7) gives 1^1 (^ ^ 5 ^) = 42"'^^ ' h (0 (4B-8) Starting with these relations the relations in the table enable us to find an expression for iFi{n + h; m; z) where n and m are integers. A number of these are given in Bennett's paper. In particular, using (4B-2), lF^ (-^ ; 1; -z) = e-'" [(1 + z)h (0 + zh (|)] . (4B-9) APPENDIX 4C The Power Spectrum Corresponding to ^" Quite often we encounter the integral Gn(f) = f [rP{r)T COS iTfrdr (4C-1) where \P{t) is the correlation function corresponding to the power spectrum w(/). From the fundamental relation between w{f) and \P{t) given by (2.1-5), Gi(/) = ""-^ (4C-2) The expression for the spectrum of the product of two functions enables us to write Gn{f) in terms of w{f). We shall use the following form of this expression: Let Fr{f) be the spectrum of the function iprir) so that ^r(r) = f^riDe'^^'-df, r = l,2 J—ao 00 —2irifT dt MATHEMATICAL AX A TVS IS OF RA.XDOM NOISE 153 Then f ^i{r)P(t)'s Table 1 Filter w{f) for/> 0 >p{r) a b ^0 e-(/-/0)2/2 > + > + + > CD.I CM + V •^ V Oil es ;i c5 "o ej ^ o e? s ^ ^ o =2 £ ;5 S S v^ fa G" 156 BELL SYSTEM TECHNICAL JOURNAL The expression for G2(/) given in Table 2 corresponding to Filter c is exact. The expressions for Filters a and b give good approximations around / = 0 and/ = 2/o where Gi{f) is large. However, they are not exact because terms involving / + 2/o have been omitted. It is seen that all three G^?. behave in the same manner. Each has a peak symmetrical about 2/o whose width is twice that of the original iv{f), is almost zero between 0 and 2/o, and rises to a peak at 0 whose height is twice that at 2/o . GsU) is obtained by cubing the i/'(t) given in Table 1 and using cos 2x/oT = f cos 27r/oT + \ cos ^ttJqt. From the way in which the cosine terms combine with cos 27r/T in (4C-1) we see that Gz{f), for our relatively narrow band pass filters, has peaks at /o and 3/o , the first peak being three times as high as the second. The ex- pressions given for Gz{f) and Gi{f) are approximate in the same sense as are those for G^if). It will be observed that the coefhcients within the brackets, for Filters a and b, are the binomial coefficients for the value of n concerned. Thus for w = 2, they are 2 and 1, for » = 3 they are 3 and 1, and for n = 4 they are 6, 4, and 1. The higher GnifYs for Filters a and b may be computed in the same way. The integrals to be used are I e cos 27r/r dr Jo I e cos 1-KjT dr = 2a\/2mr na 'o •' 2xw2«2_|_y2 In many of our examples we are interested only in the values G„(/) for / near zero, i.e., only in that peak which is at zero. It is seen that G„(/) has such a peak only when n is even, this peak arising from the constant term in the expansion cos'^^ = _!_ [cos 2kx + 2k cos 2{k - \)x + (^^)(^^ " ^) cos 2{k - 2)x + ...+ ^_(?«i-^ cos 2x + Mil Abstracts of Technical Articles by Bell System Authors Historical Background of Electron Optics} C. J. Calbick. The discov- ery of electron optics resulted from studies of the action, upon electrons or other charged particles, of electric and magnetic fields employed for the purpose of obtaining sharply defined beams. The original Braun tube (1896) employed gas-focusing, as did the low-voltage cathode-ray oscil- loscope developed by Johnson in 1920. It was early discovered that an axial magnetic field could be used to concentrate the electrons into a beam, and this method came into wide use in the field of high-voltage cathode-ray oscillography. In 1927 Busch published a theoretical study of the action of an axially-symmetfic magnetic field upon paraxial electrons, showing that the equation of the trajectories of the electrons was similar to that of the paths of light rays through an axially symmetric optical system. He concluded that such magnetic fields constituted lenses for electrons and pre- sented experimental confirmation. In 1931 Knoll and Ruska presented a large amount of additional experimental material and used the words "elec- tron optics" to describe the analogy. In 1932 Bruche and Johannson pub- lished the first electron micrographs. The Davisson and Germer electron diffraction experiments (1927) em- ployed electron beams formed by electron guns consisting of a thermionic cathode emitting electrons which were accelerated by potentials applied to a series of plates containing aligned apertures. The resultant beam was quite divergent. Davisson and Calbick made a theoretical and experimental study of the forms of such beams. They concluded that the distorted elec- tric field in the vicinity of an aperture in a charged plate constituted a lens for charged particles (1931). The optical analogy was either a cylindrical or a spherical lens, according as the aperture was a slit or a circular hole. The theory was confirmed by photographing the forms of electron beams, and by construction of an electrostatic electron microscope whose experi- mental magnification agreed with the theoretical. Coaxial Cables and Associated Facilities r J. J. Pilliod. {Summary of Talk before St. Louis Electrical Board of Trade, October 17, 1944.) Coaxial cables provide means of transmitting frequency bands several million cycles in width over a metal tube a little larger than a lead pencil, with a copper wire extending along its axis. Several of these tubes can be placed in a lead sheath. The frequency band transmitted over coaxial cables may be split up so as to provide several hundred telephone circuits or, without such division, ^Jour. Applied Physics, October 1944. '^FM and Television, November 1944. 157 158 BELL SYSTEM TECHNICAL JOURNAL coaxial cables will provide for broad-band transmission service such as is required for television. A cable is now being installed between Terre Haute and St. Louis which contains six coaxial tubes to provide telephone circuits, and which may, in the future, find use in connection with the provision of intercity television networks. The structure of the tubes used with coaxial cables consists of a central copper conductor within a copper tube about \ in. in diameter, made from flat copper strip which is formed around the insulating discs. Around each copper tube are two steel tapes which supplement the shielding of the copper tube in preventing interference between tubes in close proximity. The cen- tral conductor is separated from the outer conductor by slotted insulating disks which are forced onto the wire. The cables are formed with an appro- priate number of these tubes along with some small gauge pairs used for control and operating purposes. In the case of underground cables buried directly in the earth, jute or plas- tic protective coverings are used to assist in reducing sheath corrosion. In some parts of the country it is essential to add a metal covering outside the lead sheath and the plastic or jute to protect the cables against the operations of ground squirrels or pocket gophers. In certain areas these animals have been found to carry away long sections of the jute covering and will chew holes in the lead sheath unless other metal protection is pro- vided. Copper is sometimes used for this metal covering to assist in light- ning protection. Repeaters in the coaxial system are now located at intervals of about five miles. Power for repeaters in the auxiliary stations is supplied from the adjacent main stations located at something over 50 miles at 60 cycles over the coaxial conductors thernselves. Coaxial cables are in regular operation between New York and Philadel- phia and between Minneapolis and Stevens Point, Wisconsin, a total dis- tance of nearly 300 miles. A network of such cables totaling about 7,000 route miles and including a second transcontinental cable route is being planned over additional routes. The requirements of the armed forces, general business conditions, the volume and distribution of long distance telephone messages, the availability of the necessary manufactured cable and equipment, and other factors may modify the extent of this construc- tion, the time of starting, and the routes which will be undertaken. Western Electric Recording System — U. S. Naval Photographic Science Laboratory} R. 0. Strock and E. A. Dickixsox. This paper describes the complete 35-mm film and ?)i\ or 78 rpm. disk recording and re-recording equipment installed for the U. S. Navy at the Photographic Science Labora- tory, Anacostia, D. C. Modern design, excellent performance, and ease of operation are features of the installation. ' Jour. Soc. Motion Picture Engineers, December 1944. Cqntributors to this Issue William A. Edsox, Kansas University, B.S. 1934; M.S. 1935. Harvard University, D.S. 1937. Bell Telephone Laboratories, 1937-1941 and 1943-. Assistant Professor of Electrical Engineering 1941-1942 at Illinois Institute of Technology, Chicago. Prior to 1941 Dr. Edson was concerned with carrier telephone terminal devices. At the present time he is engaged full time on war projects. Richard C. Egglestox, Ph.B. 1909 and M.F. 1910, Yale University; U. S. Forest Ser\-ice, 1910-1917; Pennsylvania Railroad, 1917-1920; First Lieutenant, Engineering Div., Ordnance Dept., World War I, 1918-1919; American Telephone and Telegraph Company, 1920-1927; Bell Telephone Laboratories, 1927-. Mr. Eggleston has been engaged chiefly with prob- lems relating to the strength of timber and with statistical investigations in the timber products field. S. O. Rice, B.S. in Electrical Engineering, Oregon State College, 1929; California Institute of Technology, 1929-30, 1934-35. Bell Telephone Laboratories, 1930-. Mr. Rice has been concerned with various theoretical investigations relating to telephone transmission theor}% 159 VOLUME XXIV APRIL, 1945 NUMBER 2 THE BELL SYSTEM ^m.^^ TECHNICAL JOURNAL DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION Piezoelectric Crystals in Oscillator Circuits . . I.E. Fair 161 The Measurement of the Performance Index of Quartz Plates C. W. Harrison 217 Lightning Protection of Buried Toll Cable . E. D. Sunde 253 Abstracts of Technical Articles by Bell System Authors 301 Contributors to this Issue 303 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 EDITORIAL BOARD M. R. Sullivan O. E. Buckley O. B. Blackwell M. J. KeUy H. S. Osborne A. B. Clark J. J. Pilliod S. Bracken ■■■■■■■■■■■ SUBSCRIPTIONS Subscriptions are accepted at $1.50 per year. Single copies are 50 cents each. The foreign postage is 35 cents per year or 9 cents per copy. Copyright, 1945 American Telephone and Telegraph Company PRINTED IN U. S. A. CORRECTIONS FOR ISSUE OF APRIL, 1945 M Page 207: Abscissa for Fig. 12.29, P = 1 — Ct/Co should be P = M 1 + Ct/Co Page 240: Equation (15.60), T^'o = 7 0 should be IFo =7 0 a 7 J 5J r— — 7 + 7 Lo l] a + 7 J sJ 7 + 7 L<) ^J The Bell System Technical Journal Vol. XXIV April, 1(^45 No. 2 Piezoelectric Crystals in Oscillator Circuits By I. E. FAIR 12.00 Introduction A STUDY or an explanation of the performance of a piezoelectric crystal in an oscillator circuit involves a study or explanation of oscillator circuits in general and a study of the crystal as a circuit element. Nicolson^ appears to have been the first to discover that a piezoelectric crystal had sufficient coupling between electrical electrodes and mechanical vibratory movement so that when the electrodes were suitably connected to a vacuum tube circuit, sustained oscillations were produced. In such an oscillator the mechanical oscillatory movement of the crystal functions as does the electrical oscillatory circuit of the usual vacuum tube oscillator. His circuit is shown in Fig. 12.1. Cady independently though later made the same discovery, but he utilized it somewhat differently and expressed it differently. He found that when the electrodes of a quartz crystal are connected in certain ways to an electric oscillator circuit, the frequency is held very constant at a value which coincides with the period of the vibrat- ing crystal. He made the further discovery that due to the very sharp resonance properties of the quartz crystal, the constancy in frequency to be secured was far greater than could be obtained by any purely electric oscillator. The development of analytical explanations of the crystal controlled oscillator came along rather slowly. Cady explained the control in terms of operation upon the electrical oscillator to which the crystal was attached. He said that the "capacity" of the crystal changes rapidly with frequency in the neighborhood of mechanical resonance, even becoming negative. This "capacity" connected across the oscillator tuned circuit or in other places prevented the frequency from changing to any extent, as any fre- quency change caused such a "capacity" change in the crystal as to tend to tune the circuit in the other direction. Cady, however, devised one circuit, Fig. 12.2, in which no tuned electrical circuit was used, but he confined his explanation to "a mechanically tuned feedback path from the plate to the grid of the amplifier". Pierce came along later with a two-electrode crystal connected between plate and grid, and no tuned circuit, and also with a 161 162 BELL SYSTEM TECHNICAL JOURNAL Fig. 12.1 — Nicolson's crystal oscillator circuit Fig. 12.2 — Cady's oscillator circuit using a crystal as a "mechanically tuned feedback path" Fig. 12.3 — Equivalent electrical circuit of a piezoelectric crystal near its resonant frequency two-electrode crystal connected between grid and cathode and no tuned cir- cuit, where the operation would not be satisfactorily explained by Cady's method. His circuits would require the crystal to exhibit inductive react- PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 163 ance, rather than the capacitance Cady spoke of. Miller^ also produced a circuit with a two-electrode crystal connected between grid and cathode but with a tuned circuit in the plate lead, which circuit required the crystal to provide inductive reactance. It was not until after Van Dyke showed that the crystal could be repre- sented by the circuit network of Fig. 12.3 that it was possible to explain these various phenomena. With this view of the crystal, and using the differential equation method of circuit analysis, Terry pointed out that, as with electrical oscillators, the frequency is not completely governed by the resonant element, in this case the crystal, but is influenced somewhat by the circuit elements. The circuit as a whole is quite complex and the equations are difficult to use. Wright and Vigoureux also made analyses of the Pierce type oscillator. Because of the complexity of the equations, the frequency, amplitude, or activity are not computed directly, but the effects of the circuit variables are analyzed in a qualitative manner and the results compared with experimental data. Oscillators employing crystals may be classified in a number of ways. One classification is based upon whether or not the circuit without the crystal is in itself an oscillator. If it is, the oscillator is called a "crystal controlled" oscillator. If it is not, it is called a "crystal" oscillator. All of Cady's oscillator circuits, except the one shown in Fig. 12.2, are of the first named class. This type of circuit will oscillate at a frequency deter- mined by the tuned circuit if the crystal becomes broken or disconnected, or if high resistance develops in the crystal, or if the electric tuned circuit should become tuned too far from the resonant frequency of the crystal. This property at times is an advantage and at other times a disadvantage. This type of circuit will oscillate under control of the crystal with much less active crystals than most of the other types. Nicolson's, Pierce's, Cady's of Fig. 12.2 and Miller's oscillators belong to the second named class. They will cease oscillating if the crystal breaks, develops high resistance or is disconnected. Failure of the oscillator to function at all then serves as a warning that something has happened to the crystal. This second named class of crystal oscillators has been used much more than the first named. The crystal is the principal frequency determining element in the circuit. Often there are required only resistances, or re- sistances and an inductance, as the other elements to embody along with the vacuum tube and crystal. The simplicity, low costs, and usually no tuning, have made this class attractive. Most analytical studies of oscilla- tor circuits have been made upon this class. For that reason the discussion in this chapter will be limited to this class. An analytic study of the crystal oscillator can readily start by looking 164 BELL SYSTEM TECHNICAL JOURNAL upon the oscillator as consisting only of inductances, capacitances, and resistances, along with the vacuum tube. The crystal is replaced by the proper circuital elements arranged as in Fig. 12.3. This circuit or equivalent of the crystal is that of a series resonant circuit having capacitance parallel- ing it. The circuit will show both phenomena of series resonance and parallel resonance, the two frequencies being very close together. By making suitable measurements on a crystal, the magnitudes of the in- ductance, resistance, and the two capacitances can be determined. It is usually found that the series inductance is computed as hundreds or thou- sands of henries, and the series capacitance is a small fraction of a micro- microfarad. The magnitudes of the inductance and capacitance are beyond what it is possible to construct in the usual forms of building inductances and capacitances. This accounts for its superior frequency control properties. Although reducing the crystal to an equivalent electrical circuit provides one notable step in understanding the performance of the crystal oscillator, it does not readily lead to a full understanding. The electric oscillator in itself is not fully and completely analyzed in all its ramifications, although it has been under study for over 25 years. These studies have been mathe- matical and experimental in character, but in all cases it appears there have been approximations of some kind, made because the variable impedance characteristics both of the plate circuit and the grid circuit of the tubes did not lend themselves readily to a rigorous analysis. The earlier investi- gations assumed a linear relation between grid voltage and plate current and assumed constant plate impedance. Later investigations brought in further elements and further variables, the different investigators attacking the problem in different ways and attempting to prove different points. By this means a large number of factors in oscillators have been ascertained to a first degree of approximation so that a qualitative review of the per- formance of the'electric oscillator is very well known. It is the quantitative view upon the first order magnitude which is still difficult or uncertain. This is particularly true of the crystal oscillator because of the slightly different circuit. It is proposed, therefore, in this paper to cover briefly a number of the studies on crystal oscillators so as to point out the different modes of attack and the different behavior points in the oscillators which the various investi- gators have studied. After covering these points, there will be discussed the frequency control properties of the crystal and the frequency stability of crystal oscillators. The performance of the crystal in the oscillator with respect to activity is then treated. There will be introduced two new yard- sticks for measuring or indicating crystal quality, one called "figure of merit" and the other called "performance index." These are related to PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 165 the crystal constants and paralleling capacitances which are usually involved. They will be defined and their method of use and application in oscillators will be pointed out, 12.10 Solution by Differential Equations The most direct method of determining the oscillating conditions in a circuit is to analyze the differential equation for the current in some particu- lar branch of the circuit. The relations existing between the coefficients determine whether the current builds up, dies out, or is maintained at a constant value and frequency. Unfortunately the equations resulting from the application of this method to the crystal oscillator circuit are quite complicated. However, lower order differential equations result from the r \ \^ J (0) (b) ^ ''c -> 3 1 1 L 'A = ^2. - ^ R, < > > ■ < < > > Jm^ L — *- 11 -^ 11 Fig. 12.4 — Equivalent circuit of oscillator with crystal connected between grid and plate application of this method to similar electric oscillator circuits, and certain qualitative information obtained from the latter is applicable to crystal oscillators. Thus Heising's analysis of the Colpitts and Hartley circuits gives much information directly applicable to the Pierce and Miller types of crystal oscillators. From this the circuit conditions necessary for oscilla- tions to exist and the effect of certain circuit variables upon the frequency are ascertained. The more complex qualitative view is given by Terry^ who shows the relations of the coefficients of linear differential equations of the 2nd, 3rd, and 4th orders, and applies them to the analysis of three common types of crystal oscillator circuits. The resulting equations, together with certain qualitative information regarding their interpretation, are repeated here. In making this analysis the grid current is disregarded and the static tube characteristic is considered linear. 166 BELL SYSTEM TECHNICAL JOURNAL The equation is the same for the three types of circuits considered and is derived for the current ii , in Figs. 12.4 and 12.5, although it may be set up in terms of any of the currents or voltages existing in the circuit. It is of the form d ii d ii d ii dii . /n i\ The P coefficients are functions of the circuit elements and are defined for each type of circuit in the following sections. The solution of (12.1) normally represents a doubly periodic function arising from the two coupled antiresonant meshes (a) and (b). The normal Fig. 12.5 — Equivalent circuit of oscillator with crystal connected between grid and cathode modes of oscillation consist of two currents in each mesh with frequency and damping factors fi\ and ai , ^2 and a2 respectively. The conditions for undamped oscillation as derived from the general equation (12.1) are expressed in terms of the coefficients by Pg ^ Pa ± VpI - 4P4 Pi 2 (12.2) and the angular frequencies are ^ _ P2 ± Vpl - 4P4 (12.3) where the plus sign gives the condition for one damping factor to be zero and the minus sign that for the other to be zero. PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 167 The frequency at which oscillations are maintained is determined by the required phase relation of voltages applied to the tube. With crystal from grid to plate, as in Fig. 12.4, the phase difiference of grid and plate voltages is such that the circuit oscillates at only one of the normal modes, and with crystal connected between grid and cathode, as in Fig. 12.5, it oscillates at the other only. 12.11 Crystal Between Grid and Plate With the crystal connected between the grid and plate of the tube, as in Fig. 12.4, the coefficients of the general equation (12.1) are _ ^1 ^2 Ji_ Li Lo Rp Cb 1 Pz = Pi = L\Ca + ?4' + L\ Li LiCb R ?p \Li Li) Cb Li Lo Ca L Li LiCaCb L1L2G; Ri J_ / 1 R1R2 _ 1 \ 1 Lo Cb Rp \Li Ca Cb L\ Li Cb Li Cm Cm/ R2 / 1 _ 1 \ Rp \L1L2CaCb LiLiCmCmJ (12.4) where Ca Cb 1^ Cb Ci Co cl c. C2 1 cl fJiCx Cb C2 Cs v-'W c' C2C0 Cm Co Ci _ 1 11 Co C2 C3 H = the amplification factor of the tube, dCp Rp = TT- {Cg constant) dtp The uncoupled damping factors, «„ and ab , the uncoupled undamped angular frequencies, ^a and ^b , and the coupling coefficient r may be intro- duced as follows: OCb = 2Li' R^ 2L2' 0l = i = 1 L\Ca 1 LiCb 2 CaCb T = c 168 BELL SYSTEM TECHNICAL JOURNAL Note that Ca is the total capacitance across Li and Ri , and Cb is the total capacitance across L2 and i^o- The coefficients of (12.4) become Pi = 2(aa + aft) + 1 RvC p^b 1 P2 = 0a -\- 4Q!a «& + /Sfc + " («« + ttfe) „/ i?. c P3 = P4 = 0. 1 r o 1 2(a6/3; + aa/^b) + — (iS; + 4aa«6) ^/ 1 J-'l^m^v (12.5) The coefficients as given by (12.5) satisfy (12.2) and (12.3) only when the plus sign is used. The equations are simplified by dividing through by /S^ thus (12.6) P3 tV(SJ- 4P4 ~ 0t ISlPi 2 /32 ^2 - iVSJ- 4P4 (12.7) which gives the ratio of driven frequency of the crystal to its undriven value. The common variable Rp must satisfy both (12.6) and (12.7). The method of computing the frequency would be to solve for Rp in (12.6) and substitute in (12.7). However, the equations are too complicated a function of Rp for this to be practical. Terry solved them graphically by plotting (12.6) and (12.7) as functions of Rp for assigned values of the circuit, and the inter- section of these curves gave the frequency for the different circuit conditions. The results are shown in Fig. 12.6. The G-P curves show the frequency change as a function of plate circuit tuning for the grid to plate connection of the crystal. 12.12 Crystal Between Grid and Cathode With the crystal connected between the grid and cathode of the tube, the circuit is as shown in Fig. 12.5. The coefficients of equation (12.1) are as follows: PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 169 1 R\ R2 L\ Li2 Kp C 6 1 R1R2 Pi = + -^— + " jLjCo Li\ E^, E (12.8) \Ch Rp \Li Li) Cb LiLiCa LiEoCb Rp\LiCaCb LiLiCb LiCmCm/ P = ^ - ^ 4- — ( ^ - ^ !) Li L2 Ca Cb Ll L2 Cm Rp \Ll L2 Ca Cb L\ L2 Cm. Cm) _ With the substitution of uncoupled frequencies, damping factors and coupling coefficient as described in the previous section, they become Pi = 2(«a + a6) + RpCb 1 P2 = iSa + 4Q!a Oib + ^'b + '^ (tta + Oib) '^> R. Cb If" N 1 1 1 P3 = 2(a6/3; + oia^b) + "^ (/3a + ^aaab) p/ — ^———r, J\p |_ Cb EiCmCm^ p, = ^:^i\ 1 - / + fl - r^ + — (^f - ^t r^\\ |_ Rp \Cb Cm /J (12.9) Where 1 c" = 1 + Cu 1 Cb' - 1 Cb + 1 c X Cm Co Co Cb = Co + C0C3 C0+C3 Ca = Co + C0C3 )u = amplification factor of tube de Rp = —^ (Cg constant) Otv Cx Co C2 C3 C2 + C3 These equations of conditions for oscillation in this case satisfy (12.4) and (12.5) only when the minus sign is used. That is Ps ^ P2 - Vpi - 4P4 Pi 2 /3- = P2 - VpI - 4P4 (12.10) (12.11) 170 BELL SYSTEM TECHNICAL JOURNAL Again dividing by jSa to obtain the frequency as a ratio of driven to undriven crystal frequency, we have /3a Pi &l ^l f. - /(SJ 4P4 m 4P4 (12.12) (12.13) G-P / J > 0 Z y- u Z 3 < 0 1- u 1.00008 d a. w UJ _i a. < 0 1- !i > cc 1.00004 "^ 0 p f5a_ PLATE CIRCUIT TO Ptj CRYSTAL FREQUENCIES | .94 .96 MEASURED FREQUENCY COMPUTED DRIVEN FREQUENCY 1.06 Fig. 12.6 — The oscillating frequency as a function of the plate circuit frequency for the crystal connected grid to plate (G-P) and grid to cathode (G-C) The frequency change as a function of plate circuit tuning was determined graphically in the manner described in section (12.11) and the curves are shown in Fig. 12.6 as the G-C curves. 12.13 Resistance Load Circuit This is a special case of Plate-Grid connection of the crystal described in section (12.11) in which the plate circuit consists of a capacitance and re- sistance in parallel. This is a very common Pierce type of oscillator circuit and has the advantage that no tuning adjustment is necessary when using crystals of different frequencies. PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 171 Since this circuit is singly periodic, the differential equation for ii is of the third order and is derived from (12.1) by setting the plate inductance 1,2 of the P coefficients equal to zero. The general equation then becomes §■ + "'^ + "^t + ^-•' = 0 (12.14) where 1 _ ^ + RoCh 1 LiCa + Ri Ri P.-. = 1 i?2 Ll Ca Cb i?2 Ll Cb Rp Ll Cb 1 i?2 Ll Cm R (12.15) With the substitution of the uncoupled damping factors and frequencies, (12.15) becomes Pi = 2aa + 1 P2 = /3a + P3 = RiCb lag RiCh 4- + 1 RpCb 2aa R2 Cb R2 L The frequency as obtained from (12.14) is 0' = P2 with the conditions for oscillation RpCb (12.16) (12.17) ^-f: (12.18) obtained by setting the damping factor a equal to zero. The ratio of driven to undriven frequency is obtained by dividing (12.17) and (12,18) by /Sq. That is /Q2 p. p. (12.19) /3a /3' iSaPl 12.14 Interpretation of the Equations It is learned from this analysis that the frequency of oscillation while governed principally by the frequency of the crystal also depends upon all the constants of the circuit. The effect of the plate circuit impedance is 172 BELL SYSTEM TECHNICAL JOURNAL 30x10 20- Q. a: R2= 2. 1 OHMS Rg= 1.25 MEGOHMS R2= 2.1 OHMS Rg=.22 MEGOHM R2 = '2 OHMS Rg = 1.25 MEGOHMS C2 Fig. 12.7 — Calculated increase in mean plate resistance against capacitance of the oscillatory circuit Fig. 12.8 — Experimental curves, showing the influence of interelectrode capacitances on the frequency shown in Fig. 12.6. It is pointed out that the effect of the crystal resistance Ri is to decrease the frequency for the G-C connection and increase the frequency for the G-P connection. The discrepancy between the measured PIEZOELECTRIC CRYSTALS IX OSCILLATOR CIRCUITS 173 and experimental values shown on the curves is attributed to the difference between chosen and actual value of Ri . The effect of the input loss of the tube is not shown because the grid current was disregarded; however, this loss may be reduced to an equivalent Ri . The resistance of the plate cir- cuit i?2 affects the frequency in a similar manner. The effects of these resistances on frequency are less for low values of plate circuit impedances. The required value of Rp gives a measure of amplitude of oscillation because it is necessary for oscillations to build up until the internal plate resistance is equal to the calculated value. It is found that Rp increases Fig. 12.9 — Experimental curves, showing the relation between the frequency and the resistance of the oscillatory circuit gradually to a maximum as tlie common frequency for the two types of circuits is approached then abruptly drops. Vigoureux analyzes the crystal oscillator in a manner similar to Terry and correlates his interpretations of the equations with considerable experimental data, some of which are shown in Figs. 12.7, 12.8, 12.9 and 12.10. He points out that there is an optimum value of grid capacitance with the cr>'stal connected between grid and plate and a certain amount of grid-plate capacitance is required when the crystal is connected between grid and cathode. Wheeler does not assume a linear static tube characteristic but represents it by a three-term nonlinear expression. The results are more complex and it is necessary in the end to disregard certain resistance terms. 174 BELL SYSTEM TECHNICAL JOURNAL 40 fo G-C 9 2.8- UlCp = 0OL2 C2 ■ Fig. 12.10 — Experimental curves, showing the relation between the frequency of a quartz oscillator and the capacitance of the oscillatory circuit for various values of the grid leak 12.20 Solution by Complex Functions The analysis of oscillator circuits may be simplified when only steady state conditions are of interest, all circuit elements are considered linear, and certain requirements which define the conditions necessary for oscilla- tions are known. Under these conditions the common circuit equations of complex numbers give the information desired. In this method the voltage induced in the plate circuit is considered the driving voltage which produces a current in the grid circuit (see Fig. 12.11). The network be- pvg Fig. 12.11 — Equivalent circuit of Pierce and Miller types of oscillators shown in Fig. 12.12 PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 175 tween plate and grid may be of any type and oscillations are maintained when the total gain through the circuit is unity (gain of tubes = attenuation through circuit) and the phase relation between the induced plate voltage (nVg) and the grid voltage (Vg) is 180° (the phase shift is zero when fi is considered negative). The expression jujS = 1 defines these requirements. Llewellyn applies this method to oscillator circuits in general and Koga^^ uses it to study the crystal oscillator in particular. The equations are developed on the assumption that the grid-voltage vs. plate-current characteristic of the tube is linear. The fundamental equa- tion of fjLjS is given by the ratio of the voltage developed across the grid Fig. 12.12 — Circuit diagrams of crystal oscillators with crystal connected from grid to cathode (A) and grid to plate (B) circuit by the fictitious driving voltage nVg to the voltage Vg. For the general circuit, Fig. 12.11, it is I2 Z2 — M^l ^2 ^^ Vg R^Z, + Zy{Z2 + Zz) where Za = Zi-\r Z2-\- Zz It is more convenient to write this in the reciprocal form 1 ^ RpZ, + Zi(Z2 + Zz) ^ ^ n^ —fiZiZz (12.20) (12.21) In applying this to the crystal oscillator, the additional assumptions made are that the grid current is negligible and the resistance in the plate im- pedance Zi is zero. 12.21 Crystal Grid to Cathode With the assumptions made above and the crystal connected from grid to cathode of the tube according to Fig. 12. 12 A, the impedances are Zl = jXl Zi = Reg + jXcg j^z 176 BELL SYSTEM TECHNICAL JOURNAL where Reg is the effective resistance and Xcg the effective reactance of the crystal, the grid resistance Rg and the circuit capacitance Cg in parallel at the oscillating frequency. Upon substitution of these in (12.21) i_ ^ [RccRp - X.jXcg + Xs)] +j(XrRcg + RpXs) ^ J M/3 IJiXlXcg — jllXlRcg (12.22) where 1 Thus — is of the form M(3 Ns — X]_ -\- Xcg -\- Xz which means that P — 1 and () = 0. This results in the following two equations obtained from the real and imaginary parts of (12.22) both of which must be satisfied for oscillations to be maintained. The real part of (12.22) gives Xi(m + 1) (Reg + Xcg) + XcgXz ,.^ ^ s and from the imaginary part is obtained Rp. Xs = ^^^ ^^-^ (12 24) where ^c^ = — ^ (This ratio of reactance to resistance of the crystal circuit Reg will appear in various equations later.) Equation (12.24) may be said to define the oscillating frequency and is in a convenient form to examine the effect of the various circuit variables upon the frequency. The impedances -Yi , Reg , X,g and A'3 may be thought of as forming an oscillating loop (See Fig. 12.11). For oxcillations to be main- tained in such a loop the sum of the reactances must equal zero and the sum of the resistances must equal zero. But the sum of the resistances cannot equal zero since Reg is the only resistance in the loop and it is posi- tive. It is therefore necessary for the driving voltage fxVg to act upon the circuit and supply the energy dissipated by the resistance Reg (and also R^ through which the energy is supplied). This alters the frequency some- what and it is no longer determined by setting the three reactances equal to zero as may be seen by equation (12.24). Nevertheless, the right side of this equation is small and approaches zero when Reg approaches zero. It also becomes very small when the reactance Xi becomes small and Reg is not too great. This is the same condition as found by the differential PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 177 equation method and illustrated in Fig. 12.6 by the G-C curves. As the plate reactance A'l is made small the frequency increases and approaches a limiting value but does not quite reach it. This limiting value is the fre- quency at which A's = 0. The dotted G-C curve shows that Reg tends to lower the frequency and determines how close the limiting frequency is approached. The plate circuit resistance R2 (component of Zi), if con- sidered, would have a similar effect as shown by the experimental curves 12.9. The grid resistance Rg (component of Z2) has an opposite effect as shown in Figure 12.10 because increasing Rg is equivalent to decreasing the efTective resistance Reg . The effect of the various constants of the crystal and circuit upon the oscillating frequency may be obtained from (12.24) upon substitution of these constants for the reactances and resistance Reg . The equation is put in a more convenient form for this purpose by Koga.^^ Equation (12.21) is written, ^'-''^'^ 23(1 + rJ, + j^ ' ° ('"« It is assumed that the current in the grid branch is small compared to the plate current. This reduces the equation to ^ + F3 + Z3(i t^izi) = " C^-^o) The admittance expression for the crystal is ^1 - 3 1 1 uiLi — ^ coCi CO (Co + C\) \ / C4 Y . C0C4 RX + coLi - ^ _ ^ T V<^« + ^V ■^"'Co + C4 ;Ci CO (Co + C4)J (12.27) Note that Koga considers the air gap capacitance C4 as a separate factor but it may be included in the other constants of the crystal in which case the equivalent circuit is as shown in Fig. 12.3. With the crystal con- nected between grid and cathode the various circuit admittances are: 1 1 1 1 J_ Z2 Z,c JK-g 1 77 = ycoCa 178 BELL SYSTEM TECHNICAL JOURNAL After substitution of these values of the admittances in (12.26) and setting the real and imaginary parts equal to zero, the following two equations are obtained: Ri [_ coCi w(Co + Ci)^ \Co + Ca/ Rg R. — fxuC-i 1 + R (12.28) = 0 and jLi — 1 1 jCi CO (Co + 2 r _ A _ 1 1 1_ coCi a)(Co + C4J R C4) /Cn + C4Y c„ + (Co + C4 C0C4 (12.29) C0 + C4 + C3 + mG 1 + i?', Ci - ""'J. Equation (12.28) gives the conditions necessary for oscillations and (12.29) gives the oscillating frequency as explained below: 12.22 Frequency of Oscillations for G-C Connection of Crystal Equation (12.29) for frequency is simplified by the fact that over the narrow frequency range considered, the reactances of L2 and C2 do not change appreciably. Also at the oscillating frequency, - r - — - ^ T [_ ^ wCi aj(Co + C4)J With these approximations (12.29) may be written 2 1 + LiCi LiC[ 1 Co + Cj . Co Ct C4J (12.30) where C, = C, + C3 + mCs 1 -{-Rl(^ - C00C2) \CO0 i>2 / and coo is a constant approximating the oscillating frequency. PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 179 Since the frequency is a function of the internal plate resistance of the tube {Rp) and this is in turn a function of the other circuit variables, the frequency equation (12.30) is not sufficient to calculate the frequency. However, qualitative effects of the various circuit components upon fre- quency are obtained by assuming Rp an independent variable. It is readily seen that an increase in Rp increases the frequency. The effect of the air gap between crystal and electrodes, which is represented by the capacitance Ci , and the effect of the capacitance across the crystal Cg are illustrated in Fig. (12.13).* To determine the frequency change caused by tuning of O 80 CRYSTAL i^Q = r34 KC I8.3X I8.3X3.3& MM fQ= 865 KC AIR GAP=32 0.2 0.3 0.4 AIR GAP IN MM ^ 0 10 20 IN ^^^^f Fig. 12.13 — Experimental curves, showing the effect of crystal air gap and grid capacitance on the frequency of oscillations the plate circuit (variations of C2) requires the calculation of the change of the variable part of C< . This quantity is mCs (12.31) 1 +R\(^ - CO0C2) \coo Li / The plot of Cv is shown in Fig. (12.14A). The frequency decrease is pro- portional to the increase in C„. This is indicated in Fig. (12.14B). Oscil- lations stop before the point uaCi = — T is reached. The frequency thus CO0-L2 varies in the same manner as shown in Fig. (12.6) but the curve is reversed because of the fact that the independent variable is taken as C2 instead of the frequency function of C2 . The frequency change resulting from variations in the grid-plate capaci- tance C3 depends also upon the value of C^ as seen from (12.31). It is also * See also: "The Piezoelectric Resonator and the Effect of Electrode Spacing upon Fre- quency," Walter G. Cady, Physics, Vol. 7, July 1936. 180 BELL SYSTEM TECHNICAL JOURNAL seen that the smaller the value of Ci (lower the plate reactance) the less effect will the tube constants /x , Rp and C3 have upon the frequency. The circuit is therefore more stable. For this reason it has become customary to measure the frequency of crystals with the capacitance C2 reduced to a value below that which gives maximum amplitude of oscillations. 12.23 Amplitude of Oscillations for G-C Connection of Crystal A measure of the amplitude of oscillations is obtained from (12.28) which expresses the necessary conditions for oscillations to be maintained. In order for oscillations to start the expression must be negative, and, as the amplitude builds up, Rp increases which reduces the negative terms C2— ► C2— ^ A B Fig. 12.14 — The variation of grid to cathode capacitance (A) and oscillator frequency (B) with change in plate circuit capacitance. Crystal connected grid to cathode until the equality is satisfied. The difference between the positive and negative terms is therefore a measure of the amplitude of oscillations. Equation (12.28) may be written b + ^] ^ -1 $n + — I = ^ where yl is a measure of the amplitude, Rr (12.32) \p — mCsWo \ r ~" '»-'0 L'2 I \CO0 L2 / 1 + R\(-^ - W0C2) (12.33) and ^, = R 2 /Co + C4Y Co C4 Co + C4 + C„ + C3 + liCz 1 + Kp\ — CO0C2 I \coo Li / _ (12.34) PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 181 where again Rl is assumed small compared to 1 cooLi — OJoCl COo(Co I T + C.)J and Wo is considered a constant. Equation (12.32) shows that in order to obtain a large amplitude yp should be large and $o should be small. With this in mind equations (12.33) pMC-)oC3 Fig. 12.15 — Functions from which the activity variations (A) are determined as the plate circuit capacitance is varied. Crystal connected grid to cathode and (12.34) may be analyzed to determine the relation between the circuit components and amplitude. It is found that for maximum amplitude Cg and Ri should be small, C4 should be large, Cz has an optimum value, and Rg should be large. As to the plate circuit, the amplitude is maximum when R. 1 oooL'i C00C2 . A plot of ^ and $0 + — is shown in Fig. 12.15. The difference between these two curves is a measure of the amplitude and is shown by curve ,1 . Oscillations can exist only where \}/ lies over $0 + "^ • The sharp- Rg ness of \p varies considerably with the value of Rp and the resistance of the L2 — Co circuit. The latter is disregarded for simpHcity. Here again the 182 BELL SYSTEM TECHNICAL JOURNAL results can only be considered a first approximation, but agree with actual conditions sufficiently to be of considerable interest. 12.24 Crystal Grid to Plate The equation (12.20) is general and for the condition of crystal connected between grid and plate of the tube (See Figure 12.12B) Zz represents the crystal impedance which will be called Zc = Re -\- jXc , also : Zi = jXi , Zi = JX2 and Xs = Xi + X2 + Xc . Note that Rg and C3 are disregarded in this case because their eflfects are similar to those determined for the foregoing case of crystal connected grid to cathode. After substitution of these values in (12.20) the real part is found to be R, = (m+ 1)^1^2 + XiZe ^j2.35) and the imaginary part is X. = -^' (12.36) which shows the effect of the various variables on the frequency. The right hand side of equation (12.36) is comparatively small and the frequency is therefore close to a value /o which makes X, = 0. In this case the frequency is above the limiting frequency /o because the right hand side is positive since Xi is negative, whereas it was found that the frequency was below /o for the crystal connected between grid and cathode. As Re and Xi are increased the frequency will increase and as Rp increases the frequency decreases. These interpretations are verified by the G-P curves of Figures 12.6, 12.9 and 12.10. The effects of the various circuit and crystal constants are determined by Koga^^ by writing the general n^ equation as ^' + ^' + rfhz, = " ('2-3^) After substitution for*the Z's, the real and imaginary parts are respectively, \coCo/ Ri _^ _M^ iCg , r 1 1 T CO i?? + coLi - — - — |_ coLi coCoJ _ . (12.38) Rv[ ^ — WC2 ^ = 0 Ha,i:,~'^^V 1 + i?' PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 183 and J_ J_ ^ + ^ — coLi coCi coCo (12.39) a^C( -*^o "1 1 1 M - + ^ + — + Co C4 C(7 Co CL - "^v . 1 + i?p ( — wC2 .CO/.2 Fig. 12.16 — Frequency and activity change for variations in the plate circuit capacitance. Crystal connected grid to plate There are two values of co which satisfy (12.39) but only one of these co„ will satisfy (12.38). At this value of co„ 12 r _ — _ — T |_ coCi coCoJ Rt « coLi - (12.40) By introduction of this and the assumption that coo is essentially constant, (12.38) may be written RiCo '1 1 1 M ~ + — + ^ + Co C4 C(7 C(7 + 1 + R\ ( - — coo C2 ) \COoi>2 / _ jT — COo C2 J ajoi>2 / Rr (12.41) COoCg 1 + i?;("7 - C00C2) \aJo L2 / = 0 184 BELL SYSTEM TECHNICAL JOURNAL This is an approximation for the conditions for oscillation and relative amplitude. The frequency equation (12.39) becomes where G =Cl U "111m ^ + ~ + ^ + • Co C4 C^ K^g u \cx "^ Co ~ gJ (12.42) 1 + i?p I — coo C2 J \CO0 iv2 / and coq is a fixed value written in place of co„ . Figure 12.16 shows the fre- quency and amplitude changes as a function of C2 for the crystal connected between grid and plate. Fig. 12.17 — Generalized oscillator circuit in the form of a filter network 12.25 Condition miS = 1 for Circuits in General It is convenient to apply the rule m/^ = 1 as the condition for sustained oscillations to more complex oscillator circuits. The circuits may be drawn as shown in Figure 12.17 and the characteristics of the filter network between transmitting and receiving end may be analyzed by conventional filter theory to determine the conditions which fulfill the oscillation requirements. An example of this is the oscillator shown in Figure 12.18A. The equiva- lent configuration, Figure 12.18B, indicates that the crystal is part of a low pass filter and the frequency of operation is that at which the total phase shift is 180°. Oscillators involving more than one tube may also be inspected in this manner. Figure 12.19 is a two tube oscillator designed to operate at a frequency close to the resonant frequency of the crystal. The proper phase shift is obtained by a two-stage amplifier and, therefore, no phase shift is required through the crystal network. The crystal thus must operate as a resistance which it can only do at its resonant or antiresonant frequency. Since the transmission through the crystal branch is very low at the anti- resonant frequency of the crystal, it will oscillate only at the resonant PIEZOELFX'TRIC CRYSTALS IN OSCILLATOR CIRCUITS IcSS frequency. Heegner explains a number of crystal oscillator circuits by the method briefly outlined above. L2 I ^ I )MV9 I I I P -r^^ -r^9 Fig. 12.18 — The oscillator circuit (A) is equivalent to the filter circuit (B) Fig. 12.19 — Oscillator circuit in which the crystal operates at its series resonant frequency 12.30 Vector Method or Oscillator Analysis A convenient method of examining the effect of certain circuit variables on frequency and the necessary conditions for oscillation is by the vector representation of the voltages and currents in the circuit. Much of Heis- ing's^'* early work on the analysis of electric oscillators by vector methods is directly applicable to crystal oscillators. Boella analyzed the crystal oscillator circuit by this method and treated in detail the effect of the decrement of the crystal on the oscillating frequency. Since some engineers prefer this method of qualitative analysis to approximate equations it will be briefly explained. The vector diagrams for the two conditions, crystal between grid and plate and between grid and cathode, are shown in Figure 12.20A and B as applied to the circuit diagrams, Figure 12.12A and B, respectively when in the simplified form of Figure 12.11. The necessary conditions for oscilla- 186 BELL SYSTEM TECHNICAL JOURNAL tions are that Vg is in phase with and equal to y.Vg (note that /x is considered negative). Like Koga, Boella assumes the current 1 2 small compared to /i , hence the voltage drop across Zi is approximately Zi/i . The angle this makes with Vg is determined by the value of Zi and the internal plate impedance Rp . Any change in either of these requires a change in the angles \p and yp' in order that Vg shall be in phase with nVg . This means that the frequency must vary to produce this change in \j/ and 1/''. Because of the rapid change in the reactance and resistance of the crystal with frequency, these requirements are met with very little change in frequency, which accounts for the high degree of frequency stability obtained with crystals. This is described more in detail in a later section. G-C G-P -Z.I Z.I, -Z1I1 Z,Ii Fig. 12.20 — Vector diagrams of currents and voltages in the oscillator circuit Figure 12.11 with crystal connected grid to cathode (A) and grid to plate (B) 12.31 Change in Frequency with Decrement of Crystal It has been found that for the crystal connected from grid to cathode there is a maximum theoretical frequency at which the circuit can be made to oscillate by reducing the plate circuit impedance. This also corresponds to the minimum frequency which can be obtained with the crystal connected between grid and plate. This was called the limiting frequency /o . It is interesting to note that/o is determined by the intersection of the reactance curve of the crystal plotted as a function of frequency and the reactance curve of the capacitance in series with the crystal. This series capacitance is the grid-plate capacitance for one case and the grid-cathode capacitance for the other. As illustrated in the curves Figure 12.21, the limiting fre- quency/o increases as the decrement of the crystal increases. The difference between the true frequency of oscillations and/o increases PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 187 as the plate impedance is increased and as the losses in any of the circuit elements increase. This is necessary for the proper angle oi\p -\- ^p' in the vector diagram. With the G-P connections, the departure from /o and change in/o as the decrement of the quartz varies are in the same direction, while for the grid-cathode connection they vary in opposite directions, and the net result will depend upon the value of the internal plate resistance and plate circuit impedance. The curves of Figure 12.21 show that the Fig. 12.21 — The change in reactance characteristic of a crystal resulting from a change in decrement change in fa for a given change in decrement is less for smaller values of "77 (larger values of series capacitance C3). That is, the effect of the de- C0C3 crement of the crystal upon the oscillating frequency is small when the crys- tal is operated near its frequency of resonance. 12.40 Negative Resistance Method of Analysis The methods of analyzing oscillator circuits described in the previous sections define the operation in terms of the individual circuit elements and the crystal is treated as one of the circuit elements. Certain advantages result, however, by grouping all the circuit elements, except the crystal. 