THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN QUARTERLY PROGRESS REPORT NO. 10 ON IC RESEARCH IN MICROWAVE DEVICES AND QUANTUM ELECTRONICS This report covers the period August 1, 1965 to December 1 1965 Electron Physics Laboratory Department of Electrical Engineering By: B. Ho Approved by: i-I N. A. Masnari C-. Yeh J. E. Rowe Pr6Oject Engineer C. Yeh Approved by:A ERowe, Director Electron Physics Laboratory Project 05772

tfl 6I I7 7) A'-

ABSTRACT The problems of beam expansion and phase focusing in the case of cyclotron-cyclotron and cyclotron-synchronous wave interactions in a d-c pumped quadrupole amplifier are studied and compared. Although it seems that cyclotron-cyclotron wave interaction is more favorable compared to cyclotron-synchronous wave interacton as far as bea expansion is concerned, the phase focusing in the latter case is so strong that the beam never has time to expand before it is brought back into focus againo A theoretical computation of the cross-modulation products in a wideband tunnel-diode amplifier with two input signals of slightly different frequencies has been made. The effect on the output products of varying the relative aamplitudes of the input signals is shown in aseries of graphs. Results on the experimental measurements of the crossmodulat'ion products of a Crestatron power amplifier with two input signals of slightly different frequencies are shown in detail. These results indicate that the 2f and 2f harmonics are the strongest 1 2 modulation products if they happen to be within the bandwidth of the amplifier. Otherwise, the components 2f -f or 2f -f will be the 1 2 2 1 stronger. The effect on the output products of varying the relative power level of the input signals has been invostigated in great detail.

TABLE OF CONTENTS PagEe ABSTRACTii LIST OF ILLUSTRATIONS vi PERSONNEL viii ARTICLE ISSUED DURING THE LAST QUARTER ix GENERAL INTRODUCTION 1 2. STUDY OF FREQUENCY MULTIPLICATION IN AN ANGULAR PROPAGATING CIRCUIT 2. STUDY OF A D-C PUMPED QUADRUPOLE AMPLIFIER 2 1 Introduction 2.2 The Effect of Beam Expansion. The Phase-Focusing Effect 12.4 Pogram for the Next Quarter 12 4. INVESTIGATION OF THE CROSS-MODULATION PRODUCTS IN A WIDEBAND TUNNEL-DIODE AMPLIFIER 15 4. 1 Introduction 15 4.2 An Approximation 15 4.5 The Cross-Modulatilon Products 15 4.4 Program for the Next Quarter 21 5. EXPERIMENTAL CHARACTERISTIC S OF MULTI SIGNAL TRAVELING-WAVE AMPLIFIERS 21 5. 1 Experimental Investigation 21 5.2 Single-Frequency Operation 22/", 5.5 Multisignal Operation 27

Page 6. THEORETICAL STUDIESN MLTISIGNAL MICROWAVE AMPLIFIERS 40 6.1 Nonlinear 0-Type Amplifier Calculations 40.2 Multisignal Crossed-Field Amplifiers 41 6.3 Nonlinear Theory of Multisignal Crossed-Field Amplifiers 41 6.4 Program for the Next Quarter 42 7. GEN~ERAL CONCLUSIONS 42

LIST OF ILLUSTRATIONS Figure Pg 5.1 Beam Expansion for Cyclotron-Cyclotron Wave Interaction for Weak Pump Field Strength. 5 5.2 Beam Expansion for Cyclotron-Cyclotron Wave Interaction for Medium Pump Field Strength. 6 5.1 Beam Expansion for Cyclotron-Cyclotron Wave Interaction for Strong Pump Field Strength. 7 3.4 Beam Expansion for Cyclotron-Synchronous Wave Interaction for Below Critical Pump Field Strength. 8 5.5 Beam Expansion for Cyclotron-Synchronous Wave Interaction for Critical Pump Field Strength. 9 3.6 Beam Expansion for Cyclotron-Synchronous Wave Interaction for Medium Pump Field Strength. 10 5.7 Beam Expansion for Cyclotron-Synchronous Wave Interaction for Strong Pump Field Strength. 11 Phase Focusing for Cyclotron-Cyclotron Wave Interaction. 1 5.9 Phase Focusing for Cyclotron-Synchronous Wave Interaction. 14 4.1 Computed Stat'ic Characteristic of a Tunnel Diode..16 4.2 Cross-Modulation Output at Different Frequencies as a Function of a 2/a1for a0 = 0.115 and aI = 0.04. 19 4.5 Cross-Modulation Output at Different Frequenc-ies for a0 = 0.14 and a = 0.07) 20 5.1 Schematic Diagram of the UHF Crestatron. 235 5.2 Test Setup for Single-Frequency Investigations. 24 5.3a Fundamental and Harmonic Power O'utputs as a Function of the Input Power. (f'= 100 Mc,' Ik = 45 O ma,. V =1050 Volts) 25.k 5.5b Fundamental and Harmonic Gain Curves. (f = 100 Mc,

