THE UNIVERSITY OF MICHIGAN 5548-8-T AFCRL-68-0027 NETWORK THEORY PROBLEMS IN IMPEDANCE LOADING by E. Lawrence McMahon, Arthur R. Braun, Raymond S. Kasevich The University of Michigan Radiation Laboratory 201 Catherine Street Ann Arbor, Michigan 48108 January 1968 Scientific Report No. 8 Contract AF 19(628)-2374 Project 5635 Task 563502 Work Unit No. 56350201 Contract Monitor: Philipp Blacksmith Microwave Physics Laboratory Prepared for Air Force Cambridge Research Laboratories Office of Aerospace Research L.G. Hanscom Field Bedford, Massachusetts Distribution of this document is unlimited. It may be released to the Clearinghouse, Department of Commerce, for sale to the general public.

THE UNIVERSITY OF MICHIGAN 5548-8-T ABSTRACT Relationships between the real and imaginary parts of an impedance are studied. Given the real and imaginary parts in a finite frequency band, a technique is developed for finding these parts outside the band. Conditions under which the real part remains positive for all w are given. Use of a negative impedance converter for realization of a particular impedance arising from impedance loading studies is discussed. A method of compensating for collector capacitance and extending operation of a transistorized NIC into the VHF region is given. 1,

- THE UNIVERSITY OF MICHIGAN 5548-8-T TABLE OF CONTENTS Page ABSTRACT iii INTRODUCTION 1 II THEORETICAL STUDIES 3 2.1 General Method of Attack 3 2.2 Bounds on Function Behavior 7 2.3 Fourier Methods 11 2.4 Alternative Fourier Method 14 2. 5 Direct Computer Solution 20 III EXPERIMENTAL PROGRAM 24 3.1 Tunnel-Diode Network 24 3. 2 Negative Impedance Converter Development 27 3. 3 Conclusions 38 REFERENCES 39 DD 1473

THE UNIVERSITY OF MICHIGAN 5548-8-T INTRODUCTION This report is devoted to those aspects of the impedance loading problem which are concerned with network theory rather than with field theory. In seeking to control the radar cross-section of a conducting body by impedance loading, the scattering behavior of the body is first determined as afunction of loading impedance, either lumped (as for a thin wire) or distributed (as, for example, with a sphere). The equations are then solved to give the impedance, as a function of frequency, necessary to obtain the desired cross-section modification. The network-theory problem is that of synthesizing and realizing this impedance from numerical data giving its real and imaginary parts over some band of frequencies. Network synthesis actually involves two separate problems, approximation and realization. The given data must first be approximated by a rational function of frequency chosen from an appropriate class of rational functions (e. g., positivereal functions). A network realizing this rational function is then obtained by any of several fairly standard techniques. The approximation problem arising from the impedance-loading study is a rather unusual one. Both the real and imaginary parts of the impedance are specified over some relatively large, but finite, band of frequencies; the more common situation is for some one quantity, such as magnitude, to be specified over the entire frequency range. The unusual nature of the problem has prompted a theoretical study of the relations between parts of impedances specified over a finite band. A number of interesting and useful results have been obtained, and are reported in the next

THE UNIVERSITY OF MICHIGAN 5548-8-T sections. These results are expected to form the basis for techniques of handling a large class of network problems arising in connection with reactive loading. An experimental program, aimed at more immediate practical results in a specific case, has also been undertaken. This program has attempted to obtain an active-network realization of an impedance whose behavior is approximated by a pure negative reactance; such an impedance was found by Chang and Senior (1967) to be required to minimize the cross section of a sphere loaded by a slot parallel to the incident wave. The principal effort toward realization of a negative reactance has been in the area of negative impedance converters (NIC's). Considerable success has been achieved in compensating for high-frequency effects and extending the useful range of operation of the NIC into the VHF range. These results are discussed in Section 3

THE U NIVE RSITY OF M I CHIGAN 5548-8-T II THEORETICAL STUDIES The principal theoretical problem arising from the study of impedance loading can be broadly formulated in the following terms: Given the real part, R (w), and the imaginary part, X(wo), of an impedance on some frequency interval, l < K < 0 2, what can be said concerning the behavior of these functions outside the band [r1, W2]? A number of more specific questions can be based on this general statement of the problem. For example: Does there exist a function Z (p), analytic in the right-half p-plane, whose real and imaginary parts are equal to R (t) and X (to), respectively, for p = j wo, 1 < < 2? If such a function exists, is it unique? Assuming that Z (p) exists, what procedure can be used to find it, at least for p =jw? What are the conditions under which Z (p) is positive-real; that is, under what conditions will Re[Z(j )] > 0 for all w be satisfied? All the questions have been considered, some of them by several different methods. All of them have been answered at least in part; Z (p) does exist, it is unique, Z (j o) can be found for all o, and some necessary (though not sufficient) conditions for Z (p) to be positive-real are known. 2. 1 General Method of Attack The most powerful analytic tool in this investigation has been the Hilbert Trans form, which relates the real and imaginary parts of a function analytic in the closed