188 BELL SYSTEM TECHNICAL JOURNAL into a single impedance as shown in Fig. 12.22A. Here Zt represents the impedance looking into the oscillator from the crystal terminals. The requirements for sustained oscillations are that the sum of the reactances around the loop equal zero and the sum of thfe resistances equal zero as previously stated in section 12.21. These conditions are obtained when Zt is a negative resistance p in parallel with (or in series with) a capacitance Ct as shown in Fig. 12.22C. The crystal is considered to be operating as an inductance Lc and resistance Re as determined in the pre- A B C Fig. 12.22 — Equivalent fepresentations of crystal and oscillator circuit vious sections. The frequency equation has been derived by Reich from the differential equation for the current in the loop. It is /' + Ro 1 P LcCt and the condition for oscillation is shown to be \Ctp\ S (12.43) (12.44) We shall consider the crystal connected between the grid and cathode of the tube, in which case Zt is the input impedance of the vacuum tube. 1 17 ... . The expression for ^ was developed by Chaffee from which it is possible Zt to determine the circuit conditions necessary for the input resistance and reactance to be negative. The effect of the circuit variables upon the abso- lute values of p and Ci determines their effect upon the frequency and activity according to equations (12.43) and (12.44). 12.41 Input Admittance or the Vacuum Tube With the assumption that the grid current is negligible and the static tube capacitances Cp and Cg are part of the external circuit, Chaffee's equation for input conductance becomes Clo,{K + Gi) + Czi^ixKiCzoi - Bi) {K + G^Y + (C3C0 - B^y (12.45) PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 189 and for the input susceptance , CzuiixK{K + Gi) — Qw^CCsco — 5i) * = -^'" " (if + o' + (c. - B,y — ('2.46) where A' and jx are defined as follows: A' = — - I (Cy constant) — ( —^ J («p constant) and Gi and T^i are the conductance and susceptance of the plate circuit. If we let h = A = — and B= - (12.45) becomes (1 + ///>) + g = C3C0-I <^ - 3 (1 + hBf + and (12.46) becomes h = — C3C0 m(1 + /'^) 1 + - -'<^ - 1) (12.47) (1 + hBY + A (■ - -?) J (12.48) When the resistance of the plate circuit is neglected (i.e. h = 0), and /x » 1 we may write m(^ - B) K ^ \ -\- {A - BY (12.49) and A [m - iJ(.4 - ij)1 These equations are in a convenient form to determine the effect of the plate tuning J{B) and grid-plate capacitance f{A ) upon the resistance p 190 BELL SYSTEM TECHNICAL JOURNAL and capacitance C< with the assumptions of no grid current, low plate circuit resistance, and ju ^ 1. From (12.49) it is seen that in order for g to be negative, B must be positive and greater than A , since A is normally positive. That is, the plate circuit reactance must be positive and less than the grid-plate reactance when the latter is a capacitance. Under these conditions b/K and hence the input reactance will be negative according to (12.50). Curves of b/K are shown in Fig. 12.23 with B as independent variable and A as parameter. These curves indicate frequency change. On the |J=20 h = o Fig. 12.23 — Variations in the input impedance functions of an oscillator circuit for changes in plate circuit tuning same figure is plotted b/g called (i>g . This may be considered the sensitivity of the oscillator or, for a given value of uLjRc of the crystal, it represents the activity. The similarity between these curves and the actual change in frequency and activity normally experienced is apparent. It should be pointed out here that the presence of harmonics is effective in changing the input impedance of the vacuum tube and hence the fre- quency and activity of the oscillator. The presence of harmonics results from the non-linear characteristics of the vacuum tube. Llewellyn explains that a non-linear resistance may be represented by a linear re- sistance plus a linear reactance. From what has been said concerning the PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 191 frequency of the oscillating loop, it is apparent that this effective reactance will alter the frequency. However, this reactance is small when the im- pedance of the circuit is low at the harmonic frequencies and is zero when the external circuit is a pure resistance. 12.50 Efficiency and Power Output of Oscillators In many applications of erystal oscillators the efficiency and power output are important factors. These are not treated here but reference is made to the work of Heising which covers this aspect for various electric oscillator circuits. Much of the analysis is directly applicable to crystal oscillators. 12.60 Frequency Stability of Crystal Oscillators The equations for frequency show that the frequency is governed some- what by the amplification factor, the grid resistance and internal plate resistance of the vacuum tube. Since these factors are functions of voltages applied to the tube and amplitude of oscillation, they cannot be considered fixed. If the frequency change resulting from these variables is great, the frequency stability is said to be low, and if very little frequency change takes place the frequency is determined principally by the circuit constants and the frequency stability is said to be high. Llewellyn^ shows how it is possible to compensate for the change in plate resistance by the proper value of circuit elements. This was done by deter- mining the relations necessary for Rp to be eliminated from the frequency equation. It is sometimes helpful in designing very stable oscillators for frequency standards to select circuit elements which will reduce the effect of plate voltage changes on the frequency. It is more the purpose of this section, however, to show Llewellyn's derivation of the equations for fre- quency stability which have not heretofore been published and from them point out the characteristic of crystals which enable them to stabilize oscillators. 12.61 The Frequency Stability Equation The steady state oscillating condition is Mj8 = 1 (12.51) In general /3 is a function of the frequency, the amplitude of oscillations, and of some independent variable V. This independent variable is the one for which it is desired to stabilize the frequency. It may be the potential applied to the tube, or it may be a capacitance located somewhere in the circuit. (3 depends upon these three variables thus: M/3 =/(/>, a, F) (12.52) 192 BELL SYSTEM TECHNICAL JOURNAL Instead of the frequency, a more general symbol p is used and may be thought of as the differential operator d/dt which occurs in the fundamental linear differential equations taken as describing the oscillatory system. That is d , . P = -■ = a -\- iw dl (12.53) (12.54) (12.55) The function ju^ may have the form The result of taking a general variation b of (12.54) is then ^ + m = 0 A Since (12.54) is a function of the three variables p, a, and V the variational equation (12.55) may be expressed in terms of partial derivatives with respect to these three variables. That is dd „^ , „„ , hV \ -\- 1. \ ~ ()f} -+- ^ da -\- A\_dp "^ da dV dp ^ da dV -- 0 (12.56) The solution of (12.56) for the variation in p is bp =^ - 57 4- ^ 5c A\dV ^ da dv'' ^da ^ -h i dd (12.57) .1 Yp ■ dp It is a property of functions of complex variables that, provided they possess derivatives at all, then the value of the derivative is the same regard- less of the direction in which the limiting point is approached. This fact is expressed by dA dp dd _ dd _ . dd (12.58) dp ^ ^ and bp = ba -{- iSco and provides means by which the real and imaginary parts of (12.57) may be separated to yield the two equations dA da .dA doi dd da . dd dcio ba = \_A da \dV i'a j d(^\A dV , 1 5-1 . A da v. 4 6co/ \5co/ (12.59) PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 193 and — \ I 8V -\- 8a] -{- — I — -8V + — 8a] \ lAdc^\AdV ^ A da J^do:\dV ^ da /J ,,..„, 8(jO = ^ ^ 7 r-^ ^7 r-s-^ '— (12.60) /I dAV /doY \A dco) \do:/ The variable p in general may be written as the sum of a and iw. With the remembrance that p is the differential operator d/dt and that a set of linear equations expresses the transient condition, it is evident that the current will have the form /e^' which is equivalent to le" ^" . Inspection of this shows that the real part of p, namely a, determines whether the cur- rents in the system are going to build up with time, or die away with time, or remain constant, depending respectively upon whether a is greater than zero, is less than zero, or is actually equal to zero. With this in mind we see that (12.59) and (12.60) state the change in a and co respectively which would result from some change in the circuit condition. Initially the circuit was oscillating in a steady manner so that a was zero and co had some particular value. A change in V then occurred. This produced a change in the amplitude accompanied by a change in the frequency as expressed by (12.60) and a change in the transient term a. Suppose now that the change in V w'ere very small. Then in order for oscillations again to assume a steady value it is necessary for the amplitude "a" to change a sufficient amount to cause a to become zero. Thus in (12.59) we put 8a equal to zero and solve for the required amplitude change. This may then be elimi- nated from (12.60) resulting in the final expression 1 dA dd _ 1 dA dd 80. = -4aFa^ AJ^W ^^ ^2 ^j) ^dAdd_ 1 ^ cl0 A doj da A da dco which gives the frequency change Sco in terms of the change of the inde- pendent variable 8V. 12.62 Frequency Stability of Conventional Oscillator In applying this equation to the oscillator circuit, Fig. 12.24, we must first set up the conditions for oscillations. The ijl(3 equation is IjlXi X2 Rg ^^ ^ aXsRpRo - X1X2X3] - [RpX^iXi + X3) + RgXiiX. + Xs)] (12.62) The oscillating conditions /x/3 = 1 requires XgRpRg = X1A2X3 194 and BELL SYSTEM TECHNICAL JOURNAL fjiXiX^Rg + RpX2{X, + X3) + RgX,{X2 + Xs) = 0 (12.63) It will be assumed that the following relations exist: M = MV), Rg = /2(a), Xs = Xi + X2 + X3 = Mco), i?p = a constant Fig. 12.24 — Equivalent oscillator circuit analyzed for frequency stability Then we obtain from (12.63) AdV ~ fjidV IdA ^ J_dRg[ A da Rg da [_ i M = i_ ^1 r A do: Xi dw |_ dV 1 + 1 + + 1 de _ _\ BRg XoRt J da ~ ] ] X2 -\- Xs _ _ 11X2 J da Rg da nXiXz i?pX2 + Rg(X2 + X3) nRpXi Rp{X2 + X,) +RgX iiRp X\ + T j_aX2r X2 dcjo [_ 1_ 6X3 rXs{RpX2+RgX{)l ^3 5w [_ llRp X1X2 J ■] (12.64) dd do} fiX iX2\_ Xi d(ji + (X3 - X) ^ '/'I A3 000 J X2 5co PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 195 By substitution of these values in (12.61) and disregard of Xg in comparison with all other A''s the equation for frequency stability is obtained as 1 ^M V V V 7 - TT> A1A2A3 ^" - ^^^ (12.65) dV From this we learn that the values of the reactances Xi , A'2 , and X3 should be small and the values of Rp and Rg large to give small changes in CO when V is varied. These variables are more or less limited, however, by the conditions necessary for sustained oscillations according to equation (12.63). It is important to notice that the denominator of (12.65) contains functions which do not appear in equation (12.63) and hence may be of any value. These factors are the rates of change of the various reactances with frequency. For given values of circuit constants, the equation shows that the frequency stability increases as these rates of change increase. 12.63 Frequency Stability Coefficient of Crystals The rate of change of the reactance of an element is referred to as the "frequency stability coefficient"* of the element. Expressed in per cent, we have for the frequency stability coefficient of a reactance F{X) =f.^ (12.66) dco X Let us now examine the frequency stability coefficient of a crystal which is used as the reactance .Y2 when connected between grid and cathode of the tube and as .Y3 when connected between grid and plate (See Fig. 12.24). The resistance of the crystal will be assumed to equal zero due to the negli- gible effect of the resistance variations upon the reactance for crystals with average Q and operated at a frequency not too near the anti-resonant frequency. (This may be observed in Fig. 12.21.) The reactance of the crystal then is X,^-i,%^, (12.67) where /.r' 2 2 WL-o CO — a;2 CO = 27r X frequency coi = 27r X resonant frequency coo = 27r X anti-resonant frequency First suggested by N. E. Sowers. 196 BELL SYSTEM TECHNICAL JOURNAL By substitution of the relations Cl C02 — COl 1 J ^ - = 2— and - -— = Xo Co cci '*^^o into (12.67) we obtained Xc = Xo 1 — -;:; ^ ^ L Co C02 — CO J and by differentiation Xq r Cl COl "1 V ^1 r *'i^" 1 ~77 M " r ^ 2 ~ ° r r^ v C'' L Co C02 — OJ J Co L(C02 — CO ) J (12.68) dX, do3 ,2 2.2 . (12.69) (C02 — Multiply by ~ to obtain the stability coefficient Xc L Co C02 — CO J Xc do Xc Xo Cl CO 2coi Ac Co COl {032 ~ CO j (12.70) and eliminate co by substituting in (12.70) the relations obtained from equation (12.68). These are 2 2 C02 h _ ( 1 _ ^ ] ^ - co^ V ^Yo/ Cl and -. = ^' + 1 COl ^0 Cl _^ X, Xo (12.71) Thus F{Xc) = -1 -2^" (1 - 9-' X. (■ - s)' 1 + ^^- Li 1 - X, o_ (12.72) which may be written Fix.) = -^ ^0 The stability coefficient of a coil and condenser F{Xy may be obtained from (12.73) by letting Ci = oo. Then ^(X)' = -I [.+(,- I)] (12.74) PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 197 The comparative stability of the crystal and tuned circuit is given by the ratio FOQ F{xy 1 + 2 (12.75) 10,000 r 100 Co C| Fig. 12.25 — The stability coefficient of a crystal as compared to a coil and condenser for variations of the ratio of capacitances X, Co This ratio is plotted in Fig. 12.25 for — = — 1 and with — the independent X, Ci Co function. It is apparent that the value of — of a crystal is the factor which determines its frequency stability for given operating conditions. For an A T crystal in an air gap holder, the ratio of capacitances is of the order of 10 and its stability coefficient is therefore 2.6 X 10^ greater than for a simple anti-resonant circuit. Since this is so much greater than the stability coefficients for the other reactances which appear in the denominator of equation (12.65) it represents the order of magnitude of improvement of the frequency stabiUty of an oscillator obtained by the use of a crystal. 198 BELL SYSTEM TECHNICAL JOURNAL The fact that the frequency stabiUty of a crystal oscillator is a function of — explains why a BT cut crystal is in general more stable than an ^7" Ci cut. The two may be made equal, however, by adding capacitance across the A T cut crystal. Actually we have compared the frequency stability obtained by the use of one type of circuit (the equivalent crystal circuit) with one of a different configuration obtained by making Ci — cc . In practice this is usually the case since Ci must be large to obtain oscillations when using coils and con- densers. The limiting factor is therefore the value of — at which oscilla- tions stop and this is determined by the Q of the circuit elements as shown in the next section which deals with activity. It will be shown that the Q n required is proportional to — and therefore the maximum frequency sta- bility that can be obtained is directly related to Q. 12.70 Relation Between Crystal Quality and Amplitude of Oscillations The activity of a crystal is usually thought of as the relative amount of grid current produced in an oscillator circuit. This method of defining activity affords a means of comparing the quality of one crystal with another for a particular set of conditions. The disadvantages are first; it is only a relative measure, and second; it is not possible to compute the activity as thus defined by any method of oscillator analysis so far presented. Curves have been shown of amplitude of oscillations as a function of certain circuit variables, but these represent only qualitative changes associated with plate resistance variations. The first objection has been somewhat recti- fied by the use of reference oscillators* in which all the circuit elements including the tubes have been carefully matched. There is still the diffi- culty, however, of comparing crystals of different frequencies for it cannot be assumed that the measurements are independent of this variable. It would be more desirable to have some absolute measure of activity and particularly one which would lend itself to convenient computation from readily measurable constants of the crystal. 12.71 Definition op Crystal Quality for Oscillator Purposes In deriving an expression for the quality of a crystal, it is convenient to use the negative resistance concept of the oscillator as described in section 12.40. The equations are general and in a form which admit of separating * Developed by G. M. Thurston. PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 199 the crystal from the oscillator circuit. Equation (12.44) which gives the conditions necessary for oscillations to exist, may be written in the form coGp ^ "^ (12.76) In order for oscillations to start, the right side of this equation must be equal to or greater than the left side. If it is greater, oscillations build up causing p to increase until the equality is satisfied. The difference between these two terms before oscillations start is therefore a relative measure of the final amplitude for a particular oscillator. The absolute value of am- plitude cannot be obtained from equation (12.76) since we do not know the relation between p and amplitude. However the greater the magnitude of —-^ the greater will be the amplitude of oscillations for a given set of oscil- Rc lator conditions. This term may therefore be considered a measure of crystal quality. It is the effective Q of the crystal unit as measured at its two terminals and at the operating frequency. To distinguish this from the Q of the crystal as usually spoken of, it will be called (fc . In the same respect the left side of equation (12.76) may be thought of as a 1 measure of quality of the oscillator circuit, that is, pc>}Ct = — , then (12.76) becomes ^c- \^ 1 2 3 4 5 z N. Ct lij 10 -) UJ \s^ Co a 5-20 — ^**>i,^,^^ UJ — (E t- L^ U 2 1-40 UJ < - O I (E O UJ a. -60 _ Fig. 12.28 — The change in PI and oscillating frequency of a crystal as the shunt capacitance is increased tially constant over a wide frequency range. Now if we let Ct approach zero in (12.89) it becomes 1 PI = xl RiJ'Cl Ri (12.96) This equation for PI is of the same form as the anti-resonant impedance of a coil and condenser in parallel, and like this impedance it changes rapidly with frequency. The maximum value of PI is therefore A^o times M and is obtained when Ct = 0. Figure 12.28 shows a curve of % PI plotted as a (J function of — - . This curve represents the change in activity as capacitance Co is added across the crystals (increase in Ct). 206 BELL SYSTEM TECHNICAL JOURNAL 12.83 Exact Expressions for PI and Rc The error in PI caused by the assumption that the frequency is independ- ent of the crystal resistance R\ , that is, by use of approximate equation (12.86) for the frequency, may be investigated as follows: The impedance of the crystal and Ct in parallel is given by ^ = .(C. + C,)'(l + m'p-) [^ - ^'C + '"^'(» - 1))' (12.97) where „ MCo ^3 — W P = m = r -i- r 2 2 y^o ^ ^t cos — wi COS = 27r times frequency of anti-resonance of the crystal and Ct combina- tion when i?i = 0 coi = 27r times frequency of resonance of the crystal and Ct combination when Ri = 0 (ji = 2ir times operating frequency (Note that P is the figure of merit of the crystal and Ct in parallel.) The imaginary part of Z is _ 1 -f mP\m - 1) ^ - co(Co + Ct){\ + m^P^) ^^^-^^^ The condition for stable oscillations requires A^ = 0. For this condition which defines the exact frequency of oscillation. The negative sign before the radical is used since the effective resistance is greater at this frequency, thus requiring less negative conductance for oscillation. With P large {m — > 0) the frequency of oscillations coincides with cos which is the same as given by the approximate frequency equation (12.86). The real part of (12.97) is ^ = (r 4-rVi ^^^P^\ (^2.100) co(Co + Ct)\X + m P ) and when m = 0 P M , , ^ = (r A.r\ = ^ ^^2 (12.101) coCol (-8)' f PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 207 This is identical to the expression for PI of equation (12.94). PI is therefore the anti-resonant resistance of the crystal and capacitance Ct in parallel. Substitution of the value of m as given by (12.99) into (12.100) gives the anti-resonant resistance at the oscillating frequency. Thus Ro = CO (Co + Ct) 1 - -/■- p2 (12.102) P l-Ct/Co Fig. 12.29 — The error in PI resulting from the use of approximate equation (12.94) which is the exact expression for PI. The differential error resulting from the use of approximate equation (12.94) is then PI - Ro Ro V 1-1 p2 1 + ^ (12.103) The per cent error as a function of P is shown in Fig. (12.29). The error diminishes rapidly with increase in P and is negligible for crystals that are of sufficient quality for most oscillator purposes. Equation (12.91) for Re is also approximate because of the assumption that the frequency is independent oi Re . A more exact expression will be derived. The impedance of the crystal alone is ^' = .Cod + n'M') '^ - ^'C + '•^'('' - 1»1 <*2.104) 20g BELL SYSTEM TECHNICAL JOURNAL where M = the figure of merit of the crystal 2 2 0)2 — CO ^ = 1 2 C02 — Wl coi = 27r times frequency of resonance of the crystal aj2 = 27r times frequency of anti-resonance of the crystal CO = the independent variable, lir times frequency Co = the static capacitance of the crystal The resistive component of Ze is M Ri Re = coCo(l +n^M^) _1_ 4- 2 (12.105) In order to express Re in terms of Co and Cj , the quantity n must be expressed in terms of these variables which define the oscillating frequency. This is accomplished as follows: The equation of ratio of capacitances of a crystal is Cx ^ co^^I ^j2.106) t-'O COi Similarly, when the capacitance Ct is placed across the crystal ^' ''' ~ '"' (12.107) Co + Ct col where C03 is 27r times the anti-resonant frequency of the crystal and Ct in parallel. The ratio of (12.107) and (12.106) is ^ 22 Co C03 — coi Co ~r Ct C02 2 2 (12.108) COl The oscillating frequency is given by (12.99) in which m is as defined under (12.97). The oscillating frequency co is therefore given by or 2 0)3 — 2 O) 2 0)3 — OJl 2 0)1 2 — CO 2 0)3 — 0)1 i-/j/rr 4^ P2 (12.109) l+//l^ 4^ P2 (12.110) PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 209 The angular frequency cos is eliminated by multiplying this by (12.108). Thus 2 2 0)2 — COl -Co Co + Ct 1 + /- or -Co COo — COl Co + C, 1 + /' + 1 (12.111) (12.112) This is the value for ;; at the oscillating frequency and may be reduced to the form, n = V -h'* Vr- p2 (g-)J (12.113) When this value of n is substituted in (12.105), the value of Re is found to be Re = Ri M' + 1 - 4/1 -I 1 + 4/1- &:-) J (12.114) For crystals of usable quality "2 < < 1 and by this assumption the equation reduces to ' Re = Ri (12.115) This again reduces to (12.91) when if^ > > (^" + 1 j . 12.84 Frequency Change Resulting from Paralleling Capacitance It is often desirable to know how much the frequency of an oscillator may be changed by varying the capacitance Ct across the crystal. This is de- termined from (12.112) which gives the oscillating frequency as a function 210 BELL SYSTEM TECHNICAL JOURNAL 4 of Ct . For practical considerations we may assume ^ <2 — oj (o)2 — o)) (o;2 + 0)) ^ 2(a;2 — o)) then 0)i wf 0?l 0) — 0)2 — 1 2r 6: - 0 (12.118) where r is the ratio of the capacitances of the crystal. A curve of per cent frequency change multiplied by r as a function of — is shown on Fig. 12.28 for comparison with the associated PI change. Co 12.85 Relation between PI and Oscillator Activity The relation between PI and activity obtained in a particular oscillator will now be examined. Let the curves of Fig. 12.30 represent the variations of p with amplitude for two oscillators A and B, or they might be for the same oscillator at widely different frequencies. These are characteristics of the oscillator circuits and may be of any shape. However, for oscillators with grid leak bias, the curves normally have no negative slopes. The rate of change of p depends upon the rate of change of ^t and plate resistance of the vacuum tube as shown by (12.45) for input conductance.* Since p builds up to a value equal to PI we may plot PI for p. The grid current I g is usually taken as a measure of amplitude. Therefore, Fig. 12.30 may be plotted as shown in Fig. 12.31 where PI is the independent variable. These curves are the characteristics of the oscillator circuits A and B with PI defining the quality of the crystal when used with a particular value of Ct . It is characteristic of oscillators to "saturate" as shown by the curves. * It is also a function of grid resistance but this does not appear in the approximate equation (12.45) Ijecause of the assumption of no grid current. See Chaffee's'' complete equation for input admittance. PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 211 AMPLITUDE- Fig. 12.30 — Hypothetical curves illustrating the normal relation between the negative resistance of oscillator circuits and the amplitude of oscillations Fig. 12.31 — By interchanging the coordinates of Figure 12.30 the curves will represent the relation between PI and oscillator grid current 212 BELL SYSTEM TECHNICAL JOURNAL Some oscillators saturate very rapidly and completely according to curve B and no further output is obtained regardless of the improvement in the crystal quality. For this reason it has not been possible in the past to separate the performance of the oscillator and the crystal since both were based upon the grid current as a measure of quality. By defining crystal activities and oscillator sensitivity in the manner outlined, the crystal and circuit can be studied separately. The per cent of crystals obtainable with PI above a certain value will be known and the design and improvement of oscillator circuits will be facilitated. 12.86 Use of PI in Crystal Design The expression of PI in terms of the crystal constants and Ct as given by equations (12.89) or (12.92) assists in the design of crystals. As an ex- ample, the effect of changing the area of the crystal electrodes will be com- puted. The () of a crj^stal is defined as e = ^. (12.119) By introduction of the ratio of capacitances of the crystal ^ = ^ equation (12.119) becomes Q = — ^ (12.120) or - = -4r^ = M (12.121) Assuming Q and r do not vary, that is, disregarding effects such as secondary modes, change in damping produced by the mounting etc., and substituting (12.121) in (12.94) we obtain PI = ^- fr^ir^' (12.122) r co(Co + Ct) where — is considered constant. Differentiating (12.122) with respect to r Co we find that PI is a maximum when Co = C< . Since Co is proportional to the area of the electrodes this establishes the optimum area for a par- ticular value of circuit capacitance. The capacitance of BT-cut plates is 1.68 mmf. per square centimeter per megacycle.* Substitution of this for Co in (12.122) gives ^ j68 X lO^M^ ^^ (1.68 Af -f QV ^ • ^^ * All frequencies are referred to the time interval of one second throughout this paper; i.e. megacycles per second is called simply megacycles as is customary in the radio field. 1 PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 213 where A = the area of the crystal in square centimeters. / = the frequency in megacycles Ct = circuit capacitance in mmf . M = figure of merit of the crystal (assumed constant) Thus for crystals of a given area, the performance index should decrease as the frequency increases. Figure (12.32) shows the theoretical variations of PI as the function of the diameter of the electrodes of three frequencies and PI S 12 16 ELECTRODE DIAMETER IN MM Fig, 12.32 — -Theoretical curves showing the relations of PI, electrode diameter, and crystal frequency for BT crystals and a circuit capacitance of 50 n/xi for a circuit capacitance of 50 mmf. The activity of a 4-megacycle crystal with 11-mm. diameter electrodes is about the same as a 10-megacycle crystal with 18 mm. electrodes. It must be remembered in making this comparison that it is assumed that the damping introduced by the mounting is the same in both cases. Actually the damping is much greater for low- frequency crystals of this type than for high-frequency ones and maximum PI occurs at some intermediate frequency as shown by the curves of Fig. 12.33, These curves show that the damping caused by the particular mounting used was small for frequencies above 6 megacycles but increases rapidly below this value. 214 BELL SYSTEM TECHNICAL JOURNAL CALCULATED 5 MEGACYCLES Fig. 12.33 — Calculated and measured values of PI for BT crystals. The discrepancy is a measure of mounting loss 12.87 Measurement of PI and M In all the discussions so far regarding the performance of crystals in oscillator circuits, the crystal has been represented by the equivalent circuit of Fig. 12.3 in which all the elements were considered constant. It is possible to obtain crystals in which this is essentially the case, but in general there are three secondary effects which complicate the picture. These are, first, the effect of other modes of vibration of the crystal, second, variations in the crystal constants resulting from variations in the amplitude of vibra- tion, and third, the leakage or dielectric loss in the crystal holder. These factors will be considered in the order named. Secondary modes of vibration affect the crystal for oscillator purposes only when the frequencies of these modes are sufficiently close to the princi- pal one to alter its impedance characteristic in the frequency range of oscillation; that is, to alter the reactance as shown in Fig. 12.27 between the frequency /i and/o and the corresponding effective resistance between these two frequencies. With interfering modes present, the equivalent crystal circuit is so complicated as to make it impractical to compute PI or M from such measurable quantities as resonant resistance Ri , series resonant frequency /i , anti-resonant frequency /2 , etc. For this reason it is necessary to measure the reactance and effective resistance of the crystal at the operating frequency in order to obtain a measure of crystal quality which will correlate with the crystal performance. For the same reason it is important when comparing oscillator circuits that the crystal should be operated at the same frequency in each case. It is believed that the non-linear effect noticed in crystals when used as oscillators is produced by the changes in the mounting as the amplitude of PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 215 vibration varies. The PI of some clamped or pressure-mounted crystals has been found to vary as much as 50% with change in drive. Noticeable frequency change also occurs. A change in the nature of secondary modes as the amplitude is varied has also been observed. Some secondary modes which interfere with large amplitude of vibrations practically disappear when the amplitude is reduced. This may be explained by the fact that certain modes are damped out by the pressure of the mounting and with large amplitude of vibration the effect of the pressure is reduced. The dielectric loss of the holder was considered negligible in the theory but it is found that certain phenolic holders have equivalent high-fre- quency leakage resistances less than 100,000 ohms. This resistance is in parallel with the crystal and will therefore reduce the PI according to the equation ^^ - WTTR. (^"^« where PI = resulting PI Pic = calculated PI Rl — equivalent high-frequency leakage resistance Because of these secondary effects which are not negligible it is essential in measuring crystal activity that the frequency and voltage across the crystal be known. Standard test circuits should simulate operating condi- tions in this respect. With these considerations, a crystal PI meter has been developed in which the frequency and amplitude may be adjusted to correlate with various oscillators. The principle of operation and perform- ance of this meter is described by C. W. Harrison.* Bibliography 1. A. McL. Nicolson, U. S. Patent Nos. 1495429 and 2212845, filed in 1918. 2. W. G. Cad}', The Piezo-Electric Resonator. I.R.E., Vol. 10, p. 83, April, 1922. 3. G. W. Pierce, Piezo-Electric Crystal Resonators and Crystal Oscillators Applied to the Precision CaHbration of Wavemeters. Proc. Amer. Acad. Arts & Sci., Vol. 59, p. 81, 1923. 4. A. Crossley, Piezo-Electric Crystal Controlled Oscillators. I.R.E., Vol 15, p. 9, Jan., 1927. 5. K. S. Van Dyke, The Electrical Network Equivalent of a Piezo-Electric Resonator. Physical Rev., Vol. 25, p. 895, 1925. The Piezo-Electric Resonator and Its Equivalent Network. I.R.E., Vol. 16, p. 742, June, 1928. 6. E. M. Terrv, The Dependence of the Frequency of Quartz Piezo-Electric Oscillators Upon Circuit Constants. I.R.E., Vol. 16, p. i486, Nov., 1928. 7. J. W. Wright, The Piezo-Electric Crystal Oscillator. I.R.E., Vol. 17, p. 127, Jan. 1929. * "The Measurement of the Performance Index of Quartz Plates," this issue of the B.S.TJ. 216 BELL SYSTEM TECHNICAL JOURNAL 8. P. Vigoureux, Quartz Resonators and Oscillators. Published by H. M. Stationery Office, Adastral House, Kingsway, London, W.C. 2. 9. R. A. Heising, The Audion Oscillator. The Physical Review, N . S., Vol. XVI, No. 3, Sept. 1920. 10. L. P. Wheeler, An Analysis of a Pieze-Electric Oscillator Circuit, I.R.E., Vol. 19, p. 627, April 1931. 11. F. B. Llewellyn, Constant Frequency Oscillators. I.R.E.. Vol. 19, p. 2063, Dec. 1931; B.S.T.J., Jan. 1932. 12. Issac Koga, Characteristics of Piezo-Electric Quartz Oscillators. I.R.E., Vol. 18, p. 1935, Nov. 1930. 13. K. Heegner, Gekoppelte Selbsterregte Kreise und Kristallozillatoren. E.N.T., Vol. 15, p. 364, 1938. 14. R. A. Heising, The Audion Oscillator, Joiirnal of the American Institute of Electrical Engineers. April and May, 1920. 15. M. Boella, Performance of Piezo-Oscillators and the Influence of the Decrement of the Quartz on the Frequency of Oscillations. I.R.E., Vol. 19, p. 1252, July 1931. 16. H. J. Reich, Theory and Application of Electron Tubes. Page 313. 17. E. L. Chaffee, Equivalent Circuits of an Electron Triode and the Equivalent Input and Output Admittances, I.R.E., Vol. 17, p. 1633, Sept. 1929. 18. W. P. Mason, An Electromechanical Representation of a Piezo-Electric Crystal Used as a Transducer. I.R.E., Vol. 23, p. 1252, Oct. 1935. The Measurement of the Performance Index of Quartz Plates By C. W. HARRISON 15.00 Introduction THE theory of the general behavior of crystals in oscillator circuits has been described by I. E. Fair\ In Fair's paper as well as in others^, it has been pointed out that in the neighborhood of the operating frequency a crystal is equivalent to the circuit shown in Fig. 15. lA. The crystal possesses two resonant frequencies, a series resonant frequency determined by the efifective inductance, Li , and effective capacitance, Ci , and an anti- resonant frequency determined by these same elements plus the paralleling capacitance, Co . This paralleling capacitance is the static capacitance between electrodes of the crystal and any capacitance connected thereto by the crystal holder and lead wires within the holder. The dotted resistor, Rl , shunting the equivalent crystal circuit represents the effective shunt loss of the holder. In the ideal case and in many practical instances this loss is negligible. It is rather difl&cult to express the circuital merit of a crystal quantitatively in a single term such as has been found useful for inductances and capacitances. It is customary to express the circuital merit of these two elements in the form of the ratio of reactance to resistance. That is, for an inductance (15.1) a: and for a capacitance (15.2) For filter purposes, the () of a crystal involving only the inductance, Zi, and resistance, Ri , of Fig. 15. lA is adequate to express its usefulness in certain parts of a filter network, but for oscillator purposes it is insufficient. At frequencies other than the series resonant frequency the paralleling capacitor Co together with the associated shunt loss of the holder enters into the determination of a crystal's performance. The term Q therefore is not com- pletely indicative of the crystal performance. There has been devised, as pointed out in Fair's paper\ a term called "figure of merit" for a crystal ^I. E. Fair, "Piezoelectric Crystals in Oscillator Circuits," this issue of the B. S.T.J. -K. S. Van Dyke, "The Electrical Network of a Piezo-Electric Resonator", Physical Review, Vol. 25, pp. 895, 1925. 217 Q (j)L ~ ~R Q = 1 coCi? 218 BELL SYSTEM TECHNICAL JOURNAL which involves all the elements in the effective crystal circuit, and this term is much more expressive of the quality of a crystal. The figure of merit is: M='^^=^ (15.3) Ai Co r where "r" is the ratio of the paralleling capacitance to the series branch capacitance. Figure of merit is useful for expressing the quality of a crystal 1 o t o R|< Ci=l= ♦ 0 Lc Rc! (A) (B) (C) Fig. 15.1 — Electrical equivalent circuits of a piezoelectric crystal — (A) At any fre- quency between the resonant frequency coi, and anti-resonant frequency ur, (B) and (C) at any specific frequency between coi and W2- GENERATOR INSERTED HERE IN P I METER (IMPEDANCE=Zi_) Fig. 15.2 — Generalized oscillator circuit of the Pierce or Miller type. in its holder or mount; however by definition it is independent of the value of Rl , and does not permit a ready evaluation of the performance of the crystal in an oscillator of the type that may be represented by Fig. 15.2. Any oscillator that operates the crystal in the positive region of the reactance vs. frequency characteristic exhibits capacitive reactance and negative resistance paralleled across the terminals to which the crystal is connected. The operation of the crystal when connected to an oscillator will be influenced by the magnitude of these two terms, and the combination must operate at PERFORMANCE INDEX OF QUARTZ PLATES 219 such a frequency that the total reactance is zero and at such an ampHtude that the total resistance is zero. The performance of the crystal will therefore not depend solely upon its figure of merit, but will involve the impedance of the remainder of the oscillator. Up to the present time circuit design engineers have not devised standards or units to express the quality of their oscillator circuits without the crystal, so there are no corresponding circuital units of quality with which to correlate figures of merit of crystals to ascertain the suitability of one for the other. It was a practice for many years for manufacturers to test crystals in a model of the oscillator in which the crystal was to be used. This required manufacturers to keep on hand models of all oscillators for which they expected to make crystals. To avoid the mounting number of such test oscillators a special test set was developed which could be adjusted to simu- late any oscillator. By correlating various oscillator circuits to a set of adjustments on the test set, the actual model of the oscillator can be dis- pensed with. This special test set usually referred to as the ' "D" spec, test set', eliminated the "file" of oscillators, and substituted a file of adjustment readings that would be their equivalent. However, the ' "D" spec.testset' is still inadequate to the development engineer since it defines "activity" in terms of oscillator grid current rather than in terms of the electrical equiva- lent circuit of the crystal. The activity as expressed by grid current is a purely arbitrary standard and serves only as a means of determining the relative activity as against other crystals of the same frequency operated under the same circuit conditions. The need for a system of measurement using units that are fundamental and not empirical has led to the proposal of "Performance Index". An instrument to make such measurements is to be described in this paper. Specifically the Performance Index is PI='^ (15.4) where Ct is the paralleling capacitance that is found in the oscillator circuit to which the crystal is attached, and L and R represent the effective induc- tance and resistance of the crystal as measured at the operating frequency indicated in Fig. 15.1 C which is its equivalent at thai frequency. If the loss in the holder is so low that the resistance, Rl, may be neglected, then PI may be expressed in other relations that are more useful such as, _ M 1 "^" V ^ Co j ^ (15.5) or PI = P'Ri 220 BELL SYSTEM TECHNICAL JOURNAL where the symbols R\ and Co are as shown in Figs. 15.1 A and 15.2 and P is expressed as M ^ = G (15.6) Co With the effective capacitance, Ct, of the remainder of the oscillator added to the paralleling capacitance, Co, in Fig. 15.2, the operating frequency will be that frequency at which the combination will exhibit a pure resistance at the terminals AB (excluding the generator "X" which is involved in the measuring technique). This leads to the definition: The Performance Index is the anti-resonant resistance of the crystal and holder having in parallel with it the capacitance introduced by the remainder of the oscillator. The Performance Index is therefore a term to express performance not in terms of the grid current of some particular oscillator, but in fundamental circuital units — -impedance. The Performance Index is a term that may be used to compare performance of crystals at different frequencies. Its value is independent of plate voltage, grid leak resistance, or of plate impedance. It provides a measuring stick that should replace the "activity" figures of grid current in so far as the crystal is concerned. It paves the way for the oscillator circuit designers to come forth with standards of measure- ment for the oscillator circuit without the crystal in the hope that the two may be quantitatively associated and lend themselves to theoretical calculation of full oscillator performance. 15.10 Theory of Measurement The problems of measurement are most readily explained by reference to Fig. 15.2. The crystal provides elements Li,Ci, Ri and Co . The circuit of the oscillator provides an effective capacitance, Ct, which is composed of grid and lead wire capacitances plus capacitance introduced from the plate cir- cuit. The frequency at which this combination exhibits anti-resonance as measured at A B is the oscillating frequency. The resistance when added to negative resistance, p, will be zero. Oscillations will start with p numeri- cally smaller than the anti-resonant resistance measured at AB, but the amplitude of oscillations will increase causing p to increase until p and Zab are equal numerically. The primary problem is to measure the anti-resonant resistance at AB at the anti-resonant frequency with p disconnected. Measurement of anti-resonant resistance directly is very difficult. The current flowing into an anti-resonant circuit is too small to measure with the usual meters. Other devices for measuring the current are likely to introduce paralleling capacitance that will vitiate the readings. The sug- PERFORMANCE INDEX OF QUARTZ PLATES 221 gested method of measurement utilizes a suitable driving voltage at "X" (Fig. 15.2) and a means to indicate the voltage at "X" as well as at points AB. From these and other measured constants, the anti-resonant im- pedance can be computed. This method of measurement has its own difficulties, but it is believed corrections can be made to allow for errors introduced. Fundamentally, the series resonant frequency and the anti-resonant frequency are the same only when the resistances in the inductive and capacitive branches are equal. When the resistance is practically all in the inductive branch, which is true in this case, the impedance between terminals AB, at the series resonant frequency will exhibit capacitive reactance, though the total impedance will scarcely be different from that at the anti-resonant frequency. In the Performance Index meter, although the voltage is introduced in series with the circuit, the frequency is adjusted to the point of maximum voltage across AB, which further minimizes this frequency difference. A second error is inherently introduced by the loss in the crystal holder. This means that the series resonant frequency is also altered by the presence of this loss. Errors of any seriousness will result from the assumption that the series resonant and anti-resonant frequencies are identical only when the resistance in the inductive branch and loss in the crystal holder approach the effective crystal reactance in magnitude. These errors will be discussed in greater detail in a succeeding section. The development and operation of a satisfactory meter to measure PI (Performance Index) depends upon a number of factors such as : 1. A method to determine capacitance, Ct, of the circuit (Fig. 15.2). 2. A generator "X" to produce the driving voltage ei having variability in frequency and negligible internal impedance. 3. A current indicator that introduces a minimum of reactance and resistance. 4. A circuit or method to indicate PI directly, or with a minimum of calculations. 5. A number of other factors associated with the above which will be mentioned at the logical times. To construct a measuring circuit to determine the anti-resonant imped- ance by means of a series circuit so as to avoid any unnecessary measure- ments and computations involves the following basic principle. Excluding p. Fig. 15.2 is essentially equivalent to the circuit used in Q meters. The ratio of voltage Cc to the driving voltage d is the voltage stepup or the Q of that part of the circuit containing the resistance. In this case, the resistance is in the crystal which at the operating frequency has an effective Q of Qi = '^ (15.7) 222 BELL SYSTEM TECHNICAL JOURNAL where L and R represent the effective values of the crystal, will be found to embody the relation (jiL PI = oiCtR = QiX, Equation (15.4) (15.8) where Xt is the reactance of Ct, the capacitance introduced by the circuit at the operating frequency. If Ci is kept constant, Cc will at all times be proportional to Qi. By insertion of an attenuator network, whose attenua- tion varies with frequency in the same manner as does the reactance of Ct, between terminals AB and the voltmeter, the meter indication will be proportional to the product of these quantities or proportional to PI. With Cp=^ep (A) pc (B) Fig. 15.3 — Measuring circuits of the performance index meter. suitable calibrations, therefore, it should be possible to get indications of PI as readily as is now done for Q. The circuit shown in Fig. 15.2 is now best redrawn as in Fig. 15.3A. The crystal embodying elements Li , Ci , Ri , and Co of Fig. 15.2 is now repre- sented in Fig. 15. 3A by the dotted rectangle and as having effective induct- ance, L, and effective resistance, R, both of which are functions of frequency. Capacitance, Ct, is simulated by capacitors d plus Cs and Ca- in series where Cx represents the capacitance of the crystal socket. Zero internal impe- dance of the generator is simulated by maintaining the driving voltage con- stant at all times and at all frequencies. To facilitate explanation, the measured voltage d at the place shown is considered to be the driving volt- age from a zero internal impedance generator. Instead of using an ammeter to indicate current in the circuit, a voltmeter is utilized to measure voltage across an element under such conditions as PERFORMANCE INDEX OF QUARTZ PLATES 223 not to introduce disturbing capacitance. Splitting the series capacitance into two parts, C. and Ca-, the latter fixed and large compared to C,,, provides the impedance element across which the voltmeter is connected. The input capacitance of the voltmeter is incorporated in the magnitude of Ca-. A capacitance attenuator, A , of known or calibrated values interposed on the input of the voltmeter enables the voltmeter to be used to indicate voltage ratios in terms of the attenuator calibration. The measuring voltmeter and a shunting capacitance, Cp , are connected in the plate circuit of the amplifier tube, V-1. This circuit provides sufficient gain to furnish an output voltage of measurable magnitude and also provides an output voltage inversely proportional to frequency. The indication of the output voltage is proportional to PI. The utilization of a vacuum tube in a circuit leading to a quantitative measuring instrument such as the voltmeter across Cp involves determination of tube constants or calibration. The determination of these constants is best evaluated experimentally. A calibrating circuit for that purpose is shown in Fig. 15.3B. A capacitance, Ca, of high impedance in series with comparatively negligible resistance, Rj, is connected across the driving volt- age terminals of d with a voltmeter measuring g; giving a reading Cic. The second subscript "c" indicates calibration conditions. By connecting the input circuit of V-1 across this resistance, the attenuation variation with frequency of the Ra — Ca network cancels the attenuation variation with frequency in the plate circuit of V-1. The ratio of dc to Cpc will then be in- dependent of frequency. In the "calibrate" circuit (Fig. 15. 3B), the capac- tior attenuator, Ac, interposed in the grid circuit is set at unity (minimum insertion loss) for a given deflection of the meter indicating Cp. In the oper- ate circuit (Fig. 15.3A), the attenuator is readjusted so that voltage eo produces the reading of Cp as obtained in the calibrate position. The quan- titative action of the amplifier then m.ay be expressed in terms of Cp, Ra, Ca and a reading from the attenuator A, as will be shown later, and it is constant and independent of frequency. By placing this resulting constant in an equation, which will also be derived later, the value of PI may be de- termined in terms of such constant, of the reading of attenuator A, and of a reading on the scale of Cs that has been calibrated in terms of Ct. To facilitate still further the operation of the PI meter, the voltage Ci is produced as shown in Fig. 15.4 by arranging for the oscillator to have its frequency controlled by the crystal through feedback from capacitor, Ck . Automatic volume control is provided such that the amplitude of d is essentially constant at all times and at all frequencies. The circuit is con- structed to oscillate at the desired frequency, and adjustment for insuring this operation is provided in the form of a phase shifting circuit with variable 224 BELL SYSTEM TECHNICAL JOURNAL capacitor, Cr . After a crystal has been inserted in its proper place, oscilla- tions will begin, but may be slightly above or below the resonant frequency of the crystal plus Ct . By adjustment of Cr the frequency can be shifted the slight amount necessary for resonance. This is observed by placing switch 5 in the PI position and making the adjustment to give maximum deflection of e„ . SWITCH "S" V.T.V.M. Fig. 15.4 — Diagram of Performance Index meter. 