Technical Memorandum No. 90 4853-11-T ON THE SYNTHESIS OF DUAL-RESONANT CAVITIES WITH A DIGITAL COMPUTER by F. M. Waltz Approved by: A. W. Naylor for COOLEY ELECTRONICS LABORATORY Department of Electrical Engineering The University of Michigan Ann Arbor Task Order No. EDG-1 Contract No. DA-36-039 sc-89227 Signal Corps, Department of the Army Department of the Army Project No. 3A99-06-001-01 December 1962

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iv ABSTRACT v 1. INTRODUCTION 1 2. DISCUSSION 2 3. ANALYSIS OF IN-LINE CAVITY 4 3. 1 Simplified Model, Neglecting Fringing 6 3. 2 Effect of Fringing Capacitance 6 3. 3 Comparison of Results for Simplified Model with Results for Complete Model 11 4. ANALYSES OF RE-ENTRANT CAVITIES 12 5. APPLICATION TO CAVITY DESIGN 18 6. EXPERIMENTAL RESULTS 19 7. RELATED PROBLEMS 23 8. CONCLUSIONS 24 REFERENCES 24 25 DISTRIBUTION LIST iii

LIST OF ILLUSTRATIONS Figure Title Page 1 Cavity configurations: (a) in-line; (b) external re-entrant; and (c) internal re-entrant. 3 2 Solutions for in-line cavity for n=2, neglecting 7 fringing capacitance. 3 Computer flow chart for in-line cavity computations. 8 4 Plot of typical computer output for in-line cavity, based on the fringing capacitance model. 10 5 Comparison of results from simplified model and fringing capacitance model. 11 6 Computer flow chart for external re-entrant cavity computations. 14 7 Plot of typical computer results for the external re-entrant cavity model. 16 8 Experimental results for in-line cavity. (a) Dimensions essentially equal to the calculated values. (b) Dimensions deviating from the calculated values. 20 9 Experimental results for external re-entrant cavity. 22 iv

ABSTRACT Mathematical analyses of multisection coaxial cavities (both in-line and re-entrant types) using a digital computer predict the possibility of shifting one of the spurious resonant frequencies (present in all coaxial cavities) to a desired frequency, thus allowing one cavity to do the work of two. "Fringing capacitance" at the conductor discontinuities is taken into account in the analysis by means of equations fitted to curves given in the references. The specific problem solved is the design of cavities to resonate a terminating capacitance (e. g., tube capacitance, "varactor" capacitance) at two harmonically-related frequencies, with the additional requirement that the two frequencies remain very nearly in the desired ratio despite wide variations (two-to-one or more) in the magnitude of the terminating capacitance. As one obvious application, a cavity meeting these requirements would make possible an inherentlyaligned single-cavity frequency multiplier. Computer flow charts and curves based on the computed results for specific cases are presented. Experimental cavities constructed according to the predicted designs have exhibited performance which is in very close agreement with the analysis, thus verifying both the validity of the method of analysis and the feasibility of the desired result. Application of the same techniques to other and more general problems (e. g., single-cavity mixers, voltage-tunable filters) are suggested. v

1. INTRODUCTION In the UHF frequency range, coaxial cavities are widely used as resonant elements because they provide excellent performance and lend themselves to straightforward design, production, and tuning methods. Disadvantages include the spurious resonant modes frequently encountered, and the excessive size, weight, and complexity of the multiple-cavity circuits needed for all except the simplest applications. Thus, the paradoxical situation exists that coaxial cavities have extra (and usually unwanted) resonant frequencies, but that in a typical application a number of cavities must be used in order to obtain resonance at the desired set of frequencies. This memorandum considers the use of multiple-section cavities to shift a spurious resonant mode to a desired frequency, so that one cavity can do the work of two. The specific problem considered in detail is the synthesis of cavities which resonate at two harmonically-related frequencies such that the resonant frequency ratio is insensitive to variations in the terminating capacitance (i. e., the desired frequency ratio is maintained within close limits in spite of wide variations in the terminating capacitance). Insensitivity to capacitance variation eliminates the need for selection or tuning of cavities to match individual tubes or allows the use of such cavities with time-varying capacitances. A digital computer was used to obtain solutions to the system of simultaneous transcendental equations which result from the assumed mathematical model and the imposed conditions.

