THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR, MICHIGAN START-OSCILLATION CONDITIONS IN MODULATED AND UNMODULATED O-TYPE OSCILLATORS TECHNICAL REPORT NO. 35 Electron Physics Laboratory Department of Electrical Engineering By Joseph E. Rowe Harold Sobol Project 2750 CONTRACT NO. AF30(602)-1845 DEPARTMENT OF THE AIR FORCE PROJECT NO. 5573, TASK NO. 55253 PLACED BY: THE ROME AIR DEVELOPMENT CENTER GRIFFISS AIR FORCE BASE, NEW YORK February, 1960

ABSTRACT The starting conditions for the O-type backward-wave oscillator are computed for large values of C, QC and d using both digital and analog methods. A general method of solving complex polynomials called the "downhill" method is applied both to the secular equation and then to the r-f voltage equation to obtain starting conditions. The analog computer is used to solve simultaneously, by trial and error method, the linear circuit and ballistic differential equations. The analog method is applied to the modulated BWO in order to determine the effects of modulations on the starting conditions. -iii

TABLE OF CONTENTS Page ABSTRACT iii LIST OF TABLES v LIST OF ILLUSTRATIONS vi INTRODUCTION 1 BALLISTIC AND CIRCUIT EQUATIONS 2 DIGITAL CALCULATION OF STARTING CONDITIONS ANALOG CALCULATION OF STARTING CONDITIONS 7 MODULATION OF A BWO 14 THE EFFECT OF MODULATION ON THE START-OSCILLATION CONDITIONS 16 CONCLUSIONS 27 APPENDIX A 29 APPENDIX B 51 REFERENCES 58 -iv

LIST OF TABLES Table Page I Comparison of Start-Oscillation Conditions Determined by Analog and Digital Methods 14 - v -

LIST OF ILLUSTRATIONS Figure Page 1 0-Type BWO Starting Conditions. (d = 0.1) 8 2 O-Type BWO Starting Conditions. (d = 0.5) 9 3 Trial and Error Runs for Determining Start-Oscillation Conditions Showing R-F Voltage vs. Length. 15 4 Impedancel and Phase Velocity2 Characteristics of (-1) Mode of Bifilar Helix. 19 5 BWO Modulation Parameters ~, and ~2 as Functions of Frequency, *3. (Bifilar Helix; at AV = 0, ka = 0.30) 20 6 BWO Modulation Parameter F as Function of Frequency, ~. (Bifilar Helix; at AV = O, ka = 0.30; Annular Beam, H = 0.6, R = 1.2) 21 7 First Estimate of Normalized Frequency, Go, Obtained from cu- Diagram. (Bifilar Helix; at AV = 0, ka = 0.0) 22 8 Trial and Error Procedure for Finding Start-Oscillation Conditions with Beam-Potential Modulation. 23 9 Variation of Start-Oscillation Frequency as Function of Beam-Potential Modulation. (C = 0.05, QCo = 0, d = O, b =1.56; Bifilar Helix; Annular Beam, H= 0.60,R = 1.2; at AV = 0, ka = 0.30) 24 10 Variation of Start-Oscillation Frequency as Function of Beam-Potential Modulation. (Co = 0.05, QCo = 0.20, d = 0, b = 1.59; Bifilar Helix; Annular Beam, H = 0.60, R = 1.2; at AV = 0, ka = 0.30) 25 11 Variation of Start-Oscillation Length as Function of Beam-Potential Modulation. (C = 0.05, d =0, Bifilar Helix; Annular Beam, H = 0.60, R = 1.2; a? AV = 0, ka = 0.30) 26 B.1 O-Type BWO Starting Conditions. (d = 0) 52 B.2 0-Type BWO Starting Conditions. (d = 0.05) 53 B.3 O-Type BWO Starting Conditions. (d = 0.1) 54 B.4 0-Type BWO Starting Conditions. (d = 0.25) 55 B.5 0-Type BWO Starting Conditions. (d = 0.5) 56 B.6 0-Type BWO Starting Conditions. (d = 1.0) 57 -vi -

START-OSCILLATION CONDITIONS IN MODULATED AND UNMODULATED O-TYPE OSCILLATORS INTRODUCTION The O-type backward-wave oscillator has found wide application as a tunable signal source where relatively low power levels are acceptable. The chief limitation in building high-power tunable oscillators is the lack of a broadband circuit which has a high characteristic impedance for the backward-wave mode. Most O-type oscillators are built using the helix operated on the minus-one space harmonic and for normal beam current values have characteristic C values near 0.02 and below. Larger values of C have been achieved at S-band by Putz and Luebkel in their work on a 100-watt oscillator using a folded-line r-f structure. Even with values of C between 0.05 and 0.08 the efficiency of their device was around 8-10 percent. In recent years the interdigital line has been used in O-type backward-wave oscillators; its higher backwardwave impedance suggests that oscillators with higher C values will be possible. Many analyses2'3'4'5 of the O-type oscillator have been made using various linear theories to compute the starting conditions, generally assuming small C values. A linear type of analysis can be used because the oscillator is operating linearly just at start oscillation. One of the purposes of this paper is to present a summary of start-oscillation conditions for the O-type backward oscillator for large

-2values of C, QC and d computed using a digital method. It is shown how they may also be calculated on an analog computer. One of the advantages of the analog computer method is that the oscillator can be studied under modulation conditions. This study of the modulated BWO is the second objective of this paper, wherein a study of the effect of modulation on the oscillator starting conditions is presented. BALLISTIC AND CIRCUIT EQUATIONS The linear theory to be used in calculating the start-oscillation conditions for large C has been described by Rowe6 previously and was developed following a procedure similar to Johnson's small-C derivation. The interaction with the backward-wave mode of the circuit is accounted for by introducing an impedance parameter into the equations which is the negative of the forward-wave impedance parameter used in traveling-wave amplifier studies. It is also convenient to define a passive-mode parameter Q which is the negative of that used in the forward-wave equations, in addition to the change in C noted above. The linearized equations for the oscillator are easily integrated to give the propagation constants. The wave amplitudes are determined from a knowledge of the input boundary conditions at the entrance to the interaction region. An oscillation condition exists for a given C, QC and d when the proper combination of b and CN is found such that the ams plitude of the r-f wave on the structure vanishes for z > O. N is the structure length in stream wavelengths. Heffner2 and Johnson4 used essentially this method in their work except that Heffner used the beam current as the driving source for the circuit equation whereas Johnson more correctly used the space derivative of the current. In Bernier's7 analysis of the forward-wave amplifier he also used the current as the driving source in the transmission-line equation.

-3The linearized ballistic equations for the BWO are identical to those for the small-signal traveling-wave amplifier. They may be written as follows: dv nET + juv+u =, (1) o dz Po d + u d + jp = 0, (2) o d-z o dZ where p = magnitude of electron charge-to-mass ratio, ET = total z-directed electric field intensity acting on the beam (sum of the circuit and space-charge fields, E + E ), c SC v = amplitude velocity perturbation, p = amplitude of charge-density perturbation, u = average beam velocity, p = average beam charge density, and all perturbed quantities are assumed to have a time variation of exp jwt. It has been assumed that v < u and that p <K p. 0 The circuit equation is the forced transmission line equation. Note that the righthand side of this equation has the opposite sign to that of the corresponding TWA equation. d-V -c + j Z- V = - (Z o + jR) p, () 2 0 c o c 0o0 dz o where V r-f circuit potential (potential from which the slow-wave field E is derivable), P = radian wave number of circuit, 0 Z = characteristic impedance of circuit, and R = series loss of circuit (ohms/meter).

-4The above formulation of the interaction equations is convenient for both the digital and analog methods to be used in solving the equations for the start-oscillation conditions. The form of the space-charge field expression used is that given by Pierce. E - jP2i (4) sc - c' 1 where C is the capacitance defined by Pierce. In order to facilitate digital solution of the above equations it is convenient to assume that all r-f quantities vary exponentially with distance as exp -rz in addition to their exponential time variation. Since it is expected that the actual propagation constant of the system will be only slightly different from the system cold propagation constant, Pierce's8 method will be used and the following terms defined A - i - Je, + eC6,, (5) - - Je - JP Cb + b eCd, (6) where n = x +jyn, the perturbation propagation constant and the remainder of the symbols have the usual meanings (see Pierce8 or Rowe6). The r of Eq. 6 is the cold-circuit propagation constant and the rn are the actual wave propagation constants with the stream present. The interaction equations may now be combined as algebraic equations in n using the definitions given in Eqs. 5 and 6. The resultant is a fourth-degree polynomial with complex coefficients and of course complex roots. This is given below for the BWO.

