Technical Report ECOM-0138-12-T Reports Control Symbol OSD- 13 66 November 1970 ITERATIVE SYNTHESIS OF TEM-MODE DISTRIBUTED NETWORKS C.E.L. Technical Report No. 199 Contract No. DAAB07-68-C-0138 DA Project No. I HO 62102 A042 01 02 Prepared by S. Mahdi COOLEY ELECTRONICS LABORATORY Department of Electrical Engineering The University of Michigan Ann Arbor, Michigan for U.S. Army Electronics Command, Fort Monmouth, N.J. DISTRIBUTION STATEMENT This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of CG, U. S. Army Electronics Command, Fort Monmouth, N. J. Attn: AMSEL-WL-S.

ABSTRACT A procedure is described for synthesizing transmission networks which are interconnections of uniform line sections. An iterative, digital computer algorithm is developed which achieves a dominant pole synthesis. The line lengths and the characteristic impedances are controlled individually, which gives design flexibility not found in synthesis procedures based on Richards' transformation. Thus, the characteristic impedances may be restricted by upper and lower bounds when there is no restriction on the line lengths. The procedure is detailed for a TEM mode structure of alternating open stubs and connecting lines. The method uses a Newton-Raphson iterative scheme to adjust the characteristic impedances and lengths of the transmission lines for a prescribed set of dominant transmission poles. By controlling the stub line lengths, the dominant pole positions, the principal transmission zeros, and bounded characteristic impedances can be achieved simultaneously. The occurrence of nondominant poles has been analytically investigated. Frequencies at which each transmission line element is a quarterwave long or an odd multiple thereof divide the s-plane imaginary axis into halfwave frequency bands. In every semi-infinite s-plane strip, which these frequency bands subtend parallel to the imaginary frequency axis, one and only one nondominant pole is present. A numerical technique has been outlined which locates these poles. iii

ABSTRACT (Cont.) A nonlinear programming problem has been formulated, and a method for its solution presented, which aligns jw-axis transmission zeros opposite the nondominant transmission poles in or near the stopband. This approximate cancellation of these poles extends the frequency range over which the network characteristics can be controlled. iv

FOREWORD This report is directed to the engineering problem of designing networks for a prescribed frequency domain characteristic using distributed circuit elements. The specific elements considered are sections of uniform, lossless transmission lines, but the methods presented are applicable to a broader class of elements including RC transmission lines and a mixture of lumped capacities and distributed lines. The frequency domain specification used is the complex frequency (s-plane) location of the poles and zeros of the desired network function. The report treats the problem of selecting the distributed circuit element parameters (characteristic impedances and electrical lengths for lossless lines) to achieve a specified set of dominant poles and zeros while simultaneously restricting the impedance levels to practical ranges. The distributed elements considered in this report are sections of lossless, uniform transmission lines which are adequately described by the TEM propagation mode. Practical realization of such elements is achieved with coaxial, strip, and microstrip transmission line sections. Generally, the assumption of only the TEM mode is not valid at element junctions, but the principal effect of the junctions can be taken into account by a modification of the element lengths. This is illustrated for the specific example presented in Chapter 5. v

FOREWORD (Cont.) The use of a pole-zero specification presumes that the circuit designer has solved an appropriate approximation problem before trying to realize the circuit with distributed elements. For the standard filter types (lowpass, bandpass, etc., ) the required poles and zeros can be determined from a number of tabulations which are available. In Chapter 4, this report gives a specific method for designing a low-pass distributed filter consisting of open circuited stub lines alternating with connecting lines and terminated at both ends by fixed resistors. Chapter 5 illustrates the method by carrying through a 9-pole design and experimentally verifying the design in stripline form. Appendix II gives the FORTRAN IV program required to carry out such a typical design. A practical concern in the design of distributed circuits is the unavoidable resonances which occur because of the multiple resonant properties of each distributed element. These resonances give rise to poles and zeros of the network response which are not directly specified and which may disturb the desired response. The question of locating the nondominant poles and zeros is treated in Chapter 6. 1L. Weinberg, Network Analysis and Synthesis, McGraw-Hill, New York, 1962, pp. 600-631. R. Saul and E. Ulbrich, "On the Design of Filters by Synthesis, " IRE Trans. on Circuit Theory, Vol. CT-5, No. 4, Dec. 1958, pp. 284-327. vi

FOREWORD (Cont.) There, a method for finding the pole and zero locations is developed; Appendixes III and IV give FORTRAN IV programs which carry out this process for the particular filter form of Chapter 4. Chapter 7 presents one method of utilizing some of the available degrees of freedom in the distributed network to minimize the effect of non-dominant poles on the performance of a lowpass filter. This should be considered as a fruitful first step toward the optimization of a filter design; it appears that much more can be done in the future. Finally, Chapter 8, makes a start on the interesting theoretical question of the existence of solutions of the problem: Given line impedances, and load impedances, what line lengths are required to realize a specified set of dominant poles? vii

viii

Table of Contents Page Abstract iii Foreword v List of Illustrations xii List of Appendices xvii List of Symbols xviii Chapter 1: Introduction Chapter 2: Literature Survey 4 2.1 Filter with Cascade Elements 5 2.2 Filters with Parallel-Coupled Lines 12 2.3 Filters with Stubs and Richards' Transformation 17 Chapter 3: Mathematical Derivations 20 3.1 General Circuit Parameter Matrix 20 3.2 A System of Nonlinear Equations 25 Chapter 4: Realization of Prescribed Poles and Zeros 29 4.1 Generation of the System of Nonlinear Equations 29 4.2 Newton-Raphson Method 32 4.3 Approximate Solution 35 4.4 Refinement of the Solution by Newton-Raphson Method 40 4. 5 Realization of Characteristic Impedances Within a Prescribed Bound 40 4.6 The Realization of Transmission Zeros 44 Chapter 5: Design of a Nine-Element Transmission Line Network 47 5.1 Determination of Dominant Poles and Zeros 48 5.2 Realization of Zeros and Poles 50 5.3 Modified Design 58 5.4 Construction 60 ix

Table of Contents (Cont.) Page Chapter 6: Nondominant Poles 64 6. 1 General Consideration 6.2 Transmission Poles and the Zeros of Modified GCP 69 6. 2.1 ju-Axis Zeros of G1 Am + G2 D 71 6. 2. 2 jw-Axis Zeros of G1 G2 Bm + Cm 72 6.2.3 jw-Axis Zeros of G1 Am + G2Dm and G1 G2 Bm + Cm Are Their Only Zeros 72 6.3 No Repetition Interval is Free of Zeros of Am, Bm, Cm, Dm 74 6.4 Exactly One Zero of Each GCP in Each Repetition Interval 83 6. 5 Numerical Evaluation of Transmission Poles 91 6. 5. 1 Derivatives of the Zeros of (6. 58) with Respect to K 93 6.5.2 Determination of WAD or wBC 95 6. 5.3 Transmission Poles 96 6.6 An Example 97 6.7 Sensitivity of the Zeros of Am, Bm, Cm and Dm with the Variation of Line Lengths 101 Chapter 7: Optimization 105 7.1 Mathematical Formulation 105 7.2 Solution 108 7.3 Calculation of Derivatives 112 7.4 Example 116 Chapter 8: Existence Consideration for the Solution of the Two-Element Transmission Line Networks 123 8.1 Variation of the Input Admittance of a ResistanceTerminated Transmission Line 126 8.2 Variation of the Admittance of an Open Shunt Stub in Shunt with a Resistor 130 8.3 Condition for Existence of Solution 132 8.4 Example 134 Chapter 9: Summary 138 x

Table of Contents (Cont.) Page Appendices 140 Bibliography 170 Distribution List 176 xi

List of Illustrations Figure Title Page 2.1 The periodic nature of the frequency response of transmission line filters illustrated by the Butterworth attenuation of a half-wave filter 6 2. 2 A bandpass filter containing admittance inverters Ji (i=1,2,...,n) 6 2.3 A bandpass filter containing impedance inverters Ki (i=l,....,n+1) 8 2.4 Broad-band impedance-inversion networks 8 2. 5 Direct coupled-resonator filters (a) in strip transmission line (b) in waveguide 9 2.6 Ouarterwave stepped impedance transformer 9 2.7 Halfwave filter 11 2. 8 Equivalent circuit of an impedance step 11 2.9 Shielded parallel coupled stripline 13 2.10 Parallel coupled strip transmission line filters (a) with open-circuited sections (b) with short circuited sections 13 2.11 Effect of folding a X0/2 resonator to make a 0 /4 resonator 15 2.12 Interdigital filter with shortcircuited lines at the end 15 2. 13 Capacitively loaded interdigital filter with ungrounded end resonators 16 2.14 A comb-line bandpass filter 16 xii

List of Illustrations (Cont.) Figure Title Page 3. 1 General structure of the transmission line networks considered in this study 21 4.1 (a) A transmission line terminated by the impedance j-l _ E + jF in G+ jH (b) An impedance zj-1 =E + jF shunted by an in G + jH open stub 31 4. 2 Input impedance zin looking into the port 1 - 1' is determined recursively 31 4.3 Approximate lumped equivalent of the distributed network in Fig. 4.2 under the assumptions (4. 18) and (4.19) 38 4.4 A nine-element transmission line network 38 4. 5 The lumped network having 9th order Butterworth transmission poles 38 4. 6 Variation of line lengths necessary to maintain Butterworth poles (n 9) as the ratio of z.. to z is decreased 43 Connecting Line to Stub i deceased 5.1 The calculated and measured attenuation of the designed nine-element transmission line filter 54 5. 2 The synthesis procedure outlined in Sections 4.3 through 4. 6 summarized 59 5.3 The physical dimensions of the nine-element filter 63 6.1 A doubly-terminated lossless 2-port 65 6.2 (a) Typical curves of modified general circuit parameters Am, Bm, Cm and Dm (b) Indicates the sections of jc-axis where the zeros of G1Am + G2Dm and G1G2Bm + Cm occur for the curves in (a) 73 xiii

List of Illustrations (Cont.) Figure Title Page 6.3 Repetition intervals and principal interval of a transmission line of length f 75 6.4 Two lossless transmission line 2-ports joined by a connecting line 75 6. 5 -x(w) in Eq. 6. 26 and tan f for one repetition v interval of the connecting line in Fig. 6.4 80 6.6 Two lossless transmission line 2-ports having an open shunt stub at their junction 80 6.7 (a) A lossless 2-port satisfying the theorem I in Section 6.4 (b) A connecting line terminated by the lossless 2-port in (a) (c) An open shunt stub in parallel with the 2-port in (a) 84 6.8 -x'(w) in (6. 56) for the part I11- 12 of a repetition interval of the 2-port in Fig. 6.7 (a) 88 6.9 Graphical representations of x'(w) + tan wc/v = 0 illustrating the three distinct relative positions of -x'(w) and tan wo/v 89 6. 10 Zeros OwA wB, wC, wD of modified general circuit parameters of either a stub or a connecting line of length I in one of its repetition intervals, verifying the occurrence of exactly one zero of each general circuit parameter in such an interval 92 6.11 A repetition interval and the associated repetition strip 92 6. 12 Typical graphs of G1Am(w) + G2Dm(w) and G1G2Bm(w) + Cm(@) 98 6. 13 A five-element transmission line network 98 xiv

List of Illustrations (Cont.) Figure Title Page 6.14 The zeros of G1Am+G2Dm and G1G2Bm+Cm are identified by WAD and cBC respectively and the nondominant poles of the 5-element network in Fig. 6.13 are identified by NP's, together with the repetition interval of each line 100 6. 15 The computed attenuation of the 5-element network in Fig. 6.13 102 7.1 A 2-element transmission line network 117 7.2 Attenuation of a 2-element distributed filter 122 8.1 A doubly terminated 2-element transmission line network 125 8. 2 A 2-port network shown bisected into two 2-ports 125 8.3 The locus of the extremity of e2S c for 0 < Tc < 7r/2y 128 2sT c 2S~cT 8.4 (a) Vectors e2Sc + a and e 2S a with 0 < a < eX/Y and their associated angles 3 and a (b) Typical representation of Yc with 0 < a < e7X/y 128 8.5 (a) Vectors e2c + a and e2S a with 0 < e7X/Y < a < 1 and their associated angles f3 and a (b) Typical representation of Yc with e7X/ < a 129 8.6 (a) Vectors e 2S - a and e 2S+a with O < -a < e7X/ (b) Typical representation of Yc with 0 < -a < enx/Y for 0 TC 7r/2y 131 xv

List of Illustrations (Cont.) Figure Title Page 8.7 Typical representation of Yc with 0 < ex/Y <-a for 0 < T' r/2y 131 2ST 2ST 8.8 (a) Vectors e s+1 and e - 1 and the 2sT locus of e s for 0Z T Z 7T/2y (b) Typical representation of y for 0' T r/2y s 133 8.9 Plots of -y for 0 <'Tc' r/2y, and ys for O= C 7? TF2y 137 I.1 (a) A lossless 2-port having n-1 transmission line elements in cascade (b) The network in (a) augmented 141 II.1 The block diagram of the program 146 III. 1 The flow- sheet of the program RTAPR 157 IV. 1 The flow- sheet of the program RTLOC 163 xvi

List of Appendices Page Appendix I: Proof of The Expressions Given in Eqs. 3.4 Through 3.7 140 Appendix II: A Program to Solve The System of Eqs. 3.17 And To Decrease zConnecting Line/zStub Automatically 145 Appendix III: A Program For The Numerical Evaluation of WAD and wBC 156 Appendix IV: A Program For The Numerical Evaluation of Transmission Poles 162 xvii

List of Symbols Equation Where Symbol The Symbol Remarks Is First Used a (8. 5) Reflection coefficient ai (3. 4) Coefficients in a cosine series a. (L 1) A (3. 1) General circuit parameter A (6. 2) Modified general circuit parameter m1 A (6. 16) ~2 A (I. 1) Ab (6. 46) mN Ac (6. 50) mN At (3.3) General circuit parameter of a lossless network AT (3. 10) General circuit parameter of a doubly terminated lossless network AT (3. 10) bi (3. 4) Coefficients in a sine series bi (L 1) B (3. 1) General circuit parameter Bm (6. 2) Modified general circuit parameter xviii

List of Symbols (Cont.) Equation Where Symbol The Symbol Remarks Is First Used B (6. 16) B1 B (6. 16) B' (L 1) m Bb (6. 47) mN Bm (6. 51) mN Bt (3. 3) General circuit parameter of a lossless network BT (3. 10) General circuit parameter of a doubly terminated lossless network BT (3. 10) c. (3. 4) Coefficient in a sine series CI (I. 1) C (4. 16) Capacitance C (3. 1) General circuit parameter.th C. (4. 19) Capacitance of j element Cm (6. 2) Modified general circuit parameter C (6. 16) m1 C (6. 16) m2 C' (I. 1) m(L xix

List of Symbols (Cont.) Equation Where Symbol The Symbol Remarks Is First Used Cb (6. 48) mN C C(6. 52) mN Ct (3. 3) General circuit parameter of a lossless network CT (3. 10) General circuit parameter of a doubly terminated lossless network CT (3. 10) d. (3. 4) Coefficients in a cosine series di ( 1.1) D (3. 1) General circuit parameter D (6. 2) Modified general circuit parameter D (6. 16) m1 D (6. 16) m2 D' (I 1) Db Db (6. 49) mN Dc (6. 53) mN Dt (3. 3) General circuit parameter of a lossless network DT (3. 10) General circuit parameter of a doubly terminated lossless network xx

List of Symbols (Cont. ) Equation Where Symbol The Symbol Remarks Is First Used DT (3. 10) E (4. 1) Used to define the impedance (E + jF)/(G + jH) fo Quarterwave frequency in hertz f. (3. 16) Functions i= 1,..., n fiJ) (4. 9) Functions f. evaluated at x. f(J) (4. 10) Derivates of fi with respect to xk evaluated at xi F (4. 1) Used to define the impedance (E + jF)/(G+ jH) G (4. 1) Used to define the impedance (E + jF)/(G + jH) G1 (3. 10) Resistive termination of a network G2 (3. 10) Resistive termination of a network H (4. 1) Used to define the impedance (E + jF)/(G + jH) J (7. 1) Number of nondominant poles under consideration for optimization k (3. 9) Integer k. (7. 1) Positive weights K (6.9) K Odd integer xxi

List of Symbols (Cont. ) Equation Where Symbol The Symbol Remarks Is First Used Qim ~ (3. 1) Length of a transmission line element I. (3. 3) Length of ith transmission line ^~~~~1 ~element (i = 1,..., n) I (4. 26) Length of a stub s L (4. 14) Inductance.th L. (4. 18) Inductance of j element m (5.6) M (5.6) n Number of transmission line elements, number of dominant transmission poles R (8. 3) Resistive termination R (8. 4) Resistive termination s Complex frequency s. (3. 15) Prescribed transmission poles l^~~ ~(i = 1,..., n) Ske (5. 6) skd (5. 6) T (5. 1) Chebychev polynomial of the first n kind of order n v (3. 1) Velocity of propagation x1,..., xn (4. 5) Dummy variables that define the functions f. xxii

List of Symbols (Cont.) Equation Where Symbol The Symbol Remarks Is First Used x,..., x (4. 27) Dummy variables that define the functions f. (j) th x.) (4.6) x. at the conclusion of j iteration x(s) (6.22) Reactance function x'(s) (6. 54) Reactance function z (3. 1) Characteristic impedance z. (3. 3) Characteristic impedance of th transmission line z12 (3. 11) Transfer impedance z. (4. 3) Input impedance Zin (4. 1) zj (7. 13).th ZM. (7. 3) Upper bound on the j characteristic j impedance th z (7. 3) Lower bound on the j characteristic m. j impedance z. (7. 32) Zc (8. 3) Characteristic impedance of a connecting line z (8. 4) Characteristic impedance of a stub.th c. (7. 34) Real part of j nondominant pole.th Yj (7. 34) Imaginary part of j nondominant pole xxiii

List of Symbols (Cont.) Equation Where Symbol The Symbol Remarks Is First Used w Real frequency in rdn/sec a (5. 1) Start of stopband a oc (5. 1) Cut-off frequency A Zero of A A m CWB Zero of B B m ( C Zero of C WD Zero of D wr Zero of cos (col/r) nearest to the origin OAD (6.61) Zeroof G1A +G2Dm WBC (6.66) Zero of G1G2Bm + Cm X X = tanh (s/ 4f ) X() (4. 7)', (3. 1) ~ = sL/v (3. 3) i = si/v n n- 1 1 o (3. 4) i'= I, 2 1 A set containing elements 6, i~ 2-1 & i= 1..., j 1 (3. 12) Any transmission pole xxiv

List of Symbols (Cont. ) Equation Where Symbol The Symbol Remarks Is First Used a (6.68) Cr( ~ (6.71) Zero of x(w) + tan ({w/v).th 1. (7. 1) Imaginary part of j nondominant pole "j (7. 39) i^ (.(7. 19).th u. (7. 1) j nondominant transmission zero v.1 (7. 19) 6 (7. 21) ~~~~i i i+I i A Tk (7. 16) A7Tk k - Tk T. (7.2) 7.= 1i/v 1 1 1 i.th Tk (7. 13) Tk at the conclusion of i iteration Tp (7. 40) TC (8. 3) Delay of a connecting line T (8. 4) Delay of a stub T (8.6) xxv

List of Symbols (Cont. ) Superscript Symbols Subscript Symbols i a (j) c I i j k k d k e m p s xxvi

Chapter 1 Introduction Network functions that characterize a lossless transmission line section can be expressed in terms of a single function tanh(^-, where s = a + jw is the complex frequency, and f0 is the frequency, in hertz, at which the length of the transmission line element is a quarter wavelength. Using the transformation X = + jQ = tanh(4f) Richards [1] showed that distributed circuits composed of lumped resistors and equal length transmission line elements can be treated exactly as lumped networks in the new variable X. Thus all the power of conventional lumped parameter synthesis techniques is made available to the designers of distributed parameter networks. Richards' transformation has been successfully employed for the analytical design of distributed networks by Ozaki and Ishii [2], Horton and Wenzel [3], Wenzel [4], and many others. While this transformation has been a powerful tool for the analytic design of distributed structures, techniques based on it have certain weaknesses. Foremost among these is the requirement that all the transmission line elements have commensurate lengths. One half of the available degrees of freedom are thus fixed for analytic convenience, and all of the design control rests in the characteristic impedances of the 1

2 transmission line elements. This can lead to practical difficulty since the range of feasible characteristic impedance is much less than the element value range available with lumped inductances and capacitances. A microwave network structure with no restriction on line lengths has been considered by Kinariwala [5]. He has derived necessary and sufficient condition for a function to be the input impedance of a circuit made up of sections of unequal length transmission lines in cascade and terminated in a lumped resistor. He also gives a procedure for the synthesis of such a cascaded structure from an input impedance satisfying the given conditions. The approximation problem for such a cascade network is still unsolved, as are the synthesis and approximation methods for a more general structure. Although analytic design methods for distributed networks without the a priori restriction of commensurable line lengths are yet to be found, the high-speed computational capability of a large digital computer makes feasible the iterative design of such structures. The purpose of this investigation is to study iterative synthesis of distributed parameter networks to realize a prescribed set of dominant transmission poles and zeros. The synthesis technique has been delineated with reference to the particular uniform transmission line structure which operates in TEM mode and consists of open shunt stubs alternating with connecting lines. For this structure the lower

3 and upper bounds of the characteristic impedances are specified for the transmission line elements with no restriction on their lengths. The methods developed are also applicable to other microwave structures, such as, series shorted stubs or shunt lumped capacitances alternating with connecting lines.