2. DISCUSSION A uniform lossless coaxial cavity of length L and characteristic impedance Z0, short-circuited at one end and with a capacitance C connected across the other end, resonates at an infinite number of frequencies f, each of which satisfies the following transcendental equation (neglecting end effects): 1C = Z tan2f (1)T 2wAfC o c' where c is the velocity of propagation of electromagnetic radiation in the line under consideration. In general, these frequencies are not harmonically related. In certain applications (e. g., frequency multipliers, mixers) it is desirable to have a single cavity resonate a terminating capacitance (e. g., tube capacitance, diode capacitance) at two harmonically-related frequencies. (Cavities having this property will be said to satisfy the dual-resonance condition.) A uniform section of coaxial line will not, in general, satisfy the dual-resonance condition. It is shown below that coaxial cavities of the types depicted in Fig. 1 will produce the desired results. Consider first the in-line type [Fig. l(a)], with lengths LA and LB, diameters dl, d2, and d3, characteristic impedances ZA and ZB, terminating capaci2 tance C, and fringing capacitance CF, and assume that the conductors have infinite conducF' tivity. This cavity configuration permits the independent choice of five parameters: dl, d2, d3, LA, and LB. (CF is not independent, but depends on the cavity dimensions [1,2].) Thus, it may be possible to satisfy as many as five independent design conditions. A few of 1 Unless otherwise stated air dielectric is assumed. "Fringing capacitance" is a convenient device for representing the effect of the discontinuity in the center conductor. Whinnery, Jamieson, and Robbins have shown in Ref. 1 and Ref. 2 that the types of discontinuities considered in this memorandum can be accurately represented by one or more lumped frequency-independent capacitances, with the usual transmission-line equations applying to both sides of the capacitances. Further discussion of "fringing capacitances" and their effects is included elsewhere in this memorandum. 2

CF ( _ __ _______ Zd (a) d2 d ZA d, d, ^^^^^^^^^^^^^ --- L' —l —------------ diim & lit ~(b) _ --- --- Lc ----- LB L (c) Fig. 1. Cavity configurations: (a) in-line; (b) external reentrant; and (c) internal re-entrant. the many conditions which might be of interest in particular applications are as follows: a) Synthesize for a specified ZA b) Synthesize for a specified ZB c) Synthesize for specified diameters d) Minimize overall cavity length, LA + LB e) Synthesize so that the resonant frequencies shift as little as possible over a range of values of terminating capacitance. f) Synthesize so that the n-to-one frequency relationship resulting from the satisfaction of the dual-resonance condition is maintained within close limits over a range of values of terminating capacitance, instead of only at one value of capacitance. A cavity design which satisfied condition f) would be particularly desirable in applications where the terminating capacitance changes with time or where production or cost requirements preclude the tuning or matching of particular cavities to particular tubes. Attention is therefore directed toward cavity designs which satisfy the dual-resonance condition and condition f). 3

3. ANALYSIS OF IN-LINE CAVITY Applying the transmission line equations to the section nearest the short circuit [see Fig. l(a)], combining the resulting impedance with CF, and applying the transmission line equations again, results in an expression for the cavity impedance Z(f), as seen from the open end, as follows: 27r 2rr Ztan f + Z tan 2fL - 27rfCFZ Z tan fL tan - 2fL A c A B c B FAB c A c B Z(f) = jZ _- (2) ZB -ZAtan 2f f LAtan 2 fL 2fCFZAZBtan 2fL The condition that C be resonated at frequency f by Z(f) is j27rfC + = (3) Assume that there exists a pair of solutions to (3), f = f1(C) and f = f2(C). Ideally, f2 should equal nf1 for all values of C. This is not possible, in general, but can be approximated over a limited range of C near C if Z(f) is chosen so that f2(Co) = n f(Co) (4) and d f2(C) df (C) =n dC (5) d C C o o These conditions will now be rewritten in a more usable form: Substitution of Eq. 3 into Eq. 4 yields Z[f(Co)] = n Z[f2(Co)] (6) 4

df as an equivalent to Eq. 4. Differentiating Eq. 3 with respect to C and solving for dC yields df -j27rf dC aZ(f) * af j27TC - (f) Substituting this into Eq. 5 and using Eqs. 4 and 6 allows Eq. 5 to be rewritten as (7) az(f) aZ(f) af f=f (C ) 1 o = n2 az(f) af f=nfC) f 1n 0~o (8) Thus, the satisfaction of Eq. 3 at f = f1(C0), and of Eqs. 6 and 8, is sufficient to guarantee that Z(f) satisfies both the dual-resonance condition and condition f) at C = C. Application of these equations to the impedance function given by Eq. 2 yields tan A + r tan 0 - K tan A tan 0 X= r - tan A tan 0 - Ktan 0A A B A (9) (from Eq. 3) tan A + r tan B - Ktan A tan 0B r -tan 0 tan 0 - Ktan 0 = n A B A tan n0 + r tan n0 - Kn tan n0 tan n0 ~~~~~A B A B ((10 r - tan nA tan nO - Kntan n0A (from Eq. 6) rOA + r 20B + (rtA + anK tan2 A (1 + 0BK) - 2rB K tan 0A cos2 B(r - tan 0A tan 0B - K tan 0A)2 r2A + r2B + (rA + B)tan Atan 2n0A(1 + n2 0BK) - 2rnBK tan nA -_2 BAA D A \(11) n —— cos2n -(r - tan nGA tan nB - Kntan nA) (from Eq. 8) cos2 nB B(r - tan n 0A tan nO B - Kn tan n0 A)2 (from Eq. 8) " A B A7 ~~~~~~~(from Eq. 8) 1 whereX - 27f C ZB' 1 O A ZB r = -, A A 2VT GA c fLA 2 andK 0, f LandK = 2flCFZB B -CiB 1FB= 5

