RCT FILTER SYNTHESIS Thesis for the Degree of Ph. D, MICHIGAN STATE UNiVERSITY RONALD L. McNALLY 1969 kid/“ML -- fl ”5: Tr '. I; It} A? 14 R Y I‘Ilt'lllégzjn State 5. University "J It‘p ' V; m1: (7:. ‘7 it mum This is to certifg that the thesis entitled R CI" FILTER SYNTHESIS presented by R onald L. MCN ally has been accepted towards fulfillment of the requirements for ph. D. degree in E. E. Major progessfir . , / A” ’/) / Date MaY 15. 1969 0-169 #1“ BINDING DY ' HOME 8- SDNS’ 300K BINDERY INC. ‘‘‘‘‘ Y BINDERS I m, mom“; ABSTRACT RCT FILTER SYNTHESIS by Ronald L. McNally A general basis for the synthesis of active RCF (resistor-capacitor—gyrator) filter is presented where all gyrators are grounded. The approach requires the analysis of three term- inal RC sections cascaded through active (unbalanced) grounded gyrators. Two important theorems are established as a consequence of this analysis: 1. If the RC sections satisfy the following conditions: i) Each RC section is connected; ii) For the input and intermediate RC sections the edges corresponding to the conductances form a connected graph when the input and output termin— als (of the section) are grounded; iii) For the last or output RC section, the edges cor- responding to the conductances form a connected graph when the input terminals are short circuited; iv) All RC sections, except the output section, Ronald L. McNally contain at least two terminals in addition to the ground terminal; v) All RC sections contain one or more conductances; then the RCP filter is stable and remains stable irrespec- tive of RC component or gyrator parameter variation. 2. The minimum number of real poles of the voltage-ratio transfer function of a low-pass RCF filter can be determined from the gyrator placement. It is demonstrated that the Calahan [CA] and Horowitz [TH] polynomial decompositions can be derived one from the other. Both polynomial decomposition methods are extended to include those polynomials which contain distinct negative real zeros. It is also established that fourth degree low- pass (high-pass) voltage-ratio transfer functions of the form Tv = k/P(s) (TV = ks4/P(s)), where P(s) is strictly Hurwitz, can always be realized with two-gyrator RCr net- works. Realization procedures using a computer program are established for realizing fourth or higher degree RCF [CA] Calahan, D. A., "Restrictions on the Natural Frequen- cies of an RC-RL Network," Journal of the Franklin Institute, Vol. 272, pp. 112-133 (August 1961). [TH] Thomas, R. E., "Polynomial Decomposition in Active Network Synthesis," IRE Transactions on Circuit Theory, CT-8, pp. 270-274 (September 1961). Ronald L. McNally filters from the voltage-ratio transfer functions. Practical examples are realized and displayed in the form of tables. RCT FILTER SYNTHESIS by Ronald L. McNally A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1969 ACKNOWLEDGMENTS The author wishes to thank his major professor and thesis advisor, Dr. Y. Tokad, for his guidance and encouragement during the development and writing of this thesis. Thanks are also due to Dr. R. C. Dubes, who served as the author's academic advisor at the beginning of the author‘s doctoral program and while Dr. Tokad was on sabbatical leave. The author also wishes to thank the other members of his Guidance Committee for their interest in this work: Dr. H. E. Koenig, Dr. G. L. Park and Dr. E. A. Nordhaus. Dr. J. S. Frame also deserves thanks for his suggestions which led to the proof of Theoren 2.5.1. The author especially wishes to express his grat- itude to the Division of Engineering Research of Michigan State University for the financial support of this re~ search. Finally the author would like to thank his wife, Sarah, for being around the during the whole process. ii TABLE OF CONTENTS Chapter I. II. III. INTRODUCTION 0 O O I O O O O O O O O O O O l I l BaCkground O O O O O O O O O O O O O 1.2 Purpose . . . . . . . . . . . . . . . 1.3 Summary of Chapters . . . . . . . . . 1.4 Preliminary Definitions . . . . . . . THE DERIVATION OF SOME PROPERTIES OF RCI‘ FILTERS O O I O O I O O I O O O O O 2.1 The General Form of the Denominator Polynomial for RCF Transfer Functions O O O O O I O O O O O O O 2.2 RC Networks Cascaded Through Bridged Gyrators . . . . . . . . . . . . . RC Networks Cascaded with Gyrators . 2.4 Stability of the Transfer Function . 2.5 Minimum Number of Real Poles of T for Low-Pass RC Ladders CascadeX with Gyrators . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . DEVELOPMENT OF REALIZATION FORMULAS FOR ONE—GYRATOR FILTER o o o o o o o o o o o 3.1 Calahan's Decomposition . . . . . . . 3.2 Decompositions for Polynomials with Real Zeros . . . . . . . . . . . . 3.3 Realization Techniques Using Calahan and Horowitz Type Decompositions . 3.4 Conclusions . . . . . . . . . . . . . iii Page b»tu F'ld P‘ 15 19 26 32 4O 41 42 67 75 86 Chapter IV. V. LOW-PASS RCF FILTER REALIZATIONS . . . . . 4.1 Two-Gyrator RCF Realizations for Fourth Degree Low-Pass RCF Transfer Functions 0 O O O O I O O O O O O O 4.2 Computer Synthesis of Low—Pass Fourth Degree Filters . . . . . . . . . . Conjectured Necessary Conditions . . 4.4 Computer Realizations for Fifth and Sixth Degree Low-Pass Transfer Functions Using RCF Configurations 4.5 Band-Pass and High-Pass Filter Realizations . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . CONCLUSIONS . . . . . . . . . . . . . . . APPENDIX A O O O O O O O O 0 O O O O O O O O O O A.l Computer Algorithm . . . . . . . . . A.2 Usage . . . . . . . . . . . . . . . BIBLIOGRAPHY O O O O O O I O O O O O O O 0 iv Page 87 88 101 105 114 126 133 135 137 138 142 145 Figure 1.4.1 2.2.1 LIST OF FIGURES Gyrator representations . . . . . . . Three terminal RC networks cascaded through RC-bridged-gyrators . . . . Three terminal RC networks cascaded through gyrators . . . . . . . . . RCP network for Theorem 2.5.1 . . . . RLC and RCF equivalent forms . . . . RCr-R network representation . . . . General one— gyrator RCF filter . . . Realization of RC? filter for Example 3.3.]- I O O O I O O O O O O O O O 0 General RCNIC filter . . . . . . . . RCNIC realization for Example 3.3.2 . General RCT-R filter . . . . . . . . RCP-R realization for Example 3.3.3 . General two-gyrator RCF filter . . . Fourth degree low-pass RCP filter . . Frequency Response Butterworth Filter Frequency Response Chebyshev Filter . Two-gyrator sixth degree RCF low-pass filter . . . . . . . . . . . . . . Page 17 20 36 43 43 79 79 82 82 85 85 90 90 109 110 112 Figure Page 4.3.2 Three-gyrator sixth degree RCF low-pass filter . . . . . . . . . . . . . . . . 112 4.4.1 Fifth degree low-pass RCP filter . . . . 118 4.5.1 RLF netowrk . . . . . . . . . . . . . . . 129 4.5.2 Equivalent forms . . . . . . . . . . . . 129 4.5.3 RCr network . . . . . . . . . . . . . . . 130 4.5.4 RCF High-pass filter . . . . . . . . . . 130 4.5.5 Initial network for Theorem 4.5.2 . . . . 131 4.5.6 Final network for Theorem 4.5.2 . . . . . 131 vi LIST OF SYMBOLS resistor — resistance capacitor - capacitance inductor - inductance gyrator - either active or passive negative impedance converter negative resistor or resistance determinant of an admittance matrix determinant of admittance matrix when the ith row and jth column are re- moved determinant of an RC admittance matrix when the ith row and jth column are removed determinant of an RCF matrix when the ith row and jth column are removed vii CHAPTER I INTRODUCTION 1.1 Background In the technical literature very little theoreti- cal work has appeared concerning the synthesis of RCF filters. Calahan [CA1] has published some basic work on polynomial decomposition which is applicable to one- gyrator RCF networks. He also has shown that, whenever possible, a one-gyrator RCP filter realization of a transfer function is always less sensitive to parameter variations than an equivalent RC-NIC filter realization of the same transfer function [CA2]. To the best of the author's knowledge, no work has appeared in the litera- ture on RCF filters containing more than one gyrator. 1.2 Purpose The purpose of this thesis is to provide a theo- retical basis for RCF filter synthesis using one or more gyrators and also to develop realization procedures for synthesizing low—pass voltage-ratio transfer functions with RC networks and one or more gyrators. This problem is of practical interest since the use of gyrators provides a means of realizing complex natural frequencies without inductors. The elimination of inductors is de- sirable for the following reasons: i) the design of high quality inductors is dif- ficult and costly at low frequencies ii) inductors cannot be "grown" in integrated circuits iii) inductors are difficult to miniaturize in micro-miniature circuit technology. Two other methods for eliminating inductors in filter realizations are the use of NIC's (negative im- pedance converters) and controlled sources. Both of these methods introduce the possibility of instability into the network realization whereas ideal gyrators, being passive components cannot cause a network to be— come unstable. This property also holds for active gyrators in a restricted network topology as is shown in this thesis. The use of one—gyrator filter synthesis is not possible, in most cases, due to rather stringent require- ments on the transfer function [CA1]. By allowing more than one gyrator, these requirements can be relaxed or removed completely as is shown in Chapter IV. It is al— ways possible to replace an inductor by a capacitor loaded gyrator or by using two grounded gyrators and one capacitor [HT]. The latter, although it is a practical method, is somewhat extravagant as the realization of a fourth degree voltage-ratio low-pass transfer function would require four grounded gyrators. It is shown in this thesis that such fourth degree voltage-ratio low- pass transfer functions can be realized with at most two grounded gyrators. 1.3 Summary of Chapters In Chapter II, the basic form of the denominator polynomial of voltage-ratio transfer functions TV is es- tablished for various RCP network configurations. Sepa- rating the denominator polynomial of a given TV into these basic forms constitutes the first step in any RCF network synthesis procedure. A theorem is proved which establishes the stability of RCF filters consisting of a broad class of three terminal RC sections cascaded through active gyrators. Another theoren is proved which establishes the minimum number of real poles for a trans— fer function TV corresponding to a RCF network consisting of low-pass RC ladders cascaded through gyrators. In Chapter III, the derivation of the Calahan [CA1] and the Horowitz [HO] polynomial decomposition (separation into forms suitable for network realization) methods from each other is established. Extensions of these decomposition methods to polynomials containing distinct negative real zeros, in addition to complex zeros, are also established. The use of these decomposition methods is demonstrated with sample filter realizations. In Chapter IV, a theorem is proved which estab- lishes that any low-pass voltage-ratio transfer function TV which has a strictly Hurwitz fourth degree denomina- tor, can always be realized with a two-gyrator RCF filter. Analytic and computer realizations are given to sample problems. In addition, sample computer realizations are given for some fifth and sixth degree low-pass vol- tage-ratio transfer functions. An extended version of Calahan's angle condition is conjectured for cases where the number of gyrators is greater than one. Finally, a theorem is proved which establishes a two-gyrator RCP network realization for high-pass voltage-ratio transfer functions of the complex variable 5, Tv(s), when TV(%) has a two-gyrator RCP realization. 1.4 Preliminary Definitions Definition 1.4.1 An RC admittance function is a real rational function in the complex variable 5 of the form N(s)/D(s) where the zeros of N(s) and D(s) alternate along the negative real axis and the largest zero of N(s) is less than or equal to zero and greater than the larg- est zero of D(s). Definition 1.4.2 An RC impedance function has the pro- perties of the reciprocal of an RC admittance function. Definition 1.4.3 An RL admittance function has the pro- perties of the reciprocal of an RC admittance function. Definition 1.4.4 An RL impedance function has the pro- perties of an RC admittance function. Definition 1.4.5 A passive, ideal, (or balanced) gyra- tor is a 3 or 4 terminal network component which is represented by Fig. 1.4.1 and has the admittance matrix — _ O a -a 0 (1.4.1) where a2 > 0. Definition 1.4.6 An active or unbalanced gyrator is a 3 or 4 terminal network component which is represented by Fig. 1.4.2 and has the admittance matrix (1.4.2) where a &> 0. Definition 1.4.7 A polynomial in the complex variable 3 is called strictly Hurwitz if it has all of its zeros in the open left half of the s—plane. Four terminal case C v G Three terminal case Fig. 1.4.1 Gyrator representations. CHAPTER II THE DERIVATION OF SOME PROPERTIES OF RCT FILTERS In this chapter, the properties of Open-circuit voltage-ratio transfer functions, T for grounded RCP V’ (resister capacitor gyrator) networks are considered. The reason for developing these prOperties is to use them in synthesizing low pass filter configurations. The results, however, are applicable in the synthesis of band pass and high pass configurations, since the desired properties are developed in a more general con- text than the low-pass case. Initially an (n+1)-terminal RC network in which ideal gyrators are embedded is analyzed. Since the denominator polynomial of the trans— fer function T for such general networks does not give V any clue to the realization of T it is necessary to V' impose certain restrictions on this general network con- figuration. However, properties to be developed for this more general class of networks are applicable to the more restricted classes of RCF networks. From the practical point of view, the RCP network is restricted to RC sections cascaded through RC bridged and grounded gyrators. Further simplification is achieved by cascad— ing grounded RC sections through grounded gyrators only. A theorem is proved for a class of networks es- tablishing the invariance of the denominator polynomial of Tv when active gyrators replace ideal gyrators. An- other theoren is proved which establishes the stability of a class of RC-(active gyrator) networks. Finally, when the RCF network is restricted to low-pass RC ladder sections interconnected through active or passive gyra- tors, a theoren is established which gives the minimum number of real poles that the transfer function T can V have. 2.1 The General Form of the Denominator Polynomial for RC Transfer Functions. Consider an (n+l)-node connected RCF network in which each gyrator has three terminals. It is a known fact [KTK] that for the complete solvability of a network containing ideal gyrators, a formulation tree should exist such that both the edges corresponding to a gyra- tor are included in this tree or in its co-tree. If this topological condition is not satisfied, then the network cannot have a complete solution. In order to insure that the branch equations for the RLF network can be written in a suitable form, the following