ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR GENERAL DESIGN PROCEDURE FOR HIGH-EFFICIENCY TRAVELING-WAVE AMPL IFIERS Technical Report No. 24 Part I Electron Tube Laboratory Department of Electrical Engineering By Joseph E. Rowe Harold Sobol 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 February, 1958

-iiiABSTRACT A general design procedure is developed for the design of both low-power and high-power high-efficiency traveling-wave amplifiers. The process is based on the selection of optimum values (for highest efficiency) of the design parameters C, QC, B and b from the large-signal curves and design of an amplifier with the particular type of r-f structure specified by power and bandwidth requirements and operating parameters as near the optimum values as possible. In cases where the optimum design parameters cannot all be realized simultaneously the design engineer will be able to select the parameter that he wishes to compromise. The procedure is first developed for helix-type tubes and then correction factors are derived that permit the design of amplifiers with any type of r-f structure from the same set of curves.

-ivEXPLANATORY NOTES 1. Theoretical Development of Design Procedure. In Part I of this report a general design procedure is developed for the design of high-efficiency traveling-wave amplifiers. The procedure is first developed for helix-type tubes and is then extended to cover traveling-wave amplifiers with other types of r-f structures. The same design curves are used for these dispersive structures with appropriate correction factors. 2. Design Curves. In Part II all of the available design curves useful in the actual design of high-efficiency amplifiers are compiled. Immediately preceding each section of curves there is a list of parameters for which the particular curves have been calculated.

-VTABLE OF CONTENTS Title Page ABSTRACT iii EXPLANATORY NOTES iv LIST OF FIGURES vi INTRODUCTION 1 LARGE-SIGNAL RESULTS 2 DERIVATION OF HELIX DESIGN EQUATIONS 3 DEPENDENCE ON HELIX VOLTAGE 21 UNIVERSAL HELIX-DLF CURVE 28 XTENSION TO PERIODICALLY LOADED WAVEGUIDES AND MODIFIED HELIX STRUCTURES 30 PROCEDURE FOR USE OF DESIGN CURVES 37 General 37 Outline of Iterative Procedure 39 APPLICATION OF THE DESIGN PROCEDURE 50 CONCLUSIONS 53 ACKNOWLEDGMENTS 54 LIST OF SYMBOLS 55 LIST OF REFERENCES 57

-viLIST OF FIGURES Figure Page 1 Maximum Saturation Efficiency/C vs. Space Charge and Beam Diameter. (C = 0.1, d = O, A = 0.0225, a'/b' = 2) 4 2 Sheath-Helix Impedance Ks for Solid Electron Beam. Helix-toBeam Radius as the Parameter. 5 3 Impedance Reduction Factor F = F1 F2 for a Tape Helix. (Extension of Tien's Calculations.) 8 4 Helix Impedance Reduction Factor F for Various Values of Beam Voltage. (DLF = 90o) 9 5 Space Charge vs. Beam Diameter for Various Values of the Gain Parameter. (a'/b' = 1.4, Vo = 2 kv, DLF = 90%) 12 6 Space Charge vs. Beam Diameter for Various Values of the Gain Parameter. (a'/b' = 1.4, Vo = 3 kv, DLF = 90%) 13 7 Relative Injection Velocity vs. Space-Charge Parameter. (B = 1, d = 0) 14 8 b at xlmax. vs. xlmax. with C and QC as Parameters. (d = 0) 15 9 b at x1max. vs. x1max. with C and QC as the Parameters. (d = 1.0) 16 10 Space-Charge Correction. (C = 0.10, d = O) 18 11 Space Charge vs. Beam Diameter for Various Values of the Gain Parameter. (C = 0.10, a'/b' = 1.2, Vo = 1 kv, DLF = 90%) 19 12 Space Charge vs. Beam Diameter for Various Values of the Gain Parameter. (C = 0.10, a'/b' = 2.0, Vo = 10 kv, DLF = 90%) 20 13 Microperveance vs. Beam Diameter for Various Values of the Space-Charge Parameter. (C = 0.10, a'/b' = 1.4) 22 14 Microperveance vs. Beam Diameter for Various Values of the Space-Charge Parameter. (C = 0.20, a'/b' = 1.4) 23 15 Perveance Correction Factor. (C = 0.10, d = 0) 24

LIST OF FIGURES (Continued) Figure Page 16 Space Charge vs. Helix Voltgge for Various Values of the Gain Parameter. (a'/b' = 1.4, B = 1.00, DLF = 90%) 26 17 Space Charge vs. Helix Voltage for Various Values of the Gain Paeameter. (a'/b' = 1.4, B = 1.50, DLF = 90%) 27 18 ka' cot 4sh vs. ya' for a Sheath Helix. 29 19 Helix DLF vs. Pitch as a Function of Frequency. 31 20 Space-Charge Reduction Factor vs. Impedance Gain Factor for Dispersive Structures 35 21 Space-Charge Reduction Factor vs. Beam Diameter (R = h). (C = 0.1 —, C = 0.2 -, a'/b' = 1.2, V = 5 kv, DLF = 90o) 36 22 Flow Diagram for Traveling-Wave Amplifier Design Procedure. 38 23 Magnetic Field Required for Brillouin Flow vs. Stream Voltage with Perveance as the Parameter. 42 24, Input-Signal Level in db below CIoVo, vs. Tube Length at Saturation in Undisturbed Wavelengths. b is Adjusted for Maximum Saturation Gain. (C = 0.1, d = 0, B = 1, a'/b' = 2) 46 25 Relative Power at Saturation vs. Loss Factor with Space Charge as the Parameter. Uniform Loss Starting at CN ~ 0.3 and Continuing to Saturation. (C = 0.1, B = 1, a'/b' = 2) 47 26 Relative Power at Saturation vs. Attenuator Length. Attenuation Starts at CN - 0.3. (C = 0.1, QC = 0.125, b = 1.5, B = 1, d = 0.5, a'/b' = 2) 48 27 Po in db down from Small-Signal Output at Saturation vs. SmallSignal Space-Charge Parameter. (C = 0.1, d = 0, Ng = 5.5, B = 1, a'/b = 2) 49 -vii

GENERAL DESIGN PROCEDURE FOR HIGH-EFFICIENCY TRAVELING-WAVE AMPLIFIERS INTRODUCTION Both low-power and high-power traveling-wave amplifiers are used in a variety of applications and high-power tubes in particular must frequently operate at as high an efficiency as possible. Hence a simple design procedure is much to be desired. Heretofore, high-power traveling-wave amplifiers have been designed using Pierce's small-signal theory and the design then modified according to previously obtained experimental results in order to achieve high saturation output and a reasonable efficiency. In this paper large-signal results1'2'3 on the theoretical performance of traveling-wave amplifiers are used as the basis of a general design procedure to obtain near-maximum power output and efficiency at any frequency range. The large-signal curves are independent of frequency and apply to all forms of traveling-wave tubes presently in use, within the limitations noted in the developments. The chief limitation in the theory is that the electric field is assumed constant across the stream radius. This assumption is considered valid for values of the stream diameter parameter B up to 0.5 to 0.75 as indicated by experimental studies of traveling-wave tube efficiency reported by Cutler4. Other extensive experimental results on traveling-wave tube efficiency have been presented by Caldwell5 for a wide range of operating conditions and attenuator configurations. If it is assumed that the electric field varies as Io(yr) across the stream small-signal relationships can be used to predict that the efficiency will be reduced from the uniform field value by a factor 0.8 when B = 1. The true reduction factor is probably between 0.8 and O.9.

