A NEW ME't'l-KBD 58h“; CASCAGE SY‘ffl‘WES'IS OF 1-PORT PASSIVE NETWAZMQKS WSTH FiECiPRfifiAL MD NONRECEPRDCAL LOSSLESS 2—?0RT3 UESfifiiBED BY SCATTEREJ‘IG FMEAMETERS Thesia for the Degree of Ph. D. MEGIflGAE‘é STATE UNIVERSHY CHEH~YU MO 196'! mums This is to certify that the thesis entitled A New Method for Cascade Synthesis of 1-Port Passive Networks with Reciprocal and Nonreciprocal Lossless 2-ports Described by Scattering Parameters presented by Chih-yu Kao has been accepted towards fulfillment of the requirements for Ph. D. degree in Electrical Engineering / Major profegdf Date Mav 16, 1967 0-169 LIBRARY Michiga; 3“” University 3”" www...” EH“: n. W ”J, . ABSTRACT A NEW METHOD FOR CASCADE SYNTHESIS OF l-PORT PASSIVE NETWORKS WITH REC IPROCAI. AND NONRECIPROCAL LOSSLESS 2-PORTS DESCRIBED BY SCATTERING PARAMETERS by Chih- yu Kao The synthesis procedure presented in this thesis can be considered as an extension of the Darlington and the Talbot syn- thesis procedures. However, the synthesis procedure makes use of the scattering parameters and nonreciprocal elements are admitted in the realization. An existence theorem is stated and proved in the thesis, which serves as the basis for the synthesis procedure. This theorem states that given a reflection coefficient the corresponding l-port network (R LCTI") can be realized in terms of a lossless 2-port network (LCTF), N , called the ele- 1 mentary section, terminated on another l-port network (R LCTF), N2. The proof of the theorem is such that it establishes the existence of this configuration and also describes a new synthesis procedure. The principal features of this synthesis method are: (1) One simply deals with real polynomials rather than real rational functions. (2) (3) (4) - Z - Chih-yu Kao In each cycle of the procedure both the networks N1 and N2 are characterized simultaneously and it is not necessary to realize Nl for the application of the synthesis procedure to the next cycle. All the computations require only the division of real poly- nomials which can be accomplished by means of the modified Routh's array described in the thesis. Due to the existence of such a simple algorithm it is feasible to carry out the actual computation by digital computers. All the elementary lossless 2-port networks are fully charac- terized and given in a table. By referring to this table, the parameters obtained for N in each synthesis cycle yield an 1 immediate realization of N1. A NEW METHOD FOR CASCADE SYNTHESIS OF l-PORT PASSIVE NETWORKS WITH RECIPROCAL AND NONRECIPROCAL LOSSLESS Z-PORTS DESCRIBED BY SCATTERING PARAMETERS By Chih-yu Kao A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSO PH Y Department of Electrical Engineering . 1967 ACKNOWLEDGEMENT The author is indebted to his major professor, Dr. Yilmaz Tokad, for his guidance and constant encouragement in the prepara- tion of this thesis and for the inspiration he has given during this stage of the author's education. The author wishes to thank Drs. Joseph A. Strelzoff, Richard C. Dubes and Edward A. Nordhaus for their encouragements and the Department of Electrical Engineering and the Division of Engi- neering Research for the assistantships rendered to him during his graduate studies. To his parents, Mr. and Mrs. T. P. Kao, the author sin- cerely thanks for their constant encouragements and support. Finally, to his wife, Yen-yen, for her patience and under- standing durlng this stage of education, the author expresses cordial thankfulness. ii CHAPTER I CHAPTER II '2.2 2.3 2. 4 2. 5 CHAPTER III 3. 1 3.2 3.4 3.5 CHAPTER IV 4.1 4. 2 4. 3 CHAPTER V BIB LIOGRAPHY . TABLE OF CONTENTS INTRODUCTION ................. SCATTERING MATRICES OF PASSIVE 2- FOR T NE TWOR KS ............. Introduction . . ................. Scattering Matrix of a Passive Lossless n-port Network. . . . . ........... Scattering Matrices of Passive Lossless 2-port Networks . . . . . . ......... Elementary Lossless 2-port Networks ..... Appendix to Chapter II ............. REALIZATION OF TRANSMISSION ZEROS Transmission zeros ............... Two Useful Theorems and the Division Algorithm . . . . . . ............ Existence of Scattering Parameters Corres- ponding to a Selected Simple Set of Transmission Zeros. . . . . . ....... Construction of the Polynomials P1, 01' 2 Further Discussion on "Construction of the Polynomials P1, 01’ P2 and QB ....... SYNTHESIS PROCEDURE AND EXAMPLES . . Synthesis Procedure . . . ........... Example I .................... Example II ................... CONCLUSION AND FURTHER PROBLEMS P and 02 .................. O O O 000000 Q 0 O O O O .......... Page 10 13 18 .20 20 23 40 49 56 .61 61 63 68 .82 LIST OF FIGURES Page Fig. 1 1 ................. ‘1 Fig 2.2 l ................ 6 Fig 2.2 2 ................ 6 Fig. 3.1.1 ................ 23 Fig. 3.3.1 ................ 41 Fig. 3.3.2 ................ 45 Fig. 3.5.1 ................ 56 Fig. 3.5.2 ................ 59 Fig. 4.2.1 ................ 67 Fig. 4.2.2 ................ 68 Fig. 4.3.1 ................ 74 Fig. 4.3.2 ................ 77 Fig 4.3 3 ................ 80 Fig 4.3 4 ................ 81 LIST OF TABLES Table 1 .................. 14 iv CHAPTER I INTR ODUCTION The problem of cascade synthesis of passive electrical networks has been studied by several authors [DA 1, TA 1, HA 1, YO 1, RU 1, BE 4]. However, the formulation of the problem as well as the techniques developed by these authors differ from each other. In Darlington's synthesis method [DA 1], a driving- point immittance function is assumed to be given and this function is realized as a reciprocal lossless 2-port network terminated in a resistance. The reciprocal lossless 2-port network is then realized by cascade connected combinations of four different kind of sections, :5 o——-—————o 0 <5 0 T 4) called type A, B, C and D, as shown in Fig. 1.1. o—ETJ—o Type A Type B Type C Type D Fig. 1.1 The branches in the first two types consist of either a sin- gle inductance or capacitance, or a series-tuned circuit or a parallel-tuned circuit. Type C is called the Brune section which realizes either pair of purely imaginary or purely real transmission zeros which are symmetrically located with respect to the origin. Finally, type D is called the Darlington section which realizes com- plex transmission zeros. If the numerator of the even part of the immittance function is not a perfect square, the numerator and the denOminator of the immittance function are multiplied by a suitable Hurwitz polynomial, the so-called surplus factor. Such a factor, however, is always avoided if gyrators are admitted in the realization [HA 1]. Talbot [TA 1], on the other hand, used a chain matrix approach, as Piloty did, and presented a method for synthesizing reactance 2-port networks by factorizing the chain matrix into the product of two such matrices of desired degrees. Youla [ YO 1] generalized Richard's theorem and defined a set of indexes and a polynomial chain matrix which are related to the real and imagi— nary parts of the given immittance functions. The element values of the various sections are obtained in closed form in terms of three or six indexes depending upon the complexity of the section. In this method, the gyrator is also included to take the nonrecipro- cal realization into consideration. The scattering parameter formulation of passive n-port electrical networks and the properties of the scattering matrix have been treated in the literature by several authors [BE 1, BE 3, OO 2 and others]. The scattering parameters describe the performance of a network under any specified terminating conditions. The power transferred from a generator with a finite internal im- pedance to a resistive load is frequently best handled by scattering matrix [CA 2]. However, there are networks which are called degenerate or double degenerate which do not possess either an impedance or an admittance matrix or both [BE 1]. Rubin and Carlin [RU l, RU 2] presented a cascade synthe- sis procedure for lossless reciprocal and nonreciprocal Z-port networks. ”Nonreciprocal" transmission zeros are realized by four canonic nonreciprocal 2-port networks which are analogous to Darlington's A, B, C and D networks. Belevitch [BE 4] utilized the fact that the product of two passive scattering matrices is a passive scattering matrix. The corresponding n-port network is realized by interconnecting the component n- port networks by gyrators. The synthesis method presented in this thesis can be con- sidered as an extention of the Darlington and the Talbot synthesis procedures and utilizes the scattering parameters. In this pro- cedure nonreciprocal sections are also allowed. Each elementary section is analyzed on the basis of the necessary and sufficient conditions for a 2 x 2 matrix to be the scattering matrix of a lossless 2-port network and formulas are obtained for the determination of its element values. Since gyrators are allowed, the Darlington type D section can be considered as cascade connected two Brune sections each of which is series- series connected with a gyrator. This enables us to avoid surplus factors which may be needed in Dariington's procedure. Note that the degrees of the numerator and the denominator polynomials of the entries of the scattering matrices corresponding to these elemen- tary sections do not exceed 2. This gives great simplification in actual computations. The synthesis procedure is based on an existence theorem and the division algorithm described in the thesis. The existence theorem simply states that given a reflection coefficient, the para- meters of a 2 x 2 scattering matrix corresponding to a lossless elementary section and the reflection coefficient for the remaining l-port network do in fact exist. The division algorithm yields a simple computation for the determination of the said parameters. Once these parameters (polynomials) are established, the corresponding elementary section as well as the characterization of the remaining 1-port network are obtained simultaneously. CHAPTER II SCATTERING MATRICES OF PASSIVE 2-PORT NETWORKS 2. 