assumptions are used throughout the thesis. Assumptions: i) The RC portion of the given RCF network is connected ii) The P portion of the given RCF network con- tains no circuits. Under the above assumptions both the edges corresponding to a gyrator can be included in a formulation tree. For such a formulation tree the branch equations for the RC? network can be written in the following form: I = YV Or, in detail, 1 ”'7 F - d 7] “‘ ‘ 1 V1 -d 1 V2 I = < RC admittance + d2 L I ' matrix _a ’ 2 0 0Ln 0 -a V By assumption, the RC portion of the RC? network is connected, hence the admittance matrix Y in Eq. RC 2.1.1 is symmetric and non-singular; in fact, it is positive definite for real and poitive values of the com- plex variable "s". For this reason the quadratic form associated with the admittance matrix Y can be written as 10 T_T T_T x YX — x YRCX + x er — x YRCX (2.1.2) Since YT is skew symmetric, the quadratic form associated with Yr vanishes identically and therefore Eq. w.l.2 implies that the admittance matrix Y (although nonsym— metric) is positive definite for all real and positive values of "s". This proves that Y is non-singular. The entries in the first and the last position in the first and the last position in the first column of Y.1 are 11' An1 represent cofactors of order (n-l) of Y. In Eq. 2.1.1, All/A and Anl/A, respectively, where A = IYI and A if In = 0, then Vl = (All/A)Il’: Vn = (Anl/A)I,' and the Open circuit voltage-ratio transfer function is given by Tv = Anl/All (2.1.3) The forms for A and All are identical, as can be seen from Eq. 2.1.1, and therefore, except for specific cases, only the properties of A will be discussed. Theorem 2.1.1. If an (n+1)—terminal single gyrator RCF network satisfies assumptions i and ii, then i i+1 _ 2 A — A(RC) + a1A(i'i+l' RC) (2.1.4) where A(;,RC) denotes the determinant of the RC admit- tance matrix in Eq. 2.1.1 in which the i-th row and the j-th column are deleted. Proof: Consider the expression of the admittance matrix given in Eq. 2.1.1 11 _ _ _ l i+1 RC : : Y = admittance + .....' a1 1 matriC o o o—alo i+1 Forming the determinant of a sum of matrices, one obtains _ _ i _ 21+1 A _ lyl — A(RC) + alA(i+l,RC)( 1) 2i+1 i i+1 + a LA( ’i+1’RC) (2.105) '0. i+1 lA(l . IRC) (’1) From the symmetry property of the RC admittance matrix, = A(l:l,RC). Therefore Eq. 2.1.5 one has A(i i1,RC) takes the form i i+1 (2.1.6) This completes the proof. Note: The expression for All corresponding to Eq. 2.1.6 is l i i+1 A = A(1,RC) + a LA( 1+1,RC) (2.1.7) 11 If i = 1 the second term in equation 2.1.7 does not occur since the syrator constant cl (appearing in two places) is removed with the first row and the first column of Y. In this case All = A(i,RC) and the effect of the gyrator on the open-circuit voltage- ratio transfer function, TV = Anl/A is lost. 11' 12 It is desirable to use no more gyrators than are necessary, therefore the following assumption will be added to (i) and (ii). Assumption: iii) The RC? network will contain no gyrator branches connected directly across the im- put terminal vertices. Calahan has shown [CA1] that the zeros of the numerator of the sum of RC and RL immittance functions obey certain angle conditions. Since this Theoren is applicable to Eq. 2.1.6 and will be used in later develOp- ments, it is simply stated here. (For proof, see [CA1]). Theorem (After Calahan) Let the polynomial N(S) be of the form In (S +pi) H (S + Si)(S = 9.). N(S) = 1 i=1 1 i ":15 then N(S) is the numerator of the sum of an RC and an RL immittance function if the only if (a) for n = 0, ll MD kflfi arg(Si) s i l (b) for n > 0, pi > 0 and arg(Si) S — IIMD where the imaginary part of Si is > Im(Si) O 13 Theorem 2.1.2. In Theoren 2.1.1, let m l(s+ci)][i:l(s+ai+jbi)(s+ai-jbi)] A=[ i ll :15 with c., a. real and b. > 0, then 1 1 1 U‘ -l i m n 2 tan ——-= Z arg(a.+jb.)$ — O a 0 I l l 2 1-1 1 1:1 and ci > O for n # 0 Proof: From Thorem 2.1.1 2 . . A = A(RC) + alA(l 1+1 i’i+1'RC) Dividing both sides of the above equation by A(:,RC) one can form 1 1+1 A = A(RC) + a2 A(i’i+l'RC) ' ' l i A(:,RC) A(:,RC) A(i,RC) The right hand side of this equation can be recognized as the sum of an RC and an RL admittance function, i.e. A -w-——-= y + y A(:,RC) RC RL with Calahan's Theorem this completes the proof. Theorem 2.1.3. If an (n+l)-terminal RCF network satis- fies assumptions (i), (ii), and (iii) and contains two gyrators F1 and F2 such that the edges corresponding to P1 are the i-th and (i+l)-th edges of the formulation l4 tree while those corresponding to F2 are the j-th and (j+l)-th edges of the formulation tree; then i+1 A = A(RC) + mi A(., 1+1,RC) + azA (j jii’RC) + aza 2A(i i+1 j j+1 RC) 2 '3 la 1 'i+l’j’ j+l’ j _ lj+1 + 2010 2A(: +1: j+lIR C) 2a1a2A(L lj ’RC) (2.1.8) Proof: The form of the Y matrix is r-1 i+1 j j+1 RC -a a1 i Y = admittance + l i+1 matrix a J 2 . _ j+1 L “2 J Forming the determinant of Y, one has i+1 A = A(RC) + a A( . ,RC) 1 1 _ i j+1 alA(i+l,RC) + a2A( j ,RC) 2 1 1+1 -d 2A(. j+1’RC) + a1 A(i,i+l,RC) 2 j j+1 i +0‘2 A(j j+l’RC) + O‘1“‘2 A(j+k'i+1'Rc) j+1 i+1 _ _ i+1 +ala2 A( j , i ,RC) aldzA (j+1' i ,RC) _ j+1 i _ 2 i i+1 j O‘1"‘2 ( ' 'i+l’RC) O‘10‘2 A(i'i+1'j+1'RC) 2 i i+1 j+1 _ 2 i j j+1 +a 1“2 A(i'i+1' j 'RC) O‘20‘1 A(i+1'j'j+1'RC) 15 i+1 j j+1 l Ijlj+llRC) (1 1+1 j j+1 2 2 + alaz A i'i+1'j’j+1'RC) (2.1.9) Due to the symmetry property of Y certain terms will RC’ be cancelled and Eq. 2.1.9. reduces to Eq. 2.1.8. Hence the proof. Equation (2.1.8) is unsuitable for practical synthesis procedures due to the presence of terms cor- responding to a In the following sections further 1“2° developments will be considered for a restricted class of RC? network configurations to remove this difficulty. 2.2. RC Networks Cascaded Through Bridged Gyrators. In this section a restricted class of RC net- works is considered. As is shown in Fig. 2.2.1, 3-ter- minal RC networks are cascaded through bridged gyrators. For this class of networks the properties of the deter- minant A of the admittance matrix Y are given in the following theorem. Theorem 2.2.1. If three grounded RC sections are cas- caded through two grounded gyrators P1 and P2, where the gyrators are bridged with capacitors and/or resistors as shown in Fig. 2.2.1 and where the complete network sat- isfies assumptions i, ii, and iii, then l6 _ 2 i i+1 A — A(RC) + a1 A(i,i+l,RC) 2 j j+1 + a2 A(j'j+l'RC) 2 2 i i+1 j j+1 + O‘10‘2 A(i'i+l'j’j+l'Rc) (2.2.1) Proof: Since both the RC sections and the gyrators are grounded, a Lagrangian formulation tree in the corres— ponding network graph can be chosen such that it contains both the edges corresponding to the gyrators. The Y matrix is of the form i i+1 j j+1 - . : -’ - | I— Y I | I I a _ a l ___'Y.l_(s_’____:___ _Ll___.L_ ’+1 1 ‘Yl(5)I- I -al| I _ | I Y ' . l Yb l-Y (s) +' I“2 3 ___ ____ _L__“Ln_ j+1 ‘F‘_' y_( )‘T | _a1' I 25 | I 2I l I YC I I .L. l l __ __ l I_J (2.2.2) where Y (s) = c.s + b. and c., b. > 0 i = 1,2 1 1 1 1 1 Let . I YB I — _ a -Y (S) Yb ‘ _______ [_?__%_.____ -a2-Y2(sfl I l Yc A. l __ Then Eq. 2.2.2 can be written as l7 —’\/\/— r-JVVT R1 R2 —II— —-II—— C C 1 ___ 2 _.__ r---j r j r 1 ° I RC 4: 0‘1 T RC ( 0‘2 “f RC T—'0 I I | I i I I l I I I I I l I | I G ) , l l I L L_____i L____I L_____I' ° A B C Fig. 2.2.1 Three terminal RC networks cascaded through RC-bridged-gyrators. 18 i Y= - '—:a:”—‘1_1‘1- 1+1 1 Yl(s) If the determinant of Y is expanded about the first i rows using Laplace's expansion, one has A = Aa(RC) A (RCFZ) b 2 2 i i+1 or _ — _ 2 i - i+1 A — Aa(RC)Ab(RCP2) Yl(s) Aa(i,RC) b(i+l,RCF2) 2 1 1+1 + a1 Aa(i,RC)Ag(i+l,RCF2). This expression is equivalent to _ 2 i i+1 A — A(RCTZ) + al A(i,i+l,RCF2). (2.2.3) 1 1+1 Since (RCPZ) and A( l,RCI‘Z) correspond to matrices i’i+ of RC networks containing only one gyrator, Eq. 2.2.3 can be written as _ 2 j j+1 2 1 1+1 2 i i+1 j j+1 + a1[A(i’i+l'RC) + a2 A(i'i+1'j'j.+l,RC)] (2.2.4) This completes the proof. 19 The form of Eq. 2.2.4 is essentially suitable for synthesis procedures; however, if each term of the expansion could be identified with a specific RC section appearing in Fig. 2.2.1, then the hynthesis would be facilitated. For this purpose the bridged gyrators will be replaced by unbridged gyrators. This change also allows active or unbalanced gyrators to be used in place of passive gyrators. 2.3 RC Networks Cascaded with Gyrators In this section 3-terminal RC networks cascaded through passive or active gyrators are considered. The complete network, as shown in Fig. 2.3.1, is assumed to satisfy assumptions i, ii, and iii. The properties of these networks are contained in the following theorems. Theorem 2.3.1. If grounded RC networks a, b, c,... are cascaded through grounded gyrators only, as shown in Fig. 2.3.1, then the following hold: Case 1. For one gyrator and two RC networks _ 2 1 1+1 A — Aa(RC)Ab(RC) + a1 Aa(i,RC)Ab(i+l,RC) (2.3.1) Case 2. For two gyrators and three RC networks A = Aa(RC)Ab(RC)AC(RC) (2.3.2) 2 1 1+1 + a1 Aa(i,RC)Ab(i+l,RC)AC(RC) .mHOpmnmm ammonnu pmpmommo mxHOBpmc Um Hmaflaumw mouse H.m.m .mflm Z m ¢ 20 0m um um 21 j+1 2 A j A + 1 a(RC)Ab (3. ,RC) C(j+l,RC) 2 2 1 1+1 j j+1 + 0112 Aa(i'RC)Ab(i+1'j'RC)Ac(j+l RC) j g i+1 Case 3. For three gyrators and four RC networks A = A + Proof: Case 1. In i+1 and A = A a a(RC)Ab(RC)AC(RC)Ad(RC) (2.3.3) A a(%,Rc)A b<¥ji Q I—‘N NM l—‘N l—‘N 00M NM II-‘N ,RC)A c(RC)Ad(RC) A ab(RC)A (j, RC)AC (jil, Q RC)Ad(RC) k+1 d(k+l'RC) Q Aa(RC)Ab(RC)AC Ik.RC)A i+1 j Aa(i’ RC) Ab(k+l' j (i+1 b i+l’ Q Q NM DON DUN NN ,RC)Ac (j:11,RC)Ad(RC) k+1 d(k+l’RC) k+1 d(k+l’RC) j+1 k j+1 'k' A a(:,RC)A Q Q RC)A c(k,RC)A k Aa (RC)Ab (3.,RC)Ac (j:1,,k 0. Cl IRC)A i i+1 j k+1 Aa‘i'RC)A (' d(k+l’ b 1+l’j RC)A RC) Q Q Q LON ,RC)A C( this case the admittance matrix is 1 i+1 Y I j a G. _- —_-OL—| __l__ 1| Y 1. I 21 (RC)A (RC) + a2 A (i. ,RC)Ab Ii+1 RC) (2 3 4) b l a 1+1' ' ° 122 Case 2. Replace Yb in case 1 by . F'- | . _ 1+1 ii+l j| jt;_i Y = b j Y j+1 __ _ 2.431- "“2 Y I. l i then from Case 1 A (Rcr) = A (RC)A (RC) + a2 A (j RC)A (j+1 RC) b b c 2 b j' c j+1’ and — i+1 _ i+1 Ab(i+l,RCI‘) — Ab(i+l'RC)AC(RC) 2 1+1 j j+1 + 0‘2 Ab(i+l’j'RC)Ac j+l'RC) i+1 If now Ab(RC) and Ab(i+l,RC) are replaced respectively by i+1 i+1,RCI‘), then Eq. 2.3.4 becomes Eq. 2.3.2. Ab(RCP) and Ab( Case 3. Replace YC in case 2 by k k+1 then from Case 1 2 k k+1 and —— j+1 _ j+1 AC (j+l,RCT) _ Ac(j+l,RC)Ad(RC) 2 +1 k j k+1 3Ac(j+l’k +a d(k+l’RC) ,RC)A 23 j+1 . —— If now AC(RC) and Ac(j+l,RC) are replaced by AC(RCF) and j+1 AC (j+1 ,RCF), respectively, then Eq. 2.3.2 becomes Eq. 2.3.3 This completes the proof. Note: The above Theorem could be extended to the general case; however, the number of terms in the expansion for A is 2n which becomes prohibitively large for the number of gyrators n 2 3. Theorem 2.3.2. For the two gyrator case of Theorem 2.3.1, with the terms of Eq. 2.3.2 in the array 9 [Aa(RC)A::b(RC)C AC (RC) :Aa ( RC) %(l+l, RC)AC (RC) 6 I (2.3.5) 9 ZAba(RC)A (3°.RC)AC (3'11 1.RC) 2 2 1+1 j j+ alaz Aa(i,RC)Ab(k+1,3j,RC)AC(j1,RC 4 the sum of each indicated pair forms a polynomial n m N(S) = n (s + Ci) n (s + a + jb)(s + a - jb) i=1 i=1 with ci 2 O and Proof: Consider the pair N(S) = Aa(RC)Ab(RC)AC(RC) 2 1+1 + al A a(i,RC)Ab (i +1,RC)AC (RC) 24 which can be written N(S) = A (RC)[A (RC)A (RC) + a2 A (i+1 RC))] c b c l a i+l’ (2.3.6) From Theorem 2.1.2, the terms within brackets in Eq. 2.3.6 can be replaced by P(S) = ||=12° m (s + ci) .H (s + a + jb)(s + a - jb) 1 1: (2.3.7) with ci > 0 and AC(RC) is the determinant of an RC admittance matrix and so has non positive real zeros. Therefore Eq. 2.3.6 can be written as n 2 n N(S) = H (s + ci)[ H (s + Ci).H (s + a + jb)(s + a - jb)] i=£+l 1:1 1:1 with C1 > 0 and The proof of the remaining three pairs follows similar lines. Q.E.D. Theorem 2.3.3. Consider the three gyrator case in Theorem 2.3.1 with the terms in Eq. 2.3.3 represented by the power of ai in the following array 25 'F l 2 l W a2 2 2 _a3 7 ———a2 a2 a 1 2 l 3 2a2 2 3 (- E2 0.2 0.2 l 2 3 In this array the sum of each indicated pair (from Eq. 2.3.3) forms a polynomial (s + a + jb)(s + a — jb) l ":33 n N(S) = H (s+c.) =1 l i R g — and ci 2 0 m with 2 tan 2 Proof: The proof is similar to that used in Theorem 2.3.2 and will not be repeated here. Note: Theorems 2.3.2 and 2.3.3 establish necessary con— ditions which must be satisfied by any polynomial decom- position technique used in the synthesis of two and three gyrator RCP networks of the form shown in Fig. 2.3.1. Theorem 2.3.4. If in Theorems 2.3.1, 2.3.2, and 2.3.3 the passive gyrators are replaced by active gyrators, then the form of A remains invariant. Proof: Consider the Y matrix; 26 I E. > 0 Y = i+1 -§.T l B. l 1 1 B.,§. real 1 1 l l b( L— _ where Y; and §b have the same form as Y. If the deter- minant of Y is expanded about the first i rows, then A = A6(RCT)A£(RCF) ~ — i — i+1 + Bi Bi Aa(i,Rcr)Ab(k+l.Rcr). However taking ai = “éi Bi , one has A = A (Rcr)A (Rcr) + a2 A (1 Rcr)A (i+1 Rcr) a b i a i' b i+l’ ' Since an arbitrary active gyrator can always be replaced by an equivalent passive gyrator, A has the same form for active as well as for passive gyrators. Q.E.D. 2.4 Stability of the Transfer Function. In the preceeding section, networks such as that in Fig. 2.3.1 were shown to have equally stable trans- fer functions using either balanced or unbalanced gyra— ors. That is, A and All are invariant when passive gyrators are replaced by active ones. Complete stability can be shown for grounded RC sections cascaded with grounded gyrators (passive or active) where each RC section has the following prOperties: 27 1. Each RC section is connected. 2. The graph corresponding to the conductances contains all the nodes of the section and is either connected or is connected when the two terminal nodes of the RC section are connected (shorted) to the ground node. For the last section it is connected when the input node is connected to the ground. 3. Any RC section, except the last, has at least two nodes in addition to the ground node. 4. Each section contains one or more conductances. Three important classes of RC sections which satisfy the properties 1-4 are low and high-pass RC ladders and Twin-T or Bridged-T RC notch filters. Theorem 2.4.1. Let an RC section, with admittance mat- rix Y satisfy conditions 1-4: s! Case 1. If the section is an intermediate or leading section, then A (?,J,RC) > o for s = o s i 3 Case 2. If the section is the last or output section, then A (? RC) > o for s = 0 s i' where i is the input terminal and j is the output ter- minal of the RC section. Proof: Case 1. The YS matrix is 28 Note that Ysls=0 — From condition 2, when both the input and output terminal is connected to the ground, the conductance graph becomes connected. This corresponds to the ith row and column and the jth row and column being removed from the matrix G. In general a conductance matrix is positive semi- definite. However, for a connected graph it is positive definite, hence it is nonsingular. It follows, there- fore, that As(i,g,RC)> 0 for s = 0. If i and j are the . i j only nodes in Ys’ then As(i,j ,RC) = 10 Case 2. From condition 2, when the input terminal is connected to the ground, the conductance graph becomes connected. This corresponds to the ith row and column being removed from the matrix G. For a connected graph a conductance matrix is positive definite. Therefore A (?,RC) > o for s = 0. S]. If i = j, then AS(:,RC) I |—' Q.E.D. Theorem 2.4.2. If a network is made up of RC sections satisfying conditions 1-4 cascaded through grounded gyrators, then All is strictly Hurwitz. Proof: The proof iS established using an induction on the number of gyrators. Consider first the one gyrator case: 29 A — l — i+1 11 — Aa(l,RC)4b(RC) + 8131 Aa(b :,Rc)Ab(i+l'RC) (2.4.1) Since conductance matrices are positive semi—definite, one has (i+1 A11|s=o 2 8131 Aa(1 1'RC)Ab(11RC)Is-o However from Theorem 2.4.1 1 i (i+1 A ( p-IRC)| _ > O and ,RC) > 0 a 1 1 —0 Ab( k+1 Is: 0 (2.4.2) Therefore All] _ > 0 (2.4.3) s-O If Eq. 2.4.1 is rewritten as l i+1 All _ Aa(1’RC) + 8 § Ab(i+l'RC) 1*1 ’ 1 1 1 1 A (RC) Ab(RC)Aa(l,i,RC) Aa(l,i,RC) b (2.4.4) Then the right hand side of Eq. 2.4.4 can be recognized as the sum of RC and RL admittance functions. Consider the real part of the first term in Eq. 2.4.4 when s = jw Aa<},Rc> n him» Re 1 = Re ij + Z W + h ho, hi, hm 2 o A a(]1',RC) n 11.12 Re 1 i = hoo + )3 7;; a(1,1,Rc) s=jw 1:1 Ci + w (2.4.5 30 From condition 4, at least one hi or h0° must be nonzero. Therefore Eqs. 2.4.4 and 2.4.5 imply that l A A ( ,RC) Re 11 . 