-2The large-signal efficiency curves apply to tubes with all types of r-f structures; hence, to design a high-power amplifier of a particular structure, the multi-dimensional efficiency curves must be intersected with a surface determined by the impedance vs. ya' curve for the structure. The intersection of these two surfaces defines the maximum efficiency that can be achieved with the particular structure. Helix-type tubes have the advantage of a relatively broad bandwidth of operation (2 to 1 in frequency) and have sufficient power-handling capabilities, subject to thermal limitations, for many applications. However, when very high power levels are needed the helix cannot be used and the designer must resort to other structures such as various forms of modified helices (loaded) or periodically loaded waveguides. These high-impedance structures unfortunately have inherently narrow bandwidths of operation compared with the helix. The design procedure for the helix-type tube is based on evaluating the helix impedance from the usual reduction factors on the sheath-helix impedance. In the case of modified helix or periodically loaded waveguide structures, the structure impedance is also related to the sheath-helix impedance by appropriate factors. In order to design a traveling-wave tube with the particular type of r-f structure specified by power and bandwidth requirements, optimum (for high efficiency) values of the design parameters C, QC, B and b are selected from the large-signal curves and the design adjusted to produce operating parameters as near the optimum values as possible. LARGE-SIGNAL RESULTS The large-signal curves have been computed assuming a ratio of helixto-beam diameter a'/b' equal to 2; thus yaT = 2yb' where B = yjb'. Values of

-3a'/b' between 1.4 and 2.0 are typical in experimental tubes, depending upon the type of focusing used and the means of suppression of backward-wave oscillations. Fortunately the large-signal curves are relatively independent of a'/b'. A typical set of large-signal efficiency curves in which s is a function of QC and B is shown in Fig. 1 for C = 0.1. The value of the velocity parameter b has been optimized at each point to give maximum s. It is seen that for C = 0.1 the maximum saturation efficiency occurs for QC - 0.125 and B - 1.0. Extensive comparisons have been made between the above theoretical results and experimental results from several laboratories and reasonably good agreement has been found over a considerable range of the parameters. Tien6 has also presented large-signal calculations on the traveling-wave amplifier based on a onedimensional model. DERIVATION OF HELIX DESIGN EQUATIONS In order to develop a traveling-wave amplifier design procedure it is necessary to obtain an expression for the helix impedance as a function of frequency. The impedance of an actual helix mounted in its supporting structure is calculated by multiplying the sheath-helix impedance as calculated by Fletcher7 by the appropriate reduction factors to account for wire size, wire spacing and dielectric loading. The impedance vs. ya' curves for a sheath helix of various a'/b' ratios are shown in Fig. 27. The ordinates of the curves are related to the sheath-helix impedance by K = K ~(1 + - ), (1) s s 7 where 0 = co/c 72 = P2 _ Do2 and Ks = the sheath-helix impedance. 5

-4SMAX/C 6.0 " 5.0 4.0 ~~30~~~~.0 1.5 FIG. I MAXIMUM SATURATION EFFICIENCY/C VS. SPACE CHARGE AND BEAM DIAMETER. (C O.I, d =O, An:0.0225, a'/b': 2 )

-510 0__ ___ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 90 80 o- FIG. 2 70 - 50 SHEATH-HELIX IMPEDANCE Ks FOR 40 -- - 1 - SOLID ELECTRON BEAM. HELIX-TO-BEAM o30 i- RADIUS AS THE PARAMETER 20 10 i== 8 _ _ _ _ _ - _ _ —---------------- 6- 4ok 5 2..9, __\\ i\.8,,7.6.5 24 ------------ ^ 1 I r'\ \ 1.4 0______ _______ ______________.3.2 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~~~.09 _____.08.,0 7 -------------— _ —--.06 - --- - -— 5 ___~,. 0 3 _ _ _ _ _ _ _ _ _ _ I02 3 44 1245 ^'~~

-6Since the helix retarded wavelength is usually small compared to X, Eq. 1 may be written as K'=K (2) s s 7 The wave phase velocity may be approximated as v = vo(l+Cb)/(l-Cy), where 1 + Cb - 1 - Cy; hence the sheath-helix impedance can be written as K 506 Ks' (1 + Cb) (3) s TJV 1/2 where C = the gain parameter, b = the velocity parameter, V = the beam voltage, and y = the phase constant of the r-f wave. It is known experimentally that the impedance of an actual helix is always less than K, sometimes as small as K /2. In most instances a good approximation to the actual helix impedance may be found by multiplying the sheath-helix impedance K by the reduction factor F calculated by Tien8. s Tien's reduction factors F1 and F2 are for a sheath to a tape helix and consider the effect of both dielectric loading and space harmonic fields. He has of course made some approximations and hence there is some restriction on the range of validity of his results. However, good agreement with experimental results has been found at many frequencies and for both wire and tape helices. Tien presents the reduction factors as F vs. 3 a' curves, but they may be plotted vs. ya' with voltage as the parameter in the following way.

-7The phase constant is written as P0 v ba' = - a' a. (4) Using the approximate expression for the wave phase velocity given before Eq. 3, we can write Eq. 4 as 1.977 x 10-3 Vo1/2 ya'(l - Cy) o^ l —----- I+Cb —-------- (5) Po' = 1 + Cb' Usually both Cy and Cb may be neglected in comparison with unity. Also, as pointed out earlier (l-Cy)/(l+Cb) 1. Thus Eq. 5 may be simplified to P a' = 1.977 x 10-3 * V 1/2 * 7a (6) o 0 Tien's calculations of the helix-impedance reduction factors were not extensive enough to determine F vs. ya' curves over a wide range of helix voltages. The impedance reduction factor F = F1'F2 shown in Fig. 3 was calculated for a wide range of k.a and DLF from Tien's data using b/p = 1/5 and a value of F2 assumed independent of kia. The data shown in these figures are then used to calculate curves of the helix-impedance reduction factor vs. ya' for particular values of beam voltage and dielectric loading factor. A typical plot of F vs. ya' is shown in Fig. 4 for a DLF of 90 percent. Using the reduction factor F calculated above gives the actual helix impedance K as K = F K = 506 F Ks' (1 + Cb) (7) S V 1/2 o

-8NARROW TAPE HELIX ll I 1.5, 8/p I/5 1.0 D.L.F _ _ __I. 0.9 100 o,_I 0. 0. 95 a__. Il I I I I w O.6 9I G. _IMI R EDCIO F C I R _ I I I I I I. I 0. LL 0.3 0. kilo TIG. 3 IMPEDANCE REDUCTION FACTOR F~ F' F2 FOR A.;........... OX.:::.II4I I +I::.!;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::I::TI:II t- - ii T ~i il~ lilllilllilli lilllii iiili I I I llT..

