THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR, MICHIGAN PHEORY OF THE CRESTATRON: A FORWARD-WAVE AMPLIFIER TECHNICAL REPORT NO. 27 Electron Tube Laboratory Department of Electrical Engineering By Joseph E. Rowe Project 2750 CONTRACT NO. AF30(602)-1845 DEPARTMENT OF THE AIR FORCE PROJECT NO. 4506, TASK NO. 45152 PLACED BY: TKE ROME AIR DEVELOPMENT CENTER GRIFFISS AIR FORCE BASE, NEW YORK September, 1958

-iiiTABLE OF CONTENTS Title Page ABSTRACT iv LIST OF ILLUSTRATIONS INTRODUCTION 1 DERIVATION OF THE GAIN EQUATION 2 SMALL-SIGNAL GAIN 6 MAGNETIC FIELD REQUIREMENT 15 BANDWIDTH 19 MAXIMUM GAIN VS. C AND QC 19 LARGE-SIGNAL PERFORMANCE 30 SATURATION GAIN 30 EFFICIENCY 39 CONCLUSIONS 46 ACKNOWLEDGMENTS 49 LIST OF SYMBOLS Do

-ivABSTRACT A new type of forward-wave traveling-wave amplifier has been developed which offers moderate gain (10 to 20 db) and high operating efficiency coupled with a very short length (4 to 6 wavelengths), thus reducing the size of focusing magnet required. The operation of the device depends upon the beating effect produced when the three forward waves described in the travelingwave tube theory travel along the slow-wave structure. The device, named the Crestatron, operates with voltages sufficiently high so that there are no growing waves and the gain is determined by the beam injection velocity and not by the length of the tube. The operation of the Crestatron has been analyzed with both a large-C linear theory and a large-signal theory of traveling-wave tubes. The effect of circuit loss on gain is to reduce the beating gain at a rate of 12 db per unit d from the d = 0 gain. Large-signal calculations indicate that high saturation efficiency characterizes this mode of operation.

-vLIST OF ILLUSTRATIONS Figure Page 1 Gain vs. Length. (C = 0.05, QC = 0.125, d = O, b = 2.20) 7 2 Gain vs. Length. (C = 0.1, QC = 0.25, d = 0, b b = 2.57) 8 3 Gain vs. Length. (C = 0.2, QC = 0.125, d = 0.025, b > 3.0) 9 Xl=O 4 Gain vs. Length. (C = 0.2, QC = 0.125, d = 0.125, b > 3.0) 10 Xj=O 5 Gain vs. Length. (C = 0.10, QC = 0.125, d = 0, b = 2.34) 12 5 Gain vs. Injection Velocity for Fixed Tube Length. (C = 0.1, QC = 0.125, d = 0) 16 7 Gain vs. Injection Velocity for Fixed Tube Length. (C = O.2, QC = 0.125, d = 0) 17 8 b vs. QC with C as the Parameter. (d = 0) 20 X1=O 9 Maximum Gain vs. b - b with Space Charge as the Parameter. C = 0.05, d = 0) 0 22 10 Maximum Gain vs. b - b with Space Charge as the Parameter. (C = 0.1, d = 0) 0 23 11 Maximum Gain vs. b - b with Space Charge as the Parameter. (C = 0.2, d = 0) xl=o 24 12 Effect of Loss on Crestatron Gain. (C = 0.05) 25 13 Effect of Loss on Crestatron Gain. (C = 0.1) 26 14 Effect of Loss on Crestatron Gain. (C = 0.2) 27 15 Normalized Amplitudes of the Wave Voltages vs. Injection Velocity. (C = 0.1, d = 0) 28 16 Normalized Amplitudes of the Wave Voltages vs. Injection Velocity. (C = 0.1, d = 0) 29 17 CN to the First Gain Maximum vs. Injection Velocity. (C = 0.05, d = 0) 31 18 CN to the First Gain Maximum vs. Injection Velocity. (C = 0.1, d = 0) 32 19 CN to the First Gain Maximum vs. Injection Velocity. (C = 0.2, d = O0) 33

-viLIST OF ILLUSTRATIONS (Continued) Figure Page 20 Large-Signal Gain vs. Length. (C = 0.1, QC = 0.25, B = 1.0, d = O, b = 3.5) 34 21 Saturation Gain vs. Input-Signal Level with Injection Velocity as the Parameter. (C = 0.1, QC = 0.125, B = 1.0, d = 0, b = 2.33) 35 Xj=0 22 Saturation Gain vs. Input-Signal Level with Injection Velocity as the Parameter. (C = 0.1, QC = 0.25, B = 1.0, d = O, b = 2.57) 36 23 Saturation Gain vs. Injection Velocity with Input-Signal Level as the Parameter. (C = 0.1, QC = 0.125, B = 1.0, d = 0, b = 2.55) 37 24 Saturation Gain vs. Injection Velocity with Input-Signal Level as the Parameter. (C = 0.1, QC = 0.25, B = 1.0, d = O, b = 2.57) 38 25 Saturation Length vs. Input-Signal Level with Injection Velocity as the Parameter. (C = 0.1, QC = 0.125, B = 1.0, d = O, b = 2.33) 40 26 Saturation Length vs. Input-Signal Level with Injection Velocity as the Parameter. (C = 0.1, QC = 0.25, B = 1.0, d = 0, b = 2.57) 41 27 Comparison of Theoretical and Experimental Gain. 42 28 Variation of Saturation Level with Input-Signal Level. (C = 0.1, B = 1.0, d = 0) 43 29 Saturation Efficiency vs. Input-Signal Level with Injection Velocity as the Parameter. (C = 0.1, QC = 0.125, B = 1.0, d = 0, b = 2.33) 44 X1=O 30 Saturation Efficiency vs. Input-Signal Level with Injection Velocity as the Parameter. (C = 0.1, QC = 0.25, B = 1.0, d = 0, b = 2.57) 45 Xl=O 31 Saturation Efficiency vs. Injection Velocity with Input-Signal Level as the Parameter. (C = 0.1, QC = 0.125, B = 1.0, d = 0, b = 2.33) 47 X3=o 32 Saturation Efficiency vs. Injection Velocity with Input-Signal Level as the Parameter. (C = 0.1, QC = 0.25, B = 1.0, d = 0, b = 2.57) 48 Xl-=O