15.20 Derivation of PI Circuit Equation The following circuit relations derived from Figure 15.3 show first, that the ratio of Cp to Ci is a function of the Performance Index of the crystal, and second, that the calibration circuit permits an absolute evaluation of its magnitude. At resonance, the effective circuit Q, designated as Qo , is determined from Q2 = Cc + «0 (-S (15.9) Since the circuit Q includes the capacitance of Cx as a part of the crystal, it is necessary to express Q2 in terms of the crystal's properties (see Fig. 15.1). Since QA = -^ ) of the crystal is independent of Cx , the relationship between Qiand Q2 is readily obtained by equating the expressions for the anti-resonant impedance first, when Cx is considered to be in shunt with the series capac- itor, Ct, and second, when Cx is considered as part of the crystal. This re- sults in Q2 =Qi Ct — Cx Ct (15.10) PERFORM A. \CE IXDEX OF QUARTZ PLATES 225 where Ct = C,-{- CaCk Cs + c. enabling Qi to be expressed as Qi = eo Ci Ck ei (Ct — Cx)^ Expressing Cq in terms of ep , we have \fieg\ = \ip\ Vr p + X tn = Xc (15.11) (15.12) (15.13) (15.14) hence and '^-V'+C^'I ('^-'^ eo (15.16) With the above equations substituted in (15.12), we may express Qi as Qi = - oi -^ A — — 1 + X, Now Therefore PI = ^ = QiX, PI = ^ A '' ei Gm (Ct — CxY 4/^ + [tI If PT = ^ 4 iJL Ci ^ G„, {Ct - CxY (15.17) (15.18) (15.19) (15.20) (15.21) The simpHtied PI expression (15.21) assumes that the reactance of Cp is small compared to the plate resistance and plate load resistance of V-1. The evaluation of PI from this expression has three obvious difficulties: 226 BELL SYSTEM TECHNICAL JOURNAL (1) Cp/Gm is a quantity that is difi&cult to evaluate numerically, (2) the magnitude of PI is measured in terms of the ratio of the two voltages, d and gp, and (3) the measurement is dependent upon the gain of a vacuum tube amplifier, V-1. These difficulties may be materially reduced in their consequence by an internal calibration circuit. The internal calibration circuit (Fig. 15.3B) consists of a capacitor, Ca , and resistor, R.a. , in series. If the reactance of Ca is very much greater than Ra , and the plate resistance of V-1 is very much greater than the reactance of Cp , the calibration is essentially independent of frequency. The internal calibration circuit (Figure 15.3B) enables the evaluation of Cp/Gm to be carried out. The additional subscript, c, indicates "calibrate" conditions. ^Oc — IcR.i eicCoCAR.i 1 + _XcJ = e„ (15.22) (15.23) (15.24) Equation (15.15) remains the same for both "operate" and "calibrate' conditions with the exception of the second subscript reserved for the "calibrate" operation. Therefore, by solving for Cp/Gm we find ^Pc^ \ 1 + Xc. (15.25) Equation (15.25) may be rewritten as (15.26), if (15.23) and (15.24) are sub- stituted in (15.25) Cic Ra Ca ^^-my^-it (15.26) If (15.26) is substituted in (15.19), it is found that PI may be expressed as follows: a .(Ct - c.y. 1 v^-[a ''■'' PERFORMANCE INDEX OF QUARTZ PLATES 227 The above equation involves only the original approximation that maxi- mum current indicates resonance. If R,i and C,i are selected such that R.i < < X a and if .4 ^ equals unity, then PI Cp eu A - Ra Ca (15.28) (c, - c.y From this expression it can be seen that the PI measurement is inde- pendent of calibration of both the Cp and Ci vacuum tube voltmeters, pro- vided that the same voltmeter scale factors are used for the "operate" and "calibrate" conditions. The absolute calibration then depends on the magnitude of A, Ra , Ca , C/, , Ct and Cx . The "multiply-by" factor that 1? r' C is to apj)ear on the C,. dial is determined by the magnitude of -—^ -— . {Ct — Cx) Accurate evaluation of this quantity by capacitance and resistance meas- urements is a little difficult since the denominator represents the square of the difference of two small capacitances. When Ct is large, the evaluation of this factor is helped considerably. This "multiply-by" factor may be experimentally determined by a voltage measuring means which permits an evaluation of this factor to a higher degree of accuracy. Substituting (15.11) in (15.28) we have PI = A Ra Ca {• * !)■ (15.29) Ck \ Cs Now by shorting the crystal socket terminals (Fig. 15. 3A) and applying a voltage ei at the ei generator terminals of external origin (the crystal oscil- lator circuit itself may be used if self-excitation is provided), the current ii through the capacitors Ck and Cs is given as eicoCsCk _ eiojCk " :) ti = Cs + C, 1 + c. (15.30) Now the voltage, eo , across the series capacitor, C/. , is 62 = wCfc (15.31) The ratio of Ci/co may be expressed as given in (15.32) when (15.31) is sub- stituted in (15.30) Ck\ Ci \ Cs If (15.32) is substituted in (15.29), we find RaCa PI = ( V 6pc 6i _] -^ \A Ck \_ei_ Ci (15.32) (15.33) 228 BELL SYSTEM TECHNICAL JOURNAL The quantity — is readily determined by the attenuator, A, when the switch, S, (Fig. 15.4) is operated between "M" and "P/" for the above described conditions. The absolute calibration then depends upon A, Ra, Ca and Cic. All four of these quantities may be determined within a few per cent. 15.30 Oscillator Correlation The equivalent crystal circuit has been discussed in so far as the measure- ment of PI is concerned; however, for correlation with an oscillator, the behavior of the crystal in that oscillator must be duplicated. Correlation of the PI meter with an oscillator is a function of both amplitude and frequency. It is obviously necessary from the derivation of (15.28) that the frequency of operation be duplicated, but the necessity for ampli- tude correlation can only be explained from the practical consideration that the equivalent circuit components of Fig. 15.1 are parameters that may be functions of amplitude. Crystals having nonlinear characteristics of the type that necessitate amplitude correlation may in part be attributed to either the method of mounting the crystal or couplings to other modes of vibration whose coupling coefficients are functions of amplitude. In most oscillators the voltage across the terminals of a crystal is a func- tion of many parameters such as plate voltage, vacuum tubes, etc. With an average set of conditions, however, reasonable correlation is obtained with the PI meter for a single adjustment of the generator voltage, ei , for all crystals. The magnitude of Ci must, of course, be chosen to produce a voltage across the crystal equal to the average value obtained in the oscil- lator circuit for which the crystal is intended. Frequency correlation with an external oscillator is a function of the effective capacitance, Ct, in shunt with the crystal. In order to duplicate the oscillator frequency with the PI meter, the capacitance, Cs, (Fig. 15.4) must be adjusted until the frequency of oscillation in the PI meter is the same as that in the oscillator for a crystal having average activity. In Fig. 15.4, the capacitance, d, is variable, and its dial is calibrated in terms of both the total effective capacitance across the crystal, Ct, and the resulting R C C multiplying factor j-^ — ^ . The magnitude of Ct may be measured by means of a capacitance bridge connected across the crystal socket terminals with the generator impedance shorted. The determination of the dynamic or effective capacitance, Ct, across the crystal for an oscillator may similarly be obtained by adjusting the magni- tude of Cs in the PI meter until the frequencies of oscillation in the PI PERFORMANCE 1NDE,X OF QUARTZ PLATFIS 229 meter and in the oscillator under test are identical for the same amplitude of oscillation. By this means, the PI meter directly indicates the effective oscillator capacitance, C/. The amplitude of oscillation must be duplicated in as much as Ct is not independent of the amplitude in most oscillators. 15.40 Description of Oscillator Generating "ci" The generator plays no part in the theory of PI measurement as it could be replaced by a signal generator or any other suitable source of radio fre- quency energy. It is convenient, however, to utilize the voltage appearing across Ca as an input to an amplifier whose output represents the generator. This in effect constitutes a feedback oscillator whose frequency is controlled by the crystal under test. Initial consideration of the over-all charac- teristics of the PI meter oscillator leads to the following requirements. The oscillator must, 1. Be capable of oscillating all crystals usable in other oscillator circuits. 2. Be capable of operating the crystal over a wide range of shunting capacitances in order to duplicate all the frequencies of oscillators now in the field. 3. Be capable of permitting high degrees of AVC control in order to maintain the generator voltage constant while the frequency is ad- justed for reasonance. If the generator voltage, Ct , is constant, resonance of the crystal circuit is essentially indicated by maximum crystal current, and oscillation is maintained at that resonant frequency. The adjustment to obtain maxi- mum current is such that the phase shift throughout the oscillator loop is Itth where « = 0, 1, 2, 3, etc. As previously described the phase shift and resulting frequency of oscillation are varied by a tuned circuit. The generator voltage, d , is held constant by an automatic amplitude control similar to the automatic volume control which is often applied to amplifiers. The manual control of the magnitude of the generator, g, , is provided by an adjustment of the bias voltage of the automatic amplitude control circuit. In this way the maximum or start gain is independent of the setting of the amplitude control. Automatic amplitude control (commonly referred to as automatic volume control, A VC) of an oscillator may be applied by the separation of the limiter from the linear amplifier. This means that in order to apply a high degree of AVC to the PI oscillator (Fig. 15.4), the input voltage of the limiter must be above the threshold of limiting by an amount exceeding the variation in the j3 path caused by the AVC control. This enables the limiter to absorb the changes in the gain of the linear amplifier such that )u/3 = 1 at all times. The time constant of the limiter is fast compared to that of the AVC 230 BELL SYSTEM TECHNICAL JOURNAL circuit, a condition which permits damping of transients set up by changes in gain occurring from AVC action. The input of the Unear amphfier is held constant by the hmiter. Gain changes in the Hnear amplifier pro- duced by the variation of Cr (Fig. 15.4) are absorbed by AVC, while the variation of activity in the crystal is absorbed by the limiting amplifier. 15.50 Evaluating Performance Index From (15.28) it can be seen that the attenuator, A, the effective variable capacitance, C(, together with the vacuum tube voltmeters, d and Cp, offer a number of possible variations in the method of evaluating the con- stants used to determine the PI of quartz crystals. There are, however, two principal methods — the first provides direct reading, while the second is more accurate but requires an indirect evaluation. The first method utihzes a means of calibration of the meter scales directly in terms of PI. The attenuator is adjusted such that its indicator reading times the multiplying factor associated with the dial attached to Cs is some multiple of 10. If in the calibrate position, Ac is set at unity, and Cpc and Cic are adjusted by varying the capacitive load to some reference deflection, then the expression for PI becomes where PI = -^ K^ (15.34) ^ic . Ck Ra Ca _6pc {^t ^x) The Performance Index then is indicated by the two readings of ep and d . The absolute magnitudes of ei and ep need not be known since it is possible to use as a reference, the arbitrary calibrating deflections of e^ and epc . The magnitude of Cp indicates the significant figures while d is a multiplying factor. The second method of evaluating Performance Index eliminates any calibration errors in the two vacuum tube voltmeters, Ci and Cp . This method utilizes the attenuator to adjust == 1- In the "cahbrate" \_ei epc A operation, e.c is set to equal e. and then epc is adjusted for full scale or a convenient deflection. In the "operate" position, the attenuator is varied until ep equals Cpc . In this manner, the two readings of the attenuator Cp are used to determine the ratio of — and the measurement is independent enc PERFORMANCE INDEX OF QUARTZ PLATES 231 of the voltmeter calibration. The factor RaCa is a constant; therefore, the P.I. equation (15.28) simplifies to P.I. = (A.OA'o (15.35) where AA = change in attenuator insertion loss between the "operate" and "calibrate" conditions in terms of output voltage ratio — ■ , given A as ~. C i- Ra C .4 Ki = T^ T7T2 where C< is the effective capacitance in series with the crystal, Ck is the fixed series capacitance and Cx is the crystal socket capacitance. (See Fig. 15.3) 15.60 P.I. Meter Applications The application of this instrument can be extended to determine other properties of both crystal and oscillator. With the aid of a frequency measuring means and a capacitance bridge, the P.I. meter may be used to determine all the circuit constants designated in the electrical equivalent circuit of Fig. 15.1. If the loss in the holder is negligible then the equations are considerably simplified; however, in those instances where holder loss must be considered, the approximation that Xt <^Ru which may be allowed for most cases enables an evaluation of M and Qc that is readily computed. The dial controlling C^, that is calibrated in terms of the total capacitance, Ct, makes possible the calculation of the magnitude of the input impedance to the crystal circuit, i?, as well as Qi. X7 ^ = P.I. <2i p.i. x7 (15.36) The magnitude of Qi may also be measured directly from equation (15.12) where Qi was given as As may be seen from Fig. 15.4, eo/ei can be evaluated in terms of the attenuator calibration by enabling switch, S, to select the "PI" and "M" positions respectively and adjusting the attenuator such that the same output meter indication is obtained in the two cases. 232 BELL SYSTEM TECHNICAL JOURNAL The quantity Qi makes possible the calculation of Qc where Qc is defined as wLc Xt Qc- ^ = (-8)' ^c - ^ /, , CoV (15.37)^ It can be shown that Qc in terms of Qi is given by the following equation if RL»Xt. Rl Rl The Figure of Merit, M, defined by (15.3) at the series resonant frequency of the crystal, coi, becomes M ^ -^ = ^ (15.39) The measurement of M can be determined from Qc provided Ct and Co are known. M may be determined from the following expression M - ^' ^' {^ 4- ^«Y ^ . _ PJ. Co \ "^ cj (15.40) Rl If Ct is selected such that C< ^ Co , then for most cases Rl is large com- pared to QiXt . This means that Qi = Qc and R = Rc . With this approxi- mation, (15.40) becomes The relationship between i?c and Ri as a function of frequency may be expressed directly from the input impedance expression of the equivalent circuit. This is given as „ Rl ^. - r- 2 ,-,2 . y5^2) [0)2 — W , 1 where w is the unity power factor frequency in the P.I. meter, neglecting Rl If j. fco; - con '•H) The relationship i?c = i?i ( 1 + — ) was derived in Fair's paper*^ PERFORMANCE INDEX OF QUARTZ PLATES 233 which is a plausible assumption when Ct > Co , and we assume that — - — = We then, . _ ^?_ ^'^ ^ ""' ■ (15.43) [C02 — 0) C02 — OOlJ The relationship between Ri and Re does not involve i?i, and may be expressed in terms of Co and Ci instead of frequency. Neglecting Rl to determine the relationship between capacitance and frequency in the P.I. meter as derived in Section 15.92, we find . [-/-a where, M Pi (15.44) (15.45) (■ - f) If Pi » 2 then the expression between Re and Ri may be written R, = i?x (l + |j (15.46) The restriction that Ci > Co may be removed if the error between (15.42) and (15.46) is taken into account. This error may be expressed as Per Cent Error in R. = 100 [^] [i + Vf+VPf] ^^^'^^^ The resonant resistance, i^i, together with (15.46) provides a means of checking the P.I. meter. The magnitude of Ri may be determined by the substitution method and from this, the value of Re calculated. Fig. 15.5 represents the agreement between the P.I. meter and those expected from resonant frequency measurements. The remaining crystal constants Li and Ci (Fig. 15.1) may be evaluated from the measurement of wi , wo and Co . The resonant frequency, coi , is defined as col = ^ (15.48) The anti-resonant frequency, C02 , is defined as 234 BELL SYSTEM TECHNICAL JOURNAL CALCULATED MEASURED 50 40 25 u a. 20 10 - 5 0.1 ANTI-RESONANCE Fig. 15.5 — Typical characteristic of a quartz crystal measured by the Performance Index meter. Solving these equations simultaneously, it is found that 1 1 u = Ci = Co(c02 — COi) 2ciJeCo(c02 — COi) ^"0(^2 — cci) 2Co(a)2 — ooi) (15.50) 15.70 Experimental Data The performance of the P.I. meter may best be illustrated by experi- mental data. The following data indicate the correlation which may be obtained between the P.I. meter and various types of oscillator circuits. Experimental considerations are extended to (1) Frequency and ampUtude correlation with a "Pierce" and "Tuned- Plate" oscillator PERFORMANCE INDEX OF QUARTZ PLATES 235 (2) The measurement of the effective capacitance, Ct, of an oscillator as a function of tuning, and (3) The variation of P.I. as a function of voltage across the crystal. The results presented are not to be considered as generalized data, but are intended only to show a set of measurements obtained for a specific set of operating conditions for each type of circuit. It has been pointed out that the frequency of oscillation is a function of /?i , coi , 0)2 , Co and Ci , Since Ri , wi , co-) and Co are explicit parameters of the crystal, the capacitor, Ct , becomes the only frequency determining element in the P.I. meter. From analytical methods to be described in PI METER OSC. "7-~-^<_PIERCE OSC \ CALCULATED PERFORMANCE INDEX Fig. 15.6 — Frequency variations in oscillators as a function of Performance Index. Sections 15.80 and 15.92 the frequency difference between the P.I. meter and the generalized oscillator (neglecting R^) is given as (C02 — CUi) / Co\ ''bw^-a (U2 — Oil' (■ * I) (15.51) Since the magnitude of frequency change in the above equation is small compared to the variations caused by changes in operating conditions, the P.I. meter may be used as a frequency correlation medium. Fig. 15.6 is an example of the correlation between the "Pierce" and "Tuned-Plate" oscillator and the P.I. meter. The P.I. meter falls between these two oscillators in frequency for any crystal activity. It must be recognized that Fig. 15.6 is not conclusive to the extent of generalization; however, it is indicative of possible correlation with these two popular oscillator circuits. 236 BELL SYSTEM TECHNICAL JOURNAL Figures 15.7 (A) and (B) show the ampUtude correlation between the Performance Index meter and the grid current of the "Tuned-Plate" and 'Tierce" oscillators, respectively. The P.I. vs. grid current characteristic was arbitrarily taken at three frequencies — 4.5 mcs, 5.66 mcs and 7.81 mcs. The change in grid current of the oscillator shown in Fig. 15.7 (A) with frequency is caused by the varying L-C ratio in the plate circuit. (See curv-es A, B and C for constant crystal activity.) The curves A and D represent the effect of changing bands by switching coils, varying the L-C ratio 2 to 1 in the plate circuit for the same crystal frequency. Fig. 15.7 A AND D 7.810 MC. B 5.660 MC C 4.50 MC. (A) A 7.810 MC. B 5.660 MC. C 4.50 MC. .^^' (B) PI PI Fig. 15.7 — Typical oscillator characteristics. (B) represents the correlation between P.I. and grid current of the "Pierce" type oscillator. The results of this correlation indicate that the grid cur- rent is essentially independent of the operating frequency. The measurement of P.I. is independent of the level of crystal vibration, provided that the electrical equivalent circuit parameters of Fig. 15.1 become constants; however, in actual practice these are not constants, particularly Ri . Variations of this type, as previously discussed, make it necessary to duplicate the amplitude of oscillation of the P.I. meter with the oscillator. Fig. 15.8 represents the variation of P.I. as a function of voltage across the cr}'-stal terminals for five crystals arbitrarily selected. It is readily observed that P.I. may be a random function of amplitude. PERFORMANCE INDEX OF QUARTZ PLAlliS 237 VOLTAGE ACROSS CRYSTAL TERMINALS Fig. 15.8 — Observed variation of Performance Inde.x as a function of voltage across the crystal terminals. 60 "■ < U DC >- o t i O 1 < o 5 20 DECREASING 40 60 80 100 80 60 40 PERCENT OF MAXIMUM Ig WITH VARIATION OF PLATE TUNING P"ig. 15.9 — Observed variation of effective crystal capacitance in a Miller oscillator. 238 BELL SYSTEM TECHNICAL JOURNAL Equation (15.44) indicates that the capacitance, Ct, determines the oper- ating frequency between coi and 002 of any given crystal. The capacitance, however, may include reflected reactances from associated circuits or pos- sibly from circuits unintentionally coupled to the oscillator. Normally, crystals are adjusted to frequency for a specified value of Ct. This makes it of interest to measure the magnitude of Ct over the range of the manual oscillator tuning adjustments, as well as over the frequency range of the oscillator. Fig. 15.9 shows the circuit capacitance, Ct, plotted as a function of tuning of the plate circuit of a Tuned-Plate oscillator. Tuning of the plate circuit is expressed in terms of percentage of maximum grid current. 15.80 Circuit Analysis Involving the Accuracy of P.I. Measurements The method of P.I. measurement just described involved a number of unverified approximations. These approximations under the majority of conditions will be proven to be justified and the resulting expressions for percentage error will be obtained. It is necessary to apply a method of analysis that is most readily adaptable to the crystal circuit. The analysis described in this section involves the use of Conformal representation as a means of determining (1) the behavior of the equivalent crystal circuit, (2) the error resulting in P.I. from assum- ing operation at the resonant frequency rather than the frequency for mini- mum impedance, and (3) the comparison of frequency of oscillation in the P.I. meter with other oscillator circuits. Generally, the variations of reactance. A', and resistance, R, of the equiv- alent circuit (Fig. 15.1) are plotted as a function of frequency, and the analysis of the impedance, R + ^'A^, between the resonant frequency, wi, and anti-resonant frequency, aj2, are handled in precisely the same way as any linear passive element. This was essentially the procedure used to derive the equations in section 15.60. The analysis required to evaluate the errors leads to rather an elaborate study; however, in Section 15.81, it will be shown that it is very helpful analytically if the impedance of the crystal is plotted in the form of a circle diagram, that is, with the ordinate repre- senting reactance, A, and the abscissa representing resistance, R. 15.81 Conjormal Representation Conformal Representation or Mapping is a convenient tool which for this application enables the physical operating condition to be expressed quanti- tatively from its graphical counterpart. Physical interpretation also makes it possible to draw many other conclusions that by other methods prove clumsy and laborious. PERFORMANCE INDEX OF QUARTZ PLATES 239 The basis for this analysis depends upon the ability to utilize the following equation to represent any impedance whose frequency of operation is con- trolled by a crystal. This equation is known as a linear fractional trans- formation W = ^^±11 (15.52) The terms a, /3, 7, and 5 represent complex constants and Z represents a complex variable later to be chosen to represent a linear function of fre- quency. Since TI^ and Z represent the dependent and independent variable, they may also be considered as representing two separate planes. The abscissa and ordinate of these two planes represent their real and imaginary components respectively. The planes are linked by (15.52), that is, this equation will transform a specific pomt from one plane to the other. The Constantsa, /3,7 and 5 for the equivalent crystal circuit are determined by writing the expression for Zc in the form of (15.52). For example (neglecting Rl), Zc from Fig. 15.1 may be written as Zc= r^ ^ L \ ^^^^^/^^^_, (15.53) By substituting (15.48) and (15.49) in (15.53), this impedance may be writ- ten as jcoCo [wi?! -f _7Zi(co — C02)(C0 + CO2)] Since the operating frequency, w, represents some frequency between coi and C02 , and co ;:^ coo — wi , we can make the following approximations in this operating range. The symbol coo is defined as the average operatmg radian frequency. "' ~ ^ I (15.55) COe = CO j If (15.55) is substituted in (15.54), factor We out, add and subtract C02 from the imaginary component in both numera tor and denominator, we may write Z^ as r 1 , . 1 2Li . n1 I r 1 1 • 2^1 r ^ — TT + J —^ -^ (C02 - COl) + —- ; ^- (cO - CO2) J<1 240 BELL SYSTEM TECHNICAL JOURNAL The constants a, /S, 7 and b may be written immedately from (15.56); however, for purposes of simplification let = _L cogCo z = y — - (co — C02) 2/" M = ^- (co2 — coi) (See equation 15.50) (15.57) then, Z. = [r + jMr] + [t]Z J + JZ (15.58) Now if Zc may be represented by W in (15.52) the remaining constants must be a = r + JtM 7 = i /3 = T 5 = i (15.59) Now Z represents the frequency variable and graphically represents a line coincident with the F-axis in the Z-plane, and IF, the corresponding im- pedance variable, represents a circle in the TF-plane. The coordinates of the center of the circle, Wq , in the TF-plane is given by (15.60)= when 7 and 5 represent the conjugate functions ot 7 and 5 respectively. The radius of the circle is given by TFo 5 a 7 "7,7 J I. /3 a 7 b 7,7 b "^a (15.61)= *E. C. Titchmarch, "Theory of Functions," Oxford 1932, pp. 191-192. Note: These equations are not derived in Titchmarch; however, by taking the limit as the diameter of the circle in the Z-plane approaches infinity, (15.60) and (15.61) result. PERFOEI'IANCE IXDEX OF QUARTZ PLATES 241 Now substituting the values given in (15.59) in (15.60) and (15.61) to deter- mine the coordinates of the center of the circle and the radius respectively, we find Mr J-r (15.62) As might be expected, the radius of the circle is equal to the real component of the coordinates of the center of. the circle. This indicates that Fig. 15.10 NEAREST ORIGIN Fig. 15.10 — Circle diagram for crystal circuit combinations. graphically represents the circle diagram where 77 = 7. The impedance, Zc , is represented for a given frequency, as a vector from the origin to the corresponding point on the perimeter of the circle. From Fig. 15.10 the following crystal properties may be deduced. 1. The anti-resonant impedance designated as 0 — P4 is simply cr + s/c^ — if, and when evaluated equals Ri{M^ — 1) = RiM^. 2. The resonant resistance, 0 — P3 given by a- — -s/o-^ — ij- is equal to Ri . 3. The maximum positive reactance between coi and C02 is represented by the distance from P5 to Pe • It is given as cr — r; which equals 2 X .,). 242 BELL SYSTEM TECHNICAL JOURNAL 4. The condition which must exist when the crystal reactance is zero between wi and C02 occurs when o- = rj or M = 2. It follows that the error of measuring, say the series resistance Ri , by varying the frequency for maximum transmission and assuming true resonance when the crystal is between two low non-inductive resistors is associated with the difference between the length of the two vectors 0 — P3 and 0 — Pi . The per cent error caused by the crystal capacitor, Co, by this method of measurement of Ri is given as Per cent error of R\ = 100 100 1 - Ps ■i ■y/a^ + r?^ — a- 0" — Vo-- — T]-. (15.63) If 0- and T] are substituted in (15.63), we find Per cent error of Ri = 100 1 + 1 - /, _4_ 1 + / ' + w^ (15.64) This difference in amplitude was caused by the difference in frequency between resonance and minimum impedance. This frequency difference may be determined by transforming the points Pi and P3 into the Z plane by (15.52) and subtracting them arithmetically. In order to express the coordinates of any point in the U'-plane by its real and imaginary components let W = i?o + jXo Now the coordinates of P3 may be expressed as, i..=.[l-^l- ('-/]; Xn = 0 (15.65) (15.66) The coordinates of Pi are similarly given by 1 R, = a I - Xo = —7] 1 + 1 m 1 + &\ (15.67) PERFORM. I XCE fXDEX OF QUARTZ PLATES 243 The coordiuales of these two points represent values of II'; now (15.52) may be solved for Z. a - yW Substituting in values for «, f3, 7 and 8 given by (15.59), we find ^^'-R.^- '^•^^ = ' T^i^^r^ = -'' (r, + x.r + Rl ^''-''^ The real component of Z must be zero since the function of Z is coincident with the I'-axis. By substituting values of Ro and A'o from (15.66) and (15.67), we find _ (C02 — Wl) V/^ also M [1 - Vl - {v/<^y] (co2 — coi) v/o^ M 1 + ('J 1 Subtracting (15.71) from (15.70) to get Aco we have Aco M A where A ^ Vl - (v/a)- - Vl + (v/ay (ri/ay (15.70) (15.71) (15.72) (15.73) W2 — COl Now the lim .4=1. When- = ^, then Aw = W (15.79) + R.. . Co R ;] ('*'i)-\k-<{'*m 246 BELL SYSTEM TECHNICAL JOURNAL Substitute these values in (15.60) and (15.61) to obtain the coordinates of the center of the circle, PFo, and the radius, 6 ^ 6 \ I o ^ 4 \ \ 1- z u ^ 2 u Q. \ \ 0.8 0.6 \ 04 02 \ \ V \ \ 4 6 8 10 20 40 60 80 100 Fig. 15.12 — Inherent error of the P.I. meter due to tuning for an indication of minimum impedance rather than unity power factor. Rewriting (15.84) we have r [i - 4/1 - jsi Per cent Error of P.I. = 100 1 - ^ ' — tlj^ (15.86) [Z' + li-l If Co » C. then A = M ,+g+M{ ;) (15.87) The error given by (15.86) is plotted in Fig. 15.12 as a function of P^ 248 BELL SYSTEM TECHNICAL JOURNAL Fig. 15.13 — Exterior view of crystal Performance Index meter. 15.83 Frequency Errors As suggested in Section 15.81, conformal representation simplifies the mathematics required for the determination of frequency errors. In Section 15.81, the difference between the resonant frequency and the minimum im- pedance frequency was computed for the equivalent crystal circuit. This same procedure could be used to compute the frequency difference between the antiresonant frequency of the generalized oscillator circuit (Fig. 15.2) and the minimum impedance frequency of the P.I. meter for the same value of Ct. Comparison of the frequency of these two oscillators is plotted in Fig. 15.6 together with the measured values obtained from a 'Tierce" and a "Tuned Plate" oscillator. This frequency comparison involves setting up two circle diagrams, one for Zab (Fig. 15.2) and one for the impedance, Zi (Fig. 15.3) similar to Fig. 15.10. The impedance equations for both Zab and Zi would be arranged such that they have the same function of Z in PERFORMANCE INDEX OF QUARTZ PLATES 249 order to have a common Z plane. In this way, transformation of operating points such as the "anti-resonant frequency" operating point (P4) for the Zab impedance circle, could be subtracted from the "minimum impedance frequency" operating point (Pi) of the P.I. meter impedance circle in the common Z-plane. As in section 15.81, the frequency difference represents the arithmetic difference between P4 and Pi in the Z-plane in terms of "^ (w — coo). As an example, look at the calculated curve in Fig. 15.6. This curve was computed for the case when R^ is negligible. The deriva- tion of (15.51) given in section 15.92 precisely follows the procedure just described. It is of interest to note that (15.27) may also be derived by Conformal means. It is more laborious than the usual circuit equations of section 15.2; however, it does provide a check of the methods used. 15.84 Errors of Other Approximations Further consideration of P.I. meter errors leads to the assumptions made in the derivation of (15.27). The derivation of (15.27) assumed that the resistor, Ra, was non-reactive. While actually it can be made essentially noninductive, we have neglected the effect of the input capacitance of the attenuator that is shunted across its terminals. The error from neglecting this capacitance in (15.28) is given by the following expression Per Cent Error = 100 [l - ^ ^^^ JT ^V ^ (l^-^^) Where Qa equals the reactance of the shunt capacitance of the attenuator, Cu , divided by the magnitude of the calibration resistor, Ra . It is interesting to note in the derivation of (15.28) that Ra was assumed to be very much less than Xca which introduces an error of. .89) Per Cent Error = 100 1 - i/i 4. f— T (15 15.90 Derivation of Circuit Equations 15.91 Derivation of Equation (15.42) Other equations used in this paper may best be developed from Fig. 15.3. By analysis of the input impedance, Zi , the basis for the development of (15.42) is as follows: 250 BELL SYSTEM TECHNICAL JOURNAL From Fig. 15.2 Zi = ~ -\-^ F ^-— F ^ . /.'''^'^:!xnn (15-90) By substituting (15.48) and (15.49) in (15.90), we find jco \_Ct Co L^'^Ai + jLi{co- — CO.]) J J This may be expressed in the form, Zi CO^l If ' '^'i Ct ' Co J (15.92) — coZi(aj- — CO2) +i<^^-^i where CoC< Z? Co + C, Now adding and subtracting wo to the — — term we have Co z, = '1 I T r^"^^ ~ "2) I (w^ — CO?) (aj2 — C'Ji)"! — wLiioo — CO?) + joo Rl Rationalizing (15.93) and equating it to Re and substituting in Lx ^ (obtained from (15.50)), we find Co(co2 — col Rl Re- ^ . ... ^ . .. ^j5^^^ |_co| — COt J l_coi Mj 15.92 Derivation of Equations (15.51) c//(i (15.44) Equation (15.51) makes possible the theoretical computation of the fre- quency difference between the generalized oscillator and the minimum im- pedance frequency adjustment of the PI meter. The derivation assumes that i^L is negligible and that the total capacitance across the crystal ter- minals is lumped in series with the crystal. The impedance, Zab , in the generalized oscillator. Fig. 15.2, was given by (15.77). This equation may be expressed as follows: H +jM] + Z ZABOOeiCo + Ct) = [-^.-^]-^^ (15.95) PERFORMANCE INDEX OF QUARTZ PLATES 251 From this expression, as previously explained, (see Section 15.81), the following values for a and rj may be determined. M Co 2 Co + Ca (15.96) ri = 1 The anti-resonant impedance of Zab is represented by 0 — P4 in Fig. 15.10. The left hand term of (15.95) for the Pi operating point becomes ZABO^eiCo + Ci) = a -i- y/a^ — 1 Substituting this value in (15.95) and solving for Z, we find (15.97) = -/mT jM Ks Co L Co + c, ] (15.98) where 1 1 / 1 A3 - - + 2 y 1 - -2 If Kz is expanded and all except the first two terms are neglected, Z may be expressed as Z = -j M 1 + + (-c~:)" M (15.99) The next step is to obtain a similar expression to (15.99) only for the minimum frequency impedance of Z, in Fig. 15.3 with p disconnected. For 1 this application S = r~^ and T = x . Substituting these values, as well as jwLt those in (15.59), in (15.75), we have (15.100) j+jz From this equation, values for a and 77 may be determined as described in Section 15.81. Mt Co + Ct y] = — ^ T Ct (15.101) 252 BELL SYSTEM TECHNICAL JOURNAL By the same procedure just described for evaluating Z from the im- pedance expression Zab , the value of Z corresponding to the minimum impedance operating point, Pi , (Fig. 15.10) must be determined. For this operating condition Zi may be expressed by (15.65). The coefficients of this operating point are given by (15.67) with the above values of a and 7] (Equation 15.101). Utilizing (15.68) to solve for Z, we get M [1 + 4/^ + If] Zi = —j / 7TT ^ —- — (minimum impedance) (15.102) Zi - Z\ = ^' (Ac) where Aco = the difference in radian frequency between the frequency of oscillation in the generalized oscillator (anti-resonant frequency of the im- pedance, Zab) and the frequency of oscillation in the PI meter (minimum impedance frequency of the impedance, Zj) Aco CO2 — 0)1 .^ + t)[: + ^r7±]"''^'""('^^») (15.51) It is of interest to note that (15.102) becomes (15.44) when the value of Z (15.69) is introduced. Lightning Protection of Buried Toll Cable By E. D. SUNDE A theoretical study of lightning voltages in buried telephone cable, of the liability of such cal^le to damage by lightning and of remedial measures, together with the results of simulative surge tests, oscillographic observations of light- ning voltages and lightning trouble experience. Introduction PRACTICALLY all of the toll cable installed since 1939 has been of the carrier type and most of it has been buried in order to secure greater immunity from mechanical damage. It was realized, however, that bury- ing the cable would not prevent damage due to lightning and that, on ac- count of their smaller size, more damage was to be expected on the new car- rier cables than on the much larger voice-frequency underground cables then in use. Moreover, when damage by lightning does occur, such as fusing of cable pairs or holes in the sheath, it is not so easy to locate and repair as on aerial cables, since excavations may have to be made at a num- ber of points. Studies were therefore made of the factors affecting damage of buried cables by lightning and remedial measures were devised and put into effect in cases where a high rate of lightning failures was anticipated on new installations, or was experienced with cable already installed. Most of the cable installed was thus provided with extra core insulation, and shield wires were plowed in on many of the new routes. It was recognized early in these studies that more effective lightning pro- tection might be secured by providing the lead sheath with a thermoplastic coating of adequate dielectric strength and an outside copper shield, and that such cable might be required in territory where the earth resistivity is very high. This type of cable has recently been installed on a route in high- resistivity territory where experience has indicated that other types of con- struction would probably be inadequate and, since it has advantages also from the standpoint of corrosion and mechanical protection, it may be used also where lightning is not of such decisive importance. When lightning strikes, the current spreads in all directions from the point where it enters the ground. If there are cables in the vicinity they will provide low resistance paths, so that much of the current will flow to the cables near the lightning stroke and in both directions along the sheath to remote points. The flow of current in the ground between the lightning channel and the cables may give rise to such a large voltage drop that the breakdown voltage of the soil is exceeded, particularly when the earth 253 254 BELL SYSTEM TECHNICAL JOURNAL resistivity is high. The Hghtning stroke will then arc directly to the cables from the point where it enters the ground, often at the base of a tree. Fur- rows longer than 100 feet have been found in the ground along the path of such arcs. The current entering the sheath near the stroke point is attenuated as it flows towards remote points. Since a high earth resistivity is accompanied by a small leakage conductance between sheath and ground, the current will travel farther the larger the earth resistivity. The current along the sheath produces a voltage between the sheath and the core conductors, which is largest at the stroke point. This voltage is equal to the resistance drop in the sheath, between the stroke point and a point which is sufficiently remote so that the current in the sheath is negligible. Since the current travels farther along the sheath the higher the earth resistivity, this resist- ance drop will also increase with the earth resistivity. The maximum voltage between sheath and core is thus proportional to the sheath resistance and, as it turns out, to the square root of the earth resistivity. Carrier cables now being used are of smaller size and have a higher sheath resistance than full-size voice-frequency cables, and for this reason they are liable to have more lightning damage, particularly when the earth resistivity is high. To secure experimental verification of certain points of the theory pre- sented here, staged surge tests were made on the Stevens Point-Minneapolis cable, one of the first small-size buried toll cables to be installed. The results of these tests, which have already been published, are here compared with those obtained theoretically, on the basis of the earth resistivity meas- ured at the test location. Lightning voltages on this cable route were also recorded by automatic oscillographs and the results of these observations are also briefly discussed together with the rate of lightning failures experi- enced on this and other routes. The first part of the paper deals with voltages between the cable con- ductors and the sheath due to sinusoidal currents and surge currents. The second part deals with the liability of damage due to excessive lightning voltages and with certain characteristics of lightning discharges of impor- tance in connection with the present problem, such as the impedance encountered by the lightning channel in the ground, the rate of lightning strokes to ground and to buried structures and the crest current distribution for such strokes. In the third part remedial measures are discussed, to- gether with lightning-resistant cable. I. Voltages Between Cable Conductors and Sheath 1 . 1 General Cable installed in the ground is designated "underground" when placed in duct, and "buried" when not in duct. Buried cable is sometimes pro- LIGflTXIXG PROTECTION OF BURIEn TOLL CABLE 255 vided with steel tape armor for protection against mechanical damage. While such armor may also reduce voltages due to low-frequency induction, mainly because of the high permeability of the steel, this is not true in the case of lightning voltages. The magnetic field in the armor due to lightning current in the cable is rather high, and the corresponding permeability fairly low. The armor resistance is, furthermore, quite high compared to that of the sheath, so that the effect of the armor may be neglected in considering lightning voltages. The tape or armor is usually separated from the sheath by paper and asphalt, but is bonded to the sheath at every splice point. Strokes to ground, or to the cable, may give rise to large currents in the armor and thus to excessive voltages between the armor and the sheath some distance from bonding points. The resulting arc may fuse a hole in the sheath or dent it, due to the explosive efTect of the confined arc, and insulation failures may be experienced on this account. Such failures are not considered here since they are usually confined to a single point and are thus of less importance than insulation failures due to excessive voltages between the core conductors and the sheath, which may be spread for a considerable distance along the cable. For protection against corrosion, buried cables are usually jute-covered (asphalt, paper and jute) and in some cases have thermoplastic or rubber coating. The leakance of jute-covered sheaths is usually large enough so that the cable may be assumed to be in direct contact with the earth and the leakance is, furthermore, increased at the time of lightning strokes by numerous punctures due to excessive voltage between sheath and ground. This effect is large enough so that even rubber-covered cable may be re- garded as in direct contact with the soil in the case of direct strokes and sometimes also for strokes to ground in the vicinity of the cable, as discussed later. In order to calculate the voltage between the sheath and the core con- ductors of a buried cable, due to a surge current entering the sheath or the ground in the vicinity of the cable, it is convenient to consider at first a sinusoidal current. The voltage due to a unit step current may then be obtained by operational solution and, in turn the voltage for a current /(/) of arbitrary wave shape, by means of either one of the integrals: Vij) = [ /'(/ - t)S{t) dr h = \ J{i - T)S\r) dr (1) where Sif) is the voltage due to unit step current and S'{t) the time deriva- tive of this voltage. The second of the above integrals is more convenient in the present case. 256 BELL SYSTEM TECHNICAL JOURNAL Photographic observations indicate that a hghtning discharge is usually initiated in the cloud by a so-called "stepped leader," except in the case of discharges to sufficiently tall structures where this leader is initiated at the ground end. After the leader reaches the ground, or the cloud in the case of a tall structure, a heavy current "return stroke" proceeds from the ground toward the cloud at about yV the velocity of light. The main surge of current in the return stroke, which usually lasts for less than 100 micro- seconds, may be followed by a low current lasting for -yg- second or so. 220 200 180 160 M 140 a B cd 120 rH 100 50 40 1 pK, /oj > A0Z.0, and sinh (icoyY = \ exp {iuiy)"' expression (12) becomes r(0) = /At- -^ (/co)--e-^'"^'* (13) 260 BELL SYSTEM TECHNICAL JOURNAL For sufficiently low frequencies, so that {io^y)' < 1, /co < i/pK, iw < Ro/Lo , and sinh (icjoy)' = (/aj7)"(l + iu)y/6), expression (12) becomes / R 1 ^^^^ - 2 a^ + ^M^a;)^(/ + a;7 '6)^ ^^"^^ where a = v/2p, jS = RoCo For small values of time, corresponding to large values of /co, the function S'(t) defined before, as obtained by operational solution of (13) is (i)' 5'(/) = 7?7- ^ ( -J e-^'- (15) For large values of time, corresponding to small values of iu\ the function is obtained by operational solution of (14) and equals: (Reference 10, pair 542) where, with {Gt/yY = u: h{ii) = —ie " erf Ciu) = —j^ e ' e cIt (17) erf being the error function. Values of the function Ii{u) are given in Table I. In Fig. 2, curve 1 shows the function S'(f) calculated from (15), and curve 2 that calculated from (16), for a cable of 1.4" diameter, using con- stants as indicated in figure. The constants apply to a cable on which measurements have been made of the voltage between sheath and core conductors, at a location where the measured earth resistivity was 400 meter-ohms. The function S'(t) corresponding to equation (12) is obtained with sufficient accuracy by drawing a transition curve, 3, between curves 1 and 2. If the impedance s is taken equal to the direct-current resistance R of the sheath and if the velocity of propagation along the sheath and along the core are assumed to be infinite, so that T = {iuaY and To = {io^lHy, the following expression is obtained ^'« - 2-(?T7) fe)' (>«) In the following it will be shown that (18) is accurate enough for practical purposes. The wave shape of the current in lightning strokes may be approximated by an expression of the form: /(/) = /(e,-"' - e-''). (19) LIGHTNIXG PROTECTIOX OF BiKIED TOLL CABLE 261 With a = .013-10', b == .5-10', a current of the wave shape used m the measurements referred to above is obtained. This current reaches its crest in 10 microseconds and deca}-s to its half-value in 65 microseconds, and is fairly representative of the average wave shape of lightning stroke currents. In the following, the voltages are for convenience referred to a crest value of 1000 amperes, which is obtained when / = 1150 amperes, the latter current being the initial value of each of the two exponential component currents included in (19). / "L h(u) = c-"' Table I t f''" dr = H when u < .1 1 le erf (hi) 2u when II > 10 u Vf ^(«) u -y/y •/«(«) 0 0 1.5 .4283 .1 .0993 2. .3014 .2 .1948 2.5 2232 .3 .2826 3.0 1782 .4 .3599 3.5 1496 .5 .4244 4. 1293 .6 .4748 4.5 1141 .7 .5105 5. 1021 .8 .5321 6. 0845 .9 .5407 8. 0630 1.0 .5381 10. 0503 A more complete table for the range between u = 0 and ii = A\?> published in Bericht- everhandliingen Akademie der Wissenschaften. Leipzig, Malh-Phys. Klasse, Vol. 80, 1928, pages 217 to 223. In Fig. 3, the dashed curve shows the measured voltage and curves 1 and 2 that calculated for the above surge current for two conditions. In cal- culating curve 1, S'{i) was taken as curve 3 of Fig. 2, the voltage being obtained by numerical integration in accordance with (1); in calculating curve 2, z is taken as the direct-current resistance of the sheath and the velocities of propagation are assumed to be intinite, so that S'(t) is given by (18). In the latter case the following e.xpression is obtained for the voltage by solution of (1): Vit) = ^^f^ ^,^ [a-'^h{Vat) - b-''h{Vbt)] (20) where the function Jiin), u = -x/ at or \^bt, is defined as before. 262 BELL SYSTEM TECHNICAL JOURNAL Comparison of curves 1 and 2 of Fig. 3 shows that (20) is accurate enough for practical purposes, so that the voltage may be taken proportional to the direct -current resistance of the sheath. Since /3" is only 2.5 per cent of a\ propagation in the core-sheath circuit may be neglected in comparison with propagation along the sheath-earth circuit, so that it is permissible to take the voltage proportional to the square root of the earth resistivity. 1 T' 1' 1 Mljl !itl i;'i \K ' 1 i'' 1,, ■'I 1 llli Hit 'III !iil 't|! !lil III! \\v : 1 ij ' 11 ,ii ! :, j I'll III! '■ii 1(1 ' 1 I'll pi . J i' 1 ! ! 1 I ; 1 ; ' I ; 1 ill! 1 1 II ^f ^^^- - 1 , - - 1 — r 1 1 i 1 1 i ! , ■^ — ,Jli_ •3000 — 2 -r. ■ . — >— == . , ]f^ r s i il J X**^ y 1 i J^ s ' j,*f 1 y f\ ' \ _L ' .j^ 1 (/ 3 V jT 1 y- \ 1 /I 1 i 1 / \' - \ 1 ~/ 4^-- 1 ) 1 1 1 '' ■ { I N 1 ! i ! 2000 / i ! Tv 1 , 1 1 \ f 1 . : ' 1 N ; I "w-^ i J S'(t) 1 / ' III' 1 ^X "^ 1 i / / , ' 1 1 i ^ i // 1 1 \ 1 \, If [ ; j j : 1 . ■ II ! 1 1 1 \ '/ 1 ] 1 j ; : 1 V 1 1000 ^ ' i 1 ' t ; "S. * s. ■ 1 / 1 1 1 1 : 1 .,_ - ^, ' / Ml 1 ! ■ '^ 1 / III 1 ' : ' s ' 1/ : 1 ' ' 1 i ' ! / 1 1 1_L !!■' 1 \r. 1 1 1 1 ! 1 ! - / ' ' ■ 1 1 \ I 1 ■ ; 1 , ■ f 1 I 1 ■ ft \ ,,, „ 1 1 i 1 i ill: ■ 12 5 10 20 50 100 t - Microseconds Figure 2 — Approximate solution for S'ii). 1 : Calculated from formula for small times. 2 : Calculated from formula for large times. 3: Transition curve giving approximate solution for S'{i). Earth resistivity, p = 400 meter-ohms. Radius of cable, a = 1.75 cm. Sheath thickness, h = 2.4 mm. Sheath resistance, R = .92 ■ 10~^ ohms meter. Core-sheath cap. Co = .96- 10^^ fd meter. Core-sheath resist. Ro = R ^ .92 •lO"^ ohm meter. Velocity I'o = 2-10* meter 'sec. Velocity!' = 1 • 10^ meter sec. Furthermore, from (20) it is seen that when a and b are divided by the same factor k, so that the wave shape of the current remains the same but the duration of the current is increased k times, the voltage is increased -x/k times. Thus, if the surge current had reached its crest value in 20 micro- seconds and its half-value in 130 microseconds, the voltage would be in- creased by s/l, and the crest voltage would have been reached after 120 rather than 60 microseconds. LIGIITXIXG PROTJXTIOX OF BURIED TOLL CABLE 263 If the breakdown voltage of the core insulation is assumed to be 2000 volts, the above cable would be able to withstand a stroke current of about 30,000 amperes before the insulation is punctured. From Fig. 1 it is seen 1000 800 600 400 200 20 40 60 80 100 Microseconds 120 140 Figure 3 — Comjjarison of measured, shown tjy dashed curve, and calculated voltage between sheath and core conductors, shown h)- curves 1 and 2, for surge current as shown and cable constants as given in Fig. 2. 1 : Calculated from formula including skin-effect in sheath and finite velocity of propa- gation. 2: Calculated from formula based on d-c resistance of sheath and assuming infinite velocity of i)ropagation. that in about 50 per cent of all strokes the crest current exceeds 30,000 amperes. When there are two cables, each will provide shielding for the other, and the shielding effect may be calculated as for shield wires (Sec. i.?)). Fre- quently the cables are of equal or nearly equal size and are close together. It is then accurate enough to use the parallel resistance of the two sheaths in calculating the voltage, which will be practically the same in both cables. 