2 Re a l > O l 1 l i x=jw s=jw (2.4.6) where Ab(RC) and Aa(1'i’RC) are the determinants of RC admittance matrices and can have no zeros on the jw axis except at the origin. Therefore Eq. 2.4.3 and 2.4.6 imply that All has no zeros on the jw axis. Since All is the numerator of a sum of RC and RL admittance func- tions, it is the numerator of a positive real function and can therefore have no zeros in the right-half of the s-plane. Hence All is strictly Hurwitz. This completes the proof for the one gyrator case. Assume now that All is strictly Hurwitz for K gyrators. Let Y be the admit- tance matrix of (K+2) RC sections satisfying conditions 1-4 cascaded through (K+1) gyrators: R . ‘1 where Yb is the admittance matrix of the last (K+1) RC networks cascaded through K gyrators. From Y’All is: l = ~ 11 All Aa(l,RC)Ab(RcT) + 8181 Aa( i,RC)A 31 where, by assumption, Ab(i:l’RC) is strictly Hurwitz. Observe first that All|s=0 > 0. This can be shown as follows: Aa(l’RC)ls=0 2 0, since a conductance matrix i is positive semi-definite. Also Aa(l’1 ,RC)]s=0 > 0 by Theorem 2.4.1 and Ab(RC )Is=0 z 0, since a conductance matrix is positive semi-definite. Furthermore, i+1 . i+1 . . Ab(k+l)RCI‘)IS=O > 0 Since Ab(i+l,RCF) lS strictly Hur- witz by assumption. From the above inequalities and Eq. 2.4.7 one obtains ~ 1 1 1+1 A11ls=o B181 Aa‘l'i'RC)4b(1+1'RCF)ls=o > 0 (2.4.8) The second observation is the positiveness of Allls=jw° This can be shown as follows: Consider the ratio A A (1 1 RC) (RC?) 11 _ a 1'1' + g 8 4b 1 1+1 - i n+1 n+1 i+1 Aa(i,RC)Ab(i+l,RC) Aa(i,RC) Ab(i+l.RCP) (2.4.9) Now (RCFL/ (i+1 RC?) is positive real if all the Ab Ab i+l' gyrators are passive. However by Theorem 2.3.4. the form of A and All is invariant when passive gyrators are i+1 replaced by active gyrators. Therefore Ab(RCF)/Ab(i+l,RCF) i+1 is positive real and the real part of Ab(RCP)/Ab(i+l,RCP)'s=jw is greater than zero. The first expression on the right hand side of Eq. 2.4.9 can be recognized as an RL admit- tance function with the following real part when s = jw: OT, 94 h A) (7) 32 l i Aa(lliIRC) h- h ____+__ +. 15 Cl 5 co thb P Re i . . Aa(i,RC) s=jw 1 (2.4.10) where Ci’ hO’ hi' h 2 0. From condition 4 at least one 00 hi or hO is nonzero. Therefore Aa(i,%,RC) n hi ci Re 1 l = ho + Z 2 2 > 0 (2.4.11) From Eqs. 2.4.8, 2.4.9, and 2.4.11 it follows that Re All > O 1 1+1 Aa(i,Rc)Ab(i+l.RcP) s=jw (2.4.12) i+1 . . . . i Ab(i+l,RCT) lS strictly Hurw1tz by assumption, Aa(i,RC) is the determinant of an RC admittance matrix and so has negative real zeros except possibly at the origin. There— fore All is the numerator of a positive real function with no zeros on the jw axis. Eq. 2.4.8 shows that All can have no zeros at the origin and hence A1 is strictly l Hurwitz. This completes the induction and the theorem is now proved. 2.5 Minimum Number of Real Poles of TV for low-Pass RC Ladders Cascaded with Gyrators. Theorem 2.4.2 considered in the preceeding section establishes the absolute stability of the transfer func- tion TV for RcF networks satisfying conditions 1-4. Necessary conditions on the number of real zeros of All 33 with respect to the number of capacitors and the gyrator placement can be established if the RC sections are low— pass RC ladders. This is given in the following theorem. Theorem 2.4.1. If: i) A connected network is composed of (n+1) low pass RC ladder networks cascaded through n grounded gyrators. ii) The edges corresponding to the capacitors and the input voltage driver form a Lagrangian tree in the complete network graph. iii) The total number of capacitors in the first RC sec- tion, third RC section, and so on alternately is denoted as #C(l). The total number of capacitors (2) remaining is denoted as #C then has at least A 11 [#cm — #cm (2.5.1) real zeros. Proof: The admittance matrix is of the form Y = Cs + G + F (2.5.2) Now the admittance matrix Y' corresponding to All can be obtained from Y by removing from Y the first row and column: Y' = C's + G' + F' (2.5.3) Condition ii ensures that C is diagonal with positive diagonal entries. Conditions i and ii ensure that G' + F' is tridiagonal. Let C' + F' = H' then 34 Y' = C's + H' since C' is diagonal with positive diagonal entires, [C']-l/2 exists and is diagonal with positive diagonal entries, and the determinant of [C']-l/2 3(‘[c']"l/2 differs from A by a constant. Let Y" = 11 [C']-l/2 Y' [Qflfl/Z then Y" = U s + [C'J'l/2 Y" = U s + H" The zeros of All are therefore the eigenvalues of -H". The proof follows as an adaption of a method described 1/2 by Frame [FR]. Since the diagonal matrix [C']- has all positive diagonal entries, the sign matrix of Y" = [C']-l/2 Y' [C']—l/2 is the same as that for Y'. The rest of the proof can best be facilitated by a specific case before the general proof is completed. Consider the network in Fig. 2.5.1 where the Y matrix is — .— G1 "G1 | | _Gl ClS+Gl+G2 _GZ| I __ -G2 C25+G | 81 | —81) C3s+G3 G3 | Y = | -G C +G | B _ ________ |_______48__%_|___2____ ‘ 82 | CSS+G5 -G : ‘ -G5 C6s+G5+G 35 For this case )_s+al —bl ' I -) | I -bl s+a2 |Cl_—_-—|- ——_—- -Cl Is+a3 -b3 | Y" = '3’: 3*31‘iz____ _—{ -52 l s+a5 -b5 I ) -b5 s+a6 Multiplying the third and fourth column of Y" by -l, i.e. multiplying those columns corresponding to the second RC network the fourth RC network and so on alter- nately, one obtains ";+al —bl | I -bl s+a2 |--cl : " "' —.§l—':s:;37 "1.3— ‘r “ "— = 1..., (.2 —_ _ _ —:—__ :C: _ Isa—$5— 1.... | |"b5 S+a6_ This results in a symmetric sign matrix and at worst changes the sign of the determinant. Now forming the determinants of the principal minors one obtains 36 H.m.m Emnomne How xHOBum: gum H.m.m .mHm l J l 37 PO é 1 Pl : s + a1 P2 = (S + a2)Pl biPO P3 = -(s + a3)P2 - clclPl P4 : -(s + a4)P3 - b§P2 P5 : (s + a5)P4 - c252p3 P6 = (s + a6)P5 - b§P4 = 1 k All The polynomials P6 through PO constitute a Sturm sequence. The Cauchy index [T0] = v<-oo> - V(°°) *UI’U U1 8H8 6 gives a minimum number of real zeros of P6, where V(a) denotes the variation of sign in the Sturm sequence at S = as "oo +oo PC + + - + P1 P2 + + P3 + - P4 + + P5 - + P6 + + ‘V(-m) = 4 V(+w) = 2 Therefore for the network under consideration All must have two or more real zeros. In the general case one would have 38 Po(s) = l P1(s) (s + al)PO K capacitors 2 in the first P2(S) (S + a2)P1(s) - blP0(s) RC section 9 (s) = (s + a )P (s) - b2 p (s) K K K-l K-l K-2 PK+1(S) = "(S + aK+l)PK(s) ‘ B181 PK-1(S) 2 P (s) = -(s + a )P (s) - b P (s) K+s K+2 K+l K+l K z—k capacitors . in next RC : section _ _ _ 2 P£(s) - (s + a£)P£_l(s) bR-lP _2(s) I may be more negative terms or may not be 2 = i — .— Pn(s) k All (s aK)Pn_l(s) bn-an-2(S)' Therefore, V(-w) V(w) P0 + + - + P1 + always changes + never changes - + Pk .t +l + Pk+l ‘ - 1 Pk+2 does not change + always changes p i i P i +1 I always changes + does not change i ; : i 39 From PK to P , (2—k) sign changes were lost in V(-m) and (Rrk) sign changes were gained in V(+m). In the same way the number of sign changes lost and gained can be calculated for the 4th RC section, the 6th RC section and so on. Since g-k is the number of Capacitors in the second RC section, it follows that for n, the total num- ber of capacitors, (2) = #Cm V(-w) n — #C V(+oo) = #C(2) V(-oo) V(+oo) = #C(l) " #C(l) From which it follows that 00 P I ——-g'1 = #C(1) - #C(2) .—m n and so Pn has at least |#C(l) - #C(2)| real zeros. Q.E.D. Theorem 2.5.2. Low-pass RC ladders cascaded through gyrators form a low-pass network. Proof: The Y matrix is tridiagonal and the C matrix is diagonal. Therefore no 3 terms appear in the off diag- onal entries of Y. Anl is the determinant of the matrix obtained from Y when its first row and last column is removed. This matrix is upper triangular and if diagonal entries are the lower off diagonal entries of Y. There- fore Anl is not a function of s. The proof follows from Tv = Anl/All' 40 2.6 Conclusion. In this chapter an analysis of RCF networks was carried out which developed certain necessary conditions on RCF filters. The one gyrator case was shown to have an angle condition [CA1]. The transfer function of RCF networks composed of "notch" filters or low-pass ladders cascaded through passive or active gyrators was shown to stable. Finally a theorem was proved giving a minimum number of real poles in the transfer function for a cer- tain class of low-pass RCF filters. be CHAPTER III DEVELOPMENT OF REALIZATION FORMULAS FOR ONE-GYRATOR RCT FILTERS In the second chapter of this thesis, the general form of the denominator polynomial for the open circuit voltage ratio transfer function of RCr networks contain- ing one or more gyrators is established. Calahan [CA1] has given necessary and sufficient conditions under which a given set of complex frequencies may be realized as the natural frequencies of a network consisting of two inter- connected 2—terminal networks, one being RC and the other RL. His network configuration, however, can be reduced to the form under consideration in this thesis, as shown in Fig. 3.0.1. In this chapter, a new derivation of Calahan's polynomial decomposition [CA1] is given. This derivation also indicafiflSthe relationship between two seemingly different polynomial decomposition methods which are referred to, in the literature, as the Horowitz [HO] and as the Calahan [CA1] polynomial decompositions. That is, it is shown how these two decomposition methods can be developed from each other. Methods of extending the 41 42 Calahan and Horowitz decomposition methods to polynomials which have distinct negative real zeros are established. Examples are given to illustrate how these decomposition methods can be utilized to realize low pass voltage ratio transfer functions TV. Furthermore it is shown that all polynomials with distinct negative real zeros (if any) can be realized as the natural frequencies of two RC networks connected through a gyrator, the latter loaded with a negative resistor as shown in Fig. 3.0.2. How- ever, such a network will not be stable unless the denom- inator polynomial of TV is Hurwitz. 3.1 Calahan's Decomposition Calahan [CA1] has shown that any real polynomial N(S) with all complex zeros, m N(S) = H (s + sl)(s + si) (3.1.1) where the imaginary part of si satisfies Im(si) > 0, can be separated uniquely into 2 2 N(S) = a (s) + b (5) (3.1.2) such that the degree of a(s) is one greater than that of b(s) and the polynomials a(s) and b(s) have alternating real zeros. Any polynomial decomposition of the form in Eq. 3.1.2 which satisfied the above conditions will be hereafter referred to as Calahan's decomposition. Fig. Fig. 43 RC RC RC 3.0.1 RLC and RCP equivalent forms. 3.0.2 RC RC RCF-R network representation. 44 Calahan has also shown that the rational function a(s)/b(s) is an RC admittance function if, and only if IIMB arg(s.) 5 fl/2 (3.1.3) 1 i i As will be shown, a development of the Calahan decomposition different from Calahan's [CA1] original lengthy derivation can be obtained from the Horowitz [HO] decomposition. Therefore, Horowitz's decomposition will be described briefly. Horowitz [HO] has shown that if a real polynomial P(S) has no non-positive real zeros, then P(S) can be ce- composed uniquely as p(5) = i[A2(s) - sB2(s)] (3.1.4) such that A(s)/B(s) and sB(s)/A(s) are RC admittance func- tions. In order to establish the Horowitz decomposition, as indicated by Thomas [TH], consider the even polynomial P(sz) which can be written as P(sz) = 1 F(s)F(—s) (3.1.5) where the plus (minus) sign holds if P(s) is of even (odd) degree and the polynomial F(s) is strictly Hurwitz. In— deed, by hypothesis, P(s) has no zeros on the non-positive real axis, and hence the zeros of P(s2) have quadrantal symmetry and none can be on the imaginary axis. Thus, F(s) 45 can be formed by rejecting the right half-plane zeros of P(sz). If F(s) is now separated into even and odd parts 2 P(s) = A(sz) + sB(s ) (3.1.6) one obtains ll 1+ P(s ) F(5)F(-S) 2 2( [A2(sz) - s B 52)] (3.1.7) ll 1+ P(sz) and if $2 is replaced by s, the desired form 2 P(s) = 1 [A (s) - sB2(s)] (3.1.8) is obtained. The rational functions A(s)/B(s) and sB(s)/A(s) are RC admittance functions since A(s)/sB(sz) is a Foster function [TO]. Returning to the development of Calahan's decom- position, Horowitz's decomposition is used in establish- ing the following Theorems. Theorem 3.1.1. Given a real polynomial P(s) with all complex zeros, there exists a decomposition of the form I 2 ' 2 P(s) = [A (s)] + (8+ a)[B (8)] where the zeros of A'(s) and B'(s) alternate and -a < min{zeros of A'(s)}. Proof: First "shift" the zeros of P(s) into the right half plane with the transformation 5' = s + a where a is a sufficiently large positive constant. This yields P'(s') = P(s' — a) 46 Then "folding" the zeros of P'(s') into the left half plane by the transformation A = -s', one has P"(A) = P'(-A) = P(-)( -a) Since P(s) has no real zeros, P"(A) = P(—A -a) does not have. Therefore the Horowitz decomposition of P"(A) yields p"(1) = 212(1) - A 132(1) "Folding" and "shifting" P"(A) back to P(s) with the transformation 5 = —A-d, the desired decomposition is obtained in the following form: P(s) = A2(-s-a) + (s + a)B2(-s-a) Note that since A(A) and B(A) have negative real zeros, due to the properties of the Horowitz decomposition, P(s) can be written as n 2 2 n-l 2 P(s) = H (-s-d + z ) + b (s + a) H (-s-a + z ) . a. . b. i=1 1 i=1 1 or n 2 2 n-l 2 P(s)=n(s+a-z)+b(s+d)H(s+a-z) . a. . b. i=1 1 i=1 i From the properties of the Horowitz decomposition, one has or 47 Therefore, if we let n A'(s) = H (s + d - z ) = t A(-s-d) (3.1.9) i=1 1 and n-1 B'(s) = H (s + a - zb ) = i B(-s—a) (3.1.10) i=1 i it follows that the zeros of A'(s) and B'(s) do in fact alternate; and the minimum zero of A'(s) is greater than -a. Therefore P(s) = [A'(s)]2 + (s + 0L)[B'(s)]2 and the Theorem is proved. Note: If a - 2a is greater than zero, then the rational n functions A'(s)/B'(s) and (s +a)B'(s)/A'(s) represent RC and RL admittances respectively. However, since 2a is n a function of a, (a - za ) may not be greater than zero. n Theorem 3.1.2. Given a real polynomial P(s) with no real zeros, the decomposition R=[A'(s)12 + (s + on)[13'(s)12 considered in Theorem 3.1.1 reduces to the Calahan decom- position P(s) = K2(s) + §2(s) as a approaches infinity. Proof: The proof is established using an induction on n where P2n(s) denotes a real polynomial of degree 2n with complex zeros. 