THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR, MICHIGAN THEORETICAL POWER OUTPUT AND BANDWIDTH OF TRAVELING-WAVE AMPLIFIERS TECHNICAL REPORT NO. 32 Electron Physics Laboratory Department of Electrical Engineering By Harold Sobol Joseph E. Rowe Project 2750 CONTRACT NO. AF30(602)-1845 DEPARTMENT OF THE AIR FORCE PROJECT NO. 4506, TASK NO. 45152 PLACED BY: THE ROME AIR DEVELOPMENT CENTER GRIFFISS AIR FORCE BASE, NEW YORK August, 1959

ABSTRACT Expressions are developed to calculate the theoretical power output of traveling-wave amplifiers using any type of r-f structure. Calculations are made for helix-type tubes and it is shown how to calculate the power output of tubes using more dispersive structures in terms of calculations made for helix tubes. The principle factors accounting for higher power output of dispersive structures are presented and discussed. The gain and bandwidth of forward-wave helix amplifiers are derived from the smallsignal theory as functions of frequency and it is shown that the gain in db times the frequency bandwidth is a constant as a function of helix length for high y a' and the gain times the bandwidth squared is a constant for low y7a'. -iii

TABLE OF CONTENTS Page ABSTRACT iii LIST OF ILLUSTRATIONS v INTRODUCTION 1 DERIVATION OF POWER OUTPUT RELATIONS 2 THEORETICAL POWER OUTPUT OF MODIFIED HELIX AND PERIODICALLY LOADED WAVEGUIDE CIRCUITS 9 INTERACTION BANDWIDTH 12 General 12 Gain Equations for the Sheath-Helix Tube 13 Gain Factors 18 Gain as a Function of Frequency 20 Interaction Bandwidth 30 Gain-Bandwidth Product 33 CONCLUSIONS 55 ACKNOWLEDGMENTS 55 APPENDIX A 56 APPENDIX B 47 LIST OF REFERENCES 50 -iv

LIST OF ILLUSTRATIONS Figure Page 1 Saturation Efficiency vs. Gain Parameter. (B = 1.0, d = 0) 5 2 Theoretical Power Output for a Helix-Type Traveling-Wave Amplifier. (C = 0.05, a'/b' = 1.4, DLF = 80%, d = 0) 6 3 Theoretical Power Output for a Helix-Type Traveling-Wave Amplifier. (C = 0.10, a'/b' = 1.4, DLF = 80%o, d = 0) 7 4 Gain Factor, F1/co vs. Normalized Frequency. (b'/a' = 0.7) 19 5 Gain Factor, F2/co vs. Normalized Frequency. (C = 0.01, QC = 0.125, d = 0.125, b'/a' = 0.7) 21 6 Gain Factor, F2/%o vs. Normalized Frequency. (C = 0.1, QC = 0.125, d = 0.125, b'/a' = 0.7) 22 7 Gain Factor, F3/CDo vs. Normalized Frequency. (C = 0.01, QC = 0.125, b'/a' = 0.7. Independent of d) 23 8 Gain Factor, F3/wo vs. Normalized Frequency. (C = 0.1, QC = 0.125, b'/a' = 0.7. Independent of d) 24 9 Gain Factor, F4/ao vs. Normalized Frequency. (C = 0.01, QC = 0.125, b'/a' = 0.7. Independent of d) 25 10 Gain Factor, F4/Co vs. Normalized Frequency. (C = 0.1, QC = 0.125, bt/a' = 0.7. Independent of d) 26 11 Normalized Gain vs. Normalized Frequency. (C = 0.01, QC = 0.125, d = 0.125, b'/a' = 0.7) 28 12 Normalized Gain vs. Normalized Frequency. (C = 0.1, QC = 0.125, d = 0.125, b'/a' = 0.7) 29 13 Percentage Bandwidth vs. yoa'. (C = 0.01, b'/a = 0.7) G + IA = 40 db at | = 1. 31 14 Percentage Bandwidth vs. o7a'. (C = 0.1, b'/a' = 0.7) G + A | = 40 db at 5 = 1. 32 A.1 Theoretical Power Output for a Helix-Type Traveling-Wave Amplifier. (C = 0.05, a'/b' = 1.4, DLF = 80%, d = 0) 37 A.2 Theoretical Power Output for a Helix-Type Traveling-Wave Amplifier. (C = 0.05, a'/b' = 1.6, DLF = 80%o, d = 0) 38 A.3 Theoretical Power Output for a Helix-Type Traveling-Wave Amplifier. (C = 0.05, a'/b' = 1.4, DLF = 100%>, d = 0) 39 yv

THEORETICAL POWER OUTPUT AND BANDWIDTH OF TRAVELING-WAVE AMPLIFIERS Introduction In a previous paper' the authors have developed a general design procedure for high-efficiency traveling-wave amplifiers. The design equations and procedure were developed primarily for helix-type devices, but correction factors were also derived which permitted the design of tubes with other r-f structures. The procedure was to select one parameter such as electron stream perveance and then determine the other parameters consistent with highest possible saturation efficiency. The consequences of design compromises are readily apparent. The general curves and theoretical relations developed in that paper are used here to draw other composite curves which summarize the information and afford an opportunity to calculate the theoretical power output achievable from a traveling-wave amplifier as a function of any parameter. The power output calculations are predicted for any type of r-f propagating structure by relating its characteristics to a tube with a sheath-helix structure. They do not, however, account for thermal limitations of the r-f structure. Equations are derived from which the power output can be determined using a theoretical efficiency figure. The results indicate the principal dependence of efficiency on operating parameters and show how efficiency may be optimized. The small-signal theory of traveling-wave amplifiers is used to calculate the bandwidth of helix-type tubes. This is facilitated by expanding the growing-wave propagation constant in a power series in C,

-2QC, d and b. The effects, on gain and bandwidth, of circuit dispersion and loss as well as beam space charge are shown. The developed expressions for gain and bandwidth are then used to obtain a useful gain-bandwidth product for traveling-wave tubes. Derivation of Power Output Relations With the aid of appropriate reduction factors to relate the actual impedance of the helix to the impedance of a sheath helix, expressions have been obtained1 for the space-charge parameter and the electron stream perveance in terms of other operating parameters. These expressions for helix-type amplifiers are as follows. QC =6 (1) FB (P 1\2) 1 (+ )1/2 15.5 hR (-) 1-Cy and 132 (CB)2 (QC), (2) [Ll - 2C(QC)l/2 R2(l-Cy )2 where Rh = the radian plasma-frequency reduction factor for a helixtype tube, C = gain parameter, QC = small-signal space-charge parameter, B = yb' = ya'(b'/a'), the space-charge range parameter (B o' stream diameter), P = electron-stream microperveance, Ks = K (o/y)[(l + (po/y)2]3/2, proportional to the sheath-helix impedance,

-3F = sheath-helix impedance-reduction factor, ratio of actual helix impedance to sheath-helix impedance, b = (Uo-v )/Cv, the electron velocity parameter, and Yl = small-signal wave-velocity parameter. The electron-stream perveance can also be written as = 7905 C3 Po.106 PA FK' (l+Cb) = V 5/2 1 s o where P = r-f power output at saturation, 0 %s = saturation efficiency, and V = electron accelerating potential. Equations 1 and 3 are combined to obtain the following expression for the saturated power output of a helix-type tube in terms of the beam voltage, saturation efficiency and other operating parameters of the tube. 5/2 2.5 x 10 (CB)2 Be Vokv P (4) ~ 2(1_Cy)2 Q[()1/ - 15.5 C] where Vok = stream accelerating voltage in kilovolts. Large-signal calculations2Y3Y4 of the saturation efficiency as a function of the operating parameters are used in conjunction with Eq. 4 to compute the theoretical power output as a function of the stream diameter parameter B, stream microperveance P, accelerating voltage V, gain parameter C and the space-charge parameter QC. Calculations of the nonlinear performance of traveling-wave amplifiers have been made using a one-dimensional model of the device. The efficiency results are applicable to all forms of these amplifiers, independent of frequency range or type of r-f structure used. The principal