Figure 5.5 Fundamental and Second-Harmonic Power Outputs. (f = 140 mc, Ik = 450 ma, Vk 1050 olts) 29 5.6 Maximum Gain as a Function of Frequency for the Fundamental and Second-Harmonic Power Otput. (Ik = 450 ma, Vk = 1050 Volts) 0 5.7 Test Setup for the Multisignal Crestatron Investigations. 1 5.8 Output Powers of Various Signals when the Input Power at f2 Is Varied. (f 120 mc, f = 140 mc, P = 15 Watts, P2i Is Variable) 5.9 Output Powers as the f Input Power Is Varied. 2 (f = 140 mc, f = 160 mc, P 10.5 Watts P Is Variable) 4 22 5.10 Output Powers as the f2 Input (f = 260 mc, f = 280 mc, P 1 Watts 1 2 11~ P.Is Variable) 5 51Op Pwssh21In 5.ll Output Powers as the f2 InputPwrIVaid (f1= 140 mc,. f2= 120 mc, P.=14 Watts., P.Is Variable) 57 5.12 Output Powers as the f2 Input Power Is Varied. (f =240 mc,, f =220 Mc, p 11.Watts, 1 ~~~2 11 P.i Is Variable) 38 5.13 Output Powers when Signals Are Applied at f1 130 mc and f= 2f. (P. Is Variable, P.= 1.05 Watts) 2 1 1121 59

PERSONNIEL Time. Worked in Scientific and Egineering Personnel Man Months* J Rowe Professors of Electrical Engineering.20 C. Yeh 15. Solomon Associate Research Engineer W. Bond Associate Research Mathematician. 1.6 ~B. Ho ~Research Associate ~A.~ Cha Assistant in Research 1.51 Service Personnel 6.16

ARTICLE ISSUED DURING THE LAST QUARTER C. Yeh and B. Ho, "Electron Trajectories and Energy Relations in Transverse-Wave Amplifiers" Proc. IEEE (Letter to Editor), No. 9 pp. 1242-1244; September, 1965.

QUARTERLY PROGRESS REPORT NO. 10 ON BASIC RESEARCH IN MICROWAVE DEVICES AND QUANTUM ELECTRONICS 1. General Introduction (C. Yeh) The broad purpose of this project is to investigate new ideas in the area of microwave devices and quantum electronics. The program is envisioned as a general and flexible one under which a wide variety of ltopics may be studied. At present, the following areas of investigation are in progress: A. Study of frequency multiplication in an angular propagating structure. Redesigning of a low-frequency multiplier tube which multiplies a 600-mc input signal to a 2400-mc output signal with adjustable feedback control is in progress and the tube has been assen~bled and will soon be ready for processing. B. Study of a d-c pumped quadrupole amplifier. Before proceeding to design a d-c pumped quadrupole amplifier, other phases of the theoretical analysis will be performed. These include the problems of beam expansion and phase focusing. Cyclotron-synchronous wave interaction is compared to cyclotron-cyclotron wave interaction in -these respects. C. Investigation of the cross-modulation products'in a wideband tunnel-diode amplifier. The previously derived expression for the V-I characterilstic of a tunnel dilode is -used to compute the cross-modulation pnroduc-ts at. the ouitput, of t.he amplifier whe-n the innput, consists of t'Wo

-2D. Multisignal Crestatron operation. The experimental determination of the cross-modulation products of a traveling-wave amplifier with multiple-signal inputs is extended to cover the Crestatron mode of Loperation. Starting with two input signals, the effect of the relative power levels of the input signals on the cross-modulation products of the output is measured. Further extension of the experiments to three input signals and a band of noise as an input is planned. E. Theoretical studies on multisignal microwave amplifiers. Nonlinear calculations are being carried out for the cross-modulation p2oducts of 0- and M-type amplifiers as a continuation of the theoretical st,udy