THEORY OF THE CRESTATRON: A FORWARD-WAVE AMPLIFIER* INTRODUCTION It is well known that the operation of backward-wave devices, both O-type and M-type, depends on an interference phenomenon resulting from the beating between waves propagating along an r-f structure. It can also be shown that in forward-wave devices such as the traveling-wave amplifier gain can occur due to the beating of waves traveling on the r-f structure, providing that the length is correct. The magnitude of this gain for any given set of circuit and operating parameters is determined by the relative injection velocity b rather than by the length of the tube as in the case of the conventional traveling-wave amplifier. A voltage gain is produced in the Crestatron by a beating between the three forward waves propagating on the slow-wave structure. The device is operated with a beam velocity such that b'> b and hence there are no growing waves. The gain is achieved by adjusting the tube length to the proper value such that all the waves, one of which is out of phase with the others at the input, add together to give an r-f voltage greater than that at the input. Hence there is not voltage gain in the sense of that produced by growing waves but there is gain if the tube is considered to be a two-port network. The amount of gain is determined by the value of b and decreases as the beam velocity is raised above that at which growing waves cease to exist. The gain characteristics of the * Presented at the Sixteenth Conference on Electron Tube Research, Quebec, Canada; June 25-27, 1958.

-2Crestatron can be calculated from the small-signal theory and these calculations also give information on the CN bandwidth achievable in the device. The large-signal theory of traveling-wave amplifiers1 has been used to calculate the nonlinear performance of the Crestatron in terms of the gain achievable and the expected operating efficiency. Ample experimental evidence2 has already been offered to verify the theory and indicate that high efficiency is obtained with short length and moderate gains. During the course of this work it was learned that Mourier and Sugai have shown that a similar type of operation is possible with the forwardwave magnetron amplifier.3 They treated the special case of small C, no space charge and zero circuit loss. These conditions in the magnetron amplifier give rise to two forward traveling waves which under certain conditions can beat with one another to produce gain. DERIVATION OF THE GAIN EQUATION The three forward waves propagating on a traveling-wave tube r-f structure are known to vary in the following manner:4 erz e-j z eC, (1) e = e where B is the wave phase constant and is equal to 3 at synchronism. The normalized voltage VZ/V at any point along the r-f structure may be written in terms of the amplitudes of the three waves, neglecting space charge, as z F e V2 62Q V3 6 - = e-CV e + + 3 e (2) 1. Rowe, J. E., "A Large-Signal Analysis of the Traveling-Wave Amplifier: Theory and General Results", Trans. PGED-IRE, ED-3, No. 1, 39-56, January, 1956. 2. Caldwell, J. J., and Hoch, 0. L., "Large-Signal Behavior of High Power Traveling-Wave Amplifiers", Trans. PGED-IRE, ED-3, 6-18, January, 1956. 5. Mourier, G., and Sugai, I., private communication. 4. Pierce, J. R., Traveling-Wave Tubes, D. Van Nostrand Co., New York; 1950.

-3where 65 = x. + ji i = 1,X23; Vi/V = normalized voltage amplitude of each wave; Q _ PCz = 2tCN; N = structure length in wavelengths; and C = gain parameter. From the small-signal theory of the traveling-wave tube, the following expressions for the r-f convection current and velocity are obtained by retaining terms proportional to C: (l C v + (l+jC2) + j (1+jC3) juoC ( V1 + 2 V2 53 and (1+jC1) (l+jC`52) (l+jC63) 2V (4) V, V2+ V3 = (4)`62 622 632 ( where q = e/m, charge-to-mass ratio of the electron, v = r-f velocity in the stream, 1 = r-f convection current in the stream, V = d-c stream voltage, and I = d-c stream current. 0 If an unmodulated stream is injected at z = 0 and an r-f signal level V is applied to the r-f structure at that point, the boundary conditions require that the right-hand sides of Eqs. 3 and 4 be zero and that V = V1 + V2 + V3. (5) Equations 3, 4, and 5 may be solved simultaneously to give the normalized amplitudes of the individual waves. The general result is shown below. V l+jCBi+l ) ijC-5i i+2 - i+l (2) V =l C6 iCli+ 2 2` 6./- K) -

-4where 5. 6.. Equation 6 gives the total voltage associated with each 1 1+3 wave and in the absence of space charge also gives the circuit voltage. The effect of passive modes or space charge is to reduce the circuit voltage from the value predicted by Eq. 6. The ratio of the circuit voltage to the total voltage is found from the ratio of the second term on the right-hand side of the following familiar quartic determinantal equation to the total right-hand side5 of Eq. 6: 62 = (l+jC5)2 [l+C(b-jd)] -4Q (l+jC)2 (7) [-b+jd+j5+C bd - 2 + 2 + The general result for the circuit component of voltage is V. (1+jC5 )2 = 1 + 4QC - (8) V.62. 5 2 Equations 6 and 8 have been obtained by Brewer and Birdsall.6 Thus the voltage along the r-f structure may be written in general, including the effects of finite C and space charge QC, as V.i Q V = e ) e( 9) i=l where 65 -6. When C is small and the effect of passive modes or space charge is negligible, Eqs. 6 and 8 reduce to the following familiar form. V. 2 1'-'1 (10) (6i - i+l) (ii i+2) 5. Ibid., p. 113, Eq. 7.1 with corrections. 6. Brewer, G. R., and Birdsall, C. K., "Normalized Propagation Constants for a Traveling-Wave Tube for Finite Values of C," Tech. Memo. No. 331, Hughes Research and Development Laboratories, Culver City, California; October 1953.