-4assumptions made in the theory are that the electron stream is a confined flow and that the electric field due to the circuit and the space charge does not vary across the stream. This latter assumption is considered reasonable for values of the stream diameter parameter B up to 0.5 or slightly higher, as indicated by various experimental studies. For larger B values it is necessary to use correction factors on the efficiency. Rowe4 has investigated theoretically the first-order effects of radial electric field variations across a solid beam. The theoretical efficiency correction factors agree well with experimental observations5s6. The efficiency correction factors for solid and hollow beams are discussed in the next section. A summary of efficiency calculations made with this one-dimensional theory is presented in Fig. 1. Considerable comparison with experiment has been made, with satisfactory results when consistent experimental data can be obtained. Efficiency results such as those presented in Fig. 1 can be used in conjunction with Eq. 4 to calculate curves of theoretical power output for an amplifier as a function of such operating parameters as the stream diameter, the stream perveance, the accelerating voltage, the gain parameter and the saturation efficiency. In order to determine maximum power output the operating voltage is assumed to be adjusted for maximum saturation power output rather than for maximum gain. Typical theoretical power output curves for a helix-type traveling-wave amplifier are shown in Figs. 2 and 3. Additional theoretical power output curves are presented in Appendix A. The power output curves are restricted to a helix-type device because of the fact that the sheath-helix impedance and the appropriate impedance reduction factors were used in calculating the values of C and

-550'QC = 0, b= b (77S MAX.) -, I \ ________________ QC 0.03, b b (-S MAX.) 45 QCO. 0.06, b b(s MAX.) ___ OC 0.125, b: b (Is MAX) l / QCO. Z5, bb(, MAX.), a I /I / / 40 -— __ —--- 35. b 0T1 0 0.05 0.1 0.15 0.b 2 0.25 GAIN PARAMETER, C FIG. I SATURATION EFFICIENCY VS. GAIN PARAMETER. (B 1.0, d 0)

(0 =P' OB = -0a V'l,q/D'gO-' = )'*31j11-dl/V 3AVM-9NI13AV.1 3dAl- X113H V 80d.Lld.nO 83MOd 1V3I13803H. Z'-IO S'8313WMNVa wAV3 SJ'l ~1 I'1 6'0 L'O'-00 10 --- --- --- --- --- --- --- --- --- --- --- --- --- -- 1000'0 I I I 1 <- I O N1 1 1 1s 4 aM+\rxl=Ax I I I I U'Oy \ --- — 1 1\1_L-. __ _1 _,1$- _i-to. -~'0I f\g \ A l ~..- it m,\r / uI00 -.9 Q, lo^^^ —-7 I-e 0010 1,000 -10,'\. -9

Po, KW - | I " I"0 0 I 0 -1 9 2 o o b La0' _ _ g<O * ___ X - < rGIIIIII-1111 1|i|Js -- - m~ I'l l | | | | Ss-37.' "' < 0 > —'_ —-' - Q0 m _ __-__ __ I~ 1 1 1 " 1 "1;' " r -- - - --- -rq.- z 0.6 r _ m m'!D P~.'~_. s~ 38.0

-8QC. In a later section correction factors will be derived which permit the use of these curves in determining the power output of traveling-wave amplifiers utilizing periodically loaded waveguide r-f circuits. The helix voltage curves were carried to only 14 kv, as it is believed the higher voltages would result in a very low impedance and hence a low value of C at most frequencies due to the openness of the helix pitch. The limitation on stream perveance was taken as 3 x 10-6 owing to the impracticability of constructing solid streams with higher values of perveance. In addition, it is noted that the saturation efficiency is relatively low at high perveance, independent of the accelerating voltage. In all curves of the type shown in Figs. 2 and 3 the maximum saturation efficiency occurs at relatively low power output values. The perveance is low at high saturation efficiencies, which indicates that the high efficiency is due to high circuit impedance giving a relatively large value of C rather than to high stream perveance or current. Since each set of power curves is for a constant a'/b', the abscissa is directly proportional to ya' and the fact that the high efficiencies occur at low values of 7a' indicates a dispersive helix giving a relatively narrow-band operation. It will be shown in a later section of this paper how the bandwidth depends upon each of the operating parameters and in particular how the choice of B for a fixed a'/b' is dependent upon bandwidth considerations. The effect of radial electric field variations of the circuit field on the saturation efficiency has been investigated theoretically by solving the nonlinear equations assuming confined flow and no radial motion of the electrons. It was found that this radial circuit field dependence had a marked effect on the efficiency for large values of B. The results

-9indicate that the saturation efficiency for devices with large-diameter streams can be expressed as follows. s(B) - Tf fl/2(B) for b = b(xmax) (5a) f=l and ~s(B) - ~S fl/4(B) for b = b(smax) (5b) where rs = the saturation efficiency for no radial circuit field f=l variation, I 0(7b') f(B) = -(Ya' for thin hollow streams, and [I2(yb') - I2(yb')]1/2 f(B) = 1 for solid streams. I,(7a') A graphical presentation of this dependence is given in Fig. 8 of Ref. 4. This correction factor may then be used in Eq. 4 when calculating the power output. Theoretical Power Output of Modified Helix and Periodically Loaded Waveguide Circuits The previously developed power output curves may be used to determine the theoretical power output from a traveling-wave amplifier using any other type of r-f structure. All nonhelix structures will be described by writing their impedance characteristics as a factor times the sheath-helix impedance, as in Eq. 6. K = GK, (6) and s ahere Kd, G and Ks are all functions of frequency.

-10In many cases G is dependent upon frequency and hence point-bypoint power output calculations have to be made across the frequency band. If the structure is of the modified helix form and there is a dielectric material present that loads the helix, then the G factor (impedance gain factor) takes into account the effect of the dielectric. The ratio of the space-charge parameter for an arbitrary dispersive structure to that for a helix tube has been derived as' l+Cb R [I —~$C 1, ill h ic ( ( i) (7) 1+CbJh where 2 d()2 F V = --- and Rd = the radian plasma-frequency reduction factor for the resonant or dispersive structure. The power output for a dispersive r-f structure relative to the power output of a helix tube may be written from Eq. 4 as [( 60 y1/2 od (_2 ( (Rh (l+ch)22 fo /2L\ c) 15.5 C h oh * Ch d }sh! dd1)2 \Voh/ [ 6o0 1/2 2 L 15.5 Cj (8) where Cd/Ch = the ratio of the gain parameter for a dispersive structure to that for a helix, qsd/%sh = ratio of the saturation efficiency for a dispersive structure to that for a helix,

-11Rh/Rd = ratio of the radian plasma-frequency reduction factors for a dispersive structure and a helix, and Vod/Voh = ratio of voltages for dispersive and helix structures. It has been assumed in the development of the above equation that the stream diameter parameter, B, is kept the same for the dispersive structure as for the helix structure. Since all the parameters for a helix tube can be determined from the theoretical power output curves, any hypothetical helix design can be used as the reference tube in Eq. 8 and the power output of the dispersive structure may be computed in reference to this tube. In general the impedance of the dispersive structure will be higher than that of the helix tube but its frequency passband will be narrower. It is extremely convenient to write Eq. 8 assuming that the value of the gain parameter C is the same for the two structures. This, of course, means that the design and power output curves for a helix structure can be used. Also, for dispersive structures Rd > Rh, since the effective equivalent drift-tube radius for a periodically loaded waveguide circuit will be larger than that for a helix. Equation 8 can then be written as Pod sd) 2 od L/K(Qc)h/ - 21, C} P ^ AY /w2 I2rty *1^12 oh %hfd' oh' L(^)" -o 2 15.5 CJ Equation 9 may be simplified to the following for Equation 9 may be simplified to the following form: Pod Tsd _ _ od\5/2 1 - 2C(QC)hl/2 K-)- =.sd G o) / -, (10) oh 1sh Voh - 2C(QC) /2( where, = (Rh/Rd)2. The appropriate values of i and K for any structure may be calculated or