-9Lij 0..___....n. _. I I 1 -- - - f[ y~ / // / j / - -- --- -- N I/ ~ I/ I I / I/II~ I I L 0 ILL,g~~~c Z0 i 0 0.o o O',I NIo

-10The traveling-wave-tube gain parameter may then be written as F K I s o C3 =,, (8) 0 where I = the d-c stream current. Combining Eqs. 7 and 8 yields an expreso sion for beam perveance in terms of the helix impedance and the gain parameter: F 7905 C3 ~F K' (1 + Cb) ) ( ~~~0 106~~~~~~~~~~ where P = - 3/ * 106, beam micropervs. V, 3/' 10 The small-signal space-charge parameter QC may be expressed in terms of the gain parameter and the radian plasma frequency as 1 t/D 21) Qc = 4^C (7+ /), (10) where c q/ = (% /w) R = an effective normalized plasma frequency. The normalized plasma frequency is given by 2 Tl I =. bp o (11) ((D X "E b'2 U0 2 where r = e/m, the charge-to-mass ratio for the electron, and the beam diameter or space-charge range parameter B is B = b' = b' o(1 - Cy) b' (12) 0

-11Then the normalized effective plasma frequency w q/C may be written as ^TEp R pYh%2 (1 CY)2 (rp W 2 ICE (2)3/ TI/2 B2 where P = I /V3/2, the beam perveance. Equation 10 for the space-charge parameter thus becomes QC = (14) [B _ Ks 1-/2 (1 + Cb)1/2 +2 1.8 [-Rh C ( 1 - Cy X Again 1 + Cb - 1 - Cy; also, the second term in the denominator of the above equation is usually small compared to the first. Hence Eq. 14 may be simplified to QC. --- 60o (1 + Cb) (15) Ks' 1/2 2 KI C j) + 15.5 C] A further simplification of Eq. 15 can be made by neglecting Cb compared to unity. This approximation is usually valid for values of C up to about 0.1. Figures 5 and 6 are typical graphs of Eq. 15, with ordinates of QC/(l+Cb). Each group assumes particular values of a'/b', V and dielectric loading. Figure 7 is a plot of the optimum value of the electron velocity parameter b vs. QC both for maximum small-signal gain and for maximum saturation efficiency when B = 1, d = O. The value of the injection velocity parameter that gives maximum small-signal gain is shown vs. xl for particular values of QC and C in Figs. 8 and 9. It is noticed that the effect of increasing the loss factor is to shift the entire set of curves to the left9. The value

-120 X-. < - i z 0 II I_ I I I I I l I I I. I III as\ I00~ m to <. O / / t -/-l\ \ \ \D cc o o (qS + I )/30'338tH3 0 (q9 U. > IC cri o 0 ND ^0 < (M 0