-5and ci - 1. (11) The voltage gain is written as Gb = 10 log =v 10 log 9-. (12) It is important to note that, unlike conventional standing waves on a transmission line, each wave sees the r-f structure characteristic impedance at all points along the line. The gain that occurs when the velocity parameter b is greater than that for which the growth constant of the growing wave is zero is, as mentioned before, due to a beating effect between the three small-signal waves which are set up at the input and propagate along the r-f structure. The interaction is primarily between the circuit wave and the slow space-charge wave. It will be seen later that the fast space-charge wave is excited to a negligible extent. This is the same basic mode of operation as in the backward-wave device of both the traveling-wave tube and crossed-field types. As the three waves travel along the structure the phase relationship between the r-f current in the beam and the r-f field on the circuit changes and at certain points along the circuit the phase is such that energy is transferred to the circuit. At the same time there are certain regions along the tube where the phase relationship between the beam current and the circuit field is such that energy is fed from the circuit back to the beam. The gain in this mode of operation depends upon the relative injection velocity b for any given set of tube parameters rather than on the length as in the case of the conventional traveling-wave tube. The Crestatron is inherently more efficient than the backward-wave devices which operate on the same principle because the modulation in the stream

-6and the field on the circuit producing the modulation travel in the same direction, whereas in the backward-wave device the modulation in the beam and the circuit field travel in opposite directions. In the backward-wave device the circuit field is strongest where the modulation is weakest and vice versa. Mathematically speaking all the energy is abstracted from the beam at the input since in satisfying the boundary conditions energy is put into setting up the three circuit waves. Then the circuit length is simply adjusted so that at the end of the tube the wave energies all add in phase. It should be recalled that in this region of operation the propagation constants are purely imaginary, giving rise to real voltage amplitudes, and at the input one voltage component is 180 degrees out of phase with the other two. SMALL-SIGNAL GAIN It was pointed out above that the normalized voltage amplitudes are purely real since the propagation constants are purely imaginary when b > bx For fixed values of C, QC, and d the gain may be calculated from Eqs. 6 and 21 as a function of 9 for particular values of b. Typical gain curves are shown in Figs. 1 and 2. The above gain equations are valid for all values of b and it is seen from the figures that the normal exponential gain is obtained when xL i 0. The gain curves were plotted over a range of 6it radians to indicate their repetitive nature. The effect of circuit loss on the gain vs. length is shown in Figs. 3 and 4, and it is seen that the presence of loss reduces the gain particularly for large values of 9. However, for small values of d the gain at the first maximum is not appreciably reduced. The results plotted in Fig. 3 do not show the large negative dip in the gain curve for b = 0 that Brewer and Birdsall7 found. The fact that they 7. Brewer, G. R., and Birdsall, C. K., "Traveling-Wave Tube Propagation Constants"', Trans. PGED-IRE, ED-4, No. 2, pp. 140-144; April, 1957 (Figure 2).

10 b=l 0 A/ /b=2.0 b I / / I " / 9, ~ / e'/ / \ / - \' / -\ o9 b \ 2.5\^ b = 2 -V 0 T/2 3/2 2 5/2 3 7/2 411 9/2 S /2 6T FIG.. LENGTH. (C0.05,QC 0.125 d0 b02.20) 0 Tr/2 TT 3Ti/2 21'1 51T/2 313 7'1/ 2 41T 9Tr/2 5T II T/2 6Tr e, RADIANS FIG. I GAIN VS. LENGTH. (C = 0.05, QC = 0.125, d =0, bxl=0 = 2.20)

,'-b 2.5 16,,___-_-_b= 2.6 \\ 12 Oil _________ ____________ 0 //,- 3.0 __ __ 3.5__ _'_:2.75 _ 0 IT/2 T 3TIT/2 2TT 51T/2 31 7Tr/2 4ff 9ff11/2 5/TT 11T/2 6n 8, RADIANS FIG. 2 GAIN VS. LENGTH. (C 0O.I, QC 0.25, dO 0, bx: 2.57) x,:

~10a / b= 1.0 8 ---- -J(L, b 2.5 b: 2.0 \ /1I \Jfe',3 / FGb =:AN4.5 Ld, n T'^^^nA \ A/ ^ /!4'I -10 1 _ _ _ _ _ I _ _ _ _ _ I 0 T1/2 7T 31T/2 2T1 511/2 311 7T1/2 41T 9T/ 2 5T 111TT/2 6; 0, RADIANS FIG. 3 GAIN VS. LENGTH. (C= 0.2, QC0.125 d=0.025, b I>3.0) I 4.0 I 2 I I

b b=2.5 8 / b =I.0 b 2.0 b =3.0 44 - b_3.5\ / /. __ _I. \t\ iLv b.4o0 i /\ 0 TT/2 n 3TT/2 2n 51/2 3TT 7T1/2 4T1 9Tr/2 S51 IITr/2 6TT 9, RADIANS FIG. 4 GAIN VS. LENGTH. (C = 0.2, QC 0.125 d = 0125, b/ = > 3.0)

