Contains reports on three research projects.United States Department of the NavyLincoln Laboratory (Purchase Order DDL-B222)United States Department of the ArmyOffice of Naval Research (Contract Nonr-1841(49)United States Department of the Air Force (Contract AF19(122)-458

Bers, Abraham

Haus, Hermann A.

Contains reports on two research projects.Lincoln Laboratory (Purchase Order DDL-B222)United States Department of the ArmyUnited States Department of the NavyUnited States Department of the Air Force (Contract AF19(122)-458

Zacharias, A.

DSpace@MIT

VI. MICROWAVE ELECTRONICSProf. L. D. Smullin P. Chorney S. HollyProf. H. A. Haus K. W. Cooper, Jr. A. SaharianA. Bers H. W. Fockt A. ZachariasT. GoblickA. VARIATIONAL PRINCIPLE FOR UNIFORM AND PERIODIC ELECTRON BEAMSYSTEMSA variational principle has been derived for the propagation constant of a wave in auniform lossless system or of the propagation constant of one of the spatial harmonicsin a periodic lossless system containing an electron beam. The principle can be sim-plified somewhat when it is limited to longitudinal beams and Brillouin beams with zerocurl of the generalized momentum. We shall present the background, write down theprinciple for uniform structures, and give a brief summary of the salient results. Thedetails are presented in reference 1.The variational principle is based on the small-signal power theorem (2, 3). It is aconsequence of this theorem that the existence of a solution of the form exp(-yz) in alossless uniform system always implies the existence of a solution exp(+y z). Denotethe transverse dependence of the excitation quantities in the solution with thez-dependence exp(-yz) by the subscript (+), those pertaining to exp(y z) by the sub-script (-). The variational principle for the longitudinal beam then isS4 ( i x E + jwuoH ) • H - (VT X H+ - jwO oE + - J+) E-LT - + m V + Em -I+ ( - E )J +- (jJ - - dae + + - e+S E+ K ds 1 - X H + E X H + V J + VJ i dThe integration is carried over one cross section of the system. The vectors E and Hare three-dimensional vectors, but functions of x and y only; u is the z-directedsmall-signal Eulerian velocity; V is Chu's kinetic voltage, V = (m/e) uou; and 7 T isthe two-dimensional operatorThis research was supported in part by Purchase Order DDL-BZZZ with LincolnLaboratory, which is supported by the Department of the Army, the Department of theNavy, and the Department of the Air Force under Contract AF 19(122)-458 with M. I. T.tFrom Raytheon Manufacturing Company.(VI. MICROWAVE ELECTRONICS)- a + aVT xx x 8yThe contour integral ) E K ds involves the field of the (+) solution and the surface.+-current K of the (-) solution on the conductors on which the trial solutions may fail tosatisfy the electric boundary conditions. If these boundary conditions are fulfilled, thecontour integral is automatically equal to zero.The corresponding principle for the Brillouin beam isy (V X E + jwuoH+) H (VTX H - jwEEm- m - - -- *4+j+0 + + +-E m in+ ( + u V J - u u da - E K ds+(jwJ + T  o+ - -  K edsi E XH1+E xH +- u u  J+ Mu+ Uo)T ] da -4 z + - +  +e o e " -Here u and J are vectors with, in general, three components; u is the small-signalEulerian velocity; J is the ac volume current density supplement by the surfacecurrent rl nPouo , where r 1 ' n is the displacement normal to the beam boundary ofan electron on the boundary. Note that the denominator in both expressions is equalto the small-signal self-power of the solution if y is imaginary, or to the small-signalcross power between the growing and decaying waves when y is complex.In spite of their involved appearance, the variational principle leads to simple resultswhen appropriate trial solutions are introduced. It is reasonable to use as a trial solu-tion a linear superposition of the wave solutions in the beam, with the slow-wave struc-ture removed, and the wave solutions in the slow-wave structure, with the beamremoved. When these trial solutions are introduced and when the coefficients areoptimized, we obtain, in