initiated earlier. A nonlinear theory of a crossed-field injected-beam amplifier with multisignal input will be developed along the same line. 2. Study of Frequency Multiplication in an Angular Propagating Circuit Supervisor: C. Yeh Staff: B. Ho The redesigned tube is now in the final stages of assembly. A beam trimmer is being added in order to trim the beam diameter down to approximately 0.5 mm. The cold test on the multiple cavity and the three Cuccia couplers has. been completed. The tube will be tested as. soon as it is ready. s. Suyof a D-c Pumped Quadrupole ~Eplifier Supervisor: C. Yeh Staff: B. Ho

-3of the electron beam react differently to the pump field in the pump structure and some of the electrons may grow in orbital radius and eventually be intercepted by the structure. The latter is an automatic sorting mechanism which brings more electrons to work in favor of amplif ication..2 The Effect of Beam Expansion. When a finite electron beam passes through the quadrupole pump field structure, electrons from different locations of the beam will react differently with the pump field. Consequently., some of the electrons will growin orbit and intercept the pump structure. This beam expansion phenomenon is unavoidable in all transverse-wave amplifiers. The effect becomes more serious when the physical size of the beam is comparable to the size of the pump field structure. When the physical size of the beam is large, early'beam interception will limit the possibility of high-gain operation. The amount of beam expansion for individual electrons is very sensitive to their locations with respect to the pump field structure. The beam expansion effects in the quadrifilar helix pump field for a finite beam for both cyclotron-cyclotron and cyclotron-synchronous -wave interactions can'be studied through the use of the equations of motion derived in Quarterly Progress Reports No. 5 and No. 6. For an unmodulated fi'nite-diameter beam entering the structure., the initial condition, dO/dt = 0 at t, = 0, is used. In rotational coordinates this ini'tial condition becomes d-1 0

-4of the beam are affected most, only those electrons are considered in the analysis. Assume that the beam has a radius of p0 = 0.1. Four angular locations, = O, 45, 90 and 135 degrees, for each pump field 0 strength are used in the computation. The computer solutions of beam expansion effects for cyclotroncyclotron wave interaction with different pump field strengths arehown in Figs..1 through 3.3. For cyclotron-synchronous wave interaction, the results are shown in Figs. 3.4 through 3.7. From these computer solutions, several points are observed: 1. The beam expansion phenomenon is more pronounced in the case of cyclotron-synchronous wave interaction. 2. The amount of beam expansion is directly proportional to the pump field strength. 5. The least expanded electrons are those located at 0 = 0 degrees,. while those most affected are located 90 deg'rees from the least expanded ones. It can be concluded from these observations tha-t for cyclotroncyclotron wave interaction the beam expansion is less important than'in the case of cyclotron-synchronous wave interaction. For a medium strength pump field, as shown in Fig. 5.2 for cyclotron-cyclotron wave interaction, the beam remains reasonably well focused. In the case of cyclotron-synchronous wave interaction, the beam expansion is more pronounced and even for a medium strength pump field., as -shown in Fig. 3.6., the beam diverges gradually. However, the actual situation is not-4 as b-Ad ase it aper duez to- the, paefcsngy efec toc be dniscused

LO r%4 0 rM4 N\ 0 H 0 L 0 C\J~~~~~~~~~~~ 0~~~~~~~~ o 0~~~~~~~~~~~~ 0 H 0 r%4

~~~~~-6 0.2~~~~~~~~~Q 0 0~ ~~~0 0.1. I I I 0510 15 20 25 30 354 TIME, T FIG. 5.2 BEAM EXPANSION FOR CYCLOTRON-CYCLOTRON WAVE ThLEATION FOR MEDIUM PUMP FIELD STRENGTH.