-12obtained from the curves of Ref. 1. Since the second term in the last factor in the numerator of Eq. 10 is usually small compared to unity, a further simplification of Eq. 10 may be carried out: Poh sh - oh) 1 + (QC)hl2 (/2 -1) (11) Poh Tsh Voh L In general the last factor of Eq. 11 is near unity, since it is proportional to C, and hence the equation may be further simplified. Since the efficiency ratio is usually near one or somewhat greater, and F/G is less than one, it can be seen from Eq. 11 that the prime factor in obtaining greater power output is increased voltage. Here the voltage ratio has to be greater than approximately 2 to obtain more power from the dispersive structure than from the helix structure. The basic power-ratio equation (8) can be reduced to various simpler forms by assuming equality between some parameter of the dispersive structure and the corresponding parameter of the helix structure. Interaction Bandwidth General In an earlier section of this paper, it was mentioned that the choice of B (or ya', since B = ya'-b'/a') was in part dictated by bandwidth requirements. A discussion of bandwidth is therefore in order. The interaction bandwidth is determined only by the internal gain limitations and is not influenced by transducer cutoff or loading effects. When considering the overall bandwidth of a tube one must consider all external as well as internal phenomena; however, the following discussion will be limited to the interaction bandwidth. The interaction bandwidth is determined by the range of frequencies over which the electrons keep in step with the circuit wave. This requires

-13a consideration of circuit dispersion. The sheath helix will be the only circuit considered. When designing a helix-type traveling-wave tube, one usually refers to the approximate variation of gain with frequency given by Pierce7 and decides upon a center frequency value of o a' for wideband operation. It is one of the purposes of this paper to consider the gain as a function of frequency in a more detailed manner than in the above reference. Spacecharge forces, loss, finite values of C and nonsynchronism will be considered. While a nonlinear analysis of bandwidth characteristics would be desirable for this discussion, the complications introduced are quite formidable; hence the analysis presented here is a linearized one. The results are further limited to QC < 1/4, which is consistent with high saturation efficiency. Gain Equations for the Sheath-Helix Tube The small-signal gain as a function of frequency will be studied by choosing the center frequency values of the parameters, calculating their variations with frequency, determining the propagation constants as frequency functions and, finally, using the standard growing-wave gain equation G = exp A exp PeCxlL.(12) The gain as found by this method actually may cover a somewhat narrower range of frequencies than the actual gain, since it is possible to have a gain due to a beating of the three forward waves after growth has ceased. This is the phenomenon which has been named the Crestatron8 effect and it will be neglected here since it occurs near the band edge and the gain contributed is probably less than the minimum used in defining

-14the band. The band will usually be referred to as the range of frequencies in the vicinity of the center frequency over which the gain does not vary more than 3 db or 10 db9. Sensiperlo has derived explicit forms of the propagation constants as functions of the traveling-wave tube parameters. This particular formulation is valid only for QC < 1/4. The leading terms of Sensiper's expansion are r1l(C'\d(C d b 4 >C 2(2 x ^ 2 3 Q )3 (Q2 2 3 3) 3- b + n -( )] (13) Substitution of Eq. 13 into Eq. 12 gives the gain as a product of four terms, hereafter called the gain factors. Therefore if we define F 1(0) ^ XC C1 +) (14) F2() -1 + - + QC, (15) 535\ c3 2-75 3 ) F3(W) A - QC ( - b), (16) 7 3 and F4(c=) - 6 (bC ) (17) and take logarithms, the gain can be expressed as I G L F1 + F2 + F3 + F4 (18) The gain factor F1 is determined by the small-signal gain parameter C. This factor is the fundamental one in determining gain; the remaining factors introduce the effects due to other parameters. As will be seen later on, the factors F2, F3 and F4 actually define the operating range but their

-15effect is small in the region of active gain. F2 vanishes for no loss, F3 vanishes for no space charge, and F4 vanishes for synchronism and no dispersion. It will now be assumed that the initial loss A is constant with frequency. The results published by Brewer and Birdsallll indicate that A does vary with variation of the parameters. However, for the range of parameters chosen in this study, the variation of A will be small for large C. For small C, the loss of gain due to nonsynchronism will be so great as to outweigh all other effects. Equation 18 is equivalent to: Gdb() =15.1 ( ) [F1 + F2 + F3 + F4] + 8.68 A. (19) The gain factors are evaluated by substituting Eqs. 5, 11, 14 and 16 of Appendix B into Eqs. 14 through 17 above. The following normalizing parameters are introduced: t= - = normalized frequency, 0a0..- = normalized dispersion, C, 0, Y are the indicated quantities at the center or design frequency. X C3 exp 2 - a')o = 81 ( (7oa')2 and g = f' 7 (T (20) where f' is the initial slope of the plasma-frequency reduction factor curve (g = 0.6 Tya1 for b'/a = 0.7), and b'/a' is the beam-to-helix radii ratio. The gain factors are then given in two regions, ya' i 1 and ya' ~ 1. For the region in the vicinity of ya' ~ 1, one must

-16interpolate; however, the values in each region agree quite well near ya' = 1. For ya' < 1 Fl(() = ~-0/ e xp (X e -0.59 oa )] 1 + 1 (Xr) exp -0.59 Yoa~\ (21) co C d Hi F2() = -_ ~ 1 + (XT)1/3 exp (-0.59 Y a') {13 1 l+Cb \ 1/3 + C- KXT 3/2; 00d +i 00 p-]2 o + C2QC 1 - exp -g exp 0.59 Yoa't (22) lCQCo - exp lg ]2 exp 0.59 y oa 3 0 LI - exp -g j (3X)1/3 2 1/3 0, F3s(c) = - "~ CoQ L exp -g j {l - p (1\X3 L ~'r - 1] exp 0.59 Y a' (23) eD 1 rl+C b - F4(o) = - n 1 L(XT)1/3 exp -0.59 yoa't - 1 ()/ (-/b t n - 1)exp 0.59 Y7a'] (24) For ya' > 1

-17Fl) = ~[ ]/ 1 + 1 exp - ( - ya, exp - 2 - ) Y a (25) XW C oo 1 1 / exp - - b F2() = + - 0 exp -b a 3 1- exp - 7 o a 5d00 1 3x 1 p+C )KX 20a o1/3 3 o0 -exp -g/ a/ exp (l -,ati] (26) (CexpQC-exp - 2 /3 F3(0) 2Q -g exp ) a 3 - xexp - 4 (27)2 r-1/3 0 0 2 b,, / ex p - a.. (27) lWlllV73^0 ^ 21' 113 F4(cu) = - 1) [(X / exp - 3 ( a ) 7a I+C b ex 2 -1zi~ (~i~/3 (1+Cb 0 - i~exp ( ) a] (28) 3 X ( — - e x p-. (28)3o' The relation between r and ~ can be obtained from the characteristic equation of the structure.