THE UNIVERSITY OF MICHIGAN 5548-8-T right-half plane. If Z (p) is such a function, then (P) dp = 0 (2.1) C P-J where C is the contour shown in Fig. 2-1. Letting the radius of the small semicircle approach zero and that of the large semicircle approach infinity, and separating real and imaginary parts, we obtain 1 X (X) dX R (w) = -- + R (o) (2. 2a) -00 1 P R (X)_d_ X (w) - - 1b dX )(2. 2b).T -co -00 These integrals are improper, and must be evaluated in the Cauchy principal-value sense, that is, dX= -_ O [S0 dX +S dX (2.3) -O -00 W+E The basic Hilbert Transform pair, (2. 2a) and (2. 2b), may be manipulated into a number of other useful forms. In particular, if the integrals are expressed as the sum of two integrals, one from -oo to 0 and the other from 0 to co, one obtains, after a suitable change of variable R(w)= 2 = XX (X) d2 R R~(w) = scd2 - + R(oo) (2.4a) - 0 2 2

THE UNIVERSITY OF MICHIGAN 5548-8-T jw p-plane FIG. 2-1: CONTOUR OF INTEGRATION FOR HILBERT TRANSFORMS. FIG. 2-1: CONTOUR OF INTEGRATION FOR HILBERT TRANSFORMS.

THE UNIVERSITY OF MICHIGAN 5548-8-T X(t) = - M2 (X) = -(2. 4b) V ~0 X2 _2 where use has been made of the fact that R (tw) is an even function and X (w) an odd function. The Hilbert Transform need not be applied directly to Z (p). An especially useful set of relations is obtained by applying the transform to the function Z (p) / $/i + p. Of the four equations thus obtained, the two of greatest value in this work are R 2 2 = 1 X(X) d X R(X)dX d 0,< R(w) ( 2 - 1 (2<w< 7T 2 2 (2 2 1 0 0- - o I (X2- W o 2 - (2.5a) ~X(w): 2 R X 0 X (X) d:,I x 1-w (X _W 1(X (2. 5b) A simplification which has been used in much of this investigation is that of assuming R(w) and X (w) known in the band 0 < w < 1 rather than for wl<w<w2. This assumption results in considerable simplification of the mathematical manipulations without any significant loss of generality, since the results obtained for the " low -pass" case can be readily extended to the "band-pass" case The fundamental problem of this investigation has been the solution of the integral equations, (2. 5a) and (2. 5b), forthe unknown parts of R (w) and X (w). Although closed forms of solution have not been found, a number of numerical techniques have been developed; these are discussed in succeeding sections of this report.

THE UNIVERSITY OF MICHIGAN 5548-8-T 2. 2 Bounds on Function Behavior It is reasonable to ask what restrictions are placed on R (w) and X (w) in (0,1) under the assumption, necessary for physical realizability of a passive network, that R (Li) be non-negative for all. From (2.4b) Xw) 2R (X)dX 2+ L i R (X) dX(. X (v) =TT 2 (2 6) With R (t) > 0, the second integral in (2. 6) is a non-negative, non-decreasing function of to for 0 < to < 1. Therefore X(tw) > 2- f 2 2R(0)dX < < o 1 (2.7) 0 X2 2 dX(w) > d 27 r ()d 0 < < 1 (2.8) dL - d -- 0 2By adding and subtracting R (w) inside the integral we get 2t o R(X) dX _ 2 Li R (X) — R(tW) + R() __+ R R(o)in (2.9) 0 X L f 2. -t t where the first integral on the right is now proper. This result indicates that the bounds of (2. 7) and (2. 8) are rather weak, since the function on the right has a logarithmic singularity, approaching - oo as t — > 1 -. For example, for R (w) =, X (W) = 0 < < 1 (2.10) l+w 1+w

THE UNIVERSITY OF MICHIGAN 5548-8-T we have 0 w 0 1 p 1 ( 111) >2 -+ n2 r (2.11) 1+o 1~0 + and also that the slope of the left-hand term of (2. 11) must exceed that of the righthand term. These two functions are plotted in Fig. 2-2, which shows the weakness of the inequality near w = 1. It should be pointed out that inequalities (2. 7) and (2. 8) become much stronger if R(1) = 0, but of course this is a special case. A stronger inequality can be obtained by observing that the second integral on the right side of (2. 5a) is non-negative for 0 < w < 1. Thus we have 2, w2,12 1 XX(X) dX R(w) > - -l XX (X) dX O < < 1 (2.12) 0 (X2 - W2)-'X or more generally If 0(X2 242 -- > w - - ), 0 < w < w (2.13) Using the same pair of functions as the previous example; 1W R(w)= X(w) (2.14) 2 X() = - 1 the inequality (2. 13) is shown in Fig. 2-3 for w =- 2 w 0 1 and w = 2. It will be noted that as wo increases, the lower bound approaches the function itself, as one would expect. Letting w -0 oo in (2.13) and comparing with (2.4a), we

THE UNIVERSITY OF MICHIGAN 5548-8-T 0.2.4.6.8 1.0 -.2 -.4 \1%41~ ~X(w) X(@) -. 6 -. 8 -1.0 - FIG. 2-2: LOWER BOUND ON X(().