7 0 It to ln~~~~~~~~~~~~~~~~P 0~~~ re~~~~~~~~~~~~~~~p 00~~~~ 0~~~~~~~ 00 N4 0) C\j~~~~~

-.8PC) I F0 100 IC) ~~II ~~0 LO I) I 2 0~~~ 0 0~~~~ (0 N 0~~~~~~ o o~~~~~~~~~~~~~~~~\ S'SflIOVd Q3ZI1VV'JUON~~~~~~~~~~~PC

-90 co 0 0~~~~~~0 LO~~~~e g0 0 0~~~~ LO ~~~00 0 rz) 0o H\ Nq

-100 ~~o \ r~ ~ ~r \ 0 00 0 0 \ O N 0 \\ I

-11 — 0 OD 0 ~ ~ ~ ~ 0~~~~ 0~~~ - 0 o \ (0 0~ 0l! to r4- ~ ~ ~ ~ ~

-12(i.e., with an entrance angle of zero or 180 degrees) as soon as it enters the pump field structure. 5 he Phase-Focusing Effect. Although a solid uodulated beam has been assumed when discussing the effects of beam expansion, the actual configuration of the beam entering the pump structure of a transverse-wave amplifier is far from this state. In fact, the beam is initially modulated and IdO/dT o = 0. The angular location of the previous beam becomes the entrance angle of the beam under consideration. With this modification, trajectories of the beam show a strong phasefocusing effect as is indicated in Figs. 3.8 and 3.9 for the case of cyclotron-cyclotron and cyclotron-synchronous wave interactions, respectively. It is observed that all of the electrons that enter the pump field at different entrance angles in a very short time will align themselves to correspond'with the most favorable entrance angle., i.e.., the zero-degree entrance angle. Therefore, it is sufficient to say that as far as beam expansion due to the pump fileld is concerned, only those electrons with a zero-degree entrance angle must be considered. It can be seen from Figs. 5.1 through 5.7 that the beam expansions for all cases with reasonable pump field strengths do not exceed twice the initial value. Th'is indicates that in actual operation the beam expansion due to the pump field should not present a serious problem. 514 Program for the Next Quarte.r. While the experimental tube is being assembled a few more computer studies on the energy relatiornships and the kinetic power relationships will be investigated. -

-133300 0 300 0.5 0.4 3 000 600 2700 ~ ~ ~ ~ ~ ~~~~~~0 2400 1200 VP = 30 VOLTS 2100 1800.1500 FIG. 3.8 PHASE FOCUSING FOR CYCLOTRON-CYCLOTRON WAVE IFRC-TION.

-143300 0 300 20 24000 52600 0.2 ~~~~~~~~~~~~~~0,: ~~~~~~0. 2400 210 100050 FIG. 3.9 PHASE FOCUSING FOR CYCLOTRON-SYNCH{RONOUS WAVE INTERACTION.

4, Investigation of the Cross-Modulation Products in a Wideband TunnelDiode Amplifier (C. Yeh) 4ol Introduction. Equations for predicting the V-I characteristc and the cross-modulation products of a tunnel-diode amplifier haveQbeen derived in previ.ous reports (Quarterly Progress Report Nos. 8 and 9). During the present quarter, effort has been devoted to the use of these formulas in the computation of cross-modulation products if a tunneldiode V-I characteristic curve is given and if two signals are present at the input. 4.2 An Aproximation. It has been found that the series expansion used for the hyperbolic tangent function converges very slowly and mrany terms are needed to accurately simulate the V-I ch.aracteristic. This contradicts the statement that one must only consider up to cubic terms of V in the calculations. Fortunately the range of the V-I charac-teristic curve for a tunnel diode in amplifier applica-tions is so narrow that the approximation can be justified. It is suggested -that for the active range of amplifier application,. the.hyperbolic'tangent function'be represented by a straight l'ine, i.e., tanh(eV/2kT) m(V1 -V )+C, where the slope m =Atanh(eV/2kT)/(V -VM) C,t'anh(eVM/2kT>,) and VIis the value of V such that tanh(eV1/2kT) 1. To prov-e that such an approximation is actually tolerable, the negative resistance range of the V-I characteristic of a tunnel diode is calculated and plotted as cilrcles along with the exact characteristic curve of the diode and is shown in Fig.. 4.1. It is seen that the agreement is fairly good.

