THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN ENERGY CONVERSION CHARACTERISTICS OF CROSSED-FIELD INJECTED-BEAM AMPLIFIERS TECHNICAL REPORT NO. 67 Electron Physics Laboratory Department of Electrical Engineering By Joseph E. Rowe Klaus L. Volkholz Project 05361 CONTRACT NO. DA-36-039 AMC-00027(E) DEPARTMENT OF THE ARMY DEPARTMENT OF THE ARMY PROJECT NO. 3A99-13-001-01 PLACED BY THE U.S. ARMY ELECTRONICS RESEARCH AND DEVELOPMENT LABORATORIES FORT MONMOUTH, NEW JERSEY April, 1964

ABSTRACT The characteristics of and fundamental limitations in the energy conversion process for an injected beam crossed-field amplifier are examined. The conversion efficiency is highest at low values of the normalized cyclotron frequency wc/w for a fixed gain parameter and is critically dependent upon the coupling parameter, *(p). The optimum stream injection position varies between 0.45 < r < 0.6 and the saturated output is approximately wIoVo/21)c. Segmented depressed collectors are evaluated as a means for recovering beam kinetic energy. Stream prebunching by either positive-sole interaction or by the injection of delta function bunches is seen to materially enhance the conversion process and decrease the required interaction length. -iii

TABLE OF CONTENTS Page ABSTRACT iii LIST OF ILLUSTRATIONS v I. INTRODUCTION 1 II. CONVERSION PROCESS 3 III. NEGATIVE-SOLE AMPLIFIER RESULTS 4 35.1 General 4 3.2. Gain and Efficiency 8 5.3 Beam Collection on the R-f Structure 29 5.4 R-f Current Density in M-FWA's 29 5.5 Nonlinear Beating-Wave Amplification 29 3.6 Effects of a Circuit Sever 35 IV. TWO-DIMENSIONAL M-FWA WITH A POSITIVE SOLE 59 V. ADIABATIC EQUATIONS FOR A TWO-DIMENSIONAL M-FWA WITH A NEGATIVE SOLE 44 VI. INTERACTION OF DELTA FUNCTION BUNCHES 45 VII. COLLECTOR DEPRESSION 56 7.1 Depressed Collector Design 56 7.2 Computational Results 66 VIII. CONCLUSIONS 85 LIST OF REFERENCES 87 -iv

LIST OF ILLUSTRATIONS Figure Page 1 Schematic Diagram of Crossed-Field ForwardWave Amplifiers. 2 2 Gain Vs. Length and Drive Level. (D = 0.05, b = 0, r = 0.5, s = 0.1, %C/W = 0.5, C p/C = 0) 7 5 Beam-Circuit Coupling Function. 9 4 R-f Voltage Vs. Distance Showing the Effect of Space-Charge Forces in an M-Type Brillouin Flow. (D = 0.1, r = 0.5, b = O, 0 = - 30, o /W = 0.5, s = 0.1) 10 5 Gain and Efficiency at the 50 Percent Interception Plane. (r = 0.5, s = 0.1, 0 = - 30, b = 0. WD /WD = 0) 11 6 Gain and Efficiency at the 70 Percent Interception Plane. (r = 0.5, s = 0.1, o = - 30, b = 0, W /W =0) 0 12 7 Effect of Space-Charge Forces on Gain and Efficiency at the 50 Percent Interception Plane. (r = 0.5, s = 0.1, o = - 30, b = 0, Wop/ = o /C) 14 8 Effect of Space-Charge Forces on Gain and Efficiency at the 70 Percent Interception Plane. (r = 0.5, s = 0.1, r0 = - 30, b = 0, D /W = c/W) 15 9 Gain and Efficiency for r = 0.75 and Co /w = 0. (b = 0, s = 0.1, = - 30) p 17 10 Gain and Efficiency for r = 0.75 and Co /w = C / (b = 0, s = 0.1, = - 350) p 18 11 Gain and Efficiency at the 70 Percent Collection Plane for Constant Wc /. (b = 0, s = 0.1, Co /W = 0.5, c p c/wD = 0.5, to = - 50) 19 12 Gain and Efficiency at the 70 Percent Collection Plane for Constant o /Co. (b = 0, s = 0.1, C /C = 0, )CW/ = 1, f = - 350) 20 5 C p 13 DN Vs. W /W. (s = 0.1, b = o = - 30) 21 14 Gain Vs. Length and Beam Thickness. (D = 0.1, b = O, r = 0 5, D oc/D = 0.5, D tp/D = 0, ~o = - 30) 22 -v

Figure Page 15 Satura Lon Gain Vs. to at Two Different Collection Planes (b = 0, r = 0.5, s = 0.1, Wo / = 0.5, w / = )) 24 p 16 Effici, icy a DN Vs. 0. (b = 0, r = 0.5, s= 0., c /C = 0.5, p/W = 0) 25 C7 P 17 Gain 1,. Distance in the Presence of Circuit Loss. (D = 0.1, r = 0.5, b = 0, s = 0.1, o0 = - 30, Wc/ = 0.5, wDp/W = 0) 26 18 Saturation Level Vs. Loss Factor and Magnetic Field. (D = 0.1, r = 0.5, b = 0, s = 0.1, 4o = - 530, p/w = 0) 27 19 Effect of Attenuator Length on Saturated Output. (D = 0.1, r = 0.5, s = 0.1, b = 0, K = - 30, Wc/W = w p/w = 0.5) 28 20 Electron Collection Vs. Distance. (D = 0.05, r = 0.5, b = 0, s = 0.1,' = - 30, C p/w = 0) 350 21 Electron Collection Vs. Distance. (D = 0.5, r = 0.5, b = 0, s = 0.1, *o = - 50, Co / = c/@ 351 22 Interception Vs. Length and Drive Level. (D = 0.05, r = 0.5, b = 0, s = 0.1, C /cD = 0.5, wo 7o/ = 0) 352 p, 25 i /I0 Vs. Distance for an M-FWA. (D = 0.5, b = 0, s = 0.1, 4r0 0,o = - ) 55 o p c 24 il/I0 Vs. Distance for an M-FWA. (D = 0.1, b = 0, s = 0.1, o = - 50, c = o) 54 o.p c 25 M-FWA Gain Vs. Distance. (D = 0.075, pb' = 1.68, r = 0.33) 36 26 Beating-Wave Gain From Small-Signal and LargeSignal Calculations. 37 27 M-FWA Gain Vs. 0. 38 28 Gain for a Severed Circuit M-FWA. (D = 0.1, r = 0.5, s = 0.1, c /C = CD / = 1 = - 50) 40 c IDp 0 29 Output for a Severed Circuit M-FWA Vs. Sever Position. (D = 0.1, r = 0.5, s = 0.1, CX = co, 4o = - 5o) 41 -vi

Figure Page 30 Large-Signal Positive-Sole R-f Voltage Characteristics. (D = 0.2, b = 0, r = 0.5, s = 0.1, w /j = 0.5) 43 c 31 R-f Voltage Amplitude Vs. Distance for Adiabatic and Nonadiabatic Solutions with a Brillouin Beam. (D = 0.l,.r = 0. 75, C p/W = Wc / = 0.165, s = 1/15, b = 0, I = I = 0) 46 z y 32 R-f Voltage Amplitude Vs. Distance for Adiabatic and Nonadiabatic Equations. (D = 0.1, W /w = WC / = 0.5, b = 0, r = 0.5, s = 0.1, I = I = 0) 47 33 Phase Lead of the R-f Wave Vs. Distance for Adiabatic and Nonadiabatic Equations. (D = 0.1, WD / = w /W = 0.5, b = 0, r = 0.5, s = 0.1, I =I = 0) 48 34 Large-Signal Electron Trajectories for Adiabatic and Nonadiabatic Equations. (D = 0.1, w p/ = c/D = 0.5, b = 0, A0 = 0.1, r = 0.5, s = 0.1, I = I = 0) 49 z y 35 Bunched Beam Description. 51 36 Gain Vs. Distance and Bunch Injection Phase for a Delta Function Bunch in an M-FWA. (-D = 0.1, r = 0.5, s = 0.1, b = 0, wc/w = 0.5, / = 0, = 0, o - 30) 52 37 Distance-Phase Plots for Delta Function Bunches in an M-FWA. (See Fig. 36) 53 38 Dependence of M-FWA Prebunched Beam Gain on D c/0D and w /W at r =0.5. (D = 0.1, r = 0.5, s = 0.1, b = 0, = - 350) 54 39 Dependence of M-FWA Prebunched Beam Gain of w /W at r = 0.25. (D = 0.1, r = 0.25, s = 0.1, b = 0, Cop/ = 0, r = -30) 55 40 Depressed Collector Schemes for Crossed-Field Devices. 58 41 Electron Energy Diagram (r = 0.5). 61 -vii