264 BELL SYSTEM TECHNICAL JOURNAL 1.5 Direct Strokes — Lightning Voltages Along Cable It was shown above that with only a mmor error the impedance z may be taken equal to the direct-current resistance of the sheath and that the propagation constants may be taken as r = {iu:a)\ To = {io:^y With this modification expression (10) becomes: ^W = T ^ ioc .e - 13 e \{io)) (21) 2 a — f3 The corresponding function S' is The voltage due to a surge current J{t), as obtained from (1), may be expressed as: ^(•^^' ^) = 0/ ^ ^, [«-g(«--v, 0 - ^gi^x, /)] (23) 2{a - 13) where, with a = a'.v or j8\v, gia,t) = f J{t-r)(^\ e'-'^'d. Jo XTTT"/ (24) For a current as given by (19), the latter integral may be expressed in terms of error functions of complex arguments, for which, however, no tables are available at present. Curves for the function g, as obtained by numer- ical integration are shown in Fig. 4. When the core conductors are connected to the sheath at .v = 0, the con- stants .4 and B of (6) and (7) are obtained from the following boundary conditions: At :v = 0, 1^0) = 0 so that .4 = -B. As before, B = -()( '^ ). The voltage between core and sheath is then given by: — IP ' 2 r^ - r! (25) JRa' r -(ioict)ix _ -(lO}l3)ix-l o \& e- r - r In this case the derivative of the voltage due to unit step current is: g-'^^''^'] (26) LJGIITXLXG PROTECT ION OF BL'RIED TOLL CABLE 265 g(0',t) ^ "^ \ /o / 7 h^^ ¥ ^ / ¥ / V -^ \ / V -vy V x -^ \ /> / /? X '^'v X Af /-- y^ y ^ ,■3 ^^^ 10 20 50 100 200 t - Microseconds 500 1000 r--"(^)'-"" Figure 4 — Function g{a , I) = I Jit Jo J{1) = /(e-" - c^'") a = 1.3 -10^ 6 = 5-10* / = 1150 amperes and Fo(.r, /) Ra^ fg(a'.v, /) - ^(^\r,/)] (27) 2(a - /J) In Fig. 5 the crest values of l'(-v, /) and Fo(x, /), calculated for the cable considered before, are plotted against .v, together with those observed in the tests. When the voltage l'(0, t) at the point where current enters the sheath is great enough to break down the insulation of a core conductor, the 266 BELL SYSTEM TECHNICAL JOURNAL latter will be in contact with the sheath by virtue of arcing. Under this condition, the voltage Fo(-v, /) between this conductor and the sheath will increase with distance along the cable as shown in Fig. 5. A maximum is reached a fairly short distance from the original fault, and beyond this point the voltage slowly decreases. After a puncture of the insulation where current enters the sheath, other failures may therefore occur, not neces- sarily at the point where Voix, t) is largest, but sometimes at points nearer or much farther away where the insulation may be weaker. A single lightning stroke may thus cause insulation failures over a considerable dis- tance along the cable. 100 80 60 40 20 V / • ^ — - ^^^ 1 ^ \\ / \\ / \y ^— - \ 1 ^ ^,,— - _ — ■ .2 .4 .6 .3 Distance Along Cable - Miles Figure 5 — Comparison of measured variation of voltage between sheath and core, along cable, as shown by points and dashed curve, with calculated variation shown by curves 1 and 2. 1 : Conductor not connected to sheath. 2: Conductor connected to sheath at point where surge current enters sheath. 1 .6 Direct Strokes — I 'oltage Due to Long Duration Current As mentioned before, a current of low value and long duration may exist on the lightning channel after the main discharge. This current is usually of such long duration that the resistance of the sheath must be considered in calculating the current propagation along the cable. The propagation constant in that case becomes r - [(i? + io^QG^ = [t + P (£)' (28) R, L and G being the unit length resistance, inductance and leakance of the sheath-earth circuit. Neglecting propagation in the core-sheath circuit, LIGHTXIXG PROTECT lOX OF Bl'lUED TOLL CABLE 267 the voltage between core and sheath at the stroke point due to a smusoidal current Ji becomes, f'i(O) = Ti? = J^{£) J^(P + R/^^' (29) The corresponding voltage for a unit step current /i is: ri(0, /) = JiR(^ erf (Rt/i:)^ (30) where erf is the error function. For large values of time, when Rt/L > 1, (30) becomes The latter expression is valid when / exceeds about 2 milliseconds and thus applies for the long duration current of a lightning stroke, since the latter usually lasts for about 100 milliseconds. For a current of 1000 amperes, the core-sheath voltage for a cable of 1.4" diameter is about 700 volts. In many strokes the long duration current may be several hundred amperes, and a substantial voltage may then exist between core and sheath for .1 second or so. Thus, while this current component does not increase the crest voltage, it substantially increases the likelihood of permanent failure when the insulation is punctured by prolonging the current through the puncture. 1.7 Strokes to Ground Not Arcing to Cable Let it be assumed that the current enters the ground at the distance y from a buried cable and that conditions are such that it does not arc to the latter. The flow of current in the ground gives rise to an electric force in the ground along the cable, and thus to currents in the sheath and to voltages between core and sheath. When the earth is assumed to have uniform con- ductivity, the earth potential at the distance r from the point where current enters the ground is given by: Ve = JQoir) = Jp/Iwr (32) where p = Earth resistivity in meter-ohms r = (.V + y )' = Distance in meters The sheath current and the voltage between sheath and core may in this case be obtained from published formulas, provided propagation along 268 BELL SYSTEM TECHNICAL JOURNAL the core-sheath circuit is neglected in comparison with propagation along the sheath-earth circuit, which is permissible. The voltage between sheath and core conductor differs by the factor z/Z from the voltage between sheath and ground as given in Table II, case 3 of the paper referred to, z being defined as before and Z being the unit length self-impedance of the sheath-earth circuit. At a point opposite the lightning stroke, .r = 0, the voltage between core and sheath is in this case given by: U{0, y) = J^ f Qo(r)e-'- dx =jI^ *(ry) {3,3,) Z Jo Z ZTT where Z = T~/G and G is the unit length leakance of the sheath-earth cir- cuit. The leakance is given by the approximate expression: Vtt '' Taj G=(^^Jog-j (34) a being the radius of the sheath and log = log^. The function $(ry) is given by the approximate formula: HTy) = log ^^ (35) Inserting (34) and (35) in {ii), the latter expression may be written: r(0, y) = r(0, a)\{Ty) (36) where V{Q, a) = r(0) is the voltages when the current enters the sheath directly (y = a) and: X(ry) = (^log ~^'')/iog i/ra (37) where V = {iwv/lp)' — (/coa)"'. The rigorous solution of the time function corresponding to (36) would be rather complicated. Since, however, X is the ratio of two functions, each of which varies logarithmically with T, and thus varies only slightly with ico, an approximate solution is obtained by replacing /co with \/t in (37). For instance, the solution of an operational expression p~" is t"/nl while the solution of p~" log p is [\[/(\ -|- n) -f log l/t]t"/nl, \p being the logarithmic derivative of the gamma function. For representative values of n and t{n < 1, / < 10"'*), xp is less than 5% of log 1//, so that a good approximation is obtained by replacing p by 1// in log p, which in this illustration simulates the factor X(ry). With this approximation: r(0,y,0 -^ r(0, a, /)XLv(«//)^] (38) LIGHTXIXG PROTECTIOX OF BURIED TOLL CABLE 269 In Fig. 6 is shown the N'arialion in llie voUage calculated from (38), together with that ol:)served in the tests referred to before. That the meas- ured decrease in the voltage is smaller than calculated is due to the fact that the earth resistivity at the test location increases with depth. Earth resistivity measurements made by the four-electrode method show that the resistivity is about 400 meter-ohms for electrode spacings up to about 20 feet and then gradually increases, reaching about 700 meter-ohms at 300 feet, 1200 meter-ohms at 1000 feet and approaching 1500 meter-ohms for & o ■p a o u an 1 60 I • 40 \ ^ \j -^^i ?,0 2^ !!^3^^~^ — - =^^- 0 20 40 60 80 100 Distance from Cable - Feet Figure 6 — Reduction in voltage between sheath and core with increasing distance from cable to point where current enters the ground. 1: Measured when remote ground representing cloud is at a distance of 1000 ft. 2 : Calculated for uniformh' conducting earth with remote ground at distance of 1000 ft. 3: Calculated for uniformly conducting earth with remote ground at infinity. large electrode spacings. The measured variation in voltage with separa- tion is in substantial agreement with that calculated for an earth structure of this type in the manner outlined in Section 1.9. 1.8 Discharges Between Clouds In considering voltages due to discharges between clouds, the lightning channel is assumed to parallel the cable. Due to magnetic induction, the lightning current will give rise to an impressed electric force along the cable sheath. Without much error it may be assumed that there is no impressed force outside the exposed section of the sheath and that the electric force in the e.xposed section due to a sinusoidal current / is E (x) = 270 BELL SYSTEM TECHXICAL JOURNAL JM, where M is the unit length mutual impedance of the lightning channel and the sheath. The resulting sheath current I(x) is obtained from (6) and (7), when E is replaced by £ , To by T and A'o by K, the characteristic impedance of the sheath-earth circuit. The constants A and B are found by observing the voltage between sheath and ground is zero at .v = s/2. The electric force along the core is given by E(x) = RI(x), and the voltage between the core conductors and the sheath is obtained by a second ap- plication of (6) and (7), the constants .1 and B being determined from the condition that the latter voltage must equal zero at x = s/2. The voltage between the core conductors and the sheath at the distance x along the cable beyond one end or the other of the lightning channel projection on the cable is then: ^,, , JRMT's L{x) = r -Tflj- -vx 2Z(r- - To l^s (' - ^""'^ -T^'^'- ^""'] (^') the sign of the voltage beyond one end of the channel being opposite to that beyond the other end. Since F » To , the last bracket term may be neglected. It was shown previously, that attenuation along the core-sheath circuit within a distance of one mile, which is representative of the length s, is quite small, so that 1 — e~ °'' ~ T(tS. With these modifications: JRMT-s ^ ^ The earth-return impedances M and Z are given by the following approximate expressions: " M = '^ . ^? . (41) Itt (If + y-)(iu}a)' where a is defined as before, log = logt and: It = height of lightning channel above ground y — horizontal separation of lightning channel from cable The expression for M holds when a{Jf -(- y")" > 5, a condition which is satisfied in the important part of the frequency range. Inserting (41) and (42) in (40): LIGHTNING PROTECTION OF BURIED TOLL CABLE 271 where ^ o o -v/2 (-^4) Comparison with (21) and (23) shows that in this case: ^(•^^'^) = o/"\^ ^g(^-^^ 0 (45) 2(o! - ^) In the above solution, /j, was assumed constant. Actually it changes slightly with frequency and, for reasons mentioned before, it is accurate enough for practical purposes to replace iw with 1// when calculating /x. The maximum voltage is obtained at x = 0, i.e., at a point opposite one end or the other of the lightning channel, and comparison with (22) shows that this voltage differs from that obtained in the case of a direct stroke by the factor /x, since /3' 3) and proceeding as before, the voltage between core and sheath due to a stroke at the distance y may be written: F(0, y, t) = \ (0, a, t) — — (48) P2 A(a) -f (pi — P2)fj-[a) where V{0, a, () is the voltage for a direct stroke calculated for an equivalent earth-resistivity: p, = p2X(a) + (pi - p2)M(a) (49) and where X(v) = log [(1 + ry)/ryl (50) UGIirNING PROTECriON OF BCRIJW TOLL CABLE 273 dx (51) r" 1 / \ / ^ —ar —Vx m(>') = / - ^ « ^1 1 + 3'(to + r) = ^"g 3.(^„ + r) ~ "^^'"^ ^"^"^^^ ^^^^ ^e'^'^My) when 703/ » 1 (53) In applying the above expressions, a rough value of pe is first assumed in calculating \(o) and ^^((7) and a more accurate value next obtained from 100 50 10 2 3 .1 100 Distance from Cable - Feet 1000 Figure 7— Reduction in voltage between sheath and core with increasing distance from cable to point where current enters ground. 1 : Upper layer of 400 meter-ohms and 30 ft. depth. Lower layer of 4000 meter-ohms and infinite depth. 2: Uniformly conducting earth. 3: Upper layer of 1500 meter-ohms and 30 ft. depth. Lower layer of 150 meter-ohms and infinite depth. (49). If the value of pe thus obtained differs materially from the assumed value, a second calculation may be required. In the expressions for X and p. the resistivity pe is to be used in calculating T, the latter being taken as {v/2pjy where / is the time to crest value of the voltage as before. In Fig. 7 is shown the manner in which the voltage decreases with in- creasing separation for three assumed earth structures. The resistivities and the depth of the upper layer were selected such that the equivalent earth-resistivity, and thus the voltage in the case of a direct stroke, is the same in all cases and equal to 1000 meter-ohms. It will be noticed that in the case where the resistivity of the lower layer is high, the voltage due to a stroke at a distance of 200 ft. is 50% and at a distance of 1000 ft., 25% of the voltage due to a direct stroke. When the cable is small, insulation failures may thus be occasioned by strokes to ground at considerable 274 BELL SYSTEM TECHNICAL JOURNAL distances from the cable, although the resistivity near the surface up to ciepths of say 50 ft. is only moderately high. It is seen, however, that when the earth resistivity of the lower layer is low, failures due to strokes to ground not arcing to the cable are rather unlikely, even when the earth resistivity near the surface is rather high. On account of the higher surface resistivity, however, a greater number of strokes would be expected to arc to the cable for a given equivalent re- sistivity, than when the conductivity is uniformly distributed. On the other hand, many strokes which would arc to the cable if the earth were uniformly conducting may channel through the surface layer to the good conducting lower layer, so that the incidence of direct strokes is reduced on this account. Experience indicates that the latter factor tends to predominate, so that lightning damage is not ordinarily severe when the resistivity is low at depths beyond 20 ft. or so. In the case of discharges between clouds the coupling between the light- ning channel and the cable depends, in the frequency range of importance, to a great extent on the resistance of the lower layer. Thus, when the resistivity of the lower layer is very high the voltages may possibly give rise to insulation failures in the case of small cables, while this is not likely to occur when the resistivity of the lower layer is small or when the earth structure is uniform and of moderately high resistivity. 1.10 Cables with Insulated Sheaths Assume that a short length A.v of insulated sheath is placed on the ground and that a voltage is applied between the sheath and a remote ground. When the applied voltage is greater than the breakdown voltage of the insulation, arcing to ground will take place at numerous equidistant points, provided the insulation and the earth are assumed to be uniform. The voltage between the sheath and adjacent ground increases from zero at a point where arcing takes place to a maximum value midway between two points at which arcing occurs, the maximum value being equal to the breakdown voltage of the insulation. Midway between two arcing points the potential in the earth (referred to infinity) may with negligible error be calculated as though the leakage current through the numerous arcs were uniformly distributed along the sheath. This potential in the ground would then be AI/GAx, where A/ is the total leakage current and 1/GA.Y the resistance to ground of the sheath without insulation, G being the unit length leakage conductance. Midway between the arcing points the potential of the sheath to a remote ground is then: V = Vo + M/AxG (54) LIGHTNING PROTECTION OF BURIED TOLL CABLE 275 Let dh/dx be the leakage current through the arcs and dli/dx the leakage current due to capacity C between the sheath and the adjacent ground. For sinusoidal currents the following equations then hold, when Z is the unit length impedance and G the unit length leakance for a sheath in direct contact with the earth: '^ + '^)^^Vo=V (55) dx dx / G - (7o + h)Z = ^ (56) -^^^ = Fo (57) tooC dx In the last equation it is assumed that the voltage between sheath and ground is equal to the breakdown voltage of the insulation, although this is not true in the immediate vicinity of the arcs. Eliminating I' the following equation is obtained: where : (t^^»^) + (tp-.^)- 11,1 ,, io^CG = — + ^— , or: Y = (58) Y G ixoC ' G + ^wC Equation (58) is satisfied when: /o = Aoe~^'' + Boe (59) where V and Fi are the propagation constants for a sheath in direct contact with the ground and for an insulated sheath without breakdown, respec- tively. F = {GZy, Fi = {YZf For a sheath of infinite length the Bq and Bi terms vanish, so that: /(.v) = /„ + /i = A,e-^' -f .4ie-'^^ (60) The constants .lo and .li are obtained from the following boundary conditions: At .V = 0 /(,-) = 7(0) = A, + .li (61) As .T — > =c I{x)^A,e-'^' =^e-'^' (62) 276 BELL SYSTEM TECHNICAL JOURNAL where Ki = {Z/Vy is the characteristic impedance of the insulated sheath without breakdown. From (61) and (62) V, . .,.. Fo Ax =4 A,= no) - ^ (63) Ai Ai So that; -Vx , Vo , ~T^i -Vx\ 7(.v) =I{0)c-'^ ^'-^{e-'^' -e-^') (64) Ai For a rubber insulated cable the breakdown voltage T^o would be in the order of 30,000 volts and the characteristic impedance K\ would be in the order of 100 ohms. The maximum current which could flow on the sheath without breakdown, Fo/i^i , is then about 300 amperes, while in the case of an average lightning stroke the current 7(0) would be about 15,000 amperes (i.e. 30,000 amperes total). The first term in (6-i) gives the attenuation along a sheath in direct contact with the ground. The current given by this term would diminish from 15,000 amperes to about 2,000 amperes within a distance of \ mile or so, for a typical lightning stroke wave shape and an earth resistivity of 1000 meter-ohms. At distances of several miles from the stroke point the first term will vanish and the current will be de- termined by the second term, since exp ( — Fi.v) will vanish much more slowly than exp (— F.v). In the case of a stroke to ground the impressed electric force in the ground along the sheath is £o(-v) = -dVlx)ldx (65) where V ^ is the earth potential due to the lightning stroke current and is given by Fe(x) = . , /^ .,., (66) lr{x- -f y-y Instead of equation (58), the following equation is obtained for the cur- rents in the sheath Writing £oCv) — cEo{x) + (1 — f)£oCv), the solution of the latter equation may be written as the sum of two solutions of the form given by (6) and (7). After the constants Aq , Bo applying to the current /o and the constants Ai and Bi applying to the current /i have been determined from the boundary LIGHTNING PROTKCriON OF BURIED TOLL CABLE 111 conditions, in the same manner as before, the total sheath current may be written in the form: /(.v) ^ c-/„(.v) + (1 - c)I-Sx) (68) where I,, is the current for a grounded sheath, /,• the current entering a j^erfectlv insulated sheath by virtue of its capacity to ground and c is given by: c = TVT'" = ro/T\.(0) (69) where T' is the potential ditTerence between sheath and adjacent ground without breakdown at .v — 0, which is substantially equal to the earth po- tential. The above relationship for the constant c is obtained by applying equation (57) at .v = 0 to the general solution for I\ , V being given by: r = G + icoc ^ [ £u(-v)e~''^' dx (70) iwc Jo The voltage between core and sheath of an insulated cable may be written in a similar manner when Ig is replaced by Vg , the voltage for a cable in direct contact with the ground, and /^ replaced by F,- , the voltage for a cable insulated from ground. From (68) and (69) it will be seen that wdien F(,(0) is much greater than the breakdown voltage of the insulation, the current entering the sheath is nearly the same as for a sheath in direct contact with the ground. Thus, when the earth resistivity is 1000 meter-ohms, and the stroke current 30,000 amperes, the earth-potential at a distance of 30 meters (100 feet) is 160,000 volts. The impulse breakdown of the insulation may be in the order of 30,000 volts, so that the current entering the sheath will be substantially the same as for a cable in direct contact with the soil. When the earth-potential at .V = 0 is only slightly larger than the breakdown voltage of the insulation, however, the current entering the sheath through punctures in the insula- tion is fairly small. In the above derivation the voltages and currents were assumed to vary sinusoidally which, of course, is a rather rough approximation in a phenome- non where breakdown occurs after the voltage reaches a certain instan- taneous value. While the derivation is not accurate, it does indicate under what conditions an insulated cable behaves like a cable in direct contact with the ground. 1.11 Oscillographic Observations of Lightning Voltages To obtain data on the characteristics of lightning voltages in buried cable, five magnetic string oscillographs were installed for one lightning 278 BELL SYSTEM TECHNICAL JOURNAL season along a 50-mile section of the Stevens Point-Minneapolis route. The oscillographs, which were arranged to trip where the voltage exceeded 100 volts, recorded lightning voltages due to some 600 strokes on 38 days, the Weather Bureau average being ii thunderstorm days for this region in the same months. The character of the voltages varied widely from sharp transients of a few millisecond duration to slowly changing voltages lasting .2 seconds from one zero value to the next, voltages due to multiple dis- charges being quite common, the interval between voltage peaks in such cases being in the order of .1 second. Of the disturbances, 90% lasted for more than .1 second, 50% for more than .4 and 10% for more than 1.25 second, the maximum duration being 2.3 seconds. By way of comparison, the observed duration of discharges to a tall structure (3) were, in respec- tively 90, 50 and 10% of all cases, in excess of .08, .3 and .6 second, the maxi- mum duration being 1.5 second. The maximum voltage recorded was 940 volts and was probably due to a stroke to ground near the cable. About 2% of the voltages were in excess of 500 volts, most of these and the lower volt- ages being due to discharges between clouds, as indicated by the opposite polarity of the voltages at the two ends of the test section. The wave shape of the voltages at the ends of the section were much the same as at inter- mediate points, even for the sharpest surges recorded, the attenuation along the core-sheath circuit being quite small. It is possible that substantially higher voltages than observed may obtain in the case of severe discharges along a path parallel to and directly above the cable, and that such voltages may produce cable failure if the core insulation is below normal. While the oscillographs were arranged to trip on 100 volts, a smaller voltage was recorded in 40% of all cases, as the peaks were too fast to be recorded by the type of oscillograph used. It is also possible that for this reason fast voltage peaks in excess of the maximum given above may have escaped measurement. II. Lightning Trouble Expectancy 2.1 General In estimating the liability of a cable to lightning damage, it is assumed below that once the core insulation is punctured, as it is likely to be at several points, at least one permanent failure will occur. The lightning trouble expectancy curves presented here thus give the number of times lightning damage is likely to occur, without consideration of the extent of the damage on each occasion. Each case of lightning damage usually involves several pairs and, based on experience, repair of each such case would require about four sheath openings. Damage due both to direct strokes and strokes to ground is included. Discharges between clouds have been neglected as a LIGHTNING PKOTECTION OF BURIED TOLL CABLE 279 source of lightning damage, however, as the voltages are likely to be in- sufficient unless the insulation is below normal. The curves of lightning trouble expectancy calculated here are significant only if troubles on a long cable route are considered over a period of several years, so that the mile-years covered are in the order of 1000 or more. 2.2 Incidence of Strokes to Ground To estimate the lightning trouble expectancy it is necessary to consider the incidence of strokes to ground in the vicinity of the cable, the number Figure 8 — Map showing the average number of thunderstorm days per year. of such strokes that will arc to the cable and cause damage in this manner and the number that will give rise to failure without arcing to the cable. Magnetic link measurements (14) indicate that high tension transmission lines will be struck by lightning about 113 times per 100 miles per year, on the average, the minimum incidence in one year being about one half and the maximum about 1.6 times the average value. The above average inci- dence is based on observations covering about 1600 mile-years and applies for lines traversing areas where s 1000 meter-ohms r = .08 (Jp)^ meter r = .047 (Jp)' meter = .26 (/p)4eet = .15 (JpY ieet where / is in kiloamperes. 284 BELL SYSTEM TECHNICAL JOURNAL These values are, of course, of an approximate nature, and are only indicative of what may be expected under average conditions. In some cases the breakdown voltage of high-resistivity soil may be substantially lower than assumed, while that of low-resistivity soil may be noticeably higher. 2.4 Crest Currenl Distribution for Strokes to Ground When the earth resistivity is taken as high as 5000 meter-ohms and the breakdown voltage of the soil is taken as high as 5000 volts/cm, the re- sistance encountered by the channel on the ground for a current of 25,000 amperes is about 250 ohms. If the lightning channel were a long conductor already in existence at the initiation of the return stroke and capable of carrying the stroke current without being fused, the current would be propagated upward with the velocity of light and the surge impedance of the channel would be in the order of 500 ohms. Due to the resistance in the ground the current would then be some 30% smaller than for a stroke to an object of zero resistance to ground. However, the lightning channel may not be regarded in the above manner, but as a conductor which is gradually prolonged at about 1/10 the velocity of light, and the impedance of the channel is then much larger, perhaps 5000 ohms. The surge im- pedance of a long insulated conductor having unit length capacitance C is {\/Cv), V being the velocity of propagation. When energy is required to create the conductor, so that the velocity of propagation is reduced, the impedance is increased. Because of the high impedance of the channel, the resistance encountered in the ground may, therefore, be neglected as regards the effect on the crest current. The crest current distribution curve for strokes to transmission line ground structures may thus be used also in the case of strokes to ground, although a different distribution curve is obtained for those of the strokes to ground which arc to buried cable (Section 2.7). 2.5 Failures Due to Direct Strokes and Strokes to Ground In calculating the number of failures due to direct strokes and strokes to ground, the earth is assumed to be a plane surface. A tree placed at random may attract a lightning stroke toward a cable or it may divert it from the cable and the net effect of a large number of trees along a route of substantial length is likely to be small. This is also true for variations in the terrain. W^hen A^ is the number of lightning strokes to ground per unit of area, and 5 the length of the cable, the number of lightning strokes on both sides of the cable within y and y + dy is : dN = 2Ns dy (83) LIGHTNING PROTECTION OF BURIED TOLL CABLE 285 A lightning stroke at the distance y will cause cable failure when the crest current i exceeds a certain value which depends on the distance: i = f(y) ■ (84) The fraction of all lightning strokes which has a crest current larger than i will be designated Po(i)- The fraction of the lightning strokes dN which will cause cable failures is then: dn = dNP,{i) = 2NsPo(i) dy ' (85) The number of cable failures along the length 5 due to all lightning strokes to ground up to the maximum distance V that need to be considered is: n = 2Ns [ PS) dy (86) For the purpose of computation it is convenient to change the variable in the latter integral from y to i. With y = f~ (i) = y{i), di = dy-f\y), {q = /(O) and / = f{y), the following integral is obtained: n = 2Ns FPo(0 - / y{i)Po(i) di (87) In (87), / is the maximum stroke current that needs to be considered and may actually be replaced by infinity, as will be evident later on. The current /o , which is the minimum current that will cause insulation puncture in the case of a direct stroke, may readily be determined from the breakdown voltage of the insulation and the calculated voltage between core and sheath per kiloampere, in the manner illustrated in Section 2.4. In order to evaluate the integral of (87), it is divided as follows: n = 2Ns YPoiD - f ' Vi(OA'(/) di - f y-2{i)P',{i) di\ (88) When i < ii , failures of the cables will be due to arcing of the stroke to the cable and when i > /i , failures will occur before arcing takes place, due to the leakage current entering the sheath. Within each of the above two ranges the relationship of y to i is different and is designated yi{i) and Viii), respectively. As already shown, failures due to arcing will take place when: vi(/) < qipiV (89) where q is defined as before and p is the earth resistivity in meter-ohms. In Section 1.7 it was shown that failures due to leakage current will occur when i > io/\{y) 286 BELL SYSTEM TECHNICAL JOURNAL where Hy) = log (y^^) / log (i/ra) r being the propagation constant of the sheath-earth circuit and a the radius of the sheath. The solution of the latter equation for y is : 1 /2pA* 1 y = y^(y ^ wT^v^"^^ = [~) /"-o/i _ 1 (90) where m = log (l/cF) r ^ {v/2pty per meter z^ = 1.256-10" henries per meter t = 10^'* sec. = time to crest of core-sheath voltage. When (89) and (90) are equated, the following expression is obtained: i^i/""" - 1) = (~y (91) The value of i which satisfies the latter equation is the value ii defined above, and is shown in Fig. 9 as a function of mio for various values of When (89) and (90) are inserted in (88), the latter integral may be ex- pressed as follows: n = 2Nsf^ (q[H{io) - H{h)] + rjS G(h, miA (92) where the current is in kiloamperes and: TV = Number of strokes to ground per square meter .y = Length of cable, in meters q ^ .08 when p < 100, q ^ .047 when p > 1000. H(i) = -[ PPo(l)di (93) G{l, mto) = ^mioll _ J - j mioli _ J (94) The first term in (94) equals YPo(I) = V2(/)-Po(/). Since Po(0 is nega- tive, the above integrals will have positive values. The term q[H{io) — H(ii)] of (92) gives the portion of failures due to direct strokes while the term involving the function G gives the portion of failures due to ground strokes not necessarily arcing to the cable although LIGHTNING PROTECTION OF BURIED TOLL CABLE 287 they may do so (i.e. many of the currents in excess of ii may arc to the cable, although this is not essential in order to produce cable failure). 1000 00 u s 05 O 500 200 100 50 20 10 =1 (2ta/2 q V ' 100 500 1000 mi, Figure 9 — Solution of the equation: '"-"=:ey If strokes to ground not arcing to the cable were neglected as a source of failures, the number of failures would equal: n,i = INsp'qHii,) (95) If, on the other hand, the dielectric strength of the earth were assumed to be infinite, so that none of the strokes to ground would arc to the cable, the number of failures would equal INsp^ (^-^ G{i,, mk) (96) 288 BELL SYSTEM TECHNICAL JOURNAL 10 .5 H(i) .2 .1 .05 .02 .01 ^ \ \ \ \ \ \ \ \ \ \ \ 10 20 50 100 200 i - Klloamperes Figure 10 — Function Hiyi) when Po(0 is approximated by: Po{i) = exp(— /fei), k = .038 per kiloarapere. H{i) = k-^'limh-"' + ix^ erfc(^z-)^] From Fig. 1, it is seen that Po(0, as represented by curve 1, is nearly a straight Une on semi-log paper and may, therefore, be approximated by: Po(0 ^ e-'' (97) With k = .038 per kiloampere, a straight line is obtained which coincides with curve 1 at ? = 0 and i = 100 kiloamperes, and this value of k has been used in the following. The functions H and G obtained with this approxi- mation are shown in Figs. 10 and 11. In obtaining these integrals, the up- 10-1 5x10-2 2x10-2 10-2 5x10-3 G(l,ini ) o 2x10 -3 10 -3 i- kiloamps \o-20 5a\\ 75\ Vy ioo\ \ \ I O50 \\j 1 \ ^ \ s^OO \ \ \ 1 \ \ \ \ 5x10-4 2x10-4 10-4 100 200 500 1000 Figure 11— Funrfinn G(i, ffiio) when Poii) is approximatoH by: Po{i) = exp{ — ki); k = .038 per kiloampere. Git, mia) = k f 289 e at 290 BELL SYSTEM TECHNICAL JOURNAL per limit / may be replaced by infinity, and it is also seen that the first term in (94) then vanishes. The lightning trouble expectancy as calculated from (92) is shown in Fig. 12 as a function of the earth resistivity for various sheath resistances. The curves are based on 2.4 strokes to ground per square mile, which is approximately the number of strokes per square mile during 10 thunder- storm clays. The number of thunderstorm days per year along a given route is obtained from Fig. 8 and thus the number of times lightning failures would be expected during one year. 2.6 Expectancy of Direct Strokes The incidence of direct strokes to the cable may be obtained from (96) with 7*0 = 0 kiloamperes. The number of strokes arcing to the cable is thus Ua = 2Nsp^qH{0) (98) The cable will thus attract strokes within an effective distance. • y = p^qH(0) (99) p < 100 meter-ohms p > 1000 meter-ohms y = .365 p' meters y = .22 p' meters = 1.2 p" feet = .7 p* feet 2.7 Crest Current Distribution for Direct Strokes The fraction of the strokes to the cable having crest values in excess of i is given by: P{i) = H{i)/H(0) (100) ©^ ♦ = 2 ( - 1 e-'' + erfc (iky and is shown by curve 2 in Fig. 1. It will be noticed that a buried cable attracts a greater proportion of heavy currents than a transmission line, because of the circumstance that heavy currents to ground arc for greater distances. 2.8 Lightning Trouble Experience As mentioned before, lightning damage may be due to denting or to fusing of holes in the sheath, or to excessive voltages between the sheath and the cable conductors. Only the latter form of lightning failures have been considered here, since they predominate for cable of the size now being used, particularly in high-resistivity areas, and are likely to extend for a considerable distance to both sides of the point struck by lightning and are LIGHTNING PROTECTION OF BURIED TOLL CABLE ' in thus more difficult to repair. For full-size cable in low-resistivity areas, however, insulation failures are more likely to occur as a result of sheath denting. For instance, along the 300-mile Kansas City-Dallas full-size cable route, where the earth resistivity is in the order of 100 meter-ohms, and where there are some 50 thunderstorm days per year, failures over a period of about 15 years have occurred about .5 times per 100 miles per year. Of these troubles 85% were due to sheath denting as a result of arcing between the tape armor and the sheath. Based on (99), 100 miles of cable would attract lightning strokes within an area of .5 square mile, so that the cable would be struck about six times per 100 miles per year, when the number of strokes per square mile per year is 2.4 per 10 thunder- storm days (Section 2.2). The rate of lightning failures experienced on this route may thus be accounted for by assuming that about 7% of the strokes, i.e. currents in excess of 90 kiloamperes as obtained from curve 2, Fig. 1, will produce sheath denting severe enough to cause insulation failure, while about 1%, i.e. currents in excess of 140 kiloamperes, will cause insulation failure due to excessive voltage. The latter value is in substantial agree- ment with that calculated for a full-size cable when the earth resistivity is assumed to be 100 meter-ohms. It is evident from the above examples that in low-resistivity areas lightning troubles will not be a problem, and this is also borne out by experience on other routes installed in such terri- tory during the last few years. All cable installed in high-resistivity territory since 1942 has been provided with doubled core insulation and with shield wires, in spite of which con- siderable damage has been experienced on some routes, as between Atlanta and Macon. This appears to have been due partly to the circumstance that in many cases the insulation m splices and accessories has not been equal to that obtained in the cable through the use of extra core wrap, and that in some cases damage has been due to holes fused in the sheath due to arcing between the sheath and the shield wires. As an example, along the Atlanta-Macon route there are some 70 thunderstorm days per year and the average effective earth resistivity is about 1300 meter-ohms. The corresponding estimated rate of direct strokes to the cable is about 20 per 100 miles per year. It is estimated, in the manner outlined in section 3.0, that only stroke currents in excess of 80 kiloamperes are likely to damage the cable, so that on the basis of curve 2, Fig. 1, cable failures would be ex- pected to occur about 2 times per 100 miles per year. The actual rate of trouble experienced on this route over two years has been about five times higher, so that some 50% of the strokes to the cable; i.e., currents in excess of 30 kiloamperes or so, appear to have caused cable failures, most of which occurred in splices and accessories. 292 BELL SYSTEM TECHNICAL JOURNAL It is evident from the above examples that careful examinations of trouble records are required before the observed rate of lightning failures can be adequately compared with that obtained from theoretical expectancy curves. If the cable as well as splices and accessories actually have a di- electric strength as assumed in the calculations, it is likely that the average rate of failures due to excessive voltages experienced over a long period will not be any greater than estimated from these curves. Based on experience, an average of 4 sheath openings is required to repair damage caused by excessive voltage between the sheath and the cable con- ductors, as compared to about 2 sheath openings when the damage is due mainly to denting and fusing of the sheath, as in the case of full-size, tape- armored cable in low-resistivity territory. Although the damage may be confined to one point, it cannot usually be located by a single sheath opening. III. Remedial Measures 3.1 General From Fig. 12 it is evident that the rate of cable failures to be expected, and hence the need for remedial measures, depend greatly on the earth re- sistivity. Experience has indicated that lightning damage is likely to be encountered even when the surface resistivity is fairly low, provided the resistivity beyond depths of 10 or 20 ft. or so is very high. Considerably less trouble has been experienced where the resistivity below this depth is low, even where the surface resistivity has been high. The lightning stroke may then channel through the surface layer to the good conducting lower layer, so that direct strokes are not experienced as frequently in spite of the high surface resistivity. As a guide m applying protective measures, earth resistivity measurements are usually made along new cable routes. The curves given in Fig. 12 may also be used to find the lightning trouble expectancy when extra core insulation, shield wires or both are used. Thus when the insulation strength is doubled the effect is the same as if the sheath resistance is halved. If the shield wires reduce the voltage by a shield factor 77, the effect is the same as if the sheath resistance is multiplied by 77. Considering direct strokes only, curve 2 in Fig. 1 may be used to find the percentage reduction in lightning strokes that will damage the cable, when the stroke current which the cable is able to withstand is in- creased by extra insulation or shield wires. 3.2 Extra Core Insulation One method of reducing failures caused by lightning strokes to buried cables is to increase the insulation between the cable conductors and the LIGHTNING PROTECTION OF BURIED TOLL CABLE 293 sheath, no extra insulation being required between individual cable con- ductors. This has already been done for most new installations. The cable itself, cable stubs, loading cases, and gas alarm contactor terminals are all provided with sufficient extra insulation to double the dielectric strength between cable conductors and sheath. For a cable like that on which the measurements referred to before were made, such a measure would increase the stroke current which would damage the cable from 30,000 to 60,0000 amperes and would reduce the number of direct lightning strokes that could cause failure by direct arcing to the sheath to about 20 Z 1 .6 ^5 .25 '*j/inlle Q 9 H a n o >> 3 a \/2eV/m; (12) or dj = 0 (13) when V < \/2eV/m (14) Here jo is the cathode current density, V is voltage with respect to the cathode, T is the absolute temperature of the cathode in degrees Kelvin, and Vx, ijy, and Vz are the three velocity components; dj is the^element * This expression neglects the effects of electron collisions, which may actually make the current density smaller. 310 BELL SYSTEM TECHNICAL JOURNAL of current density carried by electrons which have velocity components about Vx, Vy, V,, lying in the little range of velocity dvx dvy dvz. The reason for restriction (12) is that if an electron starts with zero thermal velocity from the cathode, it will attain the velocity given by the right side of (12) by falling through the potential drop V. As electrons cannot have velocities smaller than this, we have (13) and (14). By integrating (11) with appropriate limits we obtain a more specialized but very useful expression J < J'n = ]■ . /, , 11600V\ sin- 6 (15) For usual values of voltage, unity in the parentheses is negligible, and we can say that if all the electron paths approaching a given point in an electron beam lie within a cone of half angle 6, the current density j at that point cannot be greater than a limiting value jm which is proportional to the ELECTRON LENS DEFLECTING PLATES CATHODE Fig. 3— Parameters important in determining spot size in a cathode ray tube. cathode current density, to the vohage, to sin-^, and inversely proportional to the cathode temperature. Let us see what this means in some practical cases. Figure 3 shows a cathode ray tube. The electron stream has a width W at the final electron lens, and is focused on a screen a distance L beyond the lens. The half angle of the cone of rays reaching the screen cannot be greater than sin e= d = W/2L (16) Suppose the spot must have a diameter not greater than d. Let the spot current be i. Then from (15), 4i ^ . 1' + U^) ^W/2Lf i < 1 + HfL^) iW/2Lf (17) PHYSICAL LIMITATIONS I IV ELECTRON BALLISTICS 311 Thus if for a given spot size we want to increase the spot current, and if we are limited to a given cathode current density because of cathode Hfe, we must make 1' larger, W larger or L smaller. Making W larger increases both lens and deflection aberrations. Making L smaller means that for a given linear deflection we must increase the angular deflection, and this too tends to defocus the spot. Because of these limitations, it is necessary to avail ourselves of the remaining variable and raise the operating voltage V'. Another illustration, perhaps a little more subtle, of the efTect of thermal velocities, lies in the analysis of the properties of a type of vacuum tube amplitier known as the "deflection tube". In such a device, illustrated in Fig. 4, an electron stream from a cathode is accelerated and focused by a lens and deiiected by a pair of deflecting electrodes so as to hit or miss an out- put electrode. Such a device may be used as an amplifier. Now it is obvious that as the output electrode on which the beam is focused is moved farther away from the deflecting plates, a given deflecting voltage will produce a greater linear deflection of the beam at the output. CATHODE /^ "t:, U output ' electron' ^-^ -ELECTRODE LENS DEFLECTING PLATES Fig. 4 — Amplifying tube making use of electron deflection. As this at first sight seems desirable; it has been seriously suggested not only that this be done, but that an elaborate electron optical system be interposed between the deflecting plates and the output electrode to amplify the deflection. The merit of a deflection tube is roughly measured by the deflecting voltage required to move the beam from entirely missing the output elec- trode to entirely hitting the output electrode, and, of course, moving the output electrode farther away or putting lenses between the deflecting plates and the output electrode doesn't reduce this voltage at all. As we improve the deflection sensitivity by these means, we simply increase the spot size at the same time. Focusing our attention on the beam between the deflecting plates, we appreciate at once that the electron paths through each point will be spread over some cone of half angle 9, and that to change from a clean miss to a clean hit we must deflect the electrons through an angle of at least 26, regardless of what we do to the beam afterwards. Returning for a moment to equation (15), we see that it says the current density can be less than a certain limiting value depending on 9. Yet 312 BELL SYSTEM TECHNICAL JOURNAL expression (15) was obtained by integrating a supposedly exact expression. What does this inequahty mean? The answer is that for the current to have the limiting value, electrons of all allowable velocities must approach each part of the spot from all angles lying within the cone of half angle 6. When the average current density in the spot is less than the limiting current density, the possibihties are (a) Electrons are approaching each point in the beam from all angles, but along some angles only electrons which left the cathode with greater than zero velocity can reach the spot. (b) Electrons leaving the cathode with all velocities can reach the spot, but at some portions of the spot electrons don't come in at all angles within the cone angle 6. Fig. 5- -Relation between nearness of approach to limiting current density and fraction of current utilized. Thus, we can have less than the limiting current either because electrons do not reach the spot with all allowable velocities or from all allowable angles. Of course both factors may operate. We can easily see how lens aberrations, which we know are present in all electron-optical systems, can prevent our attaining the limiting current density. There is a more fundamental limitation, however. It can be shown that even with perfect focusing, we must sort out and throw away part of the current in order to approach the limiting current density, and we can even derive a theoretical curve for the case of perfect focusing re- lating the fraction of the limiting current density which is attained to the fraction of the cathode current which can reach the spot. Figure 5 shows such a curve which applies for voltages higher than, say, 10 volts. Usually, the failure to approach the limiting current density is chiefly caused by aberrations, and in ordinary cathode ray tubes the current density in the spot may be only a small fraction of the limiting value. A very close approach to the limiting current density has been achieved in a PHYSICAL LIMITATIONS IN ELECTRON BALLISTICS 313 special cathode ray tube designed by Dr. C. J. Davisson of the Bell Tele- phone Laboratories. When we become thoroughly convinced that these equations expressing the effects of thermal velocities very much cramp our style in designing electron-optical devices, as good engineers we wonder if there isn't, after all, some way of getting around them. I don't think there is. The suggestion illustrated in Fig. 6 is a typical example of such an attempt. We know that in a strong magnetic field electrons tend to follow the lines of force. Why not use a very strong magnetic field with lines of force approaching the axis at a gentle angle to drag the electron stream toward the axis? An electron off axis traveling parallel to the axis certainly will be dragged inward by such a field. The catch is that the field pulls the electron in because it makes the electron spiral around the axis. As the beam con- verges and the field becomes stronger, the pitch of each spiral decreases and the angular speed of each electron increases. Finally, if the field is strong enough, all the kinetic energy of the electron is converted from forward ELECTRON^ V^ .^OT — — — MAGNETIC' PATH > ?<~ — >C^O __ LINES OF FORCE Fig. 6 — Reflection of an electron by a magnetic field with strongly converging lines of force. motion to revolution about the axis; the electron ceases to move into the field and bounces back out. It may be some small consolation to know that very high-current densities can be achieved by this means, but only because in their flat spiralling the electrons approach a spot at much wider angles with the axis than the small inclination of the lines of force. Space Charge Limitations In electron beam devices using reasonably large currents, the space charge of the electrons is a very serious source of trouble both in compli- cating design of the devices and in limiting their performance. Let us begin our consideration right at the electron gun, the source of electron flow in many devices such as cathode ray tubes and certain high- frequency tubes. Electron guns are sometimes designed on the basis of radial space charge limited electron flow between a cathode in the form of a spherical cap of radius fo and a concentric spherical anode a distance d from the cathode. It can be shown that by use of suitable electrodes external to the beam, radial motion can be maintained between cathode and anode along 314 BELL SYSTEM TECHNICAL JOURNAL Straight lines normal to the cathode surface. A hole in the anode electrode will allow the beam to emerge from the gun. Because of the change in Fig. 7 — Electron gun utilizing rectilinear flow. CATHODE ANODE SPACING, d/rQ Fig. 8 — Relation between perveance, angle of cone of flow, and cathode-anode spacing. field near the hole, the hole acts as a diverging electron lens. Figure 7 illustrates such a gun.