0 LO \~~~~~~~ z 0 Z< _< J__ I U)- to 0 0 0 6 WX~~~~~~~~~~~~~~~~C U- Cr a. ~ ~ ~ ~ C n~~~~~~~~~~~~~~~~~~~~~~c co o 7_ ~O_ Q 0H 0~~ LO LC) LO U) U5W'1

-17= a(V - V)2 [m(V - VM) + C] + g V m C + C V + C V2 + C V3 0 1 2 3 where aV2(C - mVM) 0 m CCL [amV2+ go - 2aV (C -.mVM) ] Im m 0 m C2 a[(C- mVM) - 2mV ] 2 ~~~m C = am 3 ( g )/( - V) tanh eVd/2kT o a (,TM -0 gVM) / (Vm VM)2 tanh eVM/ 2kT (4i.2) V =a + aI cos COt +a 2Cos L2t,9 (4i.3)'where a0 is ac-tually the d-c bias voltage and a Iand a2 are the armplitudes of the two input signals. By mreans of Tables 4!1. through 4A4 in Quarterly Progress Report No. 9., the modulation product contributions due to the quadratic and cubic'terms can'be written out and when they are mul-tiplied by the appropriate constants and s-ummed according to Eq. 4-i.1, with omission of the frequencies out of the bandwidth of the wideband amplifier, the following

-18and the current component at the fundamental frequency, 2 i j =a2 C + 2a C + 3C3(a2 + la2 + 1 a2)] (4.5) The current components of 2w1 -W and 2w2 -w are, respectively 1 2 2 1 J = 3 a2aC (4.6) 2W -( 4 1 2 3 1 2 and J 3 a a2C (47) 2w -w) 4 1 2 3 2 1 The d-c component, the difference frequency w-w2, the second harmonic frequency components 2w1 and 2w 2, and the sum frequency components w1 +w2 2w1+w2 and w1+2w2 are assumed to be out of the bandwidth of the amplifier. Figures 4.2 and 4.3 are the results of calculations for two operating conditions of a tunnel-diode amplifier. In each figure, the ratio of the signal amplitudes, a /a, is used as a variable. The signal. outputs at different frequencies are plotted as a function of a 2/a1. In Fig. 4.2, the bias voltage is 0.15 and the maximum ampliltude of a is 0.04. In Fig. 4.3., the bias voltage is 0.14 and the maximum amplitude of a1 is 0.07. The results of the calculations can be summarized as follows: a. The amplitude of one of the signals, w 1, decreases slightly as

-191.0 ao = 0.13 0.9 0a = 0.04 0.8 x /C dmI J2 x I0-/ 0.7 0 zO0.5 w 0.4 0.3 0.2 0.I w -0 0 0.2 0.4 0.6 0.8 1.0

-201.0 2.0o xl0.9 i. 8 0.8 1.6 Jwi Jao 0.7 1.4 JW2 0. 6 j.l 1.2 U) U~~~~~~~~~~~~~~~~~~~~~) 0 0 F~~~~J~- o)I cr_ o. 5 ~~~~~~~~~~~~~~1.0 0.2~~ 0.4s1 F- w2 c, - s cr right-hand(ighthan 0. 0.4I. 0. 0.3... 1.0 FI.45 2RS-OU T0.UPTATD4EET RQECE FRa=01 0

-21a2/a Ialmost quadratically and is larger than the amplitude of the 2 w componen't h atio J-2W1 v / 2W - varies almost linearly with 2 1 L. 2 The effect of changing the operating point does not seem to change this ratio appreciably. d. The bias and the signal amplitudes are small so that no gross nonlinear effects are observed.. Program for the Next Quarter. Computation of the modulation products will be continued to include the case of a large bias and strong signal amplitudes. The products due to three-signal inputs will also be studied. A wideband X-band tunnel-diode amplifier has been ordered and experimental measurements of the modulation products will be made and the results compared with those obtained theore tically. 5 Exerimental -Characteristics of Multisignal Traveling-Wave ~plif iers (N. A. Masnari) 5.1 Experimiental I-nvlestigation. The experimental multisilgnal investigations are concerned with the interaction phenomenon occurring in a microwave tube when several r-f signals are applied simultaneously to the tube. Because of the nonlinear nature of the interaction process in a microwave tube, various cross-modulation terms are generated. The relationship of these difference-frequency signals to various parameters is of special interest. Whenever possi-ble it is also important to study the harmonic-frequency terms which result from the beam-wave interaction pDrocess.