-564 _ - 2CQC) - j63(1 - 8C2QC) + 2[(1 + C3 - i42QC) + C(C + 10QC) + v2 - 2C3QC)] + jb v(-2C2 + 8CQC) + v2(4C2QC) - 2(C + 2QC)| - 1 + v(4QC - C) + v2(2CQC) = 0, (7) where v = b + jd. The input boundary conditions are applied to the stream convection current density and stream velocity and to the r-f wave on the circuit at z = 0 to obtain a wave-amplitude matrix. The solution of this matrix for the excited wave amplitudes on the circuit is given in reference 6 and need not be repeated here. Analogously to the TWA, the r-f voltage as a function of distance in the BWO may be written as V (0) -j X A ( = ~ e ~ j4 civi (8) V (0)' A where 0 = 2-CN_ = the radian length of the tube. The fourth root has s been eliminated since this wave does not interact with the electron stream. DIGITAL CALCULATION OF STARTING CONDITIONS In order to determine a starting condition for the BWO it is necessary to search the b - CN plane for points at which the right-hand s side of Eq. 8 becomes zero for particular values of C, QC and d. A general method known as the "downhill" method has been worked out to find both the complex roots of Eq. 7 and the start-oscillation condition from Eq. 8. The quartic polynomial given in Eq. 7 may be separated into real and

-6imaginary parts and written in the following form: 4 P(5) = j (A + jBi) s (9) i=0 The following definitions are made to simplify the coefficients of Eq. 7. R = C 2C2QC) U = b - 2QC(b -d2) S = 10QC + - d2 V = 4QC T = - 4bQC W = - 8CQC. (10) In terms of the above definitions the components of Eq. 9 become A = - (TCU) B = d(V-CT) 0 0 A = - 2CB B = - V + 2CA i O 1 o A2 = (b+CS) - C2A B2 = d(l+Cb) - C2B A3 = 0 B3 = - W A, = R B4 = 0 (11) The downhill method for obtaining roots of a polynomial P(6) consists of searching the surface over the complex plane generated by S = |R[P(6)]I + |I[P(b)]I. (12) When S becomes 0, the value of 6 is then a root of the polynomial. The nature of Eq. 12 is such that if a point 6& produces S, there is a point 62 where S < S unless S is zero. Thus for a continuous path from 6 2 2 1 1 to 62 there is a point 63 on this path such that S2 < S3 < S1. Hence it is clear that during the search for roots, one always proceeds downhill to the roots. Mechanization of this method requires a systematic scheme for searching the surface S.

-7A wheel of points is selected with a center point and the polynomial is evaluated at each of these points and the point for which S is a minimum becomes the center point for the next wheel. This process is repeated until the value of S reaches some minimum limit, usually of the order of 10 6. The value of 6 giving this minimum is considered to be the root. After the roots of the polynomial are determined for a given set of C, QC and d then the same technique is applied to the r-f voltage equation and the solution of! f (vc e = o (13) i i is sought. The initial center point of the wheel and the initial spacing of the endpoints are arbitrarily selected. Of course, there is more than one combination of b and CN which will satisfy Eq. 13, as was shown in s reference 6. The minimum b-CN combination is the desired one and two s typical plots of these are shown in Figs. 1 and 2*. The parameter (3-3 )L is proportional to the product of CN and b. The results indicate e s that the value of CN at start oscillation becomes relatively independent s of QC as C is increased beyond 0.1. The effect of both loss and space charge is to increase the required starting length at constant C. As expected the required CN decreases with increasing C while the required s b increases. ANALOG CALCULATION OF STARTING CONDITIONS An alternate method for determining the start-oscillation conditions is to program the differential equations oan analog computer * Starting conditions for a wide range of parameters are tabulated in Appendix Ao Plots of the starting conditions are given in Appendix B.

-87 > 1 //0/0.6]; | ~4!~~~~ 3 / ~o.3 C? ^ I/ /. I,. 3 - f —-c=0.2 0.3 0.4 05 0.6 0.7 CNs FIG.I O-TYPE BWO STARTING CONDITIONS. (d=O.I)

-99- 9 -9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 CN1 FIG. 2 0-TYPE BWO STARTING CONDITIONS. (d~O.5) ~IG,, -VP,W, T~lG CNITOS d0

-10and obtain the solutions by a trial and error procedure. The advantage of using an analog computer is that one can easily vary parameters merely by changing a potentiometer setting. Grows has previously determined the start-oscillation conditions for the unmodulated BWO using an analog computer. He found the r-f circuit voltage by using a fourth-order linear differential equation obtained by combining the circuit and ballistic equations and eliminating the beam charge density, electron velocity and current from the resulting expressions. The method used in this paper is to program the circuit and ballistic equations as simultaneous linear differential equations. The advantage of this program is that one can study the variation of current, charge density and velocity in addition to the circuit voltage. This leads to a better appreciation of the entire phenomenon of start-oscillation. The program is also applied to the determination of the startoscillation conditions of a modulated BWO. The interaction equations were given in Eqs. 1 through 3. An alternate form for the space-charge field expression is used in the analog computation. It can be shown that for a beam of finite radius the space-charge field is given by~ u2 /Z R\ whe re X = the plasma radian frequency, Vc = phase velocity of the circuit wave, R = plasma frequency reduction factor, and I = average beam current. Equation 14 is more amenable to analog computation than is Eq. 4. These different forms for the space-charge field result in different starting

-11 - conditions for the cases of large C and large QC, as will be noted later. This was pointed out previously by Grows. It is found convenient to introduce several normalizations before programming. A normalized distance variable is introduced according to the following definition. = Pez (15) where He is the stream phase constant. The normalized interaction equations then become dV 1 c V. 4CQC p dv v 1 c~ + - = + j (16) 2 dy (l-2c4J> )2 l-+Cb dy C dv d p + j P (17) dy, dy C and,2 + d 3c + +b 2 d L -y2 f\ J Vc + 2j - (l+Cb) V = [4C(lCb) + j8C2dp (18) Equations 16 through 18 are solved on an analog computer after separating into real and imaginary parts. This is accomplished by assuming the following forms for the dependent variables. v = VR + jVI v = vR+jVI P = R + jPI (19) The magnitude of any of the above variables may be found by using a standard analog computer resolver for converting cartesian coordinates into cylindrical coordinates. The boundary conditions must be specified before Eqs. 16 through 19 may be integrated. These are the usual

-12conditions that the beam enters the interaction region carrying no r-f information other than noise, the amplitude of the r-f circuit wave at this position is arbitrary*, and the r-f output is matched. Noise on the beam is neglected in determining the circuit wave amplitudes. The boundary conditions are v(o) = p(O) = O and V(0o) -- = z, (20) Ic(0) cO) which lead tod l+Cb -V ( ) [L C - 2d. (21) The problem can be scaled by using the following approximate maximum values, assuming a unit voltage |P -o, (22) C2 v 0.625 (23) 0 and a time scale of P = d (24) o dy The procedure to find an oscillating point for a given C, QC and d is to vary b and plot the circuit voltage V as a function of y for each value of b. A typical set of plots is shown in Fig. 3. The b that forces the r-f circuit voltage to go to zero represents a startoscillation condition. The starting length is determined from the value of y at which the line voltage vanishes. * This is true owing to the linearity of the problem.

-13z 0 z 0 0 z 0 o / / / 5 II _ u, 0 C,) o o H z "' 0.0 w LJ Ia ww r 0 Z C,)'0 1A I _" Z > i n f0 -2________________ >___ - _______ ^_______________ d ~~~~~~~~~~~~~~~~~~lL

-14A set of oscillating points as determined from the analog computer is shown below in Table I. A comparison is made with digital computer calculations in which the aforementioned implicit relation is solved. Note that good agreement is obtained between the two methods for all cases except for simultaneously large C and QC. This discrepancy at large QC for large C values is a result of the different space-charge expressions used. These results do disagree with those of Grow9. Since the results of Table I were found using two independent methods it is the authors' opinion that there may be some errors in Grow's work. TABLE I Comparison of Start-Oscillation Conditions Determined by Analog and Digital Methods b at CNs C QC d Start-Oscillation for Oscillation Analog Digital Analog Digital 0.05 0 0 1.57 1.559 0.3000 0.3013 0.05 0.10 0 1.61 1.575 0.09 0.3099 0.05 0.20 0 1.62 1.593 0.3167 0.3202 0.05 0.50 0 1.71 1.690 0.3748 0.3691 0.05 0 0.25 1.48 1.467 0.334 0.55314 0.05 0.10 0.25 1.51 1.484 0.3406 0.3433 0.05 0.20 0.25 1.52 1.503 0.3581 0.5579 0.05 0.50 0.25 1.67 1.652 0.4464 0.4351 0.10 0 0 1.62 1.594 0.2898 0.2891 0.10 0.5 0 1.83 1.872 0.3614 0.3399 0.20 0 0 1.67 1.664 0.2675 0.2662 0.20 0.5 0 2.17 2.329 0.3153 0.2826 MODULATION OF A BWO A detailed modulation study of the BWO would involve determining the change in frequency and power level of the operating oscillator as its average beam potential and current are varied by the modulating signal.

-15The characteristics of the operating* or unmodulated BWO can be determined only from a nonlinear analysis of the device, since it is the nonlinearities that ultimately determine the operating levels and the frequency. Nonlinear analyses of the unmodulated BWO have been carried out by Sedinll and by Rowe6 using the Lagrangian or particle approach. Sedin modified Nordsieck's12 TWA equations by changing the sign of the circuit impedance, while Rowe developed more general interaction equations and also changed the sign of the impedance. In each of these papers the authors experienced considerable difficulty and expense.in determining operating points. The major difficulty was that the boundary conditions at the beam entrance have to be found by a trial and error procedure. This, just as in the small-signal case, involves integrating the BWO equations for a series of trial boundary conditions until a zero line voltage is found to exist. In a low-frequency beam-modulation study of the BWO, it is necessary to include, in addition to the variable average beam parameters, a variable frequency as a function of the average beam conditions. Allowing the frequency to vary means that dispersion and impedance variation effects must be accounted for; this implies that an r-f structure must be specified. After selecting an r-f backward-wave structure, a set of quasi-stationary equations can be derived to describe the modulation characteristics. The solution of these equations would involve a difficult trial and error procedure which not only entails guessing the boundary conditions, but would further require guessing the new frequency. In view of the difficulties that would be encountered, it was decided not to attempt to solve the modulation problem using the Lagrangian * Operating refers to operation at current levels above the start-oscillation point.