^^ The curves shown in Fig. 8 relate to this sort of PHYSICAL LIMITATIONS IN ELECTRON BALLISTICS 315 electron gun. They are plots of a factor called the perveance, which is deiined as P = //F''' (18) (that is, current divided by voltage to the 3/2 power) as a function of 9, the half angle of the cone of flow, and d/ro, the ratio of cathode-anode spacing to cathode radius. In getting an idea of the meaning of the curves, we may note that a perveance of 10~^ means a current of 1 milliampere at 100 volts. It is obvious from the curves that to get very high values of perveance, that is, high current at a given voltage, 6 must be large and the cathode-anode spacing must be small. Making d large means that electrons approach the axis at steep angles; aberrations are bad and the beam tends to diverge rapidly beyond crossover. Moving the anode near to the cathode means that the hole which must be cut in the anode to allow the beam to pass through must be large, and cutting such a large hole in the anode defeats our aim of getting higher perveance; we can't pull electrons away from the cathode with an electrode which isn't there. Further, for ratios of spacing to cathode radius less than about .29, the lens action of the hole in the anode causes the emerging beam to diverge, which would make the gun unsuitable for many applications. When we build guns for small currents at high voltages, such as cathode ray tube guns, space charge causes little trouble; when we try to obtain large currents at lower voltages, we find ourselves seriously embarrassed. Suppose we now turn our attention to the effect of space charge in beams when the beam travels a distance many times its owm width. Consider, for instance, the case of a circular disk forming a space charge limited cathode. Suppose we place opposite this a fine grid, and shoot an electron stream out into a conducting box, as illustrated in Fig. 9a. We immediately realize that there will be a potential gradient away from the charge forming the beam. In this case, the gradient will be toward the nearest conductor; that is outwards, and the electron beam will diverge. How can we overcome such divergence? One way would be to arrange the boundary conditions in such a fashion that all the field would be di- rected along the beam instead of outwards; this might be done by sur- rounding the beam by a series of conducting rings and applying to them successively higher voltages as in 9b, the voltages which would occur in electron flow between infinite parallel planes with the same current density. Another way in which the same effect may be achieved is through use of specially shaped electrodes outside of the beam, as shown in Fig. 9c." In maintaining parallel flow by these means, the electric field due to the elec- trons acts along the beam, and increases continually in magnitude with 316 BELL SYSTEM TECHNICAL JOURNAL distance from the cathode. We can in fact calculate the potential at any distance along the beam by the well known Child's law equation / = 2.33 X 10~'^F''V^' a . CATHODE kl I I I I I I • .-AAAi^VNAi'V%A/^\\^^.\AiA/V\iv^/X^i'Vv/vi^A/V^iAA/4/\VvKA^Xv\A/^ -11 ' ' ' + Fig. 9 — Avoiding beam divergence by means of a longitudinal electric field. Here V is the anode voltage, x the cathode -anode spacing, / the current in amperes and A the cathode area. Suppose we take as an example .1 = 1 cm- / = .01 amp. X = 10 cm PHYSICAL LIMITATIONS IN ELECTRON BALLISTICS 317 Then V = 5,700 volts Thus to maintain parallel motion of the modest current of 10 milliamperes spread over an area of one square centimeter requires 5,700 volts. More- over, the requirement of distributing this voltage smoothly along the beam would make it very difficult to put the beam to any use. One means for mitigating the situation is to use an electron lens and direct the beam inward. Of course, the beam will eventually become par- allel and then diverge again, but by this means a fairly large current can be made to travel a considerable distance. Some calculations made by Thomp- son and Headrick^- cover this type of motion, with an especial emphasis on the problem in cathode ray tubes, in which the currents are moderate. In order to confine large currents into beams, an axial magnetic field is sometimes used, as shown in Fig. 10 Here a cathode-grid combination shoots a beam of electrons into a long conducting tube. A long coil around the tube produces an axial magnetic field intended to confine the electron CONDUCTING TUBE COIL cathodeI i ^ > iz=: I'I'hr i^k^nvyxv V v v y x x v v y x yyv y Fig. 10 — Avoiding beam divergence by means of a longitudinal magnetic field. paths in a roughly parallel beam. The radial electric field due to space charge will cause the beam to expand somewhat and to rotate about the axis. As the magnetic field is made stronger and stronger, the electrons will follow paths more and more nearly straight and parallel to the axis. For a given current and voltage, there is one sort of physical limitation in the strength of magnetic field we need to get a satisfactory beam. It is another effect that I wish to discuss. Suppose we have a very strong magnetic field, in which the electrons travel almost in straight lines. We know, of course, that the radial electric field is still present, and this means that the potential toward the center of the beam is depressed; this in turn means that the center electrons are slowed down. This slowing down of course increases the density of electrons in the center of the beam. The result is that if for some critical voltage or speed of injection we increase current beyond a certain value, the process runs away, the potential at the center of the beam drops to zero, and another type of electron flow with a "virtual cathode" of zero electron velocity at the center of the beam is estabhshed. Thus, although the magnetic field 318 BELL SYSTEM TECHNICAL JOURNAL has overcome the divergmg effect of the space charge, we still have a space charge limitation of the beam current. C. J. Calbick has calculated the value of this limiting current. ^^ If the beam completely fills a conducting tube at a potential V with respect to the cathode, the limiting beam current is independent of the diameter of the beam and is / = 29.3 X 10~°F'^^ (20) If the beam diameter is less than that of the conducting tube, the limiting current is lower. But perhaps we can completely overcome the effects of space charge. Suppose we put a very little gas in the discharge space. Then positive ions will be formed. Any tendency of the electronic space charge to lower the potential and slow up the electrons will trap positive ions in the potential minimum and so raise the potential. Thus the gas enables us to get rid of the the slowing up effect of the space charge as well as its diverging effect. Before we congratulate ourselves unduly, it might be well to make sure about the stability of an electron beam in which the electronic space charge is neutralized by heavy positive ions. Langmuir and Tonks, in their work on plasma oscillations, introduced a concept, extended later by Hahn and Ramo, which enables us to investigate this problem. The concept is that of space charge waves. It is found that in a cloud of electrons whose net space charge is neutralized by heavy, relatively immobile positive ions, small disturbances of the electron charge density produce a linear restoring force; and this, together with the mass of the electrons, makes possible a type of space charge wave which may be compared roughly with sound waves, although much of the detailed behavior of space charge waves is quite different from that of sound waves. We may express a disturbance in an electron beam in terms of these space charge waves and then examine the subsequent history of the disturbance as a function of time. This has been done'^ and the perhaps surprising result is that even when the electronic space charge is neutralized by hea\y positive ions, the flow tends to collapse if the current is raised above a limiting value / = 190 X 10""]'''' (21) It is true that this current is 6.5 times the limiting current in the absence of ions, but it is a limit nevertheless. If this limit in the presence of ions seems unnatural, perhaps we should recall a mechanical analogy. Consider a vertical long column subjected to a load F. If we subject it to a sidewise force aF proportional to F, as shown in Fig. 11a, the behavior on increasing F will be a gradual deformation (analogous to the space charge lowering of potential in the absence of ions) PHYSICAL LIMITATIOXS IX ELECT ROX BALLISTICS 319 ending in collapse. However, even if, as in lib, there is no sidewise loading and no bending during loading, we know from Euler's formula that beyond a certain loading the column will still collapse. This behavior is analogous to that of an electron beam in which the electronic space charge is neutralized by positive ions and there is no depression of potential in the beam. This space charge limitation either in the presence or absence of ions allows the passage of quite a large current through a tube, as the table below will show: Voltage Current, amperes, no ions Current, amperes, ions 1000 100 10 .927 .029 .009 6.01 .190 .060 We might therefore feel that the space charge is disposed of in a practical sense, and so it is in many cases.* f^ F I i .kf 1 A T=29.3y\0~\^^^ 1 = 190x10"^^^^ Fig. 11 — Comparison of limiting stable beam currents with and without positive ions. Power Dissipation Limitations Having talked about various limitations imposed by wave effects, aber- rations, thermal velocities and space charge on the electron flow in the beam itself, I want to close by discussing briefly a topic which seems hardly included in electron ballistics but yet is vital to any application in that field. I refer to the problems associated with power dissipation when electrons strike something and stop. This is a good deal like the problem imposed by suddenly coming down to earth while studying the sensations of a free fall. It is inevitable and may be fatal unless satisfactory provision is made for the dissipation of kinetic energy. What I want chiefly to bring out are the consequences of scaling a given electronic device down in size. If we change the size of each part of an * It appears that in many gas discharges, including those in which plasma oscillations are observed, the current is too high to allow persistence of the homogeneous flow upon which the plasma oscillation equations are based. 320 BELL SYSTEM TECHNICAL JOURNAL electron device in the ratio R, if we keep all voltages the same, and if we change all magnetic fields in the ratio 1/i?, electron current will remain the same (provided the cathode is still capable of giving space charge limited emission). Electron paths will remain exactly similar, though smaller; the power into the electron beam will remain the same, but what will happen to the power dissipation capabilities of the device and what will happen to the temperature? In a device cooled by radiation alone and with cool surroundings, the radiating area varies as Rr, and since the radiation per unit area varies as T^, the temperature will vary as JKT . In considering a case of cooling by conduction alone, think of a rod carrying a certain amount of power away. If all the dimensions of a rod are changed by a factor R, the length will be changed by a factor R, the cross sectional area will change by a factor R^, and if the thermal conductivity CURRENT, VOLTAGE, POWER TEMPERATURE, RADIATION ' COOLING LINEAR DIMENSION TEMPERATURE, CONDUCTION COOLING MAGNETIC FIELD Fig. 12 — Variation of magnetic field and temperature in scaling an electronic device. remains constant the temperature will vary as R~^. This is a faster rate of variation than in the case of cooling by radiation, and hence as the system is scaled to a smaller and smaller size, cooling by conduction will become negligible and radiation cooling only will remain eflfective and will determine the temperature. Figure 12 gives an idea of the variation of various quantities discussed. We want to make electronic devices smaller for a number of reasons; perhaps chiefly to reduce transit time and so to secure operation at higher frequencies. In doing this, we encounter the fundamental limitation of reduced power dissipation capabilities and increased temperature. What is the trouble? We have scaled everything. Or have we? The answer is, we have not. The electrons, atoms, and quanta are still the same size. Had we been able to scale these, we should have increased the heat con- ductivity and the radiating power of our device, and all would have been PHYSICAL LIMITATIONS IN ELECTRON BALLISTICS 321 well. As it is, if we make a tube for given power smaller and smaller, using the most refractory materials available we eventually reach a size of tube which will, despite our best efforts, melt, thaw, and resolve itself into a dew. CONCLUSION Perhaps after these somewhat gloomy words concerning physical lim- itations in electron ballistics, you may wonder how it is at all possible to surmount the difficulties mentioned. It certainly is not easy; all electronic devices represent compromises of one sort or another between fundamental physical limitations of electron flow on the one hand and structural com- plications on the other. In working with vacuum tubes one is perhaps troubled more by physical limitations, difficulties of construction, inade- quacy of materials and the lack of quantitative agreement between compli- cated phenomena and relatively simple theories than in any other part of the electric art. It is for this reason that a friend of mine twisted an old aphorism into a new one and said, "Nature abhors a vacuum tube". References Electron Microscopes 1. James Hillier and A. W. Vance: "Recent Developments in the Electron Micro- scope," Proc. I.R.E. 29, pp. 167-176, April, 1941. 2. L. Marton and R. G. E. Hutter: "The Transmission Type of Electron Microscope and Its Optics," Proc. I.R.E. 32, pp. 3-11, Jan. 1944. Thermal Velocities 3. D. B. Langmuir: "Theoretical Limitations of Cathode-Ray Tubes," Proc. I.R.E 25, pp. 977-991, Aug., 1937. 4. J. R. Pierce: "Limiting Current Densities in Electron Beams," Jour. App. Phys., 10, pp. 715-724, Oct., 1939. 5. J. R. Pierce: "After Acceleration and Deflection," Proc. I.R.E. 29, pp. 28-31, Jan., 1941. 6. R. R. Law: "Factors Governing the Performance of Electron Guns in Cathode- Ray Tubes," Proc. I.R.E. 30, pp. 103-105, Feb., 1942. 7. J. R. Pierce: "Theoretical Limitation to Transconductance in Certain Types of Vacuum Tubes," Proc. I.R.E. 31, pp. 657-663, Dec, 1943. Space Charge 8. C. E. Faj^ A. L. Samuel and W. Shockley: "On the Theory of Space Charge Between Parallel Plane Electrodes," Bell Sys. Tech. Jour. 17, pp. 49-79, 1938. 9. I. Langmuir and K. Blodgett: "Currents Limited by Space Chargeb etween Coaxial Cylinders," Phys. Rev. 22, pp. 347-356, 1923. 10. I. Langmuir and K. B. Blodgett: "Currents Limited by Space Charge between Concentric Spheres," Phys. Rev. 24, pp. 49-59, 1924. 11. J. R. Pierce: "Rectilinear Electron Flow in Beams," Jour, of App. Phys. 11, pp. 548-554, Aug., 1940. 12. B. J. Thompson and L. B. Headrick: "Space Charge Limitations on the Focus of Electron Beams," Proc. I.R.E., 28, pp. 318-324, July, 1940. 13. C. J. Calbick: "Energy Distribution of Electrons within Dense Electron Beams," Bull. Am. Phys. Soc, 19, No. 2, p. 14 (April 28, 1944). 14. J. R. Pierce: "Limiting Stable Current in Electron Beams in the Presence of Ions," Jour. App. Phys. 15, No. 10, pp. 721-726 (1944). 15. A. L. Samuel, "Some Notes on the Design of Electron Guns," Proc. IRE 33, pp. 233- 240, April, 1945. Electron Ballistics in High-Frequency Fields* By A. L. SAMUEL THIS, the final lecture of a series on Electron Ballistics, is not a summary of the material which has been previously presented but rather it is an attempt to show how the ballistic approach can be extended to the analysis of high-frequency devices. Much that might otherwise be said about ultra- high frequencies cannot be said because of secrecy requirements. However, there is considerable material which can be presented, within the limits of the necessary security regulations, which may be of interest to those who are not already well acquainted with the subject. I will, perforce, not be able to say anything specific about actual devices utilizing the principles to be discussed. Many of the ultra-high-frequency devices which have come into use during the last few years have employed electron beams of one sort or another. These devices can be analysed in any one of a number of ways. For example, we can write the equation of space-charge flow. This ap- proach considers the electric charge as a continuous fluid subject to Poisson's equation. The small-signal theory of Peterson and Llewellyn is an example of this type of analysis. Or if we wish we can consider the various types of wave motion which can exist in a space-charge region. The space-charge- wave analysis of Hahn and Ramo as applied to velocity- variation tubes is an example of this. In addition there is an elect ron-baUistic approach to the problem and it is with this method that we will be concerned in the present lecture. Before we become involved in the details of the analysis, we should perhaps spend a few moments considering the relationship between these various methods. If we have an interaction taking place between electric fields and moving charges, we know at once from Newton's second law that the forces acting on the electrons must of necessity be equal and opposite to those acting on the fields. It is therefore a matter of small concern whether we consider the forces acting on the electrons and the effects of these forces on the electron motion or whether we consider the alteration in fields which the electron motion produces. We can, if we wish, compute the energy transfer to an electric field by the motion of an electric charge or we can compute the change in energy of the electron which accompanies this trans- * Originally presented on April 11, 1945 as the concluding lecture of a symposium on Electron Ballistics sponsored by the Basic Science Group of the American Institute of Electrical Engineers. 322 ELECTRON BALLISTICS IN HIGH -FREQUENCY FIELDS 323 fer. I was tempted to say "which results from this transfer" but this implies a cause and an effect, a notion wliich has no place in the present discussion. The dual aspect of any energy-transfer problem must always be kept in mind. Much needless discussion frequently arises between proponents of one point of view and those preferring the other when the only difference is one of language and both groups are really saying the same thing. The electron-ballistic approach yields a simple physical picture; it is capable of being applied to widely differing situations, but it is not well suited for a determination of the reactive contributions of an electron stream. Basic Concp:pts There are several concepts which we will hnd useful in our analysis. These concepts are extremely simple, so simple in fact that one is tempted to assume that they are well known. However, these concepts are so basic to the subject, and their results so far reaching that we must pause to consider them. The first is the concept of total current, as distinguished from its com- ponents. One way of writing Kirchhoff's second law is Div. / = 0 (1) This simply says that the total current entering or leaving any differential region in space is zero. This expression must of course be generalized by including displacement currents as proposed by Maxwell if applied to alternating currents. The current / is the total current density as here defined. An important consequence of equation (1), actually only an alternate way of stating it, is that the total current always exists in closed paths. Let us take a simple case of a two-element thermionic vacuum tube connected to a batter}-. \'isualize the situation existing if but a single electron leaves the cathode and travels to the plate. The electron takes a finite time to cross from the cathode to the plate. During this time a current exists, the magnitude being given by the relationship I = ev and according to our premise this current is the same in every part of the circuit. The current begins at the instant that the electron leaves the cathode and it ceases when the electron arrives at the plate. In the appar- ently empty region ahead of the electron there must exist a displacement component, numerically equal to the conduction, or perhaps we should say convection component accounted for by the moving electron. An ammeter, were there one sufficiently sensitive and fast, connected in the external leads would read a current during this same interval of time. I have chosen to talk about but a single electron to emphasize the electron- 324 BELL SYSTEM TECHNICAL JOURNAL ballistic aspect; however, the concept is much broader than this since it is not at all dependent upon a corpuscular concept of the electron. As a result of this property of the total current, the current to any electrode within a vacuum tube does not necessarily bear any relationship to the number of electrons which enter or leave it. Obviously then, currents can exist in the grid circuit of a three-element tube even though none of the electrons are actually intercepted by the grid. This current may have any phase rela- tionship to an impressed voltage on the grid so that the grid may draw power from the external circuit, or it may deliver power to the external circuit, all without actually intercepting any electronic current. The grid-current component resulting from the electronic flow between cathode and plate may equally well bear a quadrature relationship to the impressed voltage, in which case it will either increase or decrease the apparent interelectrode capacitance. If these effects seem queer it is because one is still confusing the electronic component with the total current. A second basic concept once stated becomes self-evident. This is to the effect that the only one thing which we can do to an electron is to change its velocity, that is, if we are to confine ourselves to the classical concept of an electron. We can change its longitudinal velocity, that is, alter its speed but not its direction other than possibly to reverse it, or we can introduce a transverse component to its velocity, that is, alter its direction as well as its speed. Thought of in this light all electronic devices in which a control is exercised over an electron stream are velocity-modulated devices. It might be argued that one could equally well say that all we can do is to change the electron's acceleration {derivative of velocity) or its position {integral of velocity). The singling out of velocity is in a sense arbitrary. It does, however, have some very interesting ramifications. I might digress for a moment to elaborate on this idea. Since some of the newer devices have been labeled velocity-modulation tubes, there is a perfectly understandable tendency on the part of the uninitiated to assume that these tubes differ from earlier known devices, such as, for example, the space-charge-control tubes, the Barkhausen tube or the magnetron in the fact that they employ velocity modulation. The real difference lies else- where as we shall see in a few moments. At the same time that these newer devices were introduced, there was introduced a new way of looking at something which is very old in the art. This newer viewpoint, to my way of thinking, constitutes a far greater fundamental contribution than do the specific devices which have received so much attention. The pioneers in this new approach: Heil and Heil, Bruche and Recknagel, the Varian Brothers, Hahn and Metcalf, to mention a few, and the many other workers who lost in the race to publish their independent contributions in this field — all of these people deserve the greatest of praise for their stimulating contributions ELECTRON BALLISTICS IN HIGH-FREQUENCY FIELDS 325 to our thinking. My only point in all this discussion is to emphasize that the basic method of acting on the electron stream has not really been changed at all. The entire matter is summarized in the original statement that the only thing which we can do to an electron is to change its velocity. Before going on to the next aspect of the problem there is a closely related concept which should be mentioned. This concept is that a change in the component of the velocity of an electron along one space coordinate does not introduce components of velocity in directions orthogonal to the first. For example, if an electron beam is deflected by a transverse electric field, there will be no accompanying change in the longitudinal velocity. The difficulty in the way of doing this in a practical case has nothing to do with the concept but only with the problem of producing unidirectional fields. Analyses of deflecting field problems which ignore the longitudinal components of the fringing fields are apt to be wrong. The problem of high-frequency deflect- ing fields has been treated in great detail in the literature and frequently with more acrimony than accuracy. One further note should be added at this point. In an earlier lecture it was pointed out that the magnetic effects of an electromagnetic field are in general very much smaller than the electric effects. We will not stop to prove that this is still true at the frequencies which now interest us but will accept it without further discussion. For our next concept we leave electron flow for a moment and consider the fields within a resonant cavity. You may very properly object that this has nothing to do with electron ballistics, and indeed it does not. However, we will find it necessary to discuss problems involving cavity resonators, and a failure to understand some of the properties of these circuit elements can cause a great deal of trouble. There are two conflicting approaches to this problem which I will attempt to reconcile. The physicist when first presented with the problem of a resonant cavity is inclined to say: This is a boundary value problem. The solution consists in writing Maxwell's equations subject to the conditions that the tangential com- ponent of E must be zero along the conducting walls. While a scalar and a mag- netic vector potential can be defined, the field is not related to the former in the simple manner used in electrostatic problems. The engineer, on the other hand, is inclined to say: This looks like an extension of the usual resonant circuit. A capacitance exists between the top and bottom walls of the cavity; charging currents will flow through the single turn toroidal inductance formed by the side walls. I would like to know what voltage difference exists between the top and bottom walls, and what currents exists in the side walls. Now, actually, I am maligning both the physicist and the engineer by my statements; nevertheless, there are these two approaches. Which is cor- 326 BELL SYSTEM TECHNICAL JOURXAL rect? Well, they both are. It is not correct to speak of an electrostatic potential within a resonant cavity; nevertheless, we may and do talk about the voltage between the top and bottom of a resonant cavity. What do we mean? Simply the maximum instantaneous line integral of the electric held taken along some speciiied path. In any practical device utilizing electron beams we are naturally interested in the path taken by the elec- trons. The fact that the line integral is different for different paths is of no great concern. We are interested in but one of these paths. We shall therefore have occasion to talk about voltages in cavities but we must always remember what is meant, and we must never for one instant forget that this voltage is not unique but that it depends upon some assumed path. The second peculiarity of this voltage must also be emphasized. The line integral must be taken at a specified instant in time. In effect one takes a photograph of the field at some instant in time and then at one's leisure performs the integration. Now, of course, an electron when projected through such a cavity will perform yet another type of integration. The change in squared velocity of the electron as expressed in volts will be given by the line integral of the field encountered by the electron; that is, integrated not instantaneously but with the electron velocity. This is not a simple process, because the electron velocity is continuously being changed by the field interaction and therefore the velocity with which the integration is performed depends upon the integrated value of the field up to the point in question. This has nothing to do with the concept of voltage in a resonant cavity. The cavity voltage can, however, be considered as the maximum change in squared velocity expressed in volts which an electron could receive if its entrance velocity was very large so that the transit time was small compared with the period of the cavity field. The four basic concepts which I have chosen to recall to your mind are, by way of summary: (l)*the total current is the same in all parts of a circuit, that is div. / = 0; (2) the only way we can act on an electron is to change its velocity; (3) the changes in the velocity component of an electron along any one rectangular coordinate have no effect on the velocity components along any other coordinate; and (4) for convenience, a voltage can be defined in a resonant circuit as the line integral of the electric field taken along some prescribed path. Transit Angle Since we are to deal with the interaction of electrons and high-frequency fields, we frequently find it convenient to measure electron velocity not directly but in terms of the equivalent potential difference through which an electron must fall to obtain the velocity in question, and the unit of measure ELECTRON BALLISTICS IX HIGH-FREQUENCY FIELDS 327 will be a volt. Instead of measuring the time required for an electron to traverse any given distance in seconds, it is also convenient to use, as a unit of time, one radian of angle at the operating frequency. We frequently refer to the transit angle of an electron rather than the transit time, although both terms are used. In fact, we may on occasion measure distances in terms of transit angle, and this usage is extended to measure dimensions transverse to the direction of travel of the electron beam. When used in this fashion, we mean that the dimension in question is such that were an electron to be projected in this direction with a velocity equal to that of the electrons in the main beam, the high-frequency field would change through the stated number of radians during the transit time. The Five Functions in an Electronic Device With this preliminary discussion out of the way we can now answer the question which has probably been troubling quite a few of you. If the only thing we can do to an electron is to change its velocity, then in what basic way does the velocity-modulation tube differ from the conventional negative grid tube or from the magnetron? Well, this is an involved story. If we are to make any use at all of an electron beam we must in general perform five distinct operations or func- tions. First we must produce the beam. Then we must impress a signal of some sort onto the beam. From what I have just said this can be done only by varying the velocities of the electrons contained in the beam. The third operation consists in converting this variation into a usable form. It is in this way that the diverse forms of electronic devices differ to the greatest degree. W^e will go into this matter in more detail shortly. The fourth operation consists in abstracting energy from the beam, and the final operation consists in collecting the spent electrons. While these operations are distinct from an analytical point of view, in many actual devices they are performed more or less simultaneously and more than one operation may be performed by certain portions of the tube structure. In fact, in some devices, for example in the space-charge-control tube, the confusion is so great as to make the separation seem rather forced. This very confu- sion may partly explain why vacuum-tube engineers who were steeped in the art were so slow to realize the advantages of this new way of looking at things which I will call the velocity-modulation concept. By way of mental exercise in this new way of thinking let us see how we can analyze a simple space-charge-control triode. Well, first of all we have to identify the electron gun which produces the beam. The electrons most certainly come from the cathode, but where is the first accelerating electrode? Actually there isn't any unless we think of the combined d-c field resulting from the d-c potentials on the grid and plate as assisted by 328 BELL SYSTEM TECHNICAL JOURNAL the initial emission velocities as performing this function. The next func- tion, that of varying the electron velocities, is performed by the grid which varies the potential gradient in the vicinity of the cathode and hence the velocity of the electrons as they approach a potential minimum or virtual cathode which is formed a short distance in front of the cathode by the action of space charge. This virtual cathode performs the third function, that of conversion, by sorting out the electrons and allowing only those elec- trons with emission velocities greater than some specific value to pass. This, then, is one of the conversion mechanisms which we will call virtual- cathode sorting. In this example the virtual cathode occurs very close to the real cathode but this is not always the case. The fourth function, that of utilization, is performed by allowmg the sorted electrons to traverse an electromagnetic field between the virtual cathode and the plate. This operation is completed by the time the electrons have reached the plate. Of course in the triode the plate then performs the final operation, that of collecting the spent electrons and dissipating the remaining energy as heat. It should be clearly reaUzed, however, that this last function need not neces- sarily be performed by the same electrode which provides the output field. Indeed the so-called inductive-output tube proposed by Haeff is a space- charge-control tube in which these two operations are separated. Conversion Mechanisms But now to get back to a cataloguing of the different kinds of conversion mechanisms. The first general type involves sorting. The first kind which we have mentioned is by virtual-cathode sorting. A second kind of sorting might involve deflecting the electron beam in proportion to its longitudinal velocity instead of reflecting or transmitting it. Various deflection tubes have been proposed from time to time using this mechanism. We shall be forced to neglect this phase of the problem this evening because of time limitations but those of you who are interested wiU find the literature filled with detailed discussions. Still a third type of sorting, sometimes called anode sorting, is used in certain Barkhausen tubes when the plate is oper- ated at or near the cathode potential so that fast electrons are collected while slow electrons are reflected and caused to retra verse a high-frequency field. There are still other types of sorting mechanisms but I will not burden you with these. A second general type of conversion mechanism I will call bunching, to distinguish sorting in which electrons are separated according to their velocities from hunching in which electrons of differing velocities are brought together. Now it just happens that many of the older devices used sorting, while many of the newer devices use bunching but this is not universally the case. For example, the magnetron as used at high frequencies and the ELECTRON BALLISTICS IN HIGH-FREQUENCY FIELDS 329 cyclotron both employ a combination of sorting and bunching. A peculiar property of the motion of an electron in a magnetic field lies in the existence of the so called Larmor frequency. You will recall that the angular velocity of an electron in a magnetic field depends only upon the field-strength and not at all upon the electron's linear velocity. This time in seconds is given by 0.357 X 10"' ^ H ' or in radians 10600 = 2 TT XH Electrons of widely differing velocity can thus revolve together in spoke- like bunches with the faster electrons going around larger circles than the slow ones, but just enough larger to keep them together. This, then, is one kind of bunching, which for simplicity we shall call magnetic bunching. It is used in the magnetron and in the cyclotron. We will have more to say on this subject a little later. A second kind of bunching was used in some of the early Barkhausen tubes wdiere the plate electrode was operated at a fairly high negative poten- tial so that none of the electrons were able to reach it. Under such condi- tions a uniformly spaced stream of electrons with varying velocities is re- flected as a bunched stream, the slower electrons being reflected almost at once and the faster electrons penetrating the retarding field for a greater distance and hence taking longer to return. This same type of bunching is used in a newer form of oscillator, commonly referred to as a reflex tube which was suggested by Hahn and Metcalf in 1939, and by others at about the same time. The reflex tube differs from the Barkhausen tube, not in the basic mechanisms so much as in the fact that the conversion mechanism occurs in a different region in the tube from the region devoted to velocity modulation and to energy abstraction. A second kind of bunching is then reflex bunching. A third type of bunching was used in the diode oscillators of Muller and of Llewellyn. The mathematical research done by W. E. Benham may be mentioned as of interest in this connection. In these tubes a uniform stream of electrons becomes bunched simply through the fact that faster moving electrons overtake slower ones which precede them. In these earlier forms of tubes we again have the case where this conversion is performed simul- taneously with one or more of the other processes so that it is very difficult to separate them. However, in 1935 Heil and Heil proposed a tube in which the conversion region was separated from the other regions of the 330 BELL SYSTEM TECHNICAL JOURNAL tube. This tube, the velocity-modulation tubes of Hahn and Metcalf, and the klystron tubes of the Varian Brothers, are alike in their use of transit- time bunching in a relatively-field-free drift tube. Since this separation of functions renders these devices much easier to analyze and since the struc- tures are quite interesting in any case we will spend most of our time con- sidering them and will, I fear, rather neglect some of the other types of tubes. \\ e will, of course, keep our analysis as general as possible so that the results may be applied to a variety of different devices. Input Gap Analysis Let us begin by a small-signal consideration of a uniform electron stream entering a region in which there is a longitudinal field defined as some func- tion of the distance and of time. This can be the entire Llewellyn diode or it can be the input region of a klystron. We ask ourselves with what velocity will the electrons leave this region and what will be the net exchange of energy between the electrons and the field. At any point within the field a typical electron will experience an acceleration given by y =lE-^r,f{y)m (1) where r) is proportional to the maximum amplitude of the h.f. field, but con- tains a numerical constant so that y is expressed in centimeters per second per second. Now in the usual case f(t) will be a simple sine function but f(y) may assume a variety of forms. Again, by way of simplifying our work we will assume that it is also a sine function. Let us consider how we can go about solving this apparently simple equation. Unfortunately this expression can not be solved directly because the value of / at any plane (that is, the time of arrival of an electron at this plane) depends upon the interchange of energy between the electron and the field. Here we are forced back to the time-honored mathematical device of assuming a solu- tion in the form of a series and then evaluating these coefficients. There is a large number of ways in which this can be done, and consequently a large number of different solutions which look very different but which all give comparable answers. Usually when such solutions are published, the arith- metical work is omitted leaving one with the feeling that there is something involved that is not within the ken of ordinary mortals. The fact is that the work is usually extremely tedious but actually very simple. It will be instructive to follow through one form of such an analysis in just enough detail to see the amount of work involved. Since we are interested in the energy which is proportional to y- we will write at once {y%^a = K = Ko-^vK,-{- v'K2 + rj^K^ + ... ELECTRON BALLISTICS IN HIGH-FREQUENCY FIELDS 331 where the K^s are a function of the transit time, of the field distribution and of the entrance phase, and we will proceed to evaluate these coefficients. The average energy per unit of change as expressed in volts is then simply — at the end of the field while the gain is: F.v = mw - Zo) = ^-^^ + ^^ + ... where the bar means that we are averaging over all values of the entrance phase. It is of interest to evaluate the value of velocity y- which individual elec- trons receive as a function of the entrance phase. For small signals it is usually sufficient to evaluate y~ maximized with respect to the starting phase, then F_ = m)iK - KoU. = ^-^' + 1|^^ + . . .] max. We can further define the ratio of Fmax to the largest value it can have as a coefficient (3, sometimes called the modulation coefficient. But now to evaluate the K's. There are many ways of doing this as I have intimated. We will proceed by writing y = yo{t) + vyiit) + v'^y2{t) + mysiO + • • • where the y's are coefficients depending upon the transit time / which in itself is a function of the applied field thus t = to + vli + fh + V% + .... We can then expand each function of time into a series remembering that /(. + „) =;(,)+w+mi:... or for our particular case yo(to)[vh + V' h -^ V^ h + • • •] >(/) = yo(/o) + 1! , yo(to)[nh + 77^/2 + r?'/3 +•• •]' , 2! Now we can expand yi(t), y-iit) etc. in exactly the same way. Finally we get a collection of terms which can be grouped in like powers of ij thus y = yo(to) + V [terms in y, y, ti , h , etc.] -\- r [ ] . . . The coefficient of the y] is in fact y^{h) ti + yi{k). We will not bother to write the rest. This expression can then be differentiated to get y and then 332 BELL SYSTEM TECHNICAL JOURNAL squared. However, we still have some undetermined coefficients the ti , t2 etc. terms. These we can evaluate by noting that we wish these values at y = a, where a is a fixed distance in the actual device. At this distance the t coefiicients in the expression for y must have such values that the value of y does not change with the value of rj. This can only be true if the individual expressions multiplying each power of j? are each equal to zero. Equating these expressions to zero one can evaluate all of the ^'s. For exam- ple the first term yields yo(io)h + yiito) = 0 or >'i(^o) h = - yaik Introducing these values, differentiating and squaring, one finally gets an expression for (y-)y = 0 as a power series in y, the coefiicients all being of a form easily evaluated for any specified field distribution. Since we have by definition called these coefl&cients Ko , Ki, etc. these values are then -^0 = yl Ki = 2(yoy - yoyi) ? Ki = {yl — 2yiyi + 2yoy2) - 2yoy2 + -^ yo This then constitutes the formal solution of the problem. We must now particularize our problem to some specific field distribution and evaluate the y coefiicients. Suppose, for example, that there is a uniform d.c. field (E of equation 1) and an alternating field which varies as some cosine func- tion of distance. Then the latter is f{y) = cos (t + ^) and y = - £ -f 7? cos (o)/ -f ^) cos ( — -1- c j we must eliminate the y which appears in this expression and replace y by its equivalent y = yo + vyi + v^y2 + • • • and expanding ELECTRON BALLISTICS IN HIGH-FREQUENCY FIELDS 333 cos(^/ + c) = cos(^« + .) + ^sin (^ + c\[7)yx + -q y^ + • • •] H and as before equating like powers of -q with y defined as y = Jo + ^7 ji + ry-i + . . . we finally arrive at .. 1 ^ I yi = cos {wt + (^) COS f ^ + c\ yi = —ynr/b cos (w/ + ^) sin [~^ -\- c\. Now we need only integrate these expressions to obtain the values of the y's and the y's needed to evaluate the K's,, If we average y"^ over all values of the starting phase we can write the energy contributed by the field to the electron's velocity. When this is done one finds that the odd powers of 77 are identically zero leaving only the even powers to be considered and for small signal analysis purposes we need only consider K^ . The energy per electron expressed in volts is V = 2.49 X IQ'^E'XJid) where f (6) = oS^Ki , and the power is obtained by multiplying this expression by the beam current in amperes. The end results can be expressed as curves oi f{d) against d as shown in Fig, 1. Three examples are shown: the uniform field case and two different harmonic distributions as indicated by the smaller plot in the lower left- hand corner. You will note that there exist regions of positive f(0) where the net transfer of energy is from the field to the electron and regions in which the transfer is in the other direction ; the former portions are of con- siderable interest in connection with the input gaps in velocity modulation tubes, and for that matter in the cathode grid region of the negative grid tube although this is more compHcated than is here indicated, as this trans- fer of energy constitutes a loss to the field which loads the input circuit. The latter portions may be utilized as was done in the Muller and Llewellyn diodes to obtain sustained oscillations. If, as I have indicated, we maximize y- as a function of the starting phase we can evaluate the modulation coefi&cient. The value for the uniform field 334 BELL SYSTEM TECHNICAL JOURNAL case, as shown in Fig. 2, is simply, /? = — -—. For future reference we will 6/2, write the loss expression for this case as J{d) = 2 (1 - cos 6) - 6 sin 6. Drift Space Analysis Now let us consider the conversion region in a typical velocity-variation tube. Figure 3 is a drawing of several such devices with the conversion O t> /\ P = 2.49xl6®d^A^lffe) / ^ 2 / \ / \ 4 ^-^^^ X- \ / ^-y^ / \ \ V ^ /^ ^-V \ J\ 0 "^-^^ TT ^ \ V -^ / •'r=^^-<" — \ / X\ B^X. -4 ■ — V — j So / cv\ \ X "^ / 0 X (y)- d -8 / 1 / -12 ^ 2IT an 4TT TRANSIT ANGLE (9) Fig. 1 — The energy transfer between an initially uniform electron stream and a longitudi- nal electromagnetic field as a function of transit angle. regions indicated. We will assume for the moment that the electrons enter this region with a small variation in velocity and at a perfectly uniform rate. Since the total number of electrons entering the region must be equal to the number of electrons leaving the region we may write or ii dti ti iodto dh' ELECTRON BALLISTICS IN HIGH-FREQUENCY FIELDS 335 1,0 0.8 ^ 0.6 0.4 0.2 0.0 \ \ SIN e/2 ^ e/2 \ ^ \ \ / X ^^ 10 14 e IN RADIANS Fig. 2 — The (velocity) modulation coefficient between an initially uniform electron stream and a uniform electromagnetic field as a function of transit angle. HEIL & HEIL 1935 vo^ CATHODE 3 -''■'''''''■ •^^~ CONVERSION REGION T ^ COLLECTOR HAHN & METCALF 1939 CONVERSION REGION 1^ Jip iiT CATHODE COLLECTOR VARIAN & VARIAN 1939 CONVERSION REGION CATHODE COLLECTOR Fig. 3 — Typical velocity variation devices employing transit-time bunching. 