48 2 Part I. Let n = l. The polynomial P2(s) = (s + 201s +pi), where pi > Oi, can be "shifted" and "folded" into P§(A) by the transformation A = -s-a: 2 2 +2(a- 0)). +01 -20a+pi). Pam = <1 1 1 To obtain the Horowitz decomposition of P£(A), consider 4 95(12) = (A + 2(a — ol)A + a2 - 2011 + pi)- This polynomial can be written as P"(A2) = F (A)F (-A) 2 l l where F1(A) = (A2 + alA + b1) is strictly Hurwitz and b = /’2 2 l a 201a+pl > 0 (3.1.11) a1 = /251-7(a-ol) > 0 If F(A) is now separated into even and odd parts, then 2 A1(A2) = A +b and Bl(A2) = a Hence A1(A) = A + b l l' 1 and Bl(A) = a1. Performing the inverse transformation A = -s-a, one obtains Al(-s-a) = -S-a+bl and B1(-s-a) = al. The polynomial P2(s) can now be written as P2(s) = (s + a-b1)2 + (s + a)ai (3.1.12) A Lemma is necessary Lemma 3.1.2. Let bi = / 2 a -Zoia+p? and ai = /2bi-2(a-oi), l 49 then Lim a - b. = 0., Lim a. = 0 and Lim/d a. = w. i 1 1 1 1 Proof of Lemma: Consider the expression (a - oi). Since bi = /g2 - ZCia + pi2 , 2 20,+ p. a . /2 2 . 1 1 Lim a - a -201a + pi = Lim = o d+w d+® 20. + 2 1+1+—£ 0i a —— 2 a or Lim (d — b.) = 0. From the above result it follows that Lim a. = Lim /20. — 2(a - O.) = 0 a+oo a-roo l 1 However Lim/BL a. does exist. Indeed a+oo l Lim /E'a. = Lim /E /20. - 2(d-o.) or a+oo 1' a+oo '1 l l 2 l 2 (26) /2(o§ - oi ) / Lim /E'ai = Lim a+oo a+oo 1/2 2 2 l/2 + .- Ed. 20i0( + bi) + (o: Oi)] where 2 2 2 . p. - O. = w. . This proves the lemma. 1 i 1 Continuing with the proof, Eq. 3.1.12 can be written as 2 1 (3.1.13) _ 2 P2(s) — (s + 01) + w 50 This concludes the proof for part I. Before continuing with Part II observe that Ai(s) = (s + a - cl) and Bi(s) = a Also observe that l. the limits: I Lim A.(s) = (s + g ) = A (s) a+m l l l I Lim B (s) = Lim a = 0 a+oo l a+oo ‘1 Lim /EB'(s) = A = R (s) a+m 1 1 1 hold for n = 1. Part II. Assume now that the theorem is true for n = k-l, which is the same as assuming the limits Lim Afi(s) = Ah(s) n = k-l a-Hzo Lim B'(s) = 0 n = k-l a+oo n Lim VEB'(s) = E (s) n E k-l a+w n n hold. Consider the expression of P2n(s) for n = k, where P ( ) H ( 2 + 20 + 2) and 2 > o 2 2k 3 ._ S is pi p1 1 ' i—l Performing the transformation A2 = -s - a one has P" (A2) - E (A4 + 2( - c )12 + a2 — 20 a + 2 2k ’ i=1 a 1 1 pi ) This polynomial can be written as 51 n 2 _ 4 _ 2 2 _ 2 _ P2k(A ) - (A + 2(a 0k) + a 20k + pk)Fk_l(A)Fk_l( A) where Fk_l(A) is formed for the case n = k-l. Collecting the terms corresponding to the left half plane zeros one has Fk(A) = (A2 + akA + bk)Fk_l(A) where bk = /32 - Zoka +‘ofi ak = /2bk - 2(d - 0k)- Let the even and odd parts of Fk_l(A) be denoted as Ak_l(A2) and Bk_l(A2). Therefore the even and odd parts of Fk(A) can be written respectively as 2 2 2 2 (A + bk)Ak_l() ) + A akBk_l(A ) 2 Ak(A ) and 2 2 2 2 Bk(A ) Transforming P3k(A2) back to P s) with the transforma- 2k( tion A2 = -s - a one obtains P (s) = A2(-s - a) + (s + a) BZ(—s - a) 2k k k where Ak(-s - a) = (-s-a+bk)Ak_l(-s-a) + (-s—d)akBk_l(-s-a) and Bk(-s-a) = (-s-d+bk)Bk_l(-s-a) + a (-s-a) kAk-l 52 Now since Ai(s) = i Ak(-s-a), Bi(s) = P’Bk(-s-a) and the degree of Ak(A) is one greater than the degree of Bk(A), the polynomial P2k(s) can be written as P2k(s) = [Ai(S)]2 + (s+a> B];(s)12 where Afi(s) = [(s+a-bk)Afi_l(s) - (s+a)akB' k- l(5)] (3.1.14) and B£(s) = [(s+oL—bk)B'_l (s) + akA'_l (5)] (3.1.15) In order to find the expression for P (s) ascx approaches 2k infinity one can form the limits gig Ai(s) = gig [(s+a- -b k)Ak_ l( )-B'(S+a)ak k- 1(8)] 213 Bi(s) = gig [(s+a-bk)Bi_l(s)+akAfi_l(s)] and 313 /5Bi(s) = 212 [(s+a-bk)faB'Bk-—1(S)+/—akAk-l(S)1 The above limits reduce to Lim Ai(s) = (s+ak)Ak_l(s) -UJkBk_l(S) = Ak(S) a+oo (3.1.16) Lim Bi(s) — 0 a—mo Lim y/EBIL(S) = (s+Ok)Bk_l(S) + kak_l(s) = Bk(S) a-Pm (3.1.17) 53 using the assumption that the theorem holds for n = k-l and Lemma 3.1.1. The expression for P (s) in the limit 2k is p (s) = 212(5) + §2(s) (3.1.18) 2k k k This completes the induction. Now A'(s) and /&B'(s) have alternating real zeros for all finite real values at d. Since the zeros of a polynomial are continuous functions of the coefficients and the coefficients are continuous functions of a, the zeros of A'(s) and /&B'(s) must remain real in the limit. Suppose the zeros at A'(s) and /&B'(s) do not alternate in the limit, then either of the following cases occur: i) A'(s) and /EB'(s) have some coincident zeros in the limit ii) A'(s) and /&B'(s) have no coincident zeros but the zeros no longer alternate in the limit. Case 1 cannot happen since P(s) has all complex zeros. Case ii cannot happen as it implies that A'(s) and /EB'(s) have coincident zeros for some finite a. This completes the proof. It is interesting to note that Eqs. 3.1.14, 3.1.15, 3.1.16 and 3.1.17 establish recursion relations which can be used to calculate Afi(s), B$(s), Ah(s), and Bh(s). Collected and rewritten here for convenience they are: 54 A A$(s) é l B$(s) é l (by definition) I : _ i _. ' An(s) - (s+a bn)An-1(s) (s+a)aan_l(s) L I : _ I I Bn(s) ‘ (s+a bn)Bn-l(s) + anan-1(S) ((3.1.19) _ 2 _ 2 where bk - /C Zoka + pk ak = /2bk — 2(d-ok) and Ao(s) é l B6(s) é l (by definition) An(s) : (s+cn)An_l(s) - wan_l(s) _ _ _ J(3.1.20) Bn(s) — (s+0n)Bn_l(s) + wn n-l(s) _ 2 _ 2 where wk — pk Gk J n 2 2 2 2 Theorem 3.1.3. Given P (s) H (s + 20.s + p. ),p. > 0. 2n i=1 i 1 i 1 then the zeros of the polynomials AA(s) and A$_l(s) appearing in Eq. 3.1.19 alternate along the real axis. Proof: From the recursive relation Ag(s) = (s+d—bn)Ag_l(s) - (s+a)aan_l(s) the following list of properties can be stated, from which the proof is established. ' l I ' ' ' ' l. An(s), An-1(S)' and Bn_l(s) have pOSitive coefficients for the highest powers of s as can be seen from Eq. 3.1.13. 2. The zeros of Afi—1(s) are larger than -a. This is es- tablished in Theorem 3.1.1. 55 Ah_l(s) and BA_l(s) have alternate real zeros. This is established in Theorem 3.1.1. A$(s) has an odd number of zeros larger than the largest zero of A$_l(s). This follows since 1, 2, and 3 imply that Afi(s) is negative at the largest zero of A$_l(s). However 1 implies A$(s) is positive for real and sufficiently large values of s. Afi(s) has an odd number at zeros between each zero of AA (5). This follows since the sign of Afi(s) is -l the negative of the sign of B$_l(s) at each zero of I l I An_l(s), and An-l(s) and Bn-l(s) have alternate zeros. The degree of A$(s) is one greater than Afi—1(S)' Returning to the proof of the theorem, from proper- ties 4, 5, and 6 it follows that Afi(s) has one zero which is larger than the largest zero of Afi-1(S) and one zero between each zero Ah_l(s). The last zero must therefore be smaller than the smallest zero of A$_l(s). This ends the proof. Theorem 3.1.4. Consider the polynomials BA, An and En in Eq. 3.1.19 and Eq. 3.1.20, then i) the zeros of BA(s) and B5_ (5) alternate 1 ii) the zeros of Ah(s) and Ah_l(s) alternate iii) the zeros of Bh(s) and §h_l(s) alternate Proof: The proof follows an identical line to the proof for Theorem 3.1.3 and will not be repeated here. 56 A procedure for the derivation of Calahan's de- composition based on the Horowitz decomposition is now established through the use of the foregoing theorems. These theorems provide recursive formulas for obtaining the Calahan decomposition and they also establish some of the properties of the polynomials appearing in these recursive relations. The conditions under which AA(s)//EBA(S) and Ah(s)/Bh(s) are RC admittance functions have yet to be derived. It is possible to develOp the conditions for which Ah(s)/§h(s) is an RC admittance function without regarding the conditions for which AA(s)//_aBfi(s) is an RC admittance function. The next three theorems establish consitions for which Afi(s)//EBA(S) is an RC admittance function and also establish the fact that if Ah(s)/§h(s) is not an RC admittance function then Afi(s)//EBA(S) cannot be an RC admittance function for any value of a. 2) with ":25 _ 2 5' Theorem 3.1.5. Let P2n(s) — (s +2015 + pi i 1 pi > 012 and let the polynomials Afi(s) and B$(s) be de— rived from Eqs. 3.1.19. The rational function AA(s)//3Bn(s) is an RC admittance function if, and only if, Afi(0) > 0 and Ai(0) > 0 for k = l,2,...,n-1. Proof: Consider the if part of the theorem. Let AA(O)Z O and Afi(0) > 0, for k = 1,2,...n-l, and assume that 57 A$(S)//EB$(S) is not an RC admittance function. Since the zeros of A$(s) and B$(s) alternate along the real axis, the above assumption implies that Ag(s) or both A$(s) and Bfi(s) have some positive real zeros. The re— lation AA(0).: 0 implies A$(s) must have an even number of zeros in the right half plane or possibly one at the origin and at least one in the right half-plane. Theorem l(S) alternate along the real axis, therefore AA-1(s) must 3.1.3 established that the zeros of Afi(s) and AA_ have at least one positive real zero. However, the re- lation Ak-l(0)-3 0 implies that Afi_1(s) has an even num- ber of positive real zeros, and therefore from Theorem 3.1.3 Ai_2(s) has some positive real zeros. Continuing this reasoning, one can conclude that Ai(s) must have some positive real zeros. However sine Ai(s) is of de- gree one, it follows that Ai(0) < 0. Hence the contra- diction. Consider now the only if part of the theorem. If A$(S)//EE$(S) is an RC admittance function, then AA(s) has all negative real zeros or has at most one zero at the origin. Therefore A$(O) i 0. Since, from Theorem 3.1.3, the largest zero of Ai(s) is greater than the largest zero of Ai_l(s) k1 = 2,...,n; The zeros of Ai(s) k = l,...,n-l are all in the left half plane, and Ai(0) > O for k = l,2,...,n-l. This completes the proof. 58 In order to establish the conditions under which both AA(s)//EBA(S) and Afi(s)/§h(s) are RC admittance func- tions it is also necessary to consider the definition of 1 B 4 = tan- A as a single valued function. . . . -l B Definition 3.1.1. Let 4 = tan X-where O 3 ¢ 1 2 when B 1 0 and A 1 0 % < 4 i n when B 1 0 and A < 0 1T<<3TT — 2 when B < 0 and A i 0 2% <4k < 2n when B < 0 and A > 0 Theorem 3.1.6. Let A£(s) and Bi(s) be the polynomials defined in Eq. 3.1.13; then AA(s)//§Bh(s) is an RC admit- tance if, and only if, n O < ¢n : 2 0 < 4k < % k = 1,2,...n-1 -1 /EB£(0) where 4k = tan —XTT§T— as defined in Definition 3.1.1. k Proof: Consider first the if part of the theorem. From Definition 3.1.1 implies AA(0) l 0 and O A '6' A N|=I NIZI implies A£(0) > 0 for k = l,...,n-l. Therefore it follows from Theorem 3.1.5 that AA(s)//EBA(S) is an RC admittance function. Consider now the only if part of the theorem. If A$(S)//EBA(S) is an RC admittance 59 function then Afi(s) has non positive real zeros and B$(s) has negative real zeros. From Theorems 3.1.3 and 3.1.4 it is known that the largest zero at A$(s) is greater than the zeros of Ai(s) for k = 1,...,n-l and the largest zero of BA(s) is greater than the zeros of Bfi(s) for k = 2,...,n-l. Note that Eq. 3.1.19 yields Bl(s) = a1, where al is positive as it follows from the properties of the Horowitz decomposition. Therefore one can conclude that AA(O) : 0, 35(0) > O, and A£(0) > 0 Bfi(0) > 0 for k = 1,...,n-1. Consequently from Definitions 3.1.1 it n n follows that 0 < 4n 1 7 and 0 < 4k < 2 for k = 1,...,n-l. This completes the proof. Theorem 3.1.7. Let Ag(s) and B$(s) be the polynomials defined by Eq. 3.19. If AA(S)//EBA(S) is an RC admittance function for some a= a', then AA(s)//EBA(S) is an RC ad- mittance function for all a >cf. Conversely, if AA(s)//EBA(S) is not an RC admittance function for some a it cannot be an RC admittance function for a i a. Proof: The following Lemma is needed. Lemma 3.1.7. AA(s)//EBA(S) is an RC admittance function only if a > bi i = l,2,...,n Proof of Lemma: If a <13, then .. 5 a - b 0‘ 1 mm 60 and by Theorem 3.1.6, the rational function AA(s)//EBA(S) cannot be a RC admittance function. On the other hand, since the Horowitz decomposition is unique, Ag(s) depends only on a and not on the ordering of the complex zeros of P2n(s), i.e., n _ 2 2 2 2 P2n(s) — .H (s + 2015 + pi ), 01 > 01 i=1 where b. = /62 + 20.a + p.2 1 i 1 Therefore, a must be greater than bi' i = 1,...,n. This proves the Lemma. Returning to the proof of the theorem one observes from Lemma 3.1.7 that, only the case a > max {bi} need i be considered. Consider now the ratio I _ I /— I /EBn(0) = (a bn)/Esn_l(0) + aanAn_l(0) A'(O) (d-b )A' (0) - fine My (0) n n-l n n-l n which is formed from Eq. 3.1.19. This ratio can be re- written as /EB$‘1(0) /aan “535(0) _ Afi-1(07 + a‘bn _A:707_- — 1 - /aan /aB;_l(0) _ I (0 bn) An_1(0) 61 Introducing 1( 4n = tan.1 (v5hn(O)/Ag(0)) and en = tan— man/(0‘41”) I defined in Definition 3.1.1, the above expression can be written as tanbn_l + tanen tan 4 = n 1 tan4n_ltanen From this result it follows thatchn = ¢n-l + an and hence n n _ /E a. 4n = 2 9i = 2 tan 1 _ g . i=1 i=1 a 1 Therefore to complete the proof it is sufficient to show that /E ai is a decreasing function of a, and a-bi is an increasing function of a. Indeed, considering the derivatives ‘ 2] 2 a (C1 bi) = 8[ £2 20ia+pi = a1 3a 3a Zbi and 3/5 ai = 3[73'/2bi-2(a-ciA] = ai(bi-G) 3a 3a /a b the following observations can be made: The terms a1 and bi appearing in the above derivatives are positive as they follow from the properties of the Horowitz decomposition. (For proof see Theorem 3.1.1 and Eq. 3.1.11). Therefore (a-bi) is an increasing 62 function of a, while /5 ai is a decreasing function of a since a > max {bi}. Therefore /5 ai/(a-bi) and k —1 4n = 2 tan (/5 ai/(a-bi)) is a decreasing function of i=1 a, a > max {bi}. H and 0 < 4. <-— i 2 A»: The theorem follows since 0 <<4n : is necessary and sufficient for A'(s)//HB'(s) to be an RC admittance function. This completes the proof. Corollary 3.1.7 AA(S)//EB$(S) is an RC admittance function only if Ah(s) §h(s) is an RC admittance function. Proof: The proof follows immediately from the limits Lim A'(s) = X (s), Lim ./E B'(s) = 'B' (s) and n n n n a+oo a+oo Theorem 3.1.7. The conditions for which Ah(s)/Bh(s) represent an RC admittance can be developed directly from Eq. 3.1.20 and Theorem 3.1.4. A Theorem which establsihes these con- ditions follows. Theorem 3.1.8. Let Ah(s) and §h(s) be the polynomials defined by Eq. 3.1.20. Then Afi(s)/§h(s) represents an n RC admittance, if and only if 2 tan i=1 8 1 —£ < % where 0-1—- w. = a - oiz and Oi > 0. 