-18Gain Factors F1 is shown in Fig. 4 as a function of the normalized frequency, with CO and yoa' assuming several values at the center frequency (q = 1). These curves represent the gain for a hypothetical tube with no space charge, no loss and with the unperturbed circuit wave and the beam in synchronism at all frequencies. Figure 4 indicates that in this case an optimum selection of Y0a' for wideband operation would be approximately 1.2. This value is lower than that given in Fig. 3.6 of Pierce's book7 (for b'/a' = 0.7). Pierce's curve is a plot of the proportionality CNc ya' F(ya'), (29) where N is the number of circuit wavelengths. The quantity CN, however, is not exactly proportional to the gain. The gain is instead proportional to the number of stream wavelengths N. For a gain calculation at a particular frequency the error involved in using the circuit wavelengths in place of the stream wavelengths would be small. However, when the gain variation with frequency is being studied the error can increase significantly, since the number of stream wavelengths varies indirectly with frequency whereas, due to dispersion, the number of circuit wavelengths does not. Therefore, a better approximation to the gain in this very idealized tube would be v(y at) v(ya') Gain cpCN = ya' F(ya') v( (30) S U0 v(Yo T') The ratio v(ya')/v(Yoa') compares the velocity at a given ya' to that at the center frequency Yoa'. Pierce presents a curve in Fig. 3.2 showing this dependence. A multiplication of the curve in Fig. 3.6 with that in

-190.10 0.09 0.08 0.07 0.06 \,// 0.04, 7o.00'- 84.0.,,..,,,?-.,~- ^yooo: 2.0 o-'1.25 y' —1.00 0.02 7- C 0 0.05 0\' _ 4.0.... Z YOa' -2.0 F. b/.25 0. 0.0 1 -- -- - -- -- - -- - -5. 0.009 0.008 0.007 -- 0.006 - - C=0.01 0.004-"-''""0.003 y4.o- N o yo.2.0 oo o':.25 J.002 y~aa1.5 ~od ".0 0 I 2 3 4 5 6 7 FIG. 4 GAIN FACTOR,-o- VS. NORMALIZED FREQUENCY. (b'/a 0.7)

-20Fig. 3.2 results in a shift of the maximum toward the value shown in Fig. 4 of this paper. Representative plots of F2 as a function of frequency are shown in Figs. 5 and 6. For all cases considered, F2 becomes more negative as the frequency increases. The inclusion of space charge tends to accentuate the effect of F in decreasing the gain. The value of b used in the calculations is the one producing maximum small-signal gain at the center frequency. It is possible to get an increase in gain for Co = 0.01 at very low ya'. This occurs because of the large negative b value that results at the lower frequencies with small C. Actually, this increase is never realized, since at this negative b value F is a very large negative number and there is no gain. F' is shown in Figs. 7 and 8. The calculations of F3 as well as all 3 other factors involving QC were carried out only until QC reached 1/4. This restriction is imposed by the approximation for x1 given in Eq. 15. F is shown in Figs. 9 and 10. For low C, F restricts the gain to 4 4 a very small range of frequencies. C represents the coupling between the beam and the circuit. For very loose coupling the interaction will be negligibly small unless the beam travels approximately at circuit velocity. For tighter coupling, the beam can perturb the circuit wave sufficiently to produce interaction even though the velocities may be quite far from synchronism. Gain as a Function of Frequency The expression for the gain is given in Eq. 19. A summation of the gain factors describes the gain as a function of the center frequency parameters. The following sums are considered:

-21-0.004 —- /~o' = 1.25 4.0 s/ -0.001 or3~i IA I a- 0.00 0..I \ I I 4I'1 1V' -0.001 --- - -------- 0.00001 0 0.5 1.0 1.5 2.0 2.5 o0 3.5 FIG. 5 GAIN FACTOR, VS. NORMALIZED FREQUENCY. (C: 0.01, OC =0.125, d =0.125. bfa'=0.7)

-22-0.04....- -- -0.03 -0.02 I — \ F. / 0 00.01 z -0.009 ( -0.008 -- -0.007 -0.006 -0.005 x oa1.25 I l~rii ya1.5 -0.0023 i! -0.001 b 0 0.5 1.0 1.o5 2.0 2.5 3.0 -OC 0. 125 ba' 0.7)

(P jO lN3aN3d3aNI'LO,o/,q Om'SZI'O = 30'10'0 =3)'N3n0O38.I 3ZI1VVlWON'SA IO' V. NIV9 L'91 -Z o'z0 9'1 3'1 8'0 o'0 0 X, \ I I. ~-. —---— _.__.__._-1.000-'= -_z_~~~~ ~ 0'O~ ~~ I 0 ____ ______ ______!____ o___,:__ -,o9000 oz~~~o~>c^1~~ I^3

(P JO IN30N3d30NI Z'O = /,q'ZI'O: 30'O 3)'A3N3n03.j 03ZIlVWHON'SA O B'01OV-3 NIV9 8'91 r"Z'0 9'.1'1 8'0 O 0 ---- 0- - 0 —0 ------ -- 900'0900.0-— oo'o00 0 Ox /g,,., --------- --- ooo _ L I I I 1 I /ii - 9 000 800'0'4o-?010'0

F4 GAIN FACTOR FO 0 0 o 0 0 0 0 b b b b o b b b -- ( -o 0 t~,._________ —_0__ 0 o "- 00 > so z 0 > "0 "___,, __-..- ~"m — low —MmorZI< -- -- -- - 0M o rrl r org 0 0 0 0 0) h)~~~..

GAIN FACTOR, F4 -i i I I I I b b 4. a ~_____________________ 0 0 Li 0 0 0- I o< 0C:) __ " Z,, >_ 0'- ~ — - P.0 I __o — - -.._w —-M-.,0 o ~- ~~.... _ _ _______ _ _ ~ " __ _ __ _ __ __ ____ ___'-0 _'_;0.*. %< C D_ *() 0 o rr I I I I. I I I I..... C) go zm C cJ~ ~ 0 -n """~ ~ ~

-27I) F1 + F4(QC=0); C f 0 QC = d = 0 II) F1 + F3 + F4(QC#O);C 1 O, QC ~ 0 d = 0 III) F1 + F2(QC=O) + F4(QC=O); C O, d i 0 QC = 0 IV) F1 + F2(QC#O) + F3 + F4(QC/0); QC f O, C O 0 d 0 The variation of gain with frequency for two values of C and 0 several values of y a' is shown in Figs. 11 and 12. The ordinate of these figures can be converted to db through multiplication by Gdb(=l) - 8.68 A 4 Fi (=l) i=l The suinmation of Fi(O) is valid only so long as it is positive. This is necessary because the approximation for x1 given in Eq. 12 is valid only for positive x. For small C values (C = 0.01), curves of Type I are smooth functions in the vicinity of e = 1. Inclusion of space charge (Type II) preserves the regularity but narrows the band. However, introducing loss as in cases III and IV brings about the irregularities observed in Fig. 8 for a' = 4. The general shape of the gain curves in the vicinity of | = 1, for small C, varies from parabolic (for small y0a') to linear (for large 0a'). The gain variation for large C(C = 0.10) is more irregular near 5 = 1 (especially so for yoa' = 1.25); however, the same "average" shape is obtained as for the small-C curves. It is immediately evident that the

0.005 0.004 z 0.003 20. 0d-. ——' =4.0 0 — ~ 1\' I I I II oc II. 0I I b/a'0.002

( L'O =,/,q'93210 P'gZI'O =30'O 10 3)'A3N3n038J 03ZI-IVtON *SA NIV9 03ZlVlWVON ZI *9U1 t'-Z 0'Z 9-1 8I'0 t0 100 --- -- --- —,o0 z 0 0 XN!^___ ____ gX Z t I ~:0'0 N / s. r ~ ~ I I -- 0/D~/ 0 800 - -_ -_ -_ -__ ___ - - - -.- 800

-30bandwidth is much greater for the C = 0.10 tube. Interaction Bandwidth If a center frequency gain is specified, the ordinates of the curves shown in Figs. 11 and 12 can be converted to db, as mentioned above, and the theoretical bandwidth can be measured directly from the curves. The percent bandwidth as a function of y a', for different center frequency parameters, is shown in Figs. 13 and 14 for a tube with 30 db gain at the center frequency. Figures 13 and 14 can now be used to draw the following conclusions. In order to obtain a large bandwidth for a small-C tube, high values of yoa are required. In fact, a y0a' in the vicinity of 3 or 4 would be desirable to get 3-db bandwidths of about 10 percent, for a C = 0.01 tube, which would correspond to 50 percent with a 10-db bandwidth. However, using a high 7ya' value inherently raises the ka' value and in doing so makes possible backward-wave oscillation. If yoa' is lowered to values between 2 and 3, it is still possible to obtain 3-db bandwidths between 5 and 10 percent, but now the 10-db bandwidth is lowered to about 20 percent. For large-C tubes with space charge and loss, a y a' of 1.5 or less results in 3-db bandwidths of greater than 20 percent and 10-db bandwidths greater than 60 percent. It can be seen from Figs. 13 and 14 that loss tends to decrease bandwidth, as does space charge, except for the rather anomolous behavior in Fig. 11 of QC = 0.125, d = 0. When fabrication is considered, the result requiring a larger yoa' for low-C tubes is favorable, since the lower C values are usually associated with high frequencies.