-11assumed a QC = 0.25 whereas the above data are for QC = 0.125 is of little consequence to this discrepancy. It is believed that these results of Brewer and Birdsall are incorrect since to compute gain they used the small-C equations, which are not valid when C = 0.2 and QC = 0.25. It should be pointed out that only very small dips in the gain curves were noted in the large-signal calculations, as is predicted here. Large negative dips in the gain seem to occur at b = -1 for most values of C and QC and for positive values of b when d is large. The gain curves of Fig. 5 show that gain is also achieved for negative values of b such that xl = 0. Negative values of b correspond to operating voltages less than the synchronous voltage; and the resulting gains are less than those obtained with large values of b. When the loss parameter is zero all maxima of the gain curve are approximately equal (the slight variations and lack of periodicity will be explained later), but when d f 0 the first maximum will be highest and subsequent peaks will generally be lower. This merely emphasizes the fact that when circuit loss is significant the Crestatron length should be chosen so as to operate on the first maximum of the gain curve. It is seen from the gain curves that for fixed b and variable Q the curves are nonperiodic within the interval 6xr and exhibit periodic undulations in amplitude. The explanation of this phenomenon is contained in Eq. 9 for the voltage amplitude vs. distance. The lack of periodicity with 2jt and the undulating peak amplitudes are a result of the product of exp(-j @/C) and exp(6i.), where each represents a vector rotating about the origin as a function of Q. The rate of rotation of the first vector is related to 1/C, which is typically between 5 and 20, and the rate of rotation of the second is determined by i., which varies between 1 and 3. The first, then, perturbs the second as a modulation and the fact that 6. is nonintegral in general means that it is not

'I / / 6 2, bb-2.0 2 b. =-I.0 Z~~ic ~I i / A /' I -2-T\*/-I —I —-- -- I e I; I FIG. / GAIN VS. / Q= = 2 e, R F IG. 5 GAIN VS. LENGTH. (,'=0.10 I, I =2..-34) _,,,,, -....,i.tj..,,,~...., ~ O T 3T/ 2T S~/2 3T Tr2 4T r/2 5T 1nj r il~~~~~~~8 RAI II FI. GINVS LNTH (~010 Q..25 dOb~,2'4

-13periodic with 2t. It may be that Q must travel through 2mT radians with m very large before a periodicity is apparent, if ever. The above phenomenon indicates that there is something more than a simple beating effect occurring between the waves. In fact there is a continual slipping of one wave with respect to the others along the tube. This process accounts for the fact that there is or may be a net interchange of energy between the beam and the circuit in a device of infinite length. It is interesting to examine the condition necessary for the gain curve to be periodic with period 2mjr and also the condition required for the peak amplitudes to be constant. In fact these two conditions would result in a gain curve that is exactly reproduced every 2c radians. In Eq. 9 for Vz/V, when b is held fixed, the phase condition is determined by the product e.* ei = e (Yi, (13) since x. = 0. Thus in order for the phase to be periodic with 2mr 1 <yYi C) = 2mt. (14) Solving for yi gives 2m + 1 Yi n +' (15) where n and m are independent integers. Thus for phase periodicity the value of yi must be equal to a constant plus some integral or subintegral multiple of 2g. When C 0, yi = 1/C and all the waves travel at approximately the same rate, thus maintaining the same relative phase positions with respect to one another. This results in a constant-amplitude vector rotating about the origin with a period of 23 radians. Thus the gain curve will be periodic in amplitude also.

-14Under very restricted conditions a simplified expression for the gain of the Crestatron may be obtained from Eq. 9. When C -- 0, QC = 0, d = 0, and b is sufficiently large that the propagation constants are purely imaginary, it can be determined that the i.'s are given approximately by L6 - j/bl/2 82 2 -jb, and 5s j/bl/2. (16) Substitution of Eq. 16 into Eq. 9 yields for the gain, after some simplification, v 2.22 r 3 (Vi = (- ) 1 + b6 + (b3-1) sin2 - 2b3 os b cos. b1/2 b1/2 + b3/2 sin Qb sin. (17) bi/J Since b is usually greater than 2.5, Eq. 17 can be further simplified to Vz - 1 (18) - I = 1 + - sin2 - 2 cos Qb cos - + b3 sin Qb sin 8) Lb3 bl/2 b1/2 bl/2 A useful expression for predicting the value of CN at the first maximum of the gain curve may be obtained by differentiating Eq. 17 with respect to Q and setting the result equal to zero. In this way sin - cos - b3 cos Qb =0 (19) bl/2 L b1l/2 Thus maxima and minima in the gain occur for Q sin - = 0 b1/2 or n b1/2 CN = nb n = 0,1,2,... (20)

-15The other condition is cos bl/2 cos Qb = 0 for large b b3 or CN = -, n = 0,1,2,... (21) Equation 21 predicts quite accurately the distance to the first maximum in the gain curve when n = 1. Thus CN to first maximum =0.75.(22) b Equations 20 and 21 coupled with information on the second derivative of the gain curve may be used to determine subsequent maxima in the gain vs. 9 curve. In addition to gain vs. length curves for fixed b, a set of gain curves may be obtained for fixed tube length with variable voltage or b. Typical curves of this type are shown in Figs. 6 and 7. The main hump in the gain curve vs. b is of course due to exponential gain and the other lower peaks correspond to gain through beating waves. When C - 0, QC = 0, and d = 0 the gain vs. b curve will be symmetrical with respect to b since maximum exponential gain occurs at synchronism. MAGNETIC FIELD REQUIREMENT The limiting stream perveance which can be transmitted through a cylindrical drift tube or helix due to space-charge spreading is given by8 Perveance = 58.6 x 10 () (25) where d = helix diameter, and I = helix length. 8. Pierce, J. R., Theory and Design of Electron Beams, D. Van Nostrand Co., New York, 1954, p. 151.

