Contains reports on two research projects.U. S. ArmyLincoln Laboratory, Purchase Order DDL B-00306U. S. Navy (Office of Naval Research) under Contract Nonr-1841(49)U. S. Air Force under Air Force Contract AF19(604)-5200U. S. Nav

Bers, Abraham

Pawula, R.

Rook, C. W., Jr.

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VI. MICROWAVE ELECTRONICSProf. L. D. Smullin R. J. Briggs A. PoeltingerProf. H. A. Haus D. L. Morse C. W. Rook, Jr.Prof. A. Bers R. F. Pawula J. J. UebbingA. DENSE SPACE-CHARGE THEORY OF GAP INTERACTIONThe interaction between electromagnetic waves in dense electron-beam waveguidesand electromagnetic fields of gaps has been previously considered (1). We shall nowrepresent the interaction by a linear three-port, and present the associated matrixelements. These results are compared with those obtained under the assumption thatthe relative space-charge densities (w /w) in the electron-beam are weak (2). We shallconsider only electron beams that are focused by infinite magnetic fields, and considerthe presence of only a fast and a slow space-charge wave.The following matrix will be taken to describe the interaction:V 2 (z, r) A B a V r)I2(z, r) = C D b 1 (-, r) (1)I c d Y Vg e gwhere V1 (-, r) and Il(-f, r) are the excitations made up of both waves at the inputplane, and V2 (z, r) and 12(z, r) are the kinetic voltage and current at the outputplane.The matrix elements are tabulated in Table VI-1 for the dense and weak space-charge cases. In this table the subscripts "+" and "1" denote the fast space-chargewave; and "-" and "2", the slow space-charge wave. Also,P= 1 + el,2 (2)1, 2 P,where Pe is the total electromagnetic power flow (both inside and outside the electronbeam) for one wave, and Pk is the kinetic power flow for one wave. It can be shownthat for a propagating wave with a pure real longitudinal propagation constant,*This work was supported in part by Purchase Order DDL B-00306 with LincolnLaboratory, a center for research operated by Massachusetts Institute of Technologywith the joint support of the U. S. Army, Navy, and Air Force under Air ForceContract AF19(604)-5200; and in part by the U. S. Navy (Office of Naval Research)under Contract Nonr-1841(49). Reproduction in whole or in part is permitted for anypurpose of the United States Government.117Table VI-1.Matrix Weak Space Charge ( << Dense Space ChargeElement (One mode) (A fast and a slow space-charge wave)A [ [[ 0 1 exp[-jpl('+z)] + Z02 exp[-j 2(f+z)]A 21 [exp[-j +(£+z)]+exp[-jp_(f+z)]] Z + ZS+ z01 + 02ZO exp[-jpl(I+z)] - exp[-jp 2 (I+z)]B 2 [exp[-jp (+z)]-exp[-j_(f+z)]] Y01 + Y0Y exp[-jp1 (+z)] - exp[-jp 2 (1+z)]C 2 [exp[-jp+ (+z)]-exp[-jp_(+z)]] Z01 + Z021 Y01 exp[-jpl(f+z)] + Y02 exp[-jp2(f+z)]D 2 [exp[-jp (I+z)]+exp[-jp_(f+z)]] Y 1 + Y 022+ Y0 1 +Y 0 2F(pr) MI1C1F(p1 r )  M2C2F(P2r)a (pr) [MeC exp(-jpz)+ M_C_ exp(-jpz)] MC (pr) xp(-jpz) +M F(p 2 r exp(-jp 2 z)2 +1 2 2Y 0 F(pr) Y 0 1 MIC 1 F(p 1r) Y 0 2 M 2 C 2 F(p 2 r)b2 [M+C+ exp(-jp z)- M C_ exp(-jp_z)] exp(-jplz) - exp(-jp 2 z)2 1 2 2Y 1 M* K M2K2c M2F(pr) +K exp(-j+) - M exp(-j_) Z0 + Z0 2  exp(-jPp -K exp(-j) 201 02 F(pir) F(p 2 r)d M+K+ exp(-j - Kexp(-jjp_1) M exp(-jP 1 ) + 1 1 exp(-jp 2 )2F(pr) Y01 02 F(p1 r) F(p 2 r)G 0 M 2CK - M_ 2CK Y 01 M I2 C1K1 02 M212C2K2e4 1+ 4' 2B Im 2M+K+M K_ exp i2P d) 4 W k-WP) -4w(Wk-We+Wm)(VI. MICROWAVE ELECTRONICS)LP;-) -1a q z 2 +k- p2q22 (3)2 2 2Ap(p-p 3 ap =OP 8p A=0 --where A is the system determinantal equation arising from the boundary conditions.It is also possible to express in terms of integrals of the longitudinal electric fields.For a propagating wave with a pure real longitudinal propagation constant (3),Aa 2e-1q-e + a (4)P q4 2  Ez4 dapPIn Table VI-1, F(p nr) denotes the transverse variation of the longitudinal electricfield inside the electron beam, which can be determined from the Fourier integral solu-tion for E p (see Bers (1)), andzcrwE PY = (5)Y01,2 + e P1, 22 2op opY =  e - q (6)In Eq. 5, the upper sign is to be taken