THE UNIVERSITY OF MICHIGAN 5548-8-T 1.0.8.6-.4 R(WX.2 wo0=2 I~o= 1/2 \o =1 ~0.4.8 t 1.2 1.6 2.0 FIG. 2-3: LOWER BOUNDS ON R(w). 10

THE UNIVERSITY OF MICHIGAN 5548-8-T see that the lower bound actually approaches R (w) - R (co); this makes no difference in the example or indeed in most cases of practical interest. If the right side of (2.13) goes negative anywhere on (0, w ) (indicating that R (oD) is non-zero), a better lower bound can be obtained by adding a suitable constant to make the minimum value of the lower bound zero. 2. 3 Fourier Methods One way of approaching the problem of finding R (w) and X (w) outside the band in which they are given is by means of the change of variable w = 1-p (2.15) l+p which maps the right-half p-plane into the unit disc in the o - plane. If Z (p) is analytic for Re(p) > 0, then Z(w) will be analytic for 1wl < 1. Thus, a power series expansion around the origin Z(t) = A + A w + A2 w + (2.16) o 1 2 converges for Iwl < 1. On the unit circle, let w = -j (2.17) Then -2 j Z(c-j) A + A -j + A E-2j + O 2 (A + A1cos~ + A2 cos2~ +... ) - j(A1sin + A2sin2A + 2). (2.18) 1 _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ 11 1__ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _

THE UNIVERSITY OF MICHIGAN 5548-8-T From (2.15) and (2.17) we have w = tan 2. -r < < 7r (2.19) so that R e "() = R(tan2 ) = A cos + A2cos29 +... (2.20) X'(9) = X(tan2 ) = B1sino + B2sin29 +... =-A1 sin - A2 sin 2 + (2.21) We thus have the Wiener-Lee criterion for physical realizability A +B = 0 n =1, 2,.. (2.22) n n If R(w) and X(w) are givenfor we(0,1), then R (9) and X (0) are known for - < 0 <. (The extension to negative values of 0 comes from the ~~~2 2 fact that R and X are respectively even and odd. ) One possible method of solutio is to find a Fourier Series which gives the best fit (in the least-square sense) to R (0) and X (9), subject to the constraint (2. 22). This method was used by Redheffer (1947); unfortunately, the equations are rather difficult to generate, and the determination of R" and X" for J j> requires the summation of a Fourier Series. The present investigation made use of a different approach. Assume that the range 2 < 0 < 7 is divided into n equal intervals, and that R (9) and X''(0) in each of these intervals may be approximated by a constant. Thus we have...~., 12

THE UNIVERSITY OF MICHIGAN 5548-8-T R (0) = Rk X*)= k k-1 < 0 k < (2.23) x* (~) Xk where _ k + k 2n (2.24) The Fourier coefficients are then given by Am 7r4 R" (R)cosm do + s Rk cosmo do ki k-1 (2.25) B -- X (0)sinm d + Xksinm d. rn.7r 1 k-1 (2.26) Applying the Wiener - Lee criterion we obtain, after evaluating the right-hand integrals in (2. 25) and (2. 26) 7r/2 - 0 (R"'(0) cosm, + X' (0) sinm= d DL I sinmk - sinm k-l cosm k -cosmkl I. (2.27) For m = 1, 2,..., 2n, (2. 27) yields a set of 2n simultaneous linear equations for the 2n unknowns R1..., Rn... X. Xn The coefficient matrix is independent of R* and X*, although it depends on n. Aside from the approximate nature of the solution, this method has the disadvantage that R (w) and X (o) are found as the real and imaginary parts of an,_____________ 13