1. Introduction The scattering parameter formulation of passive n-port electrical networks and the properties of the scattering matrix have been treated in the literature by several authors [BE 1, BE 3, OO 2 and others]. Procedures for realizing the scattering matrices with reciprocal and nonreciprocal passive n—port networks are also available. However, the realization procedures for reciprocal and nonreciprocal passive n-port networks were considered separately. Recently, all of these procedures have been integrated in a book by Newcomb [NE 1]. In this chapter, the well known results on the necessary and sufficient conditions for a given matrix of a passive lossless n-port network containing positive inductors (L), positive capacitors (C), ideal transformers (T) and gyrators (F) are summarized. Since our primary interest in this thesis is the cascade realization of 2-port LCTI" networks, the relations among the entries of the corresponding 2 x 2 scattering matrix are emphasized and these basic relations are expressed in terms of real polynomials in a complex variable X = O + ju). -6- At the end of this chapter, a table of elementary 2-port sections together with their scattering matrices is given. Some of these sections are to be used as the basic sections in the cascade synthesis procedure presented in this thesis. 2. 2 Scattering Matrix of a Passive Lossless N-Port Network V Consider the n-port network in Fig. 2.2.1. Let v1, v2, . . , n and i1, i2, . . , in be the port voltages and port currents, respectively. The scattering matrix S of the n- port network is defined by S(V+I) = v-1 (2.2.1) where .1 H v: V2 and I: 12 (2.2.2) v 1 n n :— ____________ ‘I i _._ .19 I + 1 . { O4—N\,——O———I : ' I V1 , I I O————‘I I Cr‘r Crv : I . I . N 77—» : N : i ' I +o '—'- o: M 0 ° , I n . : I n I i C | t“*———2 0: 3N I _______________ J Fig. 2.2.1 Fig. 2.2.2 The network Nau in Fig. 2.2.2, obtained from the n-port network N by augmenting each of its ports by a unit resistance, has a termi- nal admittance matrix 7'), which is related to S by - 7 .. S = Un-Zn (2.2.3) where Un is a unit matrix of order n. This relation can be derived easily by a direct inspection of Fig. 2.2.2. If the original n- port has the terminal impedance matrix Z, then the following relations are immediate, s = (z - Un)(Z + Un)‘1 (2.2.4) -1 -1 z = 77 -Un = (Un+S)(Un-S) (2.2.5) Similar relations hold for the terminal admittance matrix for the original n- port network. The power input to the n- port network is given by T >:< (v +1T"‘)(Un .. sT Re(vT*I) = *S)(v +1) (2.2.6) l 4 where Re denotes “the real part of, ” the superscript asterisk and T denote the complex conjugation and the transposition, respectively. Also, as will be used later, the subscript asterisk is defined as 5*(k) = S(-)\). The terminal admittance matrix 17 , of the augmented network is a positive real matrix, whose entries are necessarily finite on the imaginary axis, because of the unit terminal resistances. There- fore, by Eq. (2. 2. 3), the entries of S are analytic in the right half plane, including jw-axis, i. e. , the denominators of these entries are strictly Hurwitz polynomials. Such matrices are called Hur- witzian [BE 1 ]. The power input to the passive network can not be negative, hence, as implied by Eq. (2. 2. 6) the Hermitian matrix _ 8 .. T’i‘ . . . . . * . Un - S S is non-negative definite. Since S (X) = SAX) for X = 30), T . . . . . then Un - 5*5 is also Hermitian for k 2 y». In general, a matrix having this property is called para-Hermitian and is said to be non-negative definite if the associated Hermitian form is non-negative . . . T . . . definite on k = 300. Hence Un - S*S lS para-Hermitian and non- negative definite. For a passive lossless network, since the power input is zero, Eq. (2.2.6) implies o s I >3 i.e. , S is a unitary matrix for k = jw. However, k = ->\ for . . T . . . . X = fig and the relation Sks = Un holds on the entire imaginary axis. T . Hence S*S 2 Un holds everywhere. Therefore, S is also called para-unitary. The realizability conditions for the scattering matrix S by means of a passive lossless n- port network containing positive inductors, capacitors, ideal transformers and gyrators can now be stated as in the following theorem. Theorem 2.2.1: [BE 3, OO 1, NE 1] The necessary and sufficient conditions for a matrix S to be the scattering matrix of an n-port network containing positive inductors, capacitors, ideal transformers and gyrators are: 1. S is Hurwitzian. 2. S is para-unitary Only a sketch of the proof of this theorem will be given here. The well known theorems listed in Section 2. 5 are needed for the -9- proof. The necessity of the conditions has been shown in the pre- ceding discussion, therefore, only the sufficiency of the conditions will be demonstrated. . . . . . . . T* Since S is Hurw1tzian and unitary for k = 30), i. e. , Un = S (jw)S(j(i)), then Un - ST*()\)S(X) _>_ 0 for O > 0, hence S is bounded-real. By Theorem 2. 5.1 it can be seen easily that (S + Un) and (Un - S) are both positive-real. If (Un - S).1 exists, then Z 2 (Un + S)(Un - S).1 is positive- real (Theorem 2. 5. 2) and is realizable as the terminal impedance matrix of an n- port network. Since S is para-unitary, we have T \‘o to -l T T" Z=(U +S)(U -S)=-(U +s‘,,)(U -s,,)=-z n n n r n >- which implies that Z is actually the impedance matrix of a lossless n-port network. If Un - S is singular and has a normal rank r, r > 0, then there exists a real constant orthogonal matrix N such that T NSN= S'f‘U n -r with 8' being bounded-real and (Ur - S') 1 existing. For 8', there exists 2' which is positive-real and has a network representation. By similar reasoning as before, we have which implies that the realized network is lossless. The transfor- mation matrix N corresponds to the turns-ratio matrix of an ideal transformer network of Zn- ports. As a result, S can be realized by a Zn-port ideal transformer with n-r of its ports being open circuited and the remaining r ports -10. being terminated with an r-port network with a terminal impedance matrix 2'. This proves that the conditions 1 and 2 are sufficient. 2. 3 Scattering Matrices of Passive Lossless Z-port Networks It was‘ shown in the foregoing section that the necessary and sufficient conditions for an nxn matrix S to be the scattering matrix of a passive lossless n-port network are:(1) S be Hurwitzian, and (2) S be para-unitary. Condition (1) merely states that the denomi- nator of each entry of S is strictly Hurwitzian and the degree of the numerator can not exceed that of the denominator. From condition (2), for the 2-port case, further relations on the entries of S follow. If s = 11 12 (2.3.1) S21 S22 where Si.'s are real rational functions of Mi, J = l, 2), then S S S «I S :'< 55:: 11 12 11' 21 = U2 (2.3.2) S21 822 S12 ‘ S22 ‘ 01' 511511>I< I 312512: ___ s s , +5 5 = o 11 214 12 22. (2.3.3) s‘Zisiich'SzzSirk : 0 s 8 +5 3 = 1 21 21* 22 22* Let Sij = sij/S’ where s is the least common denominator, then we have -11- (2.3.4) .3.5a) .3.5b) .3.5c) S’11511=:< + Sizs1z=z= ” 55 811521* + 81252295 I O s2151M I Szzslze “ 0 521621==< + Szzszze : 53* Also by considering 8:8 = U2 the following relations can be obtained. 5115119. + 512512.: = 35* (2 812811=I< + Szzszm : 0 (2 Sllsll==< _ 522322=I< (2 SIZSIM : 521521=I< (2 From Eq. (2. 3. 5b) we have S22 2 “Smells/821* If f0 = GCD(le, 521*), i.e. , s = f09* and s 2 £045 and GCD 12 21* (2. . 3.5d) 3.6) (9*, 4)) = 1, then fo cancels in Eq. (2. 3.6) and the remaining factor of 321* must diVide 511*. Therefore we have 311* : ho>¥<¢ = 9 812 f0 * 521 = fo>¥=¢>k Now, Eq. (2. 3. 5d) yields the following relation. f e f e = fo*¢*f0¢ O 3:: 0 >:< 01' q: Since GCD (9,.,.