3. 1 Simplified Model, Neglecting Fringing Capacitance The complexity of Eqs. 9, 10, and 11 prevents an easy determination of the ranges of the variables over which solutions are possible, if indeed any solutions exist at all. Most of the difficulty arises due to the complex inter-relationship of CF and the cavity dimensions. Since in the applications contemplated in this memorandum CF is usually not large compared to C, CF will be neglected in the initial approach to the problem. Neglect of CF implies that K = 0 in Eqs. 9, 10, and 11. Since X can be made to take on any positive value by proper choice of ZB, the problem reduces to finding values of 0A, 0 B and r such that Eqs. 10 and 11 are simultaneously satisfied and such that the right-hand side of Eq. 9 is positive. Solution of even this simplified problem in terms of known functions is for practical purposes not possible, but solution by numerical methods is not difficult. Figure 2 shows part of the solution to this problem in terms of the quantities r, X, 0A, and 0B for the case n = 2. Choice of a particular value for any one of the quantities, r, X, 0A or 0B immediately determines the other three. But since r and X depend on ratios of diameters, one degree of freedom still remains. That is, one of the three diameters can be chosen arbitrarily, after which the other two diameters can be determined from the definitions of r and X using the known values of fL, Co' r, and X, and the formulae 3 -1 A d = log 60 (12) and d., Z d3 -1 ZB d = loge 60 (13) 3. 2 Effect of Fringing Capacitance Analysis of the original model (hereafter referred to as the fringing capacitance model) is considerably more complicated than that of the simplified model. Also, because CF depends on actual diameters, rather than ratios of diameters, solutions must be tabulated as a function of two parameters rather than only one, as was the case above. (See Fig. 2. ) 6

* B. DEGREES Fig. 2. Solutions for in-line cavity for n = 2, neglecting fringing capacitance. 3 To solve this problem, a digital computer was programmed to find solutions to Eq. 10 for n = 2 for specified values of Co, fl' dl and d3, and for a series of values of d2. Figure 3 shows the flow chart for this computer program, from which it can be seen that, for each value of d2, the computer finds the values of OA and 0 (if they fall within the specified range) for which the desired capacitance is resonated at both the fundamental and the harmonic frequencies (i. e., at which Eqs. 9 and 10 are satisfied). The computer also calculates and prints out the value of each side of Eq. 11 for each solution, thus allowing the cavity designer to pick (by interpolation) the one value of d2 and the corresponding values of 0A and 0B which also satisfy Eq. 11. 3The computer determined the value of CF in each case from equations which were fitted to the curves given in Refs. 1 and 2. If a particular application requires more accurate calculation of CF than is possible by this method, then the complete procedure given in these references could be used. 7