63 Proof: From Eq. 3.1.20 one can form the ratio Bn(0) = An_l(0)wn+ Bn_1(0)0n An(0) An_l(0)on- wan_l(0T or (1:2 + Bn_1(0) B (0) o A (0) n _ n 11"]. An(0) 1 _ 22 Bn-lidY on An- (0) Introducing as before B (0) w _ -l k _ -l k @k — tan Ak(0) and 9k — tan 3; where tan.1 2 is defined in Definition 3.1.1 one has B (0) n n w —l n -1 k tan = X 0- = 2 tan —— An(0) i=1 1 k=1 0k Consider first the "only if" part of the theorem. n -1 wk n - Bn(0) n Assume kEltan a; > 7, then tan XgTfiT > f and from Definition 3.1.1, Ah(0) < 0 and/or §h(0) i 0. If Ah(0) < 0 then Ah(s) has some zeros in the right half plane and Ah(s)/§h(s) is not an RC admittance function. If §h(0) i 0 then Bh(s) has a zero at the origin or in the right half-plane. Since the zeros of Ah(s) and Bh(s) alternate and the degree of Aa(s) is one greater than the degree of 55(5), Ah(s) must have some zeros in the 64 right half-plane and Ah(s)/§$(s) is not an RC admittance function. Consider now the "if" part of the theorem. Assume "15(0) n _ (1). n-1 n = 2 tan 1 3i i % K (0) i=1 i n Then 55(0) > 0 and A (0) > 0. Note also that Oi > O n n_ w. i = 1,...,n, otherwise 2 tan 1 -£-would be greater than i=1 i g as can be seen from Definition 3.1.1. Therefore ml —1 k(0) tan WI 0 A R4: and Bk(0) > 0, Ak(0) > 0 k = l,2,...,n-l. Suppose Xh(0) : o and Kk(0) > o k = 1,...,n-l, but Ah(s)/Bh(s) is not an RC admittance function. An(s)/Bn(s) not an RC admittance function implies An(s) or An(s) and Bn(s) have some zeros in the right half-plane. Since An(0) : 0, An(s) must have an even number of zeros in the right half-plane or possibly one at the origin and at least one in the right half-plane. Since by Theorem 3.1.4 Ak(s) and Ak_l(s) have alternate real zeros, An_l(s) must have some zeros in the right half plane. By this process one concludes that Al(s) must have some zeros in the right half-plane. However Al(s) is of degree one and therefore Al(0) < 0, hence the contradiction. This completes the proof. 65 The preceeding theorems show clearly that the Calahan decompositions can be established from the Horo- witz decomposition. Conversely, by using the Calahan decomposition the Horowitz decomposition can be estab- lished. This is proved in the following theorem. Theorem 3.1.9. Let P(s) be a real polynomial of degree n with all the real zeros (if any) positive, then the Horowitz decomposition P(s) = : A2(s) - s B2(s) can always be obtained from the Calahan decomposition Proof: Using the transformation 5 = -A2 the polynomial P(s) takes on the form , _ _ 2 P2n(A) — P( A ) Now Pén(A) has no real zeros since the zeros zis of P(s) are not negative real or zero by hypothesis and the zeros of Pén(A) are Ai = 1 /:zis i = 1,2,...,n. Therefore the Calahan decomposition can be formed to yield 2' (A) = [A m12 + [B (1)12 2n n n From Eq. 3.1.14 the following ratio can be formed wn Bn_l(A) + Bn(A) = on+A An_l(A) An(A) wn Bn_l(A) ' o +A An_l(A) n 66 from which -1 B (A) n 4- tan n — -l _l . . Now since Pén(A) = P(-A2), Pén(A) is symmetric about the imaginary axis. It follows that tan = 2 (tan + tan _ An(A) i=1 Gi+A Oi+A where 2k complex aeros are located symmetrically about the imaginary axis and 2 zeros are on the imaginary axis. Since w w 1 2w -l i -l i _ - 1 tan o-+A + tan :ET:AI — tan 1? 2 2 1 1 A ’0- -w i i the above equation can be written as 2 B (A) k w-A 2 w- -l n -l i -l 1 tan = 2 tan + 2 tan —— AnZAS i=l A2-0.2-w.2 i=1 4 i i From which it follows that _ B (A) -1 B (m 4 (A) = tan n = — tan n n An(A) An(-A) Now An(A) equals zero for those values of A where ¢n(A) equals an odd multiple of 1 Similarly, Bn(A) equals 2. zero where 4n(A) equals an even multiple of %. Since ¢n(A) = -4n(-A), the polynomials An(A) and Bn(A) must have zeros placed symmetrically about the origin, with one of them having a zero at the origin. From the equation 67 IIMEO _ -1.._ 1 n(0)— tan ()—.9.2 j l Bn(A) has a zero at the origin if l is even and An(A) has a zero at the origin if 2 is odd. From the above dis- cussion, Pén(l) can be written as n/2 n/2 - 1 Pi (A) = n (-s-a.2)2 - s n (-s-b.2)2 n ._ i ._ 1 1—1 1—1 or n-1 n—1 “3" ‘3" P(s) = - s n (-s-ai)2 + n (-s-bi2)2 i=1 i=1 Now the zeros of An(x) and Bn(l) alternate on the real axis as implied by the Calahan decomposition. Therefore one can write for each case or Therefore P(s) can be written as P(s) = i [A2(s) - s B2(s)] where A(s)/B(s) and s B(s)/A(s) are RC admittance functions. This completes the proof. 3.2 Decompositions for Polynomials with Real Zeros In the preceeding section the Calahan decomposi- tion was shown to exist for polynomials with no real zeros. 68 The Horowitz decomposition is known to exist for poly- nomials with no nonpositive real zeros. In this section a method is developed for systematically decomposing those polynomials with negative real zeros. To the best of the author's knowledge this type of decomposition has not been considered in the literature. Theorem 3.2.1. Let P2j(s) and P2k(s) be two polynomials with the following Calahan decompositions _ — 2 — 2 P2j(s) — [Aj12 2k k k if P2n(s) = P2k(s)P2j(s) then P2n(s> = [Xn12 + [fin‘SHZ where An(s) = Xj(s)Xk(s) - §j(s)§k(s) (3.2.1) and §n(s) = Kj(s)§k(s) + §j(s)z‘ik(s) (3.2.2) Proof: Consider Eq. 3.1.19. It is repeated here for con- venience: An(s) = (s + on)Xh_l(s) - wn§h_l(s) §h(s) = (s + on)§h_l(s) + wnXh_l(s) 69 where X6(s) l and §$(s) E 0 It can be seen that An(s) and Bn(s) can be obtained directly from n 2 2 P2n(s) = H (s + Zois+pi ) = i=1 i (s+oi+jwi)(S+Gi-jwi) ":15 1 Indeed, since P2n(S) = [Kh(S) + j§5(S)J[Kh(S) - j§h(8)] one has (S + Oi + jwi) n 55(5) + j§h(s) = .21 i The recursion relation (Eq. 3.1.14) holds as can be seen from the following expressions [55(8) + j§5(5)] = [s + on+ jwn][Kh_l(s) + j_h_l(5)] (S + 0n)Xh—1(S) En-l 3’ m V II (s) -w n w m II (s + on)§h_l(s) + wan-l(s) From the associative and accumulative laws of complex numbers the proof follows: [Aj(S) + JBj(S)][Ak(S) + JBk(S)] = Aj(s)Ak(s) - Bj(s)Bk(s) +j[Xj(s)§k(s) + §j(s)Kk(s)] 70 k k Theorem 3.2.2. Let P (s) = n (s+a.) n ((s+c.)2+ w.2) n ._ 1 . i 1 1-1 1:1 where a > O a # a i f ' and g tan-1(w /o )< 1 i ' i j 3' i=1 i i 2' then there exists a decomposition such that Pn(s) = Ak+£(s)Ak(S) + Bk+£(s)Bk(s) where Ak+£(S)/Bk+£(s) and Ak(s)/Bk(s) are RC admittance functions. Proof: ConSider Ak+£(s) and Bk+£(s) from Eq. 3.1.1 and Eq. 3.1.2 Ak+£(s) = Ak(S)A£(S) - Bk(S)B£(S) _ _ _ _ _ (3.2.3) Bk+£(s) = Ak(s)B£(s) + Bk(s) A£(s) From Eq. 3.2.3 one can form the ratios §é+£(5) = Xk<8> _ §£(S) A£(s)Bk(s) Bk(s) 32(5) and §k+2(s) §k(s) §£(s) K£(S)Kk(s) + Xk(s) + 31(8) Adding the rational functions and clearing the denomina- tors one obtains Kx+2(S)Ak(S) + Bk+2 (s>§k = X£ + Bk2‘5’1 (3.2.4) Now it can be observed that the left hand side of Eq. 3.2.4 is of the desired form. To obtain a decomposition 71 of P(s) in this form, one may calculate the Calahan de- composition of k 2 2 _ — 2 —2 -E [S + oi) +pi ] — Ak (S) + Bk(S) i—l — 24 — and set A£(s) = N(s + ai). One can now select B£(S) i=1 such that A£(s)/B£(s) is an RC admittance function, the degree of §k(s) is one less than the degree of.K£(s) and E (0) _ E (o) X (0) + tan 1 “k < 1 ° 2, Ak(0) 2 This can always be done since the zeros of Kg(s) are dis- tinct by hypothesis and §£(s) can be modified, if neces- sary, by the multiplication of an arbitrary constant so that the angle condition is satisfied. Now using Eq. 3.2.3, Ak+£(s) and Bk+ is an RC admittance function since 2(s) can be calculated. Ak(s)/Bk(s) _ E (0) k _ w. tan 1 _k = 2 tan 1 —£ < % Ak(°) i=1 i by hypothesis. Ak+2 /Bk+£(s) is an RC admittance func- tion since E (0) fi (0) E (0) -l k+2 l g k N tan 31—7—5- - tan r— + tan m < as can be seen from the ratio 72 _ E'gw) + Ekm) Bk+ (0) _ Ki(0) Ai(o) Kk+£(0) “ §i(0) §k(0) 1‘ WW and Since Ak+£(s) and Bk+£(s) form the Calahan decompOSi- tion of the polynomial [Ak+£ (sn2 = {[Xi(s)12+[fiz(s)12}{[Xk+ B(Z)(A2)] F(A) = F‘l’12} 75 P‘2’(s) = i{[A(2)(S)]2 — s [B(2)(S)]2} This completes the proof. 3.3 Realization Techniques Using Calahan and Horowitz Type Decompositions. In the preceeding sections, relationships between two of the existing decomposition techniques are estab- lished. Furthermore these decomposition techniques are extended to polynomials containing distinct negative real zeros as well as complex zeros. In this section these techniques are used to synthesize low-pass RC net- works. The extended Horowitz decomposition can be used for RCNIC and RCF-R filter synthesis. Therefore the configurations corresponding to RCNIC and RCT-R networks are established as well as the configurations for RC? networks. 3.3.1 RCF Filter Realizations Given the open circuit voltage-ratio transfer function TV= kN(s) D(s) where D(s) has a Calahan type decomposition D(s) = Xk(s)§3(s) + §k(s)§5(s) It will be shown that a RCr network of the form shown in Fig. 3.3.1 can be realized. It is well known that the open-circuit voltage- ratio TV for a network as shown in Fig. 3.3.1 is [MI] 76 TV = ‘“ Yziazzib Y 2 22d + a leb Therefore Tv = kN(s)/D(s) is written in the form -aklNl(s) k2N2(s) Bj(s) Ak(s) V A.(s) Bk(s) §17T+°‘2‘17—TT j s a Ak s where klk2N1(s)N2(s) = kN(s), and the functions Y22a’ YZla’ ZZlb' and Z21b can be identified as Y = A.(s) Y = klNl(s) Z = k2N2(s) 22a Bj(s) ’ 21a Bj(s) ' 21b Ak(s) 1 Bk(s) and Z = ——- -——§—-. Note Y22a and leb are RC llb 2 A a driving point functions, as required, from the proper- ties of the Calahan decomposition. The realization is completed when the "a" network is synthesized from and Y and the "b" network from Z and Z . 'Yzia 22a 21b llb The realization of RC networks from -Y21 and Y22 or 221 and Z22 is well known and will not be explained here. A low pass filter example follows. Example 3.3.1 Given the open circuit voltage ratio TV k (s+l)[s+2)2+sz] 77 Realize a low pass RC gyrator filter of the form shown in Fig. 3.3.1. Realization: First one must decompose the denominator polynomial into the form established in Theorem 3.2.2. From P(s) = (s+1)[(s+2)2 + 22] let Ki(s) = 5+2, §i(s) = 2, and K (s) = s+l. Selecting gl(s) = %, i.e., such that §l(s)/§l(s) is an RC admittance function and tan-l Bl(0) + tan.1 Bl(0) < E Al(0) £130; 2 or -1 2 -1 1/2 1 tan 3 + tan 1 < 7 one can now calculate A2(s) and 82(5) using Eq. 3.2.1 to obtain X2(s) = s + 38 + 1, §2(s) = (2.55 + 3), and P(s) = (52+3s +1) (5 + 2) + 2(2.55 + 3). Arranging TV into the form = 1/(2.5s + 3) 2/(s+2) V (52 + 35 + 2)/(2.55 + 3) + 2/(s+2 one can detect that _ 2 Y22a — (s + 3s+2)/(2.55+3), 78 221 = 222 = 2/(s+2), and a = l. The network realization is shown in Fig. 3.3.2. 3.3.2 RCVNIC Realizations . . . . _ kN(s) Given the open Circuit voltage ratio Tv - 5(5)— where D(s) has a Horowitz type decomposition D(s) = A(3)(s)A(l)(s) + s 8(3)(S)B(l)(s) It will be shown that a RCF network of the form shown in Fig. 3.3.2 can be realized. It is known that the voltage-ratio transfer function T for a network as V shown in Fig. 3.3.3 is [CA3], 21a)(221b) i k (-Y V T = - (Y )(z ) V 22a llb V k and so TV is arranged as i klNl(s) k2N2(s) A(l)(s) A(3)(s) l _ s B(l)(s) B(3)(s) k .1 A(l)(s) A(3)(s) V kv from which one can identify —Y21a=klNl(s)/A(1)(s), Y22a = SBl(s)/A(l)(s), = k2N2(s)/A(3)(s), and z (B(3)(s)/A(3)(s))kv. 221b(5) llb = 79 RC RC Fig. 3.3.1 General one-gyrator RCF filter. 1.8. 1_8 29 25 l o———4\/\ H 0‘ N l—J .b m own 1 . , )7 1 All values are in mhos and farads. 2 v = (s+l)(s+2)2+22) T Fig. 3.3.2 Realization of RC? filter for Example 3.3.1. 80 Y22a and leb are RC driVing pOint functions from the properties of the Horowitz decomposition. Therefore using standard RC network realization techniques [GU], a network of the form shown in Fig. 3.3.3 can be realized. Note: One could also divide numerator and denominator of TV by sB(l)(s)/B(3)(s) and obtain similar results. A low pass filter example is given in the following. Example 3.3.2 Given the open circuit voltage ratio transfer function Tv k T = 2 (s+l)(s + 4s+9) V realize an RCVNIC low pass network as shown in Fig. 3.3.3. Realization: First one must decompose the denominator polynomial into the form shown in Theorem 3.2.4: P(s) = (s+l)(82+25+9) = (s+1)[(s+3)2 - 5(52)] where A(l)(s) = (s+3), B(l)(s) = 2, and A(2)(s) = s+1. One can select B(2)(s) = 2 so that (s+l)/2 and 2s/s+l are RC admittance functions. Now one can calculate A(3)(s) and 8(3)(s) using Eq. 3.2.2, to obtain A(3)(s) = 82+ 85+3, B(3)(s) = 25+8 and P(s) = (52+ 85+3)(s+3) - s(2s+8)(2) At this stage rearranging T into the form V _ .. .3 j: T _ 25+8 25 V 2 l - s +85+3 5+3 25+8 25 _ ._ 2 -— = The parameters KV—l, Y22a — (s +85+3)/(25+8), Y21a 3/(25+8), 221 = 3/25, and 211 = (s+3)/25 can be identified. The network given in Fig. 3.3.4 is the realization of TV. 3.3.3 RCF(—R) Filter Realizations Given an open circuit voltage-ratio transfer function TV = k %%§%-where D(s) has a Horowitz type de- composition D(s) = A(3)(5)A(l)(s) - 58(3)(s)sB(l)(5), real— ize an RCF(-R) filter as is shown in Fig. 3.3.5. Realization: The open circuit voltage-ratio transfer function for a network as shown in Fig. 3.3.1 is —O. _ Y12aZZlb T V 2 Y a+a Z 22 llb However T kN(s)/D(s) can be rearranged to obtain v K1N1(s) K2N2(5) T =__B(1)(S) A_(3)(S) V A‘l)(s) - 583(8) B(l)(s) A(3)(s) Now every negative RL impedance can be realized as a positive RC impedance and a negative R, as can be shown by the general expression 82 _— VNIC O RC RC A B Fig. 3.3.3 General RCNIC filter I—‘ox b.) VNIC ————Nflv 5'00“ ; .‘ |__ N“. uqm All numbers are in mhos or farads. -9 T = 2 (s+l)(s +4s+9) V Fig. 3.3.4 RCNIC realization for Example 3.3.2. 83 n ais -ZRL(S) = _ [ Ro 'F.§ s+b ] i-l i n n aibi -Z (5) + R + z a. = 2 = Z (5) RL 0 i=1 1 1:1 s+bl RC or -ZRL(S) ZRC_R that is K Nl(5) K N2(s) T = l Bl(5) 2 B2(s) V l A—(S) R + z 1 ' RC B (5) where R is selected sufficiently large so that _ 533(5) A3(S) tion. At this point the following identifications can + R = ZRC is a positive RC admittance func— be made: Y12a = K1N1(s)/B‘l)(s), Y22a = (A‘l’(s)/B‘1’(s) - R), Z21b = K2N2(s)/B(2)(s), and leb = ZRC' The network whose configuration is as shown in Fig. 3.35 can be realized with standard RC transfer function synthesis techniques. Example 3.3.3 Given the same open circuit voltage-ratio transfer function TV as in example 3.3.2 realize a RC gyrator negative R filter as shown in Fig. Realization: TV = or TV But every negative RL impedance can be realized as a 84 3.3.5. From example 3.3.2 k (52+ 85 + 3)(5+3)-5(25+8)(2) k (5+3)(25+8) 52+85 + 3 2s 25 + 8 5+3 positive RC impedance and a negative R. That is 25 _ 6 _ ”5:3‘+2‘2’§fi 2 Therefore k _ (5+3)(25+8) Tv’ 2 5 +85 + 3 + 6 _ 2 25+8 5+3 (Y )(Z ) From TV = 21a2 11b Y22a+ “ leb one obtains 2 -Y = __3__ Y = 5 +85 + 3 21a 25+8 ' 22a 25+8 Z = Z = —9— and a = 1 11b 21b 5+3 The realization is shown in Fig. 3.3.6. _ 2’ 85 T 1 RC - R RC A B Fig. 3.3.5 General RCT-R filter. .9. l3 2 l 0-—AVVF——_——‘\WV' J> 8 '22. l .2: ___ _ l g ] O‘)H—' All values are in mhos or farads. T = 18 v (s+l)(sz+45+9) Fig. 3.3.6 RCP-R realization for Example 3.3.3. 