-16approach. Due to the severe bunching effects shown theoretically by Sedin and experimentally by Gewartowskil3, it was decided that a hydrodynamical model using nonlinear equations would not be valid. The study presented below will discuss only the effect of modulation on the start-oscillation conditions. THE EFFECT OF MODULATION ON THE START-OSCILLATION CONDITIONS The equations developed above for the start-oscillation conditions of a BWO can easily be adapted to study the effect of low-frequency beam modulations on the start-oscillation conditions. A low-frequency beam modulation can be introduced by postulating that the beam average potential and current vary as' = O (1+ -) V (25) I = I ~ (1+ -), (26) where VO, Iol = the unmodulated quantities, and /AV, AI = the variations caused by the modulation. It is convenient to introduce some additional modulation parameters. Therefore, define: /= (1 3 7)-1/2 1 +M3 =, (27c) + — 01 01

-1703(M) B^(M=J) (27a) Z (M=l) 2 -(M) (27e) %s = X' (27f) 0 and F(M) R (M) 1 (27g) R2(M=1) 5 M = 1 indicates no modulation. When the interaction equations are revised to allow for the variations of Eqs. 25 and 26, the frequency is included as a function of the modulation, and the terms introduced in Eqs. 27 are used the following set of normalized equations results.,v tp.. (28) ~2 d + -, " o + Jo o (28) 1 c oC p F(M)_ dv V (29) 2 dy (+ 2C )2 C + (29) (1-2C QCo)2: t+Coo b o 0 0 00 3 and d.2 /1+C b - 2 a d<y2 CL) 003. c J C 0o. c — C -- V + = 2 (lCbb) V o) C (1f b p C(0) [L 023 53 1 where C, QCo, b and d refer to the unmodulated parameters.

-18The boundary conditions now become v(O) = p(O) = 0 1+C b C C 02 -V(0) - 2d 23 (3.1) dy /y=O L 2 3 Co ~ 2 j As mentioned above, in order to solve this set of equations it is necessary to specify a structure. The r-f circuit selected is the bifilar helix. The important properties of the bifilar helix as a backward-wave structure are given in the literaturel4,5 and are shown in Fig. 4. The normalized frequency parameters l 52' 3 may be calculated using Fig. 4. ~ and a as a function of ~ are shown in Fig. 5. 1 2 3 F as a function of g is shown in Fig. 6. The reduction factor for a hollow beam " was used and the unmodulated ka for all plots was taken as 0.50. The program to solve Eqs. 28 through 50 is only slightly different from the one for the unmodulated BWO. The procedure for finding the effect of modulation on the start-oscillation conditions (i.e., the new frequency and the new starting length) is to determine first the unmodulated operating point using the method given in the previous section. The frequency as a function of voltage modulation can then be estimated from the C-P diagram for the bifilar helix. Such an estimate is shown in Fig. 7. The problem is run on the computer first using the 3 determined from Fig. 7 for a specific AV/V;, is then varied according to 01 3 the -data for Figs. 5 and 6 as well as the other parameters until an oscillating point is found. This procedure is illustrated in Fig. 8. The effect of a beam-voltage modulation on the required normalized start-oscillation length and normalized starting frequency of a typical BWO is shown in Figs. 9 through 11. It can be seen that the initial values are quite accurate at low modulation amplitudes, but are in error as

-1970 o -0 70 _PHASE VELOCITY 1 — 60 - -— 0.4.,, 50L..... 1o8 REFERENCES >. TIEN; PROC IRE; JUNE,1954 / 2. WATKINS AND ASHi JAP; JUNE, 1954 / 40 -1.2 > 7 I I 7T XT 10 ICOT 4XOT=I|O m az.J 2IMEDANCE: O 20 iI — 2.0 10 ____ ____________ -24 0 0.1 0.2 0.3 0.4 0.5 0.6 NORMALIZED FREE SPACE WAVE NUMBER, ko FIG. 4 IMPEDANCE' AND PHASE VELOCITY2 CHARACTERISTICS OF (-1) MODE OF BIFILAR HELIX.

-201.6 -- --- 1.0 ---- 2.0 0.8 1.6 0.6 1.2 0.4 - -/ 0.8 COT kI =10 COTi=4 0.2 - - - 0.4 0 0.5 1.0 1.5 2.0 2.5 3.0 3 FIG.5 BWO MODULATION PARAMETERS at AND ^ AS FUNCTIONS OF FREQUENCY, ^. (BIFILAR HELIX, AT AV=O, ka=0.30)

-212.8............................................... COT*=4 2.8 COT* =10 2. 0 _ _ — 1.6 7 0.2 --- --- --- --- / —---- REFERENCE BRANCH AND MIHRAN, PGED; y~~/ ~ APRIL, 1955 0.8 - - -H — -: H0.,60 1.2......./ R 1.2 0.4 -- -- - -- -- - -- -- - --- 0 0.5 1.0 1.5 2.0 2.5 3.0 FIG.6 BWO MODULATION PARAMETER F AS FUNCTION OF FREQUENCY,.. (BIFILAR HELIX; AT AV=O, ka=0.30; ANNULAR BEAM, H=0.60, R=1.2)

-223.2 2.8 24 5 1.6 4 I I 1.2 0.8 0.c \ ________ ____ _____ ____ ____ ____ _____ ____ ____ ___.. 0.8 -I 0 1 2 3 4 5 NORMALIZED BEAM-POTENTIAL MODULATION, AV/Vo, FIG.7 FIRST ESTIMATE OF NORMALIZED FREQUENCY, go' OBTAINED FROM -,8 DIAGRAM. (BIFILAR HELIX; AT AV —O, ka=0.30)

-23AV/VoO \0 AV/Vo, 0.250 3 =1.0 C3 =0.930 OSCILLATES I DOES NOT OSCILLATE AV/ V, =0.250 AV/Vo- 0.50.92 =0.865 DOES NOT OSCILLATE DOES NOT OSCILLATE AV/VO, 0.250 A V/Vo, =0.50 =Co0.927 a0.870 OSCILLATES } OSCILLATES FIG. 8 TRIAL AND ERROR PROCEDURE FOR FINDING STARTOSCILLATION CONDITIONS WITH BEAM- POTENTIAL MODULATION.

-24w _ _ifl! u __w~ z-~I -J~~~~t ~~~~~~~~~~LL r \W i\ __________ ______ I Z --- 3 -T -r - uj ------ - -------------- <o c 4 \ I- r0.,\ \...I I I W o Z \o LL o> S0 Z C"0 1 0 dJ o -Io \. 0 O. ~ z _ _ —- -- -- -- ----- -- -- o e z -J...:0 0 Z 0).(0...... 0 O _ 0 0 0 ~~~0/)- - " - Z o

-25t_ _ _ I ~ w~~ O0 r':LL ~C: y II ~\ N.,,H' \~\ > Hc 0 0 0 \. ~ u. o ~,~, L< Z Q li 0.M ____ ____ ________ __\ ___________________ ____ 4 0, -Jod w < E 0J0? N.J1 f ~~~~~_ _ _~~~~~~_ _ _) _ _ _..\ o o:::: _ _ _ _ __Z Z -- - _ ~'/ * ~no u o0:' Q

-26m> I'- m c~ o 0 \ -o _ — __ __ o N~~~~~ 0 ("0 ig c Zw L 01 LI o~,I 0~~~~~~~~~ 0 o __ z.....___ _ —__ In. 02 _1 i^^ S~iw 0 __ __ I CO 0 N g m o z.. ~o -N ~ > ___ __ o (0.... o,/, HLeN3'1 a3ZllVl'ON

-27AV/V increases. This particular method of calculating the start01 oscillation conditions is recommended when the variation of starting frequency and required lengths is desired once an unmodulated operating point has been chosen. This method has the advantage of using only one set of parameter values. As the current or the tube length is increased beyond the startoscillation conditions nonlinearities rapidly set in to limit the power output for a specific d-c beam input. As mentioned above, the Eulerian model must be abandoned in a nonlinear description of the BWO because of the severe bunching that takes place. The Lagrangian equations for the modulated BWO can be derived in a manner similar to the derivation of the nonlinear TWA equations. CONCLUSIONS The start-oscillation conditions for the O-type backward-wave oscillator have been computed for large values of C, QC, and d, first using a digital technique and then an analog technique. The digital program involves the downhill method of finding the complex roots of the secular equation and this method is also applied to find points in the b - CNS plane which correspond to the first-order oscillation conditions, i.e., values of z > 0 where the r-f voltage is zero. The analog computer method involves the solution of the circuit and ballistic equations as simultaneous linear differential equations. Specific results of the digital calculation are given and a comparison between digital and analog results indicates excellent agreement. The analog computer technique is also applied to a study of the effect of beam-potential modulation on the starting conditions of the backward-wave oscillator. The unmodulated starting conditions are used

-28as initial estimates in a trial and error procedure. These results are summarized in several graphs.