Chapter 2 Literature Survey The literature on distributed networks useful as filters is vast. This survey outlines the principal developments in microwave structures operating in TEM-mode, and does not claim to be complete. The microwave filter design techniques may be classified into two broad categories: approximate procedures, and exact methods, based on Richards' transformation [1]. In the former approach, an appropriate, exact lumped prototype is modified in such a way that it can evolve into a distributed network, when sections of the modified lumped design are replaced by their approximate functional equivalents in distributed elements such as resonant cavities or irises, quarterwave transformers or other impedance inverting configurations, strip lines, parallel coupled lines. This equivalence between a lumped section and its counterpart in the distributed domain is, in general, achieved by comparing them at an important frequency, for example, the frequency at band center. In the exact methods, a distributed filter is directly synthesized starting with an appropriate network function. Even with these methods, however, the mathematical model describing transmission line elements neglects higher order modes and does not take into account the fringing at the junctions of elements. 4

5 Approximate methods are numerically less cumbersome than the methods based on Richards' transformation, which, however, have the advantage of being theoretically exact. Since microwave structures designed according to either technique consist of commensurate elements, their responses are periodic (Fig. 2. 1) in the real frequency domain, the principal period lying between -f0 and fo, where f0 is the frequency at which the length of each element is one quarterwave long. In practice, at high enough frequencies, the effect of higher order modes will be considerable and the actual response will deviate from being periodic. 2. 1 Filter with Cascade Elements A lumped bandpass filter may consist of shunt parallel-resonant branches alternating with series-resonant branches in series. To obtain a microwave bandpass structure, Fano and Lawson [6] and Mumford [7] transformed such a bandpass lumped network into a configuration where shunt anti-resonant branches are separated from each other by admittance inverters (Fig. 2. 2). The function of the impedance inverters are twofold. They separate the resonators; and convert a series resonant branch into a shunt resonator thus allowing the modified network to have only one kind of resonator. A quarterwave coupled microwave bandpass filter [6], [7] was thus obtained when admittance inverters were realized by quarterwave transformers and the shunt resonant branches by waveguide cavity resonators. A

0 f0 2f0 3f0 4f0 f 0____ Fig. 2. 1. The periodic nature of the frequency response of transmission line filters illustrated by the Butterworth attenuation of a half-wave filter R 2 R3 n+1R R1 Fig. 2. 2. A bandpass filter containing admittance inverters Ji (i = 1,2,..., n)

7 quarterwave long transmission line section is an exact inverter at only a single frequency; thus quarterwave transformer coupled filter designs are useful only for narrowband (10 percent). A somewhat similar but more general approach was taken by Cohn [8] in designing direct-coupled resonator filters. With the interposition of impedance or admittance inverters, a bandpass structure obtained from a lowpass prototype is modified into a hetwork which has either only series resonant branches alternating with impedance inverters (Fig. 2. 3) or only shunt anti-resonant branches alternating with admittance inverters (Fig. 2. 2). Instead of quarterwave transformers, suitable combinations (Fig. 2. 4) of microwave circuit elements are used as inverting networks. Cohn thus realizes direct coupled filters which are either capacitively coupled, realizable in stripline resonators (Fig. 2. 5) placed end to end, or inductively coupled, realizable in waveguide cavity resonators (Fig. 2. 5) separated by inductive irises. The impedance or admittance inverting networks used by Cohn have better properties than a single quarterwave line; thus the design procedure is useful for filters having bandwidths of the order of 30 percent. Quarterwave stepped impedance transformers (Fig. 2. 6) are widely used as impedance matching networks. Design equations for stepped impedance transformers having up to four cascade quarterwave sections were first derived by Collin [9]. An exact synthesis

8 1{1 _____________ 1n+1 R2 Fig. 2.3. A bandpass filter containing impedance inverters K. (i=1,...,n+1) C -C -C -C 1 T T-C C L -L -L o-L -L L T — r Ce -— n0o "+01F l C0 0 o-'..... Fig. 2.4. Broad-band impedance-inversion networks

9 Gen r E!! Z l J _ _ -- Load (a) Gene |.''; — Load "en-I I 1___.1 I (b) Fig. 2.5. Direct coupled-resonator filters (a) in strip transmission line (b) in waveguide X0/4 t- r-.... Z= z z zz"-R z 1F2 2.3 n n+1s Fig. 2.6. Quarterwave stepped impedance transformer

10 of stepped impedance transformers consisting of any number of elements has been given by Riblet [10]. Halfwave filters (Fig. 2. 7) that find their main application as lowpass or bandpass filters are an extension of quarterwave transformers. Structurally, halfwave filters differ from quarterwave transformers in one important aspect, namely, whereas the impedance of each quarterwave section increases or decreases monotonically along the length of the transformer, the impedances of halfwave sections oscillate about a mean value. Halfwave filters can be synthesized directly from an appropriate insertion loss function [ 11]. They can also be derived from a quarterwave transformer which is used as a prototype [12, 15]. Young [12] has shown that if, in a quarterwave transformer every other step of reflection coefficient r is replaced by its equivalent (Fig. 2. 8) consisting of a step of reflection coefficient -r flanked on either side by a 90 degree long line, the quarterwave transformer expands into a halfwave filter. Young [12, 13] has used a quarterwave transformer as a prototype for designing the direct-coupled resonator filters [8]. Both are examples of transmission lines loaded at intervals. Young [12, 13] has shown that the impedance steps in a quarterwave transformer can be replaced by inductive obstacles (or capacitive gaps) having the same reflection coefficients as those of the impedance steps. This design is also approximate since the equivalence of an obstacle and an

11 -~ 0/ 2 _ --— I z^ z Z R _ z_ Fig. 2.7. Halfwave filter r r T790~ ~900 rO Z1 z2 - Z2: C,' v, 0 Fig. 2. 8. Equivalent circuit of an impedance step Impedance ratio: z' /z- = z2/z Reflection coefficient: F' = -r

12 impedance step is exact only at a single frequency. Tables of element values for the design of quarterwave transformers have been given by Young [14] and for the design of halfwave filters by Levy [11]. 2. 2 Filters with Parallel- Coupled Lines Filters have been designed using parallel coupled transmission lines [17-23]. Mathematical expressions that describe the network of two transmission lines (Fig. 2. 9) coupled along their sides have been derived by E. M. T. Jones and Bolljahn [16]. This design differs from the usual end coupled strip configuration in that successive halfwave long strips are parallel coupled along a distance of a quarter wavelength (Fig. 2. 10). In contrast to direct coupled structure where the dimension of the gap between adjacent halfwave resonators becomes critical for relatively wide bandwidth, coupling the strips along their sides permits wider and less critical gaps. However, coupling is no longer capacitive since the overlapping lengths are one-quarter-wave long at band center and phase varies along them. Design formulas that are theoretically exact only in the limit of zero bandwidth, but give tolerably good results for bandwidths of up to about 30 percent have been derived by Cohn [17 ]. Formulas derived by Matthaei [18] for the design of parallel coupled stripline filters avoid narrowband assumptions, but are also approximate. They give good accuracy from narrow bandwidths to bandwidths of about 2 to 1. Exact design methods

13 < Fig. 2.9. Shielded parallel coupled stripline i x/4 /4 X/4 ZotZ I (a) I -~ l I n L Oz0 n+1 (b) (b) with short circuited sections 17, 18] 3 (b) Fig. 2.10. Parallel coupled strip transmission line filters (a) with open-circuited sections (b) with short circuited sections [17, 18]

14 for these filter structures, based on Richards' transformation have been presented by Ozaki and Ishii [19] and others [4, 24. Matthaei [20] has shown that for a halfwave long resonator, with both ends grounded, the voltage or current distribution along it does not undergo any change when the resonator is folded in the middle to make a quarterwave long resonator one end of which is grounded and the other open (Fig. 2. 11). Matthaei [20] thus converts a parallel coupled resonator filter into an interdigital filter (Fig. 2. 12). Such filters consist of quarterwave resonators that are parallel coupled, with alternate ends of the resonators grounded and the opposite ends open circuited. An exact design theory for interdigital filters and related structures is given by Wenzel [21] The digital resonators of an interdigital filter which are each a quarterwave long at band center can be made shorter, thus making the filter more compact, when loading capacitances are added at the open circuited ends of the resonators (Fig. 2. 13), as has been shown by Robinson [22]. The capacitive loading also moves the first spurious response further away from the center frequency of the principal passband. Similar in many ways to the capacitively loaded interdigital filter is the comb-line bandpass filter (Fig. 2. 14) developed by Matthaei [23]. The resonators in these types of filters consist of TEM mode transmission line elements that are short circuited at one end and have

15 I I _ AX0 /2 a',lb a 1' (a) o /4 (b) Fig. 2. 11. Effect of folding a X0/2 resonator to make a h0/4 resonator Line number —o- 0 1 X/4 YA YB= YA Fig. 2.12. Interdigital filter with shortcircuited lines at the end

16 C ep3 - - - C ~Cn Line number 4 =. I IC n-1 I Terminating Line Y^nA Yn B Admittances Fig. 2.13. Capacitively loaded interdigital filter with ungrounded end resonators. Typically, each line is less than X/4 long?C2 C3 Cn-I 4 C Line number -- 12 3 n nl YA YB Fig. 2. 14. A comb:-line bandpass filter [23]

17 a lumped capacitance between the other end of each resonator element and the ground. The capacitive loadings are essential for the function of the filter, since magnetic and electric coupling effects would cancel each other when the resonators are all open circuited at one end and grounded at the opposite end. The resonator elements are typically one-eighth of a wavelength long at the center frequency of the primary passband. Thus, a filter of this type is compact and has its first spurious band center around four times the frequency of the center of the primary passband. 2. 3 Filters with Stubs and Richards' Transformation Filters that make use of stubs are designed by techniques based on Richards' transformation [1]. Richards has shown that with the transformation X = tanh s/4f0, s-plane distributed networks consisting of lumped resistors and commensurate transmission line elements that are a quarterwave long at the frequency f0, can be treated exactly as a lumped network in the new variable X. A shorted or open stub is thus transformed into a X-plane inductor or capacitor respectively. A one quarterwave long connecting section of transmission line, henceforth referred to as a unit element, does not have an s-plane single lumped counterpart; it produces transmission zeros of order 1/2 at ~ 1 in the X-plane. Richards [1, 30] has derived the conditions for removing a unit element from an input impedance or admittance

18 function. In the absence of a transmission zero associated with the input impedance, at X = ~ 1, the removal of a unit element produces a phase rotation and does not result in the reduction of degree of the input impedance [29]. Ikeno [31] has given a synthesis procedure for the structure consisting of stubs and connecting line alternating. The same structure has been considered, from a different point of view, by Kuroda [34], who derived and made use of what are now known as Kuroda's identities. A detailed discussion of these identities may be found in [4 ]. The procedure developed by Ikeno and Kuroda, aided by Kuroda's identity, has been further extended by Ozaki and Ishii [2], Wenzel [4], Ozaki and Ishii [19], Grayzel [29] and many others [24, 25, 27, 28, 31-39]. In the distributed structures that result from a straightforward synthesis of transmission functions that have been developed for s-plane lumped networks and therefore do not possess transmission zeros at + 1, unit elements do not occur naturally; they must be incorporated into the structure as a means of relaxing the constraints on fabrication. Such unit elements produce only phase rotation and do not contribute to improving the magnitude of the response. In this sense, they are redundant. Transmission functions for the design of "optimum" filters whers where the connecting unit elements also contribute to the filter response have been derived by Horton and Wenzel [3] for both

19 Butterworth and Chebychev response in the passband. Synthesis of filters which realize such transfer functions are also presented. Original and independent Japanese contributions to the design of such "optimum" filters and other microwave structures are reported to be considerable. In a seminar on the design of distributed networks given at The University of Michigan, Professor Fujisawa cited works in this area by Japanese researchers [31-39].

Chapter 3 Mathematical Derivations The physical structure of the transmission line networks under investigation in this study is depicted in Fig. 3. 1. Analytic expressions for the elements of the general circuit parameter matrix of the lossless part of this network are obtained in Sec. 3. 1. Other network functions are derivable from the general circuit parameter matrix. The characteristic impedances and the lengths of the transmission line elements in the circuit of Fig. 3. 1 with a prescribed set of dominant transmission poles are related by a system of nonlinear equations presented in Sec. 3. 2. 3. 1 General Circuit Parameter Matrix The network configuration, shown in Fig. 3. 1, consists of n transmission line elements. In this structure open shunt stubs, numbered 1, 3, 5,..., n, alternate with connecting lines, numbered th 2, 4,..., n - 1. The i element is assumed to be completely described by two parameters: its characteristic impedance z. and its length f. The general circuit parameter matrices for an open stub and a connecting line are respectively given by'This assumption neglects fringing effects and higher order modes. 20

21 n-1 2 G1 7 G2 n n-2 3 Fig. 3. 1. General structure of the transmission line networks considered in this study. The length f. and the characteristic impedance z. characterize the jth transmission line

22 A B 1 0 z cosh oD 0 (3. 1) z cosh cD C D i tanh 1: I sinh z z cosh cj z and A B z cosh D z2 sinh (2 (3. 2) C D sinh w z cosh eJ where z = characteristic impedance of the element, = length of the element, v = velocity of propagation, sQ = i v The general circuit parameter matrix, At Bt Ct Dt for the lossless part of the network in Fig. 3. 1 is the product of individual matris hrtrtizing each element and therefore can be written as

23 At Bt z cosh n 0 t t n n n-1 n 1 sn1 O inh 1 z cosh n1 z2 cosh ~2 z 2 sinh c sinh <>2 Z2 cosh 2 Zn —1 COSh 1 nz cosh <1 1-1 sinh ~1 z cosh ^ where i= skei/v I i1=2in-1 n1, 1,n Let {O j} be the set of all the different sums that can be formed out of (f1/v ~ 2/v~... f Q /v) retaining all the terms, (+ 1/v, (3.3) 2 %/v,..., VI Q j/v), but choosing either of the two signs associated term be designated i i -,2,. where ^.i = s i 1, 2,,..., -1 1~'' "

24 It has been shown in Appendix I that the elements of the left hand matrix in (3.3) can be expressed as 2n-1 At = n ~ n * i a.cosh (0ns) (3.4) rI z. l cosh. i. 2n-1 Bt = z. o Z b sinh (ns) (3.5) H1n ]n.~~~ 1 i= 1 1 Z. n cosh'. i=l ii=l, 3,... n-i 2 nn Ct. n Z c sinh(Ons) (3.6) i=1 i Z. fr cosh. i=l i=l, 3,... 2n-i -Dt = ~*1 d. cosh ( s). (3.7) U z. r cosh i Ei=s i=, 3... Equations 3. 4 through 3. 7 demonstrate that At, Bt, Ct and Dt all have the same poles given by n sQ. cosh = 0 (3.8) i=1, 3,... v

25 i.e., 92ki - i 1, 13, 5,..., n S-2k- 1 (3. 9) i -k = 1, 2, 3, 4,... Thus poles of At, Bt, Ct and Dt occur at those frequencies at which the length of each stub is one quarterwave long or an odd multiple thereof. The zeros of At, Bt, Ct and Dt are not as straightforward to determine as the poles. The positions of these zeros are investigated in Chapter 6. 3.2 A System of Nonlinear Equations The characteristic impedances and the lengths of the transmission line elements that realize a prescribed set of transmission poles are the solution of a system of nonlinear equations. In this section, this system of nonlinear equations is derived. The general circuit parameter matrix for the whole circuit in Fig. 3. 1 is the product of individual matrices representing each circuit element, which may be written

26 sh- 01 Ok, (co~s vn V n z V V G___ n_____ 2 cosh n- z sinh V cosh~Z z s inh ~ v nlv v 2 v 1 ^~n-l n-l 2 sinh cosh sinh- cosh J n-IV V 2 v v L -n cosh 1 0 1 0 1 V s'i cosh ~ S S ~nV i1 osinh - cosh~ G 1 z1 v v 2 A B Al B' T T nT T i= in ech3 (ni (3.10) C D C' D' T T T T Since the transfer impedance is given by 12 C cosh 1 z,1,' s n (3.11) T C% ^ ^qp n = -Tec 3.0

27 transmission poles are the zeros of C'. To emphasize that C' is a function of lengths of the line elements, (1'.2''... ), the 1'^ n characteristic impedances of the line elements, (zI, 2,..., zn ) and the complex frequency s, it can be written as Ct (1,..., in; z 1,.., z; S) If ~ (and its conjugate) is a complex transmission pole of the network in Fig. 3. 1, one gets C(l1,...,' n; Z "'ll " z; ) = 0 (3. 12) C n b CT being a complex-valued function, Eq. 3. 12 can be separated into two real equations Re [C(1,', n; 1'z..., zn;)] = 0 (3.13) Tm[C' (B,., n Im[C( 1'; z.. z';)] = 0 (3.14) If 5 is real, C' is real, and in that case Eq. 3. 12 is a real equation. A complex pole and its conjugate results in two equations and a real pole results in one equation. Realization of n transmission poles at prescribed locations si, i = 1, 2,..., n requires that CT(1,...,;;Z1,..., Zn; Si) = O; i = 1, 2,..., n. (3.15)

28 Following the procedure outlined above the system of Eq. 3. 15 may be written in the form fi(1, - Z1... Zn)= 0; i =1,..., n. (3. 16 The functions f. are the real and imaginary parts of C? at the 1 T complex poles and C' itself at the real poles. Assuming that the prescribed set of n pole locations are given by * * * s s =S; s 5 = s;..; s s s s (real), 1' 2 1=;3' 4 = 3; n-2' Sn-l Sn-2;n(e the system of equations (3. 16) can be explicitly expressed as f = eIm[ C (,l..., n; z,..., z; s )] = 0 f Re[C(1'...,;z1..., z s; )] 0 2 T 1 n I n 3 f = Im[C'(k,...,;z...; S- 0 n- T 1' n 1 n n-2 2 0 f= -ImC ( 1 * *..n' Z' Z n; sn) O f d = Re[

Chapter 4 Realization of Prescribed Poles and Zeros In Section 3. 2 a system of equations has been outlined; a solution of these equations specifies a circuit of the type shown in Fig. 3. 1 to realize a prescribed set of dominant poles. In this chapter the generation of this system of equations is outlined in more detail in Section 4. 1. Section 4. 2 discusses a numerical technique for solving this system. Explicit iteration formulas are given. Since the successful convergence of most of the iterative numerical techniques is dependent on the point where the iteration is initiated, a good approximate solution of the system is derived in Section 4. 3. The exact solution that results from the iteration of this approximate solution is not, in general, practically realizable. A method for attaining a practically realizable solution is described in Section 4. 5. The computer program that has been developed to realize a prescribed set of poles is listed in Appendix II, together with a note on its capability and use. Section 4. 6 outlines a method for realizing the prescribed zeros. 4. 1 Generation of the System of Nonlinear Equations Functional dependence of all the f. in Eq. 3. 17 on the lengths and characteristic impedances of the line elements could be found 29

30 explicitly from the multiplication of the matrices in Eq. 3. 10. Such a determination of analytic expressions for the functions f. in terms of P. and z. would be tedious and lengthy. Since the 1 1 ability to solve this system of equations only requires that it be possible to evaluate the individual functions at given values of C. and zi, a digital computer can be programmed to accomplish this, even though explicit analytic expressions are not developed. A computer algorithm for evaluations of the functions f. is described in this section. The functions f. are the real and imaginary parts of C' at the prescribed poles. CT is calculated conveniently from the input impedance at the port 1-1', as indicated in Fig. 4. 2, because z. = A'/C. Evaluation of z. proceeds in the following way. The input impedance of a transmission line terminated in E+jF an impedance G+jH as shown in Fig. 4. l(a), can be manipulated G+jH into the form (E+jF) cosh (s /v) + (G+jH)z. sinh (sj./v) zin j (G+jH)zj cosh (sj/v)+ (E+jF) sinh (s.v) (4. 1) The input impedance of a one port shunted by an open stub as depicted in Fig. 4. l(b), is found to be

31 jZ. Zj-z E+jF j zj-I E+jF in in G+jH in in G+jH j,Zj/ (a) (b) Fig. 4.1. (a) A transmission line terminated by the impedance j-1 _ E+jF in G+jH -b) An impedance -i E+jF (b) An impedance zi = E+jH shunted by an open stub 1 n-1 2 1 ln n-I 2 1 G Zi __n Z Zi Z2 1' n 1n Fig. 4. 2. Input impedance Zin looking into the port 1 - 1' is determined recursively. Computation starts at G2, then proceeds toward the left as Z]. (j= 1, 2,..., n) are found using (4.1) or (4.2) until port 1' is reached until port 1 - 1' is reached

32 Z (Hzj ~(E+jF)z. cosh (sk./v) in (G+jH)z. cosh(sk./v)+ (E+jF)sinh (sk./v) (4.2) =3H= I1 2 Starting with E = 1/G2, F = 0, G, H=0, Zin, Zin..., z. are computed recursively by using whichever of the in expressions (4. 1) or (4. 2) is applicable as shown in Fig. 4. 2. Finally in 1(4.3) G1 + n z. in Evaluation of C' by first finding z. from Eq. 4. 3 involves in determination of z, ( = 1, 2,..., n) using either Eq. 4. 1 or Eq. 4. 2. Each of these zi is expressed by four real numbers. In in the computer program developed to solve the system of equations (3. 17), which is described in Appendix II, C' was found from the denominator of z.in. An alternate approach would be to calculate the product of the n matrices in Eq. 3. 10, where the representation of each matrix needs eight real numbers. 4. 2 Newton-Raphson Method We are now ready to consider the solution of the system of equations generated in Section 4. 1. There are a number of numerical techniques available for the solution of a system of nonlinear equations. Discussions of these methods can be found in [40 through 45].

33 The Newton-Raphson iterative algorithm has been utilized by the writer to solve the system of equations (3. 17); an outline of the method is presented here. Let the system of equations (3. 17) be written as fi(1' ( 2l' n; Z' Zn) 0, i = 1, 2,..., n. (4.4) This system possesses 2n variables: n line lengths and an equal number of characteristic impedances. Let any n of these 2n variables be assigned known values chosen arbitrarily or otherwise. Let the remaining n variables, which are, then, the unknowns in the system, be designated by X1 X2...' X' The system of equations (4.4) then becomes fi(xl, x2,..., x) = 0, i = 1, 2,..., n. (4.5) Adopt now the following notation: j) = the value of x. at the conclusion of the th iteration, (4.6) 1 1 x(j+) (j) (j) i 1 2 n (4.7) fi (x1 x2...xn), i = 1, 2,..., n (4.8) f) f (xj) (j) ) 1 n (4.9) f) - x. Xn ) =1, 2,..., n4, (.9) 1~~~~~ "

34 f a i Xj ) *2 x)] 0 (4. 10) fxk k (xfi (1 x2, x k =1 2,.., n. (j) (j) The corrections X(i (i = 1, 2,..., n) are obtained from x (i = 1, 2,..., n) by solving the linear system k + f( =,i = 1, 2,...,n, (4. 11) k=1. k 1k I (j) (j) where Xi) are the unknowns. With i 2,.., n)given, Xwhere k, n) givenx the process of solving the linear system of equations (4. 11) to determine Xj) defines the (j + 1)th iteration. The corrections iJ), thus found, are added to x() to arrive at the new values,.(j+l) _(j) (j) x( = x + X, of the variables. The iteration is then repeated until it converges to the solution of the nonlinear system. The (0) whole process is initiated at x.. Convergence considerations 1 dictate that the point x() may not be completely arbitrary. The 1 question of convergence of the Newton-Raphson algorithm has been studied in detail by Ostrowski [40], and Isaacson and Keller [ 44]. They have shown that convergence requires that x() the point where the iteration is initiated, lie in the close proximity of the solution.