86 3.4. Conclusions In this chapter, Calahan's decomposition is developed starting from the Horowitz decomposition. The development of the Horowitz decomposition from the Cala- han decomposition is also given. Polynomial decomposi- tions of the Calahan and Horowitz-type are developed for polynomials which contain some negative real distinct zeros. No systematic methods have been given in the literature for such decompositions. Sample low-pass transfer functions were synthesized by using Calahan and Horowitz-type decompositions. Finally, a low-pass RCP-R filter was synthesized using a Horowitz-type decom- position. This particular filter configuration, which has not appeared in the literature, can be utilized to synthesize any pole configuration (zeros of denominator) as long as the real poles (if any exist) are negative and distinct. CHAPTER IV LOW-PASS RC1" FILTER REALIZATIONS In Chapter II, it is established that the natural frequencies of a one-gyrator RCP network obey a rather restrictive angle condition. In Chapter III, it is demon- strated that this angle condition (satisfied with an in- equality) is, in fact, sufficient to permit realization of single-gyrator RCF low-pass filters. However, the poles of most low-pass voltage-ratio transfer functions, T do not obey this angle condition beyond the third v! degree case. This has necessitated, in practice, factor— ing the denominator of T into polynomials each of which V do satisfy the angle condition, and then realizing each polynomial with an RC? section. These sections must then be connected through isolation amplifiers not only to realize TV, but also to prevent loading effects. It is, of course, always possible to realize such filters with active devices such as NIC's, -R'5, or controlled sources along with RC networks. Such realization techniques are well represented in the literature [LI] [SK] [YO] [HA]. However, in all these techniques the possibility of in— stability is introduced into the network realization. As 87 88 is shown in Chapter II, RC networks satisfying the con- ditions of Theorem 2.4.2 cannot become unstable. In the first two sections of this chapter, it will be established that forth degree low-pass voltage- ratio transfer functions, whose denominator polynomial being strictly Hurwitz, can always be realized with a two-gyrator RCF network satisfying Theorem 2.4.2. An example realized by the proposed method is compared with a computer realization. Also computer realizations are given for three practical low-pass filter networks. In section 4.3, some necessary conditions, in the form of an extension of Calahan's angle condition, are conjectured and feasibility arguments are used to support them. In section 4.4, computer realizations of two- gyrator RCP networks are given for some fifth and sixth degree low-pass voltage-ratio transfer functions which are of practical interest. In addition, a sample three- gyrator RCP computer realization is given to illustrate extensions of the method. In section 4.5, the established methods of low- pass RCP filter realizations are extended to both the band-pass and the high-pass RCF filter. 4.1 Two-Gyrator RCr Realizations for Fourth Degree Low- Pass Transfer Functions In Chapter II the expression 89 _ 1 A — Aa(1,RC)Ab(RC)AC(RC) ll i+1 ~ 1 i +alal Aa(1,i,RC)Ab(i+l:RC)AC(RC) +1 +1'RC) - 1 j +d2a2 Aa(l,RC)Ab(j,RC)A j C(j i+1 j . ~ 1 i j+1 +dlald2a2 Aa(l,i,RC)Ab(k+1,J,RC)Ac(j+1:RC) is established for the denominator polynomial of the voltage-ratio transfer function of the network given in Fig. 4.1.1. Consider the network in Fig. 4.1.2. This network satisfied the conditions for Theorem 2.4.2, and it has the following admittance matrix " _ '7 G1 G1 —G1 ClS+Gl a1 -a1 CZS+G2 -G2 Y: -G2 C3S+G3 a3 L_ -a3 C4S+G4 (4.1.1) Let Y' denote the resultant matrix when the first row and column are deleted from Y, one has A = det(Y') (4.1.2) 11 and 9O (ll OLZ 0———) RC ( RC 0, l A B Fig. 4.1.1 General two-gyrator RCF filter. Gl a1 G2 d2 HWT Tm” ' 1 C1 C2 C3 I C4 G Fig. 4.1.2 Fourth degree low-pass RCF filter 91 '— “(I C18+Gl a1 -a C S+G -G Y' = a G2 C3S+G2 2 L "a2 C4S+G4 From the relations Y' = CS + H and Y" = C-l/ZY'C-l/z (4.1.4) one can obtain 5 + x1 'Xsal/al -szdl/al S + X2 -1/X2X3 Y" = — X2X3 S + X VX a /d 3 6 2 2 - X6a2 a2 S + X4 L _. (4.1.5) with x1 = Gl/Cl X4 = G4/C4 X2 = Gz/Cz X = O‘10‘1/C1‘32 (4.1.6) X3 = G3/C3 X = d2d2/C3C4 From Eq. 4.1.6 it follows that 92 Conversely, if all the X_ are positive, one can calculate i all the element values in terms of G1, alai, and a252' This results in the set of equations cl = Gl/Xl c3 = 52/x3 C2 = o‘1‘”‘1/X5Ci C4 = O‘20‘2/X6C3 (4.1.7) 52 = czx2 G4 = c3/x4 Forming the determinant of Y", one obtains from Eq. 4.1.4 1 det(Y") = det(C- )All (4.1.8) and from Eq. 4.1.3 det (Y") = S(S + xl)(s + X2 + X3)(S + X4) + X5(S + x4)(s + x3) + X6(S + Xl)(s + x2) + sz6 (4.1.9) Therefore the synthesis problem is now reduced to finding a set of positive Xi (i = 1,2,...,6) for a monic poly- nomial P(S) = det (Y") where the low-pass voltage—ratio transfer function is TV = k/P(S) = Anl/All = alaZGle/All (4.1.10) with All = det(C)det(Y"). Note 1: Every network realization of T = k/P(S) repre- V sents two possible network realizations since _ H _ I I I All — det(Y ) — det (U Y U ) where U' has the form 93 _. H U' = /l / / Ll’ - In terms of the network shown in Fig. 4.1.2 a second realization can be obtained by interchanging the follow- ing parameter values: and G l 4 Cl and C4 C2 and C3 a a and a a Note 2: The zero frequency gain for the network in Fig. 2.1.2 is given by D T ___ _41‘ = GiGzo‘if‘z 1 V 8:0 11 5:0 Gledla2 + G2G4alal + aldlazaz which can be calculated from the Y matrix in Eq. 4.1.1. Since the denominator polynomial is dependent on the pro- ducts aldl and a282, it is apparent that unbalanced or active gyrators can be used to improve the voltage gain. For example, a1 = 10 and d1 = %6-would give ten times the voltage gain that could be achieved for a1 = dl = 1. Theorem 4.1.1. Let P(S) be a real strictly Hurwitz poly- nomial of degree four, then P(S) can be put in the form 94 P(S) = [(s + al)(s +A2) + Kl][(s + a2)(s + a3) + K2] (4.1.11) where the parameters al,a2,a3,Kl, and K2 are all positive and 0 < a2 < a1 + a3. Proof: Consider first the polynomial 2 2 2 P(S) = [(S + o + “1][(S + o + wz] (4.1.12) 1) 2) which has all complex roots. If 01 = 02 Eq. 4.1.12 is already in the desired form. Therefore assume, without loss of generality, that 01 < 02. Hence, P(S) 2 2 [(S + 01) + wl][(s + ol)(s + 202 - cl) 2 2 + wz + (02 - Ol) ] 2 2 . . where 202 - 01 and wz + (02 - 01) are p051tive. Select- ing a1 = 01, a2 = 01. a3 = 202 - 01' K1 = mi and K2 = 2 _ 2 ' < < < w2 + (02 01) the relations 0 al S a2 . a3, 0 K1' and O < K2 hold. This proves the theorem for the case where P(S) has all complex roots. Consider now the real polynomial P(S) = (S + zl)(S + 22)(S + z3)(S + 24). Since P(S) is strictly Hurwitz, one can select a so that 2 0 < a2 < minimum {Re(z.)} (r.1.13) . i l=llooo,4 95 Now if z and z are real, one has 1 2 (S = zl)(S + 22) = (S + a2)(s + 21 + 22 - a2) + (21 - a2)(z2 - a2) and if 21 and 22 are complex conjugates, letting Zl = 22 = 01 = 3 ml, one has + (O - a 2 "a2)+‘” 1 2 (S + zl)(s + 22) = (S + a2)(s + 20 1 1 Note that all the constant terms in both expansions are positive. Indeed, this is evident from Eq. 4.1.13. The same type of representation holds for (S + 23)(S + 24) and therefore P(S) = [(S + a2)(s + a1) + Kl][(S + a2)(S + a + K2] 3) where a1,a2,a3,Kl, and K2 are positive as required. The condition 0 < a + a is satisfied since 2 <"1‘1 3 a2 < minimum 4{Re(zi)} implies a2 < Re(zl) + Re(22) - _pooo' a2 = al. This completes the proof. Theorem 4.1.2. Given the polynomial P(S) = [(s + al)(s + a2) + Kl][(S + a2)(s + a3) + K2] where all the parameters are positive and O < a2 < a + a3, then there exists Xi > O (i = 1,...,6) such that 1 96 P(S) = S(S + Xl)(S + x + x3)(s + x4) 2 +x5(s+x3)(s+x4) + X6(S + X1) (8 + X2) + X5X6 Proof: Let P(S) be written in the form p(5) = (s + a2)ZS(S + a + a3) + Kl(S + a2)(S + a3) 1 2 + K2(S + al)(S + a2) + a a3(S + a2) + K1K2 1 (4.1.14) In order to identify the Xi parameters it is desirable to put P(S) in the following form _ 2 — P(S) — (s + a2) 5(5 + a1 + a3) + hl(s + a1 + a3 a)(S + a2) + h2(S + d)(S + a2) + hlh2 (4 1 15) with hl,h2, a +a3 -d, and a positive. Director compari- son of Eq. 4.1.14 and 4.1.15 leads to the equations hl + h2 = Kl + K2 + ala3 h1h2 + K1K2 (4.1.16) h1(al + a3 - a) + h2(a) = Kla3 + Kza1 + ala3a2 That the relation (hl + h2)2 > 4 hlh2 holds follows im- mediately from Eq. 4.1.16. Therefore h1 and h2 are real and positive. In order to continue with the proof the following lemma is needed. and a a be K2' 1 3 Lemma 4.1.2. Let, in Eq. 4.1.16, Kl’ positive, then 97 max{hl,h2} > max{Kl,K2} + ala3 and A min{hl,h2} min{Kl,K2} l _. then from Eq. 4.1.16 one obtains min{K Proof of Lemma: Let h' max{K K } + a a 1' 2 1 3' hl + h2 = hi + hi hlhz < h'hé (4.1.17) From which it follows that hl(h1 + hi - hl) < hihé (4.1.18) or (hl - h1)(hl - hi) > 0 Without loss of generality, let hl be less than h2. Then from Eq. 4.1.19 and Eq. 4.1.17 it follows that I I hl < hl and h2 > h2 . This proves the lemma. Returning to the proof of the main theorem, from Eq. 4.1.16 one can write (h2 - hl)a = Kla3 + K2al + ala3a2 - hl(al + a3) (4.1.20) 98 Now the condition 0 < a < a1 + a3 is necessary if all the Xi are to be positive. Let h2 be greater than hl, then the condition 0 < (h2 - hl)a < (h2 - hl)(al + a3) is necessary if all the Xi are to be positive. First consider the condition (h2 - hl)a > O. From Eq. 4.1.20 it follows that (h-h)OL=(Kl-h +(K 2 l -hl)al + a a a > 0 1>a3 2 1 2 3 since h < min{K 1 l,K2}. Consider now the condition (h2 - hl)a < (h2 - hl) (a3 + a1). From Eq. 4.1.20 this is equivalent to Kla3 + Kzal + ala3a2 < h2(a3 _ a1). On the other hand, sinc a < a + a it follows that 2 1 3' K + K a + a a a )a 1a3 2 1 1 3 2 < (K1 + a la3)al + (K2 + ala3 3 or Kla3 + K2al + ala3a2 < max{Kl,K2} + ala3](a1 + a3). From lemma 4.1.2 h2 > max{Kl,K2} + ala3 and therefore 1 + alaza4 < h2(al + a3) or (h2 - hl)a < (h2 - h1)(al + a3) Eq. 4.1.15 is now established with 99 — 2 - P(S) - S(S + a2) (5 + a1 + a3) + hl(s + a1 + a3 a)(s + a2) + h2(S + a)(S + a2) + hth (4.1.21) where all parameters including a1 + a3 - a are positive. From Eq. 4.1.21, one choice of Xi is i 2 2 x2 = 9 x5 = hl (4.1.22) X3 = a1 + a3 - a X6 = h2 This completes the proof. Example 4.1.4. Let, in the voltage-ratio transfer func- tion TV = k/P(S), the polynomial P(S) be given as P(S) = ((s + 1)2 + 22)((s + 2)2 + 42) Notice that tan-1(2/1) + tan—1(4/2) > n/2 and so no one- gyrator realization is possible. TV is to be realized as a low-pass filter with the configuration shown in Fig. 4.1.2. Realization: Using the method established in Theorems 4.1.1 and 4.1.2, P(S) can be put in the form P(S) = [(s + 1)?‘ + 22][(S + 1)(S + 3) + 17] 100 or p(5) = S(S + 1)2(s + 4) + 20(s + 1)2 + 4(s + 1)(s + 3) + (4)(l7) From Eq. 4.1.20, it can be seen that h + h = 24 :S‘ 23‘ ll (4) (l7) h2(a) + hl(4 - 0:) = 32 or h1 = 3.28 h2 = 20.72 a = 1.082 P(S) can now be put in the following form P(S) = S(S + 1)2(s + 4) + 20.74 (s + 1) (s + 1.082) + 3.28(S + 1)(s + 2.918) + (3.28)(20.72) One choice of Xi’ for P(S), is X1 = 1 X4 = 1 X2 = 1.082 X5 = 3.28 X3 = 2.018 X6 = 20.72 with 101 G1 = 1 C1 = 1 C2 - .3048 G2 = .33 C3 = .1132 C4 = .427 G4 = .427 alal = l azdz = l where all component values are either in mhos or farads. The corresponding network configuration is in Fig. 4.1.2. 4.2 Computer Synthesis of Low-Pass Fourth Degree Filters In the preceeding section an analytical method of realizing fourth degree low-pass voltage-ratio trans- fer functions with the network configuration shown in Fig. 4.1.2 is established. It is also possible to realize low-pass filters by solving the set of nonlinear equations in Xi defined by the coefficients Indeed, since P(S) S(S + X1)(S + X + X3)(S 2 + X6(S + X1) (S + X5X6 Equating the coefficients expressions, one can obtain a set of the following form: 4 3 i of P(S) = S + Z P.S , . 1 i=1 + X4) + X3(S + X3) (S + X4) + X (4.2.1) 2) of P(S) in these two of nonlinear equations 102 P0 = Po(xi) s X5x6 + X6X1X2 + x5x3x4 3 P1 = P1(Xi) E x6(Xl + x2) + X5(X3 + X4) + xlx4(x2 + X3) ) (4.2.2) P2 = Px(xi) : x6 + x5 + x1(x2 + x3 + X4) ( + x4(x2 + x3) P3 = P3(xi) 5 X1 + x2 + X3 + x4 J Since there are four equations and six unknowns, there is no unique solution. A computor program (described in Appendix A) has been written to solve such nonlinear equations. An error criterion E is used to test the validity of the solutions. The error criterion used is taken in the form Pi(xi) P . O 3 E = Z = i 1 (max{PO,Pl,P2,P3})ll - 0 A solution is accepted when E is sufficiently small, so that each coefficient is accurate to at least 8 places. Example 4.2.1. Let Tv be the same as in Example 4.1.1. Realization: Using the computer program, with X1 = X4 = 1 ~ and aldl = a2d2 = l preset, the following results were obtained: 103 X1 = 1. G1 = 1. X2 = 1.0823371 C1 = 1. X3 = 2.9176629 G2 = .32975942 X4 = 1. C2 = .30467350 X5 = 3.2822021 C3 = .11302177 X6 = 20.717798 G4 = .42706533 C4 = .42706533 where the component values are either in mhos or farads. These results agree up to three significant figures with those found in Example 4.1.1, however, they are much more accurate, as Example 4.1.1 was calculated with a slide rule. For comparison purposes it is interesting to note that the error coefficient E is E = .065 for Example 4.1.1 8 E < 10- for Example 4.2.1 Considerable freedom is available in the choice of Xi to solve Eqs. 4.2.5. One can, to some extent, take advantage of this freedom by specifying additional con- straints. For example a useful constraint is G1 = k G4, and it can be written in terms of the parameters Xi as - ' X1X2X6 ‘ k X3x4x5 (4.2.3) or G162 2“2 = k. G2G40‘15‘1 c c c c c c c c (4°2°4) 1 2 3 4 3 4 1 2 104 where K' = kazdz/aldl. It is evident from 4.2.4 that different ratios of Gl/G4 are also possible by modifying the ratio azaZ/aldl for a given set of Xi. Computer solutions with various Gl/G4 ratios are offered in Tables 4.2.1, 4.2.2, and 4.2.3 to the Linear Delay, the Butterworth, and the Chebyshev fourth degree low-pass voltage-ratio transfer functions. The polynomial coefficients for the Linear Delay filter are from Van Valkenburg [VA2, the polynomial coefficients for the Butterworth filter are from formulas in Van Valkenburg generated to the necessary eight place accuracy with a computer program, and the polynomial coefficients for the Chebyshev filter are from formulas in Guillemin [GU] for the case 52 = 1/5. Both the Butterworth and Cheby- shev filter coefficients are for filters with a cut-off frequency of /15 radians. This cut-off frequency was selected so that the coefficients of P(S) would roughly compare to those for the Linear Delay filter. The solutions presented are by no means exhaustive, but they are representative of the several hundred solu- tions obtained while developing the computer algorithm. In order to test the practicality of the solutions, a frequency response curve for a Butterworth realization (generated with a computer program) is shown in Fig. 4.2.2. This curve is for the case G = G = l. in table 4.2.2 1 4 with all the component values rounded off to three 105 significant figures. As can be seen from the graphs (Fig. 4.2.1 and Fig. 4.2.2), the difference between the theoretical and actual frequency response is that i .02 d b from 0 to 100 radians/sec for both cases. 4.3 Conjectured Necessary Conditions In the second chapter, Calahan's angle condition [CA1] is seen to be a necessary condition for a one— gyrator RCP network. In the third chapter this angle condition, satisfied with the inequality, is also seen to be sufficient for a one-gyrator RCF low-pass filter realization. An angle condition extended to the case of n gyrators is conjectured to be as follows: Let TV = k/P(S) be the voltage-ratio transfer function of an RCF network consisting of low-pass RCF ladders cascaded through n grounded gyrators, then r w. H tan 1 —i < n 1 (4.3.1) . o. 2 i=1 i where 1 r 2 P(S) = II (S + ai) II [(S + a.) + mi] i=1 i=1 1 Arguments: The case n = l is already proved [CA1]. For the case n = 2 with P(S) of degree 4 or 5, Eq. 4.3.1 is is always satisfied since P(S) = det(C-1)All and All strictly Hurwitz. (For proof see Theorem 2.4.2). 