I I I I___ I I I /_ _ \ __ I ____ ILL I \ \.. o. Q...... kr) / A-\\. Z I I II__ I,.oI__.... \\ \_ \ _ 0 _ _..._ __,..__,, > - >I _ II I I I i I | L J L!o' o C, 0 t- N o O O o o o o (qD + I )/O0 *'39aVHO 3.VdS _ ~, a ~ 0 ii i i >

~~~3.4~~-143.2- b- ADJUSTED FOR MAXIMUM Xi -- b - ADJUSTED FOR MAXIMUM 7 0 C = 0.2 2.8 2.6 ------ 0 / c o0.I I > 2.4 o O 2.2 C= 0.2 / \- 2.0 C -- -- -- -- --- / - - -- ^\ — — C —0-05 ~~~~~~~~~~~~~~~o I o.os 0. I C=. 0 1.6 Lii z 1.4 i. 2 I C=0.05 1.0 0.8 0.4 0.2 SPACE -CHARGE PARAMETER, QC FIG. 7 RELATIVE INJECTION VELOCITY VS. SPACE-CHARGE PARAMETER. (B:I,d=O)

-153.4 4 - ----...... -------- QC I!.o i - 75 / 6'CJ 0.60l' 7 I 3.2 -- --- -- --- -- --- - -— f- t —+-~/ ~~ —-/ oc-o6o - -.. 3.0 _ _ 2.8iC = / 0.50 26 -- -- -- -- -- -- -i./1 -'/- -- 24 4__ _ -- i/ ___ _I. -- /0 /30'G.C 8 T XlMAX VS.Xl MX. ITH ANDO SPAAMEC ER0.20 2.0 Ci I, C= 0.15 1.8,,_,/Ill c o QC= 025 1.6C: O. 00 __\_ __ _00._125 c: 0.10 i/..0 C 0.08 0.8 ---— \ \ \ __ -_QC =0.05= 0.05 0.6 ~~..O\ QC\=O 0. 0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 X, MAX.'IG. 8 b AT XI MAX. VS. XI MAX. WITH C AND QC AS PARAMETERS. (d 0)

-16 - 3.6 ------ QC =1.0 I- I I I L/ I __ I_ I_ I_ __ __I 3.4 3.2 3.0 2.6 J l [ / l t | | FQC 0.5 4 / /_ C z 1.6r - I-C I I I. I I I I IOC 0.25 i 0. 0.46C-~ __AC —O 0. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.| 1.2 1.3 X. MAX. FIG. 9 b AT X1MAX. VS. X(MAX. WITH C AND OC AS THE PARAMETERS. (d: 1.0)

-17of b for maximum small-signal gain changes very little with increased loss but the value of xlmax. decreases nearly linearly with C at fixed QC. The data shown in Fig. 7 permit computation of the 1 + Cb correction factor and construction of a QC vs. QC/(l+Cb) curve (Fig. 10). It should be noticed that the correction factor is essentially independent of the velocity parameter b. In Figs. 11 and 12, QC is plotted vs. B using the approximate form of Eq. 15 and also using the exact expression given in Eq. 14. Points have been calculated using this exact expression for QC for two different values of the velocity parameter b, (1) the value of b that gives maximum smallsignal gain, and (2) the value of b that gives the maximum saturation efficiency. The 1 + Cb factor is important only for large values of B (beam diameter) and C (gain parameter) and may be omitted when B < 1.0 and C < 0.1. Comparison shows that the agreement between the exact and approximate expressions is essentially independent of a'/b' and V. It has also been found to be independent of the value of DLF. That the results are essentially independent of the value of b chosen is due to the fact that there is a factor 1 - Cy in the equation that nearly cancels out the influence of the 1 + Cb term. The y used is a negative number and varies as b does. An alternative expression to Eq. 13 may be found by solving Eq. 10: ) =.2C (QC)1/2 (16) [1 - 2C (QC)1/2] h Equating the expressions for wp/w in Eqs. 15 and 16 and solving for P gives

-181.6.4 1.2... 1.0 CO / w 0.6 0.4 MAx. / 0.2 MAX. X 0 a2 0.4 0.6 0.8 1.0 1.2 I Cb FIG. 10 SPACE-CHARGE CORRECTION. (C 0.10, d O)

-19-T 0 0 00 o a, 1~a.: z II ________ ____ _-___ \HLI.n 0; 0, < 0 T \1 I n __ ____ __ __ __ __~ —. cr C i ft m ~ 0I L CD J,I > 0 0 o oU. + 1 )/s'398 IC3(a_________ 0:_ 0._____________________ 0 C.) Ja:cr~~~~~ cr^~~ c~a 00 U0 ---- o o2 x 0 o o 0 o (qO+I)/3O'3sdUHO 33VdS

-200 LJ L3 a:: z u, -- 0z 0 LL --- -z o _ _ —' - - w IAJ o H L) C). o UC. +>+ m a. -I MEo 0 w LU > x d w ~o 13.. I'o CI W 0 U 0 w 0 < o d A x CLi (qO + I )/30'39toVHO 30VdS

-21p 132.11 (CB)2 (QC) (17) 1 - 2C (QC)1/2j R2 (1 - Cy)2 Equation 17 may be simplified by assuming that the quantity (l-Cy)2 may be replaced by (l+Cb)2: (1 + cb)2 p = 132 (CB)2 (QC) [1 - 2C (QC)1/2] 2 From the above expression for P, curves of the beam perveance vs. the beam diameter B are computed and shown in Figs. 13 and 14 for representative values of the parameters. It is seen from Eq. 17 that these curves are independent of V and dielectric loading. A correction factor for use in determining a more accurate value of the beam perveance when Cb is not small compared to unity has been calculated assuming that the quantity (l-Cy)2 in Eq. 17 may be replaced by (l+Cb)2. The perveance correction factor curve is shown in Fig. 15, where it is seen that for values of QC less than 0.5 the correction factor is critically dependent upon the value of b chosen. DEPENDENCE ON HELIX VOLTAGE The basic design curves of Figs. 5 and 6 show the variation of QC/(l+Cb) with the beam diameter for particular values of the helix voltage V. No curves of this type were calculated for values of QC > 1.0, since the saturation efficiency decreases rapidly at large values of QC. In the curves shown it can be seen that for a given helix voltage VO and a particular value of the beam diameter B the space-charge parameter increases

-22I i 4.0 3.5 3.0 2.5 ---.... -- QC O.50 U. - / > 000 / i^j I /QC 0.35 [.5 0QC= o.125 0 0.5 1.0 1.5 2.0 BEAM DIAMETER, B FIG. 13 MICROPERVEANCE VS. BEAM DIAMETER FOR VARIOUS VALUES OF THE SPACE-CHARGE PARAMETER. (C = 0.10, a'/b' 1.4)

-234.0I I I 40 - -'/, 3.5 / /X 3.0 QCO 0.15 i,' 2.5, i i i lr i I —- ^ —-- QC = 0. 125 o _ U QC=0.0 075 _ 0-5 QC:0.05 ____ 0.5 0 0.5 1.0 1. 5 2.0 BEAM DIAMETER, B FIG. 14 MICROPERVEANCE VS. BEAM DIAMETER FOR VARIOUS VALUES OF THE SPACE-CHARGE PARAMETER. (C = 0.20,'/b' 1.4)

-240.98 _ II__ ___ __ ___ IFIG. 15 0.94 __ - l l l lPERVEANCE CORRECTION FACTOR (C 0.10, d O) 0.90' " MAX. X, 0 86. 0.82 ND.'t8~~~~~~~~~~~~' +'N MAX. 7 s 0.701 - 0.66 --------- _- --- ~=eA 0.62.' 0 0.2 0.4 0.6 0.8 1.0 1.2 OC

-25rapidly as the ratio of helix-to-beam radii is increased. Since b' is fixed by B, this implies that QC increases as the helix radius is made larger. As long as the gain parameter is maintained constant the increase in QC must be due to an increase in the value of the plasma-frequency reduction factor, and hence an increase in the effective plasma frequency. In the limit, when the helix is removed to infinity, the reduction factor is unity and cq = u. i p Typical values of B for maximum theoretical efficiency in existing tubes are around 1.0, and it is not desirable to have a'/b' > 1.4 to 1.6 because of the resultant high values of QC and consequent lowering of efficiency. The lower limit on a'/b' is determined by the degree of beam interception and the tendency to backward-wave oscillate due to extremely tight coupling between the circuit and stream. The dependence of the space-charge parameter on the helix voltage can be seen by plotting QC/(l+Cb) vs. V for particular values of the beam diameter, as is done on an expanded scale in Figs. 16 and 17 for DLF = 90 percent. From these results it is apparent that the variation with voltage is most significant at large beam diameters. When B = 1.0 the space-charge parameter increases by approximately 40 percent for a 10-to-l change in voltage. As the voltage is increased the term FI in Eq. 8 must increase at the same rate in order to keep C constant. Since B is fixed Rh remains constant. The impedance reduction factor F decreases as V is increased at fixed ya'; hence I must increase at a greater rate than the voltage does to keep C constant. This increase in I results in an increasing QC with voltage. The variation of F with B is greatest at relatively high values of B. The maximum helix voltage that may be used (which determines the retarded wavelength) is dependent upon several factors and therefore cannot