essence, a set of lossless coupled-mode equations of the typederived from the mode expansion (4). There are advantages in using the variationalprinciple. First, we obtain a good idea of the approximations involved in using afinite set of coupled-mode equations. Second, the use of the variational principleassures that we obtain the best value of the propagation constant (and thus the best valuefor the gain) for the approximation used. Third, the coupled-mode formalisms derivedfrom the variational principle for the traveling-wave tube and magnetron amplifier canbe shown to reduce to the existing formalisms for these devices. Thus the principle(VI. MICROWAVE ELECTRONICS)provides the basis for a unified treatment of all microwave beam tubes.The variational principle for periodic structures is similar to that for uniform struc-tures.H. A. HausReferences1. H. A. Haus, Electron waves in microwave tubes (invited paper), Symposium onElectronic Waveguides, Polytechnic Institute of Brooklyn, April 8-10, 1958.2. H. A. Haus and D. L. Bobroff, Small-signal power theorem for electron beams,J. Appl. Phys. 28, 694-703 (June 1957).3. D. L. Bobroff and H. A. Haus, Uniqueness and orthogonality of small-signal solu-tions in electron beams, Submitted for publication.4. H. A. Haus and L. N. Howard, Theory of lossless mode coupling, paper presentedat URSI-IRE Meeting, Washington, May 1957. Also, Quarterly Progress Report,Research Laboratory of Electronics, M. I. T., April 15, 1957, pp. 40-45.B. DIODE NOISE AT LOW FREQUENCIES (1 mc-60 mc)In order to determine the behavior of shot-noise suppression in the space charge infront of a space-charge-limited cathode, we have been conducting experiments on planardiodes at frequencies below 60 mec. Experiments by North (1) on space-charge suppres-sion of shot noise at 1 mc showed that in a diode structure the noise was considerablyhigher than predicted. North attributed this difference to the re-entry of elasticallyreflected electrons from the anode to the region of the potential minimum, which con-tributes to a reduction of space-charge suppression of the full shot noise from the cath-ode. North's experiments were in excellent agreement with predicted noise output in anegative-grid triode in which he claimed the elastically reflected anode electrons werenot able to return to the potential minimum. A private communication from W. Veith (2)gave the results of two experiments involving movable element structures. Both struc-tures had provisions for preventing elastically reflected electrons from returning to theregion of potential minimum. Veith's measurements made at 1 mc showed no variationof noise output with element spacing. This is in direct contrast to North's predictedvalue of noise output, which is given (see ref. 1) by72 kT .2= 1 .29 I 1 (1)a I )swhere ia is the anode noise current, is is the full shot noise for I, I is the anodecurrent, and g = 8I/8Eb.Since Veith's experiments were made at constant anode current, the shot-noiseoutput should have been directly proportional to g, the diode conductance. In his(VI. MICROWAVE ELECTRONICS)experiments, g was varied by a ratio of 50 to 1, and yet the value of ia remained con-stant.To further investigate these phenomena, we constructed a planar diode structurewith provision for preventing elastically reflected anode electrons from returning to theregion of the potential minimum. This was accomplished by making the anode a long,tapered tube that is closed at the end farthest from the cathode and has a disc with anaperture at the end nearest the cathode. The aperture is approximately 1. 5 times thediameter of the planar cathode surface. The cathode-to-anode spacing is approximately6 mm; the cathode diameter is 2. 5 mm.The noise from this structure was measured with a low-noise (noise figure, 1. 