35 4. 3 Approximate Solution A method of evaluating an approximate solution of the system of equations (3. 17) under suitable assumptions regarding the lengths and the characteristic impedances is outlined in this section. Using the series expansions for cosh and tanh, the V V transmission matrixes (Eq. 3. 1 and Eq. 3. 2) for a connecting line and an open stub can be written in the forms 22 _2 3 1 + 2 2! + z s + 3! +.. V (4.12) f3 s3 f2 2 s + - + +... 1 s 3 3! + 1 2 2! V V and ~1 0 (4.13) f3 s3:S + If for the connecting line it is assumed that z - oo -0 _'O, (4. 14) and z- - L V

36 the matrix (Eq. 4. 12) in the finite frequency-plane approaches the limiting value 1 sL (4.15) 0 1 Similarly, if the open stub satisfies the conditions z - 0 O >, (4. 16) /v I z the matrix (Eq. 4. 13) in the limit becomes 1 0 (4.17) _Cs 1_ Therefore, in Fig. 4. 2, if all the connecting lines satisfy conditions similar to (4. 14), i. e., Z. - X QJ/v- 0, (4.18) = L. zjij/v = LJ j = 2, 4, 6,... and all the open stubs satisfy conditions similar to Eq. 4. 16, i. e.,

37 z. - /v -0, (4.19) lj/v =C. Z. j j = 1, 3, 5,... the general circuit matrix for the lossless portion of the distributed structure in Fig. 4. 2 becomes 1 0 1 L s 1I L2s 1 0 n-1 (4.20) Cns 1 0 1 0 1 C1S 1 The matrix product (4. 20) is the transmission matrix of the lossless part of the lumped network in Fig. 4. 3. Assuming that all the connecting lines have very high characteristic impedances, all the shunt stubs have very low characteristic impedances, and all the line lengths are electrically short, the distributed network in Fig. 4. 2 is approximately represented by the lumped network in Fig. 4. 3. It follows that the poles of this lumped network are very close to the dominant transmission poles of the distributed structure. To obtain an approximate solution of the system of equations (3. 17), the lumped network in Fig. 4. 3 is first synthesized for the prescribed set of transmission poles. With the values of

38 Ln_1 L2 G1 n G2 Fig. 4.3. Approximate lumped equivalent of the distributed network in Fig. 4. 2 under the assumptions (4.18) and (4.19) 8 6 4 2 G1=2 /77/ G2=1 1=2 2= /9 9 /7 /5 /3 /1 Fig. 4.4. A nine-element transmission line network L8 L6 L4 L2 G=2 TC9G1 G12$ _C. 9!C7 ~C5!C3 tC1^ 2 Fig. 4.5. The lumped network having 9th order Butterworth transmission poles. The element values are given in column I of Table I

39 inductances and capacitances thus determined, the lengths of the transmission line in Fig. 4. 2 are found from the relations L../v =, j = 2, 4, 6,... (4.21) Z. J /v = Cjzj, j = 1, 3 5,... (4.22) using, arbitrarily chosen, high values of characteristic impedances for connecting lines and low values of characteristic impedances for the stubs. For the examples that were worked out by the writer the values of 10 for the impedances of the connecting lines and 0. 1 for the impedances of the stubs were used in determining the lengths of the lines. This gave a good approximate solution of the system of equations (3. 17) when the terminations were of the order of unity. As an example, consider the nine-element distributed structure in Fig. 4. 4. It is required to find the approximate values of the lengths and the characteristic impedances [ i. e., the approximate solution of the system (Eq. 3. 17)] that realizes the ninth order Butterworth poles as the dominant transmission poles of this structure. The lumped network that has these prescribed transmission poles (and all the transmission zeros at infinity) is shown in Fig. 4. 5.

40 The element values of the distributed network found by using Eq. 4. 21 and Eq. 4. 22 and the values of the inductances and capacitances for the lumped network are given in columns 1, 2, and 3 of Table I. 4. 4 Refinement of the Solution by Newton-Raphson Method Once an approximate solution is obtained, this is refined to the desired degree of accuracy by the Newton-Raphson technique. For example, since the characteristic impedances are picked arbitrarily, the system of equations (4.4) is solved for the lengths of the lines. Thus the system of equations (4. 5) becomes i ('. ) -= 0, i = 1, 2,..., n (4.23) which is solved by the Newton-Raphson method starting from the approximate solution obtained in Section 4. 3. The result of applying this iteration method to the approximate solution of the nineelement distributed network in Fig. 4. 4 is given in column 4 of Table I. 4. 5 Realization of Characteristic Impedances Within a Prescribed Bound In the initial solution of the system of equations (3. 17) outlined in Section 4. 4, the characteristic impedances of the connecting

TABLE I Approximate Solution Solution Refined By From Eq. 4. 21 Newton-Raphson and Eq. 4. 22 Method C1 = 3.0223 z= 0.1 O /v = 0.30223 0.3536814 L2 = 0.9579 z = 10.0 J2/v = 0.09579 0.09084188 C3 = 3.7426 z3 = 0.1 3/v = 0.37426 0.3598843 L4 = 0.8565 z4 = 10.0 k4/v = 0.08565 0.08299981 C5 = 2.9734 z = 0.1 5/v = 0.29734 0.2877055 L6 = 0.6046 Z = 10.0 k6/v = 0.06046 0.05878263 C7 = 1.7846 z7 = 0.1 7/v = 0.17846 0.1730608 L = 0.2735 Z= 10.0 8 /v = 0.02735 0.0265765 C9 = 0.3685 z = 0.1 %/v = 0.03685 0.03653245 The capacitances and the inductances are given respectively in farads and henrys. Characteristic impedances are in ohms and delays f/v are in sec.

42 lines are impractically high and the characteristic impedances of the stubs are impractically low. However, this difficulty can be overcome. The characteristic impedances of the connecting lines can be gradually lowered and the characteristic impedances of the stubs can be gradually increased, and each time a change in the impedance (or impedances, in case more than one impedance is changed simultaneously) is made, the system (Eq. 3. 17) is solved again to find new lengths for the line elements, thus preserving the prescribed dominant poles. The iteration for this new solution is initiated with the old solution as the starting point. The variation of line lengths necessary to maintain the prescribed Butterworth poles for the nine-element filter as the ratio of characteristic impedance of connecting lines to that of the stubs was decreased is graphically presented in Fig. 4. 6. Decreasing the ratio zconnecting lines/zstubs results, in Zconnecting lines stubs general, in an increase of the line lengths. When all the connecting lines have characteristic impedance 3. 5 2 and all the stubs have characteristic impedance 0. 3 S2, the line lengths that realize the Butterworth transmission poles were found to be P1/v = 1.933716 Sec k4/v = 0.204529 Sec 12/v = 0.1635795 " 5/v = 0. 7253457 " (4.24) ~Q/v = 0.8731836 " p/v = 0.1454346 " o~~~~~l/ =.1444

1OL(4) IOL(2) O IOL(6) Cd~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~c t 2OL(9.)0^ ^ ^^ b~.O L(3) L(5) 1OL(8) cuJL(7) 0 II I IA. L I I III I I l i 1 10 100 Zconnecting lines /Zstubs Fig. 4. 6. Variation of line lengths necessary to maintain Butterworth poles (n = 9) as the ratio of zconnecting line to z stub is decreased

44 17/v = 0.4402347 Sec g9/v = 0.1061295 Sec (4 24 f8/v = 0.06485635 " Cont.) The computer program written for changing the characteristic impedances has the capability of automatically bringing down the ratio zconnecting line/zstub while finding the lengths of the transconnecting line stub mission lines needed to maintain the transmission poles at the prescribed positions. A typical run is initiated by specifying the prescribed poles, the approximate solution, and that the line lengths are to be varied. This program (listed in Appendix II) has been successfully applied to distributed structures having 2, 3, 5, and 9 elements when the transmission poles were prescribed to be the Butterworth poles. In each of these cases, impedances of the connecting lines were each decreased from 10 12 to 3. 333 Q and the impedances of the stubs were each increased from 0. 1 52 to 0. 3 1 automatically. 4.6 The Realization of Transmission Zeros Equation 3. 11 shows that for the transmission line structure being studied, the transmission zeros are given by 2k - I s = + 2k- i j7 (4. 25) 2 ~i i = 1, 3, 5,..., n k = 1, 2, 3,....

45 Thus the stubs alone are responsible for creating transmission zeros, their positions on the jw- axis being controlled by only the lengths of the stubs. Prescribed transmission zeros, generally, determine the lengths required for as many stubs as there are transmission zeros. The lengths of selected stubs can be gradually changed to the desired values to realize the transmission zeros. The resulting perturbations in the specified transmission poles- are offset by adjusting any n of the remaining parameters, i. e., the impedances of all the lines and the lengths of the connecting lines, thus preserving the pole positions. The required adjustments are obtained from the solution of Eq. 3. 17 in the following way. Let is be the length of the stub being modified to /s + As. Let the n parameters selected for adjustment to maintain the poles be designated x1, x2,..., x. Then, from Eq. 3. 17, one can n write fi(xi'2, xn,.; is ) = 0, i = 1, 2,..., n (4.26) With i changed to I + As the new solution of Eq. 4. 26 s S S denoted xi, x,.., x', can be obtained from 1 2'' n f (x1 x',.. x; s + A) =, i 2,..., n. (4. 27) Thus adjustments required in the n parameters (xi, i = 1,..., n)

46 to keep the pole locations unchanged in spite of a change in the length of a stub can be determined from Eq. 4. 27.

Chapter 5 Design of a Nine-Element Transmission Line Network A design technique to achieve a prescribed set of transmission poles for a transmission line network has been illustrated in Section 4. 5. In that example the values of a nine-element transmission line network were selected by iteration to realize the ninth order Butterworth poles as its dominant transmission poles with a specified ratio of characteristic impedances. For the transmission line network considered in this study (Fig. 3. 1), finite transmission zeros always are present because of the open shunt stubs in the circuit. Increased lengths of stubs that accompany the decrease in the ratio of zconnectig line/zstub (Fig. 4. 6) can bring the transmission zeros close enough to the origin of the s-plane to have considerable influence on the circuit response in the frequency range of interest. Such transmission zeros can be used to an advantage for improving stop band attenuation. However, to preserve the passband attenuation in the presence of transmission zeros, the dominant transmission poles must be moved to new locations. The purpose of this chapter is to present an illustrative design example of a nine-element filter where, in addition to achieving a prescribed set of dominant poles, several of the stub lengths are controlled to realize a set of specified transmission zeros. 47

48 5. 1 Determination of Dominant Poles and Zeros Consider the physical structure of the nine-element network in Fig. 4. 4. If it is assumed that only the first transmission zeros produced by each of the stubs (1), (3), (5) and (7) are significant, eight finite transmission zeros must be considered. Choosing, for example, the passband response to be maximally flat as a design criterion, the square magnitude of the transfer impedance, according to the theory of Inverse Chebychev filters [ 54], can be written as 2 [_tnT (wa/X)]_ z12( J) [ nT ( / )] 2 2 [c Tn, (5.1) where T = Chebychev polynomial of first kind of order n, n n = number of transmission poles in the lumped prototype, 2 1 - = cut-off frequency where z121 = 2 1 Wa = start of stop band where lz121 = 2 a. l+E T (w a/w) From Eq. 5.1, with n= 9, a= 1.3, c = 1 and e = 1, the a c transmission poles were found to be S, 2 = -0.112954 + j 1.002688 1, -0. 374950 ~j(5. 2) s3 4 = -0.374950 + j 1.016385

49 56 = -0. 749856 ~ j 0.984713 (5. 2 S7 8 = -1. 257454 j 0. 716270 Cont. ) s9 = -1.565015 ~ j0.0 and the eight zeros were ~ j 1.320055, ~ j 1.501110, ~ j 2.02241, ~ j 3.800946 (5. 3) which were placed to produce equi-ripple response in the stop band of the prototype. It should be noted that because of the nondominant transmission poles and zeros of the distributed system, which are not accounted for in this lumped prototype, Eq. 5. 2 represents approximations to the dominant transmission poles desired in the distributed system. Starting with the nine-element filter in Fig. 4. 4 characterized by the element values (Eq. 4. 24), which were selected for dominant Butterworth pole locations, the present design may be completed in, broadly speaking, two steps: (i) Control the lengths of the stubs to realize the desired transmission zeros and at the same time, adjust other parameters of the circuit to preserve the Butterworth poles. (ii) Make, in steps, appropriate changes in the parameters other than the stub lengths responsible for the desired zeros, to gradually move the dominant poles from the

50 Butterworth locations toward the desired poles until the poles of the network coincide with the desired ones. 5. 2 Realization of Zeros and Poles The lengths (/ v, to be exact) of the stubs required to produce the desired transmission zeros (Eq. 5. 3) are calculated to be 1. 19 sec, 1. 047 sec, 0. 775 sec, 0. 4135 sec. A designer has some degree of freedom in selecting one particular stub from the several at his disposal to realize a zero. For example, assuming the realization of the zero at ~ j 1. 320055 (desired length 1. 19 sec) under consideration, the length of any one of the stubs (1), (3), (5) or (7) may be modified to 1. 19 sec, thus achieving the desired zero at ~ j 1. 320055. Our procedure was to take each stub in the network realizing the Butterworth poles and a specified zconnecting li / zstub and examine all the desired lengths to pick connecting line stub the one which was closest to the length of the stub. The length of this stub was then gradually modified to the desired dimension. Thus, to attain the desired transmission zeros given in Eq. 5. 3, the lengths of the stubs (1), (3), (5), (7) were changed from their values 1. 933716, 0. 8731836, 0. 7253457, 0. 4402347 (in Eq. 4. 24) to respectively 1. 19, 1. 047, 0. 775, 0. 4135. While changing the lengths of these stubs appropriate changes could be made in any nine of the remaining variables [nine characteristic impedances, the lengths of the four connecting lines and stub (9)] to preserve

51 the Butterworth poles. In the computer program listed in Appendix II, any n of the 2n parameters defining the system of Eq. 3. 17 can be kept fixed, while the iteration is carried out on the remaining n parameters. In this particular example, while the lengths of the stubs were changed to desired dimensions, iteration was performed on z1, 2 z3,' 4'' 6' z7' 8' z9' i.e., these were the variables designated x1, x2,..., x9 in Eq. 4. 5. The element values that achieve the transmission zeros (Eq. 5. 3) and maintains the Butterworth poles were found to be El/v = 1.1900000 Sec Z = 0. 1930863 Q Q2/v = 0.2196553 " Z = 3.5000000" f3/v = 1.0470000 " Z = 0.3901989" 4/v = 0.1899826 Z = 3.5000000" f5/v = 0.7750000 " Z = 0.3310432" (5.4) 5 5 k6/v = 0. 1429601 Z = 3. 5000000" 6 6 f7/v = 0.4135000 " Z = 0.2752242" 8 /v = 0.06693098 " Z = 3.5000000" Qg/v = 0.1060000 Z = 0.2968349 Now consider shifting the poles of the transmission line circuit from the Butterworth positions to the desired locations. Observe that in the system of equations (3. 17), the poles also enter as parameters. Therefore, just as the stub lengths were modified to

52 desired dimensions, the pole location can be gradually moved to the desired positions. Every time a step toward the desired poles is made element values compatible with the new poles are found by solving Eq. 3. 17 anew. While this is accomplished, the lengths of the stubs responsible for the desired zeros are not allowed to change. Let n of remaining parameters, which will be treated as iteration variables during one step of the pole movement, be designated xl, x2,..., xn The system of equations (3. 17) can be written as fi(X' x2". Xn; Sk) = (5.5) i = 1, 2,..., n k = i if i odd k = i 1 if i even. Let Ske Skd where k= 1, 3,..., n if n odd and k= 1, 3,..., n- if n even denote respectively the existing and the desired pole locations. Choosing to move along the shortest path while shifting the poles from the existing to the desired location, the poles in an intermediate position are

53 Sk =ke + M kd k) (5.6) k = 1, 3,... where m and M are positive and may be selected as integers (m - M). The relative magnitudes of m and M depend on the size of the step one wishes to take. When m equals or nearly equals M, the desired pole locations have been achieved. The results of iteration as the poles were being shifted according to Eq. 5. 6 are shown in Table II. Finally, the element values given in Part 6 of Table II achieve simultaneously the transmission zeros (Eq. 5. 3) and pole locations mentioned in that part of the table. Transmission pole locations realized are within three percent of the desired ones. The attenuation characteristic of the network characterized by the element values in Part 6 of Table II is shown by the dashed curve in Fig. 5. 1. The above example illustrates the design of a distributed network from a lumped design that has eight finite transmission zeros and nine transmission poles. The distributed network which was designed to have Butterworth poles as aits dominant transmission poles was used as the starting point and following the realization of transmission zeros, the transmission poles were shifted from the Butterworth locations to the desired locations. An alternate

801 80 I' \ ~~~~~~70C~ I 0.5 1 60 J /xx 0. Jxxx /'I ^60- X RI ) l \\ i C 50 x x oX 0 C x Xi 140 x xXX / I 30ii^~~ Measured data I I (1-145 MHz) 20 - Initial iterated design 10 Modified design 10 0 2 3 4 5 Frequency in radians per second Fig. 5. 1. The calculated and measured attenuation of the designed nine-element transmission line filter

TABLE II Those parameters that were regarded as iteration variables are marked with asterisk. -D ^ XT -r~ i T ^- ^./~~~fv And z. That Achieve Part New Pole Locations /v And zi ThatAchieve The New Poles Ji/V z. i _________________________________________________ i __1__ _________ -0.161509 ~ j 0.988384 1.19 0.2379369* 1 -0.474990 ~ j 0.896094 0.2046626* 3.5 2 -0.762807 ~ j 0.711173 1.047 0.3960831* 3 -1.003245 ~ j 0.416870 0.186637* 3.5 4 -1.113003 ~ j 0. 0 0.775 0.3396033* 5 0.1389825* 3.5 6 0.4135 0.2831522* 7 0.06494685* 3.5 8 0.106 0.303761* 9 -0.149371 ~ j 0.991960 1. 19 0. 3014739* 1 -0.449980 ~ j 0.926169 0. 1966559* 3.5 2 -0.759569 ~ j 0.779558 1.047 0.4008054* 3 -1.066797 ~ j 0.491720 0.1836065* 3.5 4 2 -1.226006 ~ j 0. 0 0.775 0.3479070* 5 0.1353706* 3. 5. 6 0.4135 0.2907403* 7 0.06314698* 3. 5 8 0.106 0.3102937* 9

TABLE II (Cont. ) ^, XT ^ i T ^- ~J^l./v And zi That Achieve~ Part New Pole Locations /v And z That Achieve The New Poles J^/v z. i -0. 137232 ~ j 0. 995536 1. 19 0. 3998215* 1 -0.424970 ~ j 0.956241 0. 1975815* 3. 5 2 -0.756331 ~ j 0.847943 1. 047 0.4055608* 3 -1. 130349 ~ j 0. 566570 0.1813539* 3.5 4 3 -1.339009 ~ j 0. 0 0.775 0.3552969* 5 0.1323241* 3.5 6 0.4135 0.2975375* 7 0. 06160388* 3. 5 8 0.106 0.3161074* 9 -0.125093 ~ j 0.999112 1. 19 0. 5865872* 1 -0.399960 ~ j 0.986313 0.2152370* 3.5 2 -0.753094 ~ j 0.916328 1.047 0. 4105986* 3 -1.193902 ~ j 0. 641420 0.1814096* 3.5 4 4 -1.452012 ~ j 0. 0 0.775 0.3594772* 5 0.1305629* 3.5 6 0.4135 0.3020097* 7 0.06059191 3.5 8 0.106 0.3200997* 9

TABLE II (Cont. ) Part New Pole Locations i/v And z. That Achieve The New Poles Qi/v zi i -0.118417 ~ j 1.001079 1.19 0. 8554185* 1 -0.386204 ~ j 1. 002853 0. 250467* 3.5 2 -0.751313 ~ j 0.95394 1. 047 0.4096821* 3 -1. 228856 ~ j 0.682587 0. 1863467* 3. 5 4 5 -1.514164 ~ j 0. 0 0.775 0.3545361* 5 0. 1320657* 3. 5 6 0. 4135 0. 2996928* 7 0.06099037* 3.5 8 0. 106 0. 3187789* 9 -0.116232 ~ j 1.001723 1.19 1.119369* 1 -0.381702 ~ j 1.008266 0.250467 3.931361* 2 -0.750730 ~ j 0.966249 1.047 0.3983251* 3 -1. 240295 ~ j 0.696060 0.1863467 3.643610* 4 6 -1.534504 ~ j 0. 0 0.775 0.3439001* 5 0.1320657 3. 587569* 6 0.4135 0.2936443* 7 0.06099037 3. 564882* 8 0.106 0. 3144931* 9

58 approach would be to design the lumped network having the desired transmission poles and all the zeros at infinity. Then, following the procedure outlined in Sections 4.3, 4.4 and 4. 5 a distributed network having the desired transmission poles could be synthesized. The lengths of the stubs would then be adjusted for prescribed transmission zeros. The block-diagram in Fig. 5. 2 summarizes the synthesis procedure. 5. 3 Modified Design The transmission delay caused by the connecting lines combined with the repetitive nature of the transmission zeros produced by the shunt stubs produces transmission poles in addition to the dominant poles which are being controlled. The dip in the attenuation near w = 3. 25 rdn/sec is caused by such a nondominant pole. It is sometimes possible to approximately nullify the undesirable effect of a non-dominant pole by placing opposite it a transmission zero on the real frequency axis. A general strategy for accomplishing this can be formulated as a nonlinear programming problem. This nonlinear programming problem may then be approximately solved by successive linearization; the linear problem thus obtained at each step being solved by simplex algorithm. For this See Chapter 7.

59 Specified: n, poles, zeros, and bound on the characteristic impedances Realize the lumped prototype with all the transmission zeros at oo I Find approximate solution of the system of equations (3. 17). Refine this solution by the Newton-Raphson iteration with n lines lengths as variables 1________I | | lSolution of the system of Change the characteristic impedances equations (3. 17) by the in steps, bringing them within the Newton-Raphson iteration bound. Execute Newton-Raphson, Of the 2n parameters iteration on n line lengths to in the system, any n perserve the poles ~ can be specified as....iteration variables Determine desired lengths of the stubs Adjust the lengths of the stubs to give specified zeros. Modify any n of the remaining parameters by the NewtonRaphson iteration to maintain the poles Fig. 5. 2. The synthesis procedure outlined in Sections 4. 3-4. 6 is summarized above

60 particular example, it is observed that stub (1), in addition to creating a desired transmission zero at j 1. 32, places a second transmission zero at j 3. 96. This latter zero is very close to the desired transmission zero at j 3. 800 produced by stub (7). Stub (7) can, therefore, be utilized elsewhere. When the length of the stub (7) was adjusted to place a transmission zero at j 3. 25, the changed element values were found to be (1/v = 1.19 Sec z1 = 0.9373091 2 22/v = 0.250467 z2 = 3.609804!3/v = 1.047 z3 = 0.4040285 " -4/v = 0.1863467 " z = 3.544972 " 25/v = 0.775 " z = 0.3478956 " (5 7) 26/v = 0.1320657 " z = 3.51709 " 27/v = 0.4835 z7 = 0.3755778 " 28/v = 0.1 " z = 1.976803 " e/v = 0.106 z = 0.3387325 " The attenuation of the filter after this adjustment is shown by the solid curve in Fig. 5. 1. 5. 4 Construction The filter having the modified design was constructed in strip lines. A few details which might be of interest are given hereo The

61 frequency and impedance normalizations were with respect to 145 MHz and 25 ohm respectively. The velocity of propagation in air was assumed v = 109 x 11. 80283 in/sec and dielectric constant of circuit board material e = 2. 32. Using double stubs at each junction, the characteristic impedances and the effective lengths of the transmission line elements were computed to be 2 = 10. 121424 inch Z = 46. 865 ohm 1 1 2 = 2.130322 Z = 90. 245 " = 8.905152 Z = 20.201 " 1.584953 Z = 88.624 " 4 4 5 = 6.591684 Z = 17.394 " 5 5 = 1.123271 " Z 87.927 " 6 6 2 4 = 4.112360 " Z = 18.778 7 7 =0.850540 Z = 49. 420 8 8 2 = 0.901572 Z = 16.936 9 9 The centre-to-centre physical length of a connecting line between two stubs differs from its electrically effective length. Corrections to be added to the effective lengths to determine the physical lengths of the connecting lines were found from curves for T-junctions in [ 55]. These corrections were also verified from mathematical expressions for the parameters of the equivalent circuit of a T-junction given by Altschuler and Oliner [56]. The

62 centre-to-centre physical length of the connecting lines were thus found to be 12 = 2. 512 inch 4 = 2.190 " 2 = 1.749 6 8 = 1.407 " The physical dimensions of the filter are shown in Fig. 5. 3, which has not been drawn to scale. The circuit was etched on 1/ 8-inch polyguide circuit board deposited with 2 oz. copper. The measured attenuation of the filter is indicated by crosses in Fig. 5. 1, along with the computed attenuation presented by the solid curve.