Figure Page 42 Determination of Collector Segment Potentials. Shown Is an Example of a Cumulative Distribution of the Number of Electrons I such that V > V mi m2 >... Vml, where Vmk Is the Maximum Potential Reached by the kth Electron in Region B of Fig. 41. Energy Recovered by Each Additional Segment Is Maximized with the Tangent Hyperbola Shown, so 4 that.Z n.V.Is Maximized for p = 4. 64 j31 J cj 43 Electron Positions at the Exit Plane q = 4.8 (16 Percent Electron Collection), for a Device with r = 0.5, s = 0.1, Wc/D = 0.5, to = - 30. 68 44 Transverse Velocities at the Exit Plane q = 4.8 (16 Percent Electron Extraction), for a Device with r = 0.5, s = 0.1, Wc/D = 0.5, 4o = - 30. 69 45. z-Velocities at the Exit Plane q = 4.8 (16 Percent Electron Extraction) for a Device with r = 0.5, s = 0.1; Wc/W = 0.5, of = - 30. 70 46 Cumulative Electron Energy Distribution at Different q-Planes. (r = 0.5, s = 0.1, D = 0.05, @W/' = 0.5) 71 47 Efficiency Vs. Interaction Length, for n Collector Segments. (D = 0.05, r = 0.5, s = 0.1, b = 0, ad/@ = 0.25, o = - 30) 73 48 Efficiency Vs. Interaction Length, for n Collector Segments. (D = 0.05, r = 0.5, s = 0.1, b = O, e /@ = 0.5, o = - 30) 74 49 Efficiency Vs. Interaction Length, for n Collector Segments. (D = 0.05, r = 0.5, s = 0.1, b = 0, ~c/@ = i, ~ = - 50) 75 50 Efficiency Vs. Collector Segmentation. (D = 0.1, r = 0.5, s = 0.1, b = 0, X c/w = 0.25, 0o = - 30) 76 51 Efficiency Vs. Collector Segmentation. (D = 0.1, r = 0.5, s = 0.1, b = O, C /C) = 0.5, *o = - 350) 77 52 Efficiency Vs. Collector Segmentation. (D = 0.1, r = 0.5, s = 0.1, b = 0, w /W = 0.75, o = - 30) 78 55 Single-Stage Collector. (r = 0.5, s =- 0.1, b =0, g = - 30) 80 -viii

Figure Page 54 Optimal Segment Potentials for Two-Stage Collector for D = 0.05. (r = 0.5, s = 0.1, b = 0, ro = - 50) 81 55 Three-Stage Collector for D = 0.1. (r = 0.5, s = 0.1, b =0, o r = - 30) 82 56 Two-Stage Collector for D = 0.1. (r = 0.5, s = 0.1, b = O, r0 = - 350) 83 57 Optimal Segment Potentials for Three-Stage Collector for D = 0.05. (r = 0.5, s = 0.1, b = 0, Ao = - 350) 84 -ix

ENERGY CONVERSION CHARACTERISTICS OF CROSSED-FIELD INJECTED-BEAM AMPLIFIERS I. INTRODUCTION An important class of d-c to r-f energy conversion devices is composed of the injected-beam forward-wave amplifier (M-FWA), the injectedbeam backward-wave oscillator (M-\BWO) and the injected-beam drift region (M-Kly). This report deals with the M-FWA incorporating both positive and negative potential sole electrodes as illustrated in Fig. 1. Their distinguishing characteristic is the fact that electron stream potential energy is converted by movement normal to the primary direction of electron flow and further that the stream kinetic energy is undiminished after travel through the interaction region. Potential energy conversion is found to be a more efficient process than is kinetic energy conversion and hence the reason for the great interest in crossed-field devices. The high stream kinetic energy can be partially recovered through a segmented depressed collector. One of the principal limitations in efficiency of energy conversion and average power handling ability is associated with the fact that a high percentage of the beam electrons are collected on the r-f structure over the last half of the interaction length. The material of this report relates to the characteristics of and fundamental limitations in the energy conversion process for an injected-beam crossed-field amplifier. The effectiveness of segmented depressed collectors in recovering beam kinetic energy is also evaluated and thus the overall device efficiency is computed. The effectiveness of stream prebunching by positive-sole interaction and by the injection of delta function bunches are also evaluated.

-2R-F INPUT R-F OUTPUT ANODE R-F STRUCTURE t {~~~0 E C COLLECTOR r1 —( SOLE ELECTRODE CATHODE a) NEGATIVE-SOLE M-FWA ANODE SOLE ELECTRODE E) Bo " _. _,. __._ CATHODE 1 R-F STRUCTURE R-F INPUT R-F OUTPUT b) POSITIVE-SOLE M-FWA FIG. 1 SCHEMATIC DIAGRAM OF CROSSED-FIELD FORWARD-WAVE AMPLIFIERS.

-5II. CONVERSION PROCESS As mentioned above the energy exchange mechanism involves a displacement of the stream normal (y-direction) to the direction of primary stream flow (z-direction) with little change in the electron kinetic energy. The electrons are essentially phase focused relative to the wave and simply give up energy to the r-f wave as they move towards the positive electrode, which is the r-f structure in this case. These favorably-phased electrons are coupled more tightly to the circuit fields as they move toward the structure, and those which are in unfavorable phase positions take energy from the wave and move towards the sole electrode. Fortunately as they do, their coupling to the circuit wave decreases and they thus extract less energy from the wave. This sorting mechanism is partially responsible for the high efficiency of operation characteristic of these devices. A number of interesting analyses of this crossed-field interaction problem have been made and a brief review of the salient features and results is given here as they relate to the present study. Feinstein and Kino' developed an approximate treatment of the quasi-nonlinear M-FWA using equivalent circuit methods. They assumed stream-wave synchronism and that the circuit field decayed exponentially in moving from the circuit to the sole, which was thus moved infinitely far away. In order to make the ballistics equations tractable they neglected space-charge fields and assumed the adiabatic equations of motion. Sedin2 studied both the M-FWA and the M-BWO also using equivalent circuit methods but including space-charge fields, nonlaminar motion, and finite stream thickness. He calculated the induced current at the r-f structure from the rate of change of the y-directed space-charge

-4field. The space-charge field was calculated for infinite line charges between infinite, parallel, and infinitely conducting plates. His space-charge model only allowed calculations to be made for relatively thin streams. For purposes of calculation he also assumed the adiabatic equations of motion, which implies D << 1. A basically similar analysis, yet different in details, has been made by Kooyers and Hull3. Another study, different in several respects (space-charge fields calculated from the Green's function) but also similar in many, is due to Gandhi and Rowe4-7. The theory of Gandhi-Rowe is used here and since the mathematical details have been given elsewhere they will not be repeated. The emphasis is on the calculated results and the fundamental characteristics of the conversion process. In all of the calculations the injected beam satisfies Brillouin conditions for space-charge flow in crossed fields. III. NEGATIVE-SOLE AMPLIFIER RESULTS 3.1 General The general nonlinear interaction equations are solved by digital computer methods and a complete evaluation of the M-FWA performance requires a study of the dependence of gain and efficiency on such device parameters as /'-) IZ \-/o D A ( t ~2V)~ amplifier gain parameter, C /a A e_ B normalized radian cyclotron frequency, o = normalized input r-f signal level, d = r-f circuit loss parameter, z

-5/ ep0 \i/2 Dp/a - ( M 2Eo: ) normalized radian plasma frequency, 0 r = /b'b beam injection position, and A s = w/y~ beam thickness parameter. The assumption of an entering laminar Brillouin beam has been discussed and for purposes of calculation the stream is divided into a minimum of 5 layrers each containing a minimum of 52 representative charge groups. Before outlining the general results we will consider the method of gain calculation and derive expressions for the electronic efficiency and the percentage of available potential energy converted. The normalized r-f voltage along the structure was defined as (p,cqi) A Re L 0 A(qc) i(p) e- j, (l) where q l- z = 2DNs 0 A / p = y/w O(zt) ( z t)+ e(z), zo Z = r-f structure impedance, I = d-c stream current, 0 V __A d-c voltage equivalent of the mean stream velocity, 0 ^(p) __A stream-circuit coupling function. The power output is calculated from

-6p (zt) (2) out = Z2 0 avg so that the interaction efficiency is given by A Pout le = electronic efficiency = I,V (5) where I V indicates the d-c input power. Combining the above o a equations gives the electronic efficiency as I V A2 (q) r - A2> (1) 0 o max0 max e I= (V = " r (1 - ( )' 0 a c c Note that ne of Eq. 4 depends upon the beam entrance position r. A possibly more useful efficiency is obtained by calculating the percentage of available potential energy whidh has been converted to r-f. Such an efficiency is A e A2 (q) ax ( a = r = 1 ( -c/) (5) As illustrated in Fig. 2, the phenomenon of saturation in M-type devices is quite different from that in 0-type devices, being spread out over a relatively long distance. As shown this is independent of the initial r-f signal amplitude. Hence, we see that in order to calculate overall efficiencies some criterion for saturation is needed. Rather arbitrarily this has been selected as the displacement plane'where 70 percent of the beam has been collected on the structure. These displacement planes have been noted on Fig. 2 as a function of the initial input-signal level'K.