THE UNIVERSITY OF MICHIGAN 5548-8-T impedance whose behavior is restricted. From (2.15) and (2.16) we have Z(p) ~ A + A1 + + A (+P) = a~ + alp +'".. +anp [O + a1p=+..+ap + *@(2.28) (l+p )n Since all the poles of Z (p) are at p = -1, Z (p) lacks the approximating power of a general, nth order function. The computational difficulties of this method are avoided by using an alternative Fourier method discussed in the next section. 2.4 Alternative Fourier Method Equations (2. 5a) and (2. 5b) may be written in the form 7rR (w) + XX (X) dX XR(X) dX 21-w0 (2) 1 ox (X2 _2) 1 (2. 29a) rX(w) X 1 R(X) dX X(X) dX, (Li),0 22 7 J1 2 2< 17 2w 1-w2 0 (X2X_-w,2 )-X2 U (2 _ )i2 2(2.29b) where f(w) and g(w) are known functions for 0 < w < 1. Since X (X) is an odd function, it may be written as X E (X),-where E (X) is even. This substitution makes the right-hand sides of (2. 29a) and (2. 29b) formally identical, so that it will suffice to consider only one of them. 1________________ _ ~14

THE UNIVERSITY OF MICHIGAN 5548-8-T Making the change of variable X = sec 0 (2.30) in (2. 29a) we have 2 2d f(w) =,J / R'''(O~dO (2.31) 0 1-w cos 0 where R" (0) = R(sece). (2.32) Note that if R':(0) > 0 for 0 < 0 < 7r/2, the right-hand side.of (2. 31), and all its partial derivatives with respect to w, are non-negative for 0 < w < 1. Therefore, if f (w) is expanded in a power series about the origin 00 f(w) = K 2n (2.33) n=0 the K's must all be non-negative. n 2 2 -1 The term (1 -w cos O) appearing on the right side of (2. 31) may be expanded by the binomial theorem into an infinite series which converges absolutely for w < 1. Thus we have OD 2 7/2 O 2n 2n f (W) = K 2 = / R (0) 2 n cos O d O (2.34) n=0 0 n =0 Since the series converges absolutely, we may interchange the order of integration and summation and equate coefficients of like powers, obtaining 15

THE UNIVERSITY OF MICHIGAN 5548-8-T er/2 2n K = R':* (0) cos 0 d 0. (2.35) 0 Since 2(n+l) 2n (2.36) Cos < cos 0, <0< (2.36) it follows that if R (0) > 0 for 0 < 0 < 7r/2, then K > K > 0, all n. (2.37) n n+1 This inequality is a necessary (but by no means sufficient) condition for the existence of a pair of functions R (w) and X (w), satisfying Eqs. (2. 29a) and (2. 29b), with R (o) > 0 for all a. Such a pair of functions we will call "compatible". The necessary conditions for compatibility may be summarized by the relations anf (c) > 0 n = > 1~ o0,1,2,.... (2.38) a(2)n O<w< 1 an+ 1f (W) (n+1) nf() > (n+ 1) > 0, all n. (2.39) w=O w=0 In order to solve for the unknown part of R (w), we make the further change of variable 0 = 0/2 in (2.35), obtaining K RI-no R'()cos - 2 d (2.40) nK 2 | acos0 2d~ 16

THE UNIVERSITY OF MICHIGAN 5548-8-T where RI R (sec-) (2.41) Since R'(0) is an even function it can be expanded in a Fourier cosine series 00 R'(I) = A cosmO. (2.42) m=O Making use of the trigonometric identity cos 2n i- = -- 2 ( n) cos(n-j) ~ + ( 2n ) (2.43) and substituting (2.42) and (2.43) into (2.40) we obtain K 22n+1 A cosm 2 n)cos (n - j)0 + (2n) d n 2n+ 1 2 0 MT= j=0 (2.44) From the orthogonality relation 7r, m =n-j =0 cosmO cos(n-j) do = 4 7, m =n-j / 0 (2.45) 0, m n-j we obtain finally n K n 2n+1 A (2n) (2.46) n 2 j= 17,

THE UNIVERSITY OF MICHIGAN 5548-8-T This relation is solved recursively to obtain 22n+ 1 n A = K n - A n-(2n) (2.47) n z n _ (n-j) It would appear that by making use of the relation n cosn b= Co (2.48) k=0 nk 2 where b = 22n1 (2.49) nn bnk = (-1) n-k n 2k- 221, k n, (2. 50) in (2.42), one could obtain an expression of the form 00 R'B cos 2k = (2.51) k 2 which corresponds to 0D R(w) B k /w. (2.52) k=0 However, such a form, since it is a power-series expansion about infinity, will be meaningful only if Z (p) has no singularities outside the unit circle. Even for cases satisfying this condition, serious convergence difficulties were encountered, most probably due to the truncation of Eq. (2.42). 18