¢) ; 1, it follows that 9.9 = ¢*¢ (2.3.7) 4) : i9 (2.3.8) -12- Therefore, S12 : foe»: _—_ ifo¢.,. (2.3.9) 821 = ifoees = forts (2.3.10) 811 : ihoe>i< = halt). (2.3.11) S22 : -ho*e* z ¥h0*¢* (2.3.12) As a conclusion, than: is a common factor of s“, 312, 521 and 322, and by Eq. (2. 3. 5a) ¢¢$ must divide 55w As <1>* and s can not have common factors, (otherwise 3 would not be the least common denomi- nator) ¢* must divide 5*. Therefore, ct divides s and is a strictly Hurwitz polynomial. Let s = ¢go, then S : bod) :1: :- f0¢>k _1_ ¢go f ":¢:' ; h0:'<¢ >‘< ._ __L_ Ph «In, if 4’45] ’ ¢2g ° " ° “ (2.3.13) 0 f O*¢¢ :k ¥ h0*¢¢ >k or 1 p iR S = 6 (2.3.14) R T- P In this expression, 0 is a strictly Hurwitz polynomial. Note that, due to Eq. (2. 3.5a), the polynomials P, Q and R satisfy the relation PPk + RR), = 00* (2.3. 15) -13- 2.4 Elementary Lossless 2-Port Networks In this section, the elementary lossless 2-port networks which constitute the basic sections for a cascade realization are tabulated together with the corresponding scattering matrices. As can be seen, these matrices are of the form 1 5‘s with 00* - szk : RRrs Note that any more complicated section can be obtained by cascading some of these elementary sections. Hence it is not necessary to include such complicated sections in the tabulation. For example, the Darlington-D section does not appear in the fol- lowing list for it could be obtained by cascading two elementary sections NC3. -14- TABLE I Type Lossless 2-Port Scattering Matrix Trans. Zeros F— '1 o 56 O L L —h 1 Al 1. 1 2 x = 00 _'>. + 1 ° 2 L)‘ o o 1 ‘2' C O _ _ - 9-). 1 1 2 X z 00 A2 C C o —>\ + 1 T 2 1 9.. C} . <3 ' 2 L- _ I! _ '1 0— ]) 4) 1 3% )\ Bl C *1 + 1 x0 = o 26 , _.1_ cs 4:) ZC L. -1 —o C 1 ' "zli A ' _.___. ' X : B2 L k l o 0 J“ 7-1—- , .1. ea 0 " 2L L _ . X : + ° Cl " 2 1 1 ZC LC 0 Jul—1'5. C X +—>\+-- 2C LC kZ-I- 1 l or 4‘) LC 2C -, -15- or o _ x 12 17 L l -2—L +LE X __ +. 1 (:2 2 1 1 o ' —-J N’LC x +— x +— (3 2L LC >\2+_1_ _>\_ G T O _ LC 2L_ 1 X nZ-I-1>\2+ L(n-I)2+C)\ + 1 C3 2n ZnLC nLC " T l-nz 2Jr L(n-1)2- C X k2+ 1 2n ZnLC nLC x24. 1 _1.n‘Z 12+ L(n-1)2- C nLC 2n ZnLC 1 — + ——1-——-- o — kl-nLC 1 1m 2 2 X o _L xz+ (11-1) L+Cx 11 +1 C4 C I 2LC 2LC L (n-l)2L-C l-nZ 2 n __ k 1 +— 2LC 2LC LC X pm 46 (2,2; (2.-.1229 l-nz LC 2LC " 2LC-J -16- Y 2 W -—7 '1 x-l NBl 1Z 2L L x :+1L L 7 +1 2 O _ 1+ 2L will 7 -1 G O _ L 2L4 A P .1 2 y 7.1 ->\+_1_ NBZ 2 1 27 7C X :i—% C 7 +1 1 2 o 7 x+_ 1 7 -1 C , O _ 7 7 _ A — .- 1’ ——72'1 2 7 1 X X-—>\+_ 1 21" L LC NCl 2 Z 7 +1 1 2 L +—)‘+— 2 “y 1 7_ 2L LC 1 +L +LC A J TC " CF * o +1+ 2 _4_ ‘L— L(2"LC x = 0 2 ch +1 + 1 4 —I- — 2-—— x = 7C 726 LC -17- 2 4 NC3 4...!— + 7 —nL— 2 2 nLC n L X : o 2 1 X nz+1 2 (n-1)2L +(72+1)C l ———X + X +——- 2n ZnLC nLC P 2 2 2 -1 iififlml) L1” ‘11C x x2-1— x + 1 2n ZnLC nL nLC X 2 2 2 2 'y 1 l-n 2 (n-l) L +(7 -1)C X +—X —— -——X X L nL +nLC 2n + ZnLC NC4 2 +1 + v :19. — C — C2 " LC X : o 2 1 >: 2 2 2 2 (n-l) L +('y +1)C n +1 X + X + 2LC 2LC — 2 2 2 1 (n-1)L+(~y -1)C)\+1-n X2 1X+i >< 2LC 2LC C LC 2 'y n (n-l)2L + (72-1)C l-n2 x +— x +— x - —— L C LC 2LC 2LC -18- 2. 5 Appendix to Chapter II The definitions and theorems used in the foregoing sections are stated in this section. The proofs for these theorems and the statements of the definitions are found in [NE 1]. Definition 2.5. 1. Bounded-real Matrix: (Def. 4.1 in [NE 1]) An n x 11 matrix S(X) is called bounded-real if it satisfies all the following conditions: 1. S(X) is holomorphic in O > 0. 2. S*(X) = 500*) in o > O. T* . 3. Un - 5 (>450): o m o> 0. Definition 2. 5. 2. Positive-real Matrix: (Def. 4. 2 in [NE 1]) An 11 x 11 matrix A(X) is called positive-real if it satisfies all of the following conditions: 1. A(X) is holomorphic in O > O. 2:: a 2. A (X)=A(X )inO>O. 3. AH(X)Z O in O> 0. where AH(X) is the Hermitian part of A(X). Theorem 2.5. 1 (Theorem 5. 12 in [NE 1]) If an n x n matrix S is bounded-real, then the two matrices B and C defined by S = U - 2B = 2C - U n n are both pos itive-re a1. Theorem 2.5.2 (Theorem 5.14 in [NE 1]) If an n x n matrix S is bounded-real, with Un-S of rank r I: 0, _19- then there exists a reaLconstant,orthogonal matrix N such that NTSN = S' 1Un with S' bounded-real and Ur-S' non-singular. Further, I 1-1 A = (Ur+s)(Ur-S) is po sitive-real. CHAPTER III REALIZATION OF TRANSMISSION ZER OS 3.1 Transmission Zeros If the reflection coefficient of a l-port RLCTF network is written in the form _ PM) 51m _ _Q(>\) (3.1.1) then the transmission zeros of this network are defined as the zeros RR):< of the real rational function — , where RR* = 00* - PP*. In other QQ words, the zeros of RR*, after the cancellations of common factors with 00. and 2(GQ-OR) zeros at infinity are called transmission zeros, where 5 denotes "the degree of. ” Consider the scattering matrix of a lossless Z-port LCTF network, ”-5 s _ F‘i + :13- 11 12 Q1 — C21 = R p (3.1.2) 12* — 1* S21 S22 (21 + ‘51— The transmission zeros of this lossless Z-port network are defined as the zeros of the real rational function $12821. Thus, the trans- mission zeros consist of 2(501-5R12) zeros at infinity plus the zeros -20- -21- of RlZRlzfs’ except for possible cancellation with Q: . When this 2—port network is terminated in a l-port RLCTF network with the ELL-l- : i , then the reflection coefficient g+1 Q2 of the resultant l-port network is reflection coefficient S2 = S S S 2 12 21 5:8 + (3Lm 1 11 1 - 52522 or .13 _ 13102—01»: 2 (3 1 4) Q QIQZEPMPZ Since RR* - (20* - pr. _ R12R12*(QZQZ* - PZPZ’“) (3 1 5) - _ + P 2 . . QQ QQ (0102— P1* 2) the transmission zeros of the original lossless Z-port network, in general, are contained in those of this l-port network. The transmission zeros of a l-port RLCTI" network can also be defined by considering the even part of the given driving- point impedance (or admittance) function Z1 (or Y1). Indeed, since Z_1+Sl _o+p (316) 1‘1-.sl ”0-13 " and Ez—i(z +2) V1’2 1 1* RR, >,< : 2(o - P)(o* - 11,) (3’1'7) it becomes evident that the transmission zeros of a l-port RLCTT‘ network are the zeros of Ev Zl except those of RR» which are also the zeros of Oz. -22- For a lossless 2-port network, if (U2 + S) and (U2 - S) non-singular, the corresponding terminal impedance matrix Z are and admittance matrix Y exist and have the following forms. le Z12 z = Z21 Z22 71+5 5 s 5 +5 5 2 s .- 11 22 11 22 12 21 12 1'511'5221511522'512521 1‘511'5224'511522'512521 : (3.1.8) 1- + - _ 2’521 S11 S22 S11522 S12521 - -. + ... .. .. - 1 S11 S22 S11522 S12521 1 S11 S221511522 S12521 Yll le Y : 1Y21 Y22 r_ _ l”511+522'5115221512521 '2 S12 - + + + - 1+S1115221511522 S12521 1 S11 s22 S11522 S12521 .‘ (3.1.9) - - - + 2 821 1+511 S22 S11522 S12521 + - 1+S111522 S11522 S12521 + + - 1+S11 S22 S11522 S12521 It is clear from the above equations that the zeros of 212 (Y12) and and S , respectively. Z21 (Y21) are contained in S12 21 In order to give a physical interpretation of transmission zeros, consider a lossless 2-port network, with the terminal impedance matrix Z, terminated in a l-port RLCTI‘ network g, called load, as -23- shown in Fig. 3.1.1 The following relation is evident. Z 21 i = --————— i (3.1.10) 3 g + Z22 1 1 i i + V1 2 V2 V3 4 0-—-—— 13- Fig. 3.1.1 If Z21 vanishes at a real frequency (1)0, i.e. , X0 = 3030, then 13 = 0 for a sinusoidal excitation with an angular frequency of (1)0 applied at port 1. This indicates that the average power transmitted to the load is zero at frequency (1)0. Similarly, when a load is connected to port 1 and a sinusoidal excitation with an angular frequency of (1)0 is applied at port 2, if 212(jw0) = 0 no power is transmitted to the load. Same discussion can be applied to the terminal admittance matrix of the lossless 2-port network. Therefore, the physical mean- ing of the transmission zeros for real frequencies is that the power transmission from one port to the other is zero. 3. 2 Two Useful Theorems and the Division Algorithm In this section, two theorems and a division algorithm are presented which are important for the synthesis procedure discussed in the several later sections. These theorems deal with the exis- tence of a second and a first degree polynomials passing through some fixed points given in the complex plane. On the other hand, the division algorithm provides computational facilities in the -24- synthesis procedure. Although the proofs of these theorems as well as the proof of the procedure for division algorithm are ele- mentary, because of their importance, this section is devoted to a fairly complete discussion of these proofs. Theorem 3. 2. 1. Let E- be a real rational function in which Q P P * 513 < 50 and Q is strictly Hurwitz. If—'-— = l at X = X = " Q 0* 0 (To + jcoo, where 0’o > 0 and 0110 > 0 are finite, then there exist and C21 with 6P1 _<_ 601 = 2 such that polynomials P1 F110) mo) 131,00) 13,00) —— = —— and ——-—-—-- :—.