-3180, i -0 QC - - 0.1-25 10 d- - 70 _..____. I I I I I I z I z 00 —'0 QC:O, d: O(10 db) 1 I I/ w 7.. QC =0,25, d 0 (10 db) Q:: j | ~// (10 db) 20 2r/ QC 0, d=0/l3rd 35 I I -II I/ FIG13PEREN_______ BNIT____H V __O. (C 0.01 b) 0.7) 20 7LA -oQ 0 C 153db) /// / /I QC=-,d: 0.125 (3db) -- -- -- / / —. QC:0.\25, d:0 (3db: I0 /QC =0. 1 25, d = 0.125 (3 db) 0 2 3 4 5 6 you FIG. 13 PERCENTAGE BANDWIDTH VS. yo0. (C:0.01, b'/o' 0.7) G+ IAI = 40 db AT ( =1.

-32160.... — - --- 140 120 [LLL<Itm~~ IIJI o I I I 100o 40 \L QC 0, d 0.(10 db) _ _I I \\ _ __ __-'~~C 0, QC 0.125 (d0 Ib) --- I / t F5 - I \ I Q =0.125 d:0.125 (10 db) ____ ____ ____ ____ ____ QC 0=, d=0 (3 db) ^w V \ QC0, d - 0.125 (3 db) QC = 0.125,d = 0.125 (3 db) 0 1 2 3 4 5 6 roQ FIG. 14 PERCENTAGE BANDWIDTH VS. yrQ'. (C 0.1, b'/a' 0.7) G+ |AI 40 db AT C I.

-33External circuitry such as attenuators, couplers and loads may be used to flatten the gain characteristics by dissipating power in regions of high gain but passing the low-gain frequencies. Gain-Bandwidth Product The concept of a constant gain-bandwidth product has been proven to be of great importance in the applications of low-frequency tubes. It is worthwhile to give some consideration to a possible similar product for traveling-wave tubes. The gain of a traveling-wave tube was expressed in Eq. 12. If we once again neglect the frequency variation of A, the gain may be written as In 2 A) = f()) L, (1) kexp A where f(Xo) = e Cxl. The factor f(w) is determined by the applied signal frequency, beam voltage, current and cross section, transverse dimensions, periodicity and loss of the circuit. The quantity In(G/exp A) can be considered as a "pseudo" gain (in db). fn(G/exp A) varies linearly with the helix length at a fixed frequency. The band edge frequencies will be defined as those frequencies at which the gain differs from the center frequency gain by a specified number of db (3 or 10 as used in this report). This may be expressed mathematically as: f( - f(l) = M (2) A l= ----- — (3e) A db difference where M and 8.68 L = structure length in meters.

-34The solutions of Eq. 32 nearest 1 = 1 are the desired ones. Expanding f( ) in a Taylor series about ~ = 1 yields a ( - 1)n (33) L n=l The form of f(,) and hence that of a is determined partly by the selection of yoa'. From Figs. 8 and 9 it is seen that if ya' is large the gain varies linearly with frequency in the vicinity of ~ = 1; hence for this case one need only use the n = 1 terms of the above series. Solving Eq. 33 for the band edge frequencies and taking their difference gives the bandwidth as: =2M EM (34) a1L The "gain"-bandwidth product is therefore 2 f(o) M G bBW = a (35) db a Hence, for a helix designed with high ya', varying the gain by adjusting the helix length gives a constant "gain"-bandwidth product, since none of the variables in the right side of Eq. 35 are functions of length. We must remember that this gain-bandwidth product corresponds to a product of a frequency band and a gain in db. Unfortunately, a similar product does not yield the same results for lower y0a' values. Consider a hypothetical case where f(cu) may be approximated by a parabola centered at ~ = 1. Under these circumstances only the n = 2 term appears in Eq. 33. The bandwidth in this case is M\t =,4 (36)

-35and the gain-bandwidth product db = 2a () MrL (37) BW =2 fa 2 This product varies as the square root of the gain as the helix length is adjusted. A constant may be obtained by using the product of the gain and the square of the bandwidth: Gd(BW)2 =.f(W) (38) db a 2 As a rough approximation, one might say that the product of the gain in db and the bandwidth is a constant for large yoa', while the product of the gain in db and the squared bandwidth is constant for small yoa. Conclusions Analytical expressions have been derived and theoretical power output curves have been presented for helix-type traveling-wave tubes as a function of the principal operating parameters. It has been shown that highest efficiency occurs at relatively low perveance and that the saturation efficiency falls off markedly at large values of B. Also expressions are developed for determining the power output of tubes with high-impedance dispersive structures. An investigation of traveling-wave tube bandwidth has indicated that the gain-bandwidth product is a constant with respect to length for high y0a' and the gain times the bandwidth squared is a constant for low yoa'. The gain and bandwidth functions are plotted for representative values of the operating parameters. Acknowledgments The authors wish to express their appreciation to Messrs. R. Y.Y. Lee and N. Masnari for their work in calculating and preparing the curves.

APPENDIX A. THEORETICAL POWER OUTPUT FOR A HELIX-TYPE TRAVELING-WAVE AMPLIFIER C a'/b' DLF 0.05 1.4 80 1.6 80 1.4 100 1.6 100 0.10 1.4 80 1.6 80 2.0 80 1.4 100 1.6 100 2.0 100 -36

(o=p'%08 = dla l,q/D'SO'O = 3)'83lI1dWl/V 3AVM-9NIn3AV8l 3dAi-XI13H V lOd IndinO 13MOd 1VOIi33803HI I'V'91. 8'8313YVO1 WIV38'l ~'1 61 6'0 L'0 9'0 ~'0 1'0 0t~10~000'0 t'I _A - - U' - (,J~, ro ^ r\ 1Y\. I T \ //1-1 \.4p 10000:?Z^Z:-;*T-P5;?5Zt — --- 10 ^n^-rP'~~ 00 ID 4%od l^0'l' I^-^A I! J ^f'^;'^^ ^'^K'V;/,. JO'O0'0. ^^' I^r -

(O = P % 08 =:"1'9'1,q/,o'SO':3)'31.Jl1dlWV 3AVM-9NI"13AVI1 3dAl-XI13H V 80I indinO 83MOd -IV3113803Hl Z'V'91 8'I.313WVIGl WlV38 9'1 ~'1 1'1 6'0 L'0'o ~' 0 r 1000'0 I00'0 _I ___1. ___ _ ~o.o 0' I', _-,~~~~~~~~~001 _ _, __ Sl ~000~100 t a, I'1 0~~~~-g~000rr~lp1ST~t7*z:^r=! =-'*-r==-So^^^ i~~ ~

-391.0 ----— _ -- -~~~~~~~#I - 0..' F I I I I II I~s1+li'l H9- [ ~I I~~~~~- I 001 ___ _ _o 0.1 0. 0.5 0.7 0-9 1.1 1 TRVLNGWV A F (-0 I,,"- "-. L -00'r e.0 0. o0 0. / -0II 0.1 0.3 0.5 0.7 0.9 I.I 1.3 1.5 0.001DLF = I %,- _______^__ -- __ dt~ t: -- -:__ - -- 0.0001 - 0.1 0.3 0.5 0.7 0.9 1 TRAVELING-WAVE AMPLIFIER. (C:0.05, a/b' = 1.4, DLF = 100%,d0o)

0 PF, KW o b 0 o o -,- 0. 0 00 C) - - - -.'t > m " sl I iQ.O4 G O. t0-o so \x \ \\' % \ - I _ -- - _ _______ I I I = ^- — L —;:::, - -.-M w-a, -% — 0 3 *0 )95QPA 05^ ^"* "* ""'" *\ -V ^ _ rtl CD ms' -'