5 db)amplifier whose bandwidth is approximately 10 mc at 30-mec center frequency. Theoutput from the amplifier was fed to a Rada-Node (Kay Electric Co. ), which essentiallydetected and averaged the output from the low-noise amplifier. The anode of the testdiode and the anode of a temperature-limited tungsten diode were simultaneously con-nected to the input of the amplifier. Stray capacities were resonated with a shuntinductor. The circuit, which is shown in Fig. VI-1, is the same as the circuit usedby North (1) except for the operating frequency. Measurements were made by applyinganode voltage to either the test diode or to the reference diode. Since the anode resist-ance of the reference diode was infinite, its conduction or nonconduction did not affectthe input circuit impedance. The anode resistance of the test diode varied with thecurrent drawn, but its minimum value was more than 50, 000 ohms. Since the amplifierinput impedance was only 400 ohms, conduction or nonconduction of the test diode didnot affect the input circuit impedance. Adjustment of the reference diode current madethe output of the amplifier the same for the test diode input_and the reference diodeinput. The reference diode noise current is thus equal to i , Eq. 1, for the test diode.The ratio of is/i2 (usually called I') is seen to be the factor by which the action ofs athe potential minimum reduces the anode current fluctuations from full shot noise.OUTPUT TEST DIODEANODECATHODE-- ELECTRODEFig. VI-1. Test circuit.(VI. MICROWAVE ELECTRONICS)0n\Ib 025 ma20 5.0 10.0FREQUENCY - mcO - Ib 0 5 maA - Ib = 0.25 maSLOPE 0 9\ b= 0.5 maUPPERT BOUND ATIb = 1.0 ma50 100Ratio of equivalent shot-noise current toanode current as a function of frequency.Since both reference and test diodes have the same load impedance and the same band-widtl, the experimental value of F 2 isI2 nexp Iwhere I is the reference diode current, and I is the test diode current. The pre-n 2 odicted value of F is the bracketed quantity in Eq. 1,; that is,2 kT g= 1. 2 9  e I (3)Measurements at 30 mc showed F Z  to be about 500 times the predicted value. Byexpusing a sensitive radio receiver in place of the 30-mec amplifier and Rada-Node, meas-urements were taken down to 2. 5 mc. They are shown in Fig. VI-2. A 6 0-mc meas-urement was taken by the same technique except for the use of 60-mc components. Asthe plot shows, the output exhibits the 1/f characteristic of flicker noise. Lindemannand van der Ziel (3) investigated the same phenomenon and attributed it to the fact thations were partially neutralizing the space charge in the potential minimum region.Since these ions have a varying lifetime in the space charge, they produce a fluctuationof anode current with a spectral density proportional to 1/f. By inserting a negativeelectrode between cathode and anode, Lindemann and van der Ziel were able to elimi-nate this excess noise.The diode structure used in our experiments is a parallel-flow Pierce gun, with an0.50 -0.20O 00 050.02Fig. VI-2.(VI. MICROWAVE ELECTRONICS)insulated cathode electrode normally run at cathode potential. By biasing the electrodenegative with respect to the cathode, we hoped to drain off positive ions. This wasattempted, with the anode voltage increased to produce a constant anode current, andthe noise output was measured. No change was observed for an auxilliary electrodevoltage variation from zero to -15 volts. This result does not entirely rule out ions asthe cause of the noise output; the variation of the cathode-anode field may not be suffi-cient to cause noticeable deflection of ions from the potential minimum region.In future experiments we shall attempt to eliminate the ion source. It is believedto be the anode surface, which receives a deposit of cathode material during the conver-sion and activation process. By constructing the anode from titanium, this deposit willbe bound more strongly to the anode surface, and the production of ions by electronbombardment may be reduced.A. ZachariasReferences1. D. O. North, RCA Rev., Vol. IV, No. 4, p. 441 (April 1940) and Vol. V, No. 1,p. 106, (July 1940).2. W. Veith, Siemens and Halske, Munich.3. W. W. Lindemann and A. van der Ziel, J. Appl. Phys. 25, 1410 (1952).