106 IGOU MHO3U mz 038 Himmvmomma.v HIommmvmho.m HivmmMNNOH.v anomamammv.m Hlmhmmnmmo.v Hlomhma¢ov.m Himmmmmmmo.m Himmhmmmoo.m Himmdmahmm.m Hlvmmmmmmm.m so Amoa + mmoa + N mm¢ + .N.H.v .mHm cw ma soaumusmflm m o+oooooooo.H Himmmmoomv.a Himmmmommm.H o+oooooooo.H o+oooooooo.a o+oooooooo.H o+oooooooo.H o+oooooooo.a o+oooooooo.a o+oooooooo.a o+oooooooo.a o+oooooooo.a vo filoomsmmms.a Husmmmmsss.a H-mmmsommm.a Huoommsmms.a Hummamsamm.a Hummmmsmmm.a anomommmsm.a Humamssoms.a Hummmqamas.a mo Himmmmmomm.H Hummaomomm.a Himmmmmmma.a Hivvmmmmmm.d HImNonNNH.H Himmmvvmom.a Himammamha.a HlvamH0ha.H HIvNNmHvNH.H No moa + v mv\x u e Hl¢mmmmmom.v Himavmmmmm.v alommmommm.v Hlmvhhmmma.v Himomoomom.v Himommmmma.¢ Himmmmmamm.v Himmmvmomm.v Hlahmmmbmm.v aloamammam.w mo o+hmmmmmmw.m o+mmmmomhm.o o+mooammom.m o+vmmomomm.m o+ovomhmma.m o+mmhmmoom.v o+avmm¢hmm.v o+mmmommom.m o+mmammm¢m.m o+HHhhmmHH.m Ho .mcoflumNHHmmm Hmuaflm mmmm Boa hmama ummcflq mmummo cumsoman.a.m.v magma H+oooooooo.a o+ooooooco.m o+oooooooo.m o+oooooooo.h o+oooooooo.m o+oooooooo.m o+oooooooo.v o+oooooooo.m o+oooooooo.m o+oooooooo.a Ho 107 RHOBsz one HIHomHmwmm.m Hlmamhmgmm.m Himmmvmama.m Himmomomhm.m Alomvbdvmh.m almvhmmmna.h Himmmhmmam.m Himamhammm.m Almmmamawo.m Himmmmhmma.m vo .Huv o+oooooooo.a o+oooooooo.a o+oooooooo.a o+oooooooo.a o+oooooooo.a o+oooooooo.H o+oooooooo.a o+oooooooo.a o+oooooooo.a o+oooooooo.a «o m msmmsmmm.mnmm Hlmoomwmmm.a Huhmomammo.a mlwmammmmm.h mlhmmmmmhm.m mlmmvmmamm.m mlhmmommmm.m anamovnama.a Himaamamha.a Huhmwmmvoo.a HIMHNNONNN.H mo .N.H.¢ .mflm CH ma coflumusmflmcoo Gmmamsa.4mum mnsmmmmsms.s muomaammma.m ~1404mso¢m.m muommmsoas.m musmmmomam.m mnmsssakss.m muaammsmmm.m musmsassom.m mumssmmaom.m musmmmmwom.m mo msmmsmo.mmnam HIthwoomm.N Hloomhmmoq.m Hummmmvmmm.m HImeHmamw.N Hlomnmmmom.m HIwmoaammm.N Himmmhammh.m HivaNHmHm.N almmmommvm.m Himommomoo.m mo .H .ooanom o+mmvmaovm.m o+momwmamm.m o+mmbhmmmm.m o+mmomhmma.m o+mmmnomvm.¢ o+hvwmammm.v o+vmmmmmoa.v o+aomommmn.m o+mHHhomHv.m o+mmmvmmmm.m Ho .mcoflumNflHmmm Hmuaflm mmmm 3oq zuHOSngusm mmumma sunsomll.m.m.q magma mmma 3;} u a H+oooooooo.H o+oooooooo.m o+oooooooo.m o+oooooooo.h o+oooooooo.w o+oooooooo.m o+oooooooo.v o+oooooooo.m o+oooooooo.m o+oooooooo.H Ho 108 mna o+mmmmmna.m o+hHmHNom.H o+ovmmvv~.m o+oav~m¢m.a o+mmmmvmm.a o+mmmnmmm.a o+HmHmonv.H o+oammmvn.a o+mmmom~o.m o+wmommmm.a OH" m w o+ooooooo.H o+ooooooo.d o+ooooooo.H o+ooooooo.H o+ooooooo.H o+ooooooo.H o+ooooooo.H o+ooooooo.H o+ooooooo.H o+ooooooo.a mammonm.mu m .N.H.q .mflm CH ma coaumnsmflmcou xnozumz .mcoflumNHHmmm umuaflm mmmm 304 Amxaumwv >mgmmnmgo mmnmmo nunsomuu.m.m.s magma m mloommomm.v mlmmmmmfim.w mlmavamvv.¢ mloaammmv.m NIMthva.m mlmavaahv.m Nlmommmmm.m mummhamhh.m Nimhmvamm.v mlmwmmmmw.m smmmsm.maumm Nlmvhvmmm.m mlmhhmmma.m Himvmoomo.a Humomovma.a Himahvmvm.a Huhmammom.a lemmqahm.a HINmNHHmm.N lemoammm.m Hlaommman.n smomom.omnam Nnmmmmmmm.m mlmmmvmmo.m mlmmommmm.m mnmnahomm.m mlooamhhm.m mlmmohomm.m NIHvHHmmm.H Nlmhhvomm.m Nlmoommam.m NIHhvhmmN.m .H o ommmaw.omu m o+mommaam.v o+mommmam.v o+mnommmm.m o+hmmmvom.m o+mmhmmmh.m o+mmNHNom.N o+ommovom.a o+mo>mm¢m.a alomwomhm.m anamwhmhm.v Nde H HBHU > EXQM u .H. H+ooooooo.a o+ooooooo.m o+ooooooo.w o+ooooooo.b o+ooooooo.w o+ooooooo.m o+ooooooo.v o+ooooooo.m o+ooooooo.m o+ooooooo.H 109 Hmsuom mscHE Havaumuomca .mocmswmum UmNHHmEHoc momum> :35: CH mmcommmn .kuaflm cuHOSHmuusm omcommmm mocmovmum a.~.v A.IOS\3 .IV .mflm No.l Ho.l Ho.+ No.+ 110 .moqmsvmum pmNHHmEHoc mSmHm> =Q©= 5H mmcommmu .nmuawm >m£m>Qm£U mmcommmm mocmsqmnm A1035) H. Hoopom mscfle HMOHuoHomsa H L b ( r 1 h D { J) “ 1‘) N.N.v .mflm No.l ) ( OOH HOI Ho. No. 111 For the cases where n 2 2 and P(S) is of degree greater than five, it appears impossible to supply an analytic proof. For this reason a random analysis pro— gram was carried out on the computer for the case where P(S) was of sixth degree using the two networks shown in Fig. 4.3.1 and Fig. 4.3.2, respectively. It is not necessary to consider the other possible configurations which yield P(S) of sixth degree, since by Theorems 2.4.2 and 2.5.1 such P(S) must be strictly Hurwitz and have at lease two real zeros thereby satisfying the con- jecture. For networks 1 and 2 the parameters Xi are defined as: X1 = Gl/Cl X6 = G4/C4 1 x2 = Gz/Cl x7 = G4/C5 x3 = GZ/Cz X8 = Gs/Ce > NjZWZr:)l X4 = G3/C3 X9 = 1 1/C2C3 . . X5 = G3/C4 X10 = O‘25‘2/Csce J X1 = G1/C1 X6 = Ge/Cs 1 X2 = Gz/Cz X7 = O‘15‘1/‘31‘32 x3 = 52/c3 x8 = azaz/c3c4 > szw3r:)2 x4 = G4/C4 X9 = O‘25‘3/‘35‘36 . . x5 = G4/C5 J 112 G1 G2 0‘1 G3 G4 0‘2 0—“NAVT—J\Av -—N/v——- so c1 c2 c3 c4 c5 €61 G6 6 0 Network 1 Fig. 4.3.1 Two-gyrator sixth degree RCF low-pass filter. G a G d 2 2 4 3 -[_ ___ C2 C3 ‘ C4 C5 l C6 G6 - , 0 Network 2 Fig. 4.3.2 Three-gyrator sixth degree RCF low-pass filter. 113 From each set of X1 the corresponding set of element values can be calculated in terms of G1 and the para- meters .~.. 0Li 0‘1 A description of the method used in the random analysis program is as follows: Each Xi is selected randomly using a uniform distribution (computer library random number generator) and then the polynomial r 2 2 (S + a1) n [s + Gi) + wi] 2 P(S) = n =1 i=1 1 is obtained from which the summation -1 tan (wi/Gi) (4.3.4) 6' ll IIMH i l is formed by computing the zeros of the polynomial. In using this program, the range for each Xi was adjusted after some preliminary runs, so as to maximize 6 in Eq. 4.3.4. It was observed that the conjecture was not violated in a sample of several hundred runs for each network. Networks 1 and 2 were also analysed by a program which starts from a randomly selected set of Xi and ad— justs the X1 sequentially so that 4, in Eq. 4.3.4, is maximized within a bounded set of Xi‘ Due to the ex- treme time requirements (each iteration required the solution for the zeros of a sixth degree polynomial) only a few runs were made using this program. In no 114 case, however, was the conjecture violated. In addition it was also observed, through several examples, that the proposed angle condition could not have been made stricter. Finally attempts in realizing T = k/P(S) with V network configurations which do not satisfy the conjec- ture have not been successful, whereas when the net- work configuration satisfied the conjecture and Theorem 2.5.1, realizations have been found on the computer in a straightforward manner. (See Appendix A for the com- puter algorithm.) Some examples follow in the next section. 4.4 Computer Realizations for Fifth and Sixth Degree Low-Pass Transfer Functions Using RCF Configura- tions. Although an analytic proof of the existence of network realizations satisfying TV = k/P(S), where P(S) is strictly Hurwitz and has degree greater than four, has not been established; realizations have been ob- tained, using the computer algorithm discussed in Appen- dix A to solve the nonlinear equations (similar to Eq. 4.2.2) derived from P(S) = det(Y"). In the following examples, computer realizations for three practical low- pass voltage-ratio transfer functions, are given. Example 4.4.1. Let TV = k/P(S)S where P(S) = S5 + 1554 + 10583 + 42082 + 9458 + 945 (4.41.) 115 The coefficients for P(S) are for a Linear Delay (Thomson) filter and are taken from Van Valkenburg [VA]. Consider the network in Fig. 4.4.1 which has an admittance matrix G1 "G1 -G1 C S+Gl a1 'a1 CZS+G2 -G2 = -G2 C3S+G2 a2 -d2 C4S+G4 -G4 -G C S+G +G 4 5 4 5 L _J The matrix obrained by deleting the first row and column of Y is —ClS+Gl a1 -61 C25+G2 -G2 —G2 C3S+G2 a2 Y' = -a2 C3S+G4 -G4 L_ -G4 CSS+G4+G§_ The determinant All = det(Y') has a minimum of one real zero (for proof see Theorem 2.5.1). Other possible two— gyrator five-capacitor RCP low-pass filter configurations have minimums of two or more real zeros in All and they cannot be used since P(S) is known to have only one real 116 zero [VA]. The matrix Y' can now be pre and post mul- 1/2 tiplied by c‘ to obtain S + Xl Vx7al/al -¢x al/al S + X - VX X 7 2 2 3 y" = '- V 2X3 S + X3 VX8a2/d2 - VX8a2/d2 S + X4 - VX4X5 - VX4X5 S + X + X .L 5 3 (4.4.2) where All = det(C)det(Y") and X1 = Gl/Cl X5 = G4/C5 x2 = GZ/Cz x6 = GS/CS (4.4.3) X3 = Gz/C3 X7 = O‘151/‘31‘32 X4 = G4/C4 C8 = O‘232/C3C4 The parameters Xi must be positive since the capacitors, conductances, and the parameters Gigi i = 1,2 are all p051tive. Conversely, if Xi > 0 and alal,a202 and G1 are known then the network elements are 117 C1 = Gl/Xl C4 = O‘2‘”‘2/C3x8 C2 = O‘1‘3‘1/‘31X7 G4 = C4x4 (4.4.4) 52 = czx2 c5 = G4/X5 c3 = 52/x3 G5 = C5X6 If a set of positive X. can be found such that 1 det(Y ) = P(S), then the transfer function TV = A4l/All k/P(S) has been realized since All = det[(C)(Y")]. The nonlinear equations in Xi' defined by det(Y") = P(S), can be solved by the computer (see Appendix). Since there is considerable freedom in the choice of Xi (det(Y") = P(S) yields five equations in eight unknowns) additional constraints can be added such as G1 = k G5. Expressing this constraint in terms of Xi’ one has _ I X1X2X5X8 — K X3X4X6X.7 , _ ~ ~ where K — k dzdz/alal. Computor realizations for three different values of K are Hmuflflm sum mmmmnon mmummw shuns H.4.4 .mflm o o 8 l l m m so m m a 0 0 HH Irv UII w L[ O O )—! \/\/\||| .lL/\/\ I\/\/\llo so as No Ha 119 G1 = 1- G1 = 1- G1 = 1. C1 = .19772492 C1 = .16552402 C1 = .16965992 G2 = 2.1143563 G2 = 1.2299518 G2 = 2.8729672 C2 = 1.1606858 C2 = .97598661 C2 = 1.3117841 C3 = 1.0804736 C3 = .36074834 C3 = 1.4740298 G4 = .13470185 G4 = .17139153 G4 = .075790874 C4 = .049224786 C4 = .11972908 C4 = .02652635 G5 = 1. G5 = .1 G5 = .01 C5 = .33105995 C5 = .094977605 C5 = .040669095 6131 = 6252 - l. 6161 = 6252 = l. 6151 = 6262 = 1. All the element values given above are in mhos. or in farads. The realized network is given in Fig. 4.4.1. Example 4.4.2. Let TV = k/P(S) where 6 5 4 3 P(S) = s + 215 + 2105 + 12605 + 47252 + 103958 + 10395. (4.4.5) The coefficients for P(S) are for a sixth degree linear delay filter and are taken from Van Valkenburg [VA]. Consider the network in Fig. 4.3.1. This network satis— fies the necessary conditions established in Theorem 2.5.1 and so is suitable to realize T . The admittance V matrix for this network is Y": 1 1 -G1 ClS+Gl+G2 -G2 -G2 CZS+G2 01 -&l C3S+G3 -G3 -G3 C4S+G3+G4 -G4 -G4 CSS+G4 62 -&2 C6S+G (4.4.6) Forming Y" directly from this matrix, one obtains s+xl+x2 “565765 —. "(X2X3 S+X3 ”X9ai/a1 ”(2531751 S+X4 ‘/§4§6 “/5456 S+X5+x6 '/§6§7 -/x S+X where det(Cl/ZY"C 1/2 120 ) = All (4.4.7) 10“2/“2 8 121 x1 = Gl/Cl x6 = G4/C4 x2 = 52/01 x7 = G4/C5 x3 = 02/c2 x8 = G6/C6 (4°4°8) x4 = G3/C3 x9 = “1&1/C2C3 x5 = G3/C4 X10 = O‘25‘2/‘35Cs The relation Xi > 0, (i = 1,...,10) clearly holds since all the network parameters are positive. Conversely, if a set of positive Xi is found such that det(Y") = P(S), then the network parameters can be solved in terms of G1, 6161 and 0202 giVing the realization of TV. The equations for the network parameters are: C1 = Gl/Xl C4 = G3/X5 G2 = C1X2 G4: C4x6 C2 = G2/X3 C5 = G4/X7 (4.4.9) C3 = O‘10‘1/‘32x9 C6 = O‘20‘2/‘35x10 G3 = C3X4 G6 = C6x8 The nonlinear equations in Xi defined by det(Y") = P(S) (4.4.10) can be solved by a computer program to realize TV = k/P(S). Since the relation det(Y") = P(S) yields six equations in ten unknowns, considerable freedom in the choice of 122 Xi is possible. A useful constraint which can be added 1 = k G6' Expressing this constraint in terms of the Xi one has is G X X X X X = K'X X X X X (4.4.11) 1 3 4 6 10 2 5 7 8 9 where K' = ka & /aldl. Computor solutions for three ratios of Gl/G6 are: G1 = 1. G1 = 1. G1 = 1. C1 = .30561664 C1 = .39875898 C1 = .26994128 G2 = .54369763 G2 = .58810805 G2 = .18377659 C2 = .099232055 C2 = .21466229 C2 = .031468068 G3 = 1.0379997 G3 = .47470337 G3 = 3.6152459 C3 = .37053873 C3 = .16927982 C3 = 1.7360973 G4 = .28599644 G4 = 1.0119594 G4 = 5.8383642 C4 — .4818776 C4 = .35106470 C4 = 2.5482323 C5 = .11172470 C5 = 3.5776846 C5 = 18.287234 G6 = 1. G6 = .1 G6 = .01 C6 = .42353046 C6 = .014376383 C6 = .021445128 6161 = 6262 = 1. 6161 = azdz = 1. aldl = azdz = l. The above element values are in mhos or farads, and the network configuration is given in Fig. 4.3.1. Example 4.4.3. Consider Example 4.2.2 when the poly- nomial P(S) = PiSl has the coefficients 1 “MON 0 The coefficients for P(S) with eight digit accuracy Butterworth filter [VA]. a cut-off at w 123 15625. 12074.073 4665.0635 1142.7025 (4.4.12) 186.60254 19.318516 1. were calculated on a computer for a sixth degree low-pass This was frequency scaled for 5, so the coefficients would roughly match those in Example 4.4.2. Again the network con- figuration in Fig. 4.3.1 is used since P(S) has no real zeros. Computer solutions for three ratios of Gl/G6 are: G1 = 1.0 G1 = C1 = .24379647 C1 = G2 = .11840130 G2 = C2 = .020970984 C2 = G3 = 2.9354488 G3 = C3 = 1.6751027 C3 = G4 = .070774711 G4 = C4 = .63416227 C4 = C5 = .061371579 C5 = G6 = 1.0 G6 = C6 = .69490787 C6 = Q H 92 1 1.0 G1 = 1.0 .33142963 C1 = 1.0010717 .30864407 G2 = 2.6430899 .043334407 C2 = .50873174 .81051727 G3 = .18560747 .72286262 C3 = .064120396 .64364059 G4 = 1.3184473 .32733992 C4 = .30267577 .71763355 C5 = 9.8780364 .1 G6 = .01 .055954909 C6 = .0040216537 = azdz = 1. aldl = dzdz = 1. 124 The above parameters are in mhos. and farads, and the network Example Note tha which is figurati TV = k/P contains mum numb TV = k/P L Forming configuration is given in Fig. 4.3.1. 4.4.4. Consider Example 4.4.2 when 12(5) = [(s + 1)2 + 2213 (4.4.13) t Conjecture 4.3.1 yields -1 2 n 3 tan 1 < n 7 (4.4.14) satisfied only if n s 3. Therefore the con- on shown in Fig. 4.3.1 cannot be used to realize (8). Consider the network in Fig. 4.3.2 which 3 gyrators. By theorem 2.5.1, taking the maxi- er of real poles of TV as zero, one can realize (S). The admittance matrix is ...G1 1 ClS+Gl dl - dl C28+G2 -G2 -G2 C3S+G2 dz -a2 C4S+G4 -G4 -G4 C S+G4 3 -d3 C6S+G6 Y" directly from Y, one can obtain . 7,- was. ~ J, 125 S+Xl VX7al/dl -¢Xzal7al S+X2 -/X2X3 -VX2X3 S+X3 VX8a2/az Y = - X862 a2 S+X4 -/X4X5 -./X4X5 S+X5 VngB/d3 L. VX9d37d3 S+X6 where det(Y") = det(C-lMll and x1 = Gl/Cl X6 = G6/C6 x2 = 52/c3 x7 = alal/clc2 _ _ ~ (4.4.15) X3 ‘ Gz/C3 X8 ‘ 0‘2"‘2/‘33‘34 x4 = G4/C4 X9 = O‘3"‘3/C5C6 The Xi i = 1,...,9 must be positive; conversely if they are positive Eq. 4.4.15 can be used to solve for the network components in terms of G1, aldl, a262, and 6353. The solution of the six nonlinear equations in Xi derived from P(S) = det(Y") with positive Xi realizes T = k/P(S). The computer V solution is 126 G1 = 1. G4 = .33940651 C1 = 1.3115185 C4 = .54931147 G2 = .43199501 C5 = 3.0489273 C2 = .41741739 G6 = .098963398 C3 = .25824894 C6 = .054960727 0.161 = azaz = 03543 = 1' where all the elements are in mhos. or in farads. The realized network is given in Fig. 4.3.2. 