(0 = P'SZI0 30'1'0 = 3) *'H9N33 38nl 03X1J 80_I A113013A N01133PNI *SA NIV9 9'91 L 9.S t ~ Z I 0 I- Z- t — G- 9- 9- - ~- - —, —— ~~~~ 3. ~OI1----- -I r - -83I -V -1 — ^ —'91~~ O*~-Ix JIZ II?/ O*' 81 /lE Ie- / I 1 II _I LI - __/ ___ 9 1 8 — -9'

(0 = P'9ZIO = 30'3'0 = 3)'H19N31 38nl1 3XIJ I8OI A113013A N01133PNI SA NIV9 L' 91 q L 9 9S t ~ I I 0 - Z- ~- t- 9- 9- L- -I~~OI9Ifa~I I I I I I II___!I11'V1I /IMI Z 1 i 11 I,, Z X~~~~~~eXX<1~~~~~~~~. -Li---- f /1 - - -90 --- --- --- --- - - --- --- — _, - - -- - - -----------— L —---— ^ —---------------— 021 -- - --- --- --- -- V - - --- -j -- --- --- --- --- --- --- 2 3/U~=9^l K~ ~~~~~~~O --- --- - - --- --- -^ - f - --- ---- --- -- --- --- 8 -------- -- ] - - -- - -- - -- - o -ii- ~ ~ ~ ~ ~ ~ ~ 9

-18The following familiar relations for traveling-wave tubes are recalled /FK I \1/3 CN A Lf (24a) s k4Vo/ uo = (2Vo0)1/2 = xf, (24b) U = 1 + Cb, (24c) 0 and ya = 2Aa/\. (24d) The helix impedance FK used in Eq. 24a may be written as 506 FK' (l+Cb) FK - -, (25) ~S T 1/2 where F = the impedance reduction factor due to dielectric loading and space harmonics and K' = sheath-helix impedance as given in Fig. A6.5 in Pierce.9 s Combining Eqs. 23 through 25 yields (FK' )1/3 M/3 CN = 0.04 (ya). (26) ( l+Cb)2/3 From Eq. 26 can be calculated the maximum length of structure which can be used when no magnetic focusing field is present and the stream is allowed to spread under the influence of space-charge forces. Of course, the stream must be injected in a convergent manner aimed toward the center of the structure and then it is allowed to expand, resulting in a divergent stream at the output end. Allowance will also have to be made for stream spreading due to the presence of r-f fields on the structure. The following typical values of the parameters in Eq. 26 are used to evaluate CN: 9. Pierce, J. R., Traveling-Wave Tubes, D. Van Nostrand Co., New York} 1950.

-19ya = 1.5 (FK*)1/3 = 2.2 Using the above values gives CN = 0.34. s This value of CN is compatible with the lengths required for Crestatron operas tion. Hence under some conditions it may be possible to operate the Crestatron with little or no magnetic focusing field. Electrostatic focusing systems are sometimes possible depending upon the structure type. BANDWIDTH The bandwidth of the Crestatron may be determined from the gain curves shown in Figs. 1 through 4. The CN bandwidth can be as large as +50 percent between 3-db points on the gain curve. If the impedance of the structure remained constant over this range then the Crestatron would have a frequency bandwidth of 5:1. In general impedance variations will limit this figure to some lower value. MAXIMUM GAIN VS. C AND QC It was pointed out earlier that the voltages are purely real in the Crestatron and at the input one voltage is 180 degrees out of phase with the other two. Thus the maximum gain will occur when the three waves all add in phase. The value of b for which xl = 0 has been computed from the quartic determinantal equation and is shown in Fig. 8 for a range of values of C and QC with zero loss. When d ( 0 x1 is not exactly zero anywhere but does drop to less than one percent of its maximum value for large b.

-204.8 C 0.3' / 0.2 4.6., 4.4 3 __ Z __C 0./1 4.2 J 4.0 3.4 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.1 1.2 C = 0.08 2.01/ /c oo5 SPACE-CHARGE PARAMETER, QC FIG. 8 bx o VS. QC WITH C AS THE PARAMETER. (d O)

-21Maximum-gain curves for various values of C, QC, and d are shown in Figs. 9, 10, and 11. The gain decreases with increasing voltage and for very high voltage approaches zero asymptotically. As would be expected the gain increases with C and decreases as QC is increased. The effect of loss on the circuit is also seen to reduce the gain. The greatest gain occurs for a b slightly greater than that for which growing waves cease to exist. There is a smooth transition from the region of exponentially growing waves to the region in which the gain is a result of the beating of the three waves. The effect of circuit loss on Crestatron gain is illustrated in Figs. 12 through 14'for a wide range of parameters. It is seen that the rate of decrease of gain is relatively independent of b,.C and QC and is approximately 12 db per unit d. The loss in the absence of a stream is given by Loss in db= LN = 54.6 CNd. (27) Since typical Crestatron lengths are in the vicinity of CN w 0.4 to 0.5 these data show that the signal decreases at the approximate rate of 24 db per unit d with no stream present. Hence the effect of circuit loss is to reduce the gain at only one-half the rate when the stream is present as compared with the rate of decrease of the signal in the absence of the stream. This compares favorably with the one-third figure for conventional traveling-wave tubes. The gain for large positive values of b is a result of the combining of Vcl/V and Vc2/V, since Vc3/V is usually negligible. The magnitudes of these normalized voltages for a typical range of parameters are shown in Fig. 15. For large values of b it is seen from Fig. 15 that Vcl/V approaches zero and Vc3/V is still small, so that almost all the energy is in the second wave (circuit wave) Vc/V. On the other hand for negative values of b (voltage below synchronism), the third wave (fast space-charge wave)Vc3/V predominates, as shown in Fig. 16.