Microwave Electronics

VIII. MICROWAVE ELECTRONICSProf. L. D. Smullin A. Bers A. SaharianProf. H. A. Haus H. W. FockT A. ZachariasS. HollyA. PERTURBATION FORMULA FOR FERRITE-FILLED CAVITYConventional perturbation formulae for cavities containing ferrites (1) usually startwith the unperturbed cavity free of ferrite. The perturbing effect of the ferrite on thecavity modes is taken into account solely through the use of effective permeabilities(which include the demagnetization factors). Such an approach is adequate for all sys-tems in which the stored energy is changed only by a small amount from the value com-puted from the empty cavity modes in the absence of the ferrite. In cases in which theferrite stores substantial amounts of energy, for example, in a magnetostatic resonanceof the ferrite, the conventional perturbation approach fails.In an attempt to derive Suhl's expression (2) for the semistatic and static operationof the parametric ferrite amplifier by perturbation theory the author developed the per-turbation formula reported here.Denote by E , h , and m the electric field, small-signal magnetic field and mag-netization of the mode o of the unperturbed lossless system, enclosed by a perfect elec-tric conductor, that resonates at the frequency 2 . These fields satisfy the equationsVX E = - j 0 L ° + m ) (1)VX t. = j0 EE (2)The electric and magnetic fields satisfy the boundary conditions on the conductornX E0 = 0; n . h = 0 (3)where n is the normal on the conductor. The magnetization m fulfills the small-signalferrite equationjo m o = - y0l(Mo X ho + mo X Ho) (4)The dielectric constant E will be assumed to be a scalar. Consider now the same sys-tem perturbed by an electric current distribution J and a magnetic current distributionJM. The fields now satisfy the equationsV E = - jo(h + m) - JM (5)This research was supported in part by Purchase Order DDL-BZZ2 with LincolnLaboratory, which is supported by the Department of the Army, the Department of theNavy, and the Department of the Air Force under Contract AFl9(122)-458 with M. I. T.TFrom Raytheon Manufacturing Company.(VIII. MICROWAVE ELECTRONICS)7 X = jwcE + J (6)and the magnetization m still fulfills the relationjm =- y(M 0X h + mX H) (7)The boundary conditions on E and h on the perfectly conducting enclosure remainunchanged.We now dot-multiply Eq. 5 by ho , Eq. 6 by -Eo, the complex conjugate of Eq. 1by h, and that of Eq. 2 by -E. Adding the resulting equations, integrating over thevolume r of the perfectly conducting enclosure, using Gauss' theorem and the boundaryconditions on E o , ho , E, and h, we obtain0 -j(-o) (~h h + EE Eo) dTf-*-- :'M o 0Now, we transform the second and third terms in Eq. 8. Dot-multiplying Eq. 7 byho , the complex conjugate of Eq. 4 by h, and adding the two equations, we obtainjom • h - jo m  h= -YI [m x H h +m XH •h] (9)On the other hand, multiplying Eq. 7 by mo, the complex conjugate of Eq. 4 by m, andadding the two equations, we obtainj(.- [)m mo - [Mo x h m +M Xh 0m] (10)But Mo is parallel to H ando -Mo H (11)oH o0Introducing this relation into Eq. 10 and comparing the results with Eq. 9, we find thatS_ , H -jwm h0 - j2 m - h - j(c - 0 )m mo (12)0Finally, if we introduce this expression into the perturbation formula (Eq. 8) and solvefor w - Q2, we obtainS EE E + h + om - m dSo o o oo(VIII. MICROWAVE ELECTRONICS)If the perturbation is small and we are studying a resonance in the neighborhood of 2,O0we can replace E, h, and m in the denominator of Eq. 13 by Eo, ho, and mo andobtainw- -j J h dr+ J -E dIH rh d(14)S( , -° H0 o EE E + h + m m dT0 o o o o M oo o0Equation 14 shows that in the analysis of magnetostatic operation, the stored electro-magnetic energy must be supplemented by the termi H omo " of M 4 * m