-7OI I1 0 ~.,_, S I I^~ I I^2 0 II~~~~~~~I ~ 0 ~? —ICQ~~~~ ~ I I LL::o C~ - z i' -------- r —--- Lo H IItLz. I i ii i 3-! H 0 —o-0 -' 0 1. 0 0 0 0 q p'NIVO

-8The initial signal level applied to the r-f structure is denoted by A and is specified by the input signal-level parameter 4o in db relative to (WI oV /2wc ). The data of Fig. 2 indicate that the device saturates at a level corresponding to WI V /2w independent of 4o. 0 0 c In considering the physical significance of the results presented in the following sections it is helpful to keep in mind the importance of the coupling function r(p) and how it depends upon such parameters as ac / and r. The coupling function is shown in Fig. 3 as a function of the radian interaction width f3b'. The beam or individual electron position between the sole and the circuit is specified by 3y/3b' and we see that x(p) decreases as b1' increases. The dependence of Pb1 on beam position is expressed as b' - { i (6) -b(6) which indicates that for a fixed beam position b1' decreases with increasing CD/C and vice versa. This fact has an important influence on the variation of efficiency with cD /o. In all cases the beam and circuit wave are made synchronous, i.e., b = 0 since this leads to the greatest output. 3.2 Gain and Efficiency A minimum 3 layer Brillouin beam is studied and as shown in Fig. 4 the effect of r-f space-charge forces in such streams is not particularly significant. The gain and efficiency at both the 50 and 70 percent interception planes for a beam injected midway between the sole and circuit are shown in Figs. 5 and 6. These calculations were made assuming that CD / -> O, i.e., that beam space-charge forces could be neglected. Under conditions in which D > 0.05

-91.0 0.8 0.6b'=1.0 0.4 2.0 0.2 - - - 0 0 0.2 0.4 0.6 0.8 1.0,By/O9b' FIG. 5 BEAM-CIRCUIT COUPLING FUNCTION.

-10co \ -- ", n oDL o \ 0 0 0 \ \ -- -OEH \ ~~~0 \\^^g ~~~~~o0 co -H 1 0 0. D10 d) o o 0. o 0 (b)v

-1140 100 GAIN 90 35.. —- EFFICIENCY D= 0.2 80 30 70 D-=0.05 25 >\ _ 60 z I2 \ \ Lj. 20 D - -=0.2 1 0 X -40 I 15 - \ D=0.1O \| D=0.05 i0\ I \ 0 ~~~~\ -20 5 _ 10 FIG. 5 GAIN AND EFFICIENCY AT THE 50 PERCENT INTERCEPTION PLANE. (r = 0.5, s = 0.1, 0 = - 50, b = 0 / = 0) pc/o

-1240 1 \ 100 \ \ —- GAIN. 90 35 - \ \ \..... EFFICIENCY 30 L0 2:0 \ \\ xD=OI | } e -\D \ 0 6 i \ \ I t8 2' \ \ \5\ DD==0.205 z 40 a_ 15 I k7-D=0. DI 30, \:'- \ 20 5 10 0 I I I I I 0 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 wc/w FIG. 6 GAIN AND EFFICIENCY AT TEE 70 PERCENT INTERCEPTION PLANE. (r = 0.5, s = 0.1, 4'o = - 50, b = 0, up/ = 0) z" \ ~.~~~~~~~~~~~

-13the gain and efficiency are not significantly changed by the inclusion of space-charge forces, as shown in Figs. 7 and 8. There is, however, an appreciable lowering of both the gain and efficiency due to spacecharge forces when D = 0.05. The significant influence of space-charge forces at low interaction parameter values in undoubtedly due to the fact that the circuit field forces are too weak to effectively modulate the beam or contain the bunches in the presence of strong debunching forces. Note that the available efficiency, a' has been plotted; in the cases shown the electronic efficiency is one-half of a*. The variation of efficiency with magnetic field as measured by CDW /C is quite interesting in that low values of (C/D are indicated to be optimum, although there is a wide range of oC/w for which the gain and efficiency are relatively constant and near the maximum. The interaction parameter wo I Z -2 c O o0 Ca 2V and the coupling parameter _ sinh Py must be considered in explaining the variation of efficiency with CD /XD. Consider as a reference position in the preceding figures a value of ODc/D 0.75. Now as CD /CD is reduced from this value while D is maintained constant by increasing the circuit impedance Z, it is expected that the r-f output and hence the efficiency would increase.

-1440 100 GAIN 90 35.-. — EFFICIENCY ~~~~~~~~~~\ - ~80 30 30 D=0.2 - 70 | —r""^~ ~ \ ^~ e uD=O.I 25 60 z _ 20\\- \\ 0 D=0.2 50 15 - I \\ \ \\\,^D-0.05 30 o\ \L o 5 \\\ D=0.05 0 K —---- I --- I ---- - I ------ 0 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 Wc/) = OUp/. ) FIG. 7 EF'ECT OF SPACE-CHARGE FORCES ON GAIN AND EFFICIENCY AT THE 50 PERCENT INTERCEPTION PLANE. (r = 0.5, s = 0.1, itr = - 50,

-1540 \ I00 \ I\ ------ GAIN 90 35 \ \ \ \.... EFFICIENCY ta \ \ 1 Z80 \ \ 14030 10 \-D=0.2 70 \ _-___ ^-D=0.2_ 25 \ -20 50 Z \\ \I~ \\oD 5 I 40 m 2o \ I 3o p/@~~~\.30 10 \ = \ \ 4 o - ~.20 ~~~0 0o 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 w /w = w /W p c

-16This change, however, is modified by the fact that Pb' is increasing and hence the coupling function *(p) is decreasing for a fixed beam position. As W /a is increased from this mid-range value, the circuit impedance c must be decreased in order to maintain D constant and hence a lowering of the r-f conversion efficiency is expected. Again this action is tempered by the fact that Fb' is decreasing and thus x(p) is increasing, which should lead to an increased output and efficiency. The beam injection position, of course, should have a significant effect on efficiency since the coupling function varies rapidly over the interaction region. The efficiency at the 70 percent plane is shown in Figs. 9 and 10 for r = 0.75. The lower gain and efficiency are due to the fact that the beam electrons, starting so close to the structure, are rapidly collected on the structure before the r-f level has built up to the full saturation capability of the interaction process. On the other hand, if the beam has too low an initial position the modulation by the circuit wave is ineffective due to the low value of 4(p); thus there must be an optimum injection position which results in a balance of these two effects and should lead to a maximum output. The effect of initial beam position at two different values of w /M is illustrated in Figs. 11 and 12. A value of r; 0.5 is seen to be near optimum. The variation of the normalized length DN to the 70 percent collection plane with various parameters is illustrated in Fig. 13. The stream thickness as measured by s = w/y does not significantly affect the output or saturation characteristics as shown in Fig. 14. It should be noted that these results are for Brillouin flows.

40 " 50 45 35 - GAIN 40 —. EFFICIENCY 30 35 25 -' D=0.2 - 20 25 15 vo | D0.2 __0~ _-~ __ ^'~~I 0 _^~ ^0~-D=0.1 D=0.05 0 ---— 0 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 C /( FIG. 9 GAIN AND EFFICIENCY FOR r = 0.75 AND cu/ = 0. (b = 0. s = 0.1, air = - 50)

-1840 50 1',~~~~~~~~~~45 35 I\ --- -- GAIN 40.....- EFFICIENCY 30 35 25 -0 3 \ \ \ D=0.2 30 30 z Lu P20 25 15t \ \ ^z\ z \ \ S 5 - \ \ \\D=O.i \ \o ICS~ s sN N N=0.0. 2 5L I I\ \,Il \ \ 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 Wc/W = WUp/w FIG. 10 GAIN AND EFFICIENCY FOR r = 0.75 AND CD /aw = w) /CD. (b,s= O s =.1, = - 50)

-1940 100 ~ -'~~ ----- = *GAIN |/~ \ -- = EFFICIENCY 90 80 30 /,,20 2~~~~1 0 =01 25 60 0 1/ \\ D= 0.05 L | Z 20 - 50 1z \I~~~~~ \ \~~a. D 0. 0 FIG.~ D=0.11 GANADEFCECTTE740 P to 30 10 D =0.05 -20 0 0 0I4 0I 0 0.1 0.2 0.4 0.6 0.8 FIG. 11 GAIN AND EFFICIENCY AT THE 70 PERCENT COLLECTION PLANE FOR CONSTAINT w- a. (b = o0 s = 0.1, D = 0, CD /:o/ =o 0.5, = - 50) 0

-2040 100 GAIN -- EFFICIENCY 90 35 80 30/ 5 ~I / \ \ ~~D =0.17 70 25 / 1 / \ \ (, 20 50 W 0 / D= 0.05 0.1~ 0.2 \ - 0.4 0.6 40 15NT / \ Io = -~ \ \/ -30 20 -/ \ ~\10 I.O- D=0.05-/ 0 I 0 0.1 0.2 0.4 0.6 0.8 FIG. 12 GAIN AND EFFICIENCY AT THE 70 PERCENT COLLECTION PLANE FOR CONSTANT o 1/C. (b = 0, s = 0.1~, c)/a = oD/C _ 1, 4- = - 50) 0

-211.6 1.4 r =0.5, D=O.I 1.2 r=0.5, D=0.05 /.0 0.8 0.8- r=0.75, D=0.I 0.6 0.6F Hi\., -~~~r =0.75, X^ ~ D=0.05 0.4 / 0.2 I I I 0 0.25 0.5 0.75 1.0 FIG. 15 DN VS. o, /co. (s = O.1, b = O, o = - 30) s c0