APPENDIX A: O-TYPE BWO STARTING CONDITIONS Parameter Range C 10-5 to 0.2 QC 0 to 1.0 d 0 to 1.0 -29

-30Go H NCA\ COC O Ln _t CO N-I cCd c C o — OK \0 \!0 N O' OO OOr\ OO~\ ^ rl \ KL\ -O-\ n \D CO I tK\ n N -A- _-_ C.-t 10 h -H- O L -i N c Cd 0 H' - O O\ H H -C'C -' O0 cO cOco 0 O 0 Cd NC\ NC> COCO -O 0> > Co Nc c CC-\Z OC O ~-,-) *',-rJ., —>', -r',-',*-.r-D r- "-' "-'.r- j. *r * *r- *- r -.r - *r-) *I I I I I I I I I III I I I I I I I -CO CdJ N-C> o LC r-I \ CN J HO 60~0 H~~cC) t \0 L- Hr —f- LC\-t C J Cd - CI- C o C \,O CD d cO O cO —' In - C\j o [-LC\nNLC\\0 LrNLN >NC> NrC Cd Cd Cd Cd Cd Cd Cd 00000000000 ddddddd0d0d ir ~\^~Ooo o o t o OOO O O OOO 0 OOO OJ oc N C Cd ~ N- O NC> r ~-zt~ i ~- O: ONr N 0 O- 0 \3O'-, 0 0 Cd ~1-Z Co C \ONNLN 0 Cd OI- N C dj Cd O H-l OC-O o Cd O d Cd CdI 0 O l O\ \ \ Ln \'\ )o O o t- cO 0 Cd - N n cO Ln cO 0 co _G o 0 Cd 1C> H LCC)O H Cd cZ- IC -G\ o ON H H n \o H N C _:zi- LC> co ON r-D 0r o o r-'r- Hr'D -r-" -.r.r r-H r-r- r-.r- r* r- *i-> * ( 3 * r- *i- *i- *f- *i- I' -3 *f-> *~- o ~) r *r- ) - - *r-3 *13 *r ) *r:3 + + + + + + + + + + + -I- + CM rC H Go OO >-CrY,\ n N -- --- Cd e c tO Cdo ooo( C o O\oo-d —j- - C\-t N IN o Ln 0 \ o-i Ln \ o Ou Cr <O N K \\ 0 O\ G\ -:-r- r-l C-d t C Cd C rd n 0 ICo Nr-z r- NCOH C12 00 00 * 00000000 ~ O i i O O O O o, o00000.r- I I I 0 II 0 cO ONO 00 OO O O o cO H-i C rd -— t — I- H NCO >Go \ CNONCd ON O Cd K >\ C\KN~> \Od LCO ON h. 0 \DO \ N cO cN H L> 0O\ LN0 n L K t —- 0 —O O Ln OO c-,o\,D O'\ (,\ n C~j \,D co 0....... n Co \,,O CT\ LC,\ Cn 0 Cr\ O*HH HF H H- I-O t- NH Cd NC> LCN HS H H-zI-rO C- ON Cd NC-I rO O LO N^ O-^^ P Ol OOOO O-POO 0 c 0- 0 0- 0 H H H H 1 O C0 0 0 0F 0-H H CH 4 O-0 H + + C'I0 + + 1 I I I I I I I I Co *0 to CH 0 )O H o — O -z LCoo \O H Co ON 0 CJ o 0 -- \0O >- >-n -o 0o oH K o,\ SII C j 0 N>r-Z O -- O n Lfn \ O L\nI LN >n pq9 ~ [^ —~ l>-z1- NC>Cd Cd Cj Cd Cd Cd C Cd ~ ~ *0 00000000 (D 00 O' -C 0 C 00 0 000000O 0 0o O rlO c>O c O Cd H CO O-to o 00 >- H Ozl- NC>0 O O N7\ C H H-rD\10Hm \CNCOn ca 0 0 H M L>o- H;0 0C> O Cdj K \ nC>I\ \ 0 Cd n- IC> OH Cd C\ ti- Cd 0 — H C> OW)- -\H O t\ tC —H-z_:- Cd cOt c 3 —I-\,0 H H OC)"\ ONH OH0>0 NC>0 H Cd ON1\ t I CN-\.oordddoooo~ OiojKdadoo C\J r< 0 | n KO \rn 0l\ ON l^- \n \r0 r\ i ^ rd 4~ \oJ CdH- 0O00 C-d\ C ON\ cc)r- 0 O ONm ON\ N\ C- CC)O \1O 0C IC I IC> IC IC> IC\K0 h- GoO 0 I zn Cj- -zi- r-z ICN 0 h -Gso ON 0 o) H Cd N* L>\* CoO NO.Cd. I\ -Go O 0000000 d oH 0o0 0d0d0d 0 Hd

-31Hr - CM 000- 0J l On [ 0 [ - \- NC OCM 0(MO 0 co WHc\ O co0 [- N1\ H \ K\-_t L-\_:- 0 ON\ i\ oN\ _ H r-i'C\ 0 -CO Co -- LN CK c — HI CO CO \ b-co c co CO O H0 1 C - _d- Lc\ L cO r- l 1 0 r- CO r-i \ c ID -l>-c 00cO OC O H CM c\ -- H H H H H CM CM CM: CM CM CM CrO I I I I I I I I I I I. r-3.r - -.r-3.-'r3',' "*r' * "-' * r -3 - 3.r" 3 0 O -t't H r-' l 0 \ O j ON-_ J I I I I I I I I I I I O P n cO' c r"-C H — \ N C 0 O' ON' CO- t —- O O VKN Ln H CM ONC H C O LCn cO0-=i- [ -H CM- 0 C'C- — 0'-- 0 Ln \ -t -I O r -Ir 0 C O\O -- c 0 0 0 cO O'O i[- - O in C n n n Lr Ln r\ Lf\ N Co t- L c - CM- CM I Cr j COj CMj 00000000000 000000000 -- - r- r 0 — i r-I r r- rododoodd doJd odddoddodo \O ON b- kO O C k \O \'O - O'N CM O \ N ) (N- ON 0 nCo Ln _ -: Ln \0 CO _:I- _>-t \) Ln ON b\OCO [O -OH CMJ 0CM ON \,OO O -\ — H ONH r M CMN \ \oN \o HO- In O O C LO -O O c C cO -H-.C \ O cH CM c O OC r00 i0n0 Ho r CH Hi HTN o, r- I OJn o00 r- C - cO \ f r — j.r-j'' D.r-' * -' f.'.r". r- D. "- D *"D.r —D, —.r-D.- --. r —'i -D r -. ). r"', —,3 *.-3 1-3 + + + + + ++ + + + + + + + + + + + + + + + (0 \O _- - Hr- cO -- [NO ON o\ -o \o [o -, O.\ - r-i - 1O n \9O (K cO OC -[ —O CM H0 \ 0 W -- t-N N cO CMco H C -H J rCM H o-CO V O O o r-0 C\ o r- CO ['- -t CM H [-L Nn CNCM rC -lH > — [-C HUr-I ON-[- Wn -Z- _ 000000000 * 000000000 d00 *O 00000, - o o. 000000 I I00.'l00001 10000000 IllIOOIOO10 0 CM 0 H - - CMO- - O 0 0\ ON \O CO [-'l\ ~II O'Xc l cO CMJ -:Nd- C CM O [ — O CM rHl CM LON O -i O CJ N - O >- - - II r- H- -_i 0 KO 0 NC -J- C\ 0 d -t [-d — - - I- OI ON \0 C 0CO O CM 0o -- in - r- O\ - >- ONrH rlH -ft-O - ONH CM NI -- rd r-l rH - H - \CO O Cr- JK LN Hr-i d0 r0 r0HH HH ddd00d 00 dd r- r rO. )..*.'-3.,-3 - 3.,'" 1-r 3.. --.- 3'*'*'"'3 3'3'"-3' -3 "r-.r3 "r3 0 - + +- I I I I I I I I I 0 + + I I I I I I O 60 \O CO COO CM 0 O. Lt LLn n \O 10 -c rC \O L r- \O \ --- \O O 0 CO \1- 0 \ n n',"l -H I- 0 0 -:- N r- Cj _- 0C O 0 IHr Ln Zc t OCO K O \LNrO 0 C \-JO WC- -- - OSN —N 1>-CN - O -zt LNM HOU CM-zCM 000 H *OONO Kc CM CMCMCMCM [ — \d' o L L L.\i LO\ LN 0 Lr Lr o [ --- r-i r- t:.r- -i H- - H H H r -r 00. II 00000000000dod o 000000000 0 \01 K f \-0 WN 0_- - 0 CQH CO LC\0o n ['-H cO ONcO CM\C) 0 [n __-: Ln \H- 00 Ic'CM CC I- Ln 0 H -H t- --- -- CO IC\ c O0 MI- \O HCOO — - O \Co H CM-[ \10 \10 \ON \0n 1O H [L \[\N-t OC- o \-[ — W\~ o O- o o cOH O Co CO c O Cr[-H C OIV-Lf\LI\OCOO HW\ c O\O H \ o r-l Ln C _-_:- CMk -(N[ z CM J ln \o0 0OCMj\ O ONH CMor- J -- I \'-O- rN t- H C[n -ON 0 K\ r-c N r'c\ r <C Nt - it - iz- Lp ir \ N [N LN\ - -f -z Li\ [N n Nir\ in C\J CO 1- > C\- C M \ -i- CO CO 1-'-\ rzt \0 \ rN KN LN N C O t0-' cd cCO — co C co co O- -d- I-\ m i n WI\ \ ooL ['-OO \O o~ t t ~ ~ ~.. ~..... o H CM V-t oI\oo [-CooN 0 o CM o~t-i \[ o >-O O O