Fig. 3. Computer flow chart for in-line cavity computations. 8

DEFINITIONS and FUNCTIONS: (MKS units) A A XF Left side of Eq. 10, AXF Left side of Eq. 11 XH Right side of Eq. 10, AXH h Right side of Eq. 11 F1(a,T) = rd3 x 1010(. 334 a2 -.410 a+. 1372 +.00285 T) (fitted to curves of [1,2]) F2(TB) = r(X-TB)/[TB(X + K/X) + K + 1] (determines TA tan 0A so that X = X) F(T T = 4TA(1-TB2) + r TB(-TA2) - 4K TATB/X determines XH, using Eq. 10 F3(TA TB)= 4 3 A) B ~ r(1-TA 2)(1-TB2 )-4 TA[TB+K(1-TATB)/X] land trigonometric identities F4(TB' TBXH0,XH) = [TBO(XH-X) + TB(X-XH0)] (XH-XI0) (interpolates to find value of TB where XI. =X) F5( Ai' Bi' Ai' Bi) Ai rX2(1+TAi )+ Bi(r2X2 +TA X2 -2rKX TAi+K2TAi2 )+KX T Eq. 1 Ai rX(+T A )2+ Bi[rX (1-TAi)2+4 TAi2X-8 rKX Ai-T )+16KAi2+4KXT = 4(I+T-/) 4 + Bi {(1-T Bi2)[r(1-TAi )X-4K TAi] - 4 TAi TBi X} from Eq. 11 & trig. identities/ INSTRUCTIONS: input data: fl, C0, al, d2m' Ad, d 2ma, d3, Bmin' AO, Bmax' T r = d3/dl; ZB = 60 log (d3/dl); X = 1/(2f fl C0 ZB) ( a = (dd/(3-d) );ZA =6og(d3/d2);r = ZBZA CF =Fa,-). K = CF/CO; TBO = tan 0Bmin; TAO = F (TBO); XHO = F3 (AO TBO) B Tg = tan B; TA = F2 (TB); XH = F (TATB). ~ IXH +XH -2X| - IXHO ~ XH (curve intersection criterion) I HO H - -1 -1 () Tan (TA); =Tan (TA); Ti = F4 (T0, Tg, XI, XH); TAi F(TBi); XHi = F3(TAi Ti); Ai = Tan-TAi) i = Tan-(TBi) LA = 2.998 x 108 0Ai/2rf1 LB = 2.998 x 108 OBi/2rfl; AXF = FS(OAi, Bi TAi, Ti); AXH = F6 (Ai, Bi' TAi, TBi). output data: A0,' Ai',A' LA' 0B0' Bi' B Lg, ZA' Z, r, CF, d2, XH0 XHi, XH' X, AXF, AXH. ] 0 0 B B max ) d2 d2 max XH0 =XH; TBO = TB, TAO= TA 9

68 64ii I 624- -_ 5 -60 - _ - -- -- -- -'" - --- I 58 I 03 W V) UJ (C W 56 cQ 54 N n 2 f -775 Mc C 1.75 pf d|= 0.3414 in. d3 0.936 in. I I I — C. I I 9 4fBz(t 7,__ _ -_Z() __ CONDITION f) _ _f f1 SATISFIED HERE;73.70___ _ _._____ __ e A'"~ CONDITION f)ABA I u z cV 28 30 32 Be DEGREES B' Fig. 4. Plot of typical computer output for in-line cavity, based on the fringing capacitance model. The program was written in the MAD language, and included statements (not shown in Fig. 3) which provide for headings, output data format, overflow prevention, etc. Compilation of the program on an IBM 709 computer required 0. 42 minute. Execution time varied with the ranges and increments of B and d2. A typical run, using 61 values of 0B and 25 values of d2, required 0. 54 minute on the IBM 709. Figure 4 is a plot of the computer output for the particular case n = 2, f = 775 Mc, C = 1. 75 pf, d1 = 0. 3414 inch, and d3 = 0. 936 inch. Arrows indicate the required values of 0 A, B' and d2. 10

AVITY SIZE DETERMINED FROM SIMPLIFIED MODEL AVITY SIZE DETERMINED FROM FRINGING CAPACITANCE MODEL ------- ~ ~2.584" ~ii -' —H-'-I —II ^ —-fr -'!!; *!f --— i',-/, I III ] r_ - I [!l 5893.3414".703".710".936" IC —- ~I:j —I I- (II 5 I ^ 1.230" -------- 2.832" Fig. 5. Comparison of results from simplified model and fringing capacitance model. To satisfy one (or two) conditions in addition to the dual-resonance condition and condition f), a series of solutions for various values of d1 and d3 could be obtained from which the desired solutions (or solution) could be selected. 3. 3 Comparison of Results for Simplified Model with Results for Complete Model To get some estimate of the error introduced by the neglect of fringing capacitance, a comparison was made for the particular case covered by Fig. 4. Figure 2 was used to determine the corresponding solution of the simplified problem for the same values of d and d3 (i. e., ZB = 60. 5136 ohms), and for CO = 1. 75 pf, and f = 775 Mc (which result in X = 1. 939). The arrows on Fig. 2 indicate the resultant values of 0 A 0 B and r. Figure 5 shows how the two results compare. The conclusion that can be drawn, in this case at least, is that, while the simplified model can be used to determine rather closely the ranges of d2, 0 A and 0 B over which solutions based on the fringing capacitance model can be expected, it is not sufficiently accurate to be used as a basis for actual cavity construction. 11