-2220 18 16 --- 14'QC=O < u ~Q0.125 s P.I 6 QC= 0.25 "/ \ //QC 0.5 4 0 0.5 1.0 1.5 2.0 2.5 3.0 3a5 b-bx =o FIG. 9 MAXIMUM GAIN VS. b-bxo WITH SPACE CHARGE AS THE PARAMETER. (C- 0.05, d- O)

(O P'I'O = 3)'*313VtUVd 3H1 SV 39IDVH3 33VdS HIIM lq-q'SA NIV9 IwnwVIXVWI 01 *91d O=IXq.q I-g 02 Z -Z 9I1 T-'O I o =: /o tG T T T. A' I' \1, =11 r v r I 91'1 -A;8

(0= P' Z'O =3)'8313NWttVd 3H1 SV 398VH3 33VdS HIM o=lxqq'SA NIV9 WINIXVw II'o91 OslXq_q g'~ 0' S'Z O'Z' S1 01 gSO 91'O 30/ 0 0 0 < X SI~~~~~ol X 1X1 I Oz

-251 —-— 1-1QC - 0.25 ---- -- b = 2.5. ->cr 4 QC=0.125 x r w 2: b-2 3.0 QC4 0.25 i 1 b b'3.0 o - -QC c0o. 125 QC =0.25 3.5 QC1 0.125 0 0.05 0.1 0.15 0.2 0.25 LOSS FACTOR, d FIG. 12 EFFECT OF LOSS ON CRESTATRON GAIN.(C= 0.05)

-26I0 3S~r I I I. b= - 2.5 _1 -6 —- --- -r 1 /QC:0.125 b 33.0' QC = 0.25 2,3,,,,,b= 3.0 <- -- - - Q-C = 0.12 5 w ~ so.~s-2 -- --- -C 0.125 0 0.05 0.1 0.15 0.2 0.25 LOSS FACTOR, d FIG. 13 EFFECT OF LOSS ON CRESTATRON GAIN. (C 0.1)

-2710 IO -;- - -- -- x zI 2___ i_________ ___"QC = 0.25 (DC~I ~~~ ~ -~ b=3.5 ___ ___ ___ ___ ___ ___~___ _OQC 0.125 0 -2 - - ~ 0 0.05 0.1 0.15 0.20 0.25 LOSS FACTOR, d FIG. 14 EFFECT OF LOSS ON CRESTATRON GAIN. (C= 0.2)

-282.5 __ —-_ e- QCO 2.0 --- QC =0.125...... — -- ---- QC = 0.5 i.5 vc2 v 1.0 > 1 0.5 1 Vc3 0.5 -2.5 __. ________ v 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 FIG. 15 NORMALIZED AMPLITUDES OF THE WAVE VOLTAGES VS. INJECTION VELOCITY. (C=0.1,d=O)

(0 = P'I' 0 3) *A113013A N01133PNI'SA S39V-10A 3AVM 3HI. JO S3IanilldWl V 03ZI"lVWJON 91'91z I o 1- q z- E- -____-I I ____ _ A 1|0 - ~-/ I -- ^'9DA| OA - -iO - - I' l - ~ o N C'oL. 1:1: L IILLV' 1 ~-r'0X "A 1 A+X' —-"-~ —--— _ —-~ o 4 - W.___ ___ —z-- -- -V -- ~V -- ^^-J^T I — 3,0 = 0 -90 - - - - - -\ - IY r - - 1'0=00 - -'O ~oA —-— l --- ^ —- ^ —, I'0 —e~ l~~ ~ ~~~ ~ 1 ^&faA'0 I I -60

-30Typical plots of the distance to the first maximum of the gain curve are shown in Figs. 17, 18, and 19 for several values of C and space charge. The theoretical relation, Eq. 22, is also shown. LARGE-SIGNAL PERFORMANCE Saturation Gain. The large-signal performance of the Crestatron has been evaluated using the large-signal theory of the traveling-wave amplifier.1 Significant large-signal gain is found for large values of b > bx when the input-signal level, ~, is appreciable compared to CI V. A typical gain curve O O is shown in Fig. 20 with the input-signal level to the r-f structure as the parameter. A periodicity in the large-signal gain vs. distance curves is found similar to that previously shown with the linear theory. The gain is seen to decrease as the drive level is increased, but the power output and efficiency increase as the drive level is increased. The composite results of the large-signal calculations may be plotted in summary form to show several interesting facets of the large-signal operation of the Crestatron. As seen in Figs. 21 and 22, the saturation gain as a function of the drive level to the r-f structure goes through a maximum at a value of r dependent upon the value of b. When the tube is operated at a "b" value which gives maximum small-signal gain or maximum saturation efficiency, an increase in the drive level results in a smooth transition to the Crestatron type of operation and the gain decreases smoothly. It is useful also to plot the saturation gain vs. the velocity parameter with the drive level as the parameter. These results are shown in Figs. 23 and 24. For very high drive levels the gain is low and relatively independent of the injection velocity. This relationship is also in line with that predicted from the linear theory. The value of CN to the first maximum of the gain curve can also be plotted as a function of both the drive level and the

(0= P'O'00 3)'A113013A N01103PNI'SA fnlnlIXVY NIV9 I1SSI1 3H1 01 NO LI'91 0=lXq -q j~ 0'~ S-'Z O'Z Sl 0'1 -'o 0 | ~ ~ ~,,, I,,0 — _ —- -.xqe/gO =N3 -IsQO/,l t -1 — ~ ~ ~ ~ I - 2'OsO /! | ~ - - - - - - - ---- - -t8'0 0cj'0=30/ 1 " t ~^'2 --- --- --- -- --- --- -- --- --- -- --- --- -- --, G 3 --- --- --- --- --- --- --- --- --- --- --- --- --- --, 9 -~~ ~01 -- -- - -- - -- - -- - -- - -- - - ^..8 -1. _ —_ —-- ho;~~~ -n-~~~~~~~~~~~~~~~~~~~~~~~~~~~~

-522.0 1.6 < 1.2z 1. I.. 0.2 L. - = —-/ /-QC= 0.25 0 0.5 1.0 1.5 2.0 2.5 30 3.5.Z _ _ _ _ - QC - _ 0.6 — Il ___ _ 0,.4 C-N 0.75 /b o, _ 0.5 1.0 1.5 2.0 2.5 3.0 3.5 b-bV, 0 FIG. 18 CN TO THE FIRST GAIN MAXIMUM VS. INJECTION VELOCITY. (C=O. 1, d=O)