0 Po, KW _' b C) o o0 - ^i\"^^^S::- 0 0 — (Jl.~ -IN] 0 > o -o-I 0o ____ -OO ID -, ~> 3'8. 9'S ~ _.. _____ -r Is38.O 4 - 46I I I 38. ~, ___________._.....~~~~~~~~~~~~~ poo

1 P0 KW 0 0~~~~~~~~~, 0 b _ Gom P OD r- r 8. n IN.o 7 z 3 - 0 M O% ) sf ~:C:3 IL I L m >6Pz'~ 00 0 t L a> _ _ _ _ 60 0\" 63 63 =35.5~~~17SPi z0. 6 ml -U r PL 9sS~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ apo p 1.4535. 0, 30 pP. m a % 36 ) a IIIII IP,~1. m L. ------- ------- L-6

(o = P'% o08 = -a'0' =,q/o'01'O= 3)'831.J-dllWV 3AVM-9NI-13AVa1 3dAl-X-13H V 80O indinO 83MOd "V3113803H. L'V'91:1 8'U3l3wVia WtV38 G'l ~'i 1' 6'0 L'0 S' 0'o I'0 100'0.:1 -- -- -- -- - - -- -- -- --.^I I -- - o (SI b) 3"" 1 10'0 tO b~~~~~

p 0 PoIP KW 0 0llzlllX Tm | or _ 4.0 %It _ 0-~- - - _ - b —N-I - >o n,s s043.0 -—. — ~Io p_ — - _ —-^, - - - _ _ fr ~l slZO I i iS1OM *n -"8-5 m -o L ago — 0er 1 3 ~e 39~ 40.0'. ~ E., ~P rN IO~" C,,',,', -I ~' ~....-~ ri~~~~~~~~~~~~ -—. —. \...~, —-~~ ~,-,.,.,.. 1 1~ _ ~\~t rr b

-45I0.0'P-oI F I I I I I |s \1 \\ t \\l \^ X \ y\ n 1.0 E 1 )1'si1 \WeiR w- s o- I VlJ x -IT JB D' I / E, 1 ~ r tsia l w TAEI-.o W AVE AMPLI l F L0.0D' d:30) cSRr l l \ \\I\ \ I I I I I 0.1 0.3 0.5 0.7 0.9 I.I 1.3 1.5 TRAVELING- WAVE AMPLI FI ER. I. (C-.,b 1 D 1- O16 ----- - /. --- Q " T ^ ---- ---- ---- ----- ---- ---- ---- -M ^\j~ —---- -ST'-a o; ------ --- -OP — T^ -2-" P" ---- --- ---- --- --- --- --- DLF s 100%X, d =0)

(O = P'%001=Ja'0'Z =,q/,'10 =3) a*31JlldkJV 3AVM-9NIn3AVa. 3dAI-XI13H V 8O.J indlnO 83MOd 1V3113803HI 01V'O91 8'U3313IVIa WrV38 S' 1 i'1 6'0 ~L0 S'O'0 I'0 1 1000 Loo IVI I I I E I 1A I -,1,'I o __ " 1:.o -- -o- o --— I- - o Ad Ac e " II.i W I0 I I [ / 0.98_ 11~~~~~~~~.-o'~~~~~~~` O~~., i

APPENDIX B. TRAVELING-WAVE TUBE PARAMETERS AS A FUNCTION OF FREQUENCY The impedance of a sheath helix of radius a' surrounding a solid beam of radius b' is given as: K (a2 [IOb' - I(7 (4 [ (1 + Kl(ya) IO(7a')) (Il(a') - I (ya') (.)) + (Io(ya 2 0 K(ya') I (ya -1 (1 + 0y(aa.)a Ka2ya) a K2(a) -') (B. For large ya' the asymptotic expansions of the Bessel functions may be used. Therefore K = (a)2 (b exp [-2ya'(l -, for large ya'. (B.2) For regions of small ya', the curve of the impedance of a sheath helix as given by Fletcher12 can be approximated by the following equations. K = 181 P exp (-1.77a'), for small ya'. (B.5) Now a value of the gain parameter Co at the design frequency 0o is defined. The definition will be based on the large-ya' expression {0 0 )2 k a' b' 0 0 ( oat)2 ]0a I at Introducing the normalizing parameters as defined in Eq. 19 of the text gives the gain parameter C as: -47

-48C3 = exp ( -2 a-) o, for ya' > 1 C3 = XT exp (-1.77 a' ), for ya' < 1. (B.5) The space-charge parameter QC is given approximately as R2 )2. C -4C2 * (.6) The plasma-frequency reduction factor R2 may be approximated as13 -Bf R2 (1 - e )2 (B.7) where f is the initial slope. Now we define g = oa'f (f )' (B.8) The plasma-frequency reduction factor at any frequency is then R2 = [1 - exp (_g)]2 (B.9) The center frequency space-charge parameter is QC (1 - exp -g)2 (B.) QC - (B.10) 0 4C 2 0 Therefore the space-charge parameter at any frequency is QC C= QC Q.- 2/a (1g -) exp 3 g e-a for2 a'l1 o C o ~ (1 - exp -g a' Qo LC l-e< QC C o L - exp - exp (1.77 a'). for a' 1. (3XT)2/3 L1 - exp -g ] ~~~~(B.11)

-49The velocity injection parameter b is defined as 1 i 0 1 u VP b = - - i =.....1 (B.12) p po p The velocity ratios are u = + C b v o o po and v =. ( B(B.13) v p Therefore, b = L - 1 (1 ) exp ( 3 - a' a] for 7a' 1.b = - 1 -x./3Ya ari) for ya' 1 (B.14) If TL(To) is defined as the total cold-circuit loss at Uo, and the loss is assumed to vary as the square root of frequency, the loss parameter is given as 0.1155 TL i L d 0= (B.15) C -- u 0 which becomes d = - exp L K1 - - 7a'IT for ya' l 1 (XW/ 3 ) // dC d = exp (0.59 oa't) for ya' 1. (B.6) (XT13/2)1/3

LIST OF REFERENCES 1. Rowe, J.E., Sobol', H., "General Design Procedure for High-Efficiency Traveling-Wave Amplifiers", Trans. PGED-IRE, vol. ED-5, No. 4, pp. 288-300; October, 1958. 2. Rowe, J.E., "A Large-Signal Analysis of the Traveling-Wave Amplifier: Theory and General Results", Trans. PGED-IRE, vol. ED-3, pp. 39-57; January, 1956. 3. Rowe, J.E., "Design Information on Large-Signal Traveling-Wave Amplifiers", Proc. IRE, vol. 44, No. 2, pp. 200-211; February, 1956. 4. Rowe, J.E., "One-Dimensional Traveling-Wave Tube Analyses and the Effect of Radial Electric Field Variations", Tech. Rpt. No. 30, Electron Physics Lab., The Univ. of Mich.; July, 1959. (to be published in the Trans. PGED-IRE). 5. Caldwell, J.J., Hoch, O.L.S, "Large Signal Behavior of High Power Traveling-Wave Amplifiers", Trans. PGED-IRE, No. 1, pp. 6-18; January, 1956. 6. Cutler, C.C., "The Nature of Power Saturation in Traveling-Wave Tubes", BSTJ, vol. 35, No. 4, pp. 841-876; July, 1956. 7. Pierce, J.R., Traveling-Wave Tubes, D. Van Nostrand, New York; 1950. 8. Rowe, J.E., "Theory of the Crestatron: A Forward-Wave Amplifier", Proc. IRE, vol. 47, No. 4, pp. 536-545; April, 1959. 9. Bandwidth usually implies 3-db gain variation; however, for a wideband microwave device, the 10-db bandwidth is also of interest. 10. Sensiper, S., "Explicit Expressions for the Traveling-Wave-Tube Propagation Constants", Memorandum No. 53-16, Hughes Research and Development Laboratories; December, 1953. -50

-51LIST OF REFERENCES (Continued) 11. Brewer, G.R., and Birdsall, C.K., "Normalized Propagation Constants for Traveling-Wave Tubes with Finite Values of C", Hughes Aircraft Company, Tech. Memo. No. 331; October 29, 1955. 12. Fletcher, R. C., "Helix Parameters Used in Traveling-Wave-Tube Theory", Proc. IRE, vol. 36, No. 4, pp. 413-417; April, 1950. 13. Rowe, J.E., "A Large-Signal Analysis of the Traveling-Wave Amplifier", Technical Report No. 19, Electron Tube Laboratory, The University of Michigan; April, 1955.