-22device under study is a UHF Crestatron developed through the combined efforts of the Electron Physics Laboratory and the Bendix Corporation The tube is capable of generating in excess of 100 watts of cw power with a beam current of 450-500 ma. The beam is formed bya hollow cathode and is confined by magnetic focusing. A helix is used for the slow-wave circuit. The large operating powers require the use of a water-cooled collector. Figure 5.1 is a schematic diagram of the tube which, in general, has an operating beam voltage of 1150 volts. 5.2 Single-Frequency Operation. The purpose of the initial investigation was to evaluate the Crestatron operation with only a single frequency applied at the input. The investigation was concerned both with the fundamental frequency output and with the second harmonic (and the third harmonic when observable). The frequency and the power level of the input signal were varied and the output was evaluated over the operating range. Figure 5.2 illustrates the test setup for the experimental evaluation of the operation. The fundamental., second-harmonic and third-harmonic power outputs are plotted in Fig. 5.3a as a function of the input power, Pi., for f =100 mc, Vk = 1050 volts and I )450 ma. Figure 5.3b illustrates the gain in db as a function of P.. The fundamental has a maximum gain of approximately 12 db at P. 5 dbw, while the second harmonic has its maximum value of 7.5 db at approximately the same input power. The maximum third-harmonic gain is approximately 1 db and occurs at higher power levels. In general the second harmonic for this case is 4-.5 db

-2350 I0 w *Li -J 0 0 ~x w. t t11F p 0~. t0 ZL Y -I w w 0~~ z o >'4- ~0 03

-240< 00 Z~~~~~~ N 0 WH a. w~~~~~m J~~~~~~ 0~~~~~~~~~~~ w~~~~~~~~~~~~ wcr 0 cr~~~~~~~~~~ a~. N*jw D 0 -ocr0 L _i 0 aw~~~I w (D Ir -I -z iowi (/w Ia. 2I

-25\~~~~~~~~~~~ 0 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~Ii I 0 0 1t/2 (0 0~~ 0 o n 0 Hd

14 12 1 0 8 f n6 z 4 ~~~~~~~~~~~2f 2 3 f 0 -2 0 ~~3 6 9 12 1 5 P. d bw FIG. 5.15b FUNDAMENTT AND HARMONIC GAIN CURVES. (f= i.oo MC, Ik =430 ma., Vk = OOVLS

-27below the fundamental over the entire P. range. The third harmonic is down 3-6 db with respect to the second harmonic.... Similar results are illustrated in Figs. 5.4 and 5.5 for f = 120 mc and f = 40 mc respectively. The general results are the same, with the second harmonic being down anywhere from 5-12 db with respect to the fundamental. In each case it is observed that the fundamental experiences a maximum gain at some particular P. value and then decreases. It was 1 also observed, although not illustrated here, that there were significant oscillations for an input signal of 110 mc. Figure 5.6 indicates the maximum gain for the fundamental and the second harmonic as a function of the applied r-f signal frequency. It is apparent that the fundamental has its maximum gain (in the range between 100 and 150 mc) of 15 db at f = 120 mc. The second harmonic has its maximum gain (8 db) at approximately 130 mc when the gain of the fundamental is only 12 db. 5.3 Multi, sign1al 29peration. The single-frequency investigation was followed by a study off the tube performance when two signals were simultaneously applied to the Crestatron. The frequency and the power level of each signal were independently variable., thus allowing investigation of the nonlinear operation over the entire 100-300 mc frequency range. Figure 5.7 illustrates the circuit used for ~this investigation. When two signals with frequencies f1 and f2 are applied simultaneously to the tubeJ, the output contains various components in addition to those at the applied frequencies; e.g., the cross-modulation terms

-280 0 0 C\j~~~C LO~~~~~~~~I LO~~~C 0 o 0 0 0 00

-290 0 C\j ~~~~~~~~0 0 LI\ C.) 0 0 ~~~~~ ~~~~~~~(7) H