———— (210.0) one) 01*(10) 0,110) Proof: Part 1, 0'0 > 0. Assume P(X):aX2+aX+a 1 2 1 o 2 Q(X):bX +bX+b 1 2 o 1 and X = ' P( o) 0.1 +351 one) = a2 +sz where c , <12, 81, (32, a0, a1, a , b , b , andb are real. l 2 o 1 2 Since P1(XO) : P(Xo) and P1(—Xo) : P(-Xo) —5?ol( 0') one) —ol(-'Xo”) "o('-' No”) (3. 2. 1) (3.2.2) (3.2.3) (3.2.4) taking X0 = 0'0 + jmo, from Eq. (3. 2. l) and Eq. (3. 2. 2), we have -25- 2 2 = .. + O' + ' P1(Xo) (0’0 (100 )a2 Ca1 + a0 3(20’0w0a2 +woa1) = (0.1 +5131)(k1 +ij) = (alk1 - (311:2) +j(oIlk2 + 1311(1) (3.2.5) 2 2 : _ + ' Ql(Xo) (00 (1)0 )b.Z + (rob1 b0 +J(Zooa)ob2 + wobl) =(<12+1'132)II<.1 +jk2) = (azkl - szkz) +j(u2k2 + (321(1) (3.2.6) 2 2 . P1( X0) (C70 000 )a2 Goal ao “200°0an (coal) = (a2 +1132)“ 1 +112) = (0.211 - (3212) +J(a.2£2 + (3211) (3.2.7) _ 2 2 . 01040) - (Go - 000 )b2 - 00b + bo +J(20000013 - wobl) l 2 =(a1 +1slx11 +122) = (0111.1 .. (3112) +j(11112 + (3111) (3.2.8) where k1, k2, f 1, and! 2 are real constants. Equating the real and the imaginary parts in each of the above equations, we have - - - (OZ-(1)2) 0 1 0 0 O O O -a (3 a I] o o o 1 l 2 20 (1) <1) 0 0 0 0 0 0 -13 -o. 3'1 o o o 1 l a o o 0 (02-012) 0 1 o 0 -a2 (32 ° ° 0 b2 (3.2.9) 0 - - O O 0 200(110 (1)0 0 0 (32 e2 b1 (02.402) .0 1 o o o -a. s o o b o o o 2 2 o 1 20 co -0.) 0 0 0 0 ~43 -o. O O 1 o o o 2 2 f 2 2 2 0 0 0 (co—(1)0) -c 1 -c11 (31 0 0 k 1 0 O 0 200000 -coo 0 ..[31 -c11 0 0 k After elementary row operations, Eq. (3. 2. 9) can be reduced to o o . 3.. buooumnsp Sim-.m- N - NV-n r- I .- m u moo-inges- -aoo+ muss; took-Laos..- -uos--sos- o o o o o o “m - c 2%.... Joe 30-... $03-- -sos-~aos-- -aos-~sos- o o o o o o :- mu- me E -s.. o3- o o o o o o u rm 6.. -s Na me. o 03.. o o o 0 NC No: Nun H.o.. an: o o oQobv o o O Na :7 E- Na. Na- o o o 0303 o c on mac-segmen- -sosos~-~ao- man-30828- -sosos~--n3 o o o o cease o o... mac-303}..- -%so£--%- -aoso£+~8- -dosos~-~ae o o o o o 0303 -27- The normal rank of the coefficient matrix in Eq. (3. 2.10) is 8 and the number of unknowns is 10. It is clear from Eq. (3. 2.10) that a0, b0, a2, b2, a1, b1 and two out of £1, 12, k1 and k2 can be expressed in terms of the remaining two variables. Consider the following matrix formed by the last two rows and the last four columns of the coefficient matrix in Eq. (3. 2. 10) (woaz-OopZ) -(wo(32+00az) 403001-0051) ((1)0131+0'oa1) (3.2.11) (“Joni-00131) -((1)o(31+000.1) -(woa2-Oofi2) (wofi2+ooa2) The determinants formed by any two columns of the above matrix are given as follows. 1) columns 1 and 2. (000112- 00132) -1wop2+ 0002) = (Oimiflulfiz-filuz) (3.2. 12) (woal-oofll) -(wofil+000.1) 2) columns 3 and 4. -(c1)a-UB) ((1)13 +00) 0 1 ° 1 ° 1 ° 1 - -(Ozmzflalfiz-Bluz) (3.2.13) -(wu-OB)((1)(3+0’0.) 00 o 2 o 2 o 2 o 2 3) columns 1 and 3. (con 419) «ma 41(1) o 2 o 2 o 1 o 1 2 2 = (weal-00131) -(wan. 0052) (3.2.14) (wood-10.0131) -(woa2- 0.062) 4) columns 2 and 4. -1mo‘32+ooa2) (“1013110311 2 2 ((1) 13 +00. ) -(co (3 +00.) (3.2.15) 0 1 o 1 o 2 o 2 -(wofil+ooal) (wOBZHIan) -23- As will be seen in the following discussion that the other two possible cases are actually not necessary for further con- siderations. However they are also listed for completeness. 5) columns 1 and 4. (w a o 2 2 2 2 2 (Local-00131) ((1)0132+O’Ou2) + Oowo(uz-al+fil-BZ) (3.2.16) 2 "001321 111101311031) - (wo' 0:11azfi2'a11311 6) columns 2 and 3. -(w I3 +0a) -(w a we) =(m2-oziIa I3 +a. I3) 0202 0101 002211 2 2 2 2 -(w0131+00a1) -(c1)0c12- 00132) + Oowo(az-(32-al+fil) (3.2.17) In order to show that 601:2, we must have b2 1*- 0 for a set of f 1, f 2, k1 and k2. Thus it has to be shown that + (3121 0.112 +132k1 +02ka5 0 or 1111 “1 132 “21 11 l2 1: 0 (3.2.18) k1 k2—J A) If CL1132 ‘ 131‘12 ’£ 0 then both Eq. (3. 2. 12) and Eq. (3. 2.13) are not equal to zero. 1 and f can be obtained in terms of k and k or vice versa. 1 2 1 2 Further, left hand side of Eq. (3. 2.18) becomes -29-1 1 2 2 (00+wo)(a1132-81a2) [ 2010(00132-ODOQZNO'1fizufilazHmowofil- (1100.1 )X 2 2 2 2 x1°1+111'°2'1521 2°01wofizwoazmifiz'fli“2111110100011+wo13111°11131'“2‘13211 1 2 (3.2. 19) k k Now, we have to show these two entries do not vanish simulta- neously. If first entry vanishes, then 2 2 2 2 (Oofiz-woaz) wo(a2+sz-al-sl) (as-ma) . zo(ap-pa) (31-2-20) 0 1 o 1 o 2 1 2 1 Since 0. B -(3 a. )1: 0 and 0.2+(32-uz-(32 < O similarly if second 1 2 1 2 1 l 2 2 ’ ' entry is zero, we have 2. 2 Z 7- (“oazw’ofiz’ . wo1QZ+132'°1‘1311 (3 2 21> (ooalmopl) 200(a1s2-s1a2) From Eqs. (3. 2. 20) and (3. 2. 21) we now have (00132-400112) _ (Goa2+wo132) (oofil-woal) - (0011111110131) which is equivalent to 2 2 1°o+wo)1“152'131°21 ' 0 This result contradicts the assumption ulflz-Blaz # 0. Therefore, the entries in Eq. (3. 2. 19) can not vanish simultaneously and this proves that b2 can be chosen to be nonzero. -30- B) If -00131 16 O (11132-131112 = O and (1000.1 then Eqs. (3. 2. l4) and (3.2.15) should be considered. If Eq. (3. 2. 14) is equal to zero, then we have weal-05131 = :(moaz-Oofiz) (3.2.22) . C11 131 “liaz 131i152 Since ulfiz-Blaz = O or — = —- or —— = _— , “2 132 “2 132 Eq. (3. 2. 22) becomes + = + (1) (<11 02) 0 ([31 [32) or O u o. ..2 z .53. = _1 (3.2.23) (1)0 2 131 Similarly, for Eq. (3. 2.15) we have (1) o. a 62 = __7- = -51 (3.2.24) o 62 1 As a result, Eq. (3.2. 14) does not vanish and l 1 and k1 can be expressed in terms of! 2 and k2. In this case, the left hand side of Eq. (3. 2.18) becomes 000 2 2 [(al +131) (“1‘12 +1315,” 12 (1)001 - 00131 k2 2 It is evident that b2 can be taken as non-zero, since of + 131 it 0. — 00132 = 0' If 01132-13102 2 O and (000.1- 0051 = 0, which implies (000.2 + 0, . . . ' h (1)0131 coal at 0 and wofiz+ooa2 f- then Eq (3 2 15) does not vanis -31- and 1 and k can be expressed in terms of f and k . The 2 2 l 1 left hand side of Eq. (3. 2.18) becomes -(D o 2 2 - l o l o 1 k1 and it is evident that bZ has non-zero solutions. Since the above cases are sufficient to have b2 IE 0, the vanishing deter- minants in Eqs. (3. 2. l6) and (3. 2.17) need not be considered. Part II, 0 =0 ands) > O. o o In this case, Eq. (3.2. 9) becomes 41): 0 l O O O 0 O -<11 (31 raz 0 (130 0 O O O 0 O 431 -0.1 a1 0 O O 41): 0 l 0 0 -a2 (32 a0 0 O O 0 (no 0 O O 432 --0.2 bZ = 0 (3.2.25) «1): 0 1 O 0 0 -0.2 (32 0 0 b1 0 -(1)o 0 O O O 432 -a2 0 0 b0 0 O 0 41): O l -111 81 0 0 f 1 L O 0 0 0 «no 0 431 -0.1 0 O— f 2 k1 k2 -32- After the elementary row operations, we have F- 2 -’[- '1 ..(1) 0 O O 0 0 -al 81 1 0 a2 0 (1)0 O O O 0 -Bl -o.l O 0 al 2 O O -600 O O 0 -<1.2 (32 O 1 b2 = O (3. 2. 26) 0 O 0 (1)0 O 0 -(32 -0.2 O 0 b1 0 O 0 0 --0.2 (32 0.1 431 O 0 £1 0 0 0 O -i32 «.0.2 431 -o.:l O 0 £2 0 O 0 O «0.1 (31 0.2 432 O 0 k1 O 0 O O -(31 --0.1 -(32 -0.2 0 0 k2 a o b o Similar to the case where 00 > O, the coefficient matrix of Eq. (3.2.26) has a normal rank 8. Therefore, a2, a1, b2, b1' 2 1, f , k and k can be expressed in terms of a and b if 2 l 2 o o "°‘2 132 Cl1 ”131 -B -a -B -a Z 2 1 1 ,1 0 (3.2.27) . . _ * . . . . 2 2 _ 2 2 Since P*(on) - P (we) which implies 0.1 + (31 — 0.2 + (32, the above determinant equals to -33- '°2 132 a1 -fi1 -s -a -s -a 2 2 1 1 =-[(a:-+B:)-(cf-15:)fz= 0 -a B a -s 1 1 2 2 'B1 ‘“1 "pz ““2 Hence the rank of the coefficient matrix is at most equal to 7. Now, let us assume 0.2 It 0 and perform some elementary row operations on the matrix formed by the last 4 rows of the co- efficient matrix; the last 4 equations become F ”Q2 132 a1 '51 o (a2+pz) (a 5 +5 a ) a s a a 2 Z 12 12 12 12 :0 (3.2.28) 0 0 O 0 0 0 O 0 L _ -I Since 0.2 and (32 can not both be equal to zero, if, e. g. , 0.2 = 0, then (32 11 0 and we have 0 132 “1 -B1 21 2 -13 0 [3 (3 -u 13 f 2 1 2 1 2 z = 0 (3.2.29) 0 0 0 0 k 1 L. 0 0 O 0 _,_k2_ From Eqs. (3. 2. 28) and (3. 2. 29) we can see that the rank of the coefficient matrix is 6 and a2, a , b , b , f and 1 2 can be 1 2 l 1 found in terms of k1, k2, a0 and b0. The third equation in Eq. (3. 2. 26) gives -34- 2 wobz - -o.2k1 + 13sz + b0 and since a , 132 and b0 are not simultaneously equal to zero, b 2 2 has a non-zero solution. As a conclusion, the polynomials P and Q1 of the forms 1 given by Eqs. (3. 2. 1) and (3. 2. 2), respectively, exist and the leading coefficient of (21 can always be made non-zero. Q.E. D. Theorem 3. 2. 2 Let E- be a real rational function in which Q PP '< 6P < (SO and Q is strictly Hurwitz. If >. = 1 at X = X = - 00* o 00 where 00 Z O (000:0) then there exist polynomials P1 and Q1 with 6P1: 601 = 1 such that P1(Xo) _ P(Xo) and P1*(Xo) _ P*(Xo) 011110) Q(Xo) 01*(Xo) 0*(XO) Proof: Assume = 3. .30 P1(X) alX + a0 ( 2 ) Q(X) = bX+b (3.2.31) 1 1 o and P(Xo) = 0.1 (3.2. 32) QIXO) = a2 (3.2.33) then P(-Xo) = mo.2 (3.2.34) Q(-Xo) = mo.1 (3.2. 35) where a , a , b , b , u , o. and mare real numbers. In 1 o 1 o 1 2 Eqs. (3.2. 30) and (3.2.31), if we let X = 1: 0'0, then = + P1(O'O) aloo a0 01100) = b10.0 + b0 131(410) = -a1 011-00) = -b1 O+a o o O'+b o o -35- f Cl1 10. = k0.2 ko.1 where 1 and k are real constants. These can be written in a matrix form. — _1 Equation (3. 