-220 co II r\. II C;I r ^eoo~ II \\' I^z~H vo co (ID (Ar * z ~ l d:e-: 0W^.^' N I~ | f o o ^ <D <~~~~~P0 NI0 3 qp'NIVD

-23 - Since D was kept constant while s was varied, then the total beam current is constant and the beam current density varies. Similar results are obtained assuming a constant current density in the beam; i.e., D varies. The effect of the initial r-f signal level A on the gain of an 0 M-FWA was shown in Fig. 2, where it was evident that the saturated level is approximately equal to (WIoVo/2Lc). The saturation level at both the 70 and 90 percent collection planes is plotted vs. the input-drive level o for two values of the gain parameter D in Fig. 15. The efficiency and distance to saturation are shown in Fig. 16. In high-gain amplifiers a distributed loss along the interaction structure is required in order to insure stability in the presence of reflections. The effect of circuit loss as measured by the loss parameter, d, was included in the circuit equations as a series loss. The gain curves of Fig. 17 reveal the effect of circuit loss beginning at DN ~ 0.3 s and extending all the way to saturation. Also included are the effects of finite-length attenuators where the loss has been terminated at various signal levels below saturation. As seen from this figure, the saturation level for these uniform attenuators is lower than for the zero-loss case. The reduction in saturated output as a function of d at two different values of w /C is shown in Fig. 18. This type of variation is quite similar to the characteristics of 0-type amplifiers with distributed loss. Under high-loss conditions, i.e., d > 1, it is possible to encounter saturation effects under the attenuator which further inhibit the energy conversion process. The effect of the amount of thfree gain beyond the output end of the output end of the attenuator on the saturation power level as calculated from Fig. 18 is shown in Fig. 19 for two values of the loss parameter. These results

-2430 25 20 ^90% INTERCEPTION V~\V 20~\ 0 ~15 — \\ 2 X g D0-O.I! —- ---— D 0.05 I0 70% INTERCEPTION', -30 -25 -20 -15 -10 -5 0' db FIG. 15 SATURATION GAIN VS.'o AT TWO DIFFERENT COLLECTION PLANES. (b = 0, r = 0.5, s = 0.1, C/tCo = 0.5, O/ = 0) o~~~

-2580 70, 60 D=0.05 I 60 LL 50 L I l -35 -30 -25 -20 -15 -10 -5 0 0, db a) EFFICIENCY, 17a 1.2 1.0 ^S D -D = 0.05 0.8 D=- 0.1/ Q 0.6 0.4 0.2 L I I l l l -35 -30 -25 -20 -15 -10 -5 0 0, db b) DNs 4 FIG. 16 EFFICIENCY AND DN VS. 4. (b = 0, r = 0.5, s O = 0.1, w e/l = 0.5~, w / = 0) c~~~~

-2630 /2()d0 ---— / // =20 25 0 0 -10 0 2 4 6 8 I 0 FIG. 17 GAIN VS. DISTANCE IN THE PRESENCE OF CIRCUIT LOSS. (D = 0.1, r = 0.5, b = 0, s = 0.1, 4o 50, o/W:= 0.5y, P/W:= Q)

-270 -5 — 0 -15-0 I E =20 1 co-25 —-- ~0 ~c0.5 1.0 1.5 2.0 d FIG. 18 SATURATION LEVEL VS. LOSS FACTOR AND MAGNETIC FIELD. (D = 0.1, r = 0.5, b = 0, s =0.1, ) - - 30, p/W = 0)

-28V 01 0 0 II ~Q I n 0 3 o_ o a | z: " O s T3A31 NO~IJXI~VS~ O=P>~ V~JOUi NMOQ^~ q —p 0 I I I,,r-I 913A39 NOIJvlniVS O=P IiO_- NMOG qP H

-29indicate that as long as there is 10-15 db of saturated gain beyond the attenuator, or 15-20 db of small-signal gain, there is no appreciable effect on the saturated output level. This result is approximately the same for other operating parameters and agrees well with experimental results. 3.3 Beam Collection on the R-f Structure Significant electron collection on the r-f structure begins at a plane approximately 50 to 60 percent of the total length along the r-f structure and rises rapidly thereafter as shown in Fig. 20. The corresponding interception curves when space-charge forces are included are shown in Fig. 21. The effect of the normalized cyclotron frequency is marked, although at D = 0.1 the effect of WD /w is less pronounced. The interception characteristics as a function of the drive level are shown in Fig. 22, and as expected strong input signals produce a large beam modulation which results in early and large electron collection. 3.4 R-f Current Density in M-FWA's In O-type forward-wave amplifiers, which operate on a kinetic energy conversion process, the fundamental to d-c current ratio varies from 1.2 to 1.6, depending upon the operating parameters. In M-type amplifiers we find that the maximum values of i /I are somewhat lower 1 o than in 0-type amplifiers. Typical results for two values of the gain parameter D and two values of the initial beam location parameter r are shown in Figs. 23 and 24. The greatest values for i I occur for 1 o low values of wC /C which yielded the highest conversion efficiencies as shown earlier. 5.5 Nonlinear Beating-Wave Amplification It is easily shown, using a small-amplitude theory, that when b > b x, i.e., there is no growing-wave amplification, amplification X 0

-500 I ~I "s~~~~~~~I LC\ $ ~ - C) ~ 0ld33N L) 0 0o' o0 ~ o 0 0 ^\. S0, 0ss^^ 1'"- 0 0 Q (0 I0 H NOid3Z83iNl iN3083d

-31o ~ If C\ \^1H 0o \ rec O O 0 c \- cd L\ c N C0j 2 \\ \ I11 1 4- ) \ \ Il\ O\II \ \ \Ss. (Lc Lr\S A [ 0 ~\ v >n- ^^,~~ (0 or I" z.r -.-..x C)'"x _t 0 0' a C_ * - o > 0o ii-1L) H-\LcNO H 0~ ~~~~~H ~ @JH 0* 0 0 0 0 OC)~~~~~~ r1 O tL-\ Lr\ O O 0 0 0D3 0 qN O a) (D9 0 <\ ~ NOlid3O831NI iN3083d < 3 H

-320 o II rO 0 I oL 8 \ d33N 0 NoI1d3:OI31NI lN3Oa3d

-331.0 0.8 c / w=0.25 0.6 0.5 0.4 0.2 0 I 2 3 4 5 6 7 q a) r=0.5 0.5 0.4 0.5FIG. 2 VS. DISTANCE FOR AN M.5 0.2 0.I 0 I 2 3 4 5 6 7 q b) r =0.75 FIG. 25 i1/I0 VS. DISTANCE FOR AN M-FWA. (D = 0.5, b = 0, s = 0.1, -= - 50, c = W) 0o P c

-34I1.0 0.8 C /w =0.25 06 0.4 0.2 0 I 2 3 4 5 6 7 q a) r=0.5 0.6E 0.5 - Wc /W I.0 0.4 - c / =w 0.25 0.30.2 0.1 0 1 2 3 4 5 6 7 q b) r 0.75 FIG. 24 i /I VS. DISTANCE FOR AN M-FWA. (D = 0.1, b = 0, s = 0.1, -o = - 0, o = iA ) o p c

-55does occur due to a beating-wave process. This beating occurs between matched r-f waves traveling at different phase velocities on an r-f transmission line. A similar process takes place in O-type amplifiers and it has been determined that the amount of beating-wave gain is greater in O0-type devices than in M-type devices. Several largesignal solutions are shown in Figs. 25 and 26, where the adiabatic assumption has been made since Dc/CD > 1. Space-charge forces do not significantly affect the process and have been neglected in these calculations. It is quite apparent from these results that the largesignal performance is considerably enhanced over the predicted smallsignal performance. The increased interaction in the linear region of the gain curve of Fig. 26 (q P 10) is primarily due to the better beamcircuit coupling which results from electrons moving closer to the r-f circuit as a consequence of the large initial perturbation in their motion (large A ). The eventual saturation occurring beyond q - 12 is primarily due to the collection of an appreciable fraction of the stream on the r-f structure. An illustration of the effect of varying the input signal level for a larger circuit-sole separation is given in Fig. 27, where the beam is located below the midway position. At or near saturation the percentage of the stream collected on the r-f circuit is approximately 50 percent. 3.6 Effects of a Circuit Sever The effect of a sever in the r-f circuit on the gain and output of a M-FWA is easily evaluated by simply setting the r-f voltage to zero at various displacement planes from the output. P convenient indicator of the sever location is the r-f level at the sever relative to the saturated power output in the absence of a sever. The gain characteristic for a severed-circuit amplifier is

-3614 2.5 0.25 \/~ILARGE SIGNAL(Ao=0.45) 12 I0 8 2 ~ / /I / - b 2.25 _ /./ / LARGE' /"-/ -- SIGNAL \ _. 6' (AO=0.15) \ -2 0 200 400 600 0= 27rDNs, DEGREES FIG. 25 M-FWA GAIN VS. DISTANCE. (D = 0.075, pb' = 1.68, r = 0.55)

-3720 LARGE- SIGNAL RESULTS (D=0.05, wuc/l=l, Ao=0.15, wp/o= 0, Bb' =2) 16 SMALL- SIGNAL 12 / RESULTS ~/ I (r —O, b 2) - // z 8 4- / 0 8 16 24 NORMALIZED DISTANCE, (2-rDNs) FIG. 26 BEATING-WAVE GAIN FROM SMALL-SIGNAL AND LARGE-SIGNAL CALCULATIONS.