DISTRIBUTION LIST No. Copies Agency 2 Commander, Rome Air Development Center, ATTN: RCLRR-3, Griffiss Air Force Base, New York 1 Commander, Rome Air Development Center, ATTN: RCSST-4, Griffiss Air Force Base, New York 2 Commander, Rome Air Development Center, ATTN: RCSSL-1, Griffiss Air Force Base, New York All add. Armed Services Technical Information Agency, Documents Service copies Center, Arlington Hall Station, Arlington 12, Virginia 1 Commander, Air Force Cambridge Research Center, ATTN: CRQSL-1, Laurence G. Hanscom Field, Bedford, Massachusetts 1 Director, Air University Library, ATTN: AUL-7736, Maxwell Air Force Base, Alabama 2 Commander, Wright Air Development Center, ATTN: WCOSI-3, WrightPatterson Air Force Base, Ohio 2 Commander, Wright Air Development Center, ATTN: WCOSR, WrightPatterson Air Force Base, Ohio 1 Mr. Hans Jenny, RCA Electron Tube Division, 415 South 5th Street, Harrison, New Jersey 1 Air Force Field Representative, Naval Research Laboratory, ATTN: Code 1010, Washington 25, D. C. 1 Chief, Naval Research Laboratory, ATTN: Code 2021, Washington 25 D. C. 1 Chief, Bureau of Ships, ATTN: Code 312, Washington 25, D. C. 1 Commanding Officer, Signal Corps Engineering Laboratories, ATTN: Technical Reports Library, Fort Monmouth, New Jersey 1 Chief, Research & Development Office of the Chief Signal Officer, Washington 25, D. C. 1 Commander, Air Research and Development Command, ATTN: RDTDF, Andrews Air Force Base, Washington 25, D. C. 1 Commander, Air Research and Development Command, ATTN: IRDTC, Andrews Air Force Base, Washington 25, D. C. 1 Applied Radiation Company, Walnut Creek, California, ATTN: Mr. Neil J. Norris

DISTRIBUTION LIST (Continued) No. Copies Agency 1 Director, Signal Corps Engineering Laboratories, ATTN: Thermionics Branch, Evans Signal Laboratory, Belmar, New Jersey 1 Secretariat, Advisory Group on Electron Tubes, 346 Broadway, New York 13, New York 1 Chief, European Office, Air Research and Development Command, 60 Rue Canterstein, Brussels, Belgium 1 Prof. L. M. Field, California Institute of Technology, Department of Electrical Engineering, Pasadena, California 1 Prof. J. R. Whinnery, University of California, Department of Electrical Engineering, Berkeley 4, California 1 Prof. W. G. Worcester, University of Colorado, Department of Electrical Engineering, Boulder, Colorado 1 Mr. C. Dalman, Cornell University, Department of Electrical Engineering, Ithaca, New York 1 Mr. E. D. McArthur, General Electric Company, Electron Tube Division of Research Laboratory, The Knolls, Schenectady, New York 1 Mr. S. Webber, General Electric Microwave Laboratory, 601 California Avenue, Palo Alto, California 1 Mr. L. A. Roberts, Watkins-Johnson Company, Palo Alto, California 1 Mr. T. Milek, Hughes Aircraft Company, Electron Tube Laboratory, Culver City, California 1 Varian Associates, 611 Hansen Way, Palo Alto, California, ATTN: Technical Library 1 Dr. Bernard Arfin, Philips Research Laboratories, Irvington on the Hudson, New York 1 Columbia Radiation Laboratory, Columbia University, 538 West 120th Street, New York 27, New York 1 University of Illinois, Department of Electrical Engineering, Electron Tube Section, Urbana, Illinois 1 University of Florida, Department of Electrical Engineering, Gainesville, Florida 1 Mr. E. H. Herold, Varian Associates, 611 Hansen Way, Palo Alto, California

DISTRIBUTION LIST (Continued) No. Copies Agency 1 Dr. D. D. King, John Hopkins University, Radiation Laboratory, Baltimore 2, Maryland 1 Sperry Rand Corporation, Sperry Electron Tube Division, Gainesville, Florida 1 Dr. M. Chodorow, Stanford University, Microwave Laboratory, Stanford, California 1 Mr. D. A. Watkins, Stanford University, Stanford Electronics Laboratory, Stanford, California 1 Mr. Skoworon, Raytheon Manufacturing Company, Tube Division Waltham, Massachusetts 1 Mr. T. Marchese, Federal Telecommunication Laboratories, Inc., 500 Washington Avenue, Nutley, New Jersey 1 Mr. Donald Priest, Eitel-McCullough, Inc., San Bruno, California 1 Dr. Norman Moore, Litton Industries, 960 Industrial Road, San Carlos, California 1 Massachusetts Institute of Technology, Research Laboratory of Electronics, Cambridge 39, Massachusetts, ATTN: Documents Library 1 Sperry Gyroscope Company, Great Neck, New York, ATTN: Engineering Library 1 Dr. D. Goodman, Sylvania Microwave Tube Laboratory, 500 Evelyn Avenue, Mountain View, California 1 Dr. M. Ettenberg, Polytechnic Institute of Brooklyn, Microwave Research Institute, Brooklyn 1, New York 1 Harvard University, Cruft Laboratory, Cambridge, Massachusetts, ATTN: Technical Library 1 Dr. R. G. E. Hutter, Sylvania Electric Products, Inc., Mountain View, California 1 Mr. A. E. Harrison, University of Washington, Department of Electrical Engineering, Seattle 5, Washington 1 Mr. Robert Butman, Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, Massachusetts

DISTRIBUTION LIST (Continued) No. Copies Agency 1 Dr. J. H. Bryant, Bendix Aviation Corporation, Research Laboratories, Northwestern Highway and 10 1/2 Mile Road, Detroit 35, Michigan 1 Mr. James B. Maher, Librarian, R and D Technical Library, Hughes Aircraft Company, Culver City, California 1 Dr. Robert T. Young, Chief, Electron Tube Branch, Diamond Ordnance Fuze Laboratories, Washington 25, D. C. 1 Bendix Aviation Corporation, Systems Planning Division, Ann Arbor, Michigan, ATTN: Technical Library 1 Mr. Gerald Klein, Manager, Microwave Tubes Section, Applied Research Department, Friendship International Airport, Box 746, Baltimore 3, Maryland 1 Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota, ATTN: Dr. W. G. Shepherd 1 Director, Evans Signal Laboratory, Belmar, New Jersey, ATTN: Dr. Gerald E. Pokorney, Microwave Tube Branch, Electron Devices Division 1 Sperry Corporation, Electronic Tube Division, Gainesville, Florida, ATTN: Mr. P. Bergman 1 Mr. E. C. Okress, Electronic Tube Division, Westinghouse Electric Corporation, P. 0. Box 284, Elmira, New York 1 Power Tube Department, 1 River Road, General Electric Company, Schenectady, New York, ATTN: Dr. Bernard Hershenov 1 Microwave Electronics Corporation, 4061 Transport Street, Palo Alto, California, ATTN: Dr. S. F. Kaisel 1 Librarian, Microwave Library, Stanford University, Stanford, California