2.40) can be simplified as H00 1!: 0, then a1, b1, a0 1 and k. and b0 can be solved in terms of 021— (3.2.36) (3.2.37) (3.2.38) (3.2. 39) (3.2.40) -36- a1 a1 -0.2 b1 1 a2 -0.1 f = -- (3.2.41) a 2(To 0 a O a k 0 o l o 2 b0 Ooaz Goal 1.- 4 _ ._ Hence nonzero b1 can be obtained. Ifoo=0,thena zio. 1 and the only condition to be satis- 2 fied is b = +a 14 0 (3.2.42) 0 — 0 b1 and a1 can always be chosen arbitrarily. If 00 = 00, then 0. = + (1 also and the only condition to 1 2 be satisfied is =+ .. Io1 _al 11 o (3243) Therefore, it is always possible to find polynomials P and Q1 1 of the forms given in Eqs. (3. 2. 30) and (3. 2. 31) such that the leading coefficient of Q1 is non-zero. Q.E.D. Division algorithm: The division algorithm described here is essentially the Euclidean algorithm which has identical steps as the Routh -37- algorithm. More specifically, each cycle of division is exactly the Euclidean algorithm for polynomials [B1 1]. For each cycle, the coefficients of the quotient and the remainder polynomials are obtained by cross multiplication as in the Routh algorithm. Let P and Q be real polynomials of degree m and n, respectively, with GCD(P, Q) E 1 and n 2 m > 0, where GCD denotes "the greatest common divisor of. 1' Let P and Q be written in the following forms. III p 7 + 111 >’ + + 1» >’ + n) P(X) Q(X) "I U 7 + U 7 + + O‘ 7 + U 0 where a )6 0 and b If: 0. m n By the application of the Euclidean algorithm, from the polynomials P(X) and Q(X), we obtain a set of identities. 000 E qo(X)P(X) + r100 P(X) E qlmrlm + r200 r1(X) E q2(X)rZ(X) + r3(X) (3.2.44) ”p.11” 5 qpmrpm + rp+1 -38- where 5P > rl(X), 6ri > (Sri+1 for i = 1, 2, ..... , p and rp+1 is a nonzero constant. The above identities can also be written as r10) 5 cm - qOIMPm r20) 5 -ql(X)Q(X) + [1 + (101111111111 P00 (3- Z- 45) r30) 5 [I + q1(>~)q2(>~)]Q(M - [qom + qzm + qO(X)q1(X)q2(X)] Pm. By using the bracket symbol notation introduced by Stieltjes [ST 1], which is defined by [01 = I [<10] = <10 [qo.ql] = 1+C10ql and the recurrence formula [qo,ql, ..... ,qn]=[qo,ql,....,q n_1]qn+ [qo’ qlsoooogqn-Z] (302046) the identities given in Eq. (3. 2. 45) take on the forms r ,0) -=- IOIQIM - [40(1)] Pm r ,0) E-[q1(M]Q(X) + Iqom. q1(>~)]P(M (3. 2. 47) r30) -=- Iqlm. qZIMIQm - [40(1).q1m.q2m1pm rim 5 (-1)i+1{ [ql(>~).q,<>\). . . ..qi_1(>~)] om - Iqom.q1(x). . . ..qi_lIMI P(X)} -39- To obtain r's and q's, a modified Routh's algorithm is used. This algorithm deals with two polynomials instead of one which is used in the original Routh's algorithm. Arrange the coefficient of Q(X) and P(X) so as to form the first two rows of the array, b b oooooo’b 19 O 29 a ,a. ,am-2,oo ..... .....’a2’a1’ a0 The coefficients in third row of the array are obtained by cross multiplication exactly as in the Routh algorithm as follows. ambnol -bnam-l a'mbnu Z-bnam- 2 C = ’ C = g o o o o o 9 n-l a n-2 a m m 0- a C .- mn-i nm-i ’. n-i a m If the degree of the polynomial corresponding to the third row is greater or equal to that correSponding to the second row, a new row is generated similarly. This is repeated until the degree of the remainder polynomial becomes less than that of the divisor which corresponds to the second row of the array. This cycle yields the pair (r(X), q(X) ). Note that r(X) is formed by summing the coefficients of the last row each of which is multi- plied by the re Spective degree of X. Similarly, q(X) is obtained by first dividing each leading entry of the rows by am, then summing the leading coefficients of each row in the cycle, except those rows which are replica of the second row, which is multiplied by respective degree of X. -40- The above cycle now is repeated, if necessary, several times for the last two rows of the array. Since at the end of each cycle, the inequalities 6r1 > 6r2 > 5r3 . . . hold, there will be a final cycle for which a zero remainder is obtained. Thus, r's and q's used in the Euclidean algorithm or Eq. (3. 2. 47) can be constructed easily by the help of the array. 3. 3 Existence of Scattering Parameters CorreSponding to a Selected Simple Set of Transmission Zeros Darlington [DA 1] has shown that the driving-point impedance or admittance function, F(X), of an RLC 1-port network can be realized by a lossless 2-port network terminated in a unit resis- tance. However, in this realization procedure, it may be necessary to multiply the numerator and the denominator of F(X) by the same strictly Hurwitz polynomial, called the surplus factor. Augmen- tation of F(X) by such a polynomial will necessarily increase the number of reactive elements to be used in the realization of F(X). Hazony extended Darlington's synthesis procedure to non-reciprocal l-port networks which eliminates the use of such surplus factors. Consider a lossless Z-port network N terminated in an impedance 1; as shown in Fig. 3. 3.1. Let s = (3.3.1) -41 - be the scattering matrix of N and let S and S WithS =iS 1 2 21 12*' be the reflection coefficients,respectively, of N at port 1 when port 2 is terminated in g and of g . Then _ Li 52 — (4+1 (3.3.2) and S S S S1 = 5111 Iz—flsTZ'SZ—l 13‘3'31 2 22 o——— 0 SI S2 Fig. 3.3.1 If the terminating impedance 2; =1, i.e. , 32: 0, then Eq. (3.3.3) becomes S = S (3.3.4) 1 11 which implies that the reflection coefficient corresponding to a driving-point impedance can be considered as the entry in 1x1 position of a 2x2 scattering matrix corresponding to the network obtained by the Darlington synthesis procedure for this driving- point impedance. Let s = _ (3.3.5) where P and Q are real polynomials in X and Q is strictly Hurwitz. By Eq. (2. 3.15), we have -42- which gives all the transmission zeros other than those at infinity. It is evident that the number of transmission zeros at infinity is 2(0Q - 5R). Consider Eq. (3. 3. 3) and let F1 511 = 5- (3.3.7) 1 R 12 512 = Q— (3.3.8) 1 R 21 521 - 6?- (3.3.9) then from Eq. (2. 3. 14), we have P1* 522 = + 6]"— (3.3.10) where S22 assumes the negative Sign 1f R21 = R12* and the p061t1ve 1f R21 3 -R12*. Further, let P2 82 = 5; (3.3.11) Note that in Eqs. (3. 3. 7) through (3.3.11), P's, 0'8 and R's are real polynomials in X. Substituting Eqs. (3. 3. 7) through (3. 3.11) into Eq. (3. 3. 3), the following relation can be obtained. R12R21P2 _13 _ :1 + Q1 Q1 Q2 Q (21 l-P—ZGPH Q Q -43- PQ+QP 1 2 — 1* 2 = (3. 3. 12) + Q102 - 131’1132 where again, the upper or lower signs are used if R21 ERlZ* or R21 5 -R12*, respectively. From Eq. (3.3.12), we have 5 + P . . P PlQZ-Q1* 2 (3 3 13) E + P - - Q QIQZ—Pl* 2 (3 3 14) Although for the most general decomposition of Eq. (3. 3.12) the left hand side of Eqs. (3. 3.13) and (3. 3.14) should contain an arbitrary real polynomial, as will be seen in the proof of Theorem 3. 3. 1, without loss of generality, this polynomial can always be considered as unity. Substituting Eqs. (3. 3.13) and (3. 3.14) into Eq. (3.3.6), we have RR QQ*- PP * * (QIQ1*- P1P1*HQZQZ*- PZPZ*) (3 $15) On the other hand, for 52’ since 0202*” P2p2==< : R2112* then the following relation can be obtained immediately. RR* = R12R12*R2R2* (3.3.16) Equation (3. 3.16) clearly shows that the transmission zeros of the original driving-point impedance can always be split into two parts; the first part, R12R12*, corresponds to a lossless 2-port net- work and the remaining part, RZR 2*, corresponds to the terminating RLC network. In particular, the first part R12R12"< can be taken -44- in a relatively simple form as to correSpond to an elementary section discussed in Section2. 4. Therefore, the synthesis pro- cedure requires the proof of the fact that a simple set of trans- mission zeros can be realized by an elementary section and the information on the remaining transmission zeros are contained in a terminating impedance 1, (X), or the corresponding reflection coefficient 8 . In other words, the cascade synthesis is justi- 2 fied if, after the selection of R R the existence of the real 12 12*' polynomials P1, Q], P2 and Q2 is shown such that _1_ F1 1 R12 Q _ 1 R12* + Pl* 132 is para-unitary while 5 is a bounded-real function. To this 2 end, we shall now consider the following theorem. Theorem 3. 3. 1 Let S1 = g be a real rational function in a complex variable X = O + joo with the properties that QQ* - PP* = RR* and GCD(P, Q) E 1. If Q is strictly Hurwitz and I E I < 1 on jw-axis, then there exist polynomials P . Q . R , Q - 1 1 12 P2, 02 and R2 satisfying the relations (1) .13 — P1QZ'tQ1’1‘PZ (3 3 17) Q Qlei-PH‘P‘2 (2) R R E Q Q - P P (3.3.18) 12 12* l 1* 1 1* -45- (3) R212” 5 0202* - P2P” (3.3.19) (4) 1111* '5 R12R12*R2R2* (3.3.20) where in (1) only the upper or only the lower signs are to be used, such that 5Q1 = 1 1f 5R12-<—1 001 = 2 if 0R12=2 Pl —— <1 for h : jg) Q _ 1 P2 — <1 for X = jw Q2 — with C21 and 02 being strictly Hurwitz polynomials. Proof: Since Q is strictly Hurwitz and g I: 1 for X = jw , S is a reflection coefficient for a driving-point impedance. 