-32\H r- O CU CU O Lr O 0-o H \ O \ CO L- H -d\O 0 i r r-l O CU Lr \D CU -t H -'O -\ O - O c O )- - \ - O Lf \ - L - CO Lr\n co - cO H c'O >- - -Oo o r-H CJ O I. -- -- _ \ - CCO r-, —.- o. o'o r-.r r-. -..-.\r! 4 r f-l. r-l * * * * * ) * S *~~) *' ~ **a *~r* ~1 * * * *~' *3 * *( co I I I I I I I I I I I I I I I I I I I o LU \O C\J H r -l O CO L, C -H -\ -ON-CO Lr C N\. CO cO \- r Nl L O C- O- L \r0 L\cO u-t CO O D LU\ Lr -' - OI-o O O H \ rr\ DO'C O CO \D',OO n u\n Ln O cO - r CU rc N-C C\N CU Cu C CU C C CU Q LC L\, L\, D L N C _U o 0 H 0o Cu \ OO CU _ Ln L\ co co \9 0 L _ a O l, 0 - O 0 N CU H.CuJ \O \D c- O r o co N'- O -0 Hr 0 O O U H ON cO O o c cO H CO N - H - - C\ C0 U O L 00 CJ rr LC O H CU n c cO Oa CJU _- \O O r- \ t U\ d d d d6 r4 r4 r-I r- r-46d dd r4 r4 r *'r *i *r) * r *13 *i>.,' *r *r *i- *i *13 *1 *r^ *( *i> *I) * r + + + ++++ ++ + + + + + + + + o C- CU 0' - - CU 0 n C -O \N CO ) O N F H- C* * O, 0 O O -— Ct \O \,O C*U t — cO'H C\ 0O - o LC N J c - Ir O- C O_ O O \O- r O- - * \O 000 0ooo 0o0o 0000~ I I 000 * * 100000o 0 0. I I Lr_ - ci L 0 C\ 0'- [', - -O H- LON 0 oC O - C U-: CO \O \ 11 H CO O J CH C\JOU Ln O \ Lr\ — Oc C\OG co \OO O cO L- O CUl H I- U\ O LR - Tr CO CU N H - L -NN CO C zti O \ d N\' \ H 1- 0 H ON\ H 0 000 H H H H H H 0 O 00 H H H e~0 d *r * *n *~- *r *. *. D *r *r> *' * 0 *O.n *-.r. * * ~r3 * 0 O 0 + + I I I I I I I I I.0 * I 0 COJ -t U L O\ C C -, cO Cu O O O U H -I- O \ -O Lr\ O r L', -- O r- O cO O — O -1 l LCr Lr N U \ C O OD r 0 HlO H [ - - I -- O LU L\ -Lr C J U -I- -CU H HrcO ur'\ CU O C Od CU O Cj CU! J C \ L >- u Lr L LUrX II ddd....dddddd II dddddddd 0) |0;1 L -t CO O O'O - UcO CO Hr CU- C O O 0 -- CU (D-t 0 -CJu - L\ \O -- (\ OO co I- O CU K CCL C Lr\ t- r —- O co ) - \ H OO lO Uf\ KCO c I..........[LC-a WY O C'-OGC CONON \ — zt LC-lO NH H CU _- lO \ G zo t - o CU Cu cO O CU C H Ko O a)O Z OG co H c\ C'- CcO CU O 0 Nr- \1C -I- H CU ONON-zt L- \H o u2 \-t -: -zI- LC U-\\O CO O NC \,D0 H 000 0 00 0 0 0 O 00 0 0 O O H H H CU 4)~. C-O \cO O -Lrn C f) LrO O r - H1 J CO CON CU Cd c \Q Ln Ln -K Ln O \ \O \ > 01 O\ CU H C Ln rN-r\ - r LC\ - i O l iO r — O C rH Hl- C-U r- N - r O' r V) H CU N c Lu\ O [cO ONO H CU Lrz u0 [CO O O O 000000000 O 0. O0OOOO0 H

-33o CO n I n cN H \ cO Z cu > a Un \n Nn \ 4 o H o -- t-o - o c CO H -I \ C C u o C\ H \n r — \ n CO r- \- CO H —C O Co CcU 1 - O O L HCO CO H CU C o H CO CO OO O H 0 -- -n \oD cO oO O\ aO\ O H O r- i L \ ~., -..........-...-..... H H H H C CU C C CU CN C0 CU H H-H rH CJ CUO CU CU C\ C\l CJ *r *-3 *r. *r1 *r3 *~r *r- *. *r3 *ir *r- * * - *i- > *r- D *n- * r ) *.r *r3 - *.r *r3 *n- I II I I I II I I I I I I III c0 0 LC\ O 1< 0 Cll C N C N O", - F-L C\L C> — _ H H CU ON \- -- H 0 H Hr- O W\ C0 Co >- \ cN 000o 0 Ln \O \O \ L C\\N N CU CU CU CU CU CU CU 00 O O O OL O c — OO 0 o 0 O\ L r-1 0 \n -z rNc\ H \O O \ _:t n -Z- Z j-n \ O - >- CU ON LC rcN K C \O0 CU _ n \0 \ O O 0 O _ 0 CUO - I L>-n C0 00 CU n \o n \` r_- C n 9 \D o 00 t \ \ K'\ n \O \0 H H - Fn cO O CO r Ln \O - cO -0 H Ln 00 0 CU N n L\o - O..+...-.-.-.-......... O0000O H H H H H H H 0 0 H H H H H H H ++ + + + + + + + + + + + + co. [,- c* * O O O \cO CO C O cO O 000 00 0 00 0 O CU ONLO O *\H 00 00 * 00000000 Lr\ 0 0 c0- - CU CU O c0 *K Q 0o _O - Ln 0 CLn n LN Ct- O N-C\ \O CU -0 0 Cj r\ \0 - L\ O r H Ln \ O coo NC\o H r-L1 n L rN cO O\ - o\ \o - H - d H H CUdZ ~ f \0 0 0 NHn H H H t ~C ON 0 | r 0J r —; r<-r-4 r- 0 l r 1 - 6 O CO C 4 rl r4 rL ^ O00000 00^ H OH rl 0000 0 O OH H H H OJ H *VI * *' *1- *r 3 * * *r3 *1 3 *-3 *-3 ~ *3 *3 *r3 *3rD *r-3 *r3 *i- *r-3 *-3 *D * r-3 0 60 + + I I I I1 CU + + I I I I I I I I r H 0 o 0 r00 \ -- ONC U W U N r \ 0 0o0 * CU o L L\Hn n o N ~~ \ L \ H 0 U CU CL \0 CU CU CU rC\ \- co CII O ^ K- _t \ — O L,On \O L \r —' -- F X 11`~~ >- X t CU CU CU J U CU C C U C CU 00.0 0 -- O ~ 000000000 O O O O Oi K LfL ~C OC OOd O O n \0o cO r\N L t \ C H n rC, n L n - CJ CO- \O CO )- cO - n O I\ t — L- C n j 00 CU cO H00 Ol ON I- Cn CLo O O H t-n CO - O _: 0 > O H CU r- c\0 ONICU CJU'\ c0 CU ^VNix LC ot c' ) — d- c r —\ -- LF( \ \ 0C ON n C O -H! ON CO O CUO\ ) CO O C H1 LC CU CU CU CU Cl) 0 0 ON-t CU X 00O rKNL( It CU -- C CU \o NN-zCC)i G. - LOc'I r- LON0 CU\ 0- N CU o O CY\ L 0n -D- CO - O o..\ \..........-..\. rQ H H H H H H H H H CU C n- I H H H H HL H -I CU CO O 0 n-IO C r F X LCN [ OO C O - O 0 H CU < L 0 N-CO ON 0 Cf................... OOo0000000oo0 ooH OOOOOOOOOO H