A. 745's 10. 121. 8. 905 6.591?? 4. 112??.901" 2.512" ~ _L 2. 190" - 1. 749" 1.407 —.67"-.745 "-, 62'1.2025 Fig. 5. 3. The physical dimensions of the nine-element filter

Chapter 6 Nondominant Poles T"1 hem c,.; 3 1I k 4 ^ 4-v* I..;\ f^ e ~ - i 1-^fr* \ M4

65 G1 P Lossless G2 Fig. 6.1. A doubly-terminated lossless 2-port

66 It then follows that AT BT 1 At Bt 1 0 (6. 1) l DT G D L D 1 L T 1 D 2 where At Bt Ct D represents the general circuit parameter matrix of the lossless part of the network in Fig. 6. 1. When the lossless network is composed of open shunt stubs alternating with connecting lines, this matrix as established in Section 3. 1 is given by LAt B] A B t t m m (6. 2) 5C Dt, (z )l cosh j. C Dj where si. J v v n = number of transmission line elements,. = indicates that the product is taken over the elements which are stubs

67 and 2n-1 2n- 1 Cm D c sinh ( ) dcosh(n m i=1 di= 1sh is a modified, all-zero GCP matrix (sometimes abbreviated MGCP matrix) having exactly the same zeros as Ct D and, as an examination of Eq. 6. 2 and Eq. 3. 3 reveals, can be directly found as the product of a chain of matrices. In this chain each connecting line is represented by z. cosh 45. z. sinh Q>. z. sinh (n j, j = 2, 4,... (6.4) s inh j. z; cosh. and each open stub, by z. cosh c O0 j - 1, 3, 5y,... (6.5) sinh j. z. cosh (. J ~ ~ ~~~~~~~ Js J

68 Now, from Eq. 6. 1 CT is given by T T 1At + G2Dt + G G2Bt + Ct 1 The transfer impedance, z12 = C, can be expressed, due to Eq. 6. 2, T as n I (z ) I cosh (. ) i=l j 1 1 m+G2Dm+GiGBm+Cm (6.6) Z12 = G A + G D +G G B +C (6. 6) 1 m 2 m 1 2 m m Transmission Zeros. Transmission zeros are zeros of the numerator of Eq. 6.6 and thus satisfy si. cosh A = 0 (6.7) i.e., s = ~+ V (2k- 1) 27' (6.8) where i. designates the length of any open shunt stub and k is an integer. Expression (6.8) accounts for all the transmission zeros, since the denominator of Eq. 6. 6 contains only sums and products of sinh and cosh terms and therefore is finite throughout the finite complex frequency plane. Equation 6. 8 establishes that the transmission zeros are all on the jw-axis and are due to the stubs alone.

69 Each stub creates an infinite number of equally spaced zeros on the jco-axis at those frequencies at which the length of the stub is one quarterwave long or odd multiples thereof. 6. 2 Transmission Poles and the Zeros of Modified GCP Expression (6. 6) shows that the transmission poles are the roots of (G1A + G2Dm) + K(G1G + C ) = 0 (6. 9) for K = 1. Each of the roots of Eq. 6. 9 starts at a zero of G1 A *-I m + G2Dm when K = 0, moves as K increases, and finally coincides with a zero of G G2 Bm + C when K = co. Study of the zeros of I12 m m G1 A + G D and G1 G2 B + C is thus at the heart of the study im 2m 12m m of the transmission poles. Separation of G A + G D + G G B + C into two parts, as indicated by the parenthesis in Eq. 6. 9 is deliberate, since, as will be shown later in this section, the two parts G1 A + G D and G G B + C have their zeros all on 1m m 1 2 m m the real frequency axis and the zeros of G1 A + G D alternate with those of G1 G2 B + Cm. We begin this section with an investigation into the location and distribution of the zeros of G1 G2 Bm + Cm and G A + G D and their locations. 1m 2m Observe that Am/C, Dm/C, D/B Am/B are all reactance functions. An m m mof the reactance functions reactance functions. An important property of the reactance functions

70 is that their poles and zeros alternate on the real frequency axis. This fact alone results in the following conclusions regarding the relative distribution of the zeros of A, B C and D m ^ m ^m m (1) The zeros of A and D alternate with the zeros of B m m m and C. m (2) Of the two consecutive zeros of A and D either may m m appear before the other. (3) Of the two consecutive zeros of B and C either may m m appear before the other. Let Am(jw)= m(w), D(jw) = D (w), C (j)= jC (W), B (jw) = jB (w). Direct substitution of s = 0 in Eqs. 3. 1 and 3. 2 shows that GCP matrices for both a connecting line and an open shunt stub are unity matrices at the origin of the complex-frequency plain. This together with Eq. 6. 2 gives n A (a) = n Z, (6. 10) c-0 i=l 1 and n D() = ni z. (6.11) m =0 i=l 1 Since A (m ) and (tv) are cosine series, their slopes at = are m.e. are zero, i. e.,

71 d = 0 (6.12) mw = 0 d D () -= 0. (6.13) C= 0 Bm (w) and Cm (w) are sine series and, therefore have zero magnitude at dc. B (w)/A (w) being a reactance function, its derivative, m m d () Am dw B (w)- B (w) -d A (w) d m _ mW) da Am (W [A (w)]2 is always positive. This together with Eqs. 6. 12 and 6. 10 forces cd B (w) to be positive at co = 0. Similarly, dC() is dw m " "' dw CO = 0 positive. Typically, therefore, Am (), B (o), Cm(W) and Dm(W) have graphical representation as illustrated in Fig. 6. 2(a). 6. 2. 1 jw-Axis Zeros of G A + G2D. Since G1 and G2 are positive, at a frequency where G1A + G D = 0 A and D 1 m 2 m m m have opposite signs. Examination of the graphs of A and D in Fig. 6. 2(a) reveals that this can only happen between two consecutive zeros of A and D, not separated by any zeros of B or rmn m m C. Therefore only one zero of G A + G D will lie on the m 1 m 2 m portion of jw-axis bounded by one zero of A and one zero of D, m d Cm which are not separatedby any zeros of B and C. m Ilm

72 6.2. 2 jw-Axis Zeros of G1G2B + C. Proceeding exactly as in the case of the jw-axis zeros of G1A + GDm, it can be Im 2m concluded that one zero of G1G2B + C occurs on the portion of 12m m the jc-axis bounded by two consecutive zeros of B and C, not separated by any zeros of A or D m m For the typical curves of Fig. 6. 2(a), the intervals on the jw-axis where the zeros of G1A + G2D and G1G B + C may 1m 2m 12m m lie, are shown in Fig. 6. 2(b). Since the zeros A and D alternate with those of B and C, the zeros of G A + G D m m' m 2m alternate with those of G1G2B + C 12m m 6.2.3 jw-Axis Zeros of G1A + G2D and G GB + C 1m 2m 1 2m m Are Their Only Zeros. Consider the function G1A + G2D G G B +C (6.14) G.GZB +C 1 2 m m which, with a little manipulation, can be written as G1A +GD G1G B +C 1 1 (6.15) 1 2 m m. A D G A GD m m 1 m 2 m G2B GB C C 2m 1m M m m Since Am/Bm Dm/B, Am/ Cm are all reactance functions, it follows from Eq. 6. 15 that so is (GiAm + G2Dm )/(GG2B + C ). Therefore, its zeros must all lie on the jco-axis.

(a) A c - W WAN a wC W^]^ C oBWB i~~ l i\iK^y^ I\ C'x3 \ x^.Tl, I 1 1 1' I, A(w) —-- I I I (b)IUPI Bm (w)o-o-o (0) ~ V////'~ xxxxxx~^/I/I//~^xxxxxx' ~ WBWC WAWD B OC C) AWC WB C (w) Zeros of.XI* 0 le W o ONW 1m m+2Dm Fig. 6.2. (a) Typical curves of modified general circuit parameters Am, Bm, Cm and Dm Their zeros are designated WA, Wg, WC and WD (b) Indicates the sections of jw-axis where the zeros of GiAm + G2Dm and G1G2Bm + Cm occur for the curves in (a)

74 We have already seen that a transmission pole can be associated with one zero of G1G2B + C and one of G1A + GD each of 12m m im 2 m' which, in turn, is produced by two zeros: one each from B and C for G G2B + C, and one each from A and D for m 12 m m'm m G1A + G2D. Thus, each transmission pole is created by four zeros, one each from A, Bm C and D m' m rm m m 6. 3 No Repetition Interval is Free of Zeros of A, B,C D mm' m' m' m Let Wr be the smallest radian frequency where cos kwr/v = 0, i being the length of a component transmission line. Let a repetition interval of this line be defined as the portion of real frequency axis bounded by two consecutive frequencies which are odd multiples of r or -c. The interval between -wr and wc will r r r r be called principal interval of this line. Figure 6. 3 illustrates the repetition intervals of a transmission line element of length f. The purpose of this section is to prove that in each repetition interval of every transmission line that makes up the network in Fig. 3. 1, at least one zero of each modified general circuit parameter will appear. Consider the network in Fig. 6. 4 where a connecting line of length i (b S ) joins two lossless networks each enclosed in a box and mathematically described by their respective MGCP matrices, A B m1 m1 C D ml ml

75 Repetition Intervals -3cr w-(vr 0 (or 5wr 3,r V- Principal -^ i Interval Wr 2 /v Fig. 6. 3. Repetition intervals and principal interval of a transmission line of length i, o -- - v -A B 0,z A B mC D~ C Dm,. mo ml m2l Fig. 6.4. Two lossless transmission line 2-ports joined by a connecting line

76 and A B m2 m2 C D m2 m2 The MGCP matrix, A B mn m C D m m of the total configuration is then A B A B z cosh z sinhl IA B m m m2 m2 m1 m1 C D C D sinh D z cosh jb C D m mi m 2 mI m I (6.16) Multiplication of the right hand side of Eq. 6. 16 gives (A = A + B C ) cosh D m m1 m2 m2 m1 (6.17) 2 + (z A C +B A )sinh 4 m2 m1 m2 m1 B = z(A B +B D ) cosh ( m m2 ml m2 m1 (6.18) + (z A D +B B ) sinh 4 m2 ml in1 m2

77 C = z(A C +C D )cosh $ m ml m2 m1 m2 (6.19) 2 + (z C C +A D )sinh 4 m1 m2 m1 m2 D = z(B C +D D ) cosh m m m1 m2 m1 m2 (6.20) + (z2D C + B D )sinh b m1 m2 1 m2 From Eq. 6.17, it is seen that the zeros of A are given by the roots of z(A A +B C ) 1 2 2 1 2- + tanh = 0. (6.21) zA C +B A m2 m1 2 m1 Let A A +B C m m m m m1 m2 m2 2 1 X(S)- (6.22) zA C +i(A B ) m2 m1 Z m1 m2 Now A A +B C m1 m2 m2 m1 zA C +-(A B ) m2 m1 Z m1 m2

78 zA2 C (A B ) m2 m1 z m1 m2 A A +B C A A +B C m1 ^ m2 m2 m m1 m2 m2 m (6.23) z+ 1/z A B A C m1 m2 m2 m1 C +A B +A m1 m2 m2 m1 Since A /C B /A and their reciprocals are reactance ml' m1 m2 m2 functions, the form of Eq. 6. 23 demonstrates that x(s) is a reactance function. Thus Eq. 6. 21 becomes x(s) + tanh 4 = 0, (6.24) x(s) being a reactance function. Similarly, Eqs. 6.18, 6. 19 and 6. 20, the expressions for Bm, C and D respectively, can be manipulated to show that their m m zeros also are given by an equation having the form of Eq. 6. 24, where x(s), in these cases, will have expressions similar to Eq. 6. 22 and can be proved to be reactance functions. The zeros of B, C and m m D will not be considered in detail because the proof for the zeros of m A will be directly applicable to the other cases as well. The only property of x(s) used in the proof is that it is a reactance function. All the zeros of A are on the jw-axis. Therefore replacing s by jw in Eq. 6. 24, the zeros of A are found from m

79 x(w) + tan = 0 (6.25) V i.e., tan = -x(w), (6.26) V where x(W) = x(jw)/j. Equation 6. 26 is graphically presented in Fig. 6. 5 for one repetition interval of the connecting line. In this interval the function tan (w/v varies all the way from - o to +oo, is continuous and has positive slope. Since the negative slope of -x(w) is guaranteed by its being the negative of a reactance function, there must be at least one intersection of the curves tan iw/v and -x(w) in this interval, as indicated in Fig. 6. 5. Thus the occurrence of at least one zero of A (and also of B, C and D ) is ensured in each repetition of a connecting line. Zeros occurring in the repetition intervals of an open shunt stub will now be investigated. A shunt stub, as indicated in Fig. 6.6, connects two lossless networks having MGCP matrices, A B A B m2 m2 m1 ml and C D C D m2 m2 L m1 m1 The MGCP matrix,

80 tan -x(U) / V K0 g (K0+2)wr One Repetition Interval fw Fig. 6.5. -x(w) in Eq. 6.26 and tan-v- for one repetition interval of the connecting line in Fig. 6.4. K0 is an odd integer and wr= c i/ A B m1 m1 CA B 1 1__ okim2 mX C D m2 m 2 mD C =,C D,z m1 ml v Fig. 6.6. Two lossless transmission line 2-ports having an open shunt stub at their junction

81 A B m m C D of the total network is given by A B z cosh ( 0 A B m2 m2 m1 m1 (6.27) C D sinh 5 z cosh 4~ C D m2 m 2 1 m1 Multiplication of the matrices in Eq. 6. 27 results in (A = z(A A +B C )cosh D + A B sinh 4 (6.28) mmi m2 m2 m1 m m2 B = z(A B +B D )cosh I + B B sinh (6.29) m m2 m1 m2 m1 m m2 C = z(A C + C D ) cosh 4 + A D sinh 5 (6.30) m m1 m2 m1 m1 m1 m2 D = z(B C +D D )cosh5 + B D sinh5. (6.31) m m1 m2 m1 m2 m1 m2 The zeros of only A will be studied, since conclusions regarding the zeros of Bm C and D can be arrived at exactly in a similar m' m m fashion. From Eq. 6. 28 it is found that the zeros of A are the m roots of

82 z(A A + B C ) m1 m2 m2 m1 A B 2 1 + tanh< = 0. (6.32) m1 m2 Let z(A A + B C ) 1 m2 2 1 x(s) - B (6.33) m1 m2 Observe x(s) can be expressed as (A C \ m2 m2 m where A / B and C /A are reactance functions. m2 m2 m1 m1 Therefore the zeros under investigation are found from x(s) + tanhS = 0, (6.34) where x(s) is a reactance function. Now compare Eq. 6. 34 with Eq. 6. 24. It is clear that arguments used to prove our assertion for connecting lines are directly applicable to the present case, and thus lead to the conclusion that there must be at least one zero of A in each repetition interval of a stub. m

83 6. 4 Exactly One Zero of Each GCP in Each Repetition Interval It has just been shown that in each section of the real frequency axis bounded by two consecutive zeros of cosh (se/v), where I is the length of a stub or connecting line in the circuit, at least one zero of each of the general circuit parameters A, B, C and m^ m ^ m D appears. In situations where more than one such interval due to as many lines overlap partially or wholly, just as many zeros will appear in the overlapping intervals. In this section we prove the following. Theorem I. Exactly one zero of each modified general circuit parameter of the lossless part of the network in Fig. 3. 1 occurs within each repetition interval of every transmission line element that the network consists of. Assuming this theorem to be true for the-hypothetical case of a lossless network represented by the box in Fig. 6. 7(a), it will first be established that the theorem continues to be true with the addition of a connecting line or a stub to this circuit. Consider the circuits in Figs. 6. 7(b) and (c) which are obtained when a connecting line and an open stub respectively are added to the circuit in Fig. 6. 7(a), characterized by the modified GCP matrix A B m m mn m

84 A B m m C D m m (a) A B -_/,z mI mm.. 0 C D m m_ (b) A B m m C D m rM,z s s (c) v Fig. 6.7. (a) A lossless 2-port satisfying the theorem I in Section 6.4 (b) A connecting line terminated by the lossless 2-port in (a) (c) An open shunt stub in parallel with the 2-port in (a)

85 b b b b The modified GCP, A, B, C and D of the complete mN mN mN N N. N N network in Fig. 6. 7(b) are given by bAA sa 2 si A = A z cosh + C z sinh (6.46) mN m v m v T^^ n u s^ T^ 2 si B = B z cosh- + D z sinh (6.47) mN m v m v N C = A sinh + C z cosh (6.48) mN m v m v b sc sih D = B sinh- + D z cosh (6.49) mN m v m v and those of the complete network in Fig. 6. 7(c) are AC = A z cosh s (6.50) mN m v' Bc = B zcosh, (6.51) mN m v mN m v m s Because co B s inh + D z cosh (6.53) mN m v m v Because cosh si/v possesses exactly one zero in each repetition

86 interval the theorem is true for the zeros of AC and B. If any mN mN of the matrix elements A, B, C or D is zero as, for m m m m example, would be the case for C if the circuit in Fig. 6. 7(a) were a single open shunt stub, some of the expressions (Eqs. 6. 46 through 6. 49, 6. 52, and 6. 53) will have the form of Eq. 6. 50. In such cases, the arguments used above for the zeros of Eq. 6. 50 can be repeated. Assuming, however, that none of Am, Bm, C or *m m m Dm is zero, it is readily seen that all the zeros of each of the b b b b modified general circuit parameters A, B, C, D mN m m m mN N N CC and DC are the roots of an equation having the form mN mN x'(s) + tanh = 0 (6.54) v where x'(s), though different in each case, is a reactance function. The expression x'(s) has another important property also; notice that x'(s) is either formed dividing A or D by B or C m m m' or obtained from the reciprocal of the expression thus formed. For example, for A in Eq. 6.46 m N 1 m x'(s) = c. (6.55) z m m Since according to the assumption the theorem is true for circuit in Fig. 6. 7(a), x'(s) has exactly one pole and one zero in each repetition

87 interval of every transmission line present in the circuit. Putting s = jW in Eq. 6. 54, one gets x'(o) + tan- =, (6.56) where x'(c) = x'(j) (6.57) Equation 6. 56 will now be graphically examined. In Fig. 6. 8 -x'(o) is drawn for the portion I1 - I2 of a repetition interval that contains the pole and zero of x'(w) appearing in this repetition interval. The expression tan fw/v, due to the augmenting line, can have three distinct positions relative to -x'(o). It can completely straddle a pole of -x'(w) as in Fig. 6. 9(a), producing two zeros, one for the existing repetition interval and the other for its own repetition interval. Secondly, one repetition interval of tan fw/v may be completely contained in a section of real frequency axis where no pole of x'(w) occurs. This is shown in Fig. 6. 9(b). In such a case, tan icv/v produces a zero for its own repetition interval. Finally a pole of tan fc/v may coincide with that of x'(w), as indicated in Fig. 6. 9(c), producing a zero at a pole of x'(c), thus giving the required zero within the existing repetition interval of x'(w). We have thus established that a network for which the theorem is true, continues to satisfy the theorem even after it is augmented

88 -x?(w) ~Ii~ I 1h \ W, I2 Fig. 6.8. -x'(w) in (6. 56) for the part I - 12 of a repetition interval of the 2-port in Fig. 6.7 (a

89 tan I c x I\ (W -x(w) / I / \ /2 tans \| V \ (b) Q, l\ tan tani v -x'( \ \ / (c) Fig. 6.9. Graphical representations of x'(w) + tan aw/v = 0 illustrating the three distinct relative positions of -x'(w) and tan wc/v

90 by adding a connecting line or a stub. The proof of the theorem will be completed by demonstrating that a single connecting line or a stub satisfies the theorem. For such single elements, the zeros of the modified general circuit parameters (Eqs. 6. 4 and 6. 5) are shown in Fig. 6. 10. It is verified that the theorem is true, though this case is degenerate in that the zeros of the terms A and D lie at the m m end of the repetition intervals. Though it has been our purpose above to prove the existence of one and only one zero of each general circuit parameter in each repetition interval of a network in which connecting lines and open shunt stubs alternate, the proof given applies also to circuits consisting of m (- 0) connecting lines in cascade alternating with n (' 0) shunt open stubs in parallel. Transmission Poles We have proved in Section 6. 4. 1 the existence of exactly one zero for each general circuit parameter in each repetition interval. As a new line having length I is added to a circuit, zeros of cos wi/v repeat at regular intervals creating in their wake new zeros of A, B, C and D,one to each interval. The zeros m m m m m existing prior to the addition of this line will in general undergo rearrangement due to the addition of the new line. However, the relocation will be such that exactly one zero of each general circuit parameter of the new circuit is now assignable to and occurs within

91 each repetition interval. From Section 6.2 it therefore follows that each repetition interval contains exactly one zero of G A + G2D and of G1G2B + C. Since each transmission pole is associated 12m m with one zero of G1A + G2D and one zero of GG1 B + C, as indicated in Section 6. 2, it is reasonable to expect one nondominant pole to each repetition interval occurring, in general, in the region (henceforth referred to as repetition strip) of the left half plane bounded by two straight lines drawn parallel to the imaginary frequency axis through the two extremities of this repetition interval as shown in Fig. 6. 11. 6o 5 Numerical Evaluation of Transmission Poles Transmission poles are found from the zeros of (G Am + G2Dm ) + K(G1G2 Bm + C ) (6. 58) for K= 1 (or alternatively, from the zeros of K'(G1A + G D ) + (G G B + C ) (6. 59) ^lm 2m 12m m for K' = 1). The numerical technique used for the determination of a transmission pole is initiated by finding a zero of Eq. 6. 58 for K = 0 (or at zero of Eq. 6. 59 for K' = 0). The locus of this zero is then followed off the jw-axis as K increases, until finally, for K = 1, a transmission pole is reached. First it will be established that the

92 (2k - 1) 2v (2k + 1) 2 4 4 OA WB WA WD WC0C WD w One Repetition Interval k integer Fig. 6. 10. Zeros WA, WB, wC, wD of modified general circuit parameters of either a stub or a connecting line of length 2 in one of its repetition intervals, verifying the occurrence of exactly one zero of each general circuit parameter in such an interval jw-axis / Repetition // Repetition /// Strip // Interval Fig. 6. 11. A repetition interval and the associated repetition strip. A nondominant pole will, in general, occur in each repetition strip