4. ANALYSES OF RE-ENTRANT CAVITIES A major portion of the overall length of the cavity shown in Fig. 5 is taken up by the low-impedance section of coaxial line. Further study shows this to be generally true of the in-line type of cavity. The re-entrant designs shown in Fig. l(b) and l(c) make possible a considerable saving in both length and volume. Another advantage of the re-entrant designs is that, since these cavities have seven parameters (i. e., dl, d2, d3, d, LA, LB, and Lc), it may be possible to satisfy seven design conditions, as opposed to the five conditions that were possible with the in-line type. The use of a dielectric other than air in the re-entrant section4 adds an additional parameter to the system, bringing the total to eight. The analyses of the re-entrant configurations are orders of magnitude more difficult than that of the in-line configuration, due to the additional parameters and also due to the fact that three fringing capactiances, CA, CB, and CC, located as shown in Figs. l(b) and l(c), must be taken into account [1, 2] The expression for Z(f) in this case is (A + B tan 0A)(cot 0C - 27fCcZc) - ABZc Z(f) = ZC (A + B tan 0A)(1 + 2w1fCcZccot9 ) + ABZccotC (14) A)('A~ C AC C C CL where A A (2vfCAZA tan A - 1)/ZA B = (27fCBZB - cot 0B)/ZB, 0A = fL A A fLB ~ a 2= fL. Here e is the relative dielectric constant of the maB c B-c r c c c r terial in the re-entrant line. Digital computer programs were written to analyze both the external and the internal re-entrant configurations for the case n = 2. These programs differed only in the exIf, as in the in-line case, the low-impedance section turned out to be longer than the highimpedance section, the internal re-entrant configuration could not be used in many cases, since the center conductor would not be long enough to accomodate the desired length of reentrant line. A dielectric allows the effective length of the re-entrant line to be increased. 12

pressions used to calculate the characteristic impedances and the fringing capacitances. So that program complexity and computing time could be kept within bounds, the programs work with only four parameters: d2, LA, LB, and L. The other parameters, dl, d3, d, and e r as well as f, are constant for each set of computation. Figure 6 shows the flow chart for this computer program. Here, for each value of d2, LB, and Lc, the computer seeks (in progressively smaller LA-intervals, and then by interpolation) a value of LA within the specified range which gives C(f) = C(2f1) (15) where C(f) is the capacitance resonated by the cavity at frequency f. [The function C(f) is obtained in this case by substituting Eq. 14 into Eq. 3 and solving for C. ] If and when a value of LA which causes Eq. 15 to be satisfied is found, the computer prints out LA, LB, Lc, d2, and C. Also printed out are approximations to the reciprocals of the derivatives of Eq. 5, obtained as follows: dC 2[C(2f1) - C(1.998 f1)] dC C(f1) -C(. 999f1) 2df 2 f - 1.998f f df f 999 (16) As before, it is left to the program user to determine (by a double interpolation) the values of d2, LA, LB, and LC at which Eq. 5 is satisfied, i. e., at which 2 dC ddC f =2f 1 df!f 1 Fringing capacitances CA, CB, and CC are computed from equations fitted to the curves of [1, 2]. The specified values of LA, LB, LC, and d2 need not be equally spaced. The program was written in the MAD language, and included statements (not shown in Fig. 6) which provided for headings, output data format, overflow prevention, etc. Compilation of the program on an IBM 7090 required 0. 76 minute. Execution time varied with the number of values of LA, LB, Lc, and d2 chosen. A typical run, using 31 values of LA, 5 of LB, 5 of Lc, and 6 of d2, required 0. 81 minute on the IBM 7090. Figure 7 presents the results for the external re-entrant cavity for the special case n = 2, f1 =500 Mc, d1 =.28125 inch, d3 = 1. 436 inches, d =.01 inch, and = 1. 00. This figure can be applied to the design of particular cavities by choosing the capacitance that is to be resonated and a value for one other parameter (LA, LB, LC, or d2), and then 13

COMPUTE @ =i | I I' ----— [ ~ COMPUTE ( O I NO(S () TRUE Sa: z ISi,P NQ? i = i - - - 1 J u M YES < > b _=O N COMPTE 0COMPUTE~,, COMPUTE ~ S TRE i 0 U - z 3:: - z lYES I --- I uYES U COMPUTE ~ OP- - n F. 6. Co r fw c t fr ernal re-entant( cavity computations. < li=i+ - PcQ Wr PRINT ~ L ---- ( IS j F I N No z -r COMPUTE ( ) CFI =CF2 C CHI =CH2 JE NO P.Q COMPUTE YES PRINT (g) -NO IS Ns? IS(TRUE?.N YES ES STOP Fig. 6. Computer flow chart for external re-entrant cavity computations. 14