- THE UNI:VERSITY OF MICHIGAN 5548-8-T This difficulty is easily circumvented by summing the Fourier Series, (2.42), directly, and then changing the abcissa for 0 to w by use of the relation w = sec (2.53) 2 Results obtained by applying this technique to known functions have been excellent. The real and imaginary parts of a known impedance, calculated on (1, co) using a seven-term series, agreed with the actual values to three significant digits. The error due to truncation can be estimated rather easily. We note from (2.46) and (2.47) that the relationship between the first (N+1) A's and K's is n n independent of the higher-order terms. Thus, truncating the Fourier Series (2.42) at the Nth term is equivalent to truncating the power series (2. 34) at the Nth term. A measure of the error is thus given by 2n 2n e (w) -f K K 2n (2. 54) n=O n=N+l where e(w) is positive for all w, since the K's are all positive. To avoid difficulty with singularities, we define 7roI 2r XX(X) dX F (w) =- R(c) + /-w (2.55) ~2 0 (o 2 2 2i' and e "(~o) = F (~) W- (.62nl 2 e(L) F(L) K w -i) (2.56) where e'(w) > 0, 0 < w < 1. 19 1

.THE UNIVERSITY OF MICHIGAN 5548-8-T We now define the error by f e((w) dw E: (2.57) F (w) d Defining I = J F () do (2.58) and carrying out the integration in the numerator of (2.57) we obtain Xff N 1 (2n- 1) (2n- 3)... (3) (1) 2 K n (2n+2)(2n) (4)(2) E= 1 (2.59) 2. 5 Direct Computer Solution A number of attempts were made to obtain a direct solution by means of the computer. Given R o(w) and X (o) for wE (woa, Wob), the objective is to develop a computer program which will determine either the coefficients of a positive-real rational function or the element values of a passive network, such that the real and imaginary parts of the corresponding impedance approximate, in some sense, R (w) and X (t). The most general approach is by way of a positive-real rational function. The method which was used may be summarized by the following steps: 20

, THE UNIVERSITY OF MICHIGAN 5548-8-T 1) As a starting point, pick the coefficients of a rational function, Z (p), of degree n: n a p +.. + alP+ a z (p) = (2.60) b.n + + blP + bO The program will reject Z (p) if it is not positive-real. 2) Evaluate the real and imaginary parts of Z (p), R (w) and X (w) respectively, at the m+ 1 frequencies wi, where ~o w= <t < < < = w (2. 61) a o 1 m b 3) Compute the weighted error E = i [(R(.i) - R(wi)) + (X( ) -X(o i )2( (2.62) where the w.'s are the weights. 4) Determine the partial derivatives of E with respect to the coefficients ao, al,..., an, bo, bl..., b of (2. 60) and compute the gradient direction in the (2n + 2) - dimensional coefficient space. 5) Move in the negative-gradient direction in coefficient space until one of the following happens: a) The error reaches a minimum. In this case, compute the new gradient and proceed as above. b) The function ceases to be positive-real. In this case, retreat along the gradient line back to the positive-real region, compute the gradient again, and proceed as above. I_____________ 21 er r s~_

THE UNIVERSITY OF MICHIGAN 5548-8-T 6) When a point is reached from which no further progress can be made (the error cannot be reduced further while remaining in the positive-real region) the search terminates and results are printed. In practice, this procedure was found to be very slow. This seemed to be due to the fact that the boundaries of the positive-real region in the coefficient space are quite irregular, requiring frequent backtracking and gradient computation. In addition, the process of checking the function for positive-real character is quite time consuming. In addition, since the error is a very complicated function of the coefficients, it was felt that local minima would make the process quite dependent on the initial values of the coefficient. This line of investigation was therefore not pursued further. A simpler, although less general, procedure is to choose anetwork configuration and adjust the element values. The condition for physical realizability is now much simpler; all the element values must be non-negative. The procedure which was used was a gradient search (steepest descent method) similar to that outlined above. The most significant difference was that the boundary of the region of physical realizability is much more easily defined in the present case. Whenever movement in the negative-gradient direction caused one or more element values to become negative, they were simply set back to zero before the next step was taken. Thus, the search technique was actually a modified gradient search, since the direction of search deviated from the gradient line near the boundaries at the positive - coefficient region. This "direct synthesis" technique had much better convergence properties than the rational function approximation. The improvement appeared to be due to the simple nature of the constraints in this case (element values non-negative) compare to those of the previous case (function positive-real). A number of difficulties 22

, THE UNIVERSITY OF MICHIGAN 5548-8-T remained, however. The method is obviously dependent on the configuration chosen for the network; a ladder network was used in this study for its computational simplicity, but this may not be the best choice from a practical standpoint. The problem of local minima and dependence on initial values still remained. Further, it became apparent that, due to the high dimensionality of the problem and the extremely nonlinear dependence of the error function on the element values, the gradient search technique was not a sufficiently powerful method. A quadratic search was then tried. In contrast to the gradient search, which essentially linearizes the function to be minimized, the quadratic search retains another term of the power series expansion. This technique will minimize a quadratic form in a single step. Unfortunately, the problem at hand did not involve a quadratic form but a much more complicated function, to which the method was totally unsuited. In fact, the procedure was unstable, making large jumps in element-value space without getting any closer to the minimum point. No further effort was devoted to this area of direct computer solution. It was quite obvious that much more powerful search techniques would be required involving much longer computation times. Judging from the results obtained thus far, prospects for success are poor, even with the use of more powerful search techniques. 23