1 By Eq. (3.2.6) RR* 5 QQ* - PP]: (3.3.21) is an even polynomial whose zeros are the transmission zeros which lie symmetrically about both the real and the imaginary axes as shown in Fig. 3. 3. 2. All the jw-axis zeros are necessarily of even multiplicity including those at infinity which will exist when jco GO > 5R . o o X-plane C) CD 11 q Fig. 3.3.2 -46- Since RR* is obtained directly from 00* - PP*, once R12R12* is selected, R2R2* follows immediately. Note that if 5Q > 6R, then R12R12* can be selected as a constant (polynomial of zero degree). Due to the distribution of the zeros of RR* in the complex X-plane, RRk can be factored as follows. = 2! 2 2 2 22 2 2 RR K(-X ) 11' (-X +am) 11' [(X +bn) -ch ] 3'5 m n (3.3. 22) where 1 , m and n are non-negative integers and K is a positive constant. Therefore, it is always possible to take R12 as of degree two, one or zero. By Theorems 3. 2.1 and 3. 2. 2, there exist polynomials P and Q1 with 0P1 _<_ 0Q1 : 2 such that l P (11 ) P(X ) 1 o o = (3.3.23) 01030) 0&0) and P SAX ) PMAX ) 1' ° ' 0 (3.2.24) 01,80) 0,110) where X is a zero of R R , which is also a zero of RR . o 12 12* * Equation (3. 3. 23) implies that PQl - PIQ is divisible by R R 12 12*° Let the quotient be P2, i.e. , R12R12*P2 = i(PQ1- PlQ) (3.3. 25) Since Q1Q1* - PIP”: = O at X : X0. we have P1(Xo) — 01*(XO) (3 3 26) k -' (A O 0 C21( 0) P1,. 0) -47- By substituting Eq. (3. 3. 26) to Eq. (3. 3. 23), we obtain Q1*(1\O) _ P1110] (3 3 27) - {x O O 131,00) Q. 0) This implies that 01*0 - PHcP is diViSible by R12R12*. Calling the quotient Q2, we have R12R12*Q2 = : (Q1*Q - P1>§I<’(QQ* " 1313*) (3. 3. 31) 12 12* By Theorems 3. 2.1 and 3. 2. 2, Q1 and P1 can be obtained to satisfy E - P P R12111231c QIQH‘ l 1* Since QQ. -PP;,, ‘R R RR \. 7" -48.. Eq. (3. 3. 31) becomes 0202* -psz = RZRN which is essentially that in Eq. (3. 3.19). From Eqs. (3.3.18) and (3.3.19), we have p1 — <1 for X231.) Q _ 1 p2 . — l for ijw 02 '- To show that Q1 and 02 are strictly Hurwitz polynomials, we multiply Eq. (3. 3. 25) by P and then add it to Eq. (3. 3. 28) 1 J; . "N multiplied by 01' i. e. , iR12R12=:<‘Qioz ipikpz) = "QiQir-z: " P1P1=:<)Q = R1.2R12=-:~=Q or E ' 3 o 0 Q i ((2le i PMPZ" (3 3 32) Similarly, P = _+_ (F102 1 015.3132; (3.3. 33) It is to be noted that the signs appearing in front of the parentheses in Eqs. (3. 3. 32) and (3. 3. 33) are to be taken so that both are positive or negative. Similariy, the signs appearing in the paren- theses must be taken to be both either positive or negative, or finally P l'Z- 1* 2 Q QQ +p:?:p In Eq. (3. 3.32), since Q QZ is regular in the right half X-plane and Q does not vanish on yin-axis, hence by Rouché's theorem (2102. has the same number of zeros in the right half X-plane as Q does. Therefore, Q1 and Q2 are strictly Hurwitz polynomials. Q. E. D. 3.4 Construction of the Polynomials P 01' P2 and 02 1’ As is seen in the previous section that Theorem 3. 3.1 Q,Pand establishes the existence of the polynomials P1, 1 2. 02. However, the computation of these polynomials require further considerations. For this: reason we shall first consider the following pair of equations whale are obtained in the proof of Theorem 3. 3.1. R12R12>é:= 2 1)Q ( 44) Since by assumption GCD(P, Q) E l, the polynomials (R12R12:,< - - '. YP - ‘3 ' ' ' l XP2 Q1) and (RIZRIZ’i‘ 2 Pl) mu t be d1v131b e by Q and P, respectively, hence the quotients are equal to a poly- nomial, say I. Therefore, II! R XPZ - QJ Q (3.4.5) 1 R.12R12;::~< qu r1 Q E qlrl + rZ r E + r r. E c. r. + r 1-2 1i--l i-l i r. ‘=’ ,r. + r. 1-1 q]. 1 1+1 : '=' . r, + 1 q1+1 1+1 1+2 r E e r + r -51.. with6C2>6r1>6rz>...>6ri>6ri >6r. >...>6r +1 1+2 Assume that one of the remainder polynomials, say ri, has the degree which is equal to that of 01’ i. e. , Gri = 601 0+1 0 (3.4.8) Also consider the following expressions for the remainder poly- nomials r. r, and r, . 1' 1+1 1+2 1+1 1'1 ‘ ('1) {[qi’qz’""q1-1]R12R12*X - [(10, ql, .. .’q1-1]Q} r1+1 ('1) {[‘li’qz’""‘11]R12R1.2*X - [qo’ q1”"9qi]Q} r1+2 : ('1) ”(11' qz’"”qi+1]RizR1.2*x '[qo’ ql""’qi+1]Q} If ri is a strictly Hurwitz polynomial, we take where k is a real constant. From Eq. (3. 4. 9) we have i+1 P2 — ('1) qu1:q2:-o-:qi_1] Note that r . i-l Q " ri[q1’q2"”’q1-1' r. 1 and therefore- OQ + 5P2 bri-l Since 51‘. = qu + Gri (3. (3. (3. (3. (3. (3. (3. .9), .10) .11) .12) .13) .14) .15) -52.. we have (SQ 6P2 + qu + bri 6P2 + qu + 501 (3.4.16) On the other hand, 602 = (SQ - 601 and Eq. (3.4. 16) implies 602 = 6P2 + (Sq.1 > 6P2 (3.4. 17) If ri is not a strictly Hurwitz polynomial, then a linear combination of ri with r. 1+1 is required to obtain 01. Note that, in general, several remainder polynomials will be necessary for the construction of 01' However, as stated in a theorem from algebra which is given in the following without proof, it will be sufficient to consider only 601 + 1 remainder polynomials of different degrees. Theorem 3.4.1 Consider the set of real polynomials {Ai(X)| 5A1: i, i = 0, 1, . . . , n}. Then there exist real numbers ai such that every polynomial B0.) of degree m 5 n can be ex- pressed as a linear combination of the fir st m polynomials in the set, i.e., B0,) 5 avo(X) + a1A1()\) + . . . + amAmu). In particular, when 601 5 2, in general, consideration , and r '1 ' , ., of ri ri+1 1+2 W1 1 be required where the degrees of r1 -53- ri+l and r1+2 are assumed to be 2, l and 0, respectively. How- ever, it will be shown in the following that the strictly Hurwitz polynomial Q1 can be constructed from the polynomials ri and r. alone. The linear combination of r. and r. implies that 1+1 1 1+1 the degree of P is given by 2 6P2 : 6[q1’q2'°°"qi_l'qi] = 6[q1,q2,...,qi_1] +6qi (3.4.18) Since 60 = 6[q11q29° 0°9q1_1]+ Gri-l = 5[q19q29 - --.qi_l] + éqi + Ori = 5P2 +501 (3.4.19) hence 6P2 = 6Q - 501 = 60 (3.4.20) 2 On the other hand if we consider the linear combination of three remainders ri, r. 1+1 and ri+2, then 5P2 = 6[q1,qz,...,qi.qi+l] (3.4.21) and consequently we have 6P2 + 601: 6[q1.q2. . . °'qi'qi+1]Jr Gri 6[q1' q2’ ‘ ’ ’ ’ qi] + éq1+1 + 5’1 (SQ + 6qi+1 (3.4. 22) which implies 6P2 > GQZ (3.4.23) -54- This is a contradiction to the existence theorem stated in the previous section. As a conclusion we state that, if bri = 601, then the strictly Hurwitz polynomial 01 can always be obtained as a linear combination of the polynomials ri and r. 1+l° In Eq. (3.4.7), if Gri > 601 and 51-1 < 501’ then (5in 2 1 for 601 = 2 or 1. 1) If 601 = 2 and Gri = 1, then Eq. (3.4. 16) becomes OQ = (5P2 + oqi + 5ri > 6P2 + OQI (3.4. 24) By multiplying the remainder polynomial ri by X, the corresponding degree of P2 is increased by l, i. e., 5P2 = 5[q1,q2, . . . , qi_1] + l 5 60 - GQl (3.4.25) Therefore, by considering the linear combination of ri and Kri, the strictly Hurwitz polynomial Q1 can be constructed. 2) If 501 = 2 and 53:1 = 0 which implies qu Z 3, then in order to construct the second degree polynomial ql’ ri has to be multiplied by X2. In this case, however, the degree of the corresponding P will exceed that of 02' This contradicts 2 the existence theorem given in the previous section, hence the case under consideration cannot occur. 3) If 601 = l and Ori = 0, then ri must be multiplied by -55.. X and the degree of the corresponding P2 becomes 6P2 = 6[ql,qz, . . .,qi_1] +1 Since 5‘11: 2, then 6Q : 6[q19q21 o o o ,qi—1]+ 6q1+ 61.1 3. 6P2 + 501 (3.4.26) Thus the strictly Hurwitz polynomial 01 can be obtained from the linear combination of ri and Kri. When the polynomials P and J obtained from Eq. (3. 4. 5) 2 are substituted into Eq. (3. 4. 6) we obtain PQl- R12R12*P 2=QP1 (3.4.27) a) If 5(PQl )> 6(R12R12*P2), then 6(PQI) = 5(QP1) and since 50 3 GP, we have 601 _>_ (5P1 (3.4.28) b) If 6(PQ1 )— - 6(R12R12*P2), then 6(QP1)_<_ {5(PQl ) and conse- quently 601 Z 5P1 c) If <‘5(PQ1 )< 6(R12R12*P2), then 6(QP1)= 6(R12R12*P2) and since GRIZ _<_ 601 and 6Q: 6P2 + 6Q1_>_ 6(R12P Z), we have 6P1< 6R12_<_ 6Q1 Thus the polynomial P1 obtained from Eq. (3.4. 6) satisfies the degree condition for the elementary section to be realized. It is then demonstrated that the polynomials P1, 01' P2 and Q2 can be obtained from Eqs. (3.4.1) and (3.4.2) by the - 56 .. application of the division algorithm. 3. 5 Further Discussion on ”Construction of the Polynomials ll P1, Q1, P2 and Q2 _— The polynomials P and (21 obtained in the previous section 1 are based on the conditions that Q1 is strictly Hurwitz and 501 ibPl. This does not guarantee that PI and Q1 will be of the forms as those corresponding to the elementary sections given in Section 2. 