-3820 LARGE SIGNAL - / 40% COLLECTION / (Ao= 0.025) D=0.075 /Ao.;5 -3~~b': 2.1 2 15 / -50% COLLECTION b=2 / IAo: 0.05) 10 SMALL SIGNAL\ ( b= 2, r=0.33) / \ 5 0 200 400 600 8, DEGREES FIG. 27 M-FWA GAIN VS. i.

-39shown in Fig. 28 for several different sever positions. The saturated output is taken as that at the plane of 70 percent collection of the beam on the circuit. The r-f output in db relative to the unsevered output is shown in Fig. 29 as a function of the sever location. Thus we see that the sever must be located at least 16 db below saturation and since there is approximately 9 db compression in a M-FWA then there must be 25 db small-signal gain beyond the sever in order to develop the full output. This figure is essentially the same as that for an O-FWA. IV. TWO-DIMENSIONAL M-FWA WITH A POSITIVE SOLE To create a forward-wave positive-sole interaction region it is only necessary to interchange the positions of the sole and structure electrodes, leaving the voltage and field arrangements without change as illustrated in Fig. 1. Since the electrons will still continue to move towards the sole electrode, the principal collection will be on the heavy-sole electrode rather than on the circuit. Unfortunately though, the circuit r-f fields decay rapidly as one moves toward the sole and hence the coupling between stream and circuit is weak. Those electrons which are sorted out and move towards the circuit continue to extract energy from the wave due to their unfavorable phase positions. This suggests that the positive-sole FWA could serve as an effective means of modulating a high-density stream. Gandhi4 has studied the small-signal performance of positive-sole interaction regions and found that such a region supports three purely propagating waves, which beat to yield a strong velocity modulation of a stream passing in close proximity to the circuit. An additional advantage is that this interaction occurs over a very short distance of approximately DN - 0.15. s

-400O II II o ( 00 I I Cr 0 cO H \ 3 H I. co ^rO N.-.qP'NIV9

0 -41-2 I> I / I0. CL 3 0 IJW w 0 -12 w 0 - 4 -8 -12 -16 -20 -24 -28 SEVER LOCATION IN db BELOW UNSEVERED OUTPUT FIG. 29 OUTPUT FOR A SEVERED CIRCUIT M-FWA VS. SEVER POSITION. ( =.1, r = 05, s 30) z o 0 -12 0 -4 -8 -12 -16 -20 -24 -28 SEVER LOCATION IN db BELOW UNSEVERED OUTPUT FIG. 29 OUTPUT FOR A SEVERED CIRCUIT M-FWA VS. SEVER POSITION. (D = 0.1, r = 0.5, s = 0.1,W =' Y = - 50)

-42In analyzing the positive-sole interaction region the stream and circuit equations may be considered separately. Since the d-c field configurations have not been changed, only the circuit field coupling function must be changed in the force equations. The circuit equations are also modified by the new coupling function. The circuit field now decays in the positive y-direction and the coupling function is written as *(y) ( Y _ sinh D(b' - y) sinh Pb' or sinh n w (-4- - P L j(p) = rw sinh rs The appropriate large-signal equations for the positive-sole interaction region are then obtained directly from those for the negative-sole device. A typical calculation of the r-f voltage variation along the structure in a positive-sole device is shown in Fig. 30 for a typical set of device operating parameters. The electron stream characteristics illustrate the large modulation produced in Fig. 30. To illustrate the magnitude of nonlinear effects and also the effects of cyclotron waves included in the nonlinear theory, the small-signal r-f voltage characteristic is also plotted in Fig. 30. In designing an amplifier incorporating a'positive-sole interaction region as a prebuncher the output modulated stream is taken from the positive-sole interaction region and used as the input stream to either a drift region or the normal negative-sole interaction region.

-45-. \ SMALL- SIGNAL 0.8 RESULT 0 V./ 1 0.6 / ~~0.4 /0 I I. 0.2 0 1 2 3 4 5 NORMALIZED DISTANCE, q FIG. 30 LARGE-SIGNAL POSITIVE-SOLE R-F VOLTAGE CHARACTERISTICS. (D = 0.2, b = 0, r = 0.5, s = 0.1, acsCo = 0.5)

-44A similar analysis may be carried out for a backward-wave configuration, the only changes in the nonlinear interaction equations being a change in the signs preceding the right-hand sides of the circuit equations. Such a device, by analogy with the forward-wave behavior, yields backward-wave amplification for proper voltage (b) values. V. ADIABATIC EQUATIONS FOR A TWO-DIMENSIONAL M-FWA WITH A NEGATIVE SOLE The nonlinear negative sole crossed-field equations include the effects of cycloidal motion of the electrons and some effects of cyclotron waves. In the case of weak interaction between the stream and wave the acceleration terms in the equations of motion could be neglected. Such an approximation can be introduced in the case of the M-FWA, providing that the growth of the r-f wave in a cyclotron wavelength x = 2tru /cD is not too large. The condition to be satisfied when using the adiabatic equations of motion is that c A 1. Thus if D, the adiabatic approximation, is reasonable. If this Thus if D < 1, the adiabatic approximation, is reasonable. If this assumption is made, the dependent variable and circuit equations are unchanged and the Lorentz force equations may be considerably simplified. The equivalence of the adiabatic results with the nonadiabatic results is in terms of the r-f voltage vs. distance characteristic. The electron trajectory plots will, however, not be the same since the cycloidal component of motion has been eliminated. The effect of making the adiabatic approximation is easily illustrated by making computations both for coD DD 1 and for cD /ODD ~ 1.

-45The normalized r-f voltage vs. distance curve for w /cD = 1.65 is shown in Fig. 31 where comparison is made between the adiabatic and nonadiabatic results and the small-signal calculation. Note that the r-f signal growth rate is approximately the same for the adiabatic and nonadiabatic cases, although the actual circuit amplitude in the more exact nonadiabatic case is less in the linear region. This difference is due to the appreciable excitation of cyclotron waves in the nonadiabatic case, which are not accounted for in the other cases. Notice also that the saturation amplitude is appreciably more in the nonadiabatic case, which is in agreement with calculations based on energy changes. For C /wD = 5 we would expect that the two sets of equations would yield essentially the same results. This is illustrated in Fig. 32 where A(q) vs. q is shown. The r-f wave phase shift is compared in Fig. 33, where an appreciable difference is noted. Under the adiabatic assumption the beam loading on the circuit is evidently small, since 0(q) = 0. The electron trajectories for the two cases are compared in Fig. 34. Thus we conclude that for D << 1 the adiabatic approximation may be utilized in calculating gain and efficiency, although in order to determine detailed electron motion and stream loading the nonadiabatic equations must be utilized. VI. INTERACTION OF DELTA FUNCTION BUNCHES A crossed-field Brlllouin beam may be prebunched through the use of a positive-sole interaction region or one of several varieties of cavity circuits. To what extent can the gain, efficiency, etc. of M-FWA's be improved through various prebunching schemes? The performance is readily evaluated using the nonlinear theory by injecting near delta function bunches into the interaction region. The prebunched beam model

-460.35 - SMALL- SIGNAL THEORY 0.3 _v^~ | | / NONADIABATIC C3 0 /:-! =-I I. 0.1 0 2 3 4 5 NORMALIZED DISTANCE, q FIG. 31 R-F VOLTAGE AMPLITUDE VS. DISTANCE FOR ADIABATIC ANT NONADIABATIC SOLUTIONS WITH A BRILLOUIN BEAM. (D = 0.1, r = 0.75, p/ = o = 0.165, s = 1 15, b = 0, I = I = 0)' p' c z y

-47o Cl) H o EI IC) I I m )N MO~C < ( OH < m z| ^ z ~ ua H Nz -.- C; 0 o 3 0 000 H Q N<, LU.I I H c/

-48o 0 H H Ei II OD o H!\ H ^ LJ ~r < 0 U W c Ck.)~~~5 L 0 C) P — z N co 0 Q:.~ ~QT~ F. < II 0 z 0 0 0 0 0 I II Fr 3 SNVIOVd'(b)6'aOV3J 3SVHd J-' H

-4920 — I st LAYER, ADIABATIC -— 2nd LAYER, ADIABATIC 18....- Ist LAYER, NONADIABATIC --- 2nd LAYER, NON- 16 -ADIABATIC /._/ C17 / IS 14 17// i / l^ 0I ~2.~~ 17,7 I_ I I 1 2 0 I 2 3 4 5 6 b = 0, A = 0.1, r = 0.5, s = 0.1, = Iy = 0) 0 I02 3 4 5y

-50is illustrated in Fig. 35 and a \/20 bunch width is assumed which corresponds to i /Io i 1.98. The maximum decelerating phase of the circuit wave occurs at t/2 and it is therefore logical to expect that the optimum injection phase would also be a = t/2. The dependence of output on bunch injection phase for an M-FWA is shown in Fig. 36 for three values of injection phase. As expected the dependence on phase is not critical; the corresponding gain curve for the amplifier without prebunching is also shown. Thr striking result is the large reduction in length required to reach saturation. The calculations were terminated at a displacement plane corresponding to collection of 90 percent of the entering stream current. The ultimate level reached is approximately the same in all cases. The focusing of the bunches at or near the rt/2 point is depicted in Fig. 37 where the trajectory coordinates for the bunches are displayed. Since 3 charge layers were used in the computation 3 bunches were inserted and space-charge forces were neglected. Along with the indicated change in phase there is a conversion of approximately 4-6 percent of the bunch kinetic energy to r-f. The prebunched M-FWA characteristics are most dependent upon Co /C and r as shown in Figs. 38 and 39. The unbunched gain curves are also shown for comparison. The saturation levels are again virtually the same with and without bunching in r = 0.5 cases independent of /CD. In the r = 0.25 case where the beam is launched near to the sole electrode the r-f levels reached by the prebunched amplifier are considerably higher than those of the corresponding unbunched amplifier. The reason for this is that in the unbunched beam case the electrons move away from the sole quite slowly with the result that a large number are focused in unfavorable phase positions and are eventually collected on the sole electrode.