93 root locus of Eq. 6. 58 as K varies, which originates at a zero of G A + G D and terminates at a zero of G G B + C is Im 2m 12mm' perpendicular to the jo-axis at both these points. 6. 5. 1 Derivatives of the Zeros of (6. 58) with Respect to K. On any point,, on the root-locus of Eq. 6.58, {G1 Am ( ) + G2D ( ) +K[G1G2Bm ()+C ( )]}=0. Taking total differentials, d [GAm () +G2Dm ()] d + Kd- [G2GlBm( )+D ()] ~ d (6.60) + [G1G2B ()+ Cm ()] dK = 0 Let jWAD be any zero of G1 A +GD = 0. At this point K=0. Am m Thus Eq. 6. 60 gives d _ G1 G1 Bm (Q)+C M() ~~- 1 1 m m ~~~(6.61) dK d d G d Am ()+ G2 D ( ) j-AD as the value of d / dK at jWAD. Recall from Section 6. 2 that Bm(jwAD ) = jBm (WAD) C m(jAD ) jC (m AD)

94 Also notice that d d Ads (s) = - j Am (co) (6.62) ds m dw m S = jcW and d d Eds (s) -ji do (c) *. (6.63) ds m " d^ m s = jW Therefore, d | 1 2 m AD m AD (6.64 dR d d' - [G1 Am (wC) + G2Dm ()| JWAD c = WAD This quantity is real and, as an examination of typical curves of Am() + G D () and G1 G2 B () + C ( shown in Fig. 6.12 reveals, is also negative since the slope of G1 A G2D and G1 G2 B + C always have opposite signs at a zero of the former function. Studying K'[GAm()+ G2Dm( + [G1G2B ( C)+ C ()] 0 (6.65) exactly in the same way, one finds that

95 d5 GIAm (Q )+G2Dm(f) KIT =- A(d d (6.66) dK' G1G2 B (a)+ d C (t) 1 G2 d m " + m " j - BC at any zero, jWBC, of G1G2B + C. Or ^BC' 2l^m m d_ | G1 Am (WAB) + G2 Dm (wAB)6 4 d [G1 G2Bm () + C ()] - joBC c = WBC which is real, and as the curves in Fig. 6. 12 show, is always negative. The realness of d4/dK and d4/dK' indicate that the locus is perpendicular to the jc-axis at wAD and BC 6. 5. 2 Determination of WAD or cvBC. The zeros G1G2B + Cm, designated oBC and the zeros of G1A + G2D, called AD are all on the jw-axis. This makes them relatively easy to locate, since only one line, the jw-axis, has to be examined, instead of the entire complex frequency plane. To compute the approximate location of zeros in the interval 0 < cv' W, perhaps the simplest technique is to divide this interval into n subintervals (not necessarily equal), 0< o< Wi1 1 W,2.,' n-1 -'5 W by points wl, 2 w... W n_1 W. The value of the function (for example, G Am + GDm ) is evaluated at the points w = O, w1,..., Wn I W. The presence of a zero in the interval cv. < cv' a. is indicated if the value of the function andthat at have opposite signs. Using this1 function at ci-1 and that at cv. have opposite signs. Using this 1

96 technique a computer program has been written to find the approximate locations for the zeros G A + GD and G G D + C. This 1m 2m 1 2 m m program is listed in Appendix III together with a note on its use. 6. 5. 3 Transmission Poles. With the approximate value of WAD thus known, the approximate location for a zero of G Am + G2Dm + K(G1 G2B+C), for K small (for example, 0. 1) can be written as a ~ jAD, (6.68) where, from Eq. 6. 64, - K (6.69) J iWAD A transmission pole is now evaluated in the following way. Start with a small value of K (for example, 0. 1 or 0. 2) and an estimate + jWAD for a zero of G G B + C + K(G A + G D ) AD i m2 m m 1m 2 m Refine this estimate to a desired degree of accuracy by the NewtonRaphson method. Gradually increase K to unity and every time K is changed the zero is refined by the Newton-Raphson iteration with the zero for the previous value of K being used as an estimate. A listing may be found in Appendix IV, of the computer program written to evaluate the transmission poles following the technique just outlined.

97 6.6 An Example The computer technique delineated above is illustrated here with an example. In this example the nondominant poles of the fiveelement transmission line network indicated in Fig. 6. 13 have been found. The elements of this network chosen for dominant Butterworth transmission poles were Ql/v = 1.52761 Secs z = 0.3 Ohms Q2/v = 0.1789117 " z2 = 3333 "!3/v = 0.712483 " z3 = 0.3 Q4/v = 0.1233977 " z4 = 3.333 " 15/v = 0.1940887 z = 0.3 The approximate locations of cAD and cwBC which are the zeros of G1A + G D and G G2 B + C respectively, were WAD (rdn/sec) WBC (rdn/sec) 0.5 0 1.2 0.9 2.2 1.7 4.0 2.6 4.8 4.3 5.8 4.8 7.4 6.3 8.9 8.3 9.1 8.9

98 f(w) f(w) G1Am(w) + G2Dm() f(w)) = GiG2Bm () + Cm(w),/BC WA WBC AD /BC W - o Fig. 6. 12. Typical graphs of GiAm(w) + G2Dm(w) and GlG2Bm(w) + Cm(w) 4 2 0. 5 ft t l n1.0 0 /5 /3 /1 Fig. 6. 13. A five element transmission line network

99 The dominant poles due to the zeros at 0, 0.9, 1. 7 of G G2 B +C and at 0.5 and 1.2 of G1A + G D were verified to m I m 2m have the fifth order Butterworth locations. The Butterworth pole closest to the jw-axis, marked BP, together with several nondominant poles, identified with NP, NP,..., NP, is shown in Fig. 6. 14. To find the nondominant pole, NP, that resulted from oAD = 2. 2 rdn (see Fig. 6. 14) the starting point for iteration with K = 0. 2 was calculated from Eq. 6. 68. The point -0. 0292 + j2. 2 thus obtained was used to initiate the iteration which converged to exact zero at -0. 032 + j2. 16 for K = 0. 2. Changing K several times, the required transmission pole (K = 1) was located at -0.175 ~ j2.204 Starting from IBC = 2. 6, the iteration found the transmission pole at the same location -0.175 + j2.204 Initiating the iteration at the other zeros, AD, or o BC and proceeding in the same manner, several other nondominant transmission poles near the origin were computed to be NPII: -0.191 + j4.157

100 ijW VI s- plane WBC 9 VI s= a+ cjw NPV WAD NP ^ -.V -0C - VI 7 XNPIV ^IV XNpItV wBC IV WAD III 5- WJ BC NPIII WA II II XNP'I WBC -^ 4 II AD 3 WBC XNPI Ir 2 WAD ~2 I WBC Responsible WAD XB for dominent ( W poles WAD -0.3 -0.2 -0.1 1 2 3 4 5 Repetition Intervals of Line Fig. 6.14. The zeros of G1Am+G2Dm and G1G2Bm+Cm are identified by WAh and wBC respectively and the nondominant poles orthe 5-element network in Fig. 6. 13 are identified by NP's, together with the repetition interval of each line. Each repetition interval, W D and WBC occurring in it, and the nondominant pote in its repetition strip are marked with identical Roman numerals

101 NPI: -0.013 ~ j4. 744 IV NP: -0. 198 ~ j6.204 NPV: -0.197 ~ j8.222 VI NP: -0.00142 ~ j8.886 Figure 6. 14 displays these nondominant poles on the s-plane together with the zeros of G1A + G D and G G B +C. The repetition 1 m n 1 2 m m intervals for all the five transmission lines are also shown in this figure. Our earlier assertion that there is exactly one zero of G1 A 1m + G2 Dm and G1 G2 B + C in each repetition interval can be Zm m m verified from the figure where each such interval and the zeros associated with it are marked with identical Roman numerals. Notice also that the nondominant poles occur, as expected, one in each repetition strip. The presence of the nondominant poles of the network in Fig. 6. 13 can also be detected from the attenuation versus frequency plot presented in Fig. 6. 15. 6. 7 Sensitivity of the Zeros of A, Bm C and D with the m m' m m Variation of Line Lengths It has been established in Sections 6. 2 and 6. 3 that for the network configurations in Figs. 6. 4 and 6. 6, the zeros of Am Bm C and D which are all on the real frequency axis, are obtainable from the roots of

50 40 0 1 2 3 4 5 6 7 8 9 10 co 3 Fig. 6. 15. The computed attenuation of the 5-element network in Fig. 6. 13

103 x(w) + tan v = 0, (6. 70) V where x(w) is a reactance function and I is the length of either the stub, as indicated in Fig. 6.6, or of the connecting line as indicated in Fig. 6. 4. In this section we determine the sensitivity of a zero designated C of Eq. 6. 70 with respect to C. Since ~ is a zero of Eq. 6. 70, one can write x() + tan = 0, (6.71) which implicitly relates C and ~. We want to find dC/di. Taking total differential of Eq. 6. 71, one gets dx(d ) + -sec v d +- sec. d = 0 (6.72) dCV v v \v / which yields -C sec2 dC - vdx 2 * (6. 73) d +- sec v \v Since dx(C)/dC is always positive, dC/dk is positive for C negative and dC/ dd is negative for C positive. This means that with the increase in Q, any zero of Am, Bm C or D moves toward the origin. This result is not different from m toward the origin. This result is not different from what one would

104 anticipate judging from the fact that any repetition interval due to either a stub or a connecting line which contains these zeros slides down the real frequency axis toward the origin when the length is increased. Since nondominant poles result from these zeros, it would be logical to expect the nondominant poles to move toward the origin with the increases in the lengths of the component line elements.

Chapter 7 Optimization It has been pointed out in Chapter 5 that the undesirable effect of a particular nondominant pole may be eliminated, to an extent, by placing a transmission zero on the real frequency axis opposite the pole. In this chapter a strategy is devised to accomplish this without violating other requirements that the network must satisfy. 7. 1 Mathematical Formulation A transmission zero, in general, will not naturally occur opposite a nondominant pole. Positioning a zero opposite a pole may be viewed as minimizing the projection on the real frequency axis of the straight line that joins a specified nondominant pole and a selected transmission zero. As this minimization progresses, the locations of the specified dominant poles must be maintained and the characteristic impedances have to be held within the prescribed bounds. Such minimization of a function under nonlinear constraints is termed a nonlinear programming problem [46]..th Let.j be the imaginary part of the j nondominant pole, and let the transmission zero being utilized to nullify the effect of this pole be designated VI v. The function to be minimized, 105

106 also called the cost function, can be expressed as the weighted sum J C kjlpj- I, (7.1) j=l where J = the number of nondominant poles being considered, k. = nonnegative weights. These weights k. are necessary since the importance of a nondominant pole being nullified increases as it approaches the passband. There are two constraints. (1) The n dominant poles must be preserved at the prescribed locations. Thus, the characteristic impedances z. and the delays Ti (- i/v) of the component transmission lines must always be so chosen as to satisfy Eq. 3. 17, i. e., fi(Ti,', Tn; Z l", Zn) = 0, i = 1, 2,..., n. (7.2) (2) The characteristic impedances z. must be kept within a prescribed bound, i. e., z. = 1, 2..., n (7 3). = _,

107 where th z. = the characteristic impedance of the j transmission line, and zM and z are respectively the upper and lower bounds I m. J J within which z. must lie. Since the system of Eqs. 7. 2 can be J solved for z. in terms of T1, 2., 7n constraints expressed in Eqs. 7. 2 and 7. 3 can be combined as ZM. (i1'n) - Z, j = 1, 2,.., n. (7.4) M. j 1m J j Therefore the nonlinear programming problem can be formalized as J min k.j I V.VI, (7.5) j=1 subject to ZM. - z(7,..X Tn)' Zm j = 1, 2,..., n (7.6) J j where it is understood that z.j(T,..., n) are the solution to the system of Eqs. 7.2 with,..., Tn given. For computational convenience, the cost function (Eq. 7. 5) can be replaced by the following J min m k.x., (7.7) j=l J J

108 subject to x. - 0 j = 1, 2,..., J (7.8) x. - (.- v.) = 1 2,... J (7. 9) x. > ( - v) j= 1, 2, J (7 10) J J J 7. 2 Solution A number of methods are available for the solution of nonlinear programming problems, a survey of which may be found in [47]. Some of these techniques, for instance, the cutting plane method [48] and the small step gradient method [49] are direct extensions of linear programming problem [50, 51]. Large step gradient methods [47], also known as methods of feasible directions [ 52], work with linear subproblems and make use of procedures developed for unconstrained minimization. Fiacco and McCormick [ 53] have developed a method that transforms a minimization problem with constraints into an unconstrained one. The solution that is being delineated here for the nonlinear problem posed in Section 7. 1 is a small step method where complete linearization takes place at each step and no old information is retained. This, together with predetermined step size make this suitable for application to both convex and nonconvex problems. The constraints denoted by Eq. 7.6 can be expressed by two inequality constraints

109 ZM. j 2' T) 1 2 n (7. 11) J e and Z (T,' ) j = 1, 2, n (7. 12) ^M. ^ ^1 2'" rn'' "'' 0 ~ Expand zj(Tr, 2'... Tn) in the Taylor series about the point (71 72,..., n) to get i n az. 17 i i. Z+ = 1, 2,.., n(7. 13)' ] k=l a8k where i i z. = z (T'. ) (7.14) azi 1 = [Z.(7T.., )] (7. 15) ark ak [ j1'"'' n aT k i i+ 1 i ATk =k - k (7. 16) Here superscript i indicates that the linearization is being carried i i out at the solution (Tk zk k = 2,..., n) reached at the conclusion of the iter of the 1 iteration.

110 Substituting this Taylor's series in Eqs. 7. 11 and 7. 12 results in n iz n az. Ar k ZM. z. j = 2,., n (7. 17) k=l a'k j J'" n az. Similar substitution of the Taylor's Series expansions of p. and v. in Eqs. 7. 9 and 7. 10 gives n /a8l( avi\ k \.- A A/k (-,j = 1 2,... (7 19) -X. + T (eu. - ( -), j = 1, 2,..., J k= azk kaT J (7. 20) For the Taylor Series expansions to be accurate, differential increments ATk must be small. This consideration leads to the further constraints where 6 is small: -a' k' j- T 6, k= 1,..., n (7.21) J ~~ )~~

Putting Ak J+2k-l J+2k k = 1,.., n, (7.22) Eq. 7. 21 can be expressed as XJ+2k- 1 xJ+2k 0 (7. 23) XJ+2k-1 + XJ+2k < 6 k = 1 2,.., n Substitution of Eq. 7. 22 in Eqs. 7. 17, 7. 18, 7. 19 and 7. 20 and combining them with Eqs. 7. 7 and 7. 23 finally gives the following linear programming problem: J min m k.x. (7.24) j=1 3 J Subject to IAj3ji i n (al a. _xI i - ai ( J+2k- 1 XJ+2k < i ^iJ~ i il'T4.2 I T-i-9^^ (j j k=J \ k a k/ (7.25) =- 1, 2,.... J

112 ki ( n /all av\ i. i ( a (Xi) x+2k-1 x+2k - ( v.) k= lT aTk (7.26) j 1, 2,..., J n azi J+ 2k- XJ2k) zM., j = 2, -., n, (7 27) k=1 a Tk n az. L (xJ+2k-1l XJ+2k) m. k=1 ak j J+2k-l +XJ+2k 6, k = 1, 2,..., n (7.29) x; 0, j = I, 2,...,J+ 2n. (7.30) This linear programming problem is amenable to solution by one of the several established methods [50, 51]. This linear problem is now solved repeatedly. After each solution the increments: i - i+ = Ak are added to k to get T, and the nonlinear problem is linearized again at this point. This is continued until the increments become so small that no further improvement in cost function results. 7. 3 Calculation of Derivatives (i) az. / ak (i = 1,..., n; k = 1,..., n) can be expressed in terms of two Jacobian determinants as

113 a(f,l' f2' f az. a(z z,. T Z z ) ] 1' 2""' j-l''k' j+l1''' n a'k f' 2""''f ) a(z, z2,.., z ) where af1 af af 1 1 1 az az2 az a(fl, f2 fn) _ af2 af2 af2 n n n az az2 an and the functions f. are defined in Eq. 3. 17. An alternative way to find these derivatives is to perturb the delay of the k transmission line k by a small amount ATk and then solve the system of Eqs. 7. 2 to find zj, j =, 2,..., n. If z., j = 1, 2,..., n was the solution of the system to begin with, the partial derivatives zj / Dk are approximately expressed as az. z-z. j = 1, 2,..., n a3~~k~ T ~ k = 1, 2,, n.(7.32) rk' k k= 1,2,..., n

114 This method was used to determine the derivatives in the example given later in this chapter to illustrate the method of solution put forth earlier in Section 7. 2. (ii) Any transmission pole 5 of the transmission line net-.th work satisfies Eq. 3. 12. The j nondominant pole, a. ~ jy. t3j jYj may, therefore, be computed as the solution to Re[C (T 1.. Tn; zl''z;'Y)] = (7.34) T 1""' ~n; Z' n j' Im[ C (r,..,T Z;n Z,...,z;,j)] =0. (7.35) yj thus determined may be written as yj (Tl.' Tn; Z1..., n) However, since each z. is a function of (T.,j = 1,..., n), it is possible to express y. as a function of delays alone. Designate this Lj (T1,., T ). Therefore, j(T1,'" ) yj(l,., Tn; 1..., ZnT z n;Z1'' ZZ Z..., )' ^j~l^*^ ^ " ^^l^^ \^l'^n 1 n1 " in' (7.36) Z = Z,(7T, T ). n n 1n"l"' n'

115 Equating total differentials on both sides of Eq. 7.36, nan 8aa. n. n __ l Z k AT j (7.37) k k a k k=1 aTk k k=l azk k Thus au ay. n az j = 1, 2..., J, _ k^Z~, (7. 38) aT aT az aT m m k=l m m = 1, 2,..., n. The derivatives a.j/ a- may be computed making use of either side of Eq. 7. 38. In the example given to illustrate the minimization technique we used Eq. 7. 38 in the following way. j. was found at T1, T 2''' n and z1, z2,., zn where z1, z2..., zn were the solution of Eq. 7. 2 with 1,.., T given. T was then perturbed by AT n m m The system of Eqs. 7. 2 was solved for z' z' z'!i 2""' n with T,,..., T + ATm,, T specified. Finally the imaginary part, designated ji, of the h nondominant pole was again determined from Eqs. 7. 34 and 7.35 using T1' "Y' 2 + AT,..., T and z, z2, 1"'m m rn 1n2..., z' for the delays and the characteristic impedances. n 8,j/ T was then evaluated from mn

116 ag.'- A. J 1 1 0 (7. 39) aT - AT m m th (iii) If the j transmission zero, ~ jvj, being associated.th with the j nondominant pole, is assumed to be produced by the stub whose delay is T, then COS (V.T ) =. (7.40) J P This allows the derivative a v. / aT to be expressed as ad. v. I p aB~T T v^~ ^(7. 41) aT I P P and all the other derivatives av. -T,= 0 k p. (7. 42) ak 7. 4 Example An example of the optimization procedure outlined above is herein presented. The initial design is a two-element (n = 2) transmission line network in Fig. 7. 1. The element values of this network which were selected for dominant Butterworth poles are z1 0.3 1 = 0.253895 2 = 3.0 T2 = 0. 5621879

117 cab- 2 2 0.5/ / 1.0 0 Fig. 7.1. A 2-element transmission line network Fig. 7.1. A 2-element transmission line network

118 The first nondominant pole was found to be 1 = 5.615151 and the transmission zero created by the stub was at v = 6. 18677. The optimization problem is to minimize the distance between the transmission zero and the nondominant pole (J = 1) without violating the constraints on characteristic impedances 0.3 _ z1 3.0 and 0.3 = z2 3 3.0 It is also necessary to maintain the pole locations at the Butterworth poles. This results in the constraints f1(T1' T2' Z1' z2) = 0 and f2(T1' T2' Z 1 z2) = 0 These functions fl and f2 are the real and imaginary parts of the denominator of the input impedance looking into either end of the distributed network. The derivatives needed were azo azo azo azo 1 1 2 2 O = 1.2082,. = 0.05195 =0.5813 = -5.4208 0' 0, 0 a71 aT2 ah1 aT2 aol a a0 a P a ~ 9~1 2 0=-0.94, -8. 3123, = -24.4 0 aTa0 -aT1'1 a2 2

119 Substituting these derivatives in Eqs. 7. 24 through 7. 30, (constraints zl - 3.0 and z2 > 0.3 are not included in the tableau) the linear problem was written as min x Subject to -1 23.46 -23.46 -8.3123 8.3123 x 0. 575 -1 -23.46 23.46 8.3123 -8. 3123 2 -0. 575 0 - 1.2082 1.2082 -0.05195 0.05195 x3 0 0 0.5813 -0.5813 -5.4208 5. 4208 x4 0 0 1 1 0 0 X5 0.2 0 0 0 1 1 0.2 x.i 0 i = 1,2,,5 1 This linear subproblem was solved by LINPG, a program for solving linear programming problems available in MTS [ 57]. The solution was found to be x1 = 4.1633 x 10 -2 X2 = 2.547784 x 10 x3 = 0 3 -^-3 x4 = 2.732119 x 10 5 = 0

120 The correction to the delays of the lines were found from Eq. 7. 22. They were A 1= x2- x3 = 0.025477 A2 = x4- X5 = 0.002732 Using Eqo 7. 16, 1 and 2 were found to be T1 = 0.279372 2 = 0. 5649199 For these lengths of the transmission lines, other quantities of interest were z1 = 0.3309906 1 z2 = 3. 0001984 1 - = 5.5645 = 5.62257 v = 5. 62257

121 O O The value of the cost function, Ii1 - Vl, was reduced from I l - vl 1 1 = 0. 57162 to {I1- V11 =0.05807. 1 11 Repeating the procedure at T1, 2, zl z2, the cost function was reduced to 2 2 121 - V11 = 5.5562 - 5.5499 = 0.0063 at 2 T = 0. 283028 T2 = 0. 5653892 z1 = 0.3354596 z2 = 2.9999294 The attenuation of the two element network before and after optimization is indicated in Fig. 7. 2.

W/Wcv radian/sec -- 1 2 3 4 5 6 7 8 9 10 11 12 2 VI I ~/ I i I I I I // 4/ 6 / 8 / ^14- \ / / 12- /. 14~18 20 22 24 26 \ 28- / 30- / ---- Original design 32- 32 f Optimized design 34 36 Fig. 7.,2 Attenuation of a two-element distributed filter

Chapter 8 Existence Consideration For The Solution Of The Two-Element Transmission Line Networks The element values of the network configuration that realizes a prescribed set of dominant transmission poles are the solution of the system of equations fi(ll', in; Zl,.., n) = 0, i = 1, 2,.., n ( where n is the number of the elements in the network. It has been shown that under special conditions of very high impedance connecting lines and very low impedance stubs and all the lengths being electrically short, the solution of the system (Eq. 8. 1) can be approximately determined and this can then be refined by an iterative procedure. However, as the characteristic impedances are perturbed toward a desired range of values, solution of the system (Eq. 8. 1) for real.l.., a and zl,.. Z is not analytically guaranteed, though solutions did exist in every one of the limited number of cases tried by the author. For the relatively simple case of n = 2, two questions will be answered here. In the circuit given in Fig. 8. 1, if Rc, Rs, z, z are specified, are there values of T and T that will 123

124 realize a prescribed set of dominant poles sl 2 = x ~ jy? If there are, what approximately are their values? It is knownthatatapole of the network in Fig. 8. 2, y + Y2 = 0, where y1 and y2 are admittances as indicated in the figure. The network in Fig. 8. 1 may be divided along 1 - 1'. At a specified pole s =x jy Y + Y = 0. (8.2) As defined in Fig. 8. 1, 2sT 1 e -a YC = z 2sT- (8.3) C C e +a and 2s T 1^ e - 1 Y s R + T e2T ~ 1,(8.4) s s s sT e +1 where R -z c c a C C (8.5) C C 2s T The transformation w = e maps each of the infinite strips (2k- 1) < Im(sT)< (2k+1)j, k=...,.-2, -1, 1, 2,...