THE UNIVERSITY OF MICHIGAN 5548-8-T III EXPERIMENTAL PROGRAM Since the theoretical studies discussed above are of rather long-range applicability, an experimental study aimed at more immediate results was conducted in parallel. Efforts were directed toward finding a network realization of the impedance shown in Fig. 3-1. This impedance was determined by Chang and Senior (1967) as the required loading impedance for cross-section reduction of a sphere loaded by a circumferential slot in the plane of incidence. In the region of interest, near k a = 1, the desired impedance can be well approximated by a negative reactance, which suggests realization by an active network. Unfortunately, there exists no really coherent theory for the realization of active driving-point impedances; the task was therefore one of choosing from a number of available alternatives those which seemed to have the greatest probability of success in a reasonably short time. 3. 1 Tunnel-Diode Network Because of its size and simplicity, the tunnel diode appeared to be a likely choice for an active element. Properly biased, the tunnel diode may be modelled by a negative resistance in parallel with a capacitance, as shown in Fig. 3-2. The network in which it is embedded was designed on an intuitive basis; essentially, the inductance L paralleling the diode gives a pair of poles in the right-half plane, the RLC combination gives a mirror-image pair in the left-half plane, and L moves the poles on to the j w-axis. 24.............

THE UNIVERSITY OF MICHIGAN 5548-8-T 0. 4 R(w) _ ohms -0. 4 20 0 X(S) ohms -10 - -20 I., I 0 1.0 ka 2.0 FIG. 3-1: IMPEDANCE TO BE REALIZED. 25

THE UNIVERSITY OF MICHIGAN 5548-8-T Straightforward network analysis yields 2 iP P+ 1LC2 Z = ( L+ (3.1) 4 2 L + L _ L +2L P + P o (C R2C2) LL 2C If Lo is chosen to satisfy 1 1 1 =.. (3.2) L C 2R 2 (3.2) o 2R C RC1I7-' then (3.1) becomes 2 __LC z = ( ) (3.3) C +2 - + 2 LC R CI) For p = j w, Z is a pure reactance: W (W 2 w2) 2 a z(j ) = jx(w) = ( (3.4) b where < (3.5) a b Obviously, X(w) is positive for 0 < w < w and negative for w < w < w and a a b thus has negative slope in some region around wa; this is the desired type of behavior. 26

THE UNIVERSITY OF MICHIGAN 5548-8-T The circuit of Fig. 3-2 was constructed and tested. The predicted frequency response was not observed; the frequency response being essentially that of a passive network. It is felt that this was due to second-order effects, most importantly lead inductance, which were not included in the model of the tunnel diode. Further effort did not appear justified, and the circuit was abandoned. 3.2 Negative Impedance Converter Development Another likely active element for synthesis is the Negative Impedance Converte (NIC). Its use is readily suggested by the negative-reactance characteristic desired in the present application; in addition, a number of active-synthesis techniques using the NIC have appeared in the literature. The NIC is a two-port, active network with the property that its input impedance is the negative of its load impedance. Any linear two-port, active or passive, may be characterized by its hybrid parameters: V1 hll I1 + h12 V2 (3.6) 2 = h21 I1 + h22 V2 (3.7) If the two-port is terminated at port 2 by an impedance Z L' then the input impedance at port 1 is given by h12h21 Z_ h (3.8) in 11 h Y 22 L If hll = h22 =0, and ifh h12 h21 = then Z =-= —-Z (3.9) in Y L (3.9) 27

THE UNIVERSITY OF MICHIGAN 5548-8-T Tunnel Diode -R R L....... J.-2 T DORL FIG. 3-2: TUNNEL DIODE NETWORK. 28

-- THE UNIVERSITY OF MICHIGAN 5548-8-T which is the desired behavior. In practice, the condition h12 h21 = 1 has been satisfied by one or the other of the alternatives h h + 1. (3.10) 12 21 If h12 = h = 1, then V V1 and the device is known as a current inversion NIC, or INIC; if h = h1 - 1, the device is a voltage inversion NIC, or VNIC. Most of the circuits appearing in recent literature have been INIC's, and our work has been entirely with this type. Of the various NIC circuits available, one due to Yanagisawa (1957) was chosen for study. An idealized version of the circuit is shown in Fig. 3-3. It is rather easily shown that, for this circuit V1 V2, (3.11) I KZ 1- 1(3.12) 2 (K+1) Z + Z 2 2 1 Thus, for large K, I Z X 2 (3.12) I2 2 Yanagisawats realization of this idealized circuit is given in Fig. 3-4a, with the small-signal equivalent circuit in Fig. 3-4b; the similarity between this and the idealized circuit of Fig. 3-3 is quite apparent. The most important factor limiting the high-frequency performance of this circuit is the collector-base capacitances of the two transistors. In order to have a basis for developing techniques of compensating for these capacitances, the........______ _ ~29