-33 - 20 ie 1.6.4________ x 1.2 I,- 1.0 LL w 0.8 - - 0 ___/QC=o _______ 0.2 LCN= 0.75/b o O 0. 5 1.. 2 Q 2C5 3 3.5 VELOCITY. (C0.2, d= 0)

--— 4 —'"vw^ -'^ ~ -. — -- - 4I it.7 / -5 C t < < tt < < II~iI 7 2 -2 -4 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 y =U Z FIG.20 LARGE-SIGNAL GAIN VS. LENGTH. (C=0.1, QC=0.25, B=I.0, d=0, b= 3.5)

(~s - O=I Xq'O=P'O'I 8' ZI'O = 30'1'O= 3)'313WVt Vd 3Hi SV AJ13013A N01103PNI HJIM 13A31 VN9OIS-lindNI *SA NIV9 NOllvnivS 12'91.'A^~IO 01 3A^tV138 qp NI 13A3-1 IVN9IS-indNI' ~ 9 ~ Z I 0 I- Z- ~- fr- 9- 9- L\ \ ~; | 0'~ = q || 5 I I I \\l I I II _/'Gf_~ ~~~~~~~~~~~9 -- ~~~~~~~~~~~~~~ _____ _____ 0I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(Z.GZ O' =Xq'O=P'O'I:'gOS-O='1'0=)'3831l3WtVVd 3H1 SV A110013A N01133rNI HIIM'13A31 IVNOIS-indNI SA NIV9 NOIl.VdnliVS ZZ'91 A^~IO 01 3A11iV138 qp NI 13A31 1VNIS-lndNI' z 9 S o - Z- -- f- SPI q^ 7=q ___ _ ___ _ _2 gC q~~~~ ~, 0 T'Z.= q SZ'I =q ---, -- I --------------- I ------ I ------ I - 4.01 -9~

(~~'.Z o=' xq'O=p'01 =8'szi 0 =00'1'0=3)'U313WV8Vd 3Hi SV 13A31 "IVN9IS -indNI HIIM A1130013A N01133rNI'SA NIV9 NOIL8VnlS ~z3'O q'8313INVlw d AJ OO13A N011O3rNI O0' 9s' 0O'~ S'Z _0' O' 0I ---- ---- ---- ---- ---- --— m —m ---- ------------ --— p I.... I I I' a r t I I " I-I-I -- - -- - -- - — 1 —- II -LK. —.;0 0

-3810, ar k' l'f 1 1.0 1.5 2.0 2.5 3.0 3.5 4.0 INJECTION VELOCITY PARAMETER, b FIG. 24 SATURATION GAIN VS. INJECTION VELOCITY WITH INPUTSIGNAL LEVEL AS THE PARAMETER. (C =0.1i, OC =0.25, B 1.0, d O, bx=o - 2.57)

-39value of b. These optimum-length curves are shown in Figs. 25 and 26. A considerable amount of previously unexplained experimental information has been obtained by Caldwell and Hoch2 which verifies the existence of gain in a forward-wave amplifier due to the beating between the waves propagating on the r-f structure. A comparison of the calculated gains using the linear and nonlinear theories along with some data extracted from Caldwell and Hoch's paper is presented in Fig. 27. It is seen that the agreement is very good between the theoretical results and the experimental data. Since the value of "d" was approximately 0.025 for the experimental data it is difficult to choose a reference b where x1 is nearly zero. A value of b = 2.9 was used but had a larger value been used the agreement would have been even better. Caldwell and Hoch found in addition that the saturation power output increased as the drive level was increased, keeping the voltage constant. This phenomenon is also predictable from the nonlinear theory; a comparison is shown in Fig. 28. The amount of increase in the saturation power level for inputsignal levels comparable to and greater than CI V is some 4 db over that for O O very small input-signal levels. This increase in power output is probably due to the fact that this type of operation extracts the energy associated with all three circuit waves rather than only that of the growing wave as in the case of the conventional traveling-wave amplifier. Efficiency. The saturation efficiency also increases as the drive level is increased, resulting in a relatively high efficiency for the Crestatron. Efficiency curves as a function of the drive level for constant values of the injection velocity are shown in Figs. 29 and 30. It is seen that the saturation efficiency for high drive approaches a value relatively independent of the value of b in the Crestatron. In calculating the saturation efficiency of the Crestatron the input power is significant and hence one must consider only the energy conversion

-400.74 0.70 0.66 __ 0.62 o58se' b 2.5 0.58 b 1.5 0.54'_ 0 0.50 0.46 04 —- - 02 - __! - 0.42 0.38 b 3.0 0.34__. 0.26 b =4. 0 0.22 -7 -6 -5 -4 -3 -2 -I 0 1 2 3 4 5 6 FIG. 25 SATURATION LENGTH VS. INPUT-SIGNAL LEVEL WITH INJECTION VELOCITY AS THE PARAMETER. (C O.I, C 0.125, B 1.0, d O, bX:02 -33)

-410.74 0.70 1 0 b 2.75 0.62 0.58 __ 0.54 - 0.50 ~~~0.5o6 -____ \_b= 3.0 __ _ — -- 0.46 0.38 0.22 ____-5 -4 -3 -2 - 0 2 3 4 5 6 7 8 9 FIG. 26 SATURATION LENGTH VS. INPUT-SIGNAL LEVEL WITH INJECTION VELOCITY AS THE PARAMETER. (C - 0.1, QC = 0.25, B= 1., d - O, b-, -= 2.57)

-4210 IC\ lTCALDWELL C O. II, IQC 0.25, d 0.025 91 b x =z0 o 2.9' LARGE SIGNAL *=-5, C=0.1, QC=0.125, B=I, d=O 1LARGE SIGNAL - 4, C 0.1 QC 0.125 B=, d0= 8 i iS M*1 1 \^/LARGE SIGNAL *+-4, C=0.1, QC =0.25, B 1, d = =S l|\Y1 ARGE SIGNAL Xk'5, C 0.I., QC 0.25, B l, d=O _: \\ ^ \b \_ _ _ _I- _ _I- _ _ _ _II-I I'\ \ __ __< - -SMALL SIGNAL C 0.2 QC 0.25, d=O 2 1 _____ _ ___SMALL SIGNAL C 0.I, QC = 0.25, d =0 0 0.5 1.0 1 b5 2.0 2.5 3.0 3.5 b-bxh.o FIG. 27 COMPARISON OF THEORETICAL AND EXPERIMENTAL GAIN.