125 1 R C R C S L / A s v T =. S V C V T z S S Fig. 8. 1. A doubly terminated 2-element transmission line network. c, s are the lengths, and Zc, zs are the characteristic impedances of the connecting line and the stub respectively Y1 Y2 I I ~ n Fig. 8.2. A 2-port network shown bisected into two 2-ports, I and II

126 into the whole of the w plane. For the pole s to be in the principal 2sT 2sT strip of e and e, the maximum value of T and T canS C not exceed T, where T is given by m m 2yT = 7. (8.6) m (Here the principal strip is defined as the strip that contains the origin. ) If the admittances -y and y are plotted as functions c s of T and T respectively for 0 T _-< r/2y and 0 - Tr - r/2y, c S C s the points of intersection of the two curves will satisfy Eq. 8. 2. Thus, intersection of the curves -y and y drawn with and' c s c -T as parameters will indicate the existence of the solution of the s system of Eqs. 8. 1. Also approximate values of T and T at C s the solution can be read off at the intersection. 8. 1 Variation of the Input Admittance of A Resistance-Terminated Transmission Line This section deals with the general shape of the admittance y drawn with T as parameter. Since 2srT 2(x+jy)T C C e = e 2xT' j2yT e e c this represents, in the complex-plane, a vector that starts, at

127 T = 0, on the real axis with a magnitude of unity, rotates counterc clockwise decreasing in magnitude as T increases and finally lines c up with negative real axis at T = 7T/ 2y, the magnitude at this value 2s' C of Tc being given by -eX(' /Y). The locus of the extremity of e with T as parameter for 0 < T < T/ 2y is shown in Fig. 8. 3. c C Case I: a> 0. Since a = (Rc-zc/R +z ), the condition of a being positive means that the connecting line is terminated in a resistance greater than its characteristic impedance. (a) a < eX(V/Y). In Fig. 8. 4(a), the position of a is shown 2s T 2s T 2s T on a plot of e. The vectors e -_ a and e + a, and their associated angles a and ft are also indicated. The angle (a-3) is the angle associated with 2s T 1 e - a Yc z 2sT C C e +a 2s T Now as the point D moves from E to F on the curve e corresponding to the increase in Tc' a - f starts at zero, increases and comes back to zero. Typically, y can therefore be represented as in Fig. 8. 4(b). (b) a> e(X/Y). However, if a is greater than e(X/) as indicated in Fig. 8. 5(a), it can be concluded (a-jf) now ranges from

128 Im c= vT/4Y e2sTc Tc = - /2y e'c = -ex/y 1 Re Fig. 8.3. The locus of the extremity of e2S c for 0 < Tc < 7/2y m1 2S T e C 7ix/y a 0 < a < eaX/ <1 D e (a) r' __. a (a) F\ C(/P B/ a2F a \E -a a Re Im 0 < a < eX/ (b) ______c = T7/2y \ =c0 ex/y + a 1Re zc eX/Y-a Rc Fig. 8.4. (a) Vectors e2 + a and e2 - a with 0 < a < ex/ and their associated angles ft and a( (b) Typical representation of Yc with 0 < a < eax/y

129 a> 0O -awa-0 la> Im. 27X> / > C y c a-e (bFig) 8.5. (a) Vectors +a and a with O e a 1 an e e n Re Z(b) Typical representaton of y wh c a -e' C Fig. 8.5. (a) Vectors e c + a and e - a with 0 <e < a < I and their associated angles 3 and a (b) Typical representation of y = with e,-< a ~ x / y i Re~~~~~~\

130 zero to 180 degrees. Typical representation y for this case is shown in Fig. 8. 5(b). Case II: a < 0 This means that the resistance that terminates the connecting line is less than the characteristic impedance of the line. (a) 0 > a> -ef(X/Y). The positions of a, -a and of the 2sT 2s T vectors e c a and e + a are shown in Fig. 8.6(a). The angle j, it is seen from the figure, is greater than a except at E and F where they are equal. Thus (a-:) starts at zero degrees, decreases and then increases to zero degrees. The variation of y with T as parameter can be represented as in Fig. 8. 6(b). JC C (b) -e(x/) > a. For this case the point B in Fig. 8. 6(a) lies beyond F on the negative real axis. Therefore, (a-3) ranges from zero to -7T. The corresponding y is shown in Fig. 8. 7. 8. 2 Variation of the Admittance of an Open Shunt Stub in Shunt With A Resistor The admittance Ys is given by 2s T 1 1 e s-1 Rs z 2sT S e + 1 2sT 2sT Fig. 8. 8(a) shows the vectors e s- and e + 1 on a plot 2s of e. It is deduced from the figure that a - f ranges from

131 2sA a< O Im aIm< lal < e7/ T 2s_ D \ |1/Rc Re 2sT 2sT C / e +a Fig.8.6.(a) F C E aVectors e -a and e +a with -a e (b) Typical representation of yc with O < -a < e / for OK T K iT/2y Im a< lal > erX/Y T =2 y T = 0 1 ae Re Z eTX/ x/y\ 2sT 2sT Fig. 8.6. (a) Vectors e C - a and e C + a with 0 < -a < e7Tx/ Fig. 8.7. Typical representation of yc with O < e - < e -a for O < T < r/2y Im a< 0 lal > e~x/ T ~ 7 = 0 c= 2y C 1 a+e c a-ex/ y

132 ir/2 to vt. Therefore typically 2sT z 2s T s e S1 z5e s+1 and ys can be represented as in Fig. 8. 8(b). 8. 3 Condition for Existence of Solution Typical shapes of the curves y and y as the lengths of c s the connecting line and the stub varies are now known. Since the existence of the solution is indicated by the intersection of -y and ys, examine -yc in Figs. 8. 4(b), 8. 5(b), 8. 6(b) and 8. 7 and ys in Fig. 8. 8(b). Since ys occupies the first and second quadrants and -y lies in the third and fourth quadrants when a > 0, no intersection is possible. Therefore no solution in the range 0 < T' T and 0O<' < T can exist when R is greater s m c m c than z. For a< 0, -y is capacitive and thus -yc may intersect c c with ys. If either of the extremeties of -Y (Ys ) lies in the interval defined by the extremeties of y (-Yc) and the other outside this s (-yc interval, the solution is guaranteed; since the two curves have to intersect under this condition. If, however, this condition is not met by the element values of the network, the question of existence is decided by drawing the curves -yc and y and determining if they intersect or not.

133 2sT Im s (a) C F E Re -1 1 2sT e s 1 S 2sy e S~ Im s 1 1 1+ e /Y s 5s s I-e - 2sT 2sT Fig. 8.8. (a) Vectors e S + 1 and e S- 1 and the locus of 2sT e s for O~< T < 7/2y (b) Typical representation of or < /2y e + s ~5

134 It is also observed that when the condition is not satisfied, the two curves, if they intersect, will intersect at two points, thus giving two solutions. 8. 4 Example In Fig. 8. 1, let R = 0.5, R =0. 5 s c z = 0. 3 z = 3.0 S' C' and s = -0.707 ~ j 0.707 For T = 0, --- 5 2 s R For s= 7t/2y, 2s T 1 1 e s 1 Ys R Z 2s s e +1 2xT 2jyT 1 1 e e s R I z 2x7 2jyT S Se e +1

135 1 1 -eXY_ 1 + z x-r/ s s -e +1' 1 1 1 + erX/Y R z ~irx/y s 1-e = -1. 639 For T = 0 C 2s T e "-a -c = z 2sT C C e a 1 1-a z + a c z 2R C C 1 _ -2 R C For T =2T/2y, C -- 1 -e(X/ -a c -e +a = 0.295.

136 The two extremities of -y are -2 and 0.295 and those for ys are 2 and -1.639 Since 0 295 lies between -1.639 and 2, and -2 is outside this range, the existence of T and T that will realize the dominant C S poles -0. 707 + j 0. 707 is assured. -y and y are plotted in solid line in Fig. 8. 9 with T and r as parameters. The two C S curves intersect, as predicted from the positions of their extremities. It is possible that for the specified values of Rs, Rc, z and zc the desired condition will not be satisfied. For example, in the above example if R is specified to be 1. Q instead of 0. 5, the other quantities being left unchanged, the two extremities of ys are 1.0 and -2.639 and the condition is not satisfied. For this case, Ys is shown by the dashed curve in Fig. 8. 9. Since c and y now intersect at two positions, there are two solutions.

7C c c Im T z /^ ^^~~~~~~~~~~2 0~ S 7 Z~~~~S (R 0. s 0) YS (R 1 ss - ) I ~~ iYS Si S z C /~~~~~~~~~~~~~~~ N~~~~~~~~ IT \ s T = T I =I/2y T y\ 7= C S C S -3 -2 0 Re 1 2 Fig. 8.9. Plots of -y for O T T- 7ir/2y, and y for O T 7Ti/2y Yi.8.Poso C- fS C S

Chapter 9 Summary An iterative numerical technique has been presented for the synthesis of noncommensurate transmission line networks to realize a prescribed set of dominant transmission poles and zeros. It has been shown in Chapter 3 that the synthesis is accomplished when a system of nonlinear equations is solved. In Chapter 4, a NewtonRaphson iteration has been outlined which solves the system of nonlinear equations obtained when the distributed networks have a particular structure: open shunt stubs alternating with connecting lines in cascade. The initial solution which is required to start the Newton-Raphson iteration has been derived. In Chapter 5, the method has been illustrated with the example of a nine-element transmission line network for which nine dominant transmission poles andfour pairs of real frequency axis zeros are specified. The transmission characteristics of the realized distributed filter differ from those of a lumped element prototype because of the nondominant poles and zeros which appear. For the nine- element low-pass example the principal effect is a sharp drop in the stopband attenuation at about 1. 5 octaves above the passband cutoff frequency. This design was then modified by shifting a transmission zero along 138

139 the jw-axis to a position opposite the nondominant pole closest to the passband. The modified design maintains the stopband attenuation to about two octaves beyond the passband edge. This design was fabricated; its measured response verifies the theoretically predicted response. The periodic nature of network functions that are characteristic of distributed parameter networks gives rise to nondominant poles. In Chapter 6, it has been proved that one and only one nondominant pole is present in each repetition strip of every transmission line of a network that is made up of alternating open shunt stubs and connecting lines. A numerical technique has been delineated to compute the nondominant pole positions and illustrated with an example, which verifies the occurrence of exactly one nondominant pole in each repetition strip. The example of Chapter 5 demonstrates that the undesirable effect of a nondominant pole on the response of a network in the frequency range of interest can be nullified, to an extent, by aligning a nondominant transmission zero opposite the pole. In Chapter 7, a nonlinear programming problem has been formulated for this purpose and a method for its solution by successive linearizations has been given. An example has been added to illustrate this method. Finally in Chapter 8, the existence of solutions for the system of equations (Eq. 3. 17) developed in Chapter 3 has been analytically investigated for n = 2.

Appendix I Proof of The Expressions Given in Eqs. 3. 4 Through 3. 7 An expression for the general circuit paramater matrix for the lossless part of the network in Fig. 3. 1 is derived here. We want to prove that the general circuit parameters At, Bt, Ct and Dt of a circuit consisting of stubs and connecting lines in cascade can be expressed respectively by Eqs. 3. 4 through 3. 7. Proof: An examination of Eq. 3. 3 and Eqs. 3. 4 through 3. 7 shows that the denominators of At, Bt, Ct and Dt which are all identical are formed when factors 1/z. cosh 4. from the stubs J J (j=l, 3,..., n) (see Eq. 3. 1) and factors 1/z. from the connecting lines (j=2, 4,..., n-l) (see Eq. 3. 2) are multiplied. Thus it only remains to be proved that the numerators of Eqs. 3. 4 through 3. 7, that is, the modified GCP matrix as defined in Eq. 6. 3 is the product of a chain of matrices that is obtained when each stub in Fig. 3. 1 is represented by Eq. 6. 5 and each connecting line, by Eq. 6. 4. This will be demonstrated by induction. Let us assume that the lossless network in Fig. I. l(a) which is made up of n - 1 transmission line elements in cascade satisfy the 140

141 n- 1 Transmission Lines (a) A Connecting Line n- 1 or an Transmission Open Shunt Stub Lines (b) Fig. I. 1. (a) A lossless 2-port having n- 1 transmission line elements in cascade (b) The network in (a) augmented

142 theorem. The modified GCP matrix of this network can, therefore, be written as n- 2 2n-2 A B a! cosh (O. s) b! sinh (0n s) i i m mn j.(I. 1) C' D' 2n-1 2n-2 m m n- n m n- L J c' sinh (0 s) d' cosh (n s) s i=l i= We now prove that adding a stub or a connecting line to the circuit in Fig. I. l(a) results in an augmented circuit [Fig. I. l(b)] whose modified GCP matrix can be expressed as Eq. 6. 3. The modified GCP matrix of the augmented circuit can be written as A B z cosh l F z sinh n A' B' r m n n n n m m (Io 2) C D sinh n z cosh In C' D' m m m n n m m where sl ) = n, I being the length of the augmenting line, Zn = the characteristic impedance of the augmenting line, F = O, if the augmenting line is a stub, 1, if the augmenting line is a connecting line.

143 Therefore, one can write 2 A = A' z cosh + C' z F sinh D (I. 3) m m n n m n n 2 B = B' z cosh + D' z F sinh D (. 4) m mn n m n n C = A' sinh4~ + C' z cosh D (I. 5) m m n m n n D = B' sinhm + D' z cosh (1.6) m m n m n n A can now be manipulated in the following way. Substituting for A' and C' from Eq. I. in Eq.. 3, m m m n- 2 n- 2 n-i 2 n- 1 A = a zz cosh bn cosh (O s) + F Fn Cisinh D sinh (i s) i=1 1=1 2n- 2 2 -2 a2 z [cosh ( + cosh (n -.-1 Z IHi IIaz I +n (I. 7) n-2 2 1 2 z n-1 n-i + -e 2Fc!z' [cosh (. s+ )- cosh( s-n)] i-l i n 1 n I n 2n- 2 2 + i n- (a' +Fc Zn)zn cosh(. S+n) i=1

144 n- 2 2 z + Z -(a -Fc! zn) cosh (oi s- ) (I. 7 i~~~~~~~~2n- 1 ~Cont. ) 2n= E a. cosh ( s) i=1 where 12 n-2 a= (az + FcIzn), i=1, 2,.., 2 a n-2 2(a(zn- FcIzn), i=l,...,2 +i The expressions for B, C and D in Eqs. I. 4, I. 5 and m m m I. 6 can also be manipulated in the same manner into the forms given in Eq. 6. 3. The proof will be completed by pointing out that a circuit having a single stub or a connecting line satisfies the theorem. This is verified when matrix (Eq. 6. 3) for n = 1 is compared with Eqs. 6. 4 or 6.5.

Appendix II A Program to Solve The System of Eqs. 3. 17 And To Decrease zConecting Line /Stub Automatically It has been shown in Section 3. 2 that for the network in Fig. 3. 1 to have a prescribed set of transmission poles, the lengths and the characteristic impedances of the transmission lines in the network must be selected to satisfy Eqs. 3. 17. The generation of this system of equations and its solution by the Newton-Raphson method has been presented in Sections 4. 1 through 4. 4. A part (subroutines FITER, ZCAS, ZPAR and the main program with DMF = 1) of the program listed here solves Eq. 3. 17 following the techniques outlined in Sections 4. 1 through 4. 4. The rest of the main program (when DMF > 1 and none of the variable parameters is an impedance) uses subroutines FITER, ZCAS, ZPAR and MULFAC to bring down the ratio zconnecting line /stub automatically. The block diagram in Fig. IL 1 explains the operation of the program. For the successful convergence of iterations terminations of the network in Fig. 3. 1 must be unequal. 145

146 Given: n, poles, zi, Refine, for the given i, approximate Ii/v, the li/v estimate by the terminating resistors, Newton-Raphson iteration variables of iteration, OMF, CMF, DMF ^i^ ~< DMF = I Perturb Zi by a \^^ ^^^^ p ^^"^^ r~~'~1 ( ~'~) small amount; G^^ ~^^^^ ~ KC= o Save zi,,i/v find, byiteration ^^ ^^ ^ * ~-~~* ^ ~ J f~~~~~fi/v to preserve the poles ^\ I Compute change in \ ^~^ Decrease zconnecting line n ) ~ i/v for unit ~ (12a~ by the factor CMF and ~ chang increase instubs by same I ag in.^___ I factor Estimate new Using these estimates, a__ —-- values of Qi/v initiate the Newton\', compatible with Raphson iteration to new zi find exact li/v Zi, i/v stored in temporary location is made active zi,,i/v x. - KI= 0 I is this T Decrease rI' ____./ iteration CMF V__j ~ successful? T CMF = DMF _I~~~ I Store current T OMF CMF. values of -<^ )0I Kl- KI + I tporaryZi, /v in locations PRINT I i/v, zi 12,. \ Increase _~~ _ CMF Fig. II. 1. The block-diagram of the program

PROGRAM LISTINGS FORTRAN IV G COMPI LER MAIN 1-O9-6. 9 14:1-905 0001 DIMENSION ASAVE(15), Z F.SJ1yiALM5lhI)LT-FMP(5)t.ACHANG(1)t... 1I7INIT( 15) 0002 CCMMON X(l5),Y(l5J) A(15) l5J NE(I),_MPA15) C~*** NN=NO. OF LINES C~~~ _____R2 = RESISTANCE CN LINF(1) END___________ 0003 READ(5,500)N,'l,,R2 0)04 500 FORMAT(12,?FIO.*) _ ____ __ _ C***~ 7(1)1 I=l1,N ARE CHARACTERISTIC IMPEDANCES OF LINES; _C*** LINF I ISA STUR. C**~'A(l) ARE DELAYS ( LINE LENGTH/VELOCITY Fl PPOPAGATION, ) 0005 REAI)(^,505)( Al IN)i ( 7. (I )l I= 12 N) 0006 (505 FORMAT(8PF100) C~ ~ ~SPECIFIIED POLFS S(I)=X(l)JI)+J)U,IlJ, 3 3 C*** IF N IS onDD, LAST POLE IS REAL. _0007 READ( 5,505) ( X(I I =lll NM,?.N )(Y)iIttl N,2)___ _. __-2) C~* LIrIF(-I AND.IMP(I) SPFCI FY PARAMETERS AS _____________________C' FIX f(1=l) OP ADJUtSTAB LE(=0)._________________________________ OOOP R E A D (5,510) (L I N E( I I) t= fIl N) ( IMP( I) 1T=,N) 0009 I0) URMAT(3012) -.-.-.-___ —-. C.).IMF, CMF,, DMF CCNTRL AUTOMATIC RFOUCTION'F C.(> 7 RATIO TO (7HI/ZLC)/(DOMFDMF). INILTAL.._.._-_. C^ ~STEP CF RFnlCTION IS 1/(CMF*CMF); OMF=1.0.O _____________________ _ WHFN ANY IMP(h)=0; MUST SET DMF-=lo. 0010 READ(5960C')CF,CkF^.FDMF 00l11 60' FoC[AT(3F0)I.O), _.._ _-......-0.__ ____ __ 001? Dr 100 I=19N 0013 3I 7INIT(I)~Z(I) __ - - ____ —. —.-.- - -~ -- 0014 10 CALL FITER(R1,P?vN,8125) _______________ _~rA IF f)lF=1., WRITE RE"-SULTS AND STOP.,*________'"(^^' RFST (F PPUGRAM ITMPLFMENTS ZHI/ZLO RFDUCTION.

0015!I-(,)^.L.1. ). TOne.!.________________Tr_..._.__1_ 0016 105 K I=0 0017 O) 1 I =1,N...___-... — i1 b ASAVF( I)=A( I ) 0019 1 /SAV( I )=Z( 1 ) 0020 w2 I T E( 6,00) 0021 30 0 F t,:AT (15X,' THF I MPEDANCE VECTOR) _ __ _ __ 002? ITF(6, 30S)IZ( I), T=,N) 02A 3C0S FLRMAT(4(5X,F15.7)) )....._ 002(4 ^RITF(-6,310) Ou?') 310 FOoMAT(15X,,THF LENCTH VECTOR') 00?6 RIT6E (6, 15) (A(.I) I=1,N).00?7 31" FO ^, AT(4(5XF15.7)) _____ ____ ___ 00?8' WP I TF( 6,320 )C F 00?9. FORMAT(F15.7) _ ____ — 0030 DC 2 I =1,? N 0231 2 Z (I )=. l*ZSAVF( I ) 003) DO 3 =?2.,N,2 0033. 3 Z(I)=.'q*7SAVE I) ________ _ (00O4 DO 4 I=IN ()r) 5 4 A(. )=1.1ASAVE( I)_ 0033 WRITE(,3)00) 00 37 WRT TF {, 305) ( I I ), 1 N) 0038 W l TF (t, (19) 0039 WRITF( 315 ( A(I I1=1,N).______._ 004() ___ _WRITEJ 6 t3.0_)C MF_____.__ 0041 CALL FITER(RlR2N,N~125) 004?, WRITFf6,300)___ __ ____ __ 00L)4' W " R I T r 613053 ) 3 7( I I 11 fN) ___0044WRITE(61310) 0045 WRIT (6,315)(A() q I=,N ) 0046 WRITE ( 6,3?20) CMF_ __________ 00471 DO 5 I-1= N _ 0348 5 ACHANG ( I )=(A( )-AAVE( I) )/(Z(-ZSAVE(I) )_ __ _..,_._...._........................,.. 0049 12 DO 6 1=1,N,2 00O5 6 Z( ) =CMF*7INIT(I )

6bKPT -- 0 7 1=29N,2 00^'~7 7 (1) =7 IN INT I )/CMF _______'-" ~ 00DO3 8 l=1,N H ( A() =AS \VE ( I ) +ACHANG_(I)* 7(1 )-Z.SAVF ( I)) 00o or5 krITF(60300) 0___j0 0 PIT[r(6,3(T5h) ( 7(1)91=1 N) L 0057 W PITE(6,310) WPTI T(._ ^I IALU)1-1=,LN)___ —______________ ________________1005^ WPITH(6,3?0)CMF 0060 _~CALL FITER(PI,,N2^) _____~ _____________ 0061 IF(CMF.LT.(1)MF+.01~).AND CMF.GT. (DMF-.1)) GO TO 110 006? CI'F=CMF 0063 KFY=1 0064 KI=K+ __I~__ 0065" " DO 11I"=h^N' 0)66 6 __ _ATFM(P)11=A(r)________ ___________________ _ _ _________ _________. 0067 II ZTf'P( I )=Z( I) Cj06RH fPCALl MUt FAC(KEYDMFCMFOMF) 0009 GCO TO 12' 0 0)70 9 ___ IF(KI..0) GCTC) 120__________________________ 0071 no] 115 I=1,N __ 0072 A_(I)=ATFMP(1) I _ _____________ 0073 115 7 (1)=7 T EMP(T) 0074 _ GC TO 105 J(u075 1.20 KFY=? 0016 CAL-L MULFAC(KEYDO FCMFiOMF) _________ _._ - -_ 0077 GIJ TO 1'2~ 0078 110 WRITE(6300) ___ ___I 0079 WKI EI 6 305) Z.I)I,1 1,N) WQ. I T F (3 10lO O00ai1 WPITF- (6,315)(A(1),1=1,N)'0(182r 0 JITF(6,3?0)CMF___ ___ _____ 00P3 125 CALL SYSTFM ~0- 2, 4_FN

FORTRAN IV G COMPILER MULFAC 10-09-69 14:09,12 0001 S SUBR OUTI N _F. F OML AC __ __Y,___ 000? GO TO (5,7), KEY 0003 5 IF(()MF- CMF).GT..4) GO'I TO 9 000.4 CMF=DMF 000'0 5 RETURN 0006 9 CMF=JMF+(DMF-CMF)/2. 0007' _ RETURN_ ____________ OOOR 7 UMF=CMF- (CMF-CMF) /2. 0009 CMF=LIMF 0010 RETURN 0011 ENn cI " FORTRAN IV G COMPI LER MAIN 10-09-69 14:09.13........ -C** SUBROUTI NE F.IT ER PERFORMS__NFWTON-RAPHSON I TF PATIONS C*** TO SATISFY FF(A(I),...,AI(N;Z7(l),...,(N);s(J))=0, J=l,3,....... C.._T____IRETURN S NECESSARRY NEW PAjrAMFT. RS_._ _ _.___ ___ 0001 SUBROUTINE FITEk(RI R2Nt,*) 0002 CCMMON X( 15) Y( 15)A( 15) l ( 15)ILTNF(15),TMP( 15) 0(003 DIMENSI({N B(15 ),HSJSR(I15),RHS(15),DERIV(9,9),S(RI) 0004 20 K=O _ 0005 _ 30 CALL DENCM(PRLR2,X,YtA,rZRN) -._..............C***_ RHS IS RIGHT HAND Sf)_F _IF RF(FF'+J*~IM(FF) SYSTEM, 0006 DO 1 1=1,N 0007 1 RHS(I)=-B(I) 00108 WRITE(6,200) __ 0009 200 FORMAT('THE RHS VECTOR' ) 0010 o. WRITE(6,1iO) (RHS(T),l= lN ) ~~ o- 0 __._ 110 _ _FOPMAT.(6( E 18.7? )) __ _ 7. 2 0012 INK=O -- —... C*** DO 2 CALCULATES DERIVATIVES OF B'S WITH RESPECT TO C*** VARIABLE LENGTH PARAMETERS.