-301 8 I15 12 z (9 6 0 I 100 110 120 130 140 150 FREQUENCY, Mc FIG. 5.6 MAXIMUM GAIN AS A FUINCTION OF FREQUENCY FOR THE FUJNDAMENTAL AND SECOND-HARMONIC POWER OUTPUTS. (I = )5o ma,1 k V k = 1050 VOLTS)

-310~~~~~~~~~ J _ <!~~~~~~~ ~~0 C, j CD z www 0 o-z~~0 02 Z~~~~~~~ w.00 W~~~~~~~~~~~~~1 a a. z -0. _j 0~~~~~~ w 01- a.~~~~~~~~~~~ ~~~~~~- cr u D~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~: 0~~~~~~~~~~~~~

-32Figure 5.8 illustrates the situation in which signal 1 is applied at f 120 mc with an input level of Pli = 15 watts. A secnd variable amplitude signal is applied at f2 = 140 mc. As P. is increased, the output power at f (P ) increases as expected However, the output 2 P20 signal P at f and its second harmonic 2f decrease as P. increases. 10 1 1 This behavior results directly from the nonlinear beam-wave interaction Thch removes some of the energy from f and 2f. The cross-modulation 1 1 term at 2f -f is negligible until P. = 1 watt when it appears with a 1 2 21 value of 2 watts. The other term, 2f -f, never achieves any measurable value. Figure 5.9 illustrates a similar result when f and f2 are 1 increased to 0 mc and 160 mc respectively. P i is fixed at a level of 10.5 watts while P. is varied between zero and two watts. Again P and its second-harmonic term decrease as P. is increased. The 10 221 signal at 2f -f increases with P. until it reaches a value of four 1 2 21 watts at P. = 2 watts. The 2f -f term also appears with a power of 21 2 1 l. 5 watts for P. 2 watts. With P.i fixed at 10.5 watts., it is observed that as fI and. f2 are increased,9 the cross-modulation term 2f -f becomes larger and 1 2 eventually is comparable to P 20 This is illustrated in Fig. 5.10 for f 260 mc, f 280 mc and P. =15watts, At P. = 2 watts the power 1 ~~~2 11 21 at 2f, -f2 is approximately 6.5 watts as compared to P 6 watts. 2 ~~~~~~~~~~~~20 From the above data it is apparent that when a large signal is applied to the tube at a given frequency f 1, the application of a second

-338 0 70 6 0 -- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \ ~~~~~~~~~~~~~~~~~~~~~~~~~~~1. 60 500 4~~~0. 0- 4 ~ 2~~~~~~~~~~~~~~~~~~~~ I O. B I.0 28~~~~~~~~~I' o lZ~ inc 0 inc, P p IS I \VIT

-354C~~~~~C 0 to~~~~~~~~I 0 0 C~~~~~C to~~~~~~~~I N c'J~~~c'4-'S4- N' C\)

-355120 _. 100 f 80 C,) 60 I0~ 40 20 2f, -f2 fa A1I I --- I 0 0.4 0.8 1.2 1.6 2.0 2.4 Pa2;, WATTS FIG. 5.10 OUTPUT POWERS AS THE f2 INPUT POWER IS VARIED. (f = 260 mc, 2 = 280 mc, P i = 13 WATTS, p1 IS VARIABLE)

-36If the above experiments are repeated with the larger-power input signal P applied at a higher frequency than P2i (i.e similar results are obtained. Figure 5.11 illustrates the results for f 10 mc, f = 120 mc, P = 14 watts and a variable P2i The cross modulation term 2f -f grows with P. while the f and 2f power output 1 2 21 1 terms decrease as before. The 2f 1-f2 power is approximately the same as P. 20 Figure 3.12 indicates the results for f = 24O mc, f2 220 mc 1 and P 11 watts. In this case both of the cross-modulation terms 2f -fand 2f -f are present in the output. Again, P decreases as 2i increases, thus indicating a transfer of power from the f1 signal to the cross-modulation terms. It is apparent from the data that the strongest cross-modulation term is always 2f -f where P is greater than. P.. Thus the larger 1 2 ii 21 cross-modulation signal appears below f 1when f is less than f2 and appears above f 1whe n f1is greater than f 2 Figure 5.15 ind'icates the results when one signal is applied at f =150 mc and a second signal is applied at the second-harmonic frequency, f2 = 2f =260 mc * P. is fixed at 1.05 watts while P1 is increased up to 6 watts. P is constant at low P levels and then 20 ii decreases at higher values. The second-harmonic output P 2 increases 1 as P. is increased so that the total output at 260 mc is P 2 + P 2 1 which has a maximum value of approximately 58 watts at P 3 watts. The combined 260-inc output then decreases with further increases in P.