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-46OO.cO O LnO rl CO- O\ -C ON coO. \'O O N- O\ \' —-cO t- cO t'- cO cH Lr H K(\N o \Lr\,0\ O M r-ll-cO ONO O C \M H CM N rU' -O — 0 N-O O C k - - 00 0 ONHr-I0 K CM!~ -'O cO 0 l -.O ON 0 r-A l, -t O \ - O' _J- \'0 Hi cM c c cM CM CM CM Jt( c,\r H CM CM C C CM CM CM K rc\K *r-3 *r-3.r3 - r3 * —3 *- 3 *-r3 *-r3 *-r or-3.-'3 o —3.-3. r- 3 *r 3 *r -' -3 -D.r-3.' -' r- 3 I I I I I I I I I I I I I I I I I I I I I I Co t — Lr\ L\ -dCO- O- C O\ -I-I rC-l - - CO Lf\ tN- OCN r\UN r\ O t"O -- -.- O - — d 0 t —--' - Lr\C0 0 t-I CO H CO CM 0 KCO\ C t' - CM' -O'-CCO O OC - - - CO C0o \O C\JC L\N Hr CO -O L\ LC\ L\N L\ cO - t LTN\ - \ rc\ rc\ K\ L —t 0 \ O O u-\OL\ Lr\ Lr\ Lr\ UL H r-I r-Hl H - H H r —i - 0000000000.........d ddddoddd ddoo rc~~~\O~~~~zt H ~~~~~ o -'\O K \o HC\ ( ( u-\r _:J- r -:- O0 (J \O -:- (\\ ( r r-i Q -:d- _:I- o N -0.~ rC- CM L,\I'-..- cO C'-\ ON M -.- \0 C-.H Co O.- C\ H OJ ONL O \LO t-4O mJM 0 ON NJCMONO V O t n-^C u Mj ON CM HU t ~-ON0 H CM (N-~fE CM CNzt -O\0 H C (N t-t-t 0 0 0 00 H H H H H H 00000 H H H H H H ~ r-D.r-j.r-D. r-z.r-3.r-l. r -. — r.. -l-j.r-.r-j.fr-a.r- o..,).r —z. — j. r-.- r-. c1J + + + + + + + + + + + + + + + ++++ + + + to0 \z- O L- \ON -t -0 - c - 0 \C CO\ H0r-\_ O cO - -OTr OJ -Hz- --- 0 C b-uK( 0 ul - C O 1 r -0 r ON -J OO b -H O H C \ tl- - l- H tC HHi — t Hcr- r 00o -k - Lr rH t-\ r-i Lr-N ^- CM CMJ * 00000000 * *000000000 00 * O0000000 00 e'O000000 I IO. 0 1 0 000000'0 0 0 0 0 0 0 I I 1000 0,00 - cO -- cO C C U Ln r\ o ~0 cO oH O o \O O -H o COO o O O O HcO Lf\ o- 0 uL CO t- Hr- L u\O O J - u - OJ rH Ln- C- \ cO cO ON\ Kr- \O tr- O H-\ Hr NLr -\ 0 11O cO H-l- (u cO 0 N OO t- cO -,H- \O CO 0 u\LrN t-O\ Hrl r-i C lu CO 0 Lr\ -O\r0 dddoo6 4 4 4 r~ 4 c J 4 rdo r4'cU L(n\ 60 f - I I I I I I I I I \ I I I I I I I I I H- O CJ L L \JI C -O K\ H O-crl O CO\ - r-t 00 -CO K\ - 0\ -KCO 00 \ O \_I — CM: CM O -r-H ONOKN UNOLr C\J 0 — L —H 0 - \O H L r O 0 -- O K 0CJ Lr\C c -L\- _ — CJ — O J j- \ 0\OO \O Lr\ — U Oc \Hr- A H - r H H H rI- --- ULN -:- -- i-_ -=J-_ - -4-II _>- - r r- r- i H H r- l r *00 0000000 O 0 0 0. -000000000000 c c 0 cO Lrn CO r-H CM 0JCO o 00 COc O C\0 O\ —D tO \O CO I, l O -0-t 0 —I —I - D - 0 -C L M C O Hf_ O cC\o) CM CO M 0O \D \OtD \,O O CO ONOcO Hn \OCM OL OOdO OH OH cO J L\ -COO\ O\ON 0 CM CM 0 A CM Lr\ \ D \O 0 H Ci K\ L L- 0\ 0 \ ~ C\J~ ~NI K t ~l\ ~lW\ K K \ dooodooooooodddddddddd 4Cc- 00 Ln\O - lG ^- ^OCI- co \o - 0 o co \c; cL,; \O o MQ H H H H HC CM C MI C M C\ M - r\ H H rH H H C M- CM CM C\ 0 H CM z-^ LP O tl-CO O r0 H OJ CM I -- u \OD t -CO O 0d................. 0000000000 H O0000000000 H

-47CJ t-l > H O ao-, cO t- \ O- Ln O n O\ t \o 0o0 Lr\ cO N O \KH U HH t- C 00 o u- O 0 \h' C O O C \J _I- LC\ Ln\- Q H - O u oOL \ L L\ 0\L r \L L crO G\ 0 CU r"' c\L -O \H \ Lr\( L O \ U h H \ Lr\n' C\ r\H Kn H H C CU CU CU CU C N N N H H H H CU CU CU Cu CU NC K; *r3 *r 3 *r *r3 *r1 *T- *r1.r3 *r3 *3 *r3 *r1 *3 *T- *r1 *rr *rz *r3 *r-3 *r-3 *1 *r3 c) I I I I I I I I I I I I I I I I I I I I I I o 0' -COO H L\ O -\ 0 OCU CU O r- LrA 0 \J- -- H G\- trc\ - CU O 0 L0 1Y kC: O ) CU l O CC O -t Lc cf LLr Lr\ K Lr\ HO CU H r- 0 o r OC t-O\H N-\ I O C\ O 00 \O 0- CU O 0 t -t- - - \ C -- O \Lr - t - C\ CiJ C C Hr- H K\ n rC'J CU CtU CU CUJ CojU jo \O \ L\ L LUr\ Lr\ Lr Lr\ L L\ \ OCO O \ oc O) > w L- \L CU\J O > cO - Lr \C L C>C CU CU r. - O' l H C OJ 1 r L o H crU C\j- *r- *3 *- * *13 * * * *3 * * * 3 * * * * *r * * *. ( + + + + + + + + ++ + + + + + + + + + 6 - - 0 0 r Lr\ HL O C oO O L LOJ' I UO C u -- cO \OD O Lr \ oC LO Ln Cr n\ -J o L oN C \O CU C-I 00 - CU O OO U - O O O OCOCOCO t HO O -O - O O O \D LnL O O \ \OO l O O O 0 O O L 0 - \O _t -r — CU H r-H00O L\ -t'O H _- W CJ H Hr- r-lCO ~000- 0 *000 0 0000 I O O * * I-O O 00000 00 I"O O 0 00. 1 I 1 1 1 0 CO H o Oo * -H \C 0 0o o L 0 N CU O \O C O O H CU 0 CO CO CU ON O\ n t- -. O t - O CU 0\ \o, O O uCo \ n- O Lr\ t-H H-i i O'0 -CO- o O Co U cOl 11 C\ \ -O L \OLr L O0 0 CO U O O r-O CU irn 00 O0 Ln -O'00 VO 1" C, -O " 0\cH r- N rW..* *.... *.............. ooooo H H H H HCU 0 0o0o0ooo H H H H CU C 0 *-3 3 *r 3' -3 *r3 3 *r *r-> *r r- r3.*-3 *r-3 *-3 *-rz r3 *1rz * r- *-r rz3 r Lu\' + +1 I I I I I I I I I \ I I rH 60 M Or O( \C Lr \ 00n CUO\H O 0-^0 O LQC \ K CC\U 0 CU C, co ao H > "\ L\ r O L o0 OCu H c> Lr CO -- NCU C cu 0 Lr N-. N, o - r N\ n Hn \ CU t - H L,\ ON ON \ ON O CO LU C U CU CU C CU CU Ck U 11.\ _. U'. - - ca C U\ Lr co NO \ O CO ULrA - > n HCU \O L ro CU- H c> o * * o dooooooo ~~gddd~o dddddo oca CU co r n r zj o o3 L u t- - K o n K\1 Lr[ - - co O H H H CU -(CaO O- -H C o O CO 0o 0O- O 0o O cO H U\0 co N \ K N\ \- -- -4 N- co r- -c L LC 1\0 -- O O -:- O — t0- 0 00000 00 0 o 000 0 0 t O O r o O - - \- O O O - 0 O 0 0 0 _:-LU'-, -0 t - 0 C- CD 0 \1 C -- t-C H \ u- 0 -\> H CU'N- N \C 0 0 - \OCO 0 H CU KN Xp LfOO\ t-c (L O H CU r-CJ- L\No 0 N -C-O 0 00 0 000 0- O0 - O UO O - O — O O O O OJ 0 00 HO