4. However, it will become apparent from the follow- ing discussion that the polynomials P and Q1 of the desired forms 1 can always be obtained provided that an additional condition is imposed to the linear combination of the remainder polynomials used to generate P and (21. Note that this approach has a simple 1 network interpretation and the following discussion is actually based on this interpretation. Consider a lossless 2-port network N such that the degree of the least common denominator of the entries in the correspond- ing scattering matrix S' does not exceed 2. Let N be cascaded with an ideal transformer of turns ratio ltn as shown in Fig. N N I" ‘‘‘‘‘‘ a """I I- ------ b’""l I I I I ' lzn ' Oi m:l : ' N : I v I l 1 I : I C. : I : I : L. _________ .1 L__....._ ......1 S S' (a) 2 (b) 2 -57- The scattering matrix S' of N is of the form w I 12 I 1 -Pi* 0'1 D (3.5.1) d A simple analysis yields that the scattering matrix S of the augmented network Na is given by P1 Q1 21 21 fl 01 b (1+n2)Pi - (l-nleoi ZnR' (31+n2)Qi - (l--n2)(3l--P'1 R12 Q 1 (3.5.2) .2?” 1 q *) ZnRiz 1 (1”:sz (1 21(115' ) s 1' 'n 1* 2 , 2 - , (n -l)Ql + (n +1)(+P1*) 21 (Imam; - ”'“PWPM (anmi - (1-n2)<¥Pi,,) (3.5.3) Thus, Q — 1 2 I 1 2 “PI 3 1 -( +n )c2l -( -n )(+ 1*) (.5.4) P1 = (14ml)?!1 - (1-nz)($ovl*) (3.5.5) R12 = 2nR'12 (3.5.6) R = ZnR' (3.5.7) 21 21 -58.. From Eq. (3. 5.4), Q1 can be rewritten as 2 01 =(Q'11P'1*)+n (01:11,) (3.5.8) The driving-point impedance at port 2 of N when port 1 is termi- nated in 1 ohm resistance is equal to the ratio of the polynomials (Q'1 1 P'l*) and (Q'1 i P'1*). (These are, respectively, the numerator and the denominator polynomials of this impedance function.) These polynomials are Hurwitz. Furthermore, when one of them vanishes at a point jwo on job-axis, the other does not vanish there. Therefore, the denominator polynomial given by Eq. (3. 5.4) is strictly Hurwitz. On the other hand, due to the bounded-real property of I g—il- , the absolute values of the coefficients of P'l do not exceed the corresponding coefficients, all positive, of 01' Therefore, by proper selection of the parameter 11, one of the coefficients of P1 in Eq. (3. 5. 5) can be made zero which yields the forms appearing in the eXpressions for the elementary sections dis- cussed in Section 2.4. Consider now a l-port network g augmented by an ideal transformer of turns ratio m: 1 as shown in Fig. 3. 5.1-(b). If P! the reflection coefficient for g, is denoted by 5'2 2 6-? then it 2 follows that the reflection coefficient of the augmented network Nb is given by -59- P (1+m2)P'z - (l--mZ)Q'2 S :: — : 2 2 (3.5.9) 2 (1+m )Q'2 - (l-m )P:2 For the above reasoning S2 is also a bounded-real function, as it should. Now with the aid of foregoing discussions it becomes clear that in the realization of a given reflection coefficient S1 = g we may first extract an elementary section which is in- cluded in Table I and the remaining l-port network will now have a reflection coefficient S2 and is still bounded-real. Indeed, when the procedure described in this thesis is applied, 51 is first realized as in the form given in Fig. 3.5. 2-(a). However, in- serting two cascade connected ideal transformers of turns ratios lZn and nzl between the networks N and g, Sl remains unaltered. Considering the above discussions, now n can be selected so that the network Na in Fig. 3. 5. 2-(b) becomes identical to one of the Na Nb (- -------- 1 I------"—I 1 I o——— -—o—— 0—1— : N 2;, : N 2', l o— , i L--- ----J I SI (a) $2 1 '2 Fig. 3.5.2 elementary sections of Table I which is terminated on a new l-port network Nb whose reflection coefficient is bounded-real and com- pletely known. -60- In numerical computation of the polynomials P , Q1, P l 2 and Q2 it is advantageous to consider both division arrays corres- ponding to the pairs of polynomials, (R R X, Q) and (RlR l 1* *Y, P). 1 Since both arrays yield the same P and J, they contain the same 2 quotients up to a certain step. Note that, when only one division array is used, the step at which one should stop and determine the desired polynomials is actually the step where both division arrays deviate to having identical quotients. Therefore, simul- taneous consideration of two division arrays yields the information as to where one should stop. Once this final step is determined. the polynomials P , 01’ P2 and Q2 are constructed as described 1 in the previous section. CHAPTER IV SYNTHESIS PROCEDURE AND EXAMPLES 4. 1 Synthesis Procedure The synthesis procedure described in this section is based on the result of Chapter III. As is indicated, one always has the liberty of ordering the transmission zeros. This synthesis pro- cedure can be applied to a given reflection coefficient as well as to the driving-point immittance function of a l-port RLCTI" network. If an immittance function is given, it is first converted into the reflection coefficient and then the transmission zeros are deter- mined by Eq. (3. 3. 6). Following is the step by step description of the synthesis procedure. 1. Obtain the reflection coefficient S : This step is 1 omitted if S1 is given. However, if l-port RLCTI‘ network is characterized by the immittance function, then the reflection co- efficient of the network is - 1- 5-1.1- Y1- _ +1— + — l Z1 lY1 (4.1.1) Olru where 21 and Y1 are, respectively, the driving-point impedance and admittance functions of a l-port RI-CTP network. Since the -61- -62- numerator and the denominator polynomials of Zl or Y1 are assumed to be relatively prime, the polynomials P and Q are also relatively prime. 2. Determine the transmission zeros: By using Eq. (3. 3. 6), which is repeated here for convenience RR = QQ -PP (4.1.2) >:< 9.: >.'< the finite transmission zeros are determined since these are the zeros of the even polynomial RR*. The multiplicity of the transmission zero at infinity is given by 2(5Q - 6R). Thus the locations and the multiplicities of all transmission zeros are determined. Further, RR”: can be factored such that each factor corresponds to an ele- mentary lossless Z-port network described in Section 2. 4. More specifically, we shall take each factor to be in one of the following forms: 2 212, (1 +d2)2, (4.1.3) 1, -12, (42+ a2), (12+ b2)?” - c where a, b, c and d are real and non-zero constants. 3. Obtain polynomials X and Y: From P and Q, by using the division algorithm, the polynomials X and Y with 5P > <5Y and GO > 6X are obtained uniquely which satisfy the identity, XP- YQ l. 4. Select the transmission zeros corresponding to an ele- mentary section to be realized: Select R1R14= as one of the factors of RR* g1ven in Eq. (4. l. 3). -63- 5. Perform the division algorithm for R1R1*X and Q; and also for R1R1*Y and P: Two division algorithms are continued until different quotients show up. 6. Obtain the polynomials P Q1 and P : Take the linear 1' 2 combination of the remainders obtained in step 5 with their proper degrees. Then, together with the relation R1R1>z< = 0101* ' pip”< the polynomials P Q1 and P are obtained. 1' 2 7. Obtain the polynomial Q2: The polynomial Q2 is obtained from the following identity, R R Q E Q1*Q- P *P (4.1.4) 11*2 1 since all other polynomials are already known. This completes a cycle of realization of an elementary sec- tion. Repeating the above cycle for other selected transmission zeros, in the final cycle either both P and Q2 become constants 2 or in the cycle before the last, P and Q2 are in the forms which 2 correspond to an elementary section. For the latter case, the terminating resistance is 1 ohm. 4. 2 Example I Realize the driving-point impedance Z given by l _14+213+612+81+4 1 14+213+612+21+4 -64- in cascaded Z-port LCTl" networks terminated in a resistance. Solution: 1. The corresponding reflection coefficient is P 21" 1 3). SI = O = Zl+1 = X4+ 2X3+ 6Xz+ 5). + 4 (4.2.1) 2 Let P = 3k (4.2.2) Q = 14+ 213+ 612+ 51 +4 (4.2.3) then the transmission zeros are given by RR = QQ — PP = (12+ 2)4 (4.2.4) * >I< * i. e. , the transmission zeros are located on the imaginary axis at i j ~12 with the multiplicities of 4. 3. To obtain polynomials X and Y for the given polynomials P and Q, we have the following division array. Q l 2 6 5 4 P 3 0 3 0 6 5 4 3 O 5 4 3 O -65- From the above array we have, _1 3 2 2 5 qo—3X+3X+2X+3 (402-5) r1: 4 (4.2.6) Hence X - < 1) ~l°[ ] - - 4 q0 = -1i2(13+212+61+5) (4.2.7) Y=<-1>-l-[01 4 = .% (4.2.8) For the first cycle of realization, an elementary section with 2 R1R1* = (X + 2)2 will be extracted. Since R R is selected, we consider 1 1* R R x: -..1_(>.7 +2).6 +10).5 +13>.4 +28>.3 +28).2 1 1* 12 + 24>. + 20) (4. 2. 9) 1 4 2 : -—h - - . . R1R1*Y 4 1 1 (4 210) The division arrays for R1R1=kx and Q; and for RIR 1"\ . . qo 12 3 (4 2 11) 2 2 2 5 = -— X -—h-— . . r1 3 3 3 (4 2 12) and R R Y -1- O l 0 l 1 13 ‘ 4 " ' P 3 0 O -1 O -1 3 O O -1 which gives 1 3 l = — -—h po 12 X 3 t = _1 (4.2.13) In the above arrays, the quotients for the next steps are different. Since r is strictly Hurwitz and of desired degree, 1 then Q1 and P1 can be expressed as follows. Ql = kr1 (4.2.14) P1 = kt1 (4.2.15) Further, since 2 4 2 2 Q1Q1*- FIE-31* — k 9.