____ — 261L. e1 I. --------- < ~~I L&J ) l1l1 vto (K i i cL 0. lf~l lC| I. t(D > LJ 0C) 0 1 ~ 1 ~ J 0; o 5 0 0 s iq9+l)/90'398oc _ _ _ I I I 0: __ __- - - - --- 0 o d 0 0 o d I (q -+ ) /30'39VH0 3 )VdS

-27LJ ___-\^ __ __ 1 1 w E _0 (D <0 ~n in ^ < _X _ ) -- > -J O 0 0 \ \> II I I\ I ( O0> 0 > 0 3D < I O OXO O O ~ ~ ~ O O o C) a(qS + I )/90'~s~rti~c 39~1dS

-28be rigidly set. As the operating helix voltage and helix diameter are increased at any given frequency, the pitch of the helix increases, and it can be shown that the fundamental impedance is lowered and the effect of the harmonic fields is increased. The presence of a dielectric or metal shield surrounding the helix also influences the maximum usable helix voltage. Of course some of these difficulties can be overcome by using multi-filar or cross-wound helices. Because of the above adverse effects, the maximum useful operating voltage for a unifilar helix is between 10 and 15 kv. TNIVERSAL HELIX-DLF CURVE One of the two reduction factors used in determining the actual helix impedance from the sheath-helix impedance depends upon the DLF. The DLF is defined mathematically by Tienl~ for helices of the same radii and ya' as cot ahwd DLF = ahwd (19) sh where (ahwd) refers to the actual helix with dielectric and (sh) refers to the sheath helix in free space. Since the pitch distance for any helix is given by 2ia'vo/c, Eq. 19 may be written as 2ra' DLF = (pitch) cot sh (20) The DLF for a particular helix may be calculated from Eq. 20 using the value of cot sh found from Fig. 18 for a particular ya'. Both yat and ka' may sn

-297, __ 0 I 23 5 6 7 F I COT VS. FOR A SHEATH HEIX. O 1 2........ /_ —_ _ _ —--- I.8 -C ts S F7AH T H I

-30be calculated on the basis of a measurement of the phase velocity along the actual helix in the presence of the dielectric at a given frequency. This measurement may be accomplished by any of the standard techniques. In the design of traveling-wave tubes it is necessary to calculate the pitch or turns per unit length of the helix assuming a value for the DLF. Equation 20 can be written as DLF 2ra' ka' pitch (ka' cot s)'21 sh With the aid of Fig. 18 and Eq. 21, a universal DLF curve is plotted. This universal curve, Fig. 19, permits calculation of the DLF from experimental data on the helix phase velocity and also calculation of the helix pitch for a given DLF and (ya') ah EXTENSION TO PERIODICALLY LOADED WAVEGUIDES AND MODIFIED HELIX STRUCTURES In applications of traveling-wave tubes where the bandwidth needed is less than can be obtained with a nondispersive helical slow-wave structure but the peak or average power requirement is greater than a helix tube can yield due to thermal limitations of the helix, it is frequently desirable to employ some one of the varied forms of the periodically loaded waveguide or other dispersive structure. For many of these structures, such as the contrawound helix11 structure, the impedance vs. frequency relationship may be expressed as a constant times the sheath-helix impedance. The procedure developed here will be to relate the structure impedance to the sheath-helix impedance by an impedance gain factor, G, which in general is a function of frequency. This is similar to the determination of the actual helix impedance from the sheath-helix impedance using the impedance reduction factor.

-31__\~\ A o 0'\XfN\ \\\ \O \~~ z C c)0 0 < good CI ~ B I 4o i i Io N O -^i&Cr ^^A) — -^\^vt^ -^-^S^SS^W - t~~~~~~~~~~~'^S\S~~~~~~~~~~va^^~~~~~1 -s^- - — ^ —^^-^-^\\^A-\ \ \ \\\- — ~ 0=~~~~~~~~~~~~~~~~~~~~~~~~~~~ ^ \ W^V^VN^ -^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ " ^s, _ _\ \ ^AX \ \\l\ - ^ X ^\ \ YXV, \^\\^\ SQ: ^~~~~~~~~~~~~~~j b )hs^' d""^ " NVX"' IL " SI" -" " " ^" (M --— x^ ^ -

-32It is important to point out once again that the large-signal results apply to traveling-wave tubes independent of the r-f propagating structure as long as these structures can be represented by an equivalent circuit that is an LC transmission line even though L and C may both be functions of frequency. This is true because the circuit serves only to present a longitudinal electric field at the beam position and retard an electromagnetic wave. In the longitudinal theory the helix has been assumed nondispersive. In the event that the resonant or dispersive structure being used has an impedance curve that is not approximately parallel to that of the sheath helix, then G will be a function of ya' and hence of B. The impedance gain factor will be defined in such a way that it takes into account the effect of the mounting method on the impedance. The structure impedance is written as K GK, (22) d s where G = the impedance gain factor expressing the ratio of the structure impedance mounted in the vacuum envelope to the sheath-helix impedance. Proceeding through the derivation using the above impedance relation and assuming the same values of C and B for the dispersive structure as for the helix yields, in place of Eqs. 14 and 15, (QC)d [B,6 / + (23) B Ks' (1 + Cb) + 15.5 C and

-33(QC)d 60 (1+ cb) (2) cC)dK, -2a4) [B RdG K) + 15.5 C] For periodically loaded waveguide structures the dielectric loading factor is unity and the geometric reduction factor on the radian plasma frequency is different from that for the helix tube. The plasma-frequency reduction factor will generally be larger for loaded-waveguide structures than for the helix, since the effective circuit radius will be somewhat greater than the radius of the center hole in the structure for the beam. The ratio of (QC)d to (QC)h can be written as [ Qc 1/2 1 h [ i Cd 1/2 v F1/, (25a) 1 + Cb i 1/2 If C 11/2 [ 15.5 c R J1 Q jh v F/ where (Rd) F (25b) ^Q Ksi'A/2' (25c) Rh = the radian plasma-frequency reduction factor for a helix of equivalent B and a'/b', and R = the radian plasma-frequency reduction factor for the resonant or dispersive structure. Since the second term in the denominator is small compared to unity, Eq. 25a is expanded in a power series, neglecting higher-order terms. The binomial series is used then to give the following expression:

-34[r Qc1v 12 1 + Cb1 1 b~= [l+ 3 Q )(l )] (26) 1 + Cb The space-charge reduction factor for dispersive structures, i, defined in Eq. 25b, is shown in Fig. 20 for particular values of G and Rd/Rh. It is evident that the first-order correction term in Eq. 26 is negligible in most instances. If the radian plasma-frequency reduction factor is the same for the dispersive structure as for the helix, Eq. 26 may be written as Ll-+ — G = t * (Rd =Rh. (27) QC 1 1 + Cb h The value of G above includes the effect of dielectric loading when structures such as the contra-wound helix are used. Thus G is analogous to F G*, where in the presence of a dielectric F is the same as for a helix and G* is the impedance gain factor. In Fig. 21 is shown a graph of Eq. 25a for representative values of G and Rd = Rh. It is apparent that the F/G approximation is very good except for large values of B. The correction factor is somewhat dependent upon a'/b' but not on the voltage or DLF. The effect is also shown for large C. Since the stream perveance, as expressed in terms of the design parameters in Eq. 21, is a function of the reduction factor R, there will also be a perveance correction factor for the structures where Rd / Rh. The perveance expression for dispersive structures is, from Eq. 17, 132 (CB)2 (QC)d (28) [l - 2C (QC)l/2] Rd2 (1 - Cy)2