dTThis term can be recognized as the energy required to deflect the magnetization vectorM from alignment with M and H by the angle Im /M .If we start with Eq. 14, we can derive Suhl's formula for the semistatic and staticoperation of the parametric amplifier with little effort. In conclusion, it should be men-tioned that a relation of the form of (Eq. 12) was first discovered by Mr. D. L. Bobroff,of Raytheon Manufacturing Company, who used it for proofs of orthogonality.H. A. HausReferences1. A. D. Berk, Cavities and waveguides with inhomogeneous and isotropic media,Technical Report 284, Research Laboratory of Electronics, M. I. T., Feb. 11, 1955.2. H. Suhl, Theory of the ferromagnetic microwave amplifier, J. Appl. Phys. 28,1225 (1957).B. SMALL-SIGNAL ENERGY THEOREM FOR BEAMS WITH ZERO CURL OFGENERALIZED MOMENTUMThe energy theorem of the theory of passive electromagnetic systems (1) relates thefrequency derivative of the fields to energy storage. A corresponding theorem can bedevised for the small-signal excitation in an electromagnetic system that contains anelectron beam or an electron-space-charge cloud with zero curl of the generalizedmomentum. This theorem can be used, inter alia, to relate the group velocity of asmall-signal wave excitation to the power and energy in that wave.The fundamental equations (2) of a single velocity space-charge cloud with zero curlof the generalized momentum and a time variation ej °t are(a) the force equation,e- (1)V(v u) - wu + - E (1)o m(VIII. MICROWAVE ELECTRONICS)(b) the continuity equation,v oV J = - jwo( - po) (2)(c) and the two Maxwell equations,Vx = - jwoHF (3)Vx = jWEoE + J (4)in which no subscripts are used to indicate small-signal ac terms, and the subscript ois used to indicate the time-average components. The velocity I is the Eulerian veloc-ity. J includes the ac surface current Por l n on the beam surface (where n is thenormal to the beam surface, and rl is the small-signal displacement of an electron inthe surface). In order to derive the energy theorem, we take the derivatives ofEqs. 1, 2, 3, and 4 with respect to w. Then, we dot-multiply the resulting equationsm -* m -*-by eJ ' e u , and -E , respectively, and add them. To this result we add thecomplex conjugates of Eqs. 1, Z, 3, and 4, after having dot-multiplied them bym 8J m u 8H aEe 8w' e 8w' 8 , and - , respectively. We obtaini 8E -* -* 8H m - 8 U* m .- - 8J- X H +E X +-v - J +-v u4 8a 3W e o aw e o aw- -* m- -* m-* - m - -*4 j oH ' H + EoE - E + -- u . J +--u J-- e Po u u(5)This is the energy theorem. On the left-hand side, terms similar to power-flow termsappear, except that in each term one of the small-signal factors is differentiated withrespect to w. On the right-hand side, the sum of the electromagnetic energy and thesmall-signal kinetic energy (3), multiplied by j, appears.As one application of the energy theorem (Eq. 5) we may consider a Brillouin space-charge cloud inside a magnetron cavity under small-signal excitation (zero anode cur-rent). The cavity opens into a uniform waveguide within which a steady state is main-tained at the frequency w. The walls of the magnetron cavity, the waveguide, and thecathode are all assumed to be lossless. The waveguide field at the reference planewithin the waveguide may be characterized asE = Ve and H = Ih (6)where e and h are the frequency-independent field patterns of the dominant waveguidemode. The admittance Y is defined byI = YV (7)Now suppose that the frequency of excitation is changed by do and that I is keptconstant. Integrating Eq. 5 over the magnetron cavity up to the waveguide reference(VIII. MICROWAVE ELECTRONICS)plane, we obtaina-m Y-J + - p uSIV Iv =- j L HoH H + E0 E E +- e- (u J + u J- P) dv (8)All other terms have dropped