-48H \O E —. CM H tr< - O 0 ON r- OO H J LC C -t OC\J O l 00 C LOrnt-ON0\M \-OO (MO O O -r\ Oz-t 0 HO O _: Lr\ L'N O U\H ONt-~-O UCM \O Lnr\ CO NO 0O \O CO CM \0O Lr rH O C -t u -OCN-,\ H -: \, \ ON CJ- OLr\t H- ON\ lQ\ O CM Cu CM CU CM KN C; NC; CM CM oi KC ~3 * r-3 * 1 - 3 -j. r-3. r-3. r-.r - * r- 3 *- *r-3 r' - * -j *r-3 *r -3 * -r - r 3 *r - 3. r3. -. * r-3 CO -^ —t CO _ n — t -:- CO O r- LO\O cx K \ N OJ OJ\ CNO O O rr\ r(NUCM CM CM CM C 00000000000 lH K \ ONLR-t _:J —CO C O- OCM Or- \O0 — t 1 0 0 Hl ONCUO r-lr- t W-\\ O \ t r- O\ CO - (MO J KNO O M H -r — Lr \O K\CM ONCN\ \ (MNOO H u\CO KC\ r-O r- 0\ \OD CO C \0CO K\ONH r- 0 O \ K \ CO r-H - CO O H H C 0K OCM -=t _ -00C O H H CM KC -000000H0H0H0H0H H 00000 H H H H HH * r *3 13 613 T-3 *r 3 *r3 *r3 *1 3 *13 *f3 * 4 1 3*i- *~ * r3 ~ r3 ~ 1 * *r3 ~1 3.3 ~ 13 Cu + + + + + + + + + + + + + 60 ^0 L- \ CO 0HO k U - L -i -H O LO LNr-0 r-H- CM\ t-1 CM' COMoO OJ }-t - Hl - O CM Lc Lr\DO H OO N ~O O r- r-J r —l r- rj r — -'- CO, — C- HH O.O Lr\ -t KC\.* -000000000 00 00I000000000 I I I I... *0 00 00 0000.. LC\ H. r,> — 0 0 -, -\ \,-,\ o 0 C ooCOO \O - O O\ * 0 \ O t >-H- ON\ UCO o\ ri o o -- H-I- -\ H 0 N000 -I- C\ CO 0 \ O\0 ON \0 H, Or- C O -~ — 0 H O'\ -i-_ 0 0Q 0 - C, O Lr\H O- C) CM H r\\ O \ONC - L 0C u\ 11 CJ H NC\l CV O J-NC - t>- [ 0 O T. Mc — dddddo Io Io -ddddd4r-I II 1o I I Io OJ cO + +' l'' + + \O Lf\ CJ CMJr CM Lr O r O -t L U\ OW\ O LF\ \ N* r-H 0 K O O\ L\ 0 C0\rCJ -f l>- OC 0 t -C- Ln - O O O\0 0\CI) O r-.-O O,\ \ 0 N —r - H C r- l H l^ ^t` +~~I- > -- C r-i C J CJ Cj COJ CJ CJ o o.-,. 00000000 d-N r- L u 00000c 0000 (DO O c\O\ rct\ - -r-I l O\0 CO 00 H CUJCO H rH \O r- H c t-O C\J -J r- Lc\O L \0 \o -O CU LC CO H r-i \ CM J - Ca COjM CM N LC rz z f Lr O L O C ^ KN; ^K;C tr LC\ LrC O O CM LC-CO CO \ CM H _Jt C C O C0 t- \CO 0N L\ - H C \C COO r- CM CO —.. 0 ON 0', L\ \Nr CO _-H LC\O \0 II O \O \ -0 — COCDO OO NO ON, - CO CO O -O 000ro C CM CM CM CM CM CM C CM K-\C C M CM CM CM CM CM ^^000000000000'00000000000 0). O NHrj CM NO N\- \O O NO-\0 -C L\ CO \RO oCO rH H CO cd, o>-0 ro -C O l Lr\ - r- K Ln C -O — 0 COO O Lr 0O \O >- cO O - 0 Lr — O r\4\ \-'CO O H 0l -- t-O C - Ll\ O H CM rcN- - u — h-C- O ONO 0 - - H- C O oO NO O0000000000 H 000000000 orH

-49ON\ -- \O un Lr \O \Lr0 oo \ ON OcO C-O Lr - - ON co -\O 0 \L \0 CM0 ON 0 O u\ CO 0 0 -O H 0 0 -i HO H cO L - - L — O HO O\ L r\H -' O\ 0 o'\0 C0\ 001 N L R- 0 H\ -H _ \O \ 0 ON N ONH\ r Ki Nl\ -O O\H\ \1O ONC 0 C I~ ~- Ir i:) I I I ~ I i _ OIi I ~I Ij I J *r-3 -. --'). r -3.r -~." 3, —3. —3''0.* —>.r-'>'-'.1' ~. -.(- *r —.*"'I'r3.r'"~.1"3 T 1 —3 CO) 0 C\O L O\ \ -O CJ 0 C 0 O O\ MN r LH \r- O L\r-l O\ 1 -1- Lr\O H C \0 O-\'- cO. -- O\ HcO c -cO t0 - \O H N OcO L r-lH 0 CO Lr\ 00 - c0\ r-i1 ON rt - 0 0 H — H - NC\ 0 t- rC 0 O \ —0 \0O - cO 00 1>-O L\u - I- — f- _ d —- -tt-'- 0t- \k O LP..Lr\L L L\L\ H H r- H r-I H - r- H r- r — I r00000000000....,-i i. H ClR- - O0 0 1O - O O r O O O CI O l O NOcoOH LPO\ r- lO \ \ ( HO 00 D- O CM- 0000 C\' H \ 0 0 0 0 —I ~J' D- C 0 H r-H H H H'\ HI C 0 H H H-i HQ H"\ H' do d dod r rlrj lrcrr-Irro4 ~r4r-43r lH H ~*r->.r-D. r —_D - *r-D'r" *r i r".r"D'-D'r"D3.'D ~.1).1".r-3.*~rr.r-I.-3.1-3.r3.r-3. —3.r-3 + + + + + + + + + + + + + + + + + + + + + + 00 ON - 0c 0 cO r-I 000 rJK \ Lr\ \0 O 1\- u L -t cK NNL\ CjLr\ O O O\ r- \O Lu\0 L r- 00 -o-t- L\00 cO -L rL.- \.D OC\ \x -I- cO trx\ Lr'\ LuC C\ C ~ 0 K'\ h' O\'" h- \,D -0O\ O N L\ cO- OW CL\LNOCM -H 0\ \O tr-\O \U-N0-O 0- - Hr- O cM C H r-lr- H000r- ot H ON\0 — Kr C CJ Hrd*d 000000000 *d 000000000 00 * 00000000 00. *000 0000 ~ O O O O O O O O C 0 i I 0.00 I 1 0 0. I I I I I I 0 0 I I i I 00 CM L ON O CM rl -H - N- _J 0 0o 000 - uLr\O Lr\ H \ *o 0 t-lN- -CJr H LrN\ \-O OJ Hr t \ OK t-AO C\0[- Hco — 0 O HO C t H N-\ N\H0O r- *.3 *r *r 3.-3 o 3 1-3 *1 3 a-3 a 3 * D r- r3. 1- 3 O. - * 3. r- *z. 1-i3 r -) *ro3 *r3 C\M ~0 - + I I I I I I I I I +1 S + + I- I I CJ dt - cOoO \ LrN \-0 \0 CM ON \N 0c0 CMJ r-H00 J \O \,O O L- L rl\ - H t- r\Lr\ r-l \ *\ \. O - O O CJ LC\r OC\C j r Hf W I\ \O O C \O Hr-00 f-lt - O- 0 \O O Hr —-t - LNC\ CM HOJ H 0 II CMj N H \0,\ =tI 4 -4- N -:-00 CM 000 H H H H H II K-'\ _ - l l _~t -O _:- -4- -4- -: r- 4 -I rl N —CM CM CM N L,0 0 LD,.o1~O o CC 01CM OCM ~ a) 00 H o,,.o'c0 ~C cQ H O (N - O. H ON or L r C tC o~. N-0 C L0 0 C0 0 N- ON C\ONL H N-CM'0 0000 LO\~(\0~0~\CM CMC~f LCN00 LC~000 zt\\ LH Cl L 00 rO-N i CM 00CM -C\ H H -z- 00 LC' ON N\ LrCM LiN-\l- H cZ tN-N- 00000 ONO O Ho ONONONO OY I\H CM \ d0ddddd0 dod00000000000 o L co O, —\.C\1 co O.0 0O L -- ) C0 H O'\ 0 0 U\ C — tC\ _o O C o tNO O _ O l- 1\O L\ cO -, - O:I- J- O\ 1\ r, Kc \ r —O cl) o NC\ ON H - C C \ O -I _:-.N- l-c 0 0C 0 N L-\Lr\- \OI 0 000000000 H. 000000000 C0 CO O O O O O~ l K K K OJ J J ^ ^ \ ^ ~ O K K ft~ ~ ~ ~ ~~~~~C (lIrl Lrx Lr'X k.C), [, — CN O o.I -lOJOJO O x, —I

-50O r O G\ CU t — \0 L c \ —0 co \- 0 O\0co -Lr\- - C-U-j CU cO - co CO j LCn \ 0 - O r- OO Lr\ 0 Lr -C cO 01 0- 0 C\ CON0 r \0 r- 0\ \ -- -O. - O - C O\ K 0 CO CO O Cc\j K I. cO O ~l" O D \ - Ox rA WC' \kO c O JO - -I OC, * r -z T-r *.- " r-3.r3 r'-3.r"' 13. 1-3 1r-3.r.r"-3. i"-.r. r~.r3 *r-3.r r r-.r *r-D.r, r-. r' CO I I I I I I I I I I I I I I I I I I I I I I tO \O - t 0 - O- d-0 0 0 -- [- cO r- 0 - r-l CO 0 L\ -- L- dON r O N\ N0 o0 GN OI 0 CU C H cO \0 - CU loC \0'- N,\rO H -=I- \ \N \-OQ [\OCCO Hr-1\-CO H0 \-O j0\-\O0 0 H- C\ I -! C — h-L -[ I' —-[' —- h0 -P-1- N~ 0 C.\J C\I CQI N C Q!C!I C(I 0 1 C01 C0 101 C LrCX LrO Lu. L. \ L- r Lr\ LrN N- K~rt K^ OJ OJ OJ OJ OJ OJ OJ 0 0r OLLUrL-Lr[U' —Lr-L 0 ddddddddddd dddddddddd XJI- - N - -O — c' cOO Lr IO Lr0n L \ - O-Lr\ - [OC -- 0 0 O ~O O HO 00X O-. - O " \\H-O L -- r- H b' d- \' C O c O C, cO,, WI\ \C r: - O cO.- d- Lr'X r'\"'h, —i - b- Od, — t-t-00 C 00 0- 0 o\ \ \N c r- \O —-t \ r-H \O \ c c\O COJ - r'- O H Hr l cr-l " U'j KN rX' "C \ Lr\ -00 O Hr- U iOJ rCK\ c0ddd0d0 H H c d dddd H 00 r H r H H *r — *r-D. ir *r-..r'-.r *r-j. -r-D.r-.- *r-D,r ~..",r"D.rI -,r-D.r- D.r -, *r — *3 *r- 3 o u + + + + + + + + + + + + + + + + + + + + + + to..\0o o ['- 000 H L\N 0, \,0o Co 0 W 0\ c0 _-_ H0 Ho00-t cUOH c-4 CU o-, \o 0o0 Wc\ 0 o \0 Ltr-CU H r- H O C\ \ N0o\0 ['0 CU L\-_:- co oN \D O \o0 O 0 LO - \k O 0\> \ 0 VN H- K\ O Lr\ HO r-O 0 -\0 _- tLO - (t ON CU H -O\ - Lf\ -- N-\ O' H i' —4 CU r-l Hr 00 0 *0 00000000d ~.0OOOOO 00.. *00000 000 00 I.....100000I IlOO Ii I i Iii L 0'-0 0r\ iL0 \d-<O fO O H L rHt rH L\ \O \ \OCU\000 H [ — \O W H LrK \,O00 ['- CU CU 0 Ko CU \0 0J c'-oO ON c 00 (oO zcO Lr\ — K OII \- K r kO Od " —:- [cLn OC \ II -' I\ CQ O\ C\ L -- Hr-0 0\ \O HO'-U \ HO r"-Lr O'X H IcO u) rd.....r. 0 O H"I'DO d —'t b- k r HIO' CU CU CU 0- 0 0- H H. H CU CU CU,'rg*r-o *r-3 T- *0-3 *r ~ *1-3 *r3 *1 3 *i- *1 3 * T-) *1-3 *1 3 *~ 3 ~r3 ~-3 *13 *1 3 *13 *r3 *13 CUOJ -I -I I I I I I I I I I C -' I I I I I I I I I I 0 \OO 0 ON rH-O OH CU [0-.\J 0 L[- \O0 CUJ \O rl00 H [ -U 000 HOO 0000- \O O 0O 00 0 l\n r >c\CU rCU -=-4- o H O LNr-'"1 0r ~ |HO O \O ~O l'-! \ L O - L\ - r- CUJ 000 - c —I NCo - — t h- -\rC CU - O- Lr\'- C U\ CU 0C\JC\ CU CIU II Cl H_' C0 0 00 00 00 00 o.~.........^o.....'Lr CU CU CU CU CU CU CU CU CUC C ~ O \10- O Cj -o-cz \- o O H \ or- --'- N L OJ cu o \O ) C \ - H O, Oco - 00 [- r CO L o t - O \ CO J 0 CO co Lr iO - 0 c \ ~ 01 _ o _- O -- o mC. CJU NC"\ L\ LCn u O' C-0 u\ K \-=F-\ O -— 00\ O H OO C1 I C IU, _:- CU -Lr\ L o-O- cO o -- r ['-0\ LW r \ - C u0 ^ c[ \ i( r oV r{-r\ r~ V "\ N- -\ \j-LN L r\ Lr f\Lr\ Lr\L Lr\L dd00000000000 ddddddddd \OO K O 0NO H HO O0\ MO r\- -CU N CU L\NO O'T\I — c\ Cd C)O 0 H\ ['- rc- K' \O CU OX u\ CU -=- -00 t'- co 0co \ \ON KC\ CO'C\ LX\'- O H C\ \-00c H-i_- H CU-_:- 0 CU LC\ - rQ H H H H H CU CU CU CU r< \ ^ H rH H H rH C CU C C U O C\ K 0 H 0CU rN\-t L^ [-0 ON3 01 H CU VI -- L\0 [-00 ON 0 00000000 r 000000000 OOO rH

APPENDIX B: GRAPHS OF THE O-TYPE BWO STARTING CONDITIONS Parameter Range C 10-5 to 0.2 QC 0 to 1.0 d 0 to 1.0 -51

-52-.0C=0~~ 1.0 6' r r rl, / / 7,. 5 V-6,, ~4 ~~ 0.2 0.2 i0.3 0.4 0.5 0.6 0. 0.2 0.3 0.4 0.5 0.6 0.7 CNs FIG. B.I 0-TYPE BWO STARTING CONDITIONS. d=O

-5350.2 0.3 0.4 0.5 0.6 0.7 1.0 C9 FIG. B.2 - E B STARTING CONDITIONS. d=0.05 5~ ~~~~~0.5 3 QC =O 0.2 0.3 0.4 0.5 0.6 0.7 CN$ FIG. 8.2 O-TYPE BWO STARTING CONDITIONS. d=0.05

1.0 6 31~~~~~~~~.00.. 54cL I / I / C=0.6 4 i —y.5s —-------- 0.4 320 t 2 0 2' i 0.2 0.3 0.4 0.5 0.6 0.7 CNs FIG. B.3 O-TYPE BWO STARTING CONDITIONS. d-QI

-55^ 061 ~~~0 0~ 7 8'.! -J 0 71 1 I II l I II l-' I 5 CNs FIG. B.4 O-TYPE BWO STARTING CONDITIONS. d=0.25

-560l_ 0. 10 0^ y^0.9 0.2 0.3 04 0.5 0.6 0.8 8.C0 6FIG. B.5 -TYPE BWO STARTING CONDITIONS. d=0.5 2, _ 1 _ FIG. B.5 O-TYPE BWO STARTING CONDITIONS. d=0.5

-57~~~~20 ~~~~ 0.~~ 14 12 18 QCO=.0 QCQ=0.92 QC=O.6 /.I / C=0.4. QC =0.7 Q.C =0.7 0.4 - Q0.6 0.8 1.01.21.61. 0.4 0,6 0.8 1.0 1.2 1.4 1.6 1.8 CNs FIG. B.6 O-TYPE BWO STARTING CONDITIONS. d=I.O

REFERENCES 1. Putz, J. L., Luebke, W. R., "High Power S-Band Backward-Wave Oscillator", Tech. Rpt. No. 182-1, Stanford University Electronics Laboratory; February, 1956. 2. Heffner, H., "Analysis of the Backward-Wave Traveling-Wave Tube", Proc. IRE, vol. 42, pp. 930-937; June, 1954. 3. Walker, L. R., "Starting Currents in the Backward-Wave Oscillator", Jour. Appl. Phys., vol. 24, pp. 854-860; July, 1955. 4. Johnson, H. R., "Backward-Wave Oscillators", Proc. IRE, vol. 43, pp. 684-697; June, 1953. 5. Grow, R., Watkins, D. A., "Backward-Wave Oscillator Efficiency", Proc. IRE, vol. 43, pp. 848-856; July, 1955. 6. Rowe, J. E., "Analysis of Nonlinear O-Type Backward-Wave Oscillators", Proc. P. I. B. Symposium on Electronic Waveguides, vol. VIII, pp. 315339; April, 1958. 7. Bernier, J., "Essai de Theorie du Tube Electronique a Propagation D'Onde", Ann. Radioelect., vol. 2, pp. 87-101; January, 1947. 8. Pierce, J. R., Traveling-Wave Tubes, D. Van Nostrand Company, Inc., New York; 1950. 9. Grow, R. W., "Backward-Wave Oscillator Efficiency", Tech. Rpt. No. 80, Stanford University Electronics Laboratory; November, 1954. 10. Rowe, J. E., "A Large-Signal Analysis of the Traveling-Wave Amplifier: Theory and General Results", Trans. PGED-IRE, vol. ED-3, No. 1, PP. 39-57; January, 1956. 11. Sedin, J., I. R. E. Conference on Electron Devices, Denver, Colorado; June, 1956. -58

-5912. Nordsieck, A., "Theory of the Large-Signal Behavior of TravelingWave Amplifiers", Proc. IRE, vol. 41, pp. 630-637; May, 1953. 13. Gewartowski, J. W., "Velocity and Current Distributions in the Spent Beam of the Backward-Wave Oscillator", Trans. PGED-IRE, vol. ED-5, No. 4, pp. 215-223; October, 1958. 14. Tien, P. K., "Bifilar Helix for Backward-Wave Oscillators", Proc. IRE, vol. 42, No. 7, pp. 1137-1144; July, 1944. 15. Watkins, D. A., Ash, E. A., "The Helix as a Backward-Wave Circuit Structure", Jour. Appl. Phys., vol. 25, No. 6, pp. 782-790; June, 1954. 16. Branch, G. M., Mihran, T. G., "Plasma Reduction Factors in Electron Beams", Trans. PGED-IRE, vol. 2, pp. 3-11; April, 1955.

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1 Bendix Aviation Corporation, Systems Planning Division Ann Arbor, Michigan, ATTN: Technical Library 1 Mr. Gerald Klein, Manager, Microwave Tubes Section, Applied Research Department, Friendship International Airport, Box 746, Baltimore 3, Maryland 1 Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota, ATTN: Dr. W. G. Shepherd 1 Director, Evans Signal Laboratory, Belmar, New Jersey, ATTN: Dr. Gerald E. Pokorney, Microwave Tube Branch, Electron Devices Division 1 Sperry Corporation, Electronic Tube Division, Gainesville, Florida, ATTN: Mr. P. Bergman 1 Mr. Shelden S. King, Engineering Librarian, Westinghouse Electric Corporation, P. 0. Box 284, Elmira, New York 1 RCA Laboratories, Princeton, New Jersey, ATTN: Dr. Bernard Hershenov ~1 Microwave Electronics Corporation, 4061 Transport Street, Palo Alto, California, ATTN: Dr. S. F. Kaisel 1 Librarian, Microwave Library, Stanford University, Stanford, California ~1 Wright Air Development Division, Air Research and Development Command, Wright-Patterson Air Force Base, Ohio, ATTN: Mr. George L. Larr, WWKSC

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