-35( + QC FIG. 20 i + Cb d..' (-I +CGb)h XSPACE-CHARGE REDUCTION FACTOR n_ " QC (IhC h VS. IMPEDANCE GAIN FACTOR SPACE-CHARGE 2.4 F_______________ 2.2 1.8 5 1\\4 1 1 1; i ] i J l 1.2 1.0 0.8 0.6 ).5 0.5.O 1.5 20 2.5 3.0 3.5 4.0 G, IMPEDANCE GAIN FACTOR

-36o L I I In-'-F _* _- ii II _ < - - LL I I I I I I I I I _g,__. II, 1......_,.. __ _..... z cr n- t4 __ 3 L~ a a~ 1 ro5H-0d ^^ es~] tW ~il- _I_ I CID a: I _ I 0 0 m Q10 0 0 0 iii ni ii Is IL~~~~ - - -': o 0 0 30~ 39 VHO-30VdS SP -_ to

-37where (QC)d = ( (QC)h, the space-charge parameter for the dispersive structure, and Rd = the plasma-frequency reduction factor for a dispersive structure. Since the QC/(l+Cb) curves and the perveance curves have already been constructed for a helix structure, the design procedure will be to select the desired values of C, QC and B from the large-signal efficiency curves and then determine the equivalent space-charge parameter for the helix in order to use the existing curves. Thus the perveance curves are entered using (QC)h and the perveance correction factor is given by Plh (1 + Cb)2 Rd9 PROCEDURE FOR USE OF DESIGCT CUUvES General Usually the desired power output (pulsed or cw) and operating frequency range are determined by the proposed use of the tube. The large-signal analysis as previously discussed is independent of the operating frequency and the operating structure and hence has a wide range of application. The choice of a particular structure is usually made on the basis of the power level expected and the desired bandwidth. The design procedure is here outlined for a helix-type tube, but can be adapted for the design of other traveling-wave tube structures using the space-charge and perveance correction factors developed in the previous section. A flow diagram illustrating the design procedure using the three classes of curves, namely the efficiency, space-charge and perveance curves, is shown in Fig. 22. The procedure involves an iteration technique around

-38LJ -JI ~Z I - MICO < z U c o Om Z u i L) > > uL Cl) CY > liJ IAJ - ^ -V. L(n cr C " 0 0 o < =) $: Li -rLO I Dr 2, U -. ~< I I >cr - <::3 J - - > > Z L -O' 0 o5 __ _ _ 00.-___ I Un Q S^S a 03.W cr Cr 0 D ao I O: 13:: 0 a. —---— LL (Ir 13::.

-39the loop indicated in the above figure. From the design requirements a structure and means of mounting is selected which in the case of the helix then determines the values of a'/b' and DLF. Also a realizable value of the gain parameter C is chosen consistent with operating frequency and efficiency requirements. This value of C is then used in conjunction with the efficiency curves to tentatively decide upon values for QC and B. The space-charge and perveance curves are then used to determine if the chosen parameters can be realized. If the combination of parameters selected is not realizable for the chosen r-f structure, a new value of QC associated with the perveance is taken and the iterative loop is traversed again. This process converges rapidly to a design consistent with physical limitations and design requirements. Outline of Iterative Procedure Once the power output, frequency range and type of structure have been selected, the procedure is according to the following steps. 1. The design is based on the given power output (P 0). 2. The large-signal efficiency curves, where efficiency is presented as a function of QC, B, d and b for various values of C, are used to select the operating parameters desired and to find an efficiency. Usually the d = 0 curves are used and then values of C, QC, B and b are found. The correction for loss is included at a later point. An idea of the ideal operating point is obtained from these curves; it now remains to see if these ideal parameters can be achieved simultaneously using a particular type of r-f structure. Values of B and QC are selected from the large-signal curves to give maximum saturation efficiency. There is a fairly broad plateau

-40in the efficiency curves around the maxima so that there is some freedom of choice. High gain per unit length and high efficiency require as large a value of C as can be realized. The selection of B = yb' = ya' (b'/a') is also dependent upon the operating bandwidth desired, since the bandwidth for constant gain is dependent upon ya'. For broad-band tubes the optimum ya' is around 1.8 and for highest saturation efficiency B - 1.0. Using these values of B and ya' would dictate a b'/a' of 0.56, which would result in a weak interaction between the circuit and stream, thus lowering the impedance C and the gain. A B = 1 and bl/a' = 0.7 would give considerably stronger interaction and result in ya' = 1.43. It has been found in this laboratory, experimentally, that operation over a 1.5 to 1 frequency range can be obtained at s-band with this value of ya'. The tendency for a particular tube to backward-wave oscillate and the feasibility of using mode suppression techniques will determine the maximum usable b'/a' ratio. Values of C, QC, B, b, a'/b' and DLF are tentatively selected. Then from the large-signal curves a theoretical saturation efficiency is found. Thus the beam power may be calculated on the basis of the theoretical efficiency and the power output P. Practical values of a'/b' lie somewhere between 4/3 and 2. It was pointed out earlier that the large-signal results are relatively independent of a'/b'. 3. The next step in the design procedure is to consult a curve or nomograph of V vs. I with beam perveance and beam power as the parameters. From this curve the other beam quantities can be determined

-41using the beam power as calculated in (2) above and assuming a value of beam perveance or beam potential, since the beam power and an assumed value of perveance will automatically determine V and I. On the other hand, the beam perveance and beam current are also explicitly determined by the beam power and a chosen value of V. Whether beam voltage or beam perveance is selected as the independent variable depends upon the limitations encountered in the electron gun design. The perveance is in general limited by the type of electron gun used and the limitation on emission density imposed by the duty cycle. The beam voltage, Vo, is usually limited by power supply requirements and voltage breakdown between electron gun electrodes. In applications of traveling-wave tubes where minimum weight of the tube and its solenoid are of primary importance, the type of focusing to be employed will be influential in the choice of electron gun perveance and voltage. The minimum magnetic field required for focusing (Brillouin flow)12 is shown in Fig. 23 as a function of the voltage and perveance. For magnetic field strengths greater than that required for Brillouin flow the space-charge wave propagation could be affected and QC increased through an increase in the magnitude of the plasma-frequency reduction factor13. 4. The next step is to check the assumption on QC, B and C. Several steps are required. First decide whether the design is to be based on b for maximum small-signal gain or on b for maximum saturation efficiency. This determines the (l+Cb) correction factor. It should be recalled that ya' is equal to a'/b' times B. It is well known that ya' is selected on a compromise basis since both wideband operation and high circuit impedance are desired simultaneously.