336 BELL SYSTEM TECHNICAL JOURNAL However, a relationship exists between ti and to , / /i = /o + V Where V t'o V 1 + « sin oj/i ; / /i = /o + Vo vl + a sin co/i Now if a « 1 and and finally but so that finally /i = /o + - I 1 sin wti + — = 1 + - — cos co/i ah Vo 2 ti ( 1 + -^ cos w/i ) Vo ( 1 + y cos cc/i j ii = io I 1 + y cos cc/i This says that the velocity variation impressed on the beam at the en- trance to the drift space or conversion region has resulted in a current varia- tion at the output. For those of you who think in vacuum tube parameters it is of interest to differentiate this expression with respect to the a-c voltage and obtain the transconductance dii dVa-c rewriting dii dV~ac dio_ 2V ELECTRON BALLISTICS IX IIIGII-TREQL'ENCV FIELDS 337 This result is obtained by neglecting all of the higher order terms and is therefore only a small signal theory of a very restricted sort. Now let us consider what we have done. Well, we have followed a small interval of time through the drift tube. At the input this time ■ Fig. 7 — Kompfner's presentation of the bunching effect. V-- ' lir ■\^—Tin /^£P£LL£R P£SOA^/iTOfl OUTPUT L//V£ Fig. 8 — The elements of a modern reflex tube (Pierce) . ELECTRON BALLISTICS IX IIIGII-FREQrEXCY FIELDS 341 penetrating the tield to a greater extent and waiting, as it were, for the slower electrons which follow to catch up. The electrons which pass across the gap while the field is becoming progressively more accelerating are spread out. If the retarding field is uniform it can be likened to the earth's gravitational field and the phase-focusing paths on our time-distance plot are parabolas. Figure 9, taken from Pierce's paper, illustrates this while Fig. 10 is such a plot taken from the paper by Harrison. One interesting and. 1r>Vo PETUffA/^ //V ^/>££0 x/o />£ ri/PA/S zrTo //t^ T/zW^ To //^ 7-/M£ T< To Fig. 9 — The gravitational-field analogy to reflex bunching (Pierce). Fig. 10 — The phase-focusing diagram for a reflex oscillator (Harrison). in a way, unfortunate difference between reflection bunching and direct transit-time bunching is the fact that for reflection bunching the slow elec- trons catch up with the fast ones while the reverse is true for the other type. This means that if both types of bunching are present as shown in Fig. 11, (also taken from Harrison's paper) one will tend to undo the effect of the other. Another way of combining effects of separate bunching actions is to build 342 BELL SYSTEM TECHNICAL JOURNAL a cascade transit-time-bunching amplifier in which a series of three gaps is used together with two drift spaces. The first gap velocity modulates the beam; this modulation is converted into a current modulation in the first drift space. The beam then excites the second cavity, which again velocity modulates the beam in quadrature with the original modulation. This action of course occurs in the output gap of a two-gap tube but it is not there used. Here this second and larger velocity modulation is converted to current modulation in the second drift space. The output is finally taken ofif the beam by the third gap. A phase-focusing diagram of this sort (again taken from Harrison's paper) is shown in Fig. 12. Space-Charge-Wave Analysis This phase-focusing approach is rather intriguing as one feels that one has a physical picture of what is going on. The picture is, however, very inexact except under certain highly specialized cases, as it completely ignores t ^^^ ^^fe^ REFLECTION SPACE FiElO FBEE 1 ^^ ^^^ ""^^^^^ fe^ ^ ^ ^ TIME—" ^^ ACC£LERaTiON VOLTiGE 1 ^RESONATOR VOLTAGE Fig. 11- -Diagram showing reflex bunching combined with field-free transit-time bunching (Harrison) . space-charge effects. These space-charge effects are of two sorts: a d-c effect, if you will, and an r-f effect; that is, the presence of the electrons of the beam will alter the average velocity of the electrons at different parts of the beam, and will tend to undo the bunching action. Because of this second effect, the electrons are effectively prevented from passing each other as the graphical solution suggests. Instead, as the density of the electrons in the bunch becomes greater, the mutual repulsion forces tend to prevent a further concentration of charge. The electron bunch then tends to disperse. The action could be likened to the propagation of a sound wave in a moving column of air. While there are several approximate ways to handle this problem, Hahn was the first to propose a really satisfactor>^ theory. Inci- dentally it should be noted that the Benham, MuUer, Llewellyn and Peter- son type of theory is capable of treating this aspect of the problem in a rigorous way and including all space-charge effects, but unfortunately these theories are limited in that they have been applied only to the parallel- plane case, and of course they are only small-signal theories. ELECTRON BALLISTICS IN HIGH-FREQUENCY FIELDS 343 Hahn's analysis starts by treating an infinitely long electron beam, using cylindrical co-ordinates and is limited to a small signal theory where the a-c motions are small compared to the d-c but it does not ignore the r-f effects of the space charge forces. The electron beam is thought of as a moving dielectric rod which is capable of propagating axial waves much as a dielec- OUTPUT GAP OUTPUT GAP INPUT GAP ACCELERATION VOLTAGE INPUT GAP VOLTAGE Fig. 12 — Diagram for a cascade amplifier (Harrison). trie wave guide will do. He assumes an axial magnetic field and a stream of positive ions having the same velocity axially and the same charge density. These ions are assumed to have infinite mass. The solution is much too complicated and involved to present here even in abstract. It involves the complete solution of Maxwell's equations subjected to the stated assump- 344 BELL SYSTEM TECHNICAL JOURNAL tions as restricted by the assumed boundary conditions at the edge of the beam. It is found that two waves are possible, one traveUng sHghtly faster than the electron beam and the second traveling slower. A point where the velocity components are in phase will correspond to the input to the beam, while points where the current components are in phase correspond to the desired positions for the output. The propagation constants for these two waves in a simplified special case where the magnetic field strength is infinite are given by Hahn, as well as expressions for the optimum drift tube length. He goes on to consider the case where the magnetic field is zero and finds that for this case the density of the charge does not vary much but instead the beam swells in and out so that instead of being lumps of charge with spaces between, the lumps appear in the outer boundary. Hahn has extended his general method of analysis to consider the modulation coefficient of gaps through which the beam must pass. His results are a great deal more general than those we have presented. Ramo has reformulated Hahn's theory by means of retarded potentials for the most important case. This results in some simplification of the theory. He computes the more important design constants for a velocity modulated tube, such as the optimum drift tube length and the amount and phase of the transconductance. Those of you who are particularly inter- ested are referred to the original paper. An interesting aspect brought out rather forcibly by Ramo's analysis is the existence of higher-order waves on the beam, always occurring in pairs, one faster and the other slower than the beam velocity. The Magnetron In what time remains I want to say just a very few words about the mag- netron. This is a very complicated subject and one which cannot be ade- quately dealt with in an entire evening, and certainly not in the time remaining. As you all know, the magnetron was invented and named by Dr. A. W. Hull. Habann, Zacek, Okabe and others pioneered in the use of the mag- netron as an ultra-high-frequency oscillator. As envisioned today a magnetron is a two-element device, usually cylindrical with a centrally located cathode and a surrounding anode. The anode may be continuous or it may be split into a number of segments as suggested by Okabe, and these segments joined together either externally or internally by resonant circuits. The basic ballistic problems of the magnetron, and hence the only prob- lems which directly concern us at this time are (1) that of determining the ELECTRON BALLISTICS IN II I Gil -FREQUENCY FIELDS 345 electron paths within the magnetron and having determined these paths (2) that of getting an understanding of the mechanism whereby electrons in traversing these paths are able to deliver energy to the connected high- frequency circuits. One might think that the first problem w-ould be a relatively easy job. As a matter of fact the literature is surfeited with papers purporting to give the answer. Unfortunately almost all of the jniblished work ignores the effect of space charge. A few moments' thought will suggest that space charge may be a controlling factor because of the long electron paths which are sure to result in crossed electric and magnetic fields, and indeed more detailed computations bear this out. Nevertheless the neglect of space charge greatly simplihes the problem. There are those who believe that the no-space-charge theories have no bearing on the way actual magnetrons work and that any correspondence between the predic- tions of such theories and the actual behavior of magnetrons is simply the result of an unfortunate coincidence. In fact Brillouin points out that the simplified form in which the Larmor theorem is applied by many, is in itself an approximation which was perfectly valid as originally applied by Larmor to the electronic orbits within the atom but which does not apply to conditions as they exist in the magnetron. A number of recent workers have attempted to include the effects of space charge but have unfortunately largely restricted themselves to small signal theories while the magnetron is seldom operated under small signal conditions, at least not intentionally. Most theories are further restricted to a consideration either of the coaxial case where the cathode radius is small compared to the anode radius or of the plane case. Most practical structures are intermediate between these extremes. As an example of the difficulties involved, Fig. 13, reproduced from a paper by Kilgore, shows the electron paths as computed neglecting space charge and also show's experimental proof that these paths actually exist. This illustration has been frequently reproduced and widely accepted. The experimental picture was obtained in the presence of gas, to make the electron beam path visible, and unfortunately the ionization which makes the beam visible also tends to neutralize space charge effects. The experimental arrangement departs still further from reality in that the electron emission from the cathode was restricted to a limited region so that the space charge forces were still further reduced. Now it is probably true that some magnetrons operate with electron paths as shown; still it is not true that all magnetrons operate in this way. Contrasting with this picture which was until recently commonly ac- cepted, Brillouin, Blewett and Ramo, and others have shown that stable distributions are possible in which a space charge of almost uniform density rotates with a uniform angular velocity about the axis. Brillouin goes so 346 BELL SYSTEM TECHNICAL JOURNAL far as to label the curves due to Kilgore as wrong, and pictures the possible electron trajectories as shown in Fig. 14. One of the earliest papers to consider this newer picture of the electron paths in the magnetron was published by Posthumus in 1935. This was definitely a ballistic approach and hence suitable for discussing tonight. ELETCTRON , PATH ELECTRON PATH Fig. 13 — Typical electron paths in a two-segment magnetron showing how electrons arrive at the plate-half of lower potential (Kilgore). Posthumus limits his discussion to but one type of oscillation which can be obtained in the split-anode magnetron. Those of you who are familiar with the early literature on the magnetron will recall that two distinct types of oscillations were frequently described. One type usually called "elec- tronic" was found to occur under conditions when the magnetic field was just ELECTRON BALLISTICS IN HIGH-FREQUENCY FIELDS 347 high enough to cut off the anode current under static conditions. This field has the vahie computed by Hull: 6.72 Vf H R Hull's first computation, by the way, was made neglecting space charge, but, strangely enough, the result is not changed by space charge. These electronic oscillations were assumed to be related in frequency to the time of transit of an electron from the cathode to the anode, and at cutoff this is inversely proportional to the field strength, as expressed by the empirical relationship \H = 13,100. A Ben Electronic trajectories for different magnetic fields A — small magnetic field L^b B — moderate magnetic field L ~ 6 C — strong magnetic field L 0 were performed on the Westing- house Mechanical Transients Analyzer under the supervision of Dr. G. D. McCann, Transmission Engineer, Westinghouse Electric and Manu- facturing Company. Assumptions The procedures to be described for the analysis and design of package cushioning are based on appUcations of a few simple laws of mechanics to an idealized mechanical system representing the package and its contents. DYNAMICS OF PACKAGE CUSHIONING 355 Essentially, a package consists of 1. Elements of the packaged article which are susceptible to mechanical damage. 2a. The packaged article as a whole. 2b. A cushioning medium (excelsior, cardboard spring pads, metal springs, etc.) 3. An outer container (cardboard carton, wood packing case, etc.) The four major components are illustrated schematically in Fig. 0.2.1. The system is further ideahzed by "lumping the parameters"; for example, the outer container is considered as a single mass, the cushioning is con- sidered as a massless spring with friction losses. The result of this idealiza- tion is to lose some of the fine detail of the real distributed system such as wave propagation through the cushioning and higher modes of vibration in o- 2a ■2b Fig. 0.2.1 — Schematic representation of a package. 1. Element of packaged article 2a. Packaged article as a whole 2b. Cushioning 3. Outer container the package structure and in the packaged article. Some consideration of these details is given in Part W. The idealized system is illustrated in Fig. 0.2.2. The major components of the system are as follows: 1. A structural element of the packaged item is represented by a mass (wi) supported by a linear massless spring with or without velocity damping. The mass nti is assumed to be small in comparison with the mass of the whole packaged item. 2a. The whole packaged item is represented by a mass m-^. 2b. The cushioning is represented by a spring which may have a linear or non-linear load-displacement characteristic and which dissipates energy through velocity damping or dry friction. Permanent de- formation of the cushioning is not considered, that is, in a repetition of the drop test it is assumed that the package has the same properties as before the first test. A properly designed package will have essen- 356 BELL SYSTEM TECHNICAL JOURNAL tially this characteristic. The mass of the cushioning is assumed to be small in comparison with mo, except in Section 4.2. The outer container is represented by the mass niz . The impact of W3 on the floor is assumed to be inelastic and during contact the rela- tive displacement between m^ and the initial position of the floor is assumed to be small in comparison with the relative displacement between Wo and ntz . In other words, no spring action is assigned to the outer container and the floor is considered rigid. Element of Packaged Item ^: Packaged Item Cushion Height of drop h //////// ////////////// / / / / / (a) (b) (c) Fig. 0.2.2 — Idealized mechanical s}stem representing a package in a drop test. PART I MAXIMUM ACCELERATION AND DISPLACEMENT LI Introduction Most of Part I is concerned with the prediction of the maximum accelera- tion that the cushioning permits the packaged article (W2) to attain. In many instances this will be all the information necessary for judging the suitabiHty of a cushioning system. It will be all that is necessary if the shape and scale of the acceleration-time function satisfy certain criteria which are treated in detail in Parts III and IV. If these criteria are satisfied, the effect of the drop on the packaged article is found by multiplying the DYNAMICS OF PACKAGE CUSHIONING 357 dead load stresses (obtained in the usual manner) by the ratio of the maxi- mum acceleration to the acceleration of gravity. If the criteria for the use of maximum acceleration alone are not satisfied, then Parts II and III will supply a numerical factor (the Amplification Factor) by which the maximum acceleration should be multiplied, and the remainder of the procedure is the same as before. The determination of the maximum acceleration is founded on a knowledge of the load-displacement characteristics of the cushioning. When the cush- ioning system is simple enough, the load-displacement relation may be found or designed by purely analytical procedures. The tension spring package, discussed in Sections 1.7 and 1.8, is an example where such a treatment is possible. In many instances, as with distributed cushioning, the load- displacement relation is more easily found by test. A load-displacement test is made by applying successively increasing forces, with weights or in a load testing machine, to the packaged item completely assembled in its package, and measuring the corresponding displacements. The force is applied usually by means of a rod inserted in a hole cut through the oiiter container and the cushioning to the packaged item. It is convenient to use a low loading rate in the test, and, in doing so, the effect of resisting forces that depend on velocity is lost. These forces are often of little importance but, in certain designs, it is necessary to con- sider them. This is done for velocity damping in Sections 2.5, 2.6, 3.2 and 3.5. Most of Part I is concerned with cushioning having non-linear load-dis- placement characteristics. Linear cushioning is rarely encountered, but it will be treated first because of its simplicity and because it will be con- venient later to express the maximum acceleration in non-linear cases in terms of the maximum acceleration in a hypothetical linear case. 1.2 Derivation of Equations of Motion To introduce the method of analysis that will be used in Part I, the sim- plest possible system is considered first. The mi system is omitted entirely, the mass of the outer container (m^) is neglected, and the cushioning is assumed to have no damping or friction. There remain only the mass W2 (the mass of the packaged item alone) and the supporting spring, as shown in Fig. 1.2.1. If the spring is linear its displacement is proportional to the applied load throughout the range of use (see Fig. 1.4.1). The spring rate (^2) of a linear spring is a constant usually expressed in terms of pounds per inch. The force (P) transmitted through a linear spring is therefore given by P ^ koxo, (1.2.1) 358 BELL SYSTEM TECHNICAL JOURNAL where x^ is the displacement of m^ measured downward from its position at first contact of the spring with the floor (see Fig. 1.2.1). For a non-linear spring P will be some other function of x^ : P = P{x->). (1.2.2) To write the equation of motion for the mass mi , we consider the forces acting on it at any instant. These are (see Fig. 1.2.2(b)) the spring force P and the weight ntig, where g is the acceleration of gravity. When Xi is positive (i.e., a downward displacement of m2 from its position at first contact of the spring with the floor) the spring exerts an upward force P / floor / / /'/ y / / y / / // // /^/ //y' (a) (b) Fig. 1.2.1 — Elementary system. (c) (a) (b) Fig. 1.2.2 — Free body diagram for elementary system. (a) Spring not in contact with floor. (b) Spring in contact with floor. on the mass, opposing the weight. The total downward force on m-i is thus mig — P. By the second law of motion, the product of the mass and its acceleration at any instant is equal to the appUed force : mix% = mig — P, (1.2.3) where the symbol X2 , representing the acceleration of W2 , stands for the second derivative of displacement with respect to time {d'xi/df). Equation (1.2.3) is the law governing the motion of W2 as long as the spring is in con- tact with the floor. When the spring is not in contact with the floor, it can exert no force on the mass so that, in writing the equation of motion that DYNAMICS OF PACKAGE CUSHIONING 359 governs before or after contact, the free-body diagram of Fig. 1.2.2(a) should be used. Then X2 = g. (1.2.4) Equation (1.2.4) holds (neglecting air resistance) from the instant the package starts to fall until the instant it strikes the floor and from it we can find the package velocity at the instant of first contact. Integrating (1.2.4) with respect to time, we find Xi^gt^A, (1.2.5) where Xi is the velocity {dx^/dt) and A \s o, constant of integration whose value is found from the initial condition that when / = 0 (the instant of release) ;V2 = 0. Thus yl = 0 and X'i = gL (1.2.6) Integrating again, x^ = hgt''+B. (1.2.7) The value of the integration constant B is found from the initial condition that .T2 = — // (the height of drop) when / = 0. Hence B = —h and X2 = ig/2_/;. (12.8) At the instant of contact, X2 = 0 and, from (1.2.8), the time at first contact is given by /q = 2/?/g. Substituting this value of / in (1.2.5) we find, for the velocity at first contact, [xo]x,=o = \/2p. (1.2.9) We now have the initial conditions for finding the values of the integration constants in the solution of equation (1.2.3), which we proceed to obtain. First multiply both sides of (1.2.3) by dxi/dt and write ^2 = t ( -j^ ]: dx2 d fdxi\ dx2 _ dx2 /1 o in\ or , _, dx2 dx2 + ^-57 """^^^ Multiplying by dt and integrating once: to + / " i' dx2 = j ' niog dx2 + C, (1.2. 11; 2^2 Xi 360 BELL SYSTEM TECHNICAL JOURNAL where C is a constant of integration whose value is determined by the initial conditions that xl = 2gh and .r2 = 0 at the instant of contact. Hence C = mogh + J P dx.2. Substituting the above value of C in (1.2.11), we have >2A-2 + I P dx.2 = m.2g{h + .vo). (1.2.12) It may be observed that (1.2.12) is an energy equation in which the terms have the following meanings : hm2X'2 is the instantaneous kinetic energy of m^, Jo P dx'2 is the energy stored in the spring at any instant. It is also ''0 equal to the area under the load-displacement curve up to the displacement X2 , ni'2g{h + .Vo) is the potential energy of the mass at its initial height h + Xi above the instantaneous position .Vo . Hence (1.2.12) expresses the law of conservation of energy. Ordinarily // is very much larger than .V2 so that we may write, with good accuracy, hm^xl + [ P dx2 = m<2gh. (1.2.13) To the same approximation, equation (1.2.3) becomes ni.x2 + P = 0. (1.2.14) Equation (1.2.14) and its first integral, equation (1.2.13), are convenient forms for calculating events at any time during contact. Their use will be illustrated in Part II. For calculating only maximum displacement and acceleration, the equations become simpler. Let Wi = weight of the packaged article ( = W2g), dm = maximum displacement of the packaged article, Gm — absolute value of maximum acceleration of the packaged article in terms of number of times gravity {Gm = \ Xilg |max), Pm = maximum force exerted on packaged article by cushioning. We shall limit our study to the practical regions where P > 0 when x-i > 0. Then it may be seen from (1.2.13) that .T2 is a maximum when ±2 is zero, hence P dx.2 = WoJi, (1.2.15) / Jo DYXAMICS OF PACKAGE CUSHIONIXG 361 and, from (1.2.14), G„. = ^\ (1.2.16) where P,,, is the maximum value of P. If /"(.to) is a monotonic function, P„, may be obtained from (1.2.2) by substituting (/,„ for .vo: P,n-P{(L). (1.2.17) In the unusual case where P(.V2) is not monotonic, the maximum value of P in the interval 0 < .T2 < dm must be chosen instead of equation (1.2.17). The general procedure is to calculate dm from (1.2.15), Pm front (1.2.17) and then Gm from (1.2.16). If P can be expressed analytically in terms of x^ and if the integral in (1.2.15) can be evaluated in terms of elementary func- tions, simple formulas can be found for dm and G,„ . If this is not possible, then the integration can be performed graphically or numerically. Both of these procedures will be illustrated. In either case the maximum accel- eration and displacement are obtained in terms of the weight of the pack- aged item, the height of drop and parameters descriptive of the load-dis- placement characteristics of the cushioning. 1.3 Linear Elasticity For cushioning with a linear load-displacement relation, equation (1.2.1) applies. Substituting this value of P in (1.2.15), and performing the in- tegration, we find 2^^ (1.3.1) From (1.3.1) and (1.2.17), Pm = V2hW2h, (1.3.2) and, from (1.3.2) and (1.2.16), Grn=y^^- (1.3.3) Notice that equation (1.3.3) holds only if there is space available for a displacement dm and if the cushioning is linear and capable of transmitting aforceP„j. Also, from (1.3.3) and (1.3.1), 2h (1.3.4) V and Gn 362 BELL SYSTEM TECHNICAL JOURNAL Example: Find the properties of the hnear cushioning required so that the maximum acceleration will be 50g in a 3 ft. drop of a 20 lb. article. From (1.3.4), necessary travel, dm = — ^ — = 1.44 inches. From (1.3.5), ^ , 20 X (50)' ._ . „ ,. sprmg rate, k^ = = 694 Ibs/m. 2 X 36 From (1.2.16) Maximum force P^ = 20 X 50 = 1000 lbs. 1.4 Cushioning with Non-Linear Elasticity In practice it is rarely that a packaging system has linear spring charac- teristics. Departure from linearity may be due to Fig. 1.4.1— Linear elasticity. Class A. 1. Non-linear geometry, such as in the tension spring package described in Section 1.7. 2. Non-linear characteristics of distributed cushioning materials such as excelsior and rubber. 3. Abrupt change of stiffness such as occurs if the packaged item can strike the wall of the container. For the purpose of developing design formulas it is desirable to have analytical functions to represent load-displacement characteristics. It is not feasible to have only one family of functions with adjustable parameters to fit all possible shapes of load-displacement curves. Therefore, all the practical shapes have been divided into six general classes, most of which are associated with simple functions having one or two adjustable param- eters. The six classes are as follows: Class A — Linear Elasticity. This has already been treated. Its load- displacement function is P = hx2. (1.4.1) DYNAMICS OF PACKAGE CUSHIONING 363 Class B — Ciibic Elasticity. This includes cushioning which does not bot- tom in the anticipated range of use, but the slope of the load-displacement function generally increases with increasing displacement as in the curved full line of Fig. 1.4.2. A suitable load-displacement function is P = ko X2 + rxo (1.4.2) ko is the initial spring rate of the cushioning, as shown by the slope of the dashed straight line in Fig. 1.4.2, and r determines the rate of increase of the spring rate. The same function can be used if the slope of the curve de- creases gradually with increasing load as shown by the curved dashed line in Fig. 1.4.2. In this case the parameter r is negative. n ^2 Fig. 1.4.2 Fig. 1.4.2 — Cubic elasticity. Fig. 1.4.3 2 "b '^2 Fig. 1.4.3 Class B. Tangent elasticity. Class C. Class C^Tangent Elasticity. Cushioning that bottoms, but not very abruptly, can be represented by the load-displacement function _, 2^0 db ^ Ttxt P = tan — - TT Mb (1.4.3) Referring to Fig. 1.4.3, ^o is the initial spring rate and dh is the maximum available displacement. The figure shows hov the stiffness of the cushion- ing (i.e., the slope of the curve) increases as the displacement approaches the maximum available {db) at hard bottoming. The shape of the curve is typical of load-displacement curves for a great variety of packages with distributed cushioning. Figure 1.4.7 illustrates the wide variety of shapes of non-linear cushioning characteristics that can be obtained with the single function given by equa- tion (1.4.3) simply by varying the parameter Uq; and a similar set is given by each value of db . Although these families of curves do not include all pos- sible shapes, one of them can usually be found to fit a practical shape for cushioning of this class over the anticipated range of use. Class D — Bi-linear Elasticity. This is characterized by a load-displace- 364 BELL SYSTEM TECHNICAL JOURNAL ment curve consisting of two straight line segments. The load displacement function is (see Fig. 1.4.4) P = koXo 0 ^ X-2 ^ (Is P = kbX2 — (^6 — kn)ds X2 ^ d, (1.4.4) It is useful especially in situations where very abrupt bottoming is possible. Class E — Hyperbolic Tangent Elasticity. When the mechanism of the cushioning is such as to hmit the maximum force that can be transmitted over a considerable displacement range, the load-displacement function P = Po tanh ko X2 (1.4.5) is useful. Po is the asymptotic value of the force and ko is the initial spring rate (see Fig. 1.4.5). Fig. 1.4.5 Fig. 1.4.4 — Bi-linear elasticit}'. Class D. Fig. 1.4.5 — Hyperbolic tangent elasticity. Class E. Class F — Anomalous Elasticity. In occasional instances the load-dis- placement curve of the cushioning cannot be matched accurately enough by any of the five preceding functions. In such cases a numerical integra- tion procedure can be used, as described in Section 1.15. 1.5 Cushioning with Cubic Elasticity (Class B) Substituting (1.4.2) in (1.2.15) and performing the integration, we have: Now, let V 2W2h ko (1.5.1) (1.5.2) DYNAMICS OF PACKAGE CUSHIONING 365 that is, do is the displacement that would take place if the elasticity were linear (see equation (1.3.1)) with a constant spring rate ^o equal to the initial spring rate of the cubic elasticity. Also let B = Then, from (1.5.1), (1.5.2) and (1.5.3) (-1 + Vl + 5) B (1.5.3) (1.5.4) Equation (1.5.4) is plotted in Fig. 1.5.1 which shows graphically how the maximum displacement dm compares with the "equivalent linear displace- ment do" as the parameter B is varied. Note that B depends on the weight of the packaged item, the height of drop and the shape of the load displace- ment curve (as determined by .^o and r). Fig. 1.4.6 — Anomalous elasticity. Class F. Similarly we can compare the maximum acceleration G,„ with the maxi- mum (Go) that would obtain if the load displacement curve were linear with spring rate ^o • The latter acceleration is given by '^Jp (1.5.5) and the former is obtained by finding P^ from (1.2.17) and then, from (1.2.16), Gm 'Go V' /j/|(l + B)(-1 + Vl + 5). Equation (1.5.6) is plotted in Fig. 1.5.2. (1.5.6) 1 .6 Procedure for Fixdesg Maxevium Acceleration and Displacement for cushionlng with cubic elasticity If the load-displacement curve of a cushioning system has the general appearance of Fig. 1.4.2 (where the slope increases or decreases gradually 366 BELL SYSTEM TECHNICAL JOURNAL with displacement) the following procedure may be used for estimating the effectiveness of the cushioning. 130(0) - 100(0) - i*«t-;|;;;;|;-tn::j. / \\\] \\\ IM rihi /JWfEH:!. ■[-- n CS 0 9 Fig. 1.4.7- — Family of load displacement curves for cushioning with tangent elasticity. a. Select the point on the load-displacement curve for which the load is equal to the weight of the packaged item multiplied by the allowable Gm . Call this load P9. and the corresponding displacement d^ . DYNAMICS OF PACKAGE CUSHIONING 367 ipipi|S|f!!!!|!!i|!!i|!iifii wpiiliit|!iii|t t^-t----i-- :n;::q;^M.::4;::;l»;:l:.M.::.l:.::t- 0 12 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 „ 4W;hr Fig. 1.5.1 — Maximum displacement for cushioning with cubic elasticity. See equation (1.5.4). nWghr Fig. 1.5.2 — Maximum acceleration for cushioning with cubic elasticity. See equation (1.5.6). 368 BELL SYSTEM TECHNICAL JOURNAL b. Select another point {di , Pj) about half way toward the origin from ( several inches) is required. The decision as to whether or not a tension spring package is indicated may be made on the basis of a preliminary estimate of displacement based on the linear case. Suppose the height of drop is to be 60 inches and the allowable acceleration for the packaged item is 370 BELL SYSTEM TECHNICAL JOURNAL 20g. Then, from Equation 1.3.4, the approximate displacement that will be required is , 2 X 60 ^ . , dm = — r^r — = 6 mched (1.7.1) The actual maximum displacement in a tension spring package will prove to be somewhat more than 6 inches, but the preliminary calculation shows the displacement to be large enough to warrant the use of this type of cushioning. H G Q B Fig. 1.7.1 — Schematic diagram of a tension spring package. A schematic diagram of a typical tension spring package is shown in Fig. 1.7.1 and a photograph of one design is given in Fig. 1.7.2. The packaged item is suspended on eight identical helical tension springs which diverge to the outer frame. The analysis and design procedures described in this and the following section apply equally well if the springs converge from the packaged item to the outer frame. With a slight modification, indicated in the next section, the procedure also applies if four of the springs (say, BJ, DL, EM, OG in Fig. 1.7.1) are omitted. In all cases, however, we shall consider only systems having reflected symmetry about each of three mutually perpendicular planes through the center of gravity of the packaged article. DYNAMICS OF PACKAGE CUSHIONING 371 Fig. 1.7.2 — A tension spring package. The load-displacement characteristics of the spring system may be found by statical considerations. We shall examine, first, the displacement in the vertical direction in Fig. 1.7.1, using the following notations: P = force applied to the suspended object, .V2 = displacement of suspended object, .To = perpendicular distance (IR, Fig. 1.7.1) from inner spring support point (/, Fig. 1.7.1) to nearest plane, perpendicular to displacement direction and containing four outer spring support points (A , B, C, A Fig. 1.7.1); 372 BELL SYSTEM TECHNICAL JOURNAL li = distance (I A) between spring support points when suspended article is in equilibrium position, / = projection of /» on plane A BCD, f = -C minus length (between hooks) of unstretched spring, k = spring rate of each spring. Consider, first, the action of one pair of springs, say EM and GO of Fig. 1.7.1, independent of the remainder of the suspension. Since EM and GO lie in parallel vertical planes and the points M and 0 remain in the initial planes of their respective springs during a vertical displacement, the two springs may be considered to lie in the same plane, and to be translated hori- zontally in this plane so that their outer ends are separated by a distance 2i. Hence Fig. 1.7.3 may be used to represent the independent action of this pair of springs and it is required to find the force Q' needed to transform Fig. 1.7.3(a) to Fig. 1.7.3(b). Initially there are two springs, each of length / — / f f OVJlAJJJLJLtP CK. itojuuii ff ^ . ^ F X. ^-f Y ^ r a b Fig. 1.7.3— Diagram used in discussion of tension spring package. and spring constant ^, with no initial tension in them. One end of one spring is fixed at point E and one end of the other spring is fixed at a point G distant U from E. The springs are then stretched so that the two initially free ends are located at a point Y equidistant from E and G and distant x-i from line EG. The axis of each spring makes an angle a. with £G, where sm a = OCi ^e + xf In this state the axial force F in each spring is F = kWf^^^ - t+f] (1.7.2) (1.7.3) and the force Q\ required to equilibrate the two forces F is 2F sin a. Con- sidering the force Q' as a function of the displacement Xo , we write Q'{4) = V7T =^ [Vf + x? -i+f] (1.7.4) X2 DYNAMICS OF PACKAGE CUSHIONING (1 - b)z' where Q'(z') = 2H \z' .^•2 + 2 = Vl +2 ;] 373 (1.7.5) Consider, next, the configuration shown in Fig. 1.7.4(a), where one end of each of four springs is fixed at a corner of a rectangle of length It and width 2.ro . Each spring is again of length i — j. The four free ends of the springs are drawn together at a common point X at the center of the rectangle (see Fig. 1.7.4(b)). The system is in equilibrium in this position. A force Q is then applied at A^ in the plane of the rectangle and normal to the side It. f f uuuii) / — -V f ■ f a 2X, ckflMAJLl/p •- ^, $ - — I — • ,^ ' 1 vr ^ ♦ >*^^^^< Fig. 1.7.4 — Action of springs in a tension spring package. The common point X is displaced a distance X2 to X' (see Fig. 1.7.4(c)). Writing z = xz/t, a = Xo/t, we observe that Q(z) = Q'iz + a) + (2'(0 - a), (1.7.6) or, from equation (1.7.5), Q{z) = 2u\2z-{\-}A / ^ "^ ^ - + -^-^^^^=^11 (17 7) ^^' I ^ iVl + (z + a)2 ^ Vl + (2 - a)0/ ^ '^ The standard tension spring package has two sets of four springs so that the force P required to displace the common point X a distance Xi. is Piz) = 2Q(z). (1.7.8) If X2 is small in comparison with t (i.e., z is small in comparison with unity), equation (1.7.8) may be written approximately as P(z) = Akchz - (1 - h)z{2-i)\ Even when x^ becomes almost as large as t, equation (1.7.9) has been found, e.xperimentally, to be remarkably accurate. Writing K fl- ^-M L (1 + a2)3/2j m\ - (1.7.10) 374 BELL SYSTEM TECHNICAL JOURNAL and , = iL+4^' - 1, (1.7.U) 1 — 0 equation (1.7.9) becomes P = K((z+i\. (1.7.12) It is seen, by comparison with (1.4.2) that this is Class B cushioning (cubic elasticity). A' is the initial spring rate and c determines the rate of increase of stiffness with displacement. With the notation ^o , '' of Section 1.5, we see that h = K (1.7.13) . = ^p. (1.7.14) Hence equations (1.5.6) and (1.5.4) may again be used to calculate maximum acceleration and displacement. B has the same meaning as before (Eq. 1.5.3). To predict the performance, in the vertical direction (Fig. 1.7.1), of an existing tension spring package the same procedure as outlined in Section 1.6 may be used, except that it is not necessary to have a load-displacement curve for calculating y^o and r. Instead, these parameters may be calculated directly from equations (1.7.10), (1.7.11), (1.7.13) and (1.7.14). The remainder of the procedure is the same as in Section 1.6(d). To predict the performance perpendicular to another face, say AEHD of Fig. 1.7.1, it is only necessary, in the calculation of ^o and r, to substitute x'q for .vo , (' for ( (see Fig. 1.7.1) and, in place of b: b' = \ - j,{\ - b). (1.7.15) The initial spring rate A' for any direction of acceleration may be calcu- lated from the initial spring rates Ai , A2 , A'3 in the three directions normal to the faces of the frame by using the relation 1 s- t^ u W'^Kl^Kl^ A3 2 -I- -2 -I- -2, (1.7.16) where 5, /, u are the direction cosines of the acceleration direction with respect to the normals to the faces of the frame. It is seen, from (1.7.16), that the spring rate is given by the radius to the surface of an ellipsoid whose principal semi-axes are A'l , A2 , A3 . DYNAMICS OF PACKAGE CUSHIONING 375 The displacement direction does not necessarily coincide with the acceler- ation direction. The angle d between them is given by where A' is defined by equation (1.7.16). The spring characteristics may be made the same in all directions and the displacement direction may be made to coincide with the acceleration direction by setting . _ / _ " _ Xo — Xq — Xq — /— and C= t' = t" (see Fig. 1.7.1). This makes b = b' = b", c = 0.828 and k/K = 0.274 in the calculations of the next section. 1.8 Procedure for Designing Tension Spring Packages The design of a tension spring package, as contrasted with the analysis of one, must proceed without initial knowledge of values for the parameters ko and r, since these cannot be known until the springs are designed. There- fore equations (1.5.4) and (1.5.6) cannot be used directly. For design pur- poses they are transformed to the following set of formulas: Vc = |/^V^^ - 1 (1.8.1) ^ = vh^. - ^ //( v:v + VNTsy + f (1.8.2) :^^ = V2(-i 4- \/r+^) (1-8.3) M' = N = ^^ = I (1 + 5)(_1 + VTT^) (1.8.4) ~-b= -1 + ^i^^J^\ (1.8.5) These formulas have been converted to design curves which are given in Figs. 1.8.1 to 1.8.5. The curves are for use in connection with the following routine procedure which has been found useful in designing the springs for tension spring packages. Reference should be made to Table I. 376 BELL SYSTEM TECHNICAL JOURNAL 1. Enter, on Line 1, Table I, the weight (TF2) in pounds, of the sus- pended item. This includes the weight of the cradle or other holding arrangement and one-third the estimated weight of the springs. b=0.3 b=0.2 0.1 0.2 0.3 0.4 0.5 0 6 0.7 0.8 0 9 1.0 Xo X Fig. 1.8.1 — Tension spring package design curve. Equation (1.8.1). 2. Enter, on Line 2, the height of drop (Jt) in inches. 3. Enter, on Line 3, the maximum allowable acceleration (G,„) in units of ''number of times gravity," This should be determined before- hand from tests on the item to be packaged. 4. Enter, on Line 4, the dimension 0:0 (inches). DYNAMICS OF PACKAGE CUSHIONING 377 5. Enter, on Line 5, the dimension / (inches). For a package to have the same spring rate in all directions, ( = Xo\/2 is a necessary con- dition. 6. Enter, on Line 6, the value chosen for b. As b becomes greater than zero, the stiffness of the whole suspension increases for a given stiff- ness of individual springs. The reverse happens for b less than zero. Fig. 1.8.2 — Tension spring package design curve. Equation (1.8.2). 7. Calculate Xo/^. 8. Enter Fig. L8.1 with xo/^ and find \/c. 9. Calculate L = h/iVc^m). 10. Enter Fig. L8.2 with L and find N. n. Calculate K = (W2GJ^)/(2kN). This is the initial spring rate of the suspension in the direction of Xo . 12. Calculate/ = 3.13 (K/Wi)^ This is the natural frequency of vibra- tion (cycles per second) of the suspension for small amplitudes in the ^0 direction. This should not be close to the natural frequency of 378 BELL SYSTEM TECHNICAL JOURNAL vibration of any element in the packaged item, which should be determined by test beforehand (see, also, Section 4.2). In any case it is advisable to provide damping for the suspension. 3.4 3.2 3.0 2.8 2.6 2.4 22 2.0 1.8 1,6 1.4 1.2 1.0 08 06 04 0.2 0 0.1 0 2 0.3 0.4 0 5 0 6 0.7 0.8 0 9 10 Fig. 1.8.3 — Tension spring package design curve. Equation (1.7.10). 13. Enter Fig. 1.8.3 with Xq/ ( and find k/K. If a four-spring package is desired, instead of an eight-spring package, (see Section 1.7) the value of k/K found on Fig. 1.8.3 should be multiplied by two before entering it on Line 13 in Table I. This is the only change required in the procedure. 14. Calculate k = A'f ^, j. This is the spring rate of each of the springs in pounds per inch. w ^b = 0.0 \ b "°-'\ \ \ V\ \ V \ b = 0.2. \ \ "^ < N ^b = 0.3 ^^ ■-C> ^ -____ - ~ DYNAMICS OF PACKAGE CUSHIONING 379 15. Calculate B = (2W2h)/(Kcn). 16. Enter Fig. 1.8.4(a) orjb) with/i and lincl d,n/{VcO. 17. Calculate . G„, 35 4. xo (ins.) 5 5. I (ins.) : 7.07 6. 6 0 7. Calc. xq/( 0. 707 8. Find Vc from Fig. 1.8.1 0.91 9. Calc. h/y/c (G,n = L 0.269 10. Find N from Fig. 1.8.2 1 .265 11. Calc. W^CnyiliN = K (lbs/in.) 169.0 12. Calc./ = 3.13 (K/Wo^i (cyc./sec.) 8.9 13. Find k/K from Fig. 1.8.3 0.274 14. Calc. k = K ■ k/K (lbs/in.) 46.5 15. Calc. B = IWih/Kcf 0.368 16. Find d,„/Vc I from Fig. 1.8.4 0.575 17. Calc. djC = Vc • d,n/y/a 0.518 18. Calc. d,n = t ■ djt (ins.) 3.68 19. Calc. dm/t + xo/t 1 .220 20. Find efC from Fig. 1.8.5 and line 6 0.580 21. Calc. /^™ = ^ • e/^ • ^(Ibs) 191.0 12. Coil diameter (ins.) 1 .40 23. Wire diameter (ins.) 0.207 24. Number of turns 19 25. Fiber Stress (Ibs./sq. in.) • lO'^ 80 26. Length of Coils (ins.) 3.93 27. Length inside hooks (ins.) 7.07 1.9 Cushioning with Tangent Elasticity (Class C) This is one of the most frequently encountered classes of cushioning since it includes a very common type of bottoming (Figs. 1.4.3 and 1.4.7). The load-displacement function (equation (1.4.3)) takes into account the fact that the cushioning can be compressed only to a definite amount db . To find formulas for maximum acceleration and displacement, we pro- ceed as follows. Substitute equation (1.4.3) in (1.2.15) and perform the integration, obtaining ^Mogcos"^= -W,h, (1.9.1) IT- Mb which may be written as 382 BELL SYSTEM TECHNICAL JOURNAL Equation (1.9.2) can then be substituted into (1.4.3) to obtain the maximum force Pm in accordance with (1.2.17): 2^0 ^6 . / (ir^W^hX - 1. (1.9.3) do Fig. 1.9.1 — Curve for finding maximum acceleration for cushioning with tangent elasticit_v. See equation (1.9.4). The maximum acceleration is then obtained from (1.2.16) and may be written in the form Gm Go 2d, TTch / firdoY 1 (1.9.4) where do and Go are defined just as in (1.5.2) and (1.5.5). Go is the maxi- mum acceleration that would obtain if the cushioning did not bottom, that is, if the spring rate remained constant at its initial value ^o • do is the maximum displacement that would be reached under the same linear con- ditions. Hence Gm/Go is a multiplying factor to be applied to a hypothetical linear cushioning to take into account the effect of bottoming. The multi- plying factor depends only on the ratio (db/do) of the amount of space actually available to the amount of space that would be used under linear conditions. DYNAMICS OF PACKAGE CUSHIONING 383 The ratio G,„/Go is plotted against the ratio db/do in Fig. 1.9.1. It may be seen that the multiplying factor increases very rapidly as the displace- ment ratio (db/di)) falls below unity. For example, if the cushioning, with tangent elasticity, reaches hard bottoming (di,) when only 80% of the required displacement (do) is attained, the acceleration is multiplied by 3.5; if only 609o of the required displacement is available, the acceleration is multiplied by 11.5. Example: To illustrate with a numerical example, consider the case already discussed in Section 1.3, where we found that a spring rate of 694 lbs/in and a displacement of 1.44 inches were required to limit a 20-pound article 03 0.2 0,1 0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 24 2.6 2.8 3.0 Fig. 1 .9.2 — Curve for finding maximum displacement for cushioning with tangent elasticity. See equation (1.9.5). to an acceleration of 50g in a 3-foot drop with linear cushioning. Let us suppose that only 1.15 inches are available, instead of 1.44 inches, and that the cushioning has tangent elasticity starting with a spring rate of 694 lbs. /in. Entering the curve of Fig. 1.9.1 at db/do = 1.15/1.44 we find Gm/Go = 3.5. Hence the maximum acceleration will be 175^ instead of the required 50g. This illustrates the wide variations in acceleration that may occur as a result of minor variations of dimensions in high G packages. It is not necessarily true that the 17 5 g test is 3.5 times as severe as the 50^ test for all elements of the supported structure, since the severity de- pends also on the shape and scale of the acceleration-time relation. The factor may be more or less than 3.5 but it will be very close to this value for 384 BELL SYSTEM TECHNICAL JOURNAL all high-frequency elements of the structure. This subject is treated in detail in Parts II and III. The maximum displacement dm , in the case of tangent elasticity, may be calculated from equation (1.9.2) or, in terms of dh/dn , from d,„ 2 ^1 — - = - cos exp db TT 8 W (1.9.5) The ratio d,„/db is plotted against db/do in Fig. 1.9.2. The use of Fig. 1.9.2 can be illustrated with the example already calcu- lated, in which db/do = 1.15/1.44 = 0.8. Entering the abscissa of Fig. 1.9.2 with db/do = 0.8 we find dm/db = 0.915. Hence the maximum dis- placement will be 0.915 X 1.15 = 1.05 inches. 1.10 Optimum Shape or Load-Displacement Curve for Tangent Elasticity It is possible to choose the best shape for the load-displacement curve of the cushioning from those represented in Fig. 1.4.7. This will be, of course, not the best of all possible curves, but only the best among "tangent elasticity" curves. The best shape is defined as the one that yields the smallest maximum acceleration (Gm) for a given weight (W2), height of drop (//) and available space (db). This leaves the initial spring rate (^0) as the only remaining variable. To find its optimum value (say ^0), set equal to zero the derivative of Gm (equation (1.9.4)) with respect to ko , remembering that Go and do are functions of ko . The result is tt'-^IFs/A /tt^IFs// from which '-i4*rj-plM*r'-'^°' <^-"'-" 3AW2h ko = --^. (1.10.2 Substituting (1.10.2) in (1.9.4) we find the minimum value (G„0 of maxi- mum acceleration to be 3.9A GL=~-~. (1.10.3) db To illustrate. the application of equations (1.10.2) and (1.10.3), consider again the case of the 20-pound article dropped from a height of three feet. We found that a linear spring, with a spring constant of 694 lbs/in, would limit the maximum acceleration to 50g if 1.44 inches of displacement were available. If only 1.15 inches of displacement are available, and the initial spring rate is kept at 694 lbs/in, we found the maximum acceleration to be DYNAMICS OF PACKAGE CUSHIONING 385 17 5si if the cushioning bottoms witli tangent elasticity. Now, according to equation (1.10.2), the best initial spring rate for cushioning with tangent elasticity would be A'u = = 1690 Ibs/ni. In this case, equation (1.10.3) gives, for the maximum acceleration, „/ 3.9X36 .^- G. ^ ^^^ = 122g. Hence, confronted with a space limitation less than that required for a 50g linear spring, it is better to use an initial spring rate higher than that for the 50g linear spring in order to strike an economical balance between displace- ment and bottoming. The best balance, among cushionings having tangent elasticity, is obtained by using equation (1.10.2). If no factor of safety is considered, it would be still better not to use a bottoming type of cushion at all. From equations (1.3.5) and (1.3.3) it can be seen that a linear spring with a constant of 1090 lbs/in will give only 63g with a displacement of 1.15 inches. Such a spring, though, would bottom very sharply at a drop slightly higher than 3 ft. and would give an acceleration much greater than cushioning with tangent elasticity which bottoms more gradually. This may be important if there are high-fre- ciuency, brittle elements in the packaged article (see Part III). 1.11 Procedure for Finding Maximum Acceleration and Displacement for Cushioning w'ith Tangent Elasticity (Class C) To illustrate the use of the equations and curves for Class C cushioning, the same example used for Class B will be used, as it was observed that the experimental load-displacement curve in that example (Fig. 1.6.1) is a border line one which can be treated as either B or C. By laying a straight edge along the first part of the curve (Fig. 1.6.1), the average initial spring rate is found to be 305 lbs/in. This value is taken as k() in the present case. The next step is to find a value of db such that a graph of P/db vs Xo/db will fall slightly above the curve ko = 30(0) in Fig. 1.4.7; db must be greater than 2 inches, since that displacement was obtained in the experiment. As a trial take db = 2.25 inches and test it at one point, say the experi- mental point P = 300 lbs., x^ = | in. Then P/db = 133 and Xo/db = 0.39. The point (0.39, 133) falls below the curve ko = 30(0) in Fig. 1.4.7. Next try db = 2.5 inches. In this case, for the e.xperimental point P = 386 BELL SYSTEM TECHNICAL JOURNAL 300, .T2 = I, we find P/db = 120, x./db = .35. This falls slightly above the ko — 30(0) curve as required. The whole experimental curve is then plotted to the coordinates P/2.5 vs .V2/2.5 and is found to fit as closely as necessary. Hence the parameters are adopted as ^0 = 305, db = 2.5. We can now calculate the maximum acceleration that the tube will receive in, say, a three-foot drop test. First calculate, from equations (1) and (2), '2hW2 ^ /2 X 36 X 22.5 ^ ^ 31 2hko _ , / - ^N ^^^^^^^^^^ _ .. 22.5 Then db/do = 1.08. Entering Fig. 4 with this value we find G,n/Go = 1.82. Hence the maximum acceleration is: Grn = 31.3 X 1.82 = 57g. Finally, entering Fig. 5 with db/do = 1.08 we find d,n/db = 0.8. Hence the maximum displacement is dm = 0.8 X 2.5 = 2.0 inches. This indicates that the load-displacement test was carried far enough to cover the range up to a three-foot drop. It may be observed that the results obtained, by treating the same data as Class B or Class C cushioning, agree within a few per cent. This is because, in the example chosen, both B and C curves can be made to fit the experimental load-displacement curve. 1.12 Consequences of x\brupt Bottoming (Class D) It is useful to examine cushioning systems that can bottom more abruptly than Class C cushioning. Abrupt bottoming is possible, for example, in a tension spring package lacking a snubbing device. An estimate of the increase in acceleration can be made by studying the case of bilinear elasticity (Fig. 1.4.4). Here we have a spring rate ^0 up to a displacement ds, follow- ing which the cushioning has a different spring rate kb . ^0 represents the average spring rate before bottoming and kb can represent the much greater stiffness of the wall of the container. If do > ds, that is, if 1/ 2^^^^i>^., (1.12.1) the suspended article will bottom and the maximum displacement and acceleration are obtained by using both of the equations (1.4.4) in evaluating the integral in (1.2.15). Thus, [ ' koX2 dx2 + I "' [kbX; - {kb - ko)d,\ dx, = W,h. (1.12.2) Jo Jrl. DYNA}riCS OF PACKAGE CrSHIONLXG 387 The remainder of llie procedure for finding G,,, is the same as before. The vahie of d,,, found from (1.12.2) is suhstiluted for .vj in the second of Fig. 1.12.1 — Curves for finding maximum acceleration as a result of abrupt bottoming. See equation (1.12.3). (1.4.4) and the value of P,n , thus obtained, is used, in (1.2.16), with the result: where Go is the acceleration that would be reached if a displacement do were available: Go = /ihko (1.12.4) The ratio Gm/Go is plotted against ds/do in Fig. 1.12.1 for several values of kb/ko . Since, in practice, kb might be thousands of times as great as ^o ; it may be seen that the increase in maximum acceleration can be very large even when dg is only slightly less than do . It is apparent that a snubbing device is desirable in a tension spring suspension. This is especially true when considering high-frequency elements of the packaged article. It will be shown, in Part III, that low-frequency elements are not affected as much as might be expected from consideration of maximum acceleration alone. 