-51j \8 HM o C) z D 0 0 k ) \Q e ~^^~~0- Wa

"52Q: Ic:: a I < < Ic cr - m I::r: - LU LL LLJ L i H- H- - - azr2: 22 H LJ ULUJz I I o I3 LiJ JLd L L | 0 30^ ~~~~~~~~\ m o U CO \o~~~~~o o II H II \ 0 - O.,O O O KqP' N 1"3r qp N

-531.8 I 17 33 1.6 1 3 1.2 CENTER AT I RADIAN 2 0.8 —1 0.422 I \ 0 0.5 1.0 1.5 2.0 2.5 FIG. 537 DISTANCE-PHASE PLOTS FOR DELTA FUNCTION BUNCHES IN AN M-FWA. (SEE FIG. 36)

0 -54000 co ~ QcO 0 4 LO 34 CC)OLi,, L9 H/ 3 m m I \\ ^Sz Z P3 o 3 o I 3 3m 00 IL) Z ~-.\ I'!, ^ca co II H Ii ^ s yoo ~~~~ ~ p 3 3 b c" - o0:3:3, C)

0 \, LLJ r\ w\II 0,0 W ) — II C-) z A) p V " \ a \ "< II;It 0 II Vz I 3 3O4 I 3rcj F9 II o \ \ - ^ | ^' 3 \ \ 3. o o,.\3 \ H \ ~I\ \ \ \I 0 0 0 0 0 t Pro CJ N qP'NlVS

-56The influence of space-charge forces, i.e., p/D A 0 on performance is illustrated in Fig. 38. When D > 0.05 the inclusion of space-charge forces in Brillouin beams does not significantly alter the results. On the basis of these results one expects significant improvements in M-FWA performance are achievable through beam prebunching. The most significant advantage is the markedly reduced length and if the r-f circuit field is much greater than the space-charge debunching field there will be an increased efficiency. VII. COLLECTOR DEPRESSION 7.1 Depressed Collector Design. In the ordinary crossed-field device, electrons in the spent beam appearing at the end of the interaction region still carry considerable kinetic energy, while most of their potential energy may have been transferred to the r-f wave. This kinetic energy, which may typically account for one-half of the total initial beam energy, is normally dissipated in the form of heat when the electrons strike the collector. However, a substantial portion of this energy can be recovered by using a collector system which comprises one or several electrodes at potentials depressed below the anode potential. The objective in designing a depressed collector is to decelerate the electrons which appear at the exit plane to near-zero velocities so that as much of their energy as possible is in the form of potential energy at the point of collection. The potential energy of electrons collected by collector segments at potentials below anode potential is recovered, as compared to collection at anode potential. The overall efficiency of the crossed-field device is therefore improved to the extent that beam kinetic energy can be converted to potential energy before collection.

-57Several methods of achieving the required electron deceleration are conceivable. One possibility is that of decelerating the beam adiabatically in a slowly decreasing electric field, between divergent anode and sole electrodes. This type of collector has been studied at Litton Industries8. A controlled decrease in magnetic field would achieve similar results, i.e., cause electrons to drift adiabatically toward the sole, gaining potential energy and losing kinetic energy. A different possibility is to impose a rapid change of electric field at the exit of the interaction space, in order to force the electrons into cycloidal paths where points of high potential energy are periodically reached. Alternately, a rapid change in magnetic field might be provided by placing a magnetic shield around a collector electrode arrangement, thus avoiding the nonlaminarity of a cycloiding beam. Electrons would then be decelerated by a retarding electric field alone. The adiabatically diverging collector will generally be long and recovery of electron energy will be limited by the final drift velocity at the end of the collector. The depressed collector with an abrupt change of electric field, on the other hand, allows electrons to reach near-zero velocities in one cycloidal length; it becomes feasible to operate a number of collector segments at successively higher potentials such that groups of electrons can be collected at their respective energy levels. However, the performance of this type of collector will probably be more critically dependent on the particular energy distribution present at the exit plane. Therefore, the design of a cycloidal type of depressed collector must be carefully considered. In Fig. 40athe basic configuration of each type of depressed collector is illustrated.

-58(+)E E Bo Ey v= - EVERYWHERE Y x t/_.z a) QUASI-ADIABATIC SLOWING THROUGH A GRADUAL CHANGE IN ELECTRIC FIELD. ANODE, V=O ///////// // /// / /// /// d tE SOLE, / F//VEZ.-v 2 / 4 Y x -z b) RAPID SLOWING THROUGH AN ABRUPT CHANGE IN ELECTRIC FIELD. FIG. 40 DEPRESSED COLLECTOR SCHEMES FOR CROSSED-FIELD DEVICES.

-59In the following, a design procedure for a depressed collector as in Fig. 40b will be outlined. It will be shown how to arrive at an optimal choice of segment potentials when the exit positions and velocities of a group of electron (or line charges extending in the magnetic field direction) are known from a crossed-field interaction theory. The design procedure to be described is based on a single-trajectory approach in predicting electron motion in the depressed collector region. Space-charge forces are therefore not considered. Also, the transition in electric field from the interaction region to the collector region will be considered as a step function, such that the effect of transverse transition fields on the cycloidal motion is neglected. Once a particular collector configuration and segment potentials have been determined, one could of course evaluate actual trajectories and electron collection by solving Poisson's equation in the collector region. For design purposes, however, this approach would hardly be practical. Consider first the range of possible conditions (z,y,y) for the electrons appearing at the exit plane. The appropriate solution of the Lorentz equation for an electron injected into a region bounded by two parallel plates of spacing b' and potentials 0 (anode) and -V (sole) is y = (u - z^) sin C t + y cos u t (7) = (u z )(l - cos C t) + y sin - t + z, (8) 0 0 c o c - 0 z = u t - 2y~ i - z = u t - 2y fl - -~-'1 sin C t + 2y - (l - cos C t), (10)

-6owhere u = (Ey/Bo), = (uo/2rb')r = (Y/b') z, y = height of injection above sole. The relation 2o vy = u follows from the choice of C" O 0 u2 r V * (11) 0 2Now define sgn (uo - o) (U- )2 + y2 = u6 (12) 0 = tan 0 (13) u - 0 0 and the equations of motion in the parallel-plate region can be rewritten as follows: Ad = us sin(wc t + 0), (14) z = u - u6 cos(c t + 0), (15) o 0 y = Y + 2Y o ^ - -- - 2yO - cos c0 ^t + i)' (16) z ut + 2y - [sin - sin(c t + )]. (17) o The total energy of an electron, relative to V, is given by the sum of potential and kinetic energy, Vt y 1 2 + z2 V = 1 b' + TV' (18) 0 0 and thus depends on the triplet (y,y,z). In the absence of r-f interaction, Vt/Vo will of course be independent of z. Electrons will then travel in a (y,r,z)-space on paraboloids of total energy. A crosssection through these paraboloids for y = 0 is shown in Fig. 41. Electrons which cycloid through the interaction region will describe elliptic paths on the energy paraboloids, whose projections are shown as straight lines

-6l0 i31~~~~~~3 < ^ " ^ ^s ^~o cn W —K O zcJLOO~ ~ ~ ~ ~ ~ ~ ~ ~~~i 0. \ \'. 0 0 q o0' >/o 0> H' CL o \^ in o 6 ~ NOLSd JSASVL