OC= 0.25. b = 1.25 l 0 -- - — 25 -20 -15 -10 -5 0 + 5 — - OC= 0.50, b= 1.5 a 4~ __ CALDWELL ~^^ QC~ 0.33 bw 1.3 -30 -25 -20 -15 -10 -5 0 +5. —10 ~, db FIG. 28 VARIATION OF SATURATION LEVEL WITH INPUT-SIGNAL LEVEL. (C = 0.1, B= 1.0, d=O)

36 a. 28 b = 2.5 " ). 24 0 20-b 3. LL z 16 4' I I I / =2. = 3.5 u3 jDm / b:4.0 -7 -6 -5 -4 -3 -2 -I 0 I 2 3 4 5 6 7 4, INPUT-SIGNAL LEVEL IN db RELATIVE TO C oV0 FIG. 29 SATURATION EFFICIENCY VS. INPUT-SIGNAL LEVEL WITH INJECTION VELOCITY AS THE PARAMETER. (C =0.1, QC =0.125, B = 1.O, d = O, bxm = = 2. 33 )

36 32 ci36 i -i —-, - 2.75 ---.0| — =. - Bib.-| --. 24 Z 20,'V I RA 0 0 i, INPUT-SIGNAL LEVEL IN db RELATIVE TO CI Vo FIG. 30 SATURATION EFFICIENCY VS. INPUT-SIGNAL LEVEL WITH INJECTION VELOCITY AS THE PARAMETER. (C 0.1, QC =0.25, B = 1.0, d 0,. b =. 57 x I Xl:=O

-46efficiency and subtract out the input-power level from the calculations. For values of b just slightly greater than that for which the wave growth constants become zero the efficiency is fairly constant as the drive level is varied. Continual increase in the drive power when b is adjusted for maximum saturation efficiency with small input-signal level results in a gradual decrease in the saturation efficiency. It is also interesting to plot the efficiency vs. b for particular values of drive as shown in Figs. 51 and 32. The saturation efficiency is apparently higher at high values of drive when the voltage is adjusted for Crestatron-type operation than when the voltage is adjusted for maximum saturation output for a small input-signal level. This increase in efficiency is probably in part due to the fact that 1 + Cb is approxi~mately 0.3 or 0.4 in the Crestatron, as opposed to around 0.2 in the growing wave mode of operation, and hence has considerably more energy to give up to the wave while slowing down. CONCLUSIONS A new type of forward-wave amplifier device has been developed and analyzed with both a linear and a nonlinear theory. The Crestatron operates on the basis of gain being produced by a beating between the three forward waves set up at the input of a traveling-wave amplifier operating at a high voltage so that there are no growing waves. The gains achievable are moderate and increase with increasing C but decrease with increasing QC. The gain also decreases as the injection velocity as measured by "b" is increased. Nonlinear calculations indicate the same type of behavior as both the voltage and drive level are increased, and it has been shown that the saturation power output increased with the drive level. The saturation efficiency is high for this type of operation and the CN bandwidth can be as large as +50 percent.

40', 36 =-5 z 32 -- 0 0 4 n 28 z w 6 120 2 \ z 16 d:O, bx0: 2.33) (1 12 d=O, bxlo=2.33:)

40 36 3 2_________ ______ w -I4 0 2 ~05,...15 2.0 2.5 3.0 3.5 4.0 IJ~INJECTION VELOCITY PARAMETER, bc FIG. 32 SATURATION EFFICIENCY VS. INJECTION VELOCITY WITH INPUT-SIGNAL 120 CD -I Il.e.. 02.,57 0.5 1.0 1.5 2.0 2.5 3.0.5 4.0 INJECTION VELOCITY PARAMETER, b FIG. 32 SATURATION EFFICIENCY VS. INJECTION VELOCITY WITH INPUT-SIGNAL LEVEL AS THE PARAMETER. (C =0.1, QC=O.25, B=1.0. d=O, bx =0=2.57)

-49The chief advantage of the Crestatron over the conventional travelingwave amplifier is that its length is quite short for reasonable gains (4 to 6 wavelengths), which means that the focusing magnet can be of minimum length and weight and under certain conditions may even be eliminated. ACKNOWLEDGMENTS The author acknowledges the benefit of discussions, during the course of this work, with his associates in the Electron Tube Laboratory, and in particular the assistance of Messrs. Y. C. Lim and L. E. Stafford in solving the equations on a digital computer.

-50LIST OF SYMBOLS u -v b = v injection velocity parameter C v 0 C gain parameter d loss parameter F helix impedance reduction factor f frequency I d-c stream current 0 i r-f convection current in the beam K sheath-helix impedance s K' normalized sheath-helix impedance N structure length in wavelengths QC space-charge parameter uo stream velocity V input r-f voltage amplitude V. circuit component of wave amplitude ci Vi wave voltage amplitude V d-c stream voltage v r-f velocity in the beam v circuit characteristic wave phase velocity z distance measured from the input ( = c/v wave phase constant DB = ac/uo stream phase constant r wave propagation constant in the presence of the beam

-51LIST OF SYMBOLS (Continued) i. = x. + jyi wave incremental propagation constant cT charge-to-mass ratio of the electron Ts saturation efficiency s ~ = 27tCN radian length of the tube g guide wavelength g,x stream wavelength XC angular frequency

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