388 BELL SYSTEM TECHNICAL JOURNAL 1.13 Cushioning with Hyperbolic Tangent Elasticity (Class E) In the preceding sections, there have been considered four types of elas- ticity (hnear, cubic, tangent and bihnear) that fit the load-displacement characteristics of the more common cushioning materials and devices. There now remains the problem of finding more nearly ideal shapes of elasticity. By "more nearly ideal" is meant a shape which will result in a smaller maximum displacement for a given maximum acceleration. This is important in the packaging of very delicate articles if shipping space is limited. It may be observed (equation (1.2.15)) that the total area under the load-displacement curve is equal to the maximum energy of the system. The maximum ordinate of the enclosed area is proportional to the maxi- mum acceleration. Hence, if we wish to (1) limit the maximum acceleration (2) accomodate a given kinetic energy and (3) have as small a displace- ment as possible, the best shape for the load displacement function is P = constant, where the constant is the product of the supported mass and the maximum allowable acceleration. It is not practical to obtain this ideal shape exactly, for there will always be a finite initial spring rate and a rounding off of the load-displacement curve to the limiting maximum load. A function which represents this practical condition (and also includes the ideal case) is the hyperbolic tangent function mentioned in Section 1.4: P = Potanh^'. (1.13.1) The formulas for maximum acceleration and displacement are found in the same way as for the other classes of cushioning with the results: Po -1 (WohkA dm = r cosh exp I o j (1.13.2) or and or doPo ,-1 fWlG} "^^ = w^cr'^'" '-'n^iT^ ^'-''-'^ Gm = ~ tanh ^' (1.13.4) W 2 -i 0 G„.=i^y^l-exp(-"3p) (U3.5) DYNAMICS OF PACKAGE Ci'SIIIOXIXG 389 where, as before k = |/ Go /2hko V 1F2 • Equations (1.13.3) and (1.13.5) are plotted, in Figs. 1.13.1 and 1.13.2, against the dimensionless parameter Po/WoGo . The latter is the ratio of the maximum force, that the hyperbolic tangent cushioning will transmit, .|.:..|....i;::-|:;::|;.;.i;;;:|:. .|.;..(-,.,|.:„t^ 3 0 2.0 1.5 1.0 H— ■■ 1.0 2.0 2.5 30 1.5 Pq W2G0 Fig. 1.13.1 — ^Maximum displacement for cushioning with hyperbolic tangent elasticity. See equation (1.13.3). to the force that linear cushioning would transmit under the conditions specified. To find the value of ^0 which yields the minimum value of acceleration for a given maximum displacement, differentiate (1.13.4) with respect to ko and set the result equal to zero: •2 ^0 U„i, sech" Po 0. (1.13.6) This is satisfied by ^0 — ^ '^- , which represents the rectangular load dis- placement curve and confirms the conclusion reached from energy considerations. 390 BELL SYSTEM TECHNICAL JOURNAL Taking the limit of (1.13.4) as ^o — ^ <^ , we lind the optimum acceleration to be r' - P^ (1.13.7) The corresponding maximum displacement is found, from (1.13.2) to be W. h h ^» =-^ = ^- (1.13.8) -TO yJm W2G0 Fig. 1.13.2 — Maximum acceleration for cushioning with hj^perbolic tangent elasticity. See equation (1.13.5). 1.14 Minimum Space Requirements for Various Classes of Cushioning It is interesting to compare the minimum amount of space for displace- ment that can be attained with the various kinds of cushioning that have been discussed. Hyperbolic Tangent Elasticity dL Gm Linear Elasticity dm 2h Gm Tangent Elasticity dm 3.9h Gm DYNAMICS OF PACKAGE CUSHIONING 391 Cubic elasticity will give a (/,„ somewhat more or less than 2h/Gm depending upon whether the parameter r is positive or negative. It is seen that a factor of almost four can be gained, in the linear dimensions of the cushioning space required, by replacing the tangent type of cushioning with the hyperbolic tangent type. There are several ways of obtaining a load-displacement curve with a shape similar to the hyperbolic tangent curve. One of the most interesting is suggested by the fact that the load-displacement curve of a strut has approximately this shape. Hence a bristle brush has the proper characteristics. TABLE II 0 1 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 A(.r.)„ (■T2)„ Pn 0 0 0.500 0.500 120 0.100 0.600 150 0.100 0.700 205 ' 0.100 0.800 290 0.100 0.900 410 0.100 1.000 585 0.050 1.050 730 0.050 1.100 950 0.050 1.150 1370 0.025 1.175 1680 0.025 1.200 2240 0.0125 1.2125 2620 0.0125 1.225 3200 5 6 7 AAn = iA(Xi)n An = An-1 X (Pn + Pn-l) + AA„ 0 0 0 30 30 1.6 13.5 43.5 2.4 17.0 61.3 3.3 24.8 86.1 4.7 35.0 121.1 6.6 49.8 170.9 9.2 32.9 203.8 11.1 42.0 245.8 13.3 58.0 303.8 16.4 38.1 341.9 18.5 49.0 390.9 21.1 30.4 421.3 22.8 36.4 457.7 24.7 Gn Pn 11.1 15.7 22.2 31.6 39.5 51.4 74.0 91.0 121.0 141.5 173.0 1.15 Nltmerical Method for Analyzing Class F Cushioning The numerical method to be described is one that has been adapted from a graphical one used by the Committee on Packing and Handling of Radio \'alves of the British Radio Board. The method has advantages of sim- plicity in concept and ease of application, especially when the load-displace- ment curve of the cushioning does not resemble closely one of the Classes A to E described above. It has the disadvantage that it does not yield, directly, numerical factors by which the spring rate or depth of cushioning should be changed in the event that the analysis reveals inadequate or more than adequate protection. The method is based on the fact that the area under the load-displace- ment curve of the cushioning represents the energy stored in, or absorbed by, the cushion. The total amount of energy that must be transferred is equal to the product of the weight (IFo) of the suspended item and the height (//) of drop. By finding the abscissa (xn) and its ordinate (P) which 392 BELL SYSTEM JTECH NIC A L JOURNAL include an area Wzh, the maximum displacement is immediately .T2 and the maximum acceleration is the quotient P/W2 , in accordance with equations (1.2.15) and (1.2.16). As actually applied in the present instance, the British method was modified slightly to make the procedure a routine numerical one. The 3200 3000 2800 2600 2400 2200 «2000 o § 1800 o ^1600 < o -11400 1200 1000 800 600 400 200 L OAD 3ISPL ACEM ENT CUR\ IE (1 \ \ \ / 1 1 / A 5 ^4-<— A(X2)j A(X2)2 A(X2)3 A(X2^ X2= DISPLACEMENT V A(X2) Fig. 1.15.2 — Graphical illustration of numerical method of calculating area under load- displacement curve. See Table II. Column 5. A^„ = ^A(x2)n{Pn-i + Pn) is the area of the trapezoid with altitude A(x2)n and bases Pn-i and P„ . It is approximately the energy absorbed by the cushioning in displacing from {x2)n-i to (a;2)„ . Column 6. ^„ is the sum of all the trapezoidal areas from X2 = 0 to X2 = (.T2)„ . It is approximately the total energy the cushioning can absorb in displacing an amount (x2)„ beginning at zero dis- placement. Note that ^0 is always equal to zero. Column 7. /?„ = A„/W2 is the height of fall that will cause the cushioning to displace an amount {x2)n • In Table II, TI^2 = 18.5 pounds. Column 8. G„ = -P„/TI^2 is the maximum acceleration experienced by the suspended mass when dropped from a height hn . 394 BELL SYSTEM TECHNICAL JOURNAL 170 160 / / 150 140 130 120 1 1 / ^ inn / i U 100 (0 "^ or, / H u. 80 O ii^ 70 s: ^ 60 50 AO 30 20 10 / f / / / / y / .'^ / / z' ^ / /" X 8 10 12 14 16 18 HEIGHT OF DROP (INCHES) 20 22 24 Fig. 1.15.3 — Maximum acceleration vs. height of drop for an 18.5 pound article supported on cushioning with the load-displacement curve of Fig. 1.15.1. See Table II. From the last two columns of the table a curve of height of drop vs. the corresponding acceleration may be plotted as in Fig. 1.15.3. PART II ACCELERATION-TIME RELATIONS 2.1 Introduction In Part I we were concerned primarily with the maximum acceleration of the packaged item. In this part we shall study the details of the variation of acceleration wnth time in order to have this information available for our DYNAMICS OF PACKAGE CUSHIONING 395 study, in Part III, of its influence on the response of elements of the packaged item. The first case to be considered will be the simple single mass and linear spring example described in Sections 1.2 and 1.3. Following this the phenomenon of rebound of the package will be considered. The influence of velocity damping and dry friction will be studied; and, finally, the effects of non-linearity of the cushion elasticity on the acceleration-time relation will be investigated. 2.2 Acceleration-time Relation for Linear Elasticity Returning to the elementary example studied in Sections 1.2 and 1.3, we first write the equation of motion for the mass W2 , on its linear spring of spring rate ki (see Fig. 1.2.1.). Equation (1.2.3) becomes ni2X-i + ^2-V2 — ^Wog. (2.2.1) Using the initial conditions N.=o = 0, (2.2.2) [:v-2],=o = V2^, (2.2.3) the solution of (2.2.1) is Xo = ^^-^ — —^ — --^- sm (co2/ — a) + ^ C02 (2.2.4) or where and W2 ^ + dl, sin (co./ - a) + ^', (2.2.5) = 4 A = 2^2 = ^ (2.2.6) a = tan ^ 7^=7 = tan ^ rY' (2.2.7) W2 V 2gh ko dm 0)2 is the circular frequency, /> is the frequency and To is the period of vibra- tion of the mass W2 on its spring; d,,, has the same definition as in Section 1.3 (equation (1.3.1)). Now, Wi/ki is the static displacement of the mass nio on its spring. This is usually very small in comparison with the maximum displacement {dm) during impact. Hence 1^2/^2 will be neglected, and (2.2.5) becomes .r2 = dm sin oiot- (2.2.8) 396 BELL SYSTEM TECHNICAL JOURNAL Differentiating (2.2.8) twice with respect to /, we find, for the acceleration .^2 = —oiidm sin co2^ = — aj2 'V2gh sin 032t. (2.2.9) Hence the absolute magnitude of the maximum acceleration is ^ _ I ^2 |max _ iO^dm _ Ilkki /"l O 1 n^ g g y W-i as before. W- 9 - Fig. 2.2.1 Fig. 2.2.2 Fig. 2.2.1 — Half-sine-wave pulse acceleration. See equation (2.2.9). Fig. 2.2.2 — Oscillogram of a half-sine-wave pulse obtained with a piezo-crystal accelerometer. Equation (2.2.9) shows that the acceleration varies sinusoidally with time. It rises from its initial zero value to its maximum in a time 7r/2a)2 , at which time the displacement also reaches its maximum value. The acceleration returns to zero again at time 7r/co2 • At this time the displacement is also zero. This is the end of the range of appHcabiUty of equation (2.2.9); for when / becomes slightly greater than ir/oii , a tension in the spring is required. Since no mechanism, such as a large mass ms (Fig. 0.2.2), has been supplied, to allow a tension in the spring to develop, the system will rebound from the floor at the end of the half period 7r/cu2 . The acceleration is thus a half- sinusoidal pulse of duration 7r/co2 = T^/I and amplitude Gmg as illustrated in Fig. 2.2.1. An oscillogram of such a pulse obtained with a piezo-crystal accelerometer is shown in Fig. 2.2.2. 2.3 Package Rebound. The presence of the mass of an outer container will ajffect the acceleration after the first half cycle of displacement. The outer container is represented by the mass Ws in the general idealized system illustrated in Fig. 0.2.2 and in the simpler system (Fig. 2.3.1) that we shall consider now. DYNAMICS OF PACKAGEXUSHIONING 397 Two pairs of equations are necessary to describe the action of the system; one pair appHes during the time of contact of m^ with the floor and the second pair appKes if rebound occurs. The mass W3 is assumed to be inelastic (see Section 0,2) so that, during the interval of its contact with the floor, the equation of motion for m^ will be the same as before (2.2.1). In addition, there w ill be an equation of equili- brium for the mass W3 : R = kiXi + nizg , where R is the upward force exerted by the floor on mz . (2.3.1) h /////>///////// Fig. 2.3.1 — Two-mass system representing packaged ^^ticle, linear cushioning and outer container. Equations (2.3.1) and (2.2.1) will hold as long as i? is positive. To find out when R > 0, solve (2.2.1) for ^2-^*2 and substitute in (2.3.1): R = W2 +Ws- W2X2 (2.3.2) That is, a necessary condition for rebound is that the mass of the cushioned article, multipHed by its maximum acceleration, exceeds the total weight of the package. The condition for rebound may be written Gm > W2 + W3 W2 (2.3.3) This is a necessary, but not a sufiicient, condition for rebound because there will be energy losses as a result of damping and permanent deformation. Gm will generally have to be considerably greater than the right hand side of (2.3.3) for rebound to occur. If rebound does not occur, equation (2.2.9) continues to apply, except for damping which will be considered in Section 2.5. 398 BELL SYSTEM TECHNICAL JOURNAL 2.4 Motion After Rebound If rebound occurs, the equations of motion for the two masses, W2 and Wg , are W2i-2 + ^2(^2 — xz) = m^g, (2.4.1) nizXz - kii^Xi — X3) = nisg. (2.4.2) Multiplying (2.4.1) by W3 and (2.4.2) by m^ and subtracting, we find my + k2y = 0, (2.4.3) where y = X2 — X3 , (2.4.4) W2W3 ni2 -\- ms m = ■ . (2.4.5) Fig. 2.4.1— Oscillogram illustrating the half-sine pulse followed In- the higher frequency, lower amplitude vibration of the packaged article in a rebounding package. Equation (2.4.3) is the equation governing the vibration of the two-mass system as a simple oscillator. The circular frequency of the vibration is Vm (2.4.6) m and it may be noticed that this frequency is always greater than uo (equation (2.2.6)). This fact is important in estimating the effect of vibrations on elements of the packaged item (Section 3.5). CO is also the frequency of vibration of the packaged article during the interval of free fall. This vibration (usually of small amplitude) is initiated by the sudden release of the dead load displacement of the packaged article. As an intermediate step in obtaining the acceleration after rebound we shall find the magnitude of the relative displacement (y) of the two masses. To do this it is necessary to solve equation (2.4.3) with the appropriate boundary conditions. Calling tr the time at which nis leaves the floor, we DYNAMICS OF PACKAGE CUSHIONING 399 must find y and j at / = tr . Since Ws is motionless at / = /r , the relative displacement and velocity at that time are identical with xo and .f2 respec- tively. The former is simply the stretch of the spring necessary to just pull the mass nis off the floor, i.e., {y]t^tr = i^^i-'r = -~- (2.4.7) To find the velocity at / = /r , substitute (2.4.7) in (2.2.4) and also substi- tute Ir for / in the latter. This gives an equation for determining/,- . Then, returning to (2.2.4), differentiate it once to obtain .fo and substitute for / the value tr just found. The result is t=,, = [y],=,^ = - a/ 2gk - The solution of (2.4.3) with initial conditions (2.4.7) and (2.4 i,l,.„ = ly,,., = - ^/ 2,, _ ^-"^1<2^^^ . (2.4.8) y = -I /l/V' -^^ sin (u.( - n, (2.4.9) where f = co/r — tan k2 -1 o}[y]i^t^ [yU. We are now in a position to find the acceleration of the packaged item after rebound. Substitute y of (2.4.9) for .vo — Xs in (2.4.1) to obtain *2 = g + - i/lgh - ^^ sin (co/ - f ). (2.4.10) To obtain a simple formula for the ratio of the maximum accelerations after and before rebound, let us assume that both are much greater than gravitational acceleration. Then if Gr — maximum number of g's after rebound, (maximum of (2.4.10)) Gm = A/ Yr^ ~ maximum number of g's before rebound, we find, from (2.4.10), neglecting the term g outside the radical, G„. W2 + Ws y 2hk /- & W. ./. W3 (2,^11) il Hence, the maximum acceleration after rebound is always less than the maximum acceleration before rebound. Therefore, conditions after re- bound need only be examined when the frequency after rebound (see equa- tion (2.4.6)) is near the natural frequency of vibration of a critical element of the packaged item (see Section i.5). 400 BELL SYSTEM TECHNICAL JOURNAL The complete acceleration history of a rebounding package with un- damped Hnear cushioning is thus a half sine wave pulse of amplitude Gm = ■s/lhki/Wi and duration 7r/w2 followed by an oscillating acceleration of amplitude given by (2.4.11) and frequency given by (2.4.6). Such a wave shape is shown in Fig. 2.4.1. 2.5 Influence of Damping on Acceleration The presence of damping in cushioning is always desirable to prevent the building up of large amplitudes as a result of periodic disturbances. How- ever, damping also has an effect on the maximum acceleration that is at- tained in a drop test. From the latter point of view there is an optimum amount of damping and an amount that should not be exceeded if the maxi- mum undamped acceleration is not to be exceeded. We shall consider the case of a linear cushion with damping proportional to velocity. The system is represented in Fig. 2.5.1. With the addition 4 5 h Fig. 2.5.1^Idealization of linear cushioning with velocity damping. of the damping term the equation of motion of m^ , during contact of the package with the floor, is miX2 -f C2X2 + ^2.^•2 = 0, (2.5.1) in which C2 is the damping coefficient of the cushioning. Equation (2.5.1) is more conveniently expressed as X2 -h 2/320)2X2 + W2X2 = 0, (2.5.2) where co2 = 4/^, (2.5.3) 2w2a;2 W2 is the undamped circular frequency of vibration of W2 on its spring and ^2 is the fraction of critical damping. ^2 = 0 means no damping and 182 = 1 means just enough damping so that there will be no oscillation if the pack- aged article is displaced and released. DYNAMICS OF PACKAGE CUSHIONING 401 The acceleration solution of (2.5.2), with the initial conditions of the drop test (see (2.2.2) and (2,2.3)) is where i2 = - '^^i^M. .-^-^-' COS (co../vr^^^ + 7) V 1 - ,8i X-i tan 7 = 2/31- 1 2/32 Vl - 0 (2.5.5) (2.5.6) Gog \ \ \ V X 0 25 - 5 \ /Z ' "~-^ V Jrr V^ ^ ^ ^ S\ \ ^ ^ \ s. / /32= .0 / N ^ ^ ^ "\ \, \ \ / // 5 / ,5 1.0 :> V, \ 3.0 \ 3.5 // /4.0 45 5.0 /S2 = .75 ^/S2=1.0 "■^ Sa5= — \ -H- = "^ ^ J ' -^ \~ .- \ ^2*- — ; X V2 = .25 1 X ^ ^ Fig. 2.5.2 — Acceleration-time curves for linear cushioning with various amounts of damping (no rebound). See equation (2.5.5). The acceleration is thus a damped sinusoid with an abruptly reached initial value whose magnitude depends upon the amount of damping. For small damping, the initial acceleration is small and then the acceleration increases, but never reaches the value that would be reached without any damping. For high damping 032 > 0.5) the initial value is greater than without any damping and falls off thereafter. Figure 2.5.2 shows the shapes of the acceleration time curves for several values of 0i . All of the curves are for no rebound. It may be seen, from equation (2.5.5) and Fig. 2.5.2 that the addition of damping changes the shape of the acceleration-time relation in three ways. First, a damped sinusoid replaces the pure sinusoid; second, the frequency is reduced; and, third, the initial phase is changed. It is useful to consider in detail the effect of damping on maximum accel- eration. Let Gm = maximum number of g's with damping '2^2 -/ W-, = maximum number of g's without damping. 402 BELL SYSTEM TECHNICAL JOURNAL 2.0 1.8 / / 1.6 14 / / / 1 1.2 / / 1.0 ^1 -7^ z .8 — y^ — .6 MAXIMUM OCCUF AFTER t = 0 'S , r. ^ AXIMUM OCCURS , ATt=p , .4 .2 0 0.1 0.2 0.3 0 4 0.5 0.6 0.7 0.8 0 9 1.0 ^2 (FRACTION OF CRITICAL DAMPING) Fig. 2.5.3 — Influence of velocitj' damping on maximum acceleration. See ecjuation (2.5.5) Then, from (2.5.5), at / — 0 and, after / = 0, Go = 2)32 „—^2"2ti (2.5.7) (2.5.8) (2.5.9) where /„, , the time at which the maximum occurs, is given by tan .,/., Vl - ft = ^,(3 _ 40=) ' The largest value of G,n/Go from (2.5.7) and (2.5.8) is plotted against ^82 in Fig. 2.5.3. It is shown there that, as the damping is increased from zero, the maximum acceleration first decreases to a minimum of 80% of Go and then increases to Go at 50% of critical damping. In this interval the maxi- mum acceleration occurs after / = 0. For damping greater than /S2 = 0.5 the maximum acceleration occurs at the instant of contact and increases in direct proportion to 02 ■ 2.6 Influence of Damping on Rebound In considering rebound without damping, it was found that rebound does not occur unless the product of the maximum acceleration and the sus- DYNAMICS OF PACKAGE CUSHIONING 403 pended mass exceeds the total weight of the package. It was not necessary to distinguish between maximum acceleration on the first downstroke and first upstroke, since these are the same when there is no damping. With damping, however, the maximum acceleration on the first downstroke is 1.0 0.8 \ \ 0.6 \ V \ 0.4 \ \v \j 0.2 \ s. \ ^■^ n 1.0 Fig. 2.6.1 — Influence of velocity damping on maximum upstroke acceleration. See equation (2.5.5). greater than that on the first upstroke (Fig. 2.5.2) and it is the latter that controls rebound. Hence damping inhibits rebound. For example, with 50% of critical damping ((82 = 0.5), equations (2.5.8) and (2.5.9) and Fig. 2.5.2 show that for the first downstroke Gm/Go = 1 while for the first upstroke G,n/Go = 0.164. Hence the tendency to rebound is reduced by a factor of six when damping to the extent of 50% of critical is added to an undamped package. The ratio of the maximum acceleration on the first upstroke to the maxi- mum undamped acceleration is plotted in Fig. 2.6.1 for various values of ^o . 404 BELL SYSTEM TECHNICAL JOURNAL 2.7 Influence of Dry Friction on Acceleration and Displacement By "dry friction" is meant friction that is independent of velocity except for sign. During contact of the package with the floor the motion of W2 might be opposed by a constant friction force F. Such a force is developed, for example, in a package with corrugated spring pad cushioning by rubbing against the side and end pads in a top or bottom drop. A typical ideaUzed 2F Fig. 2.7.1 — Load vs. displacement for cushioning with dry friction. load-displacement curve is shown in Fig. 2.7.1. For the first downstroke of m^ , the equation of motion of m^ is W2X2 + ^2^*2 = —F. With initial conditions the solution of (2.7.1) is Xo = ]/dl+ (IJ sin i..J + a)- I (2.7.1) (2.7.2) (2.7.3) where do = |/^-f-^ tan a = F _ F ki do W2 Go ' Go = /2M2 DYNAMICS OF PACKAGE CUSHIONING 405 do and Go are the maximum displacement and acceleration that would obtain if no friction were present. From (2.7.2) the maximum displacement with friction is Hence, the presence of friction decreases the maximum displacement since dm < do . From (2.7.3) the acceleration is - = - yCl + ^^Y sin (co2f + a), (2.7.5) so that the maximum acceleration is G^=^Gl + {0. (2.7.6) which is greater than the maximum acceleration without friction. It would appear, at first glance, that cushioning with friction always gives a greater acceleration than the corresponding cushioning without friction. However, the reverse is actually true provided we allow the same displacement in both cases. This may be done, as may be seen from (2.7.4), by decreasing the spring rate in the cushioning with friction to k, =. k2-^. (2.7.7) do The maximum acceleration in the cushioning with friction is then, from (2.7.6), That is, for the same maximum displacement, the maximum acceleration is reduced by the addition of dry friction. 2.8 Acceleration-Time Relation for Cubic Elasticity As an example of the effect of nonhnearity of the cushioning on the shape of the acceleration-time function, the case of cubic elasticity (Class B) will be analyzed. The system to be considered is illustrated in Fig. 1.2.1, and the load-displacement relation for the cushioning is given by P = koX2 + rxl . (2.8.1) 406 BELL SYSTEM TECHNICAL JOURNAL Substituting (2.8.1) in (1.2.13) and performing the indicated integration, we find .2 ^7 «0 2 T A Xo — Igll — .T^ — .To . m-i 2ni2 Remembering that Xo = dxi/dt, we solve (2.8.2) for dt: dxt dt = \/2gh-^xl-^x\ y 1112 2nu (2.8.2) (2.8.3) \ \ 1 ! \ s .8 \ \ V Ol \ \ ^ \ 1.0 1.5 2.0 2.5 3.0 Fig. 2.8.2 — Acceleration-time curves for cushioning with cubic elasticity. See equation (2.8.14). and B and dm are as given in Part I: mthr 5 = ,2 dm = do /j/|(-l + Vl + B) (1.5.3) (1.5.4) 408 BELL SYSTEM TECHNICAL JOURNAL Then (2.8.4) becomes 1 r^ dZ dZ k \/(l - Z2)(l - yfe2Z2)' in which the integral is the elliptic integral of the first kind (see Hancock "Elliptic Integrals," John Wiley and Sons, New York, 1917). In (2.8.7), CO, = coo (1 + B)"\ (2.8.8) where coo = 's/ka/m^ is the radian frequency that would obtain if the cushion- it. g were Unear with spring rate ^o • The motion for the linear case has a half period, or pulse duration ro = tt/coo . The half-period (72) of the motion with cubic elasticity is twice the time required for X2 to increase from 0 to dm • From 2.8.5 [Z]x,=o = 0, (2.8.9) Hence, from (2.8.7), the half-period is 2 r^ dZ 2K "'^'^ci V(l - Z2)(l - ^) = ^ ^^'^-^^^ where K is the complete eUiptic integral of the first kind. The duration of the acceleration pulse is therefore 2K/uc ■ We can define a radian frequency of the acceleration by TT TTOic 7rCOo(l + -S) /T Q 1 1 \ "^ = 7. = 2^ = 2K • ^^-^'^^^ The ratio a3o/co2 (i.e., t^/tq) is plotted in Fig. 2.8.1 which illustrates how the pulse duration decreases as the parameter B increases. Hence, for a given cushioning with cubic elasticity, the pulse duration decreases as the height of drop increases. This is in contrast to the linear case in which the dura- tion is independent of the height of drop. To find the acceleration X2 , we return to (2.8.7) and write it in the form of an elliptic function: sncoc/ = Z. (2.8.12) Substituting the expression for Z given in (2.8.5) and solving for X2 , we find X2 = dmCn{oict - K). (2.8.13) Finally, differentiating (2.8.13) twice with respect to /, we find the accelera- tion to be X2 = Jcdm[2kHn\uict - K) -l]c«(co.^ - K). (2.8.14) DYNAMICS OF PACKAGE CUSHIONING 409 The ratio —Xi/Gog is plotted in Fig. 2.8.2 against a radian coordinate (woO for several values of B. It may be seen that, as B increases, the maxi- mum acceleration increases, the duration of the pulse decreases (see Fig. 2.8.1) and the acceleration-time curve becomes bell shaped. For reference, the sinusoid for the linear case {B = 0) is plotted in the figure. Figure 2.8.2 is plotted for perfect rebound. If rebound does not occur, the curves continue, mirrored in the time axis, so as to form a periodic vibration of period 2x2 . 2.9 Acceleration-Time Relation for Tangent Elasticity In this section the effect of tangent elasticity on the shape of the accelera- tion-time relation will be studied. The shape of the load displacement curve is given by 2kodb TTXi Ci A 1\ P = tan—-. (1.4.3) ■K Mb The system considered is again that shown in Fig. 1.2.1. Referring to the energy equation (1.2.13): Pdxo = m2gh, (1.2.13) 2 "^ in we substitute the above value of P and perform the indicated integration to obtain, for the velocity. ^2 = y^ X2= A/lgh + ^-^hogcos'^. (2.9.1) W2 TT^ 2db Then, as in Section 2.8, '^ dx2 C"^ dx: t r^ dx2 r^ dXi = = / «W2 (2.9.2) Jo X2 Jo ^ /n 1 t ^kodi, irX2 \/ 2gh H log cos -^ y m2ir 2db and the half-period (72) of the motion is again twice the time required for X2 to increase from 0 to dm . Hence r- r T -~^ V I V y \ \ \ N ^ *='b_^ CO / \ V \ \ \^ \, / \ s> \ V \ 1.5 CJot 20 2.5 3.0 Fig. 2.9.2 — Acceleration-time curves for cushioning with tangent elasticity. The curves of Fig. 2.9.2 were obtained by numerical integration of equa- tion (2.9.2), to obtain X2 as a function of /, following which these values were substituted in the equation W2 X2 -\ tan — - =0 TT zao to obtain .vo . It may be observed that the maximum values of the curves are the values dictated by equation (1.9.4). 412 BELL SYSTEM TECHNICAL JOURNAL In performing the numerical integrations of equations (2.9.2) and (2.9.3), it is found that the integrand becomes infinite when X2 = dm since at this point the velocity is zero. In order to avoid this difficulty, it was assumed^ that, for a small distance in the neighborhood of dm , the acceleration is constant with magnitude Gmg as obtained from equation (1.9.4). The procedure is described in further detail in Section 2.12. Figure 2.9.2 gives the acceleration-time curve for perfect rebound. If rebound does not occur, the acceleration is a periodic vibration, each suc- cessive half period having the shape shown, with alternating sign. 2,10 Acceleration-Time Relation for Abrupt Bottoming By abrupt bottoming, we mean bilinear cushioning (Class D) as treated in Section 1.12. The load-displacement relation is (see equation (1.4.4) and Fig. 1.4.4) P = koXi 0 ^ X2^ ds 1 P = kbX2 — (kb — ko)db X2 > ds] Considering, again, the system illustrated in Fig. 1,2.1, the equation of motion of W2 , before bottoming, is mzXz-h koX2 = 0 0 ^ X2 ^ ds (2.10.1) with initial conditions [x2]t=o = 0, [x2](=o = V2gh. (2.10.2) The solution of (2.10.1) is then X2 = — sin ojo /, 0 ^ X2 -^ dg, (2.10.3) OJo where _ . / kp The time (/«) at which .^2 reaches ds is found from (2.10.3); (2.10.4) 1 . _i (^ods 1 . _,ds t» = — sm /-— = — sm J , (2.10.5) coo \/2gn Wo "0 where /' A . /2W2h eta — ^ See Timoshenko, "Vibration problems in Engineering," D. Van Nostrand Co.. New York, Second Edition (1937) page 123. DYNAMICS OF PACKAGE CUSHIONING 413 i.e., do is the displacement that would have been reached if the spring rate remained constant. The velocity of W2 at time ts is [x2\t^t. = \/2pcoscoo/, = A/lghfl -^y (2.10.6) If y/lgh > coods, the displacement will exceed ds and the equation of motion becomes m2X2 + kbX2 — (^6 — kQ)ds = 0, X2 ^ d, The solution of (2.10.7), with initial conditions [X2\t^t, = dg L^2J<=<, VW^- (2.10.7) (2.10.8) is X2 = where kndi — COft/g) (2.10.9) + 1 - r W. X2 ^ d. tan" a = e-) ^6 ( To — ^6 W6 = 4/ — . m2 (2.10.10) By dififerentiating (2.10.3) and (2.10.9) twice with respect to t, the accelerations for the two regions are found to be X2 = — Gogsinuot, 0 ^ X2 ^ ds , (2.10.11) X2 > d,, (2.10.12) where Go = _ /2hko ~ V W2 (2.10.13) Typical shapes of the acceleration pulse represented by equations (2.10.11) and (2.10.12) are shown in Fig. 2.10.1. The curves are drawn for da/do = 414 BELL SYSTEM TECHNICAL JOURNAL 0.5 and for several values of kb/ko . The peak values of the curves are the same as given by equation (1.12.3). The curve marked kb/ko = 1 is the sinusoid of the linear case with duration X2 Gog TT To = — . COo (2.10.14) K A t) ^o' r \^ 5 = 10 3 1 \ / \ k - Tp = 2 1 1 V Ao ^ b' d V ^ ^ >^ ^ "^v sa'^ s a' ^^ < \^ ^\ -^3' .5 1.0 1.5 2.0 3.0 Fig. 2.10.1 — Acceleration-time curves for cushioning with bi-Unear elasticity. d,/do = 0.5. See equations (2.10.11) and (2.10.12). As before, if the package does not rebound, the acceleration shown is mir- rored in the time a.xis after each half cycle, to form a vibration of period 2x2 • It is useful to know the duration of the complete pulse {aa' in Fig. 2.10.1) and also the duration of bottoming {bb' in Fig. 2.10.1). Calling the former T2 and the latter tb , we have, from equations (2.10.11) and (2.10.12) DYNAMICS OF PACKAGE CUSHIONING 415 -X, "^ /- \ K h Tn — Z O .^ iT \ t/. / r-^ < ^C ':X / "--^ <; h^ y z' V r ./ < "^^ / ^ V ^ "^ \ ^ ,oJ / \ ^ k^._,. — ^ s \ ^ ^/' D \ A s 1.0 0.9 0.8 0.7 So |k» % 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 do Fig. 2.10.2 — Pulse durations for cushioning with bi-linear elasticity. See equations - = - sm - + To TT Co To (2.10.15) and (2.10.16) ^6 1 1 tan (2.10.15) (2.10.16) These two equations are plotted in Fig. 2.10.2 for several values of ^b/^o • 2.11 Acceleratiox-Tme Relation for Hyperbolic Tangent Elasticity The relation between acceleration and time for hyperbolic tangent elasticity is found by the same procedure that was used for tangent elasticity 416 BELL SYSTEM TECHNICAL JOURNAL in Section 2.9. The system considered is that shown in Fig. 1.2.1 and the load displacement curve of the cushioning is given by P=Potanh^^ Substituting the above expression for P in the energy equation (1.2.13), we find the velocity to be ^2 = A/lgh - ^ log cosh ^-^ . (2.11.1) Then, as before, r^ dx2 t = — (2.11.2) and the half period (72) of the motion is twice the time required for X2 to increase from 0 to dm , or C^"* dx2 T2 = 2 — , (2.11.3) Jo ^2 where, from Section 1.13, The radian frequency of the acceleration is defined as IT 032 = — T2 an J this is to be compared with the frequency TT coo = — = To Y W2 that would obtain if the cushioning had a constant spring rate equal to the initial spring rate (^0) of the hyperboUc tangent cushioning. The ratio wo/aJ2 (or T2/T0) is plotted, in Fig. 2.11.1, against the dimensionless param- eter P0/W2G0 (see Section 1.13). It may be observed that the pulse duration becomes very long when P0/W2G0 is small, i.e., when the horizontal portion of the load displacement curve (Fig. 1.4.5) comes into play. The influence on the shape of the acceleration-time curve is illustrated in Fig. 2.11.2. The curve marked Po/W^Go ^ 00 is the sinusoid for the Unear case. For small values of P0/W2G0 the curve approaches a square wave. DYNAMICS OF PACKAGE CUSHIONING 417 4.5 4.0 3.5 3.0 o Hi 2.0 \ \ 1.5 \ \ \ 1.0 .5 1.0 1.5 2.0 2.5 3.0 Fig. 2.11.1 — Duration of acceleration pulse for cushioning with hyperbolic tangent elasticity. 2.12 Numerical Procedure for Finding Acceleration-Time Relation FOR Class F Cushioning When the load-displacement curve does not resemble one of Classes A to E, the acceleration-time relation may be found by numerical integration. Combining the energy equation, m2X2 I + / P dxi = m^gh, (1.2.13) 418 BELL SYSTEM TECHNICAL JOURNAL t with the equation relating time and velocity, dx; we find ~2g Jo t = y4^ - Pdxi (2.12.1) (2.12.2) As an example, consider the problem of a 15-pound article supported on cushioning with the load-displacement curve shown in Fig. 2.12.1. The package is to be dropped from a height of 3 feet. The computations are X2 Gog 1.0 1 ^0 1 W2G0 -»-oo .8 7-^=10 LW2Go_; A \ .6 1 / \ \ Pq ^ 1 .5 \\ .4 A V r N I f Po PR \ \ \ I - W2G0 - \ \ \ .2 \ \ \ ^ f \ \ \ / W \ \ 4 Fig. 2.11.2 — -Acceleration-time curves for cushioning with hyperbolic tangent elasticity. given in detail in Tables III and IV. The headings of Columns (1) to (8) of Table III are the same as in Table II, Section 1.15. An is the integral under the radical of equation (2.12.2). Column (10) of Table III is the integrand of Equation (2.12.2), i.e., it is proportional to the reciprocal of the velocity expressed as a function of displacement. The function is plotted in Fig. 2.12.2 and its integration is performed in Table IV. In columns (11), (12) and (13), intervals of Xi are chosen to suit the shape of the curve. The values for column (14) are taken from column (10). Col- umns (15) and (16) perform the same operations on the integrand (y\l^h — Ariy that are performed in Columns (5) and (6) of Table III on the integrand P. DYNAMICS OF PACKAGE CUSHIONING 419 1 600 / 1 ' 500 t / «400 Q Z / / r 2 0^300 y y^' 200 ^ > ^ 100 / / / 1 2 X2 (INCHES) Fig. 2.12.1 — A load displacement curve for Class F cushioning. TABLE III (1) (2) (3) (4) Pn (5) (6) (7) (8) (9) (10) Mx2)n (*2)„ ^^f"(Pn +/-«-.) 0 An hn = -rrr It 2 IF2/; - An 1 n VlFj/i - An n 0 0 0 0 0 0 0 540 0.0431 1 .20 0 ?, 105 10.5 10.5 0.7 7.0 529.5 0.0435 ? .20 0 4 155 26.0 36.5 2.4 10.3 503.5 0.0446 s .20 0.6 192 34.7 71.2 4.8 12.8 468.8 0.0462 4 .20 0 8 717 40.9 112.1 7.5 14.5 427.9 0.0483 S .20 1.0 7.S7 45.4 157.5 10.5 15.8 382.5 0.0511 6 .20 1.2 757 49.4 206.9 13.9 17.1 333.1 0.0547 7 .20 1 4 777 52.9 259.8 17.3 18.5 280.2 0.0597 8 .20 1 6 ,S05 58.2 318.0 21.2 20.3 222.0 0.0671 Q .20 1 8 M2 64.7 382.7 25.5 22.8 157.3 0.0798 10 .20 2.0 397 73.4 456,1 30.4 26.1 83.9 0.109 11 .05 ?. 05 405 19.9 476.0 31.8 27.0 64.0 0.125 1? .05 ?, 10 477 20.7 496.7 33.2 28.1 43.3 0.152 n .05 7 15 440 21.6 518.3 34.6 29.4 21.7 0.215 14 .01 ? 16 445 4.42 522.7 34.8 29.7 17.3 0.240 IS .01 ? 17 450 4.48 527.2 35.2 30.0 12.8 0.279 16 .01 ? 18 455 4.52 531.7 35.5 30.3 8.3 0.347 17 .01 7 19 457 4.56 536.3 35.8 30.5 3.7 0.521 18 .01 2.20 462 4.60 540.9 36.1 30.8 0 00 420 BELL SYSTEM TECHNICAL JOURNAL TABLE IV (11) (12) (13) (14) (15) (16) (17) n A(*!)„ («2)„ 1 0 Dn = r {X2)n dX2 2g \'Wih - An Jo VWih-An 0 0 0 0.0431 0 0 1 0.4 0.4 0.0446 0.0175 0.0175 0.0024 2 0.4 0.8 0.0483 0.0185 0.0360 0.0050 3 0.4 1.2 0.0547 0.0206 0.0566 0.0079 4 0.4 1.6 0.0671 0.0243 0.0809 0.0112 5 0.2 1.8 0.0798 0.0149 0.0958 0.0133 6 0.2 2.0 0.109 0.0189 0.1147 0.0160 7 0.1 2.1 0.152 0.0131 0.1278 0.0178 8 0.05 2.15 0.215 0.0092 0.1370 0.0190 9 0.03 2.18 0.347 0.0083 0.1453 0.0202 10 0.01 2.19 0.521 0.0043 0.1496 0.0208 11 0.01 2.20 00 0.0221 05 0.4 0.3 0.2 / 0 1 / ^ ^ 1.2 Fig. 2.12.2— Plot of Column (3) vs. Column (10) of Table III. A difficulty arises because the integrand (IF2// — An)~^ becomes infinite for the maximum displacement (see Column (14)). This is avoided by assuming that the acceleration is constant in the last interval^ and has the " Timoshenko, "Vibration Problems in Engineering," D. Van Nostrand Co., New York, Second Edition (1937) page 123. DYNAMICS OF PACKAGE CUSHIONING 421 value given in Column (8), Table III, for the maximum height off drop. Then, A(:*;2)„ = ^G„.gA/2 (2.12.3) or At / 2A{X2)n Gmg (2.12.4) 30 A '^ \ 25 1 \ 1 ^ \ 20 J \ / \ V 15 / \ / \ \ 10 / \ / \ 5 / 1 .01 .02 .03 t .04 Fig. 2.12.3 — Acceleration-time curve (for the cushioning shown in Fig. 2.12.1) obtained by numerical integration. In the present instance, (AiC2)n = 0.01 inches Gmg = 30.8 X 386 = 11900 in/sec.^ Hence, from (2.12.4), At = 0.0013 sec. and the last entry in Column (17) is obtained by adding this value of At to the preceding entry. The final curve of acceleration vs. time is obtained by plotting the entries of Column (17) against the entries of Column (8), Table III, for correspond- ing values of X2 . The result is shown in Fig. 2.12.3. 422 BELL SYSTEM TECHNICAL JOURNAL PART III AMPLIFICATION FACTOR 3.1 Introduction If the maximum acceleration, of the packaged article as a whole, is reached very slowly, the severity of the disturbance experienced by a structural element of the packaged article is very nearly proportional to the maximum acceleration. Roughly speaking, "very slowly" means that the time, during which the acceleration undergoes a major change in magnitude, is long in comparison with the natural period of vibration of the element under consideration. When this is so, no transient vibration is excited in the element. The displacement response of an element under very slowly varying conditions is called the "static response". Under more rapidly varying conditions the dynamic response to the same maximum acceleration may be greater or less than the static response. The ratio (A) of the maxi- mum dynamic response to the static response is called the amplification factor. In general, for a given acceleration disturbance, very low-frequency elements have amplification factors less than unity, while the amplification factors are greater than unity for elements whose natural frequencies are near or above the disturbing frequencies. The numerical value of the amplification factor depends not only on the manner in which the disturbing acceleration varies with time, but also on the "reference acceleration", i.e., the value of acceleration for which the static response is calculated. Usually the reference acceleration chosen for calculating the static response is the maximum value (G„j) of the disturbing acceleration. However, when special circumstances are being investigated, such as the effect of damping or abrupt bottoming, the reference acceleration is taken to be Go , which is the acceleration that would be reached if the damping or bottoming were absent. In such cases the amplification factor includes both the effect of rate of change of acceleration and the effect of the special conditions. When the reference acceleration is Gm the amplification factor will be denoted by /!,„ and when the reference acceleration is Go the amplification factor will be denoted hy A^ . The symbol Ge will be used to designate the slowly applied acceleration that would produce the same maximum dis- placement as the transient acceleration, i.e., Ge = AjGm or Ge = AqGo. The symbol Gs will be used to denote the safe value of Ge , for an element of the packaged article, as determined by a strength test or by calculation. In specifying Gs some judgement is required to take into account the effects of plastic deformation in comparing tests made on greatly different time scales. Good judgement is also necessary in deciding whether or not the DYNAMICS OF PACKAGE CUSHIONING 423 assumptions listed in Section 0.2 are valid in each application. The general procedure for using amplification factors is as follows. We first find the value of the reference acceleration (in units of number of times gravity) from Part I. From Part II we find the properties of the acceleration-time rela- tion which give us the information required for entering one of the curves of Part III and finding the amplification factor. Then, the product of the reference acceleration and the amplification factor (/1,„G„, or AoGo) is a number (Ge) by which the weight of the structure is to be multiplied when calculating its deflection or stress by the usual static methods of elementary strength of materials. Alternatively, G, must be found not to exceed Gs . i^d: (a) (b) Fig. 3.2.1 — Idealized system used in calculating amplification factors for linear undamped cushioning with perfect rebound, (a) initial position, (b) first contact with floor. In the following sections the amplification factors for typical transient accelerations encountered in package drop tests are calculated. The ampli- fication factor curves that are plotted are entirely analogous to the familiar "resonance curves" for steady sinusoidal vibration, except that in this case the disturbing forces are transients of various shapes. It will be seen from the curves that the maximum acceleration, as calculated by the methods of Part I or as measured by an accelerometer, is not always a true measure of the severity of the disturbance. 3.2 Amplification Factors for a Half-Sine-Wavx Pulse acceleration The first case to be treated is the response of an element of the packaged item to the transient acceleration that would occur in a package with linear 424 BELL SYSTEM TECHNICAL JOURNAL undamped cushioning and perfect rebound. Figure 3.2.1 illustrates the idealized system, and it may be noted that the mass mz is omitted, as is required for perfect rebound (Section 2.3), At first we shall consider that the mass Wi is undamped and later we shall consider the effect of damping in this element. The mass mi is taken to be small in comparison with m^, so that the motion of the latter is the same as we found it to be in Section 2.2 where mi was not considered. Hence the acceleration of W2 is a half-sine wave pulse: x-i = — C02 s/lghsinoiit, (0 ^ / ^ 7r/oo2). (3.2.1) The equation of motion of mi is mixi + ^i(xi — x^ = 0. (3.2.2) Let X be the relative displacement of mi with respect to m^ , i.e., X = a;i — .'v:2 . (3.2.3) X is proportional to the force in the spring (^i^;) and to the acceleration of mi and hence is proportional to the deflection, strain and stress in the element which the system mi , ki represents. Substituting (3.2.3) in (3.2.2), we find: mix + kix = —miX2 . (3.2.4) This equation holds for the duration x/co2 of the pulse ^2 . The initial conditions for x are M^=o = W^=o = 0 (3.2.5) so that the solution of (3.2.4) is X = V^. [- sin C.I / - sin C02 ^1 , U^t^^. (3.2.6) It may be seen that x is composed of a forced displacement at the accelera- tion frequency co2 , on which is superposed a free vibration at the natural frequency, coi , of the element. The maximum value of the relative displace- ment is ■\/lah. 2nir •^max — ' i C Sin J^-lY'^^+l' V ^' ^co2/- (3.2.7) \W2 / W2 in which w is a positive integer chosen so as to make the sine term as large as possible while the argument remains less than tt. (3.2.7) gives the maximum dynamic response of the element mi during the interval of impact. To find the amplification factor we must compare DYNAMICS OF PACKAGE CUSHIONING 425 ^max with the "static response" i.e. with the value (.t.,«) that x would have if the acceleration x^ reached the same maximum value (co2\/2g/0 in a very long time. The resulting value may be found from (3.2.4) by omitting the transient term jhix. Then Xst = C02 or C02 / Xsi = ~2 V2gh. (3.2.8) The amplification factor for the interval 0 < / < t/co-z is then ^^ ^ x^. ^ _jo^ ^^ _2nT_ ^ (o = t^ ^M. (3.2.9) Xst ^ _ 1 ^ _1_ 1 CO2 CO2 It should be observed that Am depends only on the frequency ratio coi/co2 . That is, since coi/co2 = T2/T1 , the amplification factor depends only on the ratio of the pulse duration to the half period of vibration of the element. Thus far we have studied only the motion in the interval 0 ^ / ^ 7r/aj2 . We must not, however, overlook the possibiHty of larger displacements of nti with respect to m^ occurring after rebound. In fact, examination of (3.2.6) reveals that x has no maximum in the interval 0 ^ t ^ 7r/co2 whencoi < aj2 . It is very likely, then, that larger values will occur at later times. After rebound, nii executes free vibrations with respect to W2 . We have to compare the magnitude of .Tmax , in the interval 0 ^ / ^ t/(j02 , with the amplitude of the free vibration. Calling the relative displacement during free vibration x' and measuring a time coordinate /' from the instant the package leaves the floor, we have mix' + kix' = 0, (3.2.10) with initial conditions [X J<'=0 — [x\t^T/oi2 J (3.2.11) [X Jr=0 — [X\t=r/U2 • The solution of (3.2.10) with initial conditions (3.2.11) is 4/^' + -=-/).„(„,,+^^^), aji(aj^ - CO?) \ 2aj2/ .^ 2 12) ^ > — • 0)2 426 BELL SYSTEM TECHNICAL JOURNAL Then 4 = Xat 2 — COS -- C02 ZC02 (■ ' -■) (3.2.13) We find, on comparing (3.2.13) with (3.2.9) that for coi < wo equation (3.2.13) gives the larger value of .1,,,, while for coi > wo equation (3.2.9) 1.8 1.6 I '^' 1.4 1.0 r' .6 .4 ^ V ^,=o r^-s. ^ 0.005 -O.OI ^0.05 <^ 0.10 0.30 '0.50 *^ ^ ^ ^^^ ^ ^^ ' - / ' V /(?,=1.00 / 10 ^'/^ ^u' Fig. 3.2.2 — Amplification factors for linear undamped cushioning with perfect rebound. See Fig. 3.2.1 and equations (3.2.9) and (3.2.13). gives the larger value of Am. That is, when the duration of impact is shorter than the half-period of vibration of the element, the maximum displacement (and stress) in the element occurs after the impact is over. The curve marked ^Si = 0 in Fig. 3.2.2 is a plot of the largest value of Am from (3.2.9) and (3.2.13) with the frequency ratiocoi/co2 as abscissa. (3.2.13) was used for aji/co2 ^ 1 and (3.2.9) for co 1/0)2 ^ 1. The maximum value of Arr, is 1.76 and occurs at co]/c<;2 = 1.6. Hence, at this frequency ratio, the deformation of the element is 1.76 times as great as would be expected from a calculation using the maximum value of acceleration alone as in Part I. DYNAMICS OF PACKAGE CUSHIONINV 427 On the other hand, for frequency ratios ui/co^ < 0.5 the severity of the shock can be very much less than might be expected from the calculations of Part I. For ver>' small values of coi/a)2 the amplification factor may be seen from (3.2.13) to be equal to 2wi/w2 • For large values of wi/coo (stiff elements) Fig. 3.2.2 shows that the amplification factor is very nearly unit}' and the methods of Part I can be used without additional calculation. When damping of the element of the packaged article is considered, the amplification factors are less than without damping. The applicable equations of motion during and after impact are obtained by inserting velocity damping terms in (3.2.4) and (3.2.10): niix + Cix -\- kix ^ —miX2 , 0 ^ / ^ — (3.2.14) C02 mix' + ci:t-' + kix' = 0, / ^ -. (3.2.15) If we express the damping of the element nii as the fraction of critical damping - ^1 2 vwi^i' (as in Section 2.5) equations (3.2.14) and (3.2.15) become (3.2.16) 9 — — TT X + 2((3icoi.v + ojiX = —X2 , 0 < t < —, (3.2.17) W2 x' + 2^icoi.v' + col.v' = 0, / ^ -. (3.2.18) The amplification factors for equations (3.2.17) and (3.2.18), with boundary conditions (3.2.5) and (3.2.11), respectively, were obtained on the Westing- house Mechanical Transients Analyzer^ for ,81 = 0.005, 0.01, 0.05, 0.10, 0.30, 0.50 and 1.00. The curves are shown in Fig. 3.2.2. 3.3 Application of Half-Sine-Wave Amplification Factors As an example of the use of the amplification factor curves of Fig. 3.2.2, let us consider the following problem: ^ Arrangements lor performing these calculations were made through the courtesy of Mr. A. C. Monteith, Manager of Industry Engineering, and Mr. C. F. Wagner, Manager of Central Station Engineering, Westinghouse Electric and Manufacturing Co. Dr. G. D. McCann, Transmission Engineer, was in immediate charge of the project. For a descrip- tion of the analyzer see "A New Device for the Solution of Transient-Vibration Problems by the Method of Electrical-Mechanical Analogy" by H. E. Criner, G. D. McCann and C. E. Warren, Journal of Applied Mechanics, Vol. 12, No. 3 (1945) pp. A-135 to A-141. 428 BELL SYSTEM TECHNICAL JOURNAL It is required to judge the suitability of a proposed package for a large vacuum tube weighing 10 pounds. Strength tests have been made on the tube in a shock testing machine which produces a half-sine-wave acceleration pulse of 25 milliseconds duration. The weakest element of the tube is found to be the cathode structure, for which the safe maximum acceleration in the drop testing machine is 200g. The cathode structure has a natural vibration frequency of 120 cycles per second and has 1% of critical damping. The proposed package has essentially linear, undamped cushioning with a spring rate of 3300 pounds per inch and an available displacement of | inch. The outer container weighs much less than the tube so that the package may be expected to rebound. Is the cushioning suitable for protecting the cathode in a drop of 5 feet? First find the maximum G that the tube will experience in a 5 ft. drop of the package (equation 1.3.3): „ _ , /2hki _ , /2 X 60 X 3300 _ .^^ ^-- Vw,~ V To ^^^- The accompanying maximum displacement is, from equation (1.3.4), , 2h 2 X 60 „ , . ^- = g; = -199- ='•'"• The available displacement (| inches) is therefore sufl&cient and the maxi- mum acceleration {199 g) is slightly less than the safe maximum acceleration (200g) found with the shock testing machine. However, before the cush- ioning is approved it is necessary to investigate the frequency effects. The duration of acceleration in both the shock machine and in the package must be considered. The amplification factor for the element tested in the shock machine is found as follows. First find the frequency corresponding to the 25 milli- second pulse: U = i-— = 20 c.p.s. ■' 2 X .025 ^ The ratio of the element frequency to the shock machine frequency is /i = ^ = 1^ = 6 fi 0)2 20 Entering Fig. 3.2.2 witha)i/co2 = 6, we read, from the curve /3i = 0.01, Am = 1.14. The 200g test in the shock machine is, therefore, equivalent to a slowly applied acceleration oiGs = 200 X 1.14 = 228g. DYNAMICS OF PACKAGE CUSHIONING 429 To find the corresponding quantity for the package drop, first find the cushion frequency: / 1 . /^ 1 . /3300 X 386 „ 2ir \ mi 2ir y 10 The ratio of the element frequency to the package frequency is therefore Ti ^ coi ^ 120 ^ 2 1 /a 0)2 57 Entering Fig. 3.2.2 with coi/a)2 = 2.1 we read, from the curve 0i = 0.01, Am = 1.59. The 199g acceleration pulse in the package drop is therefore equivalent to a slowly applied acceleration of Ge = 199 X 1.59 = 316g. This is almost 40% in excess of the value (228g) found to be safe from the shock machine data. The cushioning is therefore judged to be inadequate. Ine procedure for finding the correct spring rate for the cushioning is as follows. It is known that we must have Therefore, take Now Therefore Also Ge < Gi Ge = AmGm = 228. Amf^i = 409 rad/sec. ;i = 27r X 120 = 754 rad/sec. Then, with successive trial values of W2 , we calculate wi/coo , enter Fig. 3.2.2, read the corresponding value of Am from curve ^i = 0.01 and test to see if the product AmUi = 409. The combination which satisfies the test is found to be 0)2 = 280 rad/sec. cci/iOi = 2.69 Am = 1.47. 