001~3 ~DC 2 I=1 __ )0014 IF(1.EO. LINF(i)) GC TO 2 00 1 5 P MT= 0 01*A(I) _______ 001' ASAVF=A([) 0017 A(I)=tSAVE+PERT u'018 INKZINK+1 0019 CALL DLNrM(PlPR?, XVYA,7,BSUBIN) __ __..__ 00?0 no 3 J=1,N 0 21 3 FRIV(J, INK.)=(PSU (J)-B(J) /PERT,____ _____ __ ___ ___ ___ 00 2 A(1)=ASAVE 2 CONTINUEF C*~* DO 7 SAME AS D00 2 FOR VARIABLE Z PARAMETERS 002 00O 7 I=1N__ _____________ _______ 002'5 IF(1.FO, IMP(I)) GU0 TU 7 00^ 26_ PEFRT.OO1*7(1) ___ __________________________________ 0027 ZSAVE=Z(I) ______00?7(I)=ZSAVF+PERT 0029 INK=INK+l CALL DEC-JM(RlIR2,XYA,1,tRSUn.N) 0031 Do 8 J=l,N 0o n-A O rFRIV(JtlINK)=(BSUP (J)-P(J))/PERT ~00^^ ~7(I)=7SAVE 0034 7 C O INTINUEF Cr ~ INK( IS NO.N OF VARIABLE PARAMETERS PROGRAM C^~o_ HAS CALCULATED DErIVATIVES FOR._________________ 0035 WRITF(l1115)TNK 0036 115 FO)\MAT(5X-,4HINK=, 12) __-____ _ ____ 3OJ37 00 210 I=1,N?II0 WRITF(61llO)(O[RIV(ltRJ)tJ=1N)_ 00 9 CAL=O. 0040 00 1 = l N___ 0041 4 CAL=CAL+ABS(B(I1)) 004t2 IF(K.)0O) CALSAV=CAL_.. Cc**~ APPAY( ) ANI ) SMQ( ) ARF IN *SSP; _ ______.____C SI A_ fPET)lPNS CHACGES IN A'S AND 71S F(OUND RY IT.FRATinNf 0043 CALL 1 A \IY(,NN', ),9 S, FRIV)

0044 CALL SIMo(SRHSNQ) 0045 INKO________________ ___________ C***~~ 0 D' 5 AND DO 9 CHANGE VARIABLE PARAMETERS. 0046 0 0 5 =LN I__. _____ ___ ~______ -— ~ — -- - - —. ~ ~ 0047 IF(i1.EQ. LINE(I)) GC TO 5 0' 048 fINK=INK_+1__ 0049 A( I) A( I )+RHS(INK) 0050 5 5 C ONT I NUE _ __________________________ 0051 " DCO 9 1=1,N 005?2 IF(l FQ. IMP(I)) GO TO 9 ________ 0053 INK=INK400547( I II7)= (l)-RHS( INK) 0055 9 CONTINUE 0056 ____ ___RR=O. _____________________________________________________ 0057 00 6 L=1N' 0058 __63 6 FRR=ERR+ABS(PHS(I)) 0059 WRITE(6,?30) 00_60_ 230 FORMAT('THE INCREMENTS*) 0061 IT(6 RITE(6,l10)(RHS(l),I=1,N) 006? _ _____ _WRITE(6 240)___ _______________________ 0063 240 FORMAT('THE' LENGTH VECTOR ) __ 0064 T- E 6 1 (610) (ALU4J=lA N) 0065 WRITE(69?50) h0066 __ 250 FORMAT('THE IMPEDANCE VECTOR' $ 0067 WRITE(6qllO)(Z(l)vl=lN) 00683___ W_ ____WRITE(6,110)CAL _ C***~ K= TOTAL NO. OF ITERATIONS. 0069 ___WRITF(6,1 30)K INK ____________________ 0070 130 FORMAT(5Xt2HK=,1?,SX,4HINKt12). 0071 IF(K-1)7CI75,75 0072 70 K=.K-1 0073 ____ GOTO 30 ______________________ Ct^ CAL.GTo2*CALSAV MEANS ITERATION IS DIVERGING _ 0074 75' IF(CAL.GT,2.*CALSAV) GO TO 80 0075 CALSAV=CAL _ C~~ IF CAL IS SMALL OR PARAMETER CHANGES SMALLi RETURN,

0076 10 IF(CAL.LTal.F-05.OR, ER LTr1I.E-07) GO TO 90 0077 K=K+ ____' 007 CGU TO 30 0709 90 R ETURN _.________ 0' (~180 RETUPN 1 C)3ili END U___0081____________ED________________________________~ FORTRAN IV G COMPILER MAIN _1Q0-09-69 14: 09 18.C** _SUBROUTINE DENOM TAKFS RlR2,X(J (J),A),7I, AND N__ C***~ AND CALCULATES REAL AND IMiAGINARY PARTS OF FF AS C~~ RE(PF(S(1))) + J*1M(FF(S(1))) = B(l) +J*(2), ETC._ 0001 SUBROUTINE OFNOM(PltR?,X,YA,7,R,N) O0002 DOIMENSION X ( 1 5)Y(15 A 1 5 )Zl)tR(U15) _______ 0003 DO?2 J=1,N,2 0004 E=R2 _ _ ________ -~ OW 0 h F=0. 0006 G=1 __ ____ 0007 H —. 000i 1=1 ~0009 3 CALL ZPAR(EFGHHX(J)V(J)Y(J)A(I),ZI) * 0010 1 =l+1 ___ __ __ _____ ____~_ __________ ____ ____________ ol I 0=I+ 0011 IF(I.GT.N) GO TO 1 001? _ CALLSFHJ ZCAS(E,FGH,X(J),V (J), A(I,Z(I))___________ 0013 l=l+l 0014 IF'(I.LE.N) GO TO 3 0015 I P(J)=E+Rl*G 001 ___1 _2 B(J+l)=F+l*______________________________ 0017 RETURN 0018 __FND____________________________________________________

FORTRAN IV G COMPILER MAIN. 1Q0-09-69 14: 09, 19 C*** SUBROUTINE ZPAP CALCULATES _INPUT IM1PEOANE OF AA ________ __________C^_* ______SUBBOUJIN__ PA?_C ALCULAT ES~T.. NU__lP. EDA_N__aJ A__.____~___ C*** TRANSYISSION LTNE WITH A=Vt Z=CHI IN PARALLEL _______ _C*W WITH I P._ (F+J*F)f(.+J*H _AT _X+J*Y______ C** IMPEDANCE THUS FCUND PEPLACES IMP. 0001 SUBROUTINE ZPAR(EFyGH-XXvYYvVpCHI) 0002 R=COS(V'VY)(EXP(VxXX)+FXP(-V*XX) 1/2 0003 S=Siv(V~YY)~(EX?(V*XX)-EXPf-V*XX))/2 _______________ ________ 3004 P=COS(V*YY)*(EXP(V*XX)-5rXP(-V*XX))/2o 0005 Q=SIN(V*YY)*(EXP(V*XX)+EXP(-^V*XX))/2 0006 ESAVE=CHI*(E*R-F*S) 0007 FSAVE=CHI*(FP6+E*S) 3008 GSAVE=E*P-F*~+CHI*(R*G-H*S) 0009 HSAVE=F*P*E*Q+CHI*(G*S+R*H) 0010 E=ESAVE 0011 F=FSAVE 0012 G=GSAVI 0013 H=HSAVE 0014 RETURN 0015_ END. FORTRAN IV G COMPILER- MAIN- 10-09-69 14:09.21 ~_________________^ SU3WUTINE ZCAS IS SAME AS IPARFOR CASCADEINE ___ C~* TERMINATED TN IP,. 0001-Sl _U3R - OUTI__ZC AS(EF, H XX, YYSVUCHI___ ZF_____ __H __CHI_ 0002 R=CCS(V*Yv)*(EXP(VAXX)+EXP(-V*XX) 1/2 00 I 3 S= S SI V Y Y)E X P ( V XX)- E X P-VXY) )0/ 2 __ 0004 P=CCS(V~YY)*(EXP(V*XX)-FXP(-V*VX))/2o f) 06 I 5 Q S I N V Y Y F X P ( V Xc X + EF X P V X X 2 335_._ _.~._ QQSIN(V YYj)_(FXP(V*XX)+EXP(-V~cXX))/2_ _________ 0006 ESAVE=(E*R-F*S)+CHI*(P4G-Q*H) _007 Fl_.____:FSAVE=(FRP+E^S)+C(QG+*H) _____ ______ ___ 0008 GSAVE=ChIl*(P* -S*H)"+(5^P-F*Q

0099 HSAVE=CHI ( G'S+R1H ) (FP+E*Q) _______________ 0310 E=ESAVE*CHI 0011__ F_=F SAVE*CHI,_________ ___________ 0012 G=GSAVE __0013_______ H=HSAVE___________________________________ 0014 RETURN 0015 END cJI

Appendix III A Program For The Numerical Evaluation of wAD and cBC A numerical technique has been presented in Section 6. 5. 2 for the evaluation of AD and wB. The Fortran- coded program developed to determine these zeros for the circuit in Fig. 3. 1 is listed here. The flow-sheet of the program is given in Fig. III. 1. 156

157 Read N, YMAX, YINC, T JARRAY(1)... JARRAY(N), = YAX I (l)v... I (N)v, F z(l)... z(N). 1+ jO O + jO I CHOLD =>N O + jo J +. KEY=JARRAY(I) l=I+ 1 Multiply CHOLD by the modified GCP matrix of the Ith element and return in CHOLD PRINT PRINT FAD=./ T YAD _FBC= F FADO F10^- 5 / Fig. III. 1. The flow-sheet of the program RTAPR

PROGRAM LISTINGS FORTRAN_ IG COMPILER ____ MAIN 10-09-69 14:16. 50 C*** PROGRAM RTAPR FINDS APPROXIMATE ZEROS FOR C*** Gl*A+G2*D ANO GI*G2*B+C 0001 DIMENSION JARRAY(15)tFLENG(15),FIMP(15),CHNEW(8),CHOLD(8) C*** INPUTS: C*** N=NtJMBER OF LINES C*** YMAX=MAXIMUM VALUE ON JW-AXIS C*** YINCR=SEARCH INCREMENT ON JW-AXIS C*** G1=TERMINATION AT LINE N END C***~' G2=TERMINATION AT LINE 1 END C***......VECTOR JARRAY DEFINES NETWORK TOPOLOGY. _C***__ JARPAY(I)=O MEANS ITH LINE IS A STUB oo C*'** JARRAY(I)=1 MEANS ITH LINE IS A CONNECTING LINE C*** FLENG(I)=LENGTH ( L/V ) OF THE ITH LINE C**~* FIMP(I)= IMPECANCE OF THE ITH LINE 0002 27 READ(591 END=29)NYMAX,Y INCRtG1,G2_ 0003 I FORMAT(I2,4F10.O) 0004 READ(52) ( JARRAY(I), I=1,N) 0005 2 FORMAT( 1512) 0006 READ(5,3)(FLENG I),I=1,N) FIMP(I),IN) _ __ ___ 0007 3 FORMAT(8F10.0) OOOR X=O. _ _______ 0009 Y=O. C*** Y IS LOCATION_ OF ZEROS 0010 5 IF(Y.GT.YMAX) GO TO 25 C*** CHOLD VECTOR DEFINES A UNITY COMPLEX MATRIX: C*** CHOLD(1)+J*CHOLD(2) CHOL3 CHOL( 3 HOLD( 4) C*** CHOLD 5)+J*CHOLD(6) CHOLD(7)+J*CHOLD(8) 0011 CHOLD(1)=l. 0012 CHOLD(2)0. _ _}_ _._ _______ 0013 CHOLD(3 ) =0. 0014 CHOLD(4)=0. ___ _

0015 CHOLD(5)=0 0016 CHOLD(6)=O __ 0017 CHOLD(7)=1. 0018_______ _____CHOLD(8)=O_______ _________________ C** DO 7 COMPUTES MODIFIED ABCD MATRIX FOR THE LOSSLESS C** PART OF A TRANSMISSION LINE NETWORK. 0019 00 7 11,N 0020 KEY=JARRAY( I) 0021 CALL GENCH(XYVCHNEWKEYtFIMP(I),FLENG( I)) 0022____ CALL PRODUC(CHNEWtCHOLD) _________________________ 0023 7 CONTINUE 0024 FAD=G1*CHOLD(1)+G2*CHOLD(7) 0025 FBRCG1*G2*CHOLD(4)+CHOLDO(6) 0026 IF(Y*EQO0.) GO TO 55 0027 IF(ABS(FADO).LE1dOE-05) GO TO 30 I 0028 ____ _IF(FAD/FADO)30,37,37 ___________________ 0029 30 WRITE(6,38)Y 0030 38 FORMAT(3X,4HYAD=,F12.5) 0031 37 IF(ABS(FBCO).LE.I.E-05) GO TO 75 0032 IF(FBC/FBCO)75955955 0033 75 WRITE(6,77)Y 0034 77 F ORM AT ( 3 X 4HYB CF 12 __________________ ____ 0035 55 FADOAFAD 0036 FBCO=FBC 0037 Y=Y+YINCR __ 0038 GO TO 5 0039 25 GO TO 27_ __ __ 0040 29 CALL SYSTEM 0041__ __..._END________________________

FORTRAN IV G COMPILER GENCH 10-09-69 14:16.53 0001 SUBROUTINE GENCH(X,Y,CHNEWKEYtZW) C*** GENCH(X,Y,CHNEWFItP(I),FLENG(I)) CALCULATES C*** MODIFIED ABCD MATRIX ( THAT IS, WITHOUT COSH FACTOR ) C*** OF STUB OR CONNECTING LINE AT S=X+J*Y. THIS MATRIX C***__ IS RETURNED IN C*** CHNEW( 1)+J*CHNEW(2) CHNEW(3)+J*CHNEW(4) C*** CHNEW(5 ) +J*CHNEW 6 ) CHNEW ( 7)+J*CHNEEW( 8) ___ 0002 DIMENSION CHNEW(8) 0003 WX=W*X _ ____ 0004 WY=W*Y 0005 A=COSH(WX)*COS(WY 0006 B=SINH(WX)*SIN(WY) 0007 C=SINH(WX)*COS(WY) 0008 D=COSH(WX)*SIN(WY) 0009, IF(KEY)1,2,1 _ 0010 1 CHNEW(1)=A 0011 CHNEW( 2)=B 0012 CHNEW(3)=C*Z 0013 CHNEW(4)=D*Z _ _ 0014 CHNEW(5 )=C/Z 0015 CHNEW(6)=D/Z___ ____ 0016 CHNEW(7)=A 0017 CHNEW(8) =B 0018 RETURN 0019 2 CHNEW(1.)=A 0020 CHNEW(2)=B 0021 CHNEW(3)0 _. 0022 CHNEW(4).=O0. 0023 CHNEW(5 )=C/Z 0024 CHNEW(6)=D/Z 0025 CHNEW(7)=A 0026 CHNEW(8)=B 0027 RETURN 0028 END

FORTRAN IV G COMPILER - PRODUC 10-09-69 14:16.55 0001 SUBRCUTINE PRODUC(CHNEWCHOLD) C*** PrODUC(CHNEW,CHOLD) MULTIPLIES THE MATRICES C*** CHNEW AND CHOLD, AND RETURNS THE PRODUCT IN CHOLD. _ _ 3002 DIMENSION CHNEW(8) CHOLD(8),SAVE(8) 0003 SAVE(1)=CHNEW(1)*CHOLD(1)-CHNEW(2)*CHOLD(2) +CHNEW( 3) CHOLD (5) -CHN EW( 4) *CHOLD ( 6) 0004 SAVE(2)=CHNEW(l )*HOL2)CHOLD(2+CHNEW(2)CHOLD( 1+CHNEW(3) CHOLD( 6) +CHNEW(4) *CHOLD( 5) 0005 SAVE(3)=CHNEW(1)*CHOLD(3)-CHNEW(2 *CHOLD(4I 1+CHNEW ( 3 )CHOLD( 7 )-CHNEW (4) CHO LD( 8) 0006 SAVE (4) =CHNEW (1) *CHOLD(4) +CHNEW (2)*CHOLD(3). 1+CHNEW( 3)*CHOLD( 8) +CHNEW (4)*CHOLD( 7) 0007 SAVE(5)=CHNEW(5)*CHOLD(1)-CHNEW(6)*CHOLD2)___ _______ l+CHNEW(7)*CHOLD(5 )-CHNEW( 8) *CHOLD( 6) 0008 SAVE (6) =CHNEW ( 5) *CHOLD( 2 ) +CHNEW (6) CHOLDI J) 1+CHNEW( 7) *CHOLD( 6) +CHNEW(8 ) CHO LD( 5) 0009 SAVE (7)=CHNEW ( 5)*CHOLD(3)-CHNEW(6) CHOLD( 4) 1+CHNEW(7)*CHOLD(7 )-C CHNEW (8) OCHOL(8) 0010 SAVE(8)=CHNEW (5)*CH (4)*CLD(4)CHNE(6)HOLD3) 1+CHNEW(7) *CHOLD(8) +CHNEW(8)*CHOLD(7) 0011 _ DO 3 1=1,8 _ __ 0012 3 CHOLD(I)=SAVE(I) 0013 RETURN 0014 END

Appendix IV A Program For The Numerical Evaluation of Transmission Poles An outline of a numerical technique for finding the transmission poles, i. e., the zeros of (G1 A + G2D) + K(G1 G2 B + C) has been given in Section 6. 5. 3. The Fortran program to compute the transmission poles for the network in Fig. 3. 1 according to this method is listed here. Fig. IV. 1 shows the flow-sheet of this program. 162

163 ReadN, G1, G2, y ARRAY(1)... JARRAY(N), x=0 Read J=l I | (l)v... $(N)/v, AY AK JJ0 — ZO) 00 ~~~....of the th element and Multiply CHOLD by the modified GCP matrix of the 1th element and return in CHOLD Y>^^ IF, IFAD=... ~-_J - FBC=... Y IFADSAV=... FBCSAV... YSAVE=Y. TY= 110 -^^~1 I~K 1^ DERIV=... x=.. DELX, DLYo...: ^ ~ ~ERIV. J.X, Y. DERV.... AIVR=.. REAL=.. IAIMAG=.. T JJ=.. PRINT~ DL < 10-5 x DELX,DELY~{o I~: -<~1 J, X, Y, DELX ^ 10' CAL DELY 5 10' Y= <or J > 10 Fig. IV. 1. The flow-sheet of the program RTLOC

PROGRAM LISTINGS FOQTRA\ IV r C:J4PILE:P AN 1-?.?- 13:33.11 C*: Pr(;/.l r'T!.r F I r.L A. irfp: FI - (Gl?A+Z*r;, )+ A*GC.1, 2* +C )= o(11 O I S4r t!S I(' J^, I'AV,(l' ), FL: C,( lr),F IiP( l5 t ) CtNr ( q ) CHUL O( ) C*** INPU)lS: C ** NINJ;P- [-FR.F L IN- S C* *.l=TtR [I'.ATIf; AT L INE N END C,^~* =l. =P' 1='.T:I lATIC AT I!Nf- I END c~*** Y=APPH!)XI/ATE Z-ru; nF (;lc [,1*G2r)) OR (GO1G2*q+C) C*** Ji"Ff PAY S^:TS T()UPILCGY (r EFTWt^R AS 1pN TAPP C~* F rL..G(l ) AND FI"tP(I ),S IN PTAPR C**.f AK SHrULD qF S4AL. L(u.I UR 0.2) IF Y IS A ZFRO GI*A+CG2*, C*^ AK Th i.tL'-) PF LArGF {C. Ti! 10.) IF Y IS A Z.Qo nF GC*G2*+C C**z- A. StLL t. rF AK Ct~F S SEPVES TO STEP ALONG THIE ROT LOCUS 0002 RE AD(5, 1 )i, G1 I,?,Y 00 0 3 R E4A ) ( r,2 ) [ JA r' A Y ( I ) = I =, N ) 00))4 I F OP AT( I,3F10.3) 0')I. 5 2 F CRAT ( I2) O0)J1 REA)(5,' ) (f LF NG( II=I, "J), (FlP l ), I1=,N) 0007 3 FGCATRT ( r,.O O0,38 J AV=C C**: r'r I';T PASS FRl!HM COOr TI. O.5 CLCIJLATES DEVIVATIVE C *N*i'.:- llF- PrC(lT LUCUS "AN^J FSTI:. ATES INITI L X. 00 9 X=0. 001.) 5 9EA l;( {,tN6 )A 0 1 1 4 F Cr T { F O. 0 ) 00 1 J=l 0013.JJ=() 0014?00 CIt LF 1)=l. C)0J5 C]'l WC. (DOi/. *,l 1'Ct.,( o ) =. ()701 7 C t ri.ir( ) =o. (v') I q 14r'! I'l(' ): =).