(>\ +2) —R1R1* (4.2.16) then (4.2.17) Q = 1 +1+§ (4.2.18) -67- p = % (4.2.19) Therefore, A 3 3 : -— 0 : -— O 0 I?2 2[] 2 (4220) The polynomials Q2 is A 1 Q = —— (Q ~(on 13,13) 2 R1R1* la 1' = xz+x+g (4.2-21) The elementary section described by P1, Q1 and R1, and the remaining network described by P and Q2 are shown in 2 Fig. 4.2.1. 1 1"2' o as T4 _- £2 1 C;P CL s -2 s 932 1‘Q 2'6— Fig. 4.2.1 3 A A For the remaining section, since P2 and Q2 correspond to an elementary section with 526 1’5? —().2+2)2—RR 2 2* 2 2* — " z 25:: the terminating resistance is 1 ohm. Thus, the complete realization is now given in Fig. 4. 2. 2. 4. 3 Example II Synthesize a cascade network whose reflection coefficient is given by 2X3+8x2+3X—1 = 3 2 (4.3.1) 6). + 12>. +7>.+1 P 51 ’ 6 Solution: Without indicating the steps of the synthesis procedure ex- plicitely we first consider the transmission zeros. Since 3 P 2). +812+31-1 (4.3.2) 2 Q 613+12x +7>.+1 (4.3.3) then the transmission zeros are determined from RR* = QQ - PP, = 814(1 -412) (4.3.4) 3:: 2,: It can be seen easily that the transmission zeros are located at the origin With mult1p11c1ty of 4 and on the real am at i 2- With multiplicities of 1. To obtain X and Y, we form the following division array. which gives Therefore, -69- 6 12 7 1 2 8 3 -1 -12 -2 4 23 11 7 T "1 -12 -2 4 22 if 18 9 250 -——— 4 43 22 if 18 9 4'99 432 3 1 23 -—).-— 6 36 _ 1:;18 l. + 18'250 43 4-99 432 3 1 {-1) ;-[qo:5< = l - 4X2. Since (1 - 4X2) can be factored in two different ways as (l - 2).)(1 + 2}.) or (-1 - 2).) (-1 + 2).), there are two different elementary sections corresponding to these transmission zeros: one factorization corresponds to the relation R21 = R 12* and the other to R2.1 = - R1293 These two cases will be considered separately. 2 .0 o, = " X, 1) If R21 R12*, 1 e R12 1 2 then we have 1243 4 4-17 3 29 2 174 79 = >. —— >. — >. - ——>. - — . . R1R1*X 11 + 11 +11 11 22 (4 3 7) 4°43 4 4°l44 3 71 2 144 57 = -— —— ). —— >. .. —->. - — . . R1R1>stY 11 + 11 +11 11 22 (4 3 8) The division arrays for Rllex and Q, and for R1R1*Y and P are given as follows. 12'43 4-174 29 -l74 -79 RlRH‘X 11 11 Ti 11 22 Q 6 12 7 1 -8«42 -573 -260 -79 11 ll 11 22 6 12 7 l 3 9 12 2 4 6 l 3 9 12 2 2 l 3 3 4.5. .3. 2 2 E l 3 3 _71- which yields 9X2 +12). +-:- 2 4 ql : Eh'l'g 2 1 = ..). _ r2 3 +'3 27 = ——-l\ q2 2 15 3 : -—>.+— r3 2 2 4-43 4:144 -157 -144 57 R1R14Y' 11 11 11 11 22 P 2 8 3 -1 -7-16 -415 -29-2 g1 11 11 11 22 2 8 3 -1 -5 3 10 -2_ 4 14 _ .—— -1 3 3 -5 10 -—— 3 2 E. l 9 9 ll. :2 2 2 E. .l 9 9 whichyields _ 2-43 X 22 p6 " 11 ' 11 2 5 ). >. — 1 3 +10 +2 -72- p123x+9 t2 = 7733+; Pz = 3'21). t3 = .121)”; Since the constant terms of qZ and p2 would be different if the divisions were carried one more step, both division arrays are stopped. Next, we consider the linear combinations of the remainder , r , t and t to satisfy the strictly Hurwitz character polynomials r3 2 3 2 of Q1 and the relation R1R1* = Q1Q1* - Plpl’l" Let Q1 = n(r3 - krz) = n[(l-2§-%k)>.+(%-%k)] (4.3.9) P1 = n(t3 - ktz) = nulzl -%k)1-(%+%k)] (4.3-10) Then 0101* - P1P1* = nz(§8T k2 - $1. - 4)(1- 41.2) (4.3.11) For Ql to be strictly Hurwitz, the bounds for k are 9 45 k<2 and k> 4 (4.3.12) Referring to the corresponding elementary section and noting that P1 has only X term, we have k : - 225- which is in agreement with the bounds. Thus from 2 8 2 14 16(81): --§-k-4)-1 (4.3.13) -73- wehave 2-_1_ n. - 81 or 1 n: +— (4.3.14) -9 1 Ifweletn=-§-,wehave P —i(t 14)-3). (4315) 1'9 3‘ 2 ‘2 " Q —l(r kr)--5-).+1 (4316) 1‘9 3 2‘2 " On the other hand,since n(r3 - krz) : n{R1R1*X[q1,qZ]- Ql: qo’ ql’qz] = - - . . + niR1R1,,X[q1.(qz+k)] Q[qo <11 <42 k)]l (4.3.17) hence P =n[q.(q +1.)] = 12-1-1 (4.3.18) 2 l 2 Q = 1 [(1-21)(6>.3+12>.2+7>.+1) 2 R R 2 1 1* +% ).(2>.3 + 8).2 + 3). .. 1)] = 312 + 3). +1 (4.3.19) Thus, the realization for the first cycle is shown in Fig. 4. 3.1 . .._J ——D- ”:1 G? v— s 3 5 392 1’0 2‘52 Fig. 4.3.1 In the following realization cycles, the notations X, Y, qi's, pi's, ri's and ti's are repeatedly used. To obtain X and Y from 132 and 62, we again form the division array for $2 and 62. 02 3 3 1 P2 1 -l -1 6 4 5 -3 -1 6 4 _1_ 9 which gives q0 = 3 l 5 : — X .. — q1 6 18 1. _ 1 2 9 and 2 l 9 3 = _ —— , : — X +— . 3. 0 x (1)r[qoq1]2 Z (4 2) 2 R = For R2 2* Then form the division arrays for RZR A R R *Y and P 2 2. RR 2 2 which gives R2 R2 -)\2, we have 1 3 5 (-1) r—[qll-EX-E (4.3.21) 2 = £243-99 (4.3.22) 3 3 5 2 = -—>. —x .. 2 +2 (4 3 23) 2*X and 622, and for -3 —Z 0 O 3 l 3 E 0 3 l -l l -l 3 —)\+1 2 3 —>\-l 2 2X +1 and which gives Let then Referring to the corresponding elementary section in Table I , P is either a constant or it has only the X term. However, when P -76- -3 5 — — O 2 2 0 l -1 -1 -3 1 -—Z O l -l -l -1 7 1 l -l -l _2. 1 3 p0 --2->\ +1 l -—X t1 2 p1 - 2X X - t2 1 02 = n(-rZ +krl)‘ = n[(-1--:—k)>\-(l+k)] (4 P2 = n(-t2 +kt1) = n[(-1--;-k)7\+(l+k)] (4 Q Q P P = -n2(2k + 21\Z (4 2 2*” 2 2* 2 2 . 3.24) . 3.25) . 3.26) -77.. has only X term, then (22 would not be strictly Hurwitz due to the vanishing constant term. Therefore, P can only be a constant. 2 With this conclusion, we have k : -2 (4.3.27) which yields a strictly Hurwitz polynomial Q2. From Eq. (4. 3. 26), wehave .2 -1 - 4 01‘ n- +—1- ’—2 Ifwe take nzéu we have p - i (4328) 2 ‘ ‘2 " Q - x+-1- (4329) 2 ‘ 2 " $3 = x+1 (4.3.30) A Q3 : 3x+1 (4.3.3.1) The realization, after the second cycle, is shown in Fig. 4.3.2. A 0—725 V (I; 1 Y, o o A f> £3 5 -E s — 2 s - 3 1 Q 2 6—2 3 6'3 -78.. To obtain X and Y for P3 and (33, form the division array for A A P and Q3, and we have 3 A 1 Q3 3 i3 1 1 3 --2 which gives q0 = 3 r1 = -2 and 3 X - 2 1 Y - '2' = -X For R3R3* , we have 3 2 : .—X . . R3R3*X 2 (4 3 32) 1 2 : -— X . . R3R3*Y 2 (4 3 33) Consider the division arrays for R3R3>,EY d ’13 an 3. R R x "—3 0 0 3 3* 2 A 1 Q3 3 1 — O 2 3 1 which yields 1 = —x r1 2 and RR LL 0 33* 2 R 1 1 3 1 — o 2 1 1 which yields 1 = - —- X Po 2 1 = —x t1 2 Let Q — k6 3"n”1‘ 9 n[ (% - 3k». - k] P kP 3’ MH' 3) d(%-mx-k] (4. 3. 34) (4. 3. 35) The constant term of P can not vanish, otherwise the constant term 3 2 D 1 of Q3 would vanish too. Hence it is necessary to take k = — for which Q3 is strictly Hurwitz. Since (2303* - p3p3* = n2(2k - 8kz)>\2 we have n = i 1. Taking n = -l, we further have P3: 2 Q3: 3+% = R R (4.3.36) Z) ..80- R1 4- 5-3 4- Thus the final realization is as shown in Fig. 4. 3. 3. ‘ ‘ H1 T1 . . is 5:3 szpz s—p3 54:35:1 1 Q 2 6; 3 6; 4 Fig. 4.3.3 If R21 = -R12*, i.e., R12 2 -l + 2X, then the lelSlOI‘l arrays for the first cycle are the same as those in case (1). In this case Pl has a constant term only, hence we have 9'17 1 : ———-— : +— k 4 and n _ 9 Taking n = - % and from Eqs. (4. 3. 9) and (4. 3.10), we have P - 3 (4 3 37) 1 - 4 O I 5 : A - . . 01 2 +4 (4 3 38) A 2 7 2: X + — X + Z . . 3 P2 2 (4 3 9) A 2 9 Q2 2 3x + 2 x + 2 (4. 3.40) By repeating the steps in case (1), one will have the realization as shown in Fig. 4. 3. 4. -81- Fig. 4.3.4 A A V7 V A A V V CHAPTER V CONCLUSION AND FURTHER PROBLEMS A method for cascade synthesis of l-port passive networks by means of successive extraction of Z-port elementary sections of Table I is fully discussed. Each of the elementary sections is characterized by scattering parameters. The use of nonreciprocal elements enables us to consider each of the elementary sections with not more than two reactive elements. For this reason, it is sufficient that the existence theorem stated in Section 3. 3 (Theorem 3. 3.1) is to be restricted for (SQ E 2. The synthesis procedure is based on the step by step reali- zation of the simple sets of transmission zeros of a given reflection coefficient. In each step, the realization consists of simple mani- pulation on polynomial-s, viz. , the division algorithm and the linear combination of certain polynomials. At the end of each step, informations are. obtained which are sufficient for the deter- mination of elementary section to be extracted (whose element values can be determined later), and for generating the reflection coefficient for the remaining l-port network. -82.. -83- The procedure described in this thesis is useful in the filter synthesis. In general, the filter synthesis is reduced to the realization of the reflection coefficient S1 with specified trans- mission zeros. The computation of the key polynomials are accomplished by the use of the division array in a straightforward manner. It is suggested as a further problem that one may consider complicated elementary sections. In this case, however, the existence theorem (Theorem 3. 3.1) must be extended and such an extension should follow a different approach than that con- sidered in this thesis. Another area of investigation is the extension of the present method to the n-port cascade synthesis by essentially using the idea of Belevitch [BE 3] but carrying the computation by the method described in this thesis. 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