DEFINITIONS and FUNCTIONS: (MKS units) CF = capacitance resonated by cavity at fundamental frequency CH = capacitance resonated by cavity at harmonic frequency PR = proximity factor for fringing capacitances YC (GA + G TA) ( CC CT + YC)+ CT GA G F11w' TA' CTB' CTc] - (GA + GB TA) (Y CTC - CC) -GA G where GA = CA TA - YA' and GB =, CB -YB CTBINSTRUCTIONS: (parentheses indicate integer indices) INPUT DATA: fl e, d, d, d, P, Q, LA(0),..., LA(P -), Ng, Lg(O),. LB(NB), NC, LC(0),..., Lc(Nc), ND, d2(0),... d2(ND) ( T=d3/d1, K1 = 2rif1/2. 998 x 108, K2 = 4r/(d3-dl), YA = 1/(60 logeT) W = 27rf, X =2W, Y =.999W, Z =2Y, Af =.001 f E = Ve, K3 = K1E, KA = 7(. 375 T2 - 4.8629 T + 16. 213) x 1012, AA -. 050019 T2 +. 590242 T -.972512, KB = 7r(-. 01278 T +. 19363 T -. 19864) x 102, AB =.57368 Tr + 7. 83883 7 - 25. 85208 KC = 7T(. 099911 T2 - 1. 465289 T + 7. 990667) x 1012, A =.003342 -.05993 +. 30615 K2LA(i) ( TAl(i) = tan K1 LA(i); TA2(i) = 2TAl(i)/f1 - TAlZ(i)]; PR(i) = 1 + 2/(e -1) ~ CTBl(i) cot K3 LB(i);B CT(i) = [ CT(i) - 1/CTB(i)]; CTB3(i) = cot.999 K3 LBi); CTB4(i) = 2 [CTB3(i) - /CT3(i)] ) CTCI(i) = cot K1 LC(i); CTC2(i) = [CTC(i) - l/CT2(i)] CT3(i) =cot.999 K1 LC(i); CTc4(i) = [CTc3(i) - 1/CT3(i)] YB = E/[60 loge(d3/(d2(i) + 2d))]; YC = 1/[60 loge(d2(i)/d1)]; r = dl/[d2(i) + d] CA = -d3 KA (AA + r) (r - 1/T) (1 - r)/r2; Cb = d3 Kc(r - 1/7)/(r - AC); CB = d3 KB [-AB + T/(r,-1)] [1. 1881 -.008 T - (18. 81 -.8T) (r -. 4)2] ~ CA = PR(0)Ck;CB(O)CR(0)C; CC = PR(0)C; CF1 =Fl[W,TAl(0),CTBl(j),CTC(k)] CH1= F[X, TA2(0), CTB2(j), CTC2(k)]; CA = PR(P. Q)CA; CB = PR(P Q)C(3; CC = PR(P.Q)CC CF= Fl[W, TA(P Q), CTB(j), CTCl(k)]; CH2 = Fi(X,TA2(P Q), CTB2(j), CTC2(k)] @ PRINT LB(j), LC(k), CF1, CH, CF2, CH2' C, C, C C'(-.,(-'C r l~ lr - C C'l intersection CF1 + CF2 -H CH -H21 <ICF1 CF2 - CH1 + CH2 critersction ) CA = PR(f)Ck, CB = PR(Q)CB, CC = PR(f)Cb F2 = F[W,TA1()), CTBl(j) CTCl(k)]; CH2 = F1IX,TA2()), CTB2(j), CTC2(k)] () C = PR(m)CX, CB = PR(m)C%, CC = PR(m)Cb CF2 = F [W,TAl(m), CTBl(j), CTCl(k)]; CH2 = F1[X,TA2(m), CTB2(j), CTC2(k)] ~( L = LA(m-l) + [LA(m) - LA(m-l)] (CH1 - CF)/(CF2 - CF1 - C2 + CH1) int TA5 = tan K1LX; TA6 = 2TA5/(1-TA5 ); TA7 = tan. 999 K1LA; TA8 = 2TA7/(1-TA72 K2LA PR' = 1 + 2/(e -1), C = F([W, TA5, CTBl(j), CTCl(k)] CH' = F[X,TA6, CTB2(j). CTc2(k)], C' = F1[Y,TA7, CTB3(j), CTC3(k)] CH C = FiFZ,TA8, CTB4(j), CTC4(k); ACF = (CH - C'H)/Af; CH=(C H - C )/Af ~ PRINT LA(m-1), Lk, LA(m), CF1, CF, CF2 CH1 C', CH2 ACF, ACH 15