001 q crJC L C ( L() 0. 0O)?0 Ct)l:(7) = 1. 001 I CHL[:,f'j =C. 00?32 D 7 1=1,N 0023 KL V=J "i AY(I) 00?4 CALt Gt:NC H( x.YCHNKr4,KEYFrIP I ),FLFNG( ) ) 0025 CALL P't; UC(rCFI;E,CF(Ln)) 00 26 7 CCNT!JNLIF 0?27 1F( JAY.GT.1) Gl) T0 130 0O?8 I -A='.I t CHCL! I( 1 ) +G2*CFiLD( 7) 00?). F C- r, ^2?Cr LO ( 4 ) +CJIrqLD( ) 0030 IF(JAY.C.] ) G! TO 110 0031 FAi)S.A'V = F 0032 FiKCSAV=t PC 0C33 IF(Y ) I 1, 00 4 15 YSAVFr= 0035 Y=. 09 0^' YPOYPI)=. "l 00o 7 JAY=JY+1 u 0033 GO TO 210 05.O 9 1) VSAVF=Y 00J4 Yp!=.'0 *YSAVE 0041 Y=Y 1.0 l 0042 JAY=JaY+1 004 3 GC 1(;?00 C** I 110i rFC InFS' ^llICH ENO CF THE ROOT-LOCUS ON STARTS FROCM. 00)4. 110 IF(I'K.t'1.1.) GQ TO 120 00(/^,4 PE I V= (F r C-F ACS SV )/vP OY) 4 Xs\v' = fr S AV / 1, IV 00:7 ll(0f ((,) X SAVF u 4} 1 F)5 V 0 A,tT (r X X. Al'X g/' V f F,1 5. 5'.J too X' =Y: Y SAV[ C-,: JYI:. --:ra!SI [rtll1 IS TH,: t:li-'SI TI'^- AFTFR ESTIMATINGl X. n0'5.':) JAY'V + I (00) VV/ v, 0'.)r,7 (;f. T'~'^(ji

Or 5 3 1 20 ) E I V= ( r pC-Iv- fC,-AV ) / Y P C)O4 S 4- = F A SA V/).JU I V 005 iRI F1(. I, C'.) S VY. o0 r^.,c X=-Y AVE/AK nr, A JAY= JY+1 O )-3 q5 Y=VSAV 0o05 C.(:''?o00 00o ) 13-f REtAL =^1 ICH rc ( ) +~G?*CHL n(7 ) * AK( G1IG2 CH0ln L 3 )+C(lL;)( 5 ) ) 0061 A I'A,= 1: L ) ( + G2 CL 8 ) + \K< ( G *G? CHL L n ( 4 +CIL 6 ) C*-* JJ=1'EA' S TiF NtXE X HAS #IF. rN INCREFNTF'.Tr TC. ESTIMAT F DFRIVATIVF 0062 IF(JJ.ro).) rGi rn 1. (003 A IVi = F AL 30 64 AI V= II'AG 005 XSV, r=X 0015G x=XSAV-' 1.O1 0L067- J= J + 1I 006R GJ T(I 200 0069 135 X=YSAV- 00.)70 JJ=O 0;)71 ANVK=- FAL 0072 NVI =.V I A O(73 R.=(ANVP-AIVPR)/(.0OCI'X) O0 7 S A I x= ( A r,' V I,- a I V R ) / (. C 1O!; X ) 0074 AIX=(r VII-AIVI / (.~CO 1-X) C;** PETtEEN 72 TO) 73 CALCULATF DER IVATIVFS WITHt RESPECT TO C~*.-X \: rr. Y ( bUSING CU,[JCtHY-RIF-AN C[NrITION ). 0075 RY=-AIY 0076 AIY=PX 0077 Di0L= RY A I X-FP X: A I Co~~ iF'r1 rFEN 7., Tl 77 CALCIJLATFS NF X ANn Y Ry'!FWTlN-RAPHSOF!q. C,: J =.JIJ.;.FP f'Fr'JASS TS TlCtJt r 1;r T' i - AP HS N. 007)L (, I V I Y - I VI \:'Q Y )/) F N 0079 0.LY(' I VI,:X-A IV ^ Q IX~/0 A[:! o00.30 C At.- S(AIVl ) +A S(, (A T VI) 903, X=Y+f: LX 00.2 Y=Y4!)I Y 00^ 1- (,,'I ):' [.lXt, ) vJ,

0 ^ ^i it F1'O= ET C C L X ~5,5xs5HPF L Y=,F 15.,t 5)xHJ=, 1 7?) 00^ 3 R ITR (, 1 E I05 - ) X, Cy L 003^' 105 F(A T (2HX= I- 5,, x,?fY=,FI.5, 5X,4HCAL=, F 1565'0O7 IF(rCAL.L.I.F-)S.CR. (A' S (tL X).LE.1.E-IJ7. ANO. 1lrS( rLY).iF.1.E-?7) ~^ J.G.10 ) GO TO 5 nO'48 J=J+1I 00oC Gf Tl iT Oo000 69 CALL SVS;FMf 0091 \) FORTRANA IV G Cw>3I PI-.P GENCH 1N0-2-0) -9 13:33. 3?0 0001 SU UJUTINr GENCH(XV,CHI E vKCYYZW) C*** GrNCH ( X,Y,CHK:j, FT ( I I ),FLNG( I)) CALCULAT FS C~^r; ~:r;:)nI[Ibf ^t'iC MATRIX ( THAT IS, WITHOUT CUFSH FACTOR ) C**#^ -CF STU()' C. C('J.'FCT ING L INE AT S=X+J*Y. Th IIS ATRI X C** ITS Pt:ITUPND IN C~^ (C\F ( 1 + J C I4 i ( 2~ C HN (3~FJ C F 3 +( ) Trc*~ CHj\F +.C( 5) +J jC ( 6~ CHlEW( 7)+ J CHN\W(8~ 0302 flIlF!'SlU CH'T\:W(f ) 0 0023 W X= Xt r 000" Y^^ o0!o^ *rnCSSy(WX)* rs(AY) 3 000^ I 8 S) C. SIH( X) S I N (.Y) OJ3 ^'i 7 C = S3 I r.H~r( IN X ) ~ ( C~.r. (;JY ) C00^ Ir(KCY)1,2,J 00 II Ct F( c) =. 001? C JK. C' 7 C.'3 C1' IF ^ (A) -P)07 00 I f^ (1, If -4 1 V: — C. / 7 (JO 1 5 C" -it v-: =01)-)/

ChNLJ;(7)=A 0: 7 CHPr..I ~=B C)1" IR ETUP',i C'.) 2 0r k 2 P -f Qo1^? CH d =01A CC)20 Ciirw(<?P)= 0 )?? CHNFH.: I O0. 00 3 CH"IF =C/7 00^4 C^^^(^~=n/7 00 2 5 C P "iJ-?V.( 7 ~=A 0027 5I7 002$ E r FOKT.t,'I IV G CP'iPILE? PPIDUZ C 10-?I -6c) 13:33.23 03 0001 SUW OUTI1F P(F -UC(CHNEWCHOLD) C~~ PDl'LJD C(C^ErCF "LD~ ^UJLTIPLIrS THF MATRICES C7^^ CHrrqW AND CHLI.), AND) QFTUKU4S THE PRODUCT IN CHOLI)D 000? I:AF.NSIIN CIH.W(q),CHOLD(8),SAVf(8)'0003 SAV[:(l)^CkNIS (l1)~Ci(,LO(1)-CHr'F-'(2h^CHO'LD(2) 0,.) 3 5 4 V [: ( 3 S4V1 cc I1 *CHN * ( I ): CHKL F) ( J 2 ) -C4 ( I2 ) _: 0 HOC li1L 2 6 l 4CW,.F('3)^'i CP~.)LO5)~CH^N^(^ ) "CHOLlO 6)' O004 SAV"',)=C 4-V()1 CMU1. (2)4CHWFNI )4CHOL DCI) 4I c H^'^ Y ('3 ) C i r i (6) CHN. ( 4 ) ci cH L s 0005 SAVF(1)=Ch\%F1'(1)CHnrLF3)-C( (? CHaL)(L~4 1i+V'*^.(i(3 )~^CH)L{i'(7 ) CH\F/< (A ) c~hCi ^ C ) oo0A SA(V ~ C Ii rC i (F W ( CH (:L~(4 4C I I[ 2 )0 C00 D 3) i +CH2A (3 ) ^c7tH')( ) +C!T w C ) 4 ( 1.' ( ) 0037 SAV () CH r( ) Ch f)( -C I (N ) CH.D(2 I +C'% 7):C.L( )-H ()')L(6)7 I l~1;^^t* 7):r'F^ L ( ^) *(<V ^ (2) ^(t~)^('; oo~t SAVr7):C.J -,(\ r <C II-'C -ClI ( ).CILC() I i 40 C C 7 ) ci('I n( 7 )-Ch\r.NC^) Ci')I C )

O010 SAVF ( ~)CHNFA().HCLr (I) CHN 6 CHL() I +C; r\'F; ( 7 )CI!:1 0 ( ^ ) *CnrIL7( 7) 0011 P0 3 T01,8 0o0 1? 3 CKl:Lr( I)=SAVf ( 1) 0013 QF1tJRP.N oo0014 N) f0) CD

Bibliography 1. P. I. Richards, "Resistor-Transmission-Line Circuits, " Proc. IRE, Vol. 36, February 1948, pp. 217-220. 2. H. Ozaki and J. Ishii, "Synthesis of Transmission Line Networks and The Design of UHF Filters, " IRE Trans. On Circuit Theory, Vol. CT-2, December 1955, pp. 325-336. 3. M. C. Horton and R. J. Wenzel, "General Theory And' Design of Optimum Quarterwave TEM Filters, " IEEE Trans. On Microwave Theory and Techniques, Vol. MTT-13, May 1965, pp. 316-327. 4. R. J. Wenzel, "Exact Design of TEM Microwave Networks Using Quarterwave Lines, " IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-12, January 1964, pp. 94-111. 5. B. K. Kinariwala, "Theory of Cascaded Transmission Lines," Proceedings of The Symposium on Generalized Networks, Microwave Research Institute Symposia Series, Vol. XVI, Polytechnic Press of The Polytechnic Institute of Brooklyn, New York, 1966. 6. R. M. Fano and A. W. Lawson, "Microwave Filters Using Quarterwave Couplings, " Proc. IRE, Vol. 35, November 1947, pp. 1318-1323. 7. W. W. Mumford, "Maximally Flat Filters in Waveguides," Bell System Technical Journal, Vol. 27, October 1948, pp. 684-713. 8. S. B. Cohn, "Direct- Coupled-Resonator Filters, "Proc. IRE, Vol. 45, February 1957, pp. 187-196. 9. R. E. Collin, "Theory and Design of Wideband Multisection Quarterwave Transformers, " Proc. IRE, Vol. 43, February 1955, pp. 179-185. 170

171 Bibliography (Cont.) 10. H. J. Riblet, "General Synthesis of Quarterwave Impedance Transformers, " IRE Trans. on Microwave Theory and Techniques, Vol. MTT-5, January 1957, pp. 36-43. 11. R. Levy, "Tables of Element Values for The Distributed Low-Pass Prototype Filter, " IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-13, September 1965, pp. 514-536. 12. L. Young, "The Quarterwave Transformer Prototype Circuit, " IRE Trans. on Microwave Theory and Techniques, Vol. MTT8, September 1960, pp. 483-489. 13. L. Young, "Direct Coupled Cavity Filters for Wide and Narrow Bandwidths, " IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-11, May 1963. pp. 162-178. 14. L. Young, "Tables for Cascaded Homogeneous Quarterwave Transformers, " IRE Trans. on Microwave Theory and Techniques, Vol. MTT-7, April 1959, pp. 233-237 and Vol. MTT-8, March 1960, pp. 243-244. 15. L. Young, "Stepped Impedance Transformers and Filter Prototypes," IRE Trans. on Microwave Theory and Techniques, Vol. MTT-10, September 1962, pp. 339-359. 16. E. M. T. Jones and J. T. Bolljahn, "Coupled Strip Transmission Line Filters and Directional Couplers, " IRE Trans. on Microwave Theory and Techniques, Vol. 4, April 1956, pp. 75-81. 17. S. B. Cohn, "Parallel-Coupled Transmission-Line-Resonator Filters, " IRE Trans. on Microwave Theory and Techniques, Vol. MTT-6, April 1958, pp. 223-231. 18. G. L. Matthaei, "Design of Wideband (and Narrowband) Bandpass Microwave Filters on The Insertion Loss Basis, " IRE Trans. on Microwave Theory and Techniques, Vol. MTT-8, November 1960, pp. 580-593.

172 Bibliography (Cont.) 19. H. Ozaki and J. Ishii, "Synthesis of A Class of Stripline Filters, " IRE Trans. on Circuit Theory, Vol. CT-5, June 1958, pp. 104-109. 20. G. L. Matthaei, "Interdigital Bandpass Filters, " IRE Trans. on Microwave Theory and Techniques, Vol. MTT-10, November 1962, pp. 479-491. 21. R. J. Wenzel, "Exact Theory of Interdigital Bandpass Filters And Related Coupled Structures, " IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-13, September 1965, pp. 559-575. 22. L. A. Robinson,'Wideband Interdigital Filters with Capacitively Loaded Resonators, " G-MTT Symposium Digest, 1965, pp. 33-37 23. G. L. Matthaei, "Combline Bandpass Filters of Narrow Or Moderate Bandwidth, " Microwave Journal, Vol. 6, August 1963, pp. 82-91. 24. B. M. Shiffman and G. L. Matthaei, "Exact Design of Bandstop Microwave Filters, " IEEE Trans. on Microwave Theory And Techniques, Vol. MTT-12, January 1964, pp. 6-15. 25. B. M. Shiffman, "A Harmonic Rejection Filter Designed By An Exact Method, " IEEE Trans. on Microwave Theory And Techniques, Vol. MTT-12, January 1964, pp. 58-60. 26. L. Young, G. L. Matthaei and E. M. T. Jones, "Microwave Bandstop Filters with Narrow Stop Bands, " IRE Trans. On Microwave Theory and Techniques, Vol. MTT- 10, November 1962, pp. 416-427. 27. B. M. Shiffman, "A Multi-Harmonic Rejection Filter Designed By an Exact Method, " IEEE Trans. on Microwave Theory And Techniques, Vol. MTT-12, September 1964, pp. 512-516. 28. H. J. Carlin and W. Kohler, "Direct Synthesis of Bandpass Transmission Line Structures, " IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-13, May 1965, pp. 283-297.

173 Bibliography (Cont. ) 29. A. I. Grayzel, "A Synthesis Procedure for Transmission Line Networks, " IEEE Trans. on Circuit Theory, Vol. CT-5, September 1958, pp. 172-181. 30. P. I. Richards, "A Special Class of Functions with Positive Real Part in a Half Plane, " Duke Mathematical Journal, Vol. 14, September 1947, pp. 777-786. 31. N. Ikeno, "On Distributed Filters, " Proc. Joint Conference Of Three Societies Related to Electrical Engineering, May 1952 (Japanese). 32. N. Ikeno, "A Consideration in Coaxial Filters, " Journal of The Institute of Electrical Communication Engineers, Japan, Vol. 36 June 1953 (Japanese). 33. N. Ikeno, "Design of Rod-Like Circuits, " Proc. Joint Conference of Three Societies Related to Electrical Engineering (Tokyo Chapter), October 1952 (Japanese). 34. K. Kuroda, "A Method to Derive Distributed Filters From Lumped Filters, " Proc. Joint Conference of Three Societies Related to Electrical Engineering (Kansai Chapter), October 1952 (Japanes e). 35. H. Ozaki, "Synthesis of Unbalanced 4-Pole Consisting Of Distributed Elements, " Journal of The Institute of Electrical Communications Engineers, Japan, Vol. 36, December 1953 (Japanese). 36. N. Ikeno, "Design Theory of Distributed Filters, " Progress Report of The Institute of Electrical Communications of JTT, Vol. 4, 1957 (Japanese). 37. K. Kuroda, "Exact Design Theory of Distributed Networks, " Journal of The Institute of Electrical Communications Engineers, Japan, Vol. 37, May 1954 (Japanese). 38. Y. Kasahara and T. Fujisawa, "Design of Distributed Constant Filters, " Journal of The Institute of Electrical Communications Engineers, Japan, Vol. 37, Janaury 1954 (Japanese).

174 Bibliography (Cont.) 39. Y. Kasahara, H. Ozaki and T. Fujisawa, "Design of Distributed Constant Filters, " Technological Reports, Osaka University, Vol. 4, 1954. 40. A. M. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New York, 1966. 41. V. L. Zaguskin, Handbook of Numerical Methods for The Solution of Algebraic and Transcendental Equations, Pergamon Press, New York, 1961 (Translated from the Russian by G. O. Harding). 42. J. Todd, Survey of Numerical Analysis, McGraw-Hill Book Company, Inc., New York, 1962. 43. P. Henrici, Elements of Numerical Analysis, John Wiley and Sons, Inc., New York, 1964. 44. E. Isaacson and H. B. Keller, Analysis of Numerical Methods, John Wiley and Sons, Inc., New York, 1966. 45. A. S. Householder, Principles of Numerical Analysis, McGrawHill Book Company, Inc., New York, 1953. 46. G. Hadley, Nonlinear and Dynamic Programming, AddisonWesley, Reading, Massachusetts, 1964. 47. G. Zoutendijk, "Nonlinear Programming: A Numerical Survey, SIAM Journal on Control, Vol. 4, February 1966, pp. 194-210. 48. E. W. Cheney and A. A. Goldstein, "Newton's Method For Convex Programming and Tchebycheff Approximation, " Numer. Math., Vol. 1, 1959, pp. 253-268. 49. R. E. Griffith and R. A. Stewart, "A Nonlinear Programming Technique for Optimization of Continuous Processing Systems," Management Science, Vol. 7, July 1961, pp. 379-392. 50. G. B. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, 1963.

175 Bibliography (Cont.) 51. G. Hadley, Linear Programming, Addison-Wesley PublishinL Co., Massachusetts, 1962. 52. G. Zoutendijk, Methods of Feasible Directions, Elsevier, Amsterdam, 1960. 53. A. V. Fiacco and G. P. McCormick, "The Sequential Unconstrained Minimization Technique for Nonlinear Programming, A Primal Dual Method, " Management Science, Vol. 10, January 1964, pp. 360-366. 54. L. Weinberg, Network Analysis and Synthesis, McGraw-Hill Book Company, Inc., New York, 1962. 55. Go L. Matthaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance Matching Networks and Coupling Structures, McGraw-Hill, New York, 1965. 56. H. M. Altschuler and A. A. Oliner, "Discontinuities in The Centre Conductor of Symmetric Strip Transmission Line, IRE Trans. on Microwave Theory and Techniques, Vol. MTT8, May 1960, pp. 328-339. 57o University of Michigan Terminal System (MTS), Second Edition, December 1967, pp. 171-174, 266.

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182 DISTRIBUTION LIST (Cont.) No. of copies 1 Chief Missile Electronic Warfare Tech Area EW Lab, USA Electronics Command White Sands Missile Range, N. M. 88002 I~1 ~ Headquarters U. S. Army Combat Developments Command Attn: CDCLN-EL Fort Belvoir, Virginia 22060 1 USAECOM Liaison Officer MIT, Bldg. 26, Rm. 131 77 Massachusetts Avenue Cambridge, Massachusetts 02139 18 Commanding General U. S. Army Electronics Command Fort Monmouth, New Jersey 07703 Attn: 1 AMSEL-EW 1 AMSEL-PP 1 AMSEL-IO-T 1 AMSEL-GG-DD 1 AMSEL-RD-LNJ 1 AMSEL-XL-D 1 AMSEL-NL-D 1 AMSEL-VL-D 1 AMSEL-KL-D 3 AMSEL-HL-CT-D 1 AMSEL-BL-D 3 AMSEL-WL-S 1 AMSEL-WL-S (office of records) 1 AMSEL-SC I~1 ~ Dr. T. W. Butler, Jr., Director Cooley Electronics Laboratory The University of Michigan Ann Arbor, Michigan 48105 16 Cooley Electronics Laboratory The University of Michigan Ann Arbor, Michigan 48105

Securit v Classification DOCUMENT CONTROL DATA - R & D!Sfclirity classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION Cooley Electronics Laboratory Unclassified The University of Michigan 2b. GROUP Ann Arbor, Michigan 48105 3. REPORT TITLE Iterative Synthesis of TEM-Mode Distributed Networks 4. DESCRIPTIVE NOTES (Type of report andinclusive dates) C.E.L. Technical Report No. 199 5 AU THOR(S) (First name, middle initial, last name) Mahdi, Solaimanul 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS November 1970 210 57 I 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) DAAB07-68-C-0138 01482-12-T TR 199 b. PROJECT NO. I HO 62102 A042 01 02 c. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) _d. ECOM-0138-12- T 10. DISTRIBUTION STATEMENT This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of CG, U. S. Army Electronics Command, Fort Monmouth, N. J. Attn: AMSEL-WL- S 1 1. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Army Electronics Command Fort Monmouth, New Jersey 07703 Attn: AMSEL-WL- S 13. ABSTRACT A procedure is described for synthesizing transmission networks which are interconnections of uniform line sections. An iterative, digital computer algorithm is developed which achieves a dominant pole synthesis. The line lengths and the characteristic impedances are controlled individually, which gives design flexibility not found in synthesis procedures based on Richards' transformation. The characteristic impedances may be restricted by upper and lower bounds when there is no restriction on the line lengths. The procedure is detailed for a TEM mode structure of alternating open stubs and connecting lines. The method uses a NewtonRaphson iterative scheme to adjust the characteristic impedances and lengths of the transmission lines for a prescribed set of dominant transmission poles.- By controlling the stub line lengths, the dominant pole positions, the principal transmission zeros, and bounded characteristic impedances can be achieved simultaneously. The occurrence of nondominant poles has been analytically investigated. Frequencies at which each transmission line element is a quarterwave long divide the s-plane imaginary axis into halfwave frequency bands. In every semi-infinite s-plane strip which these frequency bands subtend parallel to the imaginary frequency axis, one and only one nondominant pole is present. A numerical technique has been outlined which locates these poles. The approximate cancellation of these poles extends the frequency range over which the network characteristics can be controlled. DD FORM 1473 (PAGE 1) S/N 0101 -807-681 1 Security Classification A- 31:408

UNIVERSI OF MICHIGAN Se curity Classification 01 1846 ____ 3 9015 03466 1846 14. LINK A LINK B LINK C KEY WORDS __ ROLE WT ROLE WT ROLE WT Iterative synthesis TEM- Mode Newton-Raphson iterative scheme Synthesizing transmission networks D D. NOVe1473 (BACK) S/N 0101-807-6821 Security Classification A-31409