AD l UNCLASSIFIED AD_ NCLASSIFIED University of Michigan, Electron Physics Laboratory, Ann Arbor, 1. Power Output Relations UUniversity of Michigan, Electron Physics Laboratory, Ann Arbor, 1. Power Output Relations Michigan. THEORETICAL POWER OUTPUT AND BANDWIDTH OF TRAVELING-WAVE 2. Power Output of Helix and Loaded Michigan. THEORETICAL POWER OUTPUT AND BANDWIDTH OF TRAVELING-WAVE 2. Power Output of elix and Loaded AMPLIFIERS, by H. Sobol and J. E. Rowe. August 1959. 51 pp. incl. Waveguide Circuits AMPLIFIERS, by H. Sobol and J. E. Rowe. August 1959. 51 pp. incl. Waveguide Circuits illus. (Proj. 4506; Task 45152) (RADC-TR-59-153) Unclassified Report. 3. Interaction Bandwidths illus. (Proj. 4506; Task 45152) (RADC-TR-59-153) Unclassified Report. 3. Interaction Bandwidths I. Sobol, H. I. Sobol, H. Expressions are developed to calculate the theoretical power output of II. Rowe, J. E. Expressions are developed to calculate the theoretical power output of II. Rowe, J. E. traveling-wave amplifiers using any type of r-f structure. Calculations traveling-wave amplifiers using any type of r-f structure. Calculations are made for helix-type tubes and it is shown how to calculate the power are made for helix-type tubes and it is shown how to calculate the power output of tubes using more dispersive structures in terms of calculations output of tubes using more dispersive structures in terms of calculations made for helix tubes. made for helix tubes. The principle factors accounting for higher power output of dispersive The principle factors accounting for higher power output of dispersive structures are presented and discussed. The gain and bandwidth of for- structures are presented and discussed. The gain and bandwidth of forward-wave helix amplifiers are derived from the small-signal theory as ward-wave helix amplifiers are derived from the small-signal theory as functions of frequency and it is shown that the gain in db times the functions of frequency and it is shown that the gain in db times the frequency bandwidth is a constant as a function of helix length for high frequency bandwidth is a constant as a function of helix length for high 7 a' and the gain times the bandwidth squared is a constant for low and the gain times the bandwidth squared is a constant for low UNCLASSIFIED UNCLASSIFIED AD ____________UNCLASSIFIED ~~~~~~A~~~~~~~~~~~~~ ~ ~ ~~~~AD UNCLASASSIFIED University of Michigan, Electron Physics Laboratory, Ann Arbor, 1. Power Output Relations traveling-wave amplifiers using any type of r-f structure. Calculationstr,elin s avelapplifiers u a te o r.icaler J A bo are made for helix-type tubes and it is shown how to calculate the power traveling-wave amplifiers using any type of r-f structure. Calculations I Row J E. output of tubes using more dispersive structures in terms of calculations ar e made for helix-ty e dispersive structures in terms of calculations made for helix tubes. made for helix tubes. The principle factors accounting for higher power output of dispersiveT principle factors accounting for higher power output of dispersive structures are presented and discussed. The gain and bandwidth of forward-wave helix amplifiers are derived from the small-signal theory as structures are presented and discussed. The gain and bandwidth of forfunctions of frequency and it iseshown that the sgain in db times the lward-wave helix amplifiers are derived from the small-signal theory as frequency bandwidth is a constant as a function of helix length for high functions of frequency and it is shown that the gain in db times the fa and the gain times the andidth squared is a constant for low rquency bandwidth is a constant as a function of helix length for high l o7~' I1 ~a ~.I UNCLASSIFIED UNCLASSIFIED

AD l____ UNCLASSIFIED AD UNClASSIFIED University of Michigan, Electron Physics Laboratory, Ann Arbor, 1. Power Output Relations University of Michigan, Electron Physics Laboratory, Ann Arbor, 1. Power Output Relations Michigan. THEORETICAL POWER OUTPUT AND BANDWIDTH OF TRAVELING-WAVE 2. Power Output of Helix and Loaded Michigan. THEORETICAL POWER OUTPUT AND BANDWIDTH OF TRAVELING-WAVE 2. Power Output of Helix and Loaded AMPLIFIERS, by H. Sobol and J. E. Rowe. August 1959. 51 pp. incl. Waveguide Circuits AMPLIFIERS, by H. Sobol and J. E. Rowe. August 1959. 51 pp. incl. Waveguide Circuits illus. (Proj. 4506; Task 45152) (RADC-TR-59-153) Unclassified Report. 3. Interaction Bandwidths illus. (Proj. 4506; Task 45152) (RADC-TR-59-1553) Unclassified Report. 3. Interaction Bandwidths I. Sobol, H. I. Sobol, H. Expressions are developed to calculate the theoretical power output of II. Rowe, J. E. Expressions are developed to calculate the theoretical power output of II. Rowe, J. E. traveling-wave amplifiers using any type of r-f structure. Calculations traveling-wave amplifiers using any type of r-f structure. Calculations are made for helix-type tubes and it is shown how to calculate the power are made for helix-type tubes and it is shown how to calculate the power output of tubes using more dispersive structures in terms of calculations output of tubes using more dispersive structures in terms of calculations made for helix tubes. The principle factors accounting for higher power output of dispersive The principle factors accounting for higher power output of dispersive structures are presented and discussed. The gain and bandwidth of for- structures are presented and discussed. The gain and bandwidth of forward-wave helix amplifiers are derived from the small-signal theory as ward-wave helix amplifiers are derived from the small-signal theory as functions of frequency and it is shown that the gain in db times the functions of frequency and it is shown that the gain in db times the frequency bandwidth is a constant as a function of helix length for high frequency bandwidth is a constant as a function of helix length for high a' and the gain times the bandwidth squared is a constant for low yoa' and the gain times the bandwidth squared is a constant for low Yoa Yoa. Yoa ~ UNCLASSIFIED UNCLASSIFIED.AD |UNCLASSIFIED | AD | UNCLASSIFIED University of Michigan, Electron Physics Laboratory, Ann Arbor, 1. Power Output Relations Michigan. THEORETICAL POWER OUTPUT AND BANDWIDTH OF TRAVELING-WAVE 2. Power Output of Helix and Loaded University of Michigan, Electron Physics Laboratory, Ann Arbor, 1 Power O AMPLIFIERS, by H. Sobol and J. E. Rowe. August 1959. 51 PP. incl. Waveguide Circuits Michigan. THEORTICAL POWER OUT AND ANDWIDTH OF TRAVELINGWAVE 2. Power utput of Helix and Loaded illus. (Proj. 4506; Task 45152) (RADC-TR-59-155) Unclassified Report. 3. Interaction Bandwidths AMPLI FIERS, by H. Sobol and J. E. Rowe. August 1959. 51 P. elport. IWavegtion dt I. Sobol, N. illus. (Proj. 4506; Task 45152) (RADC-TR-59-153) Unclassified Report. 3 Intera Expressions are developed to calculate the theoretical power output of II. Rowe, J. E. in are ev pd t al at th I Sobo traveling-wave amplifiers using any type of r-f structure. Calculations trevsing-wav am p ifi t theretia power output f. are made for helix-type tubes and it is inshown how toerms of calculate the powerns are made for helix-type tubes and it is shown how to calculate the power output of tubes using more dispersive structures output of tubes using more dispersive structures in terms of calculations made for helix tubes. made for helix tubes. The principle factors accounting for higher power output of dispersive principle factors accounting or higher power output of dispersive su r a ps e n Iscussed. TheSoainandbandwidthof forThe principle factors accounting for higher power output of dispersive structures are presented and discussed. The gain and bandwidth of for-srcueaepeendaddiusd.Tegnadbnwitofor ward-wave helix amplifiers are derived from the small-signal theory as structures are presented and discussed. The gain and bandwidth of forfunctions o f frequency and it is shown that the gain in db times the ward-wave helix amplifiers are derived from the small-signal theory as aunctions of frequency mandeit is sfowr thatthelgaix-in db tyfes tions of frequency and it is shown that the gain in db times the frequency bandwidth is a constant as a function of helix length for high freuen y bandwidth is o ao tt is shown thatot gai lat e the po we and the gain the gain times the bandwidth squared is a constant for low T.e prnil'o acutingL SS hie po e tput o at.sp r L sie UNCLASSIFIED UNCLASSIFIED

UNIVERSITY OF MICHIGAN IIII E1inIi[Ititi l 3 9015 03525 1209