-3790, 80, 70 - 6O — 50 co I C 40 40 30 2.0~ ~ ~ 2l 30~~~~~~~~~~~ f I lfl 12 O | T ~ ~ t,~ —— O m

-38120 100 f 80 0 0 60 a_ 40 20~~~~~~~~~~~~~~~ 0 0.4 0.8 i.2 1.6 2.0 2.4 =220 mc, P11 =~~~~~~" 11WT'', P21 Is, - ARIABL

-3960 _ \/~~~~~~~~~~~P 50 I 1 P20 2 f1 40 P 20 30 0~ 20 P 2f I0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 1 2 3 4 5 6 7 8 Pi,WATTS

-40of either signal to be less than when only a single frequency is applied. This,, of course is directly a result of the nonlinear nature of the beamwave interaction process which causes the generation of various crossmodulation terms. Conclusions and Program for the Next Quarter Experimental results have been obtained for the multisignal operation of a Crestatron in the 100-00 mc range. The cross-modulation terms 2f-f2 and 2f -f occur with increasing amplitudes as the power levels of the input signals are increased. When the two input signals are quite different in power the larger signal determines the frequency at which the larger crossmodulation term exists; i.e., when Pi is the larger signal, then 2f -f is the larger cross-modulation signal and its frequency is either below or above f depending on whether f is below or above f 1 _ 1 n n n 1 2 The investigations will be continu-ed during the next quarter. Attempts will be made to apply two saturation-level signals to the Crestatron and to observe the various cross-modulation signals which are generated. The possibility of applying a band of noise and a signal frequency is being investigated and it is hoped that such an experiment will be completed during the next quarter. The multisilgnal operation of a tapered-helix tube will also be investilgated to determine the effect of circuit tapering on the generation of cross-modulation signals. 6. Theoretical Studies on Multi'signal Mi,crowave ~plif iers (J. E. Rowe) 6.1 ft _-1_ Nolna 0- - AmlferClultos.I reiu

-41amplifiers with multisignal r-f inputs, some difficulty was encountered in accurately determining the magnitude of the cross-modulation products. The nonllnear equations adequately describe the multisignal operation and it is felt that the difficulty results from the fact that the magnitudes of the generated signal components near the input are of the same order of magnitude as the error in the numerical procedure and thus the initial growth rates are usually in error. The principal effort during recent months has been directed toward modifying the numerical calculation procedures in order to minimize the error and insure the stability of the system. Specific calculations have been made for a TWA with a singlefrequency input and fundamental and second-harmonic outputs. This case was selected because of the availability of a quasi-nonlinear theory to predict the amplitude of the second- and third-harmonic outputs of a nonlinear 0-type TWA. These theoretical studies will continue during the next quarter. 6.2 Multi-signal Crosse_d-Field Amplifiers. The small-signal theory of an injected-beam crossed-field amplifier has been generalized to include mult'isilgnal inputs as indicated in Quarterly Progress Report No. 8. These equations have recently been programmed for solution on an IBM-7090 digital computer and the program'is presently being checked out. It is anticipated that preliminary solutions should be available during the nex:t period. This theory is particularly useful. for determining the growth rates of-'the cross-modulation products and for evaluating the am to pm conversion inherent in multisiLgnal operation of a nonlinear micro

-42mutisignal inputs can be developed along the same lines as that for the nonlinear 0-type amplifier. Further progress on this problem awaits the satisfactory solution of the numerical accuracy and stability problem described in Section 6.1. 6 Program for the Next Quarter. Plans for the next quarter include further work on the numerical accuracy and stability problem associated with the nonlinear analysis of the multisignal 0-type amplifier and the completion of check runs on the multisignal crossedfield amplifier using a generalized small-signal analysis. It is also anticipated that experimental work can be initiated on the multisignal crossed-field amplifier. 7. GeneralConclusions (C. Yeh) Beam expansion in a quadrupole amplifier has been studied by introducing a nonrotating'beam of radius p0into a quadrupole structure a-t the proper d-c potential and by computing the trajectories of the peripheral electrons at different locations in response to the structure field. Both modes of operation,'i.e., cycotron-cyclotron and cyclotronsynchronous wave interactions,. have been studied and compared. It can be concluded that the former mode is subject to much less beam expansion than the latter. However,, from the actual trajectories of the electrons in cyclotron-synchronous wave interaction,. electrons from different entrance angles of the beam are quickly brought into focus with those from the most favorable entrance angle., i.e.., they are bunched into a beam for favorable interaction. Therefore the beam expansion due to the

different frequencies have been computed. For an X-band amplifier in -wh'ich the second harmonics of the input signals are not inside the bandwidth of the amplifier, the 2f -f and 2f -f products become important. 1 2 2 1 As the ratio of the amplitudes, a /a, is increased, both of these 21 components increase, one almost linearly (2f -f ) and the other almost 1 2 quadratically (2f -f1 ). If f is the larger signal, then the component 21 ~1 2f2-fbecomes more important than the component 2f 1 -f. Also, the amplitude of the input decreases as a2/a1 is increased These are general phenomena in all nonlinear devices. It is expected that the relative importance of the 2f -f and 2f -f components will change as 2 1 i 2 the bias point and the maximum voltage swing are changed. This effect will be studied in the next quarter. The Crestatron is a power amplifier operating on the principle of beating-wave amplification. It has been found that the second-harmonic output of this amplifier is quilte high over the entire input power range and generally is only 4-5 db below the fundamental output l'evel. With multiple-signal input operation, if second harmonics are not within the operating range of the amplifier, then the'important components are again the 2f1 -f2 and 2f2 -f1 signals. If. however,. the power level of f 1is greater than that of f,2. the important contribution is from-the 2f f2 component and that from the 2f2 -f1component is negligibly small.

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