with the subscript "1" and the lower with the sub-script "2". In the dense space-charge formulationand M1, 2 is found from the relationship defining M by changing the propagation constantfrom p~ to p1, 2' andK =C 1 [F(p r)]Z da (8)1, 2 1, 2 P l'  2 d (8)Reference to Table VI-1 shows that the weak space-charge matrix elements can beobtained from the dense space-charge elements when << 1 by the following procedure:(i) Set n equal to 1 (disregard electromagnetic power flow in comparison tokinetic power flow).119(VI. MICROWAVE ELECTRONICS)(ii) Assume that the transverse wave numbers are the same for the fast and theslow space-charge waves. Then F(p 1 r) - F(p 2 r) - F(pr).(iii) Assume the same p for the fast and slow space-charge waves; that is,qThen Y 0 1 - Y 0 2 - Y 0.Under these conditions the M's and C's of the dense space-charge theory go to thoseof the weak space-charge theory.The expressions for Be are rather complicated to evaluate, and at present we haveno direct comparison between the two formulations.R. Pawula, A. BersReferences1. A. Bers, Electromagnetic waves in dense electron-beam waveguides and theirinteraction with electromagnetic fields of gaps, Quarterly Progress Report No. 58,Research Laboratory of Electronics, M. I. T., July 15, 1960, pp. 115-121.2. A. Bers, Space-charge mode theory of gap interaction, Quarterly ProgressReport No. 55, Research Laboratory of Electronics, M. I. T. , Oct. 15, 1959, pp. 65-70;Quarterly Progress Report No. 59, Oct. 15, 1960, pp. 68-71.3. A. Bers, Properties of waves in electron-beam waveguides, Quarterly ProgressReport No. 61, Research Laboratory of Electronics, M. I. T., April 15, 1961, pp. 102-105.B. KINEMATIC GAP THEORY FOR ACCELERATED ELECTRON STREAMSA small signal kinematic analysis of the interaction of an electron beam and the elec-tric field of a klystron gap has been carried out by Bers (1). This analysis can beextended to include the effects of acceleration of the electron beam.Following Bers' notation, but not introducing normalized variables, we obtain thefollowing equations.V(E, r) = e - j ' V(-oo, r) + e- E(e, r) ej dO (1)-oo P (e)I(0, r) = e I(-oo, r) + je o G (6) V(e, r) e e dO (2)00o E(O r)I = da I(8, r) dO (3)From these equations, the matrix coefficients for the equivalent linear three-portcan be determined. Of particular interest are the coefficients Y13 or M, the voltagecoupling coefficient and Y 3 3 or Ye', the electronic admittance.120(VI. MICROWAVE ELECTRONICS)00 E(0, r) -jY 1 3 (r) = - v e dOE(0, r) - ndO (e)V e6)-odndn Go(n)ccoThe real part of Y33, the electronic loading conductance, is given bydaf0 VP() e-j d Im 0-co Vgpe(6 ) -1dn- Go(n)n E(, r)-oo V pe(y)Consider the special case illustrated in Fig. VI-1. The beam passes through aV-shaped dc potential depression. The gap is gridded and the E-field is independentGAP REGIONI Iv - I I BEAMI IL d IEI/d-Fig. VI-1. Gridded gap with depressed potential resulting from spacecharge. (a) Geometry of the problem. (b) Magnitude of gapelectric field. (c) Shape of dc potential variation.of r. Let the velocity at the potential minimum be (1-a) times the velocity at entranceand exit. Then we obtain the following explicit formulas for the voltage coupling coeffi-cient and the electronic loading conductance.121dafE(y, r)dy eV gP ()GY =f dyY33 =Voltage coupling coefficient versus transit angle.05 10 15 2.0 25 30 35 4.0 4.5 50 55 60 65 70 75 80 85 90 95Electronic loading conductance versus transit angle.122Fig. VI-3.Fig. VI-2.(VI. MICROWAVE ELECTRONICS)M -1 asin 2a 1- cos)] (7)1-a6 2 0 222 1 -2 sin -cos sin" 1 cos)] (8)-2 2In Figs. VI-2 and VI-3, M and GeQ are plotted against 0 , the transit angle in the absenceof the depression, for various values of a; 0 and 6 are connected by the relatione o6= o(1-a)/2C. W. Rook, Jr.References1i. A. Bers, Klystron gap theory, Quarterly Progress Report, Research Laboratoryof Electronics, M. I. T., July 15, 1958, p. 49.123

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