430 BELL SYSTEM TECHNICAL JOURNAL Then 2 (280)' X 10 -_,„ ,, ,. k', = C02W2 = ^ -— = 2030 Ibs./in. 386 V 2M2 _ < rr 2h dm = -p:^ = -I I in. Hence the spring rate of the cushioning should be reduced from 3300 lbs. /in. to 2030 Ibs./in. and the available space should be increased to accomodatg the 0.77 inch maximum displacement before bottoming. 3.4 Special Treatment of Strong, Low Frequency Elements The product of the amplification factor (.4,„) and the maximum accelera- tion {Gm) must be equal to or less than the maximum allowable slowly applied acceleration (Gs): Ge = AmGm ^ G, . (3.4.1) For frequency ratios -^ 00 . 434 BELL SYSTEM TECHNICAL JOURNAL I- _3 o < 10 < 1 o /ff2 =0.01 1 1 1 1 \ i 1 1 . . 1 1 ^ 1 1 ■L , ^^0.01 — H—j — ' y^ ^^ 0.10 — — 1 — ' ' ■ ' — ' [ — 1 — — ' — ' I — /// w^ J s^S ^ •sSf^/ 6 ^^^^5=T=f— — d 1 1 ^ .J^-^ — • — ^^^^^^^^^^ 1 — 1 T^ — \ L-^— 1 : 1 I r-r ^ i _, -^ \ 1XX) 1 1 1 ■ 1 i 1 1 ; 1 // ''' 1 If 1 1 ; III i lil '1 I 1 Fig. 3.5.3 — Amplification factors for linear damped cushioning with no rebound. /32 = 0.01. See equations (3.5.1) and (3.5.2). 5^3 O < fi z =0.05 1 \ 1 j 1 10 I / \ 5 If \ ^ /S, =0.005 — 0.01 ^ — 0.05 1 / / "v^^ y : ^ ! 1 // ^\V/ ^ ^ 1/ y ■- 0 and /32 < 1 the amplification factor is less than four, as 051/0)2 — ^ <» , in accordance with the curves plotted in Fig. 3.5.8. Example: A 1.5-pound vacuum tube is to be packed in a container whose estimated weight will be at least 50 pounds. The cathode structure of the tube has a natural frequency of 25 c.p.s. with damping 0.5% of critical 33 /!?2 =1.0 A Ao .005 396 .01 3.94 .05 3.68 .10 3.44 .50 2.28 1.00 2.00 >ei=ooo5 /X,0.05 1 1 1 _ , — = 1 ; =- 1 / = >- ' i ' __ .^ uy ^-^^^ ^0.5 ~ _ ^ 0 ^=^^ Ji r— _J — — — ~z! --?. k^ i_ j — ^ ■ — ' ^ — ' — — — : ^ ;^ ^^:^ ' ' ' — 1 /// -^ ! i-^^^ -.' /^ ^^^ 1 / / / 1 i ^ I \W 2 Fig. 3.5.7 — Amplification factors for linear damped cushioning with no rebound. i32 = 1.0. See equations (3.5.1) and (3.5.2). and its safe acceleration, as determined in a centrifuge, is 90g. What spring rate of cushioning is suitable for protecting the cathode in a drop of five feet? It is specified that the cushioning shall have damping 50% of critical. Assuming linear cushioning, the spring rate that would be prescribed, by considering maximum acceleration alone, is ife2 = W^GL 1.5 X (90)' = lOllbs./in. 2h 2 X 60 Considering damping, Fig. 2.5.3 shows that 50% of critical damping does DYNAMICS OF PACKAGE CUSHIONING 437 not change Gm . To find the ampUfication factor we must first decide if the package will rebound. With 50% of critical damping, the maximum ac- 40 36 3.2 2.8 2.4 -^3 2.0 o ' < . 1.6 1.2 1.0 .8 .6 .4 .2 / ( /5i=o / ( / / / / / / / / 4=1 / / / / if ^.^ / / / / / / , / ;, ^.5 / / / / '[ / A // / / / /. Sri / / A / / r' V 7 A '/ / // 7 y . //. ^ /y .10 13: Fig. 3.5.8 — Limiting values of amplification factors for linear damped cushioning with no rebound. wi/c<;2— > «. See equations (3.5.1) and (3.5.2). celeration on the first upstroke is 0.164 Gm (see Section 2.6 and Fig. 2.6.1). Then, 0.164 X 90 X 1.5 = 22 lbs. which is less than the estimated weight of the outer container. The package will not rebound and Fig. 3.5.6 should be used for the amplification factor. 438 BELL SYSTEM TECHNICAL JOURNAL The frequency of vibration of the tube in its cushion will be 4/1 = /-^ X ^86 ^ J39 rad./sec. Hence coi/co2 = 2x X 25/159 = 0.99 and, from Fig. 3.5.6, ^lo = 1.4. Hence Ge = 90 X 1.4 = 126, which is greater than the allowable Gs = 90, so that the 101 Ib./in. cushion is unsatisfactory. To obtain satisfactory cushioning, set ^oGo = 90, that is 90 ^oco2 = — 7^ = 159 rad./sec. /! Noting thatcoi = 27r X 25 = 157, we find from Fig. 3.5.6 that there are two values of C02 (90 and 600 rad/sec.) that satisfy the criterion A^p^i = 159 rad/sec. The first gives C02 = 90 rad/sec. A,= 1.8 ^2 = 31.5 Ibs./in. Go = 50 Ge= 90 dm = 2.4 in. The second gives CO 2 = 600 rad/sec. Ao= 0.27 k. = 1400 Ibs./in. Go = 335 Ge = 90 dm = 0.36 in. The second solution requires less space for cushioning than the first but should be used only if the remainder of the tube can endure the high ac- celeration of 335g. Otherwise the 50^ package should be used . DYNAMICS OF PACKAGE CUSHIONING 439 3.6 Amplification Factors for the Pulse Acceleration of Cubic Cushioning In a rebounding package with undamped Class B cushioning, the pack- aged article (W2) will undergo a pulse acceleration of duration ir/w2 as given by equation (2.8.11). The shape of the pulse is illustrated in Fig. 2.8.2 and its functional form is X2 = 4K ' W2 dm r 2 2 /2K(j}2 1 „ —5 2k sn I — K TT- |_ \ ■T )-] f2Koj2t _ en I \ IT .), (2.8.14) To determine the influence of the shape and duration of this pulse on the amplification factor, we proceed as before by substituting (2.8.14) in the r ////////////////////y /// ///77T//////// (a) (b) Fig. 3.6.1 — Idealized system used in calculating amplification factors for non-linear, undamped cushioning with perfect rebound. differential equation governing the relative displacement {x = Xi — x^ between m\ and m-i (see Fig. 3.6.1): x-\-oiiX= —Xl. (3.6.1) With boundary conditions x(0) = .f(0) = 0, the solution of (3.6.1) may be written as X = — I X2 (X) sin coi (X — t) d\ COi Jo and the maximum value of x may be expressed by 1 r''" Xmax = — / XoiX) sin aji(X — tm) d\, COi Jo where /„, is the time at which the largest value of x occurs. (3.6.2) (3.6.3) 440 BELL SYSTEM TECHXICAL JOURNAL The amplification factor, in this case, will be taken as the ratio of .Tmax to the relative displacement (xst) resulting from a slow apphcation of the maximum value of .fo . From (3.6.1), X2 Gmg 2 2 ' where Gm is given by equation (1.5.6). Then (3.6.4) ■Am. — •''max -^max '*'! Xst Gmg (3.6.5: Fig. 3.6.2 — Amplification factors for undamped cushioning with cubic elasticit}'. rebound. See ecjuation (3.6.6). or Am = Wi X2(X) sin coi(X - t„) d\. Perfect (3.6.6) Am was evaluated, mostly by graphical methods, for four values of B (0, 2, 20 and cc) and the results are plotted in Fig. 3.6.2. Observing that B = 0 corresponds to hnear cushioning, it may be noted that cubic non- linearity in the cushioning does not change the amplification factor by more than 35% even in the most extreme case (B -^ oc). The severity of the shock, however, may be much greater for the cubic cushioning than for linear cushioning with a spring rate equal to the initial spring rate (^o) of the cubic cushioning. This is because A,n is multiplied by Gm to obtain Ge and, for large values of B, Gm may be much larger than the maximum acceleration for the linear case. In other wordfe, in comparing Class B DYNAMICS OF PACKAGE CLSIIIOXING 441 with Class A cushioning the difference in maximum acceleration, rather than the difference in amplification factors, is usually more imix)rtant. Example: Consider the example given in Section 1.6 and let it be required to determine the effect of pulse duration on a cathode structure with a 200 c.p.s. natural frequency of vibration. In Section 1.6 we found that B = 5.4 ko = 255 Go = 28.6 r = 108. Gm = 5^ With B = 5.4, enter Fig. 2.8.2 and find coo = 0. W2 Now .. = y/|l = ^'-^^^^ = 66.1 rad./sec. 22.5 Hence W2 = ^^ = 75 rad./sec. Then, with coi/co. = 27r X 200/75 = 16.7, enter Fig. 3.6.2 and find Am = approximately 1 .0. Hence Ge is about the same as Gm and the conclusions reached for this problem in Section 1.6 are not altered. 3.7 Amplification Factors for Abrupt Bottoming The amplification factors for bilinear elasticity have not been computed in complete detail. They can be obtained approximately by using the dura- tion curves (Fig. 2.10.2) and the amplification curves for the linear case (Figs. 3.2.2 and 3.5.2 to 3.5.7). It is useful, however, to calculate the am- plification factors for extremely abrupt bottoming {kb —^ =o) to obtain a general understanding of the accompanying phenomena. The system to be considered is illustrated in Fig. 3.7.1. It is assumed that the impact between wz2 and the base (occurring at / = /s , ^'2 = ds) has a coefficient of restitution of unity. Hence mo will strike the base with velocity tel,=, = /2,*(l-|) (see equation (2.10.8)) and leave it at a velocity of the same magnitude but opposite sign. Perfect rebound of the whole package is also assumed. 442 BELL SYSTEM TECHNICAL JOURNAL The acceleration pulse will then look like the curve marked kb/ko — > <» in Fig. 2.10.1. There will be three regions in which to consider the relative displacement x: Region 1 0 < / < /, Region 2 ts < t < Its Region 3 t > 2ts The relative displacement {x = Xi — X2) of Wi with respect to W2 will have Ti 1 j/ L rrip ^-!."" //////// Fig. 3.7.1 — Idealized system representing abrupt bottoming. the same functional form for Region 1 as in the linear case (see equation (3,2.6)), and the amplification factor is, by analogy with (3.2.9), OJo Int sm COo - 1 COo 0 < / < /, , (3.7.1) + 1 where COo = k(j/m2 For Region 2, we use the differential equation X -^ wix = coo\/2gh sin coo(^ — 2ta) (3.7.2) and, as initial conditions at ^ = /« , we use the terminal conditions for Region 1 with the sign of [x2]«=«s reversed. The amplification factor for this region is found to be (by the same method as in Section 3.2): DYNAMICS OF PACKAGE CUSHIONING 443 .2 Ao = 7=- VA^ + B' sin (coi/,„ + r? - coi/^) coo V 2g/i 1 . / , , . -i4\ (3.7, v-2 Sin I Wo /m — coo /a — Sin -y J , coo\ \ rfo/ 3) ts < t < 2t, where OJo Wo \/2g/; ' . _ /^Y \wo ^0/ \"i/ Wo Wo — 7^=^ -5 = 3 — ^„ 2-S/4/1 rf — COS I — sin - ) V2gh ^ _ /cOoV L "0 y ti^ Vc^O ^O/ J T? = tan - and tm is the root of .3 w Wo \/2g^ 7^^. VA^ + ^2 cos (w]/,„ + rj - wi/,) 1 / -fe) cos I Wo tm — 0)0 is — Sin -1:)=» that yields the largest value of Ao in equation (3.7.3). Region 3 is gov- erned by X + wix = 0 (3.7.4) and the initial conditions are the terminal conditions of Region 2. By the same method as was used in Section 3.2, we find Ao = ^^¥^ \h) 4/1 - I - cos h sin- f)] , 1 L\'«^o/ V dl \wo do/ A (3.7.5) \Wo/ t> 2ts. The largest value of Ao from equations (3.7.1), (3.7.3) and (3.7.5) is plotted against wi/wo in Fig. 3.7.2 for several values of d^/do . 444 BELL SYSTEM TECHNICAL JOURNAL Notice that the amplification factor is Aq rather than Am. That is, the reference acceleration is Go rather than Gm . This is necessary because Gm is infinite in the present instance. Hence Fig. 3.7.2 cannot be com- pared directly with Figs. 3.2.2 and 3.6.2. However, it is interesting to observe that, for on/wt < 0.5, (low frequency elements) abrupt bottoming has no harmful effect. For high-frequency elements, the severity of bottom- ing is very great even when very nearly all of the required space (do) is available. For example, if 90% of the required space is available (ds/do = 0.9) and the frequency of the element is ten times the package frequency. 30 20 ^-« ^ ^^^^ 10 =i--H - Ho=-5 ^^ -'^•"'^ ^^ ' _^^^ — - ^ ^ ■—-'■'''^ "^b _ 4 3 2 X ^^- — do-" y /^ / ^ --^^ ^^ 10 / "^ __— ao--\ 1 1 0123456789 10 (Jo Fig. 3.7.2 — Amplification factors for abrupt bottoming. See equations (3.7.1), (3.7.3) and (3.7.5). the severity of the shock is almost ten times as great as it would be if the additional 10% of space were available. 3.8 General Influence of Shape of Acceleration-Time Curve ON Amplification Factor When amplification factor curves are not available for a special shape of acceleration-time curve, an approximate value oi Am may be obtained by interpolation between or extrapolation from the curves of the preceding sections. The shape of the acceleration-time curve and its duration (72) or frequency (0)2) should be found, first, by the methods described in Part II. The shape found should then be compared with the standard shapes shown in Part II, for which amplification factors are given in Part HI. The amplification factor found in this way will generally be within 25% DYNAMICS OF PACKAGE CUSHIONING 445 of the true value because amplification curves for pulse accelerations do not diflfer greatly even for very different acceleration-time curves as long as the Fig. 3.8.1 — Dependence of amplification factor on shape of symmetrical acceleration pulse. 2. 1.8t 1.6 Fig. 3.8.2 — Effect of asymmetry of an acceleration pulse on amplification factor. amplitudes and frequencies are adjusted to the same scales. This is illus- trated in Fig. 3.8.1 where the amplification factor curves are drawn for square wave, half-sine wave, triangular and cubic pulses. 446 BELL SYSTEM TECHNICAL JOURNAL Amplification factors for small values of 001/002 may be calculated very accurately if it is observed that the initial slope of the amplification factor curve for a pulse acceleration is proportional to the area under the accelera- tion-time curve. For example, noting that the initial slope of the amplifica- tion factor curve for the half-sine wave pulse is 2, we assign the value 2 to the area under the half -sine wave. On the same scale, the area under a square wave pulse is tt and under a triangular pulse is 7r/2. Accordingly, the initial slopes of the amplification factor curves for the latter two pulses are ir and 7r/2 respectively. As an additional aid in finding amplification factors for unusual cases. Fig. 3.8.2 is given to show the effect of asymmetry of an acceleration pulse. The pulse is triangular in shape but the time (t/>) taken to reach the peak value of acceleration may have any value from zero to the total duration (T2) of the pulse. PART IV DISTRIBUTED MASS AND ELASTICITY 4.1 Introduction It is important to be aware of the conditions under which the assumption of lumped parameters is permissible. In Parts I and II the cushioning medium was assumed to be massless, so that wave propagation (or surges) through it was ignored. Such surges will contribute to the acceleration imposed on the packaged article and we should be able to predict both the magnitudes and frequencies of the additional disturbances. If this is done, the information in Part III may be used to obtain at least a rough estimate of the resulting effects. In Part III itself the effects of accelerations were determined by studying the response of a system having only one degree of freedom; that is, an element of the packaged article was assumed to be a single mass supported by a massless spring. Every real element, of course, has an infinite number of degrees of freedom, so that it is important to discover the contribution, of the higher modes of vibration of an element, to the overall response. Both of these problems (distributed parameters of mass and elasticity in the cushioning medium and in an element of the packaged article) are studied in this part. One example of each type is considered, and the choice of the example in each case was influenced by considerations of expediency, namely that the mathematical derivations be relatively simple and lead to solutions for which not too lengthy computations are necessary to yield results that can be applied practically. At the same time, the examples chosen are believed to give some insight into several of the physical phe- DYNAMICS OF PACKAGE CUSHIONING 447 nomena involved. The treatment is by no means complete, but a more detailed investigation is beyond the scope of this paper. 4.2 Effect of Distributed Mass and Elasticity of Cushioning ON Acceleration of Packaged Article Referring to Fig. 4.2.1, we consider the packaged article, of mass W2 , to be supported by distributed cushioning of mass mc and depth ^. The cushioning may be a pad, say of rubber, in which case / is the pad thickness, or it may be a helical metal spring, in which case f is the coil length. The package is dropped vertically from a height h and has attained a velocity v at the instant of contact (/ = 0) of the outer container and the floor. The outer container is assumed to be heavy enough so that there is no rebound. A horizontal plane in the cushioning is located by a coordinate x measured from the end of the cushioning attached to the outer container. The vertical -V i 1 1 1 1 1 ; / (cushion) •: Floor ////////////////////////// Fig. 4.2.1 — Packaged article of mass tni, supported on distributed cushioning of depth t and mass vie, depicted at the instant of first contact of the outer container {m^) and the floor. displacement of the plane x is designated by u. The undamped motion of the cushioning after contact is governed by the one-dimensional wave equation: b U d U a/2 ^ a^2' (4.2.1) in which a is the velocity of propagation of longitudinal waves in the cushion- ing. If the cushioning is continuous. 2 E a = — . (4.2.2) where E is the modulus of elasticity and p is the density of the cushioning. If the cushioning is a helical spring, 2 kf a = — , where k is the spring rate. (4.2.3) 448 BELL SYSTEM TECHNICAL JOURNAL The initial conditions of the system are M(=o = 0, (4.2.4) \fL = - '-•=> The boundary conditions are Nx=o = 0, (4.2.6) k(\ — \ = -mo —r . (4.2.7 Equation (4.2.7) expresses the requirement that the force on the upper end of the cushioning must balance the inertia force of the packaged article. For continuous cushioning kf should be replaced by EA, where A is the cross-sectional area of the cushioning. A solution of (4.2.1) satisfying conditions (4.2.4) and (4.2.6) is w = X] ^In sin — ^ sin co„/, (4.2.8) n=i a where co„ is the w"" root of a transcendental equation to be obtained from (4.2.7) and yln is a constant to be determined by (4.2.5). Substituting (4.2.8) in (4.2.7) and equating coefficients of like terms of the series, we obtain the transcendental equation ^^an'^^ = ^. (4.2.9) a a mz Substituting (4.2.8) in (4.2.5) we obtain, by the usual methods of expansion into trigonometric series, 2v An = ~ ^ T— A ■ (4.2.10) in [ — + I sm • 1 \ a a / Hence the complete solution of the problem is 2v sm sm co„ t = -E —7 r . (4.2.11) co„ I + ^ sm 1 \ a a / Our chief interest is in the acceleration of W2 . Making use of (4.2.7) and (4.2.9) we find, from (4.2.11), that this acceleration is = z'coo 2^ Bn sin co„/, (4.2.12) DYNAMICS OF PACKAGE CUSHIOXIXG 449 where o;o = — (4.2.13) and 2 t/^c/ml co^„fA B„ = ' '"•' v;-2 ^ ^^ / (4 2.14) W2 Wo a^ The acceleration of Wo is, therefore, a sum of sinusoids of frequency ojn and ampHtude vuoBn . Now, iwo is the maximum acceleration that niz would attain if the mass of the cushioning were negligible. Calling G„ the maximum acceleration in the n^^ mode and Go the maximum accelera- tion neglecting the mass of the cushioning, as in Part I, we have ^ = ^n. (4.2.15) But Bn depends only on the ratio mdmi , as may be seen from equations (4.2.9) and (4.2.14). Similarly the ratio of the frequency (w„) of any mode to the frequency (ojo) with massless cushioning depends only on mdmi , as may be seen from equations (4.2.3), (4.2.9) and (4.2.13). Hence, both the amplitude and frequency ratios for the acceleration in any mode depend only on the ratio of the mass of the cushioning to the mass of the packaged article. The ratios Gn/Go andco„/coo are plotted against mdmi in Figs. 4.2.2 and 4.2.3 for the first five modes. It may he seen from these figures that the accelerations in the higher modes can he very important. For example, if the cushioning weighs half as much as the packaged article the maximum acceleration in the second mode is about 40% of the acceleration in the first mode and the latter is about the same as found by the elementary method of Part I. This could have a disastrous effect on an element of the packaged article if the latter had a fundamental frequency near that of the second mode of the cushioning, the latter being found, from Fig. 4.2.3, to be about five times the fundamental frequency of the package. It must be remembered that damping has been neglected in the above investigation and damping in the cushioning will serve to mitigate the se- verity of the higher mode accelerations to a great extent. However, the danger is always present at the start of a design and the possibilities of un- favorable combinations should be studied in every case. 450 BELL SYSTEM TECHNICAL JOURNAL Go 1.0 0.9 0.8 0.7 0.6 ' 0.5 0.4 0.3 0.2 0.1 p •S: :: :::::: ^wttltmtttlttTT ' ' i ::. ::B5i S iBlfci 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Fig. 4.2.2 — Influence of ratio of mass of cushioning (m^) to mass of packaged article (mh) on acceleration ratio. The numerator of the acceleration ratio is the maximum acceleration (G„) in the n*'' mode of vibration transmitted through the cushioning. The denominator of the acceleration ratio is the maximum acceleration (Go = y/lhki/rmg) that the mass m^ would experience if the mass of the cushioning were negligible. See equations (4.2.15), (4.2.14), (4.2.9). ^ 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Fig. 4.2.3 — Influence of ratio of mass of cushioning {vie) to mass of packaged article on frequency ratio. The numerator of the frequency ratio is the frequency (w„) of the n**" mode of vibration transmitted through the cushioning. The denominator of the frequency ratio is the frequency (co„ = Vki/m^ of vibration of the mass rm neglecting the effect of the mass of the cushioning. See equations (4.2.9), (4.2.13), and (4.2.3). DYNAMICS OF PACKAGE CUSHIONING 451 4.3 Effect of Distributed Mass and Elasticity, of an Element OF THE Packaged Article, on the Amplification Factor for A Half-Sine -Wave Pulse Acceleration In this section we shall determine the contribution of the higher modes of vibration of a structural element to its total response to a half-sine-wave pulse acceleration. For the shape of the element, we choose a prismatic bar because this leads to the simplest mathematical formulation of the problem and such a bar is also a common structural element. Other con- siderations influence the choice of direction of acceleration with respect to the axis of the bar. The transverse direction (cantilever) is the most practical from a physical standpoint, but, for purposes of comparison with the one-degree-of-freedom system, the parallel (axial) direction of accelera- tion is the more logical. Both problems lead to solutions in the form of infinite series, but, in the latter case, the expression for the strain at a fixed -V / Fig. 4.3.1 — The system studied in Section 4.3 depicted at the instant of contact with the floor. end can be summed in terms of elementary functions without difficulty. Since it is necessary to determine maximum values of strain over a wide range of frequency ratios for the plotting of an amplification factor curve, an enormous reduction in the time required for accurate computations is obtained by choosing the axial case. Furthermore, the axial case appears to contain the essential features which might result in differences between the response of a one-degree-of-freedom system and a continuous one. The complete system to be studied is illustrated in Fig. 4.3.1. To the mass W2 , supported on massless cushioning of constant spring rate ^2 , is attached one end of an elastic prismatic bar, of length /, cross sectional area A, modulus of elasticity E, and density p, with its axis oriented verti- cally. The system is dropped from a height h so that its velocity is v at the instant of contact of the cushioning with the floor. The mass of the bar is supposed to be small in comparison with nh and perfect rebound is assumed, so that the motion of mo during contact is a half-sine wave of frequency 452 BELL SYSTEM TECHNICAL JOURNAL C02 = (4.3.1) The maximum acceleration of m^ is thus i'co2 . If this magnitude of ac- celeration were reached very slowly, so as not to excite transient longitudinal waves in the bar, the maximum force between the bar and m^ would be the product of the acceleration and the mass of the bar: F = voi.pAf. (4.3.2) Hence the strain at the end of the bar attached to ^2 would be VOi-2 pC fo E (4.3.3) Our problem is to find the ratio of the maximum transient strain to eo Id 1.6 1.4 1.2 / / / ^ ^ 1.0 0.8 0.6 0.4 0.2 / / / / / 0 123456789 ^1 = TTa Fig. 4.3.2 — Amplification factors for an element of the packaged article having dis- tributed mass and elasticity. The package has linear undamped cushioning and perfect rebound. See equations (4.3.15), and (4.3.14). Let u be the displacement of a transverse plane section of the bar distant x from the end attached to mo . Then, the equation of motion of the bar is d''u d'u (4.3.4) where a is the velocity of propagation of longitudinal waves in the bar: (4.3.5) E P Taking the instant of first contact of the cushioning with the floor to be / = 0, we know, from Part II, that the system will leave the floor when / = DYNAMICS OF PACKAGE CUSHIONING 453 ir/o}2 • We shall therefore treat separately, as in Part III, the motion during contact ^ z=: = T 0 < / < — 0)2 and after rebound / > -■. During the first interval, the initial and boundary conditions are [hUo = 0, (4.3.6) [a. = -V, (4.3.7) u]i^o = — sin Wo/, (4.3.8) C02 [-1 = 0. (4.3.9) The first and second conditions state that, at the instant of contact, all points in the bar are moving with the approach velocity v, without relative displacement. The third condition prescribes the half -sine wave motion of the end of the bar that is attached to W2 • The fourth condition states that the strain at the free end of the bar is always zero. By the usual methods, a solution of (4.3.4) satisfying conditions (4.3.6) to (4.3.9) is found to be V COS —{I - x) sm W2 / „ . 00 sm -— sm — y- « = -, + ^ Z. ~P7 v^ n (4.3.10) co2t- TT-a n=i,3,5-- 9 ( uray The displacement is seen to be a forced vibration at the frequency (002) of the applied acceleration, on which are superposed the free vibrations of the bar given by the series expression. The frequency of the fundamental mode of vibration of the bar isxa/2f and the frequencies of the higher modes are the odd integral multiples of the fundamental. 454 BELL SYSTEM TECHNICAL JOURNAL The strain at the attached end of the bar is = [-1 f . nirat v] Wit . , 4 -^-^ ^^ \ ,. ^ .^s / _ _ IT mra \ \ 0)2 ' 2co2^ / It may be verified that the sum of the series in (4.3.11) is given by tiTat A " ^^^ 2f at 52 —5 = tan — sin C02/ + cos u^t — 1, (4.3.12) It should be observed that the summation is vaUd only in the interval 0 < / < 21/ a. However, the series is periodic with half period 21/ a and includes only the odd terms, so that the function repeats itself with reversed sign after each interval 21/ a. Hence the summation, valid for all t, can be written rnrat 4 ^ '^^17 TT n=l,3,B- m) - ■] ("1) tan — sm C02 U - — ) + cos C02 U — — j- 1 t -"—]- \\ (4.3.13) , 2mt ^ ^ ^ 2{m + \)l k = m when < t < a a w = 0, 1, 2, 3, • • • . We may, therefore, rewrite (4.3.11) in the form ea (Jilt — = —tan — sm aj2^ + ( — 1 ) tan — sm «2 1 / — — 1 a L ^ \ CL / ('-t)-0 + cosco2(/ - — - 1 (4.3.14) DYNAMICS OF PACKAGE CUSHIONING 455 k = m when — < / < -^ ■ — ~ a a w = 0, 1 , 2, 3 • • • . The expression (4.3.14) is simple enough so that the maximum value (e^,) of the strain at the attached end can be obtained without difficulty for any ratio of the fundamental frequency (wi = ira/lf) of the bar to the frequency (coa) of the disturbing acceleration. The amplilication factor ■Am — (4.3. I5) vuzp^ iraj2 V may then be calculated. The results of these calculations are plotted in Fig. 4.3.2. The important feature of this curve is that the amplification factor is everywhere less than the corresponding amplification factor for the one-degree-of -freedom system (Fig. 3.2.2, /3i = 0). Hence the assumption of lumped parameters is on the side of safety as regards amplification factor. It is interesting to observe that the curve of A^ vs. coi/w2 , for this case, is a straight line between ui/002 = 0 and a;i/co2 = 1. This arises from the fact that, for wi/w2 < 1, equation (4.3.14) reduces to f= cos .,*-!, (^51). (4.3.16) Hence, when the duration of shock is less than the half period of the funda- mental mode of vibration, the maximum value of strain occurs at the end of impact and is equal to twice the ratio of the approach velocity to the velocity of wave propagation in the bar. The whole solution of the problem is not yet completed; for, although it is fairly evident from the fact that there is at least one maximum in the inter- val 0 < / < ir/co-i for all values of coi/co2 , it must be verified that the maxi- mum strain (and, therefore, the amphfication factor) is never greater after / = ir/a)2 than before. Defining a new time coordinate t' = t - -, (4.3.17) aj2 we have, for the initial and boundary conditions of equation (4.3.4) for t ^ 7r/w2 , 456 BELL SYSTEM TECMNtCAL JOURNAL nair- nirx rdui Idti'- TT'O » sin _ , ^ za)2t' sm 2^ (4.3.18) 2^ n=l,3,5-' 0 n C02 / /, nair nirx V COS — (^ — a C02/ cos a x) . 00 cos sin TT „=i,3,5 ■■• r/niraV "iWj) ~ M (4.3.19) [l(]i=0 = i*^', (4.3.20) du _ }. (4.3.21) The first and second conditions state that the displacement and velocity of every point in the bar must be the same at the beginning of the second inter- val as at the end of the first interval; the expressions in (4.3.18) and (4.3.19) are obtained from (4.3.10). The third condition prescribes the constant velocity of departure from the floor of the mass ntn and, therefore, of the end of the bar attached to it. The last condition states, again, that the strain at the free end of the bar is zero. It may be verified that a solution of (4.3.4) satisfying conditions (4.3.18) to (4.3.21) is ^ mrx /nirat' \ vt' -\- 2^ Cn sin — - sin I ^t" + 7„ 1 {t' ^ 0, / ^ 7r/w2), (4.3.22) where ^vt sin - ~ Cn sin 7„ = j-/^ ^v2 =1 (4.3.23) (nair\ ■K an mra DYNAMICS OF PACKAGE CUSHIONING 457 Hence, the strain at the attached end of the bar is nirat El 7ra n=l,3,5- 2€ mra 2 4v ■aV 1 sin ira ,1=1,3,5 ■ nirat' (4.3.25) (7nra \^ 2^0 The two series may be summed, as before, with the result --= (-1) V + (-1) COof . tan — sm coo a 2H a tan '^^^ sin coo I / tan — sm W2\t — ) La \ a / + cos 2k( a - 1 cosc^oU'- — )- 1 (4.3 .26) t > tt/co: t' = t- 7r/a-, ^ = w when 2wf 2(w + 1)^ < f < , a a k' = ni' when 2m'( ^ ^, ^ 2{m' + \)t a a w = 0, 1, 2, 3 • • ni' = 0, 1, 2, 3 • • • Once more, the expression for the strain at the attached end of the bar is in a form suitable for rapid calculation and it can be shown the e in equation (4.3.26) for / ^ 7r/aj2 is never greater than the e in equation (4.3.14) for 0 ^ / ^ tt/coo for the samecoi/co2 . Hence, Fig. 4.3.2 and the conclusions follow- ing equations (4.3.15) and (4.3.16) need not be modified. Notations A Cross sectional area of a bar element of the packaged article. Also, a constant of integration. Ao A^, Amplification factor when the reference acceleration is Go. Ratio of maximum dynamic response to the response to a slowly applied ac- celeration of magnitude Gog. Amplification factor when the reference acceleration is Gm. Ratio of maximum dynamic response to the response to a slowly applied accelera- tion of magnitude Gmg- In Section 1.15, the sum of all the trapezoidal areas from .V2 = 0 to Xi = (xijn. M.SO, in Section 4.2, the coefficient of the nth term of a series. The area of a trapezoid with altitude ^{x-ijn and sides P„_i and P„. xa/l in the tension spring package. Also, in Part IV, the velocity of propagation of longitudinal waves. 458 BELL SYSTEM TECHNICAL JOURNAL B A parameter of cushioning with cubic elasticity defined in equation (1.5.3). Also, a constant of integration. Bn Coefficient in the w*'' term of a series. 6 /// in the tension spring package. C A constant of integration. Cn Coefficient of the n^^ term of a series. c A constant defined in equation (1.7.11) C\ Damping coefficient of an element of a packaged article. C2 Damping coefficient of linear cushioning. cn The elliptic cosine function. do Hj^Dothetical displacement that would result if initial spring rate wer^ maintained. dh Maximum possible displacement of packaged article in cushioning with tangent elasticity. dm Maximum displacement of packaged article. d'm Value of dm when ^o = ^o- dt Displacement of bi-linear cushioning at which the spring rate changes from ^0 to ^6. E Modulus of elasticity. e In the tension spring package the stretch of a spring when the displace- ment is dm- exp ( ) e ' \ where e is the Naperian base 2.718- • • F In section 2.7, a frictional force. Fm In the tension spring package, the maximum force on a spring. / In the tension spring package, the difference between / and the distance between hooks of an unstretched spring. /i Frequency of vibration of an element of the packaged article. fz Frequency of vibration of the packaged article on its cushioning. Go Hypothetical maximum acceleration (in number of times g) that would result if initial spring rate were maintained. Ge AmGm OX AaGo, l.c. the slowly applied acceleration (in number of times g) that will produce the same maximum response as a transient acceleration of maximum value Gm or Go. Gf Maximum acceleration (in number of times g) in cushioning with fric- tion and spring rate kp. Gm Absolute value of maximum acceleration of packaged article in units of "number of times gravitational acceleration." Gm' Value of Gm when ^o = K. DYNAMICS OF PACKAGE CUSHIONING 459 Gn In section 1.15, the maximum acceleration (in number of times g) ex- perienced by the suspended mass when dropped from a height hn- In Part IV, the maximum acceleration (in number of times g) of the n**" mode of vibration. Gr Maximum acceleration (in number of times g) after rebound. Gt Safe value of Ge. g Gravitational acceleration. h Height of drop. hn In Section 1.15, the height of fall that will cause the cushioning to dis- place an amount (3:2) n. K In the tension spring package, the initial spring rate of the suspension. In Section 2.8, the complete elliptic integral of the first kind. Ki, K2, A'3 The initial spring rates in the three mutually perpendicular directions normal to the faces of the package frame. k In the tension spring package, the spring rate of a spring. In Section 2.8, the modulus of an elliptic integral. k, k' In Section 4.3, 0, 1, 2, 3, • • • . ^0 Initial spring rate of non-linear cushioning. k'a Optimum value of initial spring rate ka. ki Spring rate of lumped elasticity of element of packaged article. jfej Spring rate of linear cushioning. kb Spring rate of bilinear cushioning after bottoming. kp Spring rate defined in equation (2.7.7). L Constant defined in equation (1.8.2). / In the tension spring, the projection of h on a horizontal plane. In Section 4.2, length of cushioning. In Section 4.3, length of element of packaged article. li In the tension spring package, the distance between the two support points of a spring when the suspended article is in the equihbrium position. M Constant defined in equation (1.8.4), equal to Gm/Go. m Reduced mass defined in equation (2.4.5). m, m' In Section 4.3, 0, 1, 2, 3, • • • . m\ Lumped mass of element of packaged article. mi Lumped mass of packaged article. m-i Lumped mass of outer container. vfii Mass of cushioning. N M^. 460 BELL SYSTEM TECHNICAL JOURNAL n 0,1,2,3, ■■■ . P Force transmitted through cushioning. Po Asymptotic value of force transmissible through cushioning with hyper - bolic tangent elasticity. Pm Maximum force exerted on packaged article by cushioning. Pn In Section 1.15, the load that produces displacement {x2)n- R Force between package and floor. r Coefficient of cubic term in load-displacement function for cushioning with cubic elasticity. s, t, u The direction cosines of the acceleration direction with respect to the normals to the faces of the package frame. sn The elliptic sine function. 7*2 The period of vibration of the packaged article on its cushioning. t Time coordinate. /' t to Time of first contact of package with floor. tm Time at which maximum displacement or acceleration occurs. tr Time at which package leaves floor on rebound. t. Time at which the displacement reaches ds- u Displacement in x direction. V Approach velocity. Wz Weight of packaged article. Wz Weight of outer container. X Xi — X2; relative displacement of lui with respect to m-i. X Xi — X2. X Xi — X2- x' Relative displacement of m\ v.ith respect to mi at time t' . xo In the tension spring package, the perpendicular distance from an inner spring support point to the nearest plane, perpendicular to the displace- ment direction and containing four outer spring support points. Xi Displacement of nii. xi Velocity of nii. Xi Acceleration oinii. Xi Displacement of m2- X2 Velocity of m2- DYNAMICS OF PACKAGE CUSHIONING 461 :V2 Acceleration of wo- •Tmax Maximum value of x. ix2)n In Section 1.15, the displacement associated with the n^^' point. Xst The value x would have if the acceleration reached its maximum value in a very long time. (A.T2)n In Section 1.15, equals (x2)„ — (.T2)„_i. y X2-Xi. z xn/l (tension spring package). «) y> ^i s") '7 Phase angles. /3i Fraction of critical damping of an element of the packaged article. /32 Fraction of critical damping of package cushioning. 7„ Phase angle of n"> term of series (equation (4.3.22)). e Strain at attached end of element under transient conditions. eo Strain at attached end of element under non-transient conditions. em Maximum strain at attached end of element under transient conditions. 6 Angle between the displacement direction and the acceleration di- rection. IT 3.14159---. p Density (mass per unit of volume) TO Pulse duration of a half-sine-wave acceleration. T2 Pulse duration associated with non-linear cushioning. tb Duration of bottoming of cushioning with bi-linear elasticity. Tp Time required to reach peak vs,lue of a triangular acceleration pulse. w Radian frequency defined in equation (2.4.6). wi Radian frequency' of vibration of an element of the packaged article. wi' Radian frequency of vibration of damped element of packaged article. 0)2 Radian frequency of vibration of the packaged article on its cushioning. W2' Radian frequency of vibration of the packaged article on damped cushioning. ub A frecjuency defmed in equation (2.10.10). Wc A frequency defined in equation (2.8.8). w„ Radian frequency of nth mode. Abstracts of Technical Articles by Bell System Authors Dimensional Stability of Plastics} Robert Burns. Because of inherent insulating properties, rigid plastics play an important part in the design and manufacture of precision electrical apparatus. Almost invariably, practical design considerations require that the plastics have reasonable structural possibilities since it is rarely practicable to disassociate completely electrical and structural functions. This paper discusses one of the important factors in the successful use of plastics in precision devices, namely, dimensional stability. Since plastics are organic compounds, one must be prepared to accept a degree of insta- bility not usually encountered in metals. The measurement of this property is therefore of prime importance to the user of plastics since the data provide a basis for design adjustment which frequently is the difference between failure and success. The various t^^pes of dimensional change are reviewed. Data illustrating the separate effects of humidity, drying, and cycling procedures are sub- mitted. The influence of fabricating processes such as compression or injection molding, and sheeting, is included. Some Numerical Methods for Locating Roots of Polynomials.^ Thornton C. Fry. It is the purpose of this paper to discuss the location of the roots of polynomials of high degree, with particular reference to the case of com- plex roots. This is a problem with which the Laboratories has been much concerned in recent years because of the fact that the problem arises rather frequently in the design of electrical networks. Attention is not given to strictly theoretical methods, such as the exact solution by elliptic or auto- morphic functions: nor to the development of roots in series or in continued fractions, though such methods exist and one at least — development of the coefl&cients of a quadratic factor — is of great value in improving the accuracy of roots once they are known with reasonable approximation. Instead, the paper deals with just two categories of solutions: one, the solution of the equations by a succession of rational operations, having for their purpose the dispersion of the roots; the other, a method depending on Cauchy's theorem regarding the number of roots within a closed contour. Thermistor Technics.^ J. C. Johnson. This paper is confined to a study of how the three basic types of thermistors, namely, externally-heated or ambient temperature type, the directly-heated type, and also the indirectly- ^A.S.T.M. Bulletin, May 1945. ^Quarterly Applied Matltematics, July 1945. ' Electronic Industries, August 1945. 462 ABSTILiCTS OF TECHNICAL ARTICLES 463 heated type, are used in simple feedback amplifiers as regulation and control devices to effect the economies inherent in an entirely electrical system by eliminating such mechanical devices as motor-driven condensers, sliding contacts and rotary switches. Dynamic Measurejnents on Electromagnetic Devices.* E. L. Norton. A method is presented by which measurements of flux may be made at any desired time during the operate cycle of an electromagnet. Apparatus is described which operates the magnet cyclically at an accurately held rate, and provides a means for measuring flux either by the use of a search coil or by the operating winding of the magnet itself. When using a search coil, it is connected to a direct-current milliammeter at the time in the cycle at which the value of the flux is desired and disconnected at the end of the cycle or just before the magnet is energized for the next pulse. If proper precautions are taken, the steady reading of the instrument is an accurate measure of the difference in the flux in the coil between the time it is con- nected to the meter and the time it is removed, or, since the latter is zero except for residual flux, the reading is a direct measure of flux. The same apparatus may be used for the measurement of instantaneous current by the addition of an air core mutual inductance, and its use is extended to the measurement of armature position and velocity by the addition of a photoelectric cell and the proper amplifiers. A form of vacuum tube filter is described which effectively filters the pulses from the indicating instrument without affecting the accuracy of the measurements. Coaxial Cables and Television Transmission.^ Harold S. Osborne. Communication techniques and facilities useful to the entertainment industry have evolved naturally from the Telephone Companies' main objective — the transmission of speech. The development of carrier sys- tems for long-distance transmission and technical features involved in the latest carrier medium — the coaxial cable — are reviewed. The television transmission capabilities of this medium, both now and what may be expected shortly after the war, are mentioned. The extensive system of such cables planned for the next five years, supplemented by radio relay systems to the extent that these prove themselves as a part of a communica- tions network, will provide an excellent beginning for a nation-wide tele- vision transmission network. Planned primarily to meet telephone requirements, this network of cables will be suitable to meet the transmission needs of the television industry. The Performance and Measurement of Mi.xers in Terms of Linear-Xetivork Theory.^ L. C. Peterson and F. B. Llewellyn. This paper discusses ^ Elec. Engg., Transactions Section, April 1945. 'Jour. S.M.P.E., June 1945. « Proc. I.R.E., July 1945. 464 BELL SYSTEM TECHNICAL JOURNAL the properties of mixers in terms of linear-network theory. In Part I the network equations are derived from the fundamental properties of nonUnear resistive elements. Part II contains a resume of the appropriate formulas of linear-network theory. In Part III the network theory is applied, first to the case of simple nonlinear resistances, and next to the more general case where the nonlinear resistance is embedded in a network of parasitic resistive and reactive passive-impedance elements. In Part IV application of the previous results is made to the measurement of performance properties. The "impedance" and the "incremental" methods of measuring loss are contrasted, and it is shown that the actual loss is given by the incremental method when certain special precautions are taken, while the impedance method is in itself incomplete. A Figure of Merit for Electron-Concentrating Systems.'' J. R. Pierce. Electron-concentrating systems are subject to certain limitations because of the thermal velocities of electrons leaving the cathode. A figure of merit is proposed for measuring the goodness of a device in this respect. This figure of merit is the ratio of the area of the aperture which, in an ideal system with the same important parameters as the actual system, would pass a given fraction of the cathode current to the area of the aperture which in the actual system does pass this fraction of the cathode current. Ex- pressions are given for evaluating this figure of merit. A 60-KilowaU High-Frequency Transoceanic-Radiotelephone Amplifier.^ C. F. P. Rose. Here is described a high-frequency radio amplifier recently developed for the transoceanic-telephone facilities of the Bell System at Lawrenceville, New Jersey. In general, the ampliiier is capable of delivering 60 kilowatts of peak envelope power when excited from a 2-kilowatt radio- frequency source. It is designed to operate as a "class B" amplifier for transmitting either single-channel double-sideband or twin-channel single- sideband types of transmission. Features are described which permit rapid frequency-changing technique from any preassigned frequency to another lying anywhere within the spectrum of 4.5 to 22 megacycles. Some Notes on the Design of Electron Guns.^ A. L. Samuel. A method is outlined for the design of electron guns based on the simple theory first published by J. R. Pierce. This method assumes that the electrons are moving in a beam according to a known solution of the space-charge equa- tion, and requires that electrodes exterior to the region of space charge be shaped so as to match the boundary conditions at the edge of the beam. An electrolytic tank method is used to obtain solutions for cases which are not amenable to direct calculation. Attention is given to some of the ^ Proc. LR.E., July 1945. 8 Proc. LR.E., October 1945. » Proc. LR.E., April 1945. ABSTRACTS OF TECH MCA L ARTICLES 465 complications ignored by the simple theory and to some of the practical difficulties which are encountered in constructing guns according to these principles. An experimental check on the theory is described, together with some information as to the actual current distribution in abeam produced by a gun based on this design procedure. Microwave Radiation from the Stm}° G. C. Southworth. During the summer months of 1942 and 1943, a small but measurable amount of micro- wave radiation was observed coming from the sun. This appeared as ran- dom noise in the outputs of sensitive receivers designed to work at wave- lengths between one and ten centimeters. Over a considerable portion of the range, the energy was of the same order of magnitude as that predicted by black-body radiation theory. Attempts were made to determine the effect of the earth's atmosphere on this radiation. Measurements made near sunrise or sunset, when the path through the earth's atmosphere was relatively long, differed only slightly from those made at noon. This suggested that any absorption that may have been present was small. In this connection it is of interest that small temperature differences could be noted between points below the horizon and the sky immediately above. This also suggested that the earth's atmosphere was relatively transparent. In another kind of measurement the parabolic receiver was centered on the sun and its output was observed as the sun's disc moved out of the aper- ture of the receiver. The directional pattern so obtained indicated that at the shorter wave-lengths the sun's apparent diameter was considerably larger than that measured by ordinary optical means. This suggested that there may have been some refraction or perhaps scattering by the earth's atmosphere. Resistive Attenuators, Pads and Netivorks — An Analysis of their Applica- tions in Mixer and Fader Systems {Part Eight of a Series)}^ Paul B. Wright. In last month's discussion, the series-connected fader and the parallel-connected fader systems were considered, together with an analysis of their performance expressed both algebraically and in terms of the hyperbolic functions of a real variable. In this instalhnent, the series- parallel-connected fader system discussion is continued and equations describing the complete behavior of this type network system are developed. This is followed by further analytical work dealing with the parallel-series- connected fader and mLxer system and several lesser known systems which are quite useful to use. These are the midliple bridge and the lattice network systems which may be utilized to advantage for some applications. All of ^^ Jour. Franklin Institute, April 1945. 1' Communications, September 1945. {Preceding parts of this Scries appeared in earlier issues of Communications.) 466 BELL SYSTEM TECHNICAL JOURNAL the equations which are derived are shown in the algebraical, hyperbolical and symbolical forms. The key chart which was presented earlier in this series may be used to great advantage when checking the definitions of the symbols used which are not specifically defined in the text. This procedure also may be directly applied to the hyperbolic equations shown. It is of course necessary to take into account that, in general, subscripts are used in most of the equations in the text while the key chart does not have any subscripts. This does not, however, alter the fundamental forms nor their definitions in terms of the propagation function, theta. To avoid the neces- sity for extensive interpolation of the hyperbolic function tables to find the correct numerical values for the various functions used throughout the text, a series of tables providing all of the functions required is presented. Contributors to this Issue Raymond D. Mindlin, B.A., Columbia University, 1928; B.S., 1931; C.E., 1932; Ph.D., 1936. Assistant 1932-38, Instructor 1938-40, Assistant Professor 1940-45, Associate Professor 1945-, Department of Civil Engineer- ing, Columbia University. Consultant, Section T, National Defense Research Committee (later Office of Scientific Research and Development), 1940-45, on the development of the rugged radio proximity fuze and on mathematical problems in fire control. Consultant, Bell Telephone Labora- tories, 1943-44, Member of the Technical Staff 1944-45, Consultant 1945-. Dr. Mindlin has been concerned with the fields of mathematical and experimental mechanics. J. R. Pierce, B.S. in Electrical Engineering, California Institute of Technology, 1933; Ph.D., 1936. Bell Telephone Laboratories, 1936-, Engaged in study of vacuum tubes. A. L. Samuel, A.B., College of Emporia (Kansas), 1923; S.B. and S.M. in Electrical Engineering, Massachusetts Institute of Technology, 1926. Additional graduate work at M.I.T. and at Columbia University. Instruc- tor in Electrical Engineering, M.I.T. , 1926-28. Mr. Samuel joined the Technical Staff of the Bell Telephone Laboratories in 1928, where he has been engaged in electronic research and development. Since 1931, his principal interest has been in the development of vacuum tubes for use at ultra-high frequencies. 467 ¥