THE UNIVERSITY OF MICHIGAN 5548-8-T I1 i2 + Il KI + vi V2 - z 1 Z2 - -Z10 n -- FIG. 3-3: IDEALIZED NEGATIVE IMPEDANCE CONVERTER. 30

THE UNIVERSITY OF MICHIGAN 5548-8-T R1 R2 FIG. 3-4a: NEGATIVE IMPEDANCE CONVERTER I1 re rb R3 I I bi'I FIG. 3-4b: AC EQUIVALENT CIRCUIT. 31

THE UNIVERSITY OF MICHIGAN I 5548-8-T simplified equivalent circuit of Fig. 3-5 was used. Base and emitter resistances have been neglected, but the two capacitances have been accounted for by the additio of the capacitance C = 2C bC A fairly straightforward analysis of this network yields the hybrid parameters: h =0 h =21 11' 12 R 1+ =,h =- (3.14) h21 = CR h 21 R2) 1 R 22 pC +p CR where a has been assumed to be unity. Comparing these parameters with the parameters of an ideal NIC, it will be seen that the phase shift in h21 is the limiting factor in high-frequency operation; at sufficiently high frequencies, h21 approaches a negative real number. If a capacitance C2 is placed in parallel with R2, which is equivalent to the substitution R2~ 1+PC R2 2 (3.15) 2 2 the h-parameters now become h =0 h = 1 R 1 I P(C OR 1+ + pC2R1 21 12 = 2 2 h PC 2 (3.16) 2 1 1 Now setting R1 and C = 2C, we have [ ~~32

THE UNIVERSITY OF MICHIGAN 5548-8-T I1 I2 o - +- I, 1 2 -+ ) _c R',, R2 - FIG. 3-5: SIMPLIFIED HIGH-FREQUENCY MODEL OF NIC. 33

THE UNIVERSITY OF MICHIGAN 5548-8-T hl =0, 1, 11 1.2 h 1 h22 = 2pC (3.17) 21 22 The only departure from ideal behavior now is the non-zero value of h22; this can either be absorbed in the load or compensated for by a capacitance 2p C in parallel with the input. The final version of the circuit is given in Fig. 3-6. Resistors R3 and R4 are bias resistors; since at signal frequencies they appear in shunt across the input and output respectively, their effect is cancelled out by the NIC action. When this circuit was constructed, some difficulties of stability and reliability of operation were encountered, but these were eventually overcome. The bias resistors must be adjusted to prevent "lock-up" (both transistors cut off), but the adjustment is not critical. Oscillation is possible for certain combinations of load impedance and generator impedance (impedance seen looking to the left and the input). For most measurements, it is necessary to pad the input with a resistance in order to prevent oscillation. Figure 3-7 shows the input impedance of the NIC over the frequency range 1 - 5 MHz. The load impedance was a 270 f2 resistor; a 68002 resistor was placed in series with the input during measurement to maintain stability, but its resistance has been subtracted from the measured values so that Fig. 3-7 gives the actual input impedance at the NIC terminals. The fact that the negative resistance seen at the input is on the order of-600 Q rather than -27002 is of no consequence, since this is merely a scale factor which can easily be accommodated by changing the load. - 34

THE UNIVERSITY OF MICHIGAN 5548-8-T Vcc' R3 I-I R1.C2 CR2 FIG. 3-6: COMPENSATED NIC. 35

THE UNIVERSITY OF MICHIGAN 5548-8-T 700 -R(w) 600 x 500 400 ohms 300 - 200 x 100 - 0 1 2 3 4 5 Frequency, MHz FIG. 3+7: INPUT IMPEDANCE OF NIC; ZL = 270+j0. (n's indicate measured values; the lines are merely to guide the eye. ) 36

THE UNIVERSITY OF MICHIGAN 5548-8-T The rather large reactive component of the input impedance is a rather more serious problem, it indicates that the ratio of input impedance to load impedance is complex rather than negative-real. However, the circuit is still in the early stages of development, and considerable improvement can be expected as the development continues. Furthermore, the NIC need not be perfect in order to be useful; this point is discussed in more detail below. Measurements have thus far been restricted to the 1 - 5 MHz frequency band. This is sufficiently high for the effect of collector capacitance to be significant and low enough that the frequency variation of a may be neglected, and is thus an ideal range for testing and adjusting the compensation scheme. Since the NIC is essentially a lowpass device, the problem of increasing the bandwidth and the problem of increasing the maximum operating frequency are essentially the same. The critical assumption in the analysis above is that a = 1, which implies that operation must be at frequencies well below alpha cut-off. To a certain extent, then, improvement in NIC performance depends on improvement in transistor performance. Given the present rate of development of transistor technology, there are good grounds for optimism in this regard. There is also the possibility of compensating for the frequency dependence of a by the addition of more passive elements to the circuit. This has not been tried so far, since compensation for collector capacitance is a much more pressing problem in the lower VHF band, but there is no reason to believe that it cannot be done. Even if perfect compensation cannot be obtained, the device may still be useful if some measure of NIC action takes place. Imperfections in the NIC can be overcome by changes in the load impedance. As a trivial example, suppose that an 37