-62in Fig. 41. These ellipses must be centered about the line z = u, y = 0, which is evident from Eqs. 14 and 15. The region of electron transmission (A) is defined by all possible cycloids which do not lead to interception. Under conditions of fairly good beam injection into the interaction region and no r-f interaction, electrons will appear at the exit plane on the paraboloid of energy Vo, not too far from z = u. When interaction with an r-f wave takes place, many electrons will lose energy and move to energy surfaces between 0.5 V and V. Some will gain energy and move to surfaces between V and 1.5 Vo. Typical exitplane data derived from crossed-field interaction theory will be discussed in the next section. However, it must be true in general that the z-velocity distribution of electrons has the average u and that a great number will remain near the line z = u, y = 0 on our energy diagram. Electron motion in the collector region may be described by Eqs. 14 through 17 as well, if the drift velocity is reinterpreted as E' u' - y (19) o B where E' is the reduced d-c electric field of the collector region. y This parallel-plate description will be valid beyond a transition region, whose effect on electron motion is neglected. The optimal choice of E' for the purpose of decelerating electrons to zero velocity becomes y evident from Eqs. 12 and 15, when the average z = u, and the average y = 0 at the exit plane: E' = 1 E (20) y 2 y such that u' = uo/2. Then the average u' will also be equal to O o

-63u /2, and electron velocity will be zero at the lowest point of the collector-region cycloid. In Fig. 41, several possible cycloids are shown for the collector region; only those electrons reaching their lowest point on the line z = 0 momentarily convert all their energy into potential energy, others always have some energy in kinetic form. All electrons appearing inside region A at the exit plane must then reach the points of highest potential energy inside region B. This amount of electron energy is available for recovery and can be computed from the exit conditions (y,y,z) of each electron. Expressed in the normalized variables of crossed-field interaction theory, the total energy of an electron, relative to Vo, is t = (1prs) + (1 + 2Du)2 + 1 ( S vJ (21)?1 a v (21) O c The potential energy available for recovery at the lowest point of the collector-region cycloid is given by V V ut vm vt 1^ (T (22) Vo V 2 2 u 0 0 0 where 2 + 2Du + - v. (23) o c As we can only provide a finite, and practically a small, number of depressed collector segments, less than the sum of the available potential energy can be recovered. The n electrons appearing at the exit plane may now be ordered by decreasing magnitude of available potential energy V k. A cumulative ~ mk distribution of Z V for ~ = 1, 2,... n is shown in Fig. 42. The ink~~...

-641O 0 p~o H H o F aq X Z Cq SNO U I!-I O I-N 0O M \ a A M o M

-65maximal energy which could be recovered with an infinite-stage collector is equal to the area under the distribution. If p collector segments at potentials V. were provided, each collecting n. electrons, the total cj energy recovered equals.Z n. Vj This energy is shown by the four j= j cj rectangular shaded areas in Fig. 42. If only one segment were used, the energy recovered would be n V. If two are provided, the sum (n + n ) c4 1 2 Vc + (n3 + n )V4 would be recovered. Thus the energy additionally recovered by the second segment is only equal to (n1 + n2)(Vc2 - VC4). When two more segments are present, their potentials would optimally take the values V and V, and small additional areas would be added to total recovery of potential energy. The area of additional recovery can be maximized graphically by drawing a rectangular hyperbola tangent to the cumulative energy distribution within each sectioning (V.-V.). ci cj It can be appreciated that a small number of segments is sufficient to recover 90 percent or more of the available potential energy. The graphical procedure can be rapidly applied for p = 1, 2, 4 segments to maximize p n n. V. < ) V, J cj - i mk j=l k=l Computationally, we have to compute this sum nP times by varying V over all n Vk, then varying V over all Vmk in turn for each choice of Vc, ~mk- l ~ c2 mk ci and so forth. p may be any integer in this procedure. If the conversion efficiency of the device was To, the electronic efficiency with the p-segment depressed collector will be 1p+ c- n cj (24) p 0 N Vn j=l where N = original number of electrons injected, NVo = d-c input power.

-66The electronic efficiency which could be achieved with an infinite-stage collector is o = no + V (25) k=l This will generally be less than 100 percent, because: n n Vmk Vtk k=l k=l and n < N, i.e., some electrons have been collected in the interaction region. It is clear that the further the amplifier is driven into saturation, the smaller is the energy still to be recovered by a depressed collector. The physical arrangement of the segments will be such that the d-c electric field E' remains constant throughout the collector. Due y to the effects of transition fields, it was brought out by studies of the analogous trajectory problem in parallel-plate crossed-field guns that E' might optimally be somewhat smaller than E /2. The length y y of each collector segment should at least be one cyclotron length, 1 2ic ~ = u - = 2 rb'. (26) c 7.2 Computational Results. Electron position and velocity data from nonlinear crossed-field amplifier calculations for six different device parameter combinations were used for depressed collector design. Positions and velocities for 96 injected electrons in three layers were taken at three points from each interaction

-67runs corresponding to about 20, 50, and 70 percent electron extraction. The distributions of recoverable potential energy were computed from p, v, and 1 + 2Du according to Eq. 21. The sum n.V. was then maximized j=l j cj for p = 1, 2, and 3. Computation time is not too long up to p = 5, because the computer need only scan one eighth of the n x n x n cube of points. It is sufficient to vary Vc only over all Vc2 > Vc for any one choice of Vc; and likewise V3 only over V3 > Vc > Vc for any choice of V and V. It should be mentioned that one cannot simplify C2 C1 the search for the optimal combination of potentials by stopping at a local maximum of recovered energy, because Z n. V is generally not j=l J cj a monotone function away from the absolute maximum in p dimensional V.-space. cj - Figures 43, 44 and 45 show a set of typical position and velocity data after 16 percent of the injected electrons have been collected on the circuit. The major variation in total energy of the electrons will be due to the variation in p, i.e., in potential energy at the exit. Transverse velocities do not exceed 5 percent of u, and neither does the z-velocity deviate by more than 5 percent from u. When positions and velocities are sampled at points of heavier saturation and electron extraction, the variations are somewhat greater. Figure 46 shows the cumulative distributions of recoverable energy for three exit planes on the same interaction run. It is to be noted that a spread toward higher energies occurs as the device is driven more into saturation. As the number of electrons appearing at the exit plane decreases with increasing collection on the circuit, additional "electrons" were introduced into the position and velocity data by interpolation. This accounts for the differing step sizes in Fig. 46.

-6820 CIRCUIT / 7 o LAYER I 16 A LAYER 2 * LAYER 3 14 12' e,S // \\/I ~/10/. 8 6 ~/' 2 _ _ _ _ _ _ _ _ _ _ _lll.. 4 - -3 -2 -I 0 2 3 4 FIG. 43 ELECTRON POSITIONS AT THE EXIT PIANE q = 4.8 (16 PERCENT ELECTRON COLL.ECTION), FOR A DEVICE WITH r = 0.5, s = 0.1, c/~ = 0.5, f' = - 50o.

-691.0 0.8 o LAYER I A LAYER 2 i 0.6 * LAYER 3 ~0.4 4'// -0.2 FIG. 44 TRANSVERSE VELOCITIES AT TEE EXIT PLANE q = 4.8 (16 PERCENT ELECTRON EXTRACTION), FOR A DEVICE WITH r = 0.5, s = 0.1, - / = 0.45, = - 0. C0

-700 or r -j -i -J E4~ ~~~~o < * J0~~~~~~j- ^ ~c E- I L Q0 *^r <c- EII. 0 LO 0 a0) H *i ^ \o -g 1:0 -9- 6 (b)lnaG + I

-7190 q =4.8 (16% Extraction ) 80 70 - cr 60 m z 50 q=5.6 (51% Extraction) I / J — q =6.4 (67% Extraction) 30 20 10 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 Vm/Vo RELATIVE ELECTRON ENERGY FIG. 46 CUMULATIVE ELECTRON ENERGY DISTRIBUTION AT DIFFERENT q-PIANES. (r = 0.5, s = 0.1, D = 0.05, %c/ = 0.5)

-72The results of the optimization of collector segment potentials for the exit data of six interaction runs are given in the following figures. The device parameter combinations are: D = 0.05 oc/ = 0.25 c WC/W =0.5 (c /(C = 1.0 D = 0.1 c/cD = 0.25 o c/D = 0.5 a)/ w = 0.75 Initial amplitude is A = 0.0225 (o = - 30 db) in all cases, also 0 0 r = 0.5, s = 0.1 and b = 0. The efficiencies non, T, 2), 13 and qr are plotted as a function of the exit plane positions for the six cases in Figs. 47 through 52. The conversion efficiency q of course increases as the amplifier is driven more heavily into saturation. The efficiency ql for one depressed collector segment at optimum potential is much higher than Io, and usually almost independent of exit plane position. The use of two and three segments affords an additional, even though smaller, improvement in efficiency. It can be appreciated that T3 is not too far below the curve for qr in all the cases studied. The maximally achievable depressed collector efficiency qr decreases as the tube is more heavily saturated because electron kinetic energy is irretrievably lost when electrons are collected on the circuit. It is seen that the case for CDc/ = 1.0, D = 0.05 shows a somewhat different behavior from the others; this is mainly due to the more pronounced velocity spread which occurs in the process of very rapid saturation of this tube.