-42-o 0 ~ —V T m~W Y^\ \\\i\i lE0^"v_ _zz 0 1 > T 0 VY:-\ \ Z<_____ 0 _ I_ _ 1_11\\_ \\\_ _\_ 0\1 \ __ ___ _ 0 0 1 0 \ 0_\\_ _ 0\- - Zz _ __ c'J 0 0 0 0 0 01- 00\ - (NI-SS ) \\ \\

-43a. Denote the temporary values of the design parameters selected in (2) and (3) above by the subscript 1. If an r-f structure other than the helix is involved then the value of QC used is the value taken from the large-signal curves divided by the QC for a helix tube. Thus for nonhelix structures (Qc)1 = QC b. Use (QC)1 to find the perveance correction factor from Fig. 15. Then P P (l+Cb)2 = 1 Al. (ordinate of Fig. 15) c. Enter the appropriate curve of the type shown in Figs. 13 and 14 with P (l+Cb)2 and B1 to find (QC)2. d. If (QC)2 C (QC)1, then change to a P (l+Cb)2 such that (QC)2 = (QC)1 for the same B1. Using (QC)2 determine P2 from Fig. 15. e. Find Io2 and V02 to give Pol. f. From the appropriate curve of QC/(l+Cb) vs. B, find (QC)3/(l+Cb) for Vo2 and C1. g. If (QC)3/(l+Cb) / (QC)1/(l+Cb) then select (QC)3/(l+Cb) and find (QC)3 using the correction contained in Fig. 10. h. Find P 3(1+Cb)2 using (QC)3 and B1. Use Fig. 15 to get P3. i. Find Ss2 from the large-signal curves using (QC)3, C1 nnd B1. j. Calculate Pb2 to give Po using rS2 as the efficiency.

-44k. Find Vo3 and I03 using Pb2 and P 3. 1. Compare V03 and V02; if they differ appreciably determine (QC)4 at V03. m. Repeat the above-outlined process until Vo3 - Vo2. This is usually not necessary after some experience in using the curves is acquired. If success is not achieved after several iterations, then the values of B and C selected are not consistent with the desired power output. In this case compromises are obviously necessary. Usually the difficulty will be remedied by increasing B, which has the advantage that for a fixed value of a'/b' the value of ya' is increased. 5, The value of y corresponding to the value of b chosen earlier is found from the small-signal solutions. Since the beam voltage has been determined and the center frequency is known, the value of X and hence the value of a' the mean helix radius, can be calculated assuming that the phase velocity of the r-f wave is given by 1 + Cb o 1 - Cy From this point on the design procedure will be directed toward helix-type tubes. Other r-f structures will require different methods for determining actual structure dimensions. 6. Since a'/b' has been fixed, the beam radius b' is also determined and the stream current density is specified. 7. The value of \ found under part 5 and the value of ya' are used to g find the helix T.P.I. from Fig. 19.

-458. The helix wire diameter is usually chosen to be somewhere between onequarter and one-half the pitch distance. At s-band it has been shown experimentally and theoretically that there is a broad minimum in the attenuation curve for a ratio of wire diameter to pitch around onethird14,15. 9. The first estimate of the length of the helix structure in terms of N may be found from the large-signal curves where d = O. The optig mum circuit length as a function of the input-signal level is shown in Fig. 24. The input-signal level is referred to CIoVo and the saturation level is approximately 5 db above CIOVo. The large-signal curves for which d 4 0, shown in Figs. 25 and 26, may be used to estimate the added length in terms of N's necessary to recover the gain lost in the attenuator region. These curves not only show the loss in saturation Dower due to attenuation but also indicate the effect on the saturation power level of the placement of the attenuator. 10. An alternate method for determining the helix length for a given useful gain is to use the small-signal gain equation G = A + BCN. Since g the value of C has previously been determined and A and B may be found from Pierce's small-signal theory, the tube length for a given lossfree small-signal gain may be computed. The large-signal (saturation) gain in db down from the small-signal gain may be found from a largesignal plot of the saturation power output in db down from small signal for the particular values of C, QC and B previously selected. The reduction in saturation level due to large-signal effects is shown in Fig. 27.

-46FIG. 24 I, INPUT-SIGNAL LEVEL IN db BELOW C IVo, VS. TUBE LENGTH AT SATURATION IN UNDISTURBED WAVELENGTHS. b IS ADJUSTED FOR MAXIMUM SATURATION GAIN (C 0.1, d 0, B I, a/b' 2) 0 o!.._____.______ 1_ I L1 301 40LQC ~~QC M \ \ C 0.05 0 ____!o i'...... TUBE LENGTH, Ng, IN UNDISTURBED WAVELENGTHS, _____bl.Y5 bl.5\...... 20 _____kC=O TUBE LENGTH) Ng: IN UNDISTURBED WAVELENGTHS

-47W Z 00 U o 0':'' 0 o 0! ___ ____ ____ ____., CT__ 0d 0 ~ ~- I —- ) - / / ~~~co D r — I0H w Z W Z Z __~ _____ rN c0~~~y i N 0'A 0'A 0 0 qP'NOIJ.VfnliVS IV d3MOd 3AIlV"138

" " __ __ I_ __AX r' — Cr -3 15 L&J 0 I 2 3 4 5 6 7 8 9 0 11 12 LENGTH OF ATTENUATOR,Ng FIG. 26 RELATIVE POWER AT SATURATION VS. ATTENUATOR LENGTH. ATTENUATION STARTS AT CNN 0.3. (C OI, QC O0.125, b 1.5, B:1, d 0.5, o'/b' 2.)

-49Z 0 I0 <[ 0 I —J 2 0 0 a.o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SMALL-SIGNAL SPACE-CHARGE PARAMETER,QC FIG. 27 Po IN db DOWN FROM SMALL-SIGNAL OUTPUT AT SATURATION VS. SMALL-SIGNAL SPACE-CHARGE PARAMETER. (C 0.1, d=, Ng 5.5, B I, a'/b' 2.)

-50APPLICATION OF THE DESIGN PROCEDURE In the previous section the general procedure was outlined for the design of either pulsed or cw helix-type traveling-wave amplifiers to operate at as high an efficiency as possible. The procedure is applicable to either low-power or high-power tubes but has greatest value in the design of highpower tubes. The following is an illustration of the use of this procedure in the design of a cw s-band amplifier to operate at high efficiency over a broad frequency range. The design objectives are to achieve at least 200 watts cw output at a 30-db large-signal gain using a wire helix and a solid electron beam. The following steps outline the design of the desired tube. 1. The design is based on P = 200 watts and the objective is to obtain as high a saturation efficiency as possible. 2. From the large-signal efficiency curves the following desirable operating parameters are obtained (to give maximum efficiency): C = 0.1, QC = 0.125, B = 1.0, T = 55%, b = 1.5, d = 0. The following parameters are assumed: a. Ratio of helix-to-beam radii, a'/b' = 1.4. b. DLF = 90 for a helix supported in glass by three quartz or sapphire rods. c. It is desirable to have 1.5 < ya' < 2.1 for broadband operation. The beam power is then calculated to be 200 P = O0 = 364 watts. 3. The beam perveance is assumed to be 1 x 106, which gives the following values of V and I for Pb = 364 watts:

-51V = 2650 volts Io = 137 ma. 4. The next step is to determine whether the desired operating parameters can be achieved with a helix-type tube. Denote the assumed values by the subscript one. a. (QC)i = 0.125, C1 = 0.1, B1 = 1.0, Vol = 2650 volts and Iol = 137 ma. b. From Fig. 15 for (QC)1 = 0.125, P1,(l+Cb)2 = 5 = 1.32. c. From Fig. 13 it is found that (QC)2 = 0.25 $ (QC)1. d. Since (QC)2 i (QC)1 the perveance is changed so that (QC)2 = (QC)1 for the same B1. This give Pl2(l+Cb)2 = 0.625. Using (QC)2 to determine P!2 from Fig. 15 gives 0.625 PL2 = 0.758 = 0.825. e. For P1.2 = 0.825 and Pb = 364 watts, Vo2 = 2850, I02 = 128 ma. f. Using C1 = 0.1 and Vo2 = 2850 and Figs. 5 and 6 gives (QC) = 0.18. Note that (Q)b 0.11. l +Cb'+Cb g. Since (QC)3/(l+Cb) $ (QC)l/(l+Cb), (QC)3/(l+Cb) is selected and (QC)3 is found from Fig. 10. (QC)3 = 0.202. h. Pg13(l+Cb)2 = 1.04 from Fig. 13 using (QC)3 and B1. Thus P(3 = 1.04(0.781) = 0.811 from Fig. 15. i. From the large-signal efficiency curve shown in Fig. 1, (QC)3, C1 and B1 will give an efficiency rS2 = 45*.

-52200 j. Thus Pb2 = = 444 gives Vo3 = 3120 volts and I03 = 143 ma. k. Vo3 = 3120 volts is not equal to Vo2 = 2850 volts. However, the values of QC do not differ appreciably; hence the process is assumed to be complete. Thus the final design parameters are: C = 0.1 VO = 3120 volts QC = 0.202 I = 143 ma. B = 1.0 P = 0.811 ya' = B(a'/b') = 1.4 b = 1.32. Throughout the design procedure compromises in QC and hence efficiency were made in order to obtain C = 0.1. 5. The helix voltage corresponding to synchronism is calculated using 1 + Cb = 1.132 from the above material. V(v)= o 3120 V(o) = (1Cb = (132) = 2430 and 5.93x5 (2430)1 x 102 = 0.977 cm. g 3x108 The mean helix radius is then found from a' = = 1.4. g a' = 0.218 cm. or 0.0859 inch.

-536. The beam radius b' = 0.7 (0.0859) = 0.060 inch. The beam current density is calculated as I b = = 1.94 amps/cm2. c (bt)2 7. ka' = 2 = 0.1375 and thus - = 0 7 = 6.55. From Fig. 19 for % ka= 0.1375 ya' = 1.4 it is seen that the helix T.P.I. = 18.55. 8. The helix wire diameter is selected to be Pitch 1 ^ire = -3 = 3(T.P.. ) = 0.018 inch. No. 24 wire, which is 20.1 mils in diameter, will be used. 9. The helix length for d = 0 is estimated from Fig. 24 to be N = 18 to obtain a gain of 60 db assuming a 30-db attenuator. Thus there is a gain of 3.33 db/g. The effect of the attenuator in reducing the g saturation gain is shown in Figs. 25 and 26. If the 30-db attenuator is made approximately 2 inches long, Fig. 26 indicates there will be no reduction in the saturation level. Additional curves of this type are being calculated for other values of C, QC and d. The helix length is L = 977 (18) 7 inches. CONCLUSIONS A general design procedure has been developed that facilitates the design of both low-power and high-power large-signal traveling-wave tubes to

-54operate at high saturation efficiency. The design procedure is directed primarily toward the design of helix-type tubes, but modifications are also presented that facilitate the design of tubes with any type of modified helix structure or with any of the varied forms of periodically loaded waveguides. The procedure permits the rapid design of a tube and has been found to give very satisfactory results at varied frequencies. ACKNOWLEDGMENTS The authors wish to express appreciation to their associates for valuable suggestions during the course of the work and in particular acknowledge the aid of Messrs. Masnari, Ohtomo, Pua, and Wen in calculating the design curves.

-55LIST OF SYMBOLS a' mean helix radius b injection velocity parameter B = f b' space-charge range parameter b' stream radius c velocity of light C gain parameter F = F1 F2 helix-impedance reduction factor G impedance gain factor I d-c stream current 0 Kd impedance of a modified helix or periodically loaded waveguide structure Ks sheath helix impedance P stream perveance P power output P stream micropervs P stream micropervs for tubes with modified helix or periodically loaded waveguide structures P h stream micropervs for a helix tube QC space-charge parameter (QC)d space-charge parameter for tubes with modified helix or periodically loaded waveguide structures (QC)h space-charge parameter for a helix tube R plasma frequency reduction factor Rd plasma frequency reduction factor for modified helix and periodically loaded waveguide structures

-56LIST OF SYMBOLS (Continued) Rh plasma frequency reduction factor for a helix tube u d-c stream velocity v undisturbed circuit phase velocity V d-c stream voltage y small-signal phase constant of the r-f wave B~P ~ wave phase constant PB stream phase constant = k, = k free-space phase constant o 1 j = (p2_02) 1/2 radial propagation constant n = e/m charge-to-mass ratio for the electron T]^s ~ saturation efficiency, percent K perveance correction factor t, v space-charge parameter correction factors helix pitch angle; also input-signal level relative to CIoVo ewo radian frequency Xw radian plasma frequency p o = Rx effective radian plasma frequency q P

-57LIST OF REFERENCES 1. 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. 2. Rowe, J.E., "Design Information on Large-Signal Traveling-Wave Amplifiers", Proc. IRE, Vol. 44, No. 2, pp. 200-211; February, 1956. 3. Rowe, J.E., "Large-Signal Traveling-Wave Amplifiers", Tech. Rpt. No. 20, Electrical Engineering Department, Electron Tube Laboratory, The University of Michigan; December, 1956. 4. Cutler, C.C., "The Nature of Power Saturation in Traveling-Wave Tubes", BSTJ, Vol. 35, No. 4, pp. 841-876; July, 1956. 5. Caldwell, J.J., Hoch, O.L., "Large Signal Behavior of High Power Traveling-Wave Amplifiers", Trans. PGED-IRE, No. 1, pp. 6-18; January, 1956. 6. Tien, P.K., "A Large Signal Theory of Traveling-Wave Amplifiers, Including Effects of Space Charge and Finite Coupling between Beam and Circuit", BSTJ, Vol. 35, No. 2, pp. 349-374; March, 1956. 7. Fletcher, R.C., "Helix Parameters Used in Traveling-Wave-Tube Theory", Proc. IRE, Vol. 38, No. 4, pp. 413-417; April, 1950. 8. Tien, P.K., "Traveling-Wave Tube Helix Impedance", Proc. IRE, Vol. 41, No. 11, pp. 1617-1624; November, 1953. 9. These data were obtained from the quartic determinantal equation by Hill's method. 10. Tien, P.K., op. cit. 11. Birdsall, C.K., Everhart, T.E., "Modified Contra-Wound Helix Circuits for High-Power Traveling-Wave Tubes", Trans. PGED-IRE, No. 4, pp. 190205; October, 1956. 12. Pierce, J.R., Theory and Design of Electron Beams, Van Nostrand Company, New York, 1949; Chap. 9. 13. Brewer, G.R., "Some Effects of Magnetic Field Strength on Space-ChargeWave Propagation", Proc. IRE, Vol. 44, No. 71, pp. 896-903; July, 1956. 14. Olson, A.B., Peter, R.W., Ruetz, J.A., "Attenuation of Wire Helices in Dielectric Supports", RCA Review, Vol. 13, pp. 558-573; December, 1952. 15. Chu, C.M., "Effect of Conductor Size and Shape on the Propagation of Waves in Helical Waveguides", Engineering Research Institute Report No. 2144-193J, The University of Michigan, Ann Arbor, Michigan; 1957.

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