THE UNIVERSITY OF MICHIGAN 5548-8-T input impedance of -1 + j 0 is desired at some particular frequency. Assume further that a load impedance of 1 + j 0 produces an input impedance of - 1 - j 1 rather than the desired value. Provided the NIC is operating linearly, a load impedance -l+jO = 0.5-jo.5 ZL -1- ji should give the desired input impedance. More generally, the process of obtaining a particular input-impedance characteristic using an imperfect NIC would require determining, as a function of frequency, the load impedance needed to give the desired input impedance, and then finding a passive network realizing (or approximately realizing) the necessary load impedance. 3. 3 Conclusions The most important result of the experimental program has been the development and preliminary evaluation of an NIC with frequency compensation for operation in the VHF region. Since NIC development, as reported in the technical literature, has generally been at frequencies below 100 kHz, the results which have been obtained to date are a significant improvement. The immediate objective of continued development in this area is to obtain usable NIC action at 50 MHz or higher. This objective appears quite attainable. 38

THE UNIVERSITY OF MICHIGAN 5548-8-T REFERENCES Chang, S. and T. B. A. Senior (1967), "Study of the Scattering Behavior of a Sphere with an Arbitrarily Placed Circumferential Slot, " The University of Michigan Radiation Laboratory Report 5548-6-T, AFCRL-67-0111, AD 648717. Redheffer, R. M. (1947), "Design of a Circuit to Approximate a Prescribed Amplitude and Phase, " J. Math. PhJs., 28 (140-147). Yanagisawa, T. (1957), "RC Active Networks Using Current-Inversion Type Negative Impedance Converters, " IRE Trans. on Circuit Theory, CT-4 (140-144)........._ _ ~39 __

Unclassified Security Classification DOCUMENT CONTROL DATA- R & D (Security classification of title, body of abstract iand ildexi.nl annotlration naust be enttered when the overall report is clasallied) 1. ORIGINATING ACTIVITY (Corporate author).B. REPORT SECURITY C LASSIFICATIO The University of Michigan Radiation Laboratory, Dept. of UNCLASSIFIED Electrical Engineering, 201 Catherine Street, 2b. GROUP Ann Arbor, Michigan 48108 3. REPORT TITLE NETWORK THEORY PROBLEMS IN IMPEDANCE LOADING 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Scientific. Interim. 8. AUTHOR(S) (First name, middle it,ital, last name) E. Lawrence McMahon Arthur R. Braun Raymond S. Kasevich 6. REPORT DATE 17e. TOTAL NO. OF PACES 7b. NO. OF REFS January 1968 3 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) AF 19(628)-2374 5548-8-T b. Project, Task, Work Unit Nos. S t ~~~~~5635-02-01 |Scientific Report No. 8 5635-02-01 ~ DoD Element 61445014')h. OTHER REPOR T NO(S) (An;y other nlumbers thot may be assi.nedo 61445014 this report) d. DoD Subelement 681305 AFCRL-68- 0027 10. DISTRIBUTION STATEMENT 1 - Distribution of this document is unlimited. It may be released to the Clearinghouse, Department of Commerce, for sale to the general public. II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Force Cambridge Research Laboratories (CRI) TECH, OTHER L.G. Hanscom Field Bedford, Massachusetts 01730 13. ABSTRACT Relationships between the real and imaginary parts of an impedance are studied. Given the real and imaginary parts in a finite frequency band, a technique is developed for finding these parts outside the band. Conditions under which the real port remains positive for all w are given. Use of a negative impedance converter for realization of a particular impedance arising from impedance loading studies is discussed. A method of compensating for collector capacitance and extending operation of a transistorized NIC into the VHF region is given. DD FORM 1473 Unclassified S........u (; --..H...........j

Unclassified Security Classification 14. LINK A LINK!B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT NETWORK THEORY HILBERT TRANSFORMS INTEGRAL EQUATIONS NEGATIVE IMPEDANCE CONVERTERS IMPEDANCE LOADING Unclassified

UNIVERSITY OF MICHIGAN III IIII 11111111115111 036511 111 3 9015 03465 8222