2.50 2.60 2.70 Lg,INCHES Fig. 7. Plot of typical computer results for the external re-entrant cavity model. reading from the figure the values for the remaining parameters. As before, additional conditions could perhaps be satisfied by other choices of dl, d3, d, and er. The limited ranges of the parameters shown in this figure do not imply that no solutions exist outside of these ranges. However, in every special case that was investigated, the ranges were severely limited, (particularly the range of d2). Usually one or more exploratory computer runs had to be made before the problem could be narrowed down to the ranges in which solutions were possible. In this connection it should also be noted that all the examples considered in this memorandum involve standing-wave patterns analogous to a quarter-wave mode for the funda 16

mental frequency and a three-quarter wave mode for the second harmonic. Other mode combinations are possible but were not considered here because minimum over-all length was desired. The internal re-entrant model was investigated for the special case n = 2, fl = 600 Mc, r = 2. 10 (Teflon), d1=.28125 in., d3 =.936 in., and d =.01 in. Some solutions which satisfied both conditions were found, but all of them involved either small values of C or values of LB larger than LC (a physically-unrealizable configuration in most practical applications, as inspection of Fig. l(c) will reveal). Trends observed in the computer output indicated that, in this case at least, a larger value of e might yield more practical results. Further investigation of the internal re-entrant model was not carried out, however, because it was assumed that the validity of the approach and the approximations used could be adequately tested on the basis of results for the other two cavity types. 17

5. APPLICATION TO CAVITY DESIGN Actual coaxial cavities typically depart from the models described above in two important respects: a) some losses are present, and b) the simple open-end construction shown in Fig. 1 is usually not used. Losses are present due to the finite conductivity of the conductors, the external loading, and the shunt conductance of the terminating device (vacuum tube, "varactor," etc. ). When these losses are reasonably small, however, they principally affect the Q of the cavity and have only minor effect on the resonant frequencies. Thus the models used here can be expected to give a reasonable approximation to the performance of well-designed and wellconstructed cavities in many applications. The conditions at the open end of actual cavities can also be fitted to the model, in many cases. A typical UHF vacuum tube, such as the one shown in Fig. 5, has anything but a constant conductor diameter inside the tube. Pencil triodes even have a re-entrant section inside the tube, since the grid is coaxial with the hollow plate conductor. It has been shown [1, 2], however, that if the cavity outside diameter is large compared to these internal conductor diameters of the tube, the fringing capacitances are small. Furthermore, the many individual fringing capacitances typically present (due to the many changes in diameter and dielectric constant in and near the tube) are typically located within a region the dimensions of which are small compared to a wavelength, so that they can often be well approxi5 mated by a single fringing capacitance at a properly chosen location. Note also that, because of the satisfaction of condition f), the operation of the cavity will be insensitive to the small variation of the equivalent capacitance with frequency that does exist. Thus, it appears that the models considered here may be useful for the design of actual coaxial cavities. 5The equivalent capacitance can be determined with the methods of Refs. 1 and 2, or by direct experiment. Experiments performed with a 5893 pencil triode in a simple cavity (1" outer conductor, 5/16" inner conductor) gave a value of C which was constant within 5% over the range investigated, 600 to 1400 Mc. Corresponding results can be expected for other devices and frequencies, as long as the wavelength is not too short. 18

6. EXPERIMENTAL RESULTS To test the validity of the previously mentioned analyses, experimental cavities were constructed and tested. The lengths of the cavities were made adjustable, so that dimensions near as well as at the design dimensions could be evaluated. Figure 8 shows the experimental results for an in-line cavity with d1 =.3414 in., d2 =. 703 in., and d3 =. 936 in., which is the case discussed in Section 3. 2 and shown in Fig. 5 (solid lines). Figure 8(a) reveals that the desired result can indeed be achieved, and at dimensions very close to the calculated dimensions. The pairs of curves in Fig. 8(b) show the effects of variations in the length dimensions away from the optimum values. It is apparent from these curves that, for small variations at least, the low-frequency curves are less sensitive to dimensional changes than the high-frequency curves, and that changes in LB are more critical than changes in LA. The lowest pair of curves is included to show that dimensions which deviate significantly from the calculated values produce greatly inferior results, and hence that the obtaining of the desired behavior is indeed neither a coincidence nor an inherent property of all such cavities. The experimental external re-entrant cavity model was built to conform as closely as was practical to the special case covered by Fig. 7. Lengths LA, LB, and LC [see Fig. l(b)] were adjustable. In this case, because there are three variables and only two conditions to be satisfied, a family of calculated results, rather than a single result, is to be expected. Therefore, exploration of a region including part of the calculated family, rather than verification of a single calculated design, was undertaken. 6A.010-inch-thick copper sheet was rolled and soldered into a tube, which was used as the re-entrant wall. This construction technique resulted in some variation in diameter and in concentricity. 19