-7300 l ll. 11E-i t~C) OrcU uI II g O O t O ~ ~ ~~~~~~~o C 0 o II nr I._)~~~~~~~ ~~~ — I tAJl) A - H Lfl Cl ~ C I coo - I \ I ~ II / / LN3 3d \ A AN3I o 0.. rI I I

-74CO o 0 I' — ~I1"o z Ez 0 0 a. 0o LL I I I I I II L) z _., z0 Io 0 8 rO'j -0 L 0 lN3L383d~r H 0 a. 0 HE- II 0 0 0 0 — 0 H O iN3J39d'AO3N131J.J3

-75z o LS~ ~~~~~~~~O~~~ 1 CO 9 I M I- 8 )o - o v oZ~o Eo 0 CO z( ) a C" L) Z 0 3 H O 0H Cr \ Z 0 H 0 U k H II _ 00 ~ C0 H N d'D (3N3 1- c 3 lN33083d AON31314J3

-76) 0 0 ~ I I I II I ~~z ~~~~~~Id ~~~0 OD~~~~~~~~~~ ~~~~F~~~~~~~~~~~- HO~ o \c ( II \ LL. \ A I I I I I I I - zz ( \0 rz 0 o C H 9 ro _ \3 " Q 0ro OJ - 00 iNBii id'A N3VJSdr 0 0 0 0 IN33W3d'A3N3131J-3

-77z - C J I; w ) 1 1, ii Ld II I \ )1 L~ ~ ~ ~~~~O~ ~O IcNJ E- II o U 8F-~~~~~ H~~oH z I I I I It 0:: t-l: r\ t - O LU/ / / / \: II 0 ~ 8croJ<M0 II 3,, II II, I I,,I U O.LN30213d ft

-78L\o O Z 0 LdL, E I I I LH W II L)O 0N3083d:0 I —' (.0._ 03 L"/ IIIC~~ \C~~~~ C^~~~- II.3 03 LI I-I,-I -I n 0 CO (,0 0 0J 0 00 CD N1 lNJ333d'A3NJIc3dB

-79Finally Figs. 53 through 57 give the optimal combinations of segment potentials which must be used to achieve the efficiencies just discussed. The general tendency is for all optimal segment potentials to increase with increasing saturation of the device. This reflects the shift in energy distribution noted in Fig. 46. For a single-stage depressed collector, the optimal potential is to be found between 0.8 V and 1.0 V, as seen from Fig. 53. Only the case of wC /W = 1.0, D = 0.05 requires a low optimal potential initially. When two collector stages are used, the optimal potential of the first stage is below cathode potential in all cases. For the three-stage collector, the potential of the first stage must be as high as 1.1 to 1.3 V, i.e., up to 30 percent depression below cathode potential. The third stage then tends to require as little depression as 0.4 to 0.7 Vn 0 In examining the results of these depressed collector calculations, it should be borne in mind that the effects of transition fields and space-charge have been neglected and actual trajectories may be such as Io deteriorate the performance of a depressed collector optimally designed on the basis of given exit conditions. Moreover, the electron positions and velocities in an actual device might depart considerably from the conditions assumed in the interaction calculations, due to nonlaminarities and other imperfections in the beam optics. Then the recoverable energies V may be significantly below electron total energies Vt. whereas in the above calculations this difference is generally small. It has become clear, nonetheless, from the above results that considerable efficiency improvements can be expected from a depressed collector, even when only one stage is provided. More than three collector segments do not appear to be necessary to exploit most of the efficiency improvement

-80) - 00 0 e0 0,,O 0,\ 0 N' I \r 3. 0 II'*IO I I% 0 0 I 4I o IO!I 03 0 I! [, 0 ^/(^ C) 0 0 0 0A 3

OD 00 -8l- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~0 II 3 \0 o~~~~~~ \ ^> \ KI 3 t 0 E-i 0 U) - I 0 LO~~ ~~~~~~~ i\~~~~~~~~~~~~~~( PSII o LOLO) LC0 o3 3 I II~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3I^ I I \ r \ M 3 ~ "~~~~~~~~~~~~3 -~~~~~~~~~~~~ - -6 gn 3" [^ — **-o 6 o ~A/ 3A

-82") On C\j 0 a'\3 II C)) j H1 0 0 0 0 Crl LO 0 I \ \ I\: * _ A O O o, 0 I0 N 3 - I 3 - o o- 00

-830 OD \-00 LC) L O, \ ~ r \^ l ^\ I 0 3 1 j o H 3 II 0 II \ \ I r;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I 3 L(-4 * LO 0' 0 0 a:: L~~~~~~~~~~~~~~ ~~~~~~E-i II 6 H II~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I - -o 6 6 ~rA d~~~~~~~~~~~~ d cu q a> ~~~~~D (a~ o d o~~~~~~~~~~C OA/ 3A~~~~~~~~~~~~~~~~~~~~~~'

-8400 OD 0~~~~~~~ II \\~ \ \ N \~ F 0 0 0 Cr 0 11~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1 3c:Q rZ 0 10 0 ~ ~~~~~~~~~~~~~~~~~~~~~3 U311~~~~~~~~~~~~~tJ Q~~~~~~~~~~~~'I~~~~~~~~~~~~~~~~~~~~~~~~~~~~I 3~~~~~~~~~i 0t 3J0~.< II~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I 0~~~~~~~~~~~~~~~~~~~~ I I~~~~~~~~~~~~~~~~~~~~~~I ~A/ 3A

-85which could ideally be obtained from the spent beam. The curves do not generally exhibit a large sensitivity to such device parameters as D or c /C. Placing a depressed collector at a point corresponding to between 50 and 70 percent beam collection on the r-f circuit would seem to combine a large conversion efficiency with substantial energy recovery from the spent beam. VIII. CONCLUSIONS The energy conversion process in injected-beam crossed-field amplifiers was examined for both negative- and positive-sole configurations revealing an efficient potential energy conversion with little change in the stream kinetic energy after travel through the interaction region. The gain and conversion efficiency are principally dependent upon the stream injection position, r and the normalized cyclotron frequency, C /C. Low values of LC /C i.e., C /C < 1 and a stream c c c injection position approximately midway between the sole and r-f structure, 0.4 < r < 0.6 yield near maximum conversion independent of the gain parameter, D. Saturation of the r-f signal level is spread out over several wavelengths and is independent of the initial r-f signal level. The saturation level is approximately given by DI V /2 c. A sever in the 00 c r-f circuit on a distributed attenuator along the structure does not significantly reduce the r-f saturation level provided that there is at least 15-20 db of small-signal gain beyond the sever or the end of the attenuator. A positive-sole interaction region was shown to serve as an effective prebuncher wherein the output modulated stream is taken from the positive-sole interaction region and used as the input stream

-86to either a drift region or the normal negative-sole interaction region. The most significant advantage is the markedly reduced length and if the r-f circuit field is much greater than the space-charge debunching field there will be an increased efficiency. Segmented depressed collectors were shown to be effective in recovering beam kinetic energy and thereby can lead to a marked increase in overall efficiency. Probably not more than three collector segments are justified in view of the added complexity and the improvement is not critically dependent upon D or w /C.

LIST OF REFERENCES 1. Feinstein, T., Kino, G., "The Large Signal Behavior of Crossed Field Traveling-Wave Devices", Proc. IRE, vol. 45, pp. 1364-1373; October, 1957. 2. Sedin, J. W., "A Large-Signal Analysis of Beam-Type Crossed-Field Traveling-Wave Tubes", Tech. Memo No. 520, Hughes Research Laboratories; July, 1958. "A Large-Signal Analysis of Beam-Type Crossed-Field TravelingWave Tubes", Trans. PGED-IRE, vol. ED-9, No. 1, pp. 41-51; January, 1962. 3. Hull, J. E., Kooyers, G. P., "Experimental and Theoretical Characteristics of Injected Beam Type Forward-Wave Crossed-Field Amplifiers", Nachrichtentechnische Fachberichte, vol. 22, pp. 151-158; 1960. 4. Gandhi, 0. P., "Nonlinear Electron-Wave Interaction in Crossed Electric and Magnetic Fields", Tech. Rpt. No. 59, Electron Physics Laboratory, The University of Michigan; October, 1960. 5. Gandhi, O. P., Rowe, J. E., "Nonlinear Analysis of Crossed-Field Amplifiers", Tech. Rpt. No. 37, Electron Physics Laboratory, The University of Michigan; June, 1960. 6. Gandhi, 0. P., Rowe, J. E., "Nonlinear Analysis of Crossed-Field Amplifiers", NTF-Nachrichtentechnische Fachberichte, vol. 22, Germany; 1960. 7. Gandhi, O. P., Rowe, J. E., Crossed-Field Microwave Devices, Academic Press, New York, vol. I, pp. 439-495; 1961. 8. "Study of Depressed Collectors for Crossed-Field Amplifiers", ASD-TDR-62-989, Contract No. AF-33 (616)-7921, Litton Electron Tube Corporation, Research Laboratory, San Carlos, California; April, 1963. -87

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No. Copies Agency 1 Mr. Gerald Klein, Manager, Microwave Tubes Section, Applied Research Department, Westinghouse Electric Corporation, Box 746, Baltimore 3, Maryland 1 Dr. Om P. Gandhi, Central Electronics, Engineering Research Institute, Pilani, Rajasthan, India 1 Dr. Walter M. Nunn, Jr., Electrical Engineering Department, Tulane University, New Orleans, Louisiana 1 Hughes Aircraft Company, Microwave Tube Division, 11105 South LaCienaga Blvd., Los Angeles 9, California, Attn: Mr. A. Weglein

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