4.5 Band-pass and High-pass Filter Realizations Although the purpose of this thesis is to develop low-pass RCP realizations, it is possible, in some cases, to extend the low-pass realization into other filters. This is established by the following two theorems. Theorem 4.5.1. Let TV = k Sn/P(S), where P(S) is of degree n. Then there exists a high-pass RC filter realization of Tv if there exists an RCr low-pass filter realization for T6 = TV(5) satisfying Theorem 2.4.3 which uses exactly two gyrators. Proof: Since there is a network realizing T6 = TV(%)’ there is a set of positive Xi such that det(Y") = Phi-)8n where det(Y") = det(C-1)All. From this set of Xi one can calculate a set of components so that 127 cl = 61 = a2 = dz = l. The complex variable S can now be replaced by g to obtain realization of a RLF network corresponding to T as shown in Fig. 4.5.1. Making V’ extensive use of the equivalent forms in Fig. 4.5.2, the network is transformed into the RCP network shown in Fig. 4.5.3. Finally recognizing that the current- ratio transfer function, without the terminating gyrators, is the same as the voltage-ratio transfer functions and also using Thevenin and Norton equivalents, the desired RCr network can be obtained as shown in Fig. 4.5.4. Q.E.D. Corollary 4.5.1. Let P(S) be a strictly Hurwitz poly- nomial of degree four, then TV = k S4/P(S) can always be realized. Proof: Since P(S) is strictly Hurwitz, so is S4P(%). The transfer function T1 = k/P'(S), where P'(S) is strictly Hurwitz and of degree four, is always realiz- able from Theorem 4.1.1. Q.E.D. Note: Theorem 4.5.1 can be extended to the networks containing any even number of gyrators, however the number two is probably the practical limit. Note: A high-pass filter as shown in Fig. 4.5.4 satis- fies the conditions for Theorem 2.4.2. Therefore such a filter remains stable for R, C, or F parameter varia- tion. 128 Theorem 4.5.2. Let T = k N(S)/P(S) where N(S) = V (S2 + a2) and T6 = k/P(S) has a low pass network realiz- ation of the form shown in Fig. 4.5.5. Then, TV can be realized with an RCF network. Proof: Consider the chain matrix representation of the network shown in Fig. 4.5.5. .QA B7 _A B11 FA B.1 From which one can obtain A = A1A2 + B1C2 Since A = l/TV, A' can be formed as AlAZ + B1C2 A' = 1/T6 = kN(s) Note that Al/Bl and l/Bl define Y22 and -Y21 respectively for network 1. Similarly, AZ/CZ and l/C2 define 211 I _ I _ and 221 for network 2. Let Al — Al/K N(S) and B1 - . (l) . I u = Bl/K N(S), then A 1/B1 Al/Bl defines Y22 and l/Bi = k N(S)/Bl defines -Y(l) for the desired realiza- 21 tion. Since the degree of N(S) equals the degree of Al’ the synthesis of the network from -Y§i) and Yéé) is possible using standard synthesis techniques. The final realization is shown in Fig. 4.5.6. Q.E.D. Mir... 1 .1 W... ...)? i)” ? (1 )1 G [T ”(5) 1W 4 c 5 7 R=G C=L (a) (b) (C) Fig. 4.5.2 Equivalent forms. .Hmpaflu mmmmugmfln sum s.m.s .mflm 130 .xuogumc pom m.m.4 .mflm 011.354....41 f. K x) 131 RCF ——( ———))—~ L. _ _ ___.____J (l) (2) Fig. 4.5.5 Initial network for Theorem 4.5.2. C.) r— 11* l —l | fil l I I °--—I . ' -— 1 I c j c;LI RCI‘ l L . I I G Lfi I L_ _____________ .1 In general, either CL or GL (2) Fig. 4.5.6 Final network for Theorem 4.5.2. or both are zero. 132 Note: The network in Fig. 4.5.6 satisfies Theorem 2.4.2 and therefore remains stable for R, C, or F para- meter variation. Note: All the results of this chapter remain true if practical gyrators which have admittance matrices of the form G11 G12 Y = F ’G21 G22 L _ where G11, G12, G21, and G22 are pOSitive parameters, are used rather than ideal active gyrators. Indeed the form of Y? suggests that a practical gyrator is equivalent to an ideal active gyrator loaded by conductances at its ports. Consider TV = k/P(S) where P(S) is strictly Hurwitz. Since one can find a positive constant, y, such that P'(s) = P(S - Y) is strictly Hurwitz, the voltage-ratio transfer function T1 = k/P'(S) can be realized as a low-pass RCF filter by methods established in this chapter. In this realization re- placing S by S + y, the RC? filter is now modified and it becomes the realization of 133 TV = k/P(s) In this modified network every capacitor Ci is connected in parallel with a conductance of value yci. Since each gyrator branch is always in parallel with a capacitor, each gyrator branch can now be considered as being in parallel with a conductance. This permits one to re- place ideal active gyrators by practical ones. For the high-pass case the process is similar except each gyrator branch is in series with a resist- ance. This results in a gyrator impedance matrix of the form which represents a practical gyrator. 4.6. Conclusions In this chapter it is established that every fourth degree low-pass voltage-ratio transfer function whose denominator is a strictly Hurwitz polynomial, can be realized using a two-gyrator low-pass RCP filter. This also holds for the high-pass filter case by a simple extension established in Theorem 4.5.1. Computer 134 RCF realizations are given for some practical fourth, fifth, and sixth degree low-pass voltage-ratio transfer functions which utilize two gyrators. A proposed nec- essary condition is given and supported with feasability arguments. A computer realization using three gyrators is given to illustrate extensions of the method. Fin- ally, it is demonstrated that, for certain cases, the method can be extended to high-pass and band-pass fil- ter realizations. CHAPTER V CONCLUSIONS The purpose for this thesis is to provide a basis for RC? filter synthesis and to Specifically develop realization procedures for low-pass voltage- ratio transfer functions. A general basis for RCF filter synthesis is established in Chapters II and III, whereas synthesis procedures for low-pass voltage-ratio transfer functions are established in Chapter IV. The main contribution of this thesis can be listed as follows: 1) Theorem 2.4.2 is established. As a consequence of this theorem, it is shown that a large class of active RCF filter realizations are stable and remain stable irrespective of the variation in the RC components of the gyrator parameters. All the RCr filter realizations considered in this thesis belong to this class defined in section 2.4. ii) It is proved that the Calahan and Horowitz poly- nomial decomposition methods can be derived one from the other. 135 iii) iv) v) 136 The Calahan and the Horowitz polynomial decom- position methods are extended to polynomials which contain distinct negative real zeros. Low—pass RCP filter realizations are shown to exist for voltage-ratio transfer functions Tv = k/P(s), where P(s) is strictly Hurwitz and of degree four. Through the use of a network trans- formation, it is shown that high-pass RCF filter realizations exist for the voltage-ratio trans- fer function TV = k 54/P(s), where P(s) is a strictly Hurwitz polynomial of the fourth degree. Computer realization procedures are established to realize TV = k/P(s), where P(s) is strictly Hurwitz. Suggestions for Future Work: 1) ii) iii) To establish the necessary and sufficient con- ditions for network realizations when the degree of the polynomial P(s) appearing in TV = k/P(s) is greater than four. To establish parameter sensitivity comparisons between the general RCr filter realizations and The RCNIC filter realizations. To develop analytic techniques for the realiza- tion of TV = k/P(s) when P(s) is of degree greater than four. APPENDIX A Consider the solution of a set of nonlinear equations of the general form p1 = Pl(xl,x2,...,x ) nv P2 = P2(X1IXZI°--Ixnv) (A.0.1) Phc = Pnc(xl’x2"'°’xnv) where the subscripts nv and no are the number of vari— ables and the number of equations respectively. Several computer programs have been written and applied to the problems of the above form considered in this thesis. One of these, a direct search scheme based on Hooke and Jeeves [WI] pattern search, proved to be successful when the number of variables nv is less than seven. This program is now available from the Michigan State Univer- sity program library [MCl]. To solve the nonlinear equations in Eq. A.0.l when the number of variables is greater than seven, an algorithm was adapted based on the Taylor least squares reduction scheme [CT]. This algorithm is discussed in the following section. 137 138 A.1 Computer Algorithm Consider the error vector F where Fl = pl - Pl(x1,X2'ooo'Xnv) F2 = p2 - P2(X1,X2,...,Xnv) Fnc = pnc - Pnc(x1’x2""'xnv) Let x be the column vector with the variables xl,x2,... Xnv' and let 2 be a solution for Eq. A.0.l. A Taylor series expansion of F about 2 is BE A 0 = F(X) + 3}? (X - x) + (A.1.2) It is interesting to note at this point that, for the network problems considered in this thesis, the func- tions Pi(xl’x2""’xnv) are always of the form Pi(xl’x2"°"xnv) = gi(xl,...,xj_l,xj+l,...,xnv)xj +hi(xl"°"Xj-l'xj+l"°"Xnv) (A.1.3) This property, in general, also holds for the nonlinear equations generated from passive netowrk functions. Equation A.1.3 allows the matrix BF/Bx to be evaluated very accurately and simply by a computer program. In Eq. A.0.l, if nv equals nc, the matrix BF/ax is the Jacobian of F and neglecting higher order terms I 139 of the Taylor expansion, 2 can be calculated iteratively using the relation x(i+l)- x(i) = — [aF/ax1‘lF (A.1.4) where x(l) is the ith iteration. Equation A.1.4 represents the familiar Newton-Raphson method. If nv is less than no no solutions exist in general. However, one can use . . T T x(l+l)- x(1) = - [ %§—-%§] l %§}'F (A.1.5) which gives a least squares estimate of 2 [CT]. Finally, consider the case where nv is greater than nc. Solutions exist for this case, however, the matrix aF/ax is not square and the matrix [8F/3x]T[3F/3x] is singular. Therefore neither Eq. A.1.4 or Eq. A.1.5 can be used for this case. However, F can be augmented with a set of trivial functions such that they become zero as F becomes zero, and the relation in A.1.5 can be used for this case. Indeed let F F. = /_X(x~x(i)) then Eq. A.1.5 takes on the form 140 . . T Tl x(i+1) _ x(1) = _ [ %§__%§_+ Au] 1[%§_: WVu] F 0 or . . T T x(1+l)- x”) = - [ ga— 13% + AuJ'l gi—F (A.1.6) It is interesting to note that E . A.1.6 reduces to a Newton-Raphson reduction scheme if nv equals no and 1 equals zero. On the other hand if A is large this method has the characteristics of a steepest descent method [CT]. In general the reduction scheme represented in Eq. A.1.6 simply minimizes the step size as well as the error vector F. Let E be the error function defined by nc E = .2 IF” (A.1.7) i=1 Since the Newton Raphson scheme converges rapidly in the vicinity of a solution and the steepest descent method converges more rapidly far from a solution, a value of 1 based on E is practical. For the examples considered in the thesis the criterion 1 = minimum {E, /E} (A.1.8) has worked very well. 141 In order to equalize the accuracy among the co- efficients Pi' Eq. A.0.2 is modified so that F. = wi[pi - Pi(xl,x2,...,xnv)] (A.1.9) where 2 II (maximum {Pj})/Pi. The introduction of these weighing factors improves the performance of the program as well. Two possible heur- istic explanations for this are: the addition of the w im ro es the ond tionin of the matric [EFT 33 + Au] 19" Cl 9 7—fo ' or the addition of the wi promotes uniform reduction of the terms in the error vector F. The problem of oscillation about a solution is prevented by introducing a damping factor a into the reduction scheme. This is represented by the following equation. . . T T x(l+l)- x”) = -a[ ’35—‘33 + u] 3%}? (A.1.10) Initially a is set equal to 1. If the condition E(x(i+l)) Z E(x(i)) (1+1) recalculated. holds, a is reduced by one half and x This is repeated up to seven times before the attempt is given up. The algorithm described here converges rapidly 142 from a good initial point x, however it does not neces- sarily converge from an arbitrary initial point. For this reason a Monte Carlo random search program [WI] was written to provide a good initial point. Experience gained using this program indicates that the Monte Carlo initial point search becomes increasingly important as the number of variables increases. A flow diagram of the entire program is given in Fig. A.1.l. A.2 Usage Program documentation which describes how to use this program and the associated program source deck is available from the Michigan State University program library [MC2]. Unfortunately, a certain amount of practice is required to successfully use this program for solving a set of nonlinear equations. This is true since the boundaries for the Monte Carlo search must be set by the program user. If the boundaries are set too large the region searched becomes too large and con- versely, if they are too small, the region may not con- tain any solutions. If no previous knowledge is avail- able about a particular problem, it becomes necessary to try several exploratory runs on the computer before 143 satisfactory search boundaries are established. Fortun- ately even very "rough" results obtained by hand cal- culations or knowledge about the roots, etc., provide sufficient information for the establishment of search boundaries. 144 Read in boundaries v } Select Reset and tes 1n15lal boundaries pOint ye Error reduction no number of solutions satisfied ? error in neighborhood of solution ? no Are attempts satisfied yes Fig. A.1.l Computer algorithm flow chart. no [CA1] [CA2] [CA3] [CT] [FR] [GU] [HA] [HO] [HT] [KTK] BIBLIOGRAPHY Calahan, D. A.: "Restrictions on the Natural Frequencies of an RC-RL Network," Journal of the Franklin Institute, Vol. 272. pp. 112-133; August 1961. Calahan, D. A.: "Sensitivity Minimization in Active RC Synthesis," IRE Trans. on Circuit Theory, Vol. CT-9, pp. 38-42; March 1962. Calahan, D. A.: Modern Network Synthesis, Vol. II, Hayden, New York, 1964. Calahan, D. A. and Temes, G. C.: "Computer Aided Network Optimization," Proc. of the IEEE, Vol. 55, p. 1839; November 1967. Frame, J. 8.: "Matrix Functions and Applica- tions," IEEE Spectrum, Vol. 1, No. 7, pp. 103- 109, July 1964. Guillemin, E. A.: Synthesis of Passive Networks, John Wiley and Sons, Inc., New York, 1957. Hakim, S. 8.: "RC Active Filters Using an Amplifier as the Active Element," Proc. IEE (London), Vol. 112, pp. 901-912; May, 1955. Horowitz, I. M.: "Optimization of Negative- Impedance Methods of Active RC Synthesis," IRE Trans. on Circuit Theory, Vol. CT-6, pp. 296-303; September, 1959. Holt, A. G. and Taylor, J.: "Method of Replac- ing Ungrounded Inductors by Grounded Gyrators," Electronic Letters, Vol. 1, No. 4, pp. 105; June 1965. Koenig, H. E., Tokad, Y. and Kesavan, H. K.: Analysis of Discrete Physical Systems, McGraw- Hill, New York, 1967. 145 [LI] [MCl] [MC2] [MI] [SK] [TH] [TO] [VA] [WI] [YO] 146 Linvill, J. G. "RC Active Filters," Proc. of the IRE, Vol. 42, pp. 555-564; March 1954. McNally, R. L.: "Optimization with Hooke and Jeeves Pattern Search," Michigan State Univer- sity program library, No. 00000384. McNally, R. L.: "Network Synthesis with Optimiz- ation," Michigan State University program lib- rary, No. 00000406. Mitra, S. K.: Analysis and Synthesis of Linear Active Networks, Wiley, New York, 1969. Sallen, R. P. and Key, E. L.: "Practical Method of Designing RC Active Filters," IRE Trans. on Circuit Theory, Vol. CT—2, pp. 74—85; March 1955. Thomas, R. E.: "Polynomial Decomposition in Active Network Synthesis," IRE Trans. on Circuit Theory, Vol. CT-8, pp. 270-274; September 1961. Tokad, Y.: "Foundations of Passive Electrical Network Synthesis," part 1, (Bound Notes), Michigan State University, East Lansing, Michi- gan, 1966. Van Valkenburg, M. E.: Introduction to Modern Network Synthesis, John Wiley and Sons, Inc., New York, 1960. Wilde, D. J.: Optimum Seeking Methods, Prentice- Hall, Englewood Cliffs, New Jersey, 1964. Yonagisawa, T.: "RC Active Networks Using Cur- rent Inversion Type NIC," IRE Trans. on Circuit Theory, Vol. CT-4, pp. 140-144; September 1957. —-._q- ”(WIN [1|]I]]])][]]|][I|]][[I[ES 3 1193 0314i 5920