u \o E 820-.o > <o.; _ z w 0o-. HIGH FREQ.- 2 w 800 hr LOW FREQ. F- 0o \ Z Z 780 LA=CALC. VALUE +.001 in. 0 LB CALC. VALUE+.00I in. 760 - 1.6 1.8 2.0 2.2 2.4 TUBE CAPACITANCE, pf (0) { LA= CALC. VALUE +.006in. 840 v LB CALC.VALUE+.006in. F LA= CALC. VALUE+.011 in. 82 \'\ l ^ *LB= CALC. VALUE+.001 in. u \ \ 2 820 \ o \ LA= CALC. VALUE +.201 in. z LB CALC. VALUE +.066 in. W B o w 800. It \. \^K^<.. HIGH FREQ.-2 LL L- - \\ y\ ^ ^. -LOW FREQ. z o 780- \,- U 7 6010 22 24 1.6 1.8 2.0 2.2 2.4 TUBE CAPACITANCE, pf (b) Fig. 8. Experimental results for in-line cavity. (a) Dimensions essentially equal to the calculated values. (b) Dimensions deviating from the calculated values. 20

Figure 9 shows some of the results obtained. Note that, while the separation between the curves in Fig. 9(a) is in general greater than that shown in Fig. 8(a) the curves in Fig. 9(a) remain close together over a wider range of capacitance variation. Limitations in the experimental equipment prevented checking the performance over an even wider capacitance range, but it appears from Fig. 9(a) that the curves should remain close to each other over a capacitance range considerably beyond two-to-one. Alternatively, some of the capacitance range could be sacrificed for the sake of better midrange performance, if the highfrequency curve could be shifted upwards slightly. There is good reason to believe that a set of values for LA, LB, and LC near those used in obtaining Fig. 9(a) would accomplish this result. Figure 9(b) shows that here, as in the case of the in-line configuration, good performance is not a foregone conclusion. In general, the experimental results for the re-entrant cavity did not agree with 7 the calculated results as well as in the in-line case previously considered. Nonetheless, the analyses for both configurations yielded results very close to those determined experimentally. For optimum performance in a given application, some tuning may be needed, but the analyses allow a drastic reduction in the range of the variables over which a solution must be sought. In the case of the re-entrant design, this is especially important, since an experimental cavity involving eight or nine manipulated variables (even if it could somehow be built) would be almost impossible to investigate experimentally in a reasonable length of time. 7 This is believed to be due in part to the fact that the fringing capacitance approximations used become somewhat less accurate at the small values of LA used, and in part to the limitations of the experimental setup, which rendered impractical an exact embodiment of the mathematical model and a full investigation of all combinations of LA, LB, and LC which might have been of interest. 21

0 2 z =l 0 Z 0 w LL r. 520 500 dl. 281 in. -o d2 -. 890 in. *' o ~1IGH FREQ.-2 d321.436in. "*Z.o. /'7LOW FREQ. d 0 in. L = 2.860 in. I I l I l l l I l 480 460 1.6 1.8 20 TUBE CAPACITANCE, pf 2.2 2.4 2.6 2.8 30 (a) u 0 z w w LJ L_ Uz z 0 ir TUBE CAPACITANCE, pf (b) Fig. 9. Experimental results for external re-entrant cavity. 22

7. RELATED PROBLEMS The results of this investigation, and the success of the mathematical model in dealing with this problem, suggest that the same techniques might be effective in the solution of different but related problems, some of which are as follows: Synthesis of cavities which resonate at three or more frequencies which are in specified ratios, harmonically-related or not. Synthesis of cavities which resonate at two or more frequencies which maintain constant frequency separations (as opposed to frequency ratios) over a range of terminating capacitances (i. e., for mixer applications). Control of antiresonant frequencies in addition to or instead of resonant frequencies (i. e., for filter applications). Synthesis of tuning or modulation devices of special characteristics, using voltage-variable capacitors to obtain capacitance variation. 23

8. CONCLUSIONS This memorandum has primarily considered coaxial cavities which satisfy both the dual-resonance condition and the equal slope condition. The mathematical model which has been used predicts that the desired conditions can be met. Experimental cavities built according to computed results based on the mathematical model gave performance very close to the predicted performance, thus proving that the desired results are indeed obtainable, and demonstrating the usefulness of the method of analysis. REFERENCES 1. J. R. Whinnery and H. W. Jamieson, "Equivalent Circuits for Discontinuities in Transmission Lines," Proc. IRE, Vol. 32, February 1944. 2. J. R. Whinnery, H. W. Jamieson, and T. E. Robbins, "Coaxial Line Discontinuities," Proc. IRE, Vol. 32, November 1944. 24

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