out, since the boundary conditions have to be satisfied atthe frequencies w and w + do. Over the entire surface of integration, n • J and 8J/8aw n,where n is the normal to the surface, are zero. Equation 8 shows that 8Y/8w is pureimaginary. Its sign is determined by the sign of the total small-signal energy, electro-magnetic and kinetic. Thus 8Y/aw is negative imaginary when the energy is negative.Now we turn to another application of the energy theorem. In a uniform losslesssystem containing an electron beam, consider a wave with the dependence e - jpz, withp assumed real. We factor the common z-dependence from all field quantities, andindicate the remaining vector functions of x and y by a circumflex mark. If we applythe energy theorem (Eq. 5) to the fields of the wave, we have8 8H m -7 XH +E X +-v Jaw aw e o 8wm- * j [ ^*+-v u + E X H+e o o -/ ^  ^ ^ A* m ^ m A m u- *(=-j H  H + E EE +-e u J +- u J-e Po u (9)o e e e oIf we integrate Eq. 9 over a cylinder bounded by the perfectly conducting boundary ofthe system (or by a cylinder at infinity), and by two reference planes, separated by adistance L, the first term on the left-hand side integrates out to zero. We obtaindw P (10)d3 wwhere P is the small-signal power, electromagnetic and kinetic (2).S^ m - *P =-Re EX H + -v )J da (11)2 .e e ocross sectionand w is the small-signal stored energy per unit lengthS* * m ^* A* m (1)w= L H - H + E E ( J + u J)- po " da (1Z)o o e e oThus 8aw/8p has formally the significance of energy velocity. Note, however, thatw can now assume negative values.H. A. Haus(VIII. MICROWAVE ELECTRONICS)References1. C. G. Montgomery, R. H. Dicke, and E. M.Circuits, Radiation Laboratory Series, Vol.New York, 1948).Purcell, Principles of Microwave8 (McGraw-Hill Book Company,2. H. A. Haus, Electron waves in microwave tubes, International Symposium onElectronic Waveguides, Polytechnic Institute of Brooklyn, Brooklyn N. Y., April1958 (to be published).3. H. A. Haus, (unpublished work).C. KLYSTRON GAP THEORYThe small-signal longitudinal interaction of an electron stream with the electricfields of a gap can be represented, for convenience, by a three-port network, as shownGAP FIELD E, (z,o)DRIFT TUBE , ELE,d dFig. VIII-1. Electron-stream interactionwith electric fields of a gapand small-signal equivalentcircuit.- V ++ LINEAR +VI  V2THREE- PORTin Fig. VIII-1. The theoretical development of this problem was reported in theQuarterly Progress Report of July 15, 1958, page 49.of the evaluation of the gap matrix elements,V yll12 = YI Y31Lgi 31Here we shall give the results0 y13  VlIYZZ Y23 IY3Z Y3 VgConsider a system consisting of a gap and a gap electric field with symmetry aboutthe z = 0 plane, as shown in Fig. VIII-I. The matrix elements of Eq. 1 can beexpressed in terms of the Fourier integrals or of the normalized field (1).3 = E(O, p) ejO dOThis work was supported in part by the U.S. Navy (Office of Naval Research) under Contract Nonr-1841(49).(2)(VIII. MICROWAVE ELECTRONICS)andy= - je E(O, p) eJ 6 dO (3)00both of which are real functions. The parameters 3 and y can be evaluated if the elec-tric field at the gap, Ez(z, a), is known, or they can be measured in an actual gap systemby a perturbation technique. The matrix elements and their designations are:Drift coefficients (0d is the normalized distance of the gap field extension)-jzedYll = Y22 = e (4)Coupling coefficients-j dY13 = Y3 = e (5)Bunching admittance1 G (j ) e d21 Z Go(Zd) e dInduced-current admittanceY23 31 = Go(y + j 0 dp ) e-6d (7)Electronic-loading admittanceY 1 Go(y p + jZb)33 oZ 0ebe = Re dE(, p)J_ O d(- ) E(, p)e-in which G = I /V is the de beam conductance, and all other quantities are as pre-viously defined (1).The equivalent circuit at the gap-circuit terminals follows directly from the kineticpower theorem and has been given previously (1). When the gap circuit is terminated ina passive admittance Yc (Fig. VIII-Za), the gap terminals are loaded by the electronicloading admittance (Eqs. 8), and driven by current generators that are proportional tothe excitations present in the incoming electron stream, as shown in Fig. VIII-Zb. Atthe same time, the electron-beam kinetic voltage V 1 and beam current I I undergo atransformation that can be represented by a linear two-port, as in Fig. VIII-Zc.For convenience, letZ = (Yc + Y 3 3 ) 1 (9)(VIII. MICROWAVE ELECTRONICS)1Ge =Re (Y 3 3 ) = GoY (10)(11)andPdMaking use of Eqs. 1 and 4 - 8, and of the definitions that have been given, we obtainthe following description of the electron-beam two-port network of Fig. VIII-Zc.V A(Z)= (A2 /CB -j2ed V1e (D / 1A = D = 1 - (G+Z)(l + j)B = G (GeZ)eQGC = - (G e Z)(I + jW)Z]In the important case of a resonant circuit, we have1 1G (+ 1)where G = G c Ge is the total gap conductance; and X =Fig. VIII-2.- Vg +Y33= Ye- aVI-01I ,+ LINEAR +VI V2TWO - PORT(16)2 2s +0 is the normalized2as(a) Gap circuit terminated ina passive admittance.(b) Equivalent circuit at thegap terminals.(c) Equivalent circuit at thebeam terminals.with(12)(13)(14)(15)(VIII. MICROWAVE ELECTRONICS)frequency parameter with wo the loaded resonant frequency, and a the loaded dampingconstant. Three special cases of varying circuit loading with respect to beam loadingare:a. Lossless circuit with G = 0c(1 + j)A= 1- (17)(X+l)2B 1 (18)ef (X+1)Ge (1 + j4)ZC = j2 - (19)P (X+ 1)b. Lossy circuit with G >> GekGeA (1 + jp)A - 1 G (20)c (X+1)B zB G (21)c (X+1)P c (X+ 1)c. Loss loading = beam loading with G = Gek1 (1 + j4)A = I z (23)B = - P - (24)eG (X+ 1)Ge j, 1 (l + j ()C= - 24 ) (25)Most gaps that are of interest have (P 2/Ge) >> 1 and - 1. For such gaps, inall of the cases that we have mentioned, C 0 is a good approximation. For case2, in which the circuit conductance is much larger than the electronic conductance,we may also assume that A = 1. Only for this case can we approximate the two-port network of Fig. VIII-Zc by a series admittance that is equal to the circuitadmittance, as shown in Fig. VIII-3 (for convenience, exp (-j 2 0 d ) is omitted).(VIII. MICROWAVE ELECTRONICS)Gc>> GeeII Yc 12I  C +zz+V, V2 B _- 2 Z CC ~OFig. VIII-3. Approximate equivalent circuit forthe electron beam when electronicloading is negligible compared withcircuit loading [exp (-jZ d) phaseshift omitted].C 0  D 0 k 0 e j 29d(AO Do) x (0  kO) x (0 e-dOI 12+ LINEAR IDEAL IDEAL +VI I PHASE V2BILATERAL AMPLIFIER SHIFTERFig. VIII-4.1. ReciprocityFrom Eqs. 12 - 15, we haveAD- BC = 1 - ZGe£Z = kEquivalent circuit for electron beam interacting withpassive circuit fields.(26)The linear two-port network of Fig. VIII-Zc is reciprocal if k2 exp(-j 4Od) = 1; this willbe so when 0d = 0 and k = 1. It is interesting to note that when k = 1, but 0d * C, thetwo-port network of Fig. VIII-Zc can be considered as reciprocal with respect to areversal of the direction of flow of the electron stream. Two such cases occur whenZ = 0 (which is a drift region) and when Gef = 0. In the general case, Eq. 12 can berewritten (2) as'V( 2 o12 1 , oBD)-jZ6d Vl)k e1II(27)where Ao = A/k, Bo = B/k, and so forth; and therefore we now have A D - B C = 1.The representation that is equivalent to Fig. VIII-Zc is as shown in Fig. VIII-4. Thelocation of the ideal amplifier and phase shifter is arbitrary.A. BersReferencesi. A. Bers, Klystron gap theory, Quarterly Progress Report, Research Laboratoryof Electronics, M.I.T., July 15, 1958, p. 49.2. H. A. Haus, Equivalent circuit for passive